YES(O(1), O(n^1)) 233.14/71.67 YES(O(1), O(n^1)) 233.45/71.73 233.45/71.73 233.45/71.73 233.45/71.73 233.45/71.73 233.45/71.73 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 233.45/71.73 233.45/71.73 233.45/71.73
233.45/71.73 233.80/71.88 233.80/71.88
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233.80/71.88

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(minus(0, Y)) → mark(0) 233.80/71.88
active(minus(s(X), s(Y))) → mark(minus(X, Y)) 233.80/71.88
active(geq(X, 0)) → mark(true) 233.80/71.88
active(geq(0, s(Y))) → mark(false) 233.80/71.88
active(geq(s(X), s(Y))) → mark(geq(X, Y)) 233.80/71.88
active(div(0, s(Y))) → mark(0) 233.80/71.88
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)) 233.80/71.88
active(if(true, X, Y)) → mark(X) 233.80/71.88
active(if(false, X, Y)) → mark(Y) 233.80/71.88
active(s(X)) → s(active(X)) 233.80/71.88
active(div(X1, X2)) → div(active(X1), X2) 233.80/71.88
active(if(X1, X2, X3)) → if(active(X1), X2, X3) 233.80/71.88
s(mark(X)) → mark(s(X)) 233.80/71.88
div(mark(X1), X2) → mark(div(X1, X2)) 233.80/71.88
if(mark(X1), X2, X3) → mark(if(X1, X2, X3)) 233.80/71.88
proper(minus(X1, X2)) → minus(proper(X1), proper(X2)) 233.80/71.88
proper(0) → ok(0) 233.80/71.88
proper(s(X)) → s(proper(X)) 233.80/71.88
proper(geq(X1, X2)) → geq(proper(X1), proper(X2)) 233.80/71.88
proper(true) → ok(true) 233.80/71.88
proper(false) → ok(false) 233.80/71.88
proper(div(X1, X2)) → div(proper(X1), proper(X2)) 233.80/71.88
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3)) 233.80/71.88
minus(ok(X1), ok(X2)) → ok(minus(X1, X2)) 233.80/71.88
s(ok(X)) → ok(s(X)) 233.80/71.88
geq(ok(X1), ok(X2)) → ok(geq(X1, X2)) 233.80/71.88
div(ok(X1), ok(X2)) → ok(div(X1, X2)) 233.80/71.88
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3)) 233.80/71.88
top(mark(X)) → top(proper(X)) 233.80/71.88
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
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(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 233.80/71.88
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 233.80/71.88
active(geq(z0, 0)) → mark(true) 233.80/71.88
active(geq(0, s(z0))) → mark(false) 233.80/71.88
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 233.80/71.88
active(div(0, s(z0))) → mark(0) 233.80/71.88
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 233.80/71.88
active(if(true, z0, z1)) → mark(z0) 233.80/71.88
active(if(false, z0, z1)) → mark(z1) 233.80/71.88
active(s(z0)) → s(active(z0)) 233.80/71.88
active(div(z0, z1)) → div(active(z0), z1) 233.80/71.88
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 233.80/71.88
s(mark(z0)) → mark(s(z0)) 233.80/71.88
s(ok(z0)) → ok(s(z0)) 233.80/71.88
div(mark(z0), z1) → mark(div(z0, z1)) 233.80/71.88
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 233.80/71.88
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 233.80/71.88
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 233.80/71.88
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 233.80/71.88
proper(0) → ok(0) 233.80/71.88
proper(s(z0)) → s(proper(z0)) 233.80/71.88
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 233.80/71.88
proper(true) → ok(true) 233.80/71.88
proper(false) → ok(false) 233.80/71.88
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 233.80/71.88
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 233.80/71.88
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 233.80/71.88
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 233.80/71.88
top(mark(z0)) → top(proper(z0)) 233.80/71.88
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 233.80/71.88
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 233.80/71.88
ACTIVE(div(s(z0), s(z1))) → c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1), S(div(minus(z0, z1), s(z1))), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1), S(z1)) 233.80/71.88
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 233.80/71.88
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0)) 233.80/71.88
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 233.80/71.88
S(mark(z0)) → c12(S(z0)) 233.80/71.88
S(ok(z0)) → c13(S(z0)) 233.80/71.88
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 233.80/71.88
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 233.80/71.88
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 233.80/71.88
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 233.80/71.88
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.88
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 233.80/71.88
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.88
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.88
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 233.80/71.88
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 233.80/71.88
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 233.80/71.88
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 233.80/71.88
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 233.80/71.88
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 233.80/71.88
ACTIVE(div(s(z0), s(z1))) → c6(IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), GEQ(z0, z1), S(div(minus(z0, z1), s(z1))), DIV(minus(z0, z1), s(z1)), MINUS(z0, z1), S(z1)) 233.80/71.88
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 233.80/71.88
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0)) 233.80/71.88
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 233.80/71.88
S(mark(z0)) → c12(S(z0)) 233.80/71.88
S(ok(z0)) → c13(S(z0)) 233.80/71.88
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 233.80/71.88
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 233.80/71.88
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 233.80/71.88
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 233.80/71.88
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.88
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 233.80/71.88
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.88
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.88
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 233.80/71.88
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 233.80/71.88
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 233.80/71.88
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 233.80/71.88
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c6, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29

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(3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 233.80/71.88
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 233.80/71.88
active(geq(z0, 0)) → mark(true) 233.80/71.88
active(geq(0, s(z0))) → mark(false) 233.80/71.88
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 233.80/71.88
active(div(0, s(z0))) → mark(0) 233.80/71.88
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 233.80/71.88
active(if(true, z0, z1)) → mark(z0) 233.80/71.88
active(if(false, z0, z1)) → mark(z1) 233.80/71.88
active(s(z0)) → s(active(z0)) 233.80/71.88
active(div(z0, z1)) → div(active(z0), z1) 233.80/71.88
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 233.80/71.88
s(mark(z0)) → mark(s(z0)) 233.80/71.88
s(ok(z0)) → ok(s(z0)) 233.80/71.88
div(mark(z0), z1) → mark(div(z0, z1)) 233.80/71.88
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 233.80/71.88
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 233.80/71.88
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 233.80/71.88
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 233.80/71.88
proper(0) → ok(0) 233.80/71.88
proper(s(z0)) → s(proper(z0)) 233.80/71.88
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 233.80/71.88
proper(true) → ok(true) 233.80/71.88
proper(false) → ok(false) 233.80/71.88
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 233.80/71.88
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 233.80/71.88
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 233.80/71.88
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 233.80/71.88
top(mark(z0)) → top(proper(z0)) 233.80/71.88
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 233.80/71.88
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 233.80/71.88
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 233.80/71.88
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0)) 233.80/71.88
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 233.80/71.88
S(mark(z0)) → c12(S(z0)) 233.80/71.88
S(ok(z0)) → c13(S(z0)) 233.80/71.88
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 233.80/71.88
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 233.80/71.88
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 233.80/71.88
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 233.80/71.88
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.88
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 233.80/71.88
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.88
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.88
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 233.80/71.88
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 233.80/71.88
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 233.80/71.88
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 233.80/71.88
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 233.80/71.88
ACTIVE(div(s(z0), s(z1))) → c6(GEQ(z0, z1), MINUS(z0, z1), S(z1))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 233.80/71.88
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 233.80/71.88
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 233.80/71.88
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0)) 233.80/71.88
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 233.80/71.88
S(mark(z0)) → c12(S(z0)) 233.80/71.88
S(ok(z0)) → c13(S(z0)) 233.80/71.88
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 233.80/71.88
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 233.80/71.88
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 233.80/71.88
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 233.80/71.88
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.88
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 233.80/71.88
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.88
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.88
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 233.80/71.88
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 233.80/71.88
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 233.80/71.88
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 233.80/71.88
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 233.80/71.88
ACTIVE(div(s(z0), s(z1))) → c6(GEQ(z0, z1), MINUS(z0, z1), S(z1))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c6

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(5) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 233.80/71.88
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 233.80/71.88
active(geq(z0, 0)) → mark(true) 233.80/71.88
active(geq(0, s(z0))) → mark(false) 233.80/71.88
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 233.80/71.88
active(div(0, s(z0))) → mark(0) 233.80/71.88
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 233.80/71.89
active(if(true, z0, z1)) → mark(z0) 233.80/71.89
active(if(false, z0, z1)) → mark(z1) 233.80/71.89
active(s(z0)) → s(active(z0)) 233.80/71.89
active(div(z0, z1)) → div(active(z0), z1) 233.80/71.89
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 233.80/71.89
s(mark(z0)) → mark(s(z0)) 233.80/71.89
s(ok(z0)) → ok(s(z0)) 233.80/71.89
div(mark(z0), z1) → mark(div(z0, z1)) 233.80/71.89
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 233.80/71.89
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 233.80/71.89
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 233.80/71.89
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 233.80/71.89
proper(0) → ok(0) 233.80/71.89
proper(s(z0)) → s(proper(z0)) 233.80/71.89
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 233.80/71.89
proper(true) → ok(true) 233.80/71.89
proper(false) → ok(false) 233.80/71.89
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 233.80/71.89
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 233.80/71.89
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 233.80/71.89
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 233.80/71.89
top(mark(z0)) → top(proper(z0)) 233.80/71.89
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 233.80/71.89
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 233.80/71.89
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 233.80/71.89
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0)) 233.80/71.89
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 233.80/71.89
S(mark(z0)) → c12(S(z0)) 233.80/71.89
S(ok(z0)) → c13(S(z0)) 233.80/71.89
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 233.80/71.89
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 233.80/71.89
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 233.80/71.89
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 233.80/71.89
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.89
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 233.80/71.89
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.89
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.89
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 233.80/71.89
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 233.80/71.89
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 233.80/71.89
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 233.80/71.89
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 233.80/71.89
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 233.80/71.89
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 233.80/71.89
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 233.80/71.89
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 233.80/71.89
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 233.80/71.89
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0)) 233.80/71.89
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 233.80/71.89
S(mark(z0)) → c12(S(z0)) 233.80/71.89
S(ok(z0)) → c13(S(z0)) 233.80/71.89
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 233.80/71.89
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 233.80/71.89
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 233.80/71.89
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 233.80/71.89
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.89
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 233.80/71.89
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.89
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.89
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 233.80/71.89
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 233.80/71.89
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 233.80/71.89
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 233.80/71.89
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 233.80/71.89
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 233.80/71.89
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 233.80/71.89
ACTIVE(div(s(z0), s(z1))) → c(S(z1))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c

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(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(minus(0, z0))) → c9(S(mark(0)), ACTIVE(minus(0, z0))) 233.80/71.89
ACTIVE(s(minus(s(z0), s(z1)))) → c9(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1)))) 233.80/71.89
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)), ACTIVE(geq(z0, 0))) 233.80/71.89
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)), ACTIVE(geq(0, s(z0)))) 233.80/71.89
ACTIVE(s(geq(s(z0), s(z1)))) → c9(S(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1)))) 233.80/71.89
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0)))) 233.80/71.89
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 233.80/71.89
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 233.80/71.89
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 233.80/71.89
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 233.80/71.89
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 233.80/71.89
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
233.80/71.89
233.80/71.89

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 233.80/71.89
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 233.80/71.89
active(geq(z0, 0)) → mark(true) 233.80/71.89
active(geq(0, s(z0))) → mark(false) 233.80/71.89
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 233.80/71.89
active(div(0, s(z0))) → mark(0) 233.80/71.89
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 233.80/71.89
active(if(true, z0, z1)) → mark(z0) 233.80/71.89
active(if(false, z0, z1)) → mark(z1) 233.80/71.89
active(s(z0)) → s(active(z0)) 233.80/71.89
active(div(z0, z1)) → div(active(z0), z1) 233.80/71.89
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 233.80/71.89
s(mark(z0)) → mark(s(z0)) 233.80/71.89
s(ok(z0)) → ok(s(z0)) 233.80/71.89
div(mark(z0), z1) → mark(div(z0, z1)) 233.80/71.89
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 233.80/71.89
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 233.80/71.89
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 233.80/71.89
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 233.80/71.89
proper(0) → ok(0) 233.80/71.89
proper(s(z0)) → s(proper(z0)) 233.80/71.89
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 233.80/71.89
proper(true) → ok(true) 233.80/71.89
proper(false) → ok(false) 233.80/71.89
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 233.80/71.89
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 233.80/71.89
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 233.80/71.89
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 233.80/71.89
top(mark(z0)) → top(proper(z0)) 233.80/71.89
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 233.80/71.89
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 233.80/71.89
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0)) 233.80/71.89
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 233.80/71.89
S(mark(z0)) → c12(S(z0)) 233.80/71.89
S(ok(z0)) → c13(S(z0)) 233.80/71.89
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 233.80/71.89
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 233.80/71.89
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 233.80/71.89
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 233.80/71.89
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.89
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 233.80/71.89
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.89
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.89
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 233.80/71.89
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 233.80/71.89
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 233.80/71.89
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 233.80/71.89
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 233.80/71.89
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 233.80/71.89
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 233.80/71.89
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 233.80/71.89
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)), ACTIVE(minus(0, z0))) 233.80/71.89
ACTIVE(s(minus(s(z0), s(z1)))) → c9(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1)))) 233.80/71.89
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)), ACTIVE(geq(z0, 0))) 233.80/71.89
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)), ACTIVE(geq(0, s(z0)))) 233.80/71.89
ACTIVE(s(geq(s(z0), s(z1)))) → c9(S(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1)))) 233.80/71.89
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0)))) 233.80/71.89
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 233.80/71.89
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 233.80/71.89
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 233.80/71.89
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 233.80/71.89
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 233.80/71.89
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 233.80/71.89
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 233.80/71.89
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0)) 233.80/71.89
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 233.80/71.89
S(mark(z0)) → c12(S(z0)) 233.80/71.89
S(ok(z0)) → c13(S(z0)) 233.80/71.89
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 233.80/71.89
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 233.80/71.89
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 233.80/71.89
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 233.80/71.89
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.89
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 233.80/71.89
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.89
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 233.80/71.89
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 233.80/71.89
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 233.80/71.89
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 233.80/71.89
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 233.80/71.89
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 233.80/71.89
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 233.80/71.89
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 233.80/71.89
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 233.80/71.89
ACTIVE(s(minus(0, z0))) → c9(S(mark(0)), ACTIVE(minus(0, z0))) 233.80/71.89
ACTIVE(s(minus(s(z0), s(z1)))) → c9(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1)))) 233.80/71.89
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true)), ACTIVE(geq(z0, 0))) 233.80/71.89
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)), ACTIVE(geq(0, s(z0)))) 233.80/71.89
ACTIVE(s(geq(s(z0), s(z1)))) → c9(S(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1)))) 233.80/71.89
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0)))) 233.80/71.89
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 233.80/71.89
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 233.80/71.89
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 233.80/71.89
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 233.80/71.89
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 233.80/71.89
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9

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233.80/71.89

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts
233.80/71.89
233.80/71.89

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 233.80/71.89
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 233.80/71.89
active(geq(z0, 0)) → mark(true) 233.80/71.89
active(geq(0, s(z0))) → mark(false) 233.80/71.89
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 233.80/71.89
active(div(0, s(z0))) → mark(0) 233.80/71.89
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.90
active(if(true, z0, z1)) → mark(z0) 234.18/71.90
active(if(false, z0, z1)) → mark(z1) 234.18/71.90
active(s(z0)) → s(active(z0)) 234.18/71.90
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.90
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.90
s(mark(z0)) → mark(s(z0)) 234.18/71.90
s(ok(z0)) → ok(s(z0)) 234.18/71.90
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.90
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.90
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.90
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.90
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.90
proper(0) → ok(0) 234.18/71.90
proper(s(z0)) → s(proper(z0)) 234.18/71.90
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.90
proper(true) → ok(true) 234.18/71.90
proper(false) → ok(false) 234.18/71.90
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.90
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.90
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.90
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.90
top(mark(z0)) → top(proper(z0)) 234.18/71.90
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.90
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.90
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0)) 234.18/71.90
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 234.18/71.90
S(mark(z0)) → c12(S(z0)) 234.18/71.90
S(ok(z0)) → c13(S(z0)) 234.18/71.90
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.90
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.90
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.90
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.90
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.90
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.90
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.90
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.90
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.90
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.90
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.90
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.90
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.90
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.90
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.90
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.90
ACTIVE(s(minus(s(z0), s(z1)))) → c9(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1)))) 234.18/71.90
ACTIVE(s(geq(s(z0), s(z1)))) → c9(S(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1)))) 234.18/71.90
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0)))) 234.18/71.90
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.90
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 234.18/71.90
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 234.18/71.90
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.90
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.90
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.90
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.90
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.90
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.90
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.90
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0)) 234.18/71.90
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 234.18/71.90
S(mark(z0)) → c12(S(z0)) 234.18/71.90
S(ok(z0)) → c13(S(z0)) 234.18/71.90
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.90
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.90
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.90
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.90
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.90
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.90
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.90
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.90
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.90
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.90
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.90
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.90
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.90
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.90
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.90
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.90
ACTIVE(s(minus(s(z0), s(z1)))) → c9(S(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1)))) 234.18/71.90
ACTIVE(s(geq(s(z0), s(z1)))) → c9(S(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1)))) 234.18/71.90
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0)))) 234.18/71.90
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.90
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 234.18/71.90
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 234.18/71.90
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.90
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.90
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.90
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.90
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.90
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false)))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9

234.18/71.90
234.18/71.90

(11) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC
234.18/71.90
234.18/71.90

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.90
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.90
active(geq(z0, 0)) → mark(true) 234.18/71.90
active(geq(0, s(z0))) → mark(false) 234.18/71.90
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.90
active(div(0, s(z0))) → mark(0) 234.18/71.90
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.90
active(if(true, z0, z1)) → mark(z0) 234.18/71.90
active(if(false, z0, z1)) → mark(z1) 234.18/71.90
active(s(z0)) → s(active(z0)) 234.18/71.90
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.90
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.90
s(mark(z0)) → mark(s(z0)) 234.18/71.90
s(ok(z0)) → ok(s(z0)) 234.18/71.90
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.90
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.90
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.90
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.90
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.90
proper(0) → ok(0) 234.18/71.90
proper(s(z0)) → s(proper(z0)) 234.18/71.90
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.90
proper(true) → ok(true) 234.18/71.90
proper(false) → ok(false) 234.18/71.90
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.90
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.90
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.90
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.90
top(mark(z0)) → top(proper(z0)) 234.18/71.90
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.90
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.90
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0)) 234.18/71.90
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 234.18/71.90
S(mark(z0)) → c12(S(z0)) 234.18/71.90
S(ok(z0)) → c13(S(z0)) 234.18/71.90
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.90
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.90
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.90
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.90
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.90
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.90
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.90
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.90
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.90
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.90
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.90
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.90
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.90
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.90
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.90
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.90
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0)))) 234.18/71.90
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.90
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 234.18/71.90
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 234.18/71.90
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.90
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.90
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.90
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.90
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.90
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.90
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.90
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.90
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.90
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.90
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.90
ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0)) 234.18/71.90
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 234.18/71.90
S(mark(z0)) → c12(S(z0)) 234.18/71.90
S(ok(z0)) → c13(S(z0)) 234.18/71.90
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.90
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.90
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.90
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.90
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.90
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.90
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.90
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.90
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.90
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.90
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.90
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.90
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.90
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.90
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.90
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.90
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0)))) 234.18/71.90
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.90
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 234.18/71.90
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 234.18/71.90
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.90
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.90
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.90
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.90
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.90
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.90
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.90
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.90
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.90
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1))))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c10, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2

234.18/71.90
234.18/71.90

(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(div(z0, z1)) → c10(DIV(active(z0), z1), ACTIVE(z0)) by

ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1), ACTIVE(minus(0, z0))) 234.18/71.90
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c10(DIV(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1), ACTIVE(geq(z0, 0))) 234.18/71.91
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1), ACTIVE(geq(0, s(z0)))) 234.18/71.91
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c10(DIV(mark(geq(z0, z1)), x1), ACTIVE(geq(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1), ACTIVE(div(0, s(z0)))) 234.18/71.91
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1))) 234.18/71.91
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1))) 234.18/71.91
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.91
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.91
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
234.18/71.91
234.18/71.91

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.91
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.91
active(geq(z0, 0)) → mark(true) 234.18/71.91
active(geq(0, s(z0))) → mark(false) 234.18/71.91
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.91
active(div(0, s(z0))) → mark(0) 234.18/71.91
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.91
active(if(true, z0, z1)) → mark(z0) 234.18/71.91
active(if(false, z0, z1)) → mark(z1) 234.18/71.91
active(s(z0)) → s(active(z0)) 234.18/71.91
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.91
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.91
s(mark(z0)) → mark(s(z0)) 234.18/71.91
s(ok(z0)) → ok(s(z0)) 234.18/71.91
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.91
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.91
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.91
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.91
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.91
proper(0) → ok(0) 234.18/71.91
proper(s(z0)) → s(proper(z0)) 234.18/71.91
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.91
proper(true) → ok(true) 234.18/71.91
proper(false) → ok(false) 234.18/71.91
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.91
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.91
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.91
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.91
top(mark(z0)) → top(proper(z0)) 234.18/71.91
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.91
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.91
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 234.18/71.91
S(mark(z0)) → c12(S(z0)) 234.18/71.91
S(ok(z0)) → c13(S(z0)) 234.18/71.91
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.91
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.91
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.91
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.91
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.91
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.91
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.91
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.91
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.91
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.91
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0)))) 234.18/71.91
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.91
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 234.18/71.91
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 234.18/71.91
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.91
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.91
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.91
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.91
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.91
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.91
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.91
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.91
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.91
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1), ACTIVE(minus(0, z0))) 234.18/71.91
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c10(DIV(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1), ACTIVE(geq(z0, 0))) 234.18/71.91
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1), ACTIVE(geq(0, s(z0)))) 234.18/71.91
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c10(DIV(mark(geq(z0, z1)), x1), ACTIVE(geq(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1), ACTIVE(div(0, s(z0)))) 234.18/71.91
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1))) 234.18/71.91
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1))) 234.18/71.91
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.91
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.91
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.91
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.91
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 234.18/71.91
S(mark(z0)) → c12(S(z0)) 234.18/71.91
S(ok(z0)) → c13(S(z0)) 234.18/71.91
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.91
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.91
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.91
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.91
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.91
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.91
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.91
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.91
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.91
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.91
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0)), ACTIVE(div(0, s(z0)))) 234.18/71.91
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.91
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 234.18/71.91
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 234.18/71.91
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.91
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.91
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.91
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.91
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.91
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.91
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.91
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.91
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.91
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1), ACTIVE(minus(0, z0))) 234.18/71.91
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c10(DIV(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1), ACTIVE(geq(z0, 0))) 234.18/71.91
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1), ACTIVE(geq(0, s(z0)))) 234.18/71.91
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c10(DIV(mark(geq(z0, z1)), x1), ACTIVE(geq(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1), ACTIVE(div(0, s(z0)))) 234.18/71.91
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1))) 234.18/71.91
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1))) 234.18/71.91
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.91
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.91
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2)))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10

234.18/71.91
234.18/71.91

(15) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 5 trailing tuple parts
234.18/71.91
234.18/71.91

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.91
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.91
active(geq(z0, 0)) → mark(true) 234.18/71.91
active(geq(0, s(z0))) → mark(false) 234.18/71.91
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.91
active(div(0, s(z0))) → mark(0) 234.18/71.91
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.91
active(if(true, z0, z1)) → mark(z0) 234.18/71.91
active(if(false, z0, z1)) → mark(z1) 234.18/71.91
active(s(z0)) → s(active(z0)) 234.18/71.91
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.91
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.91
s(mark(z0)) → mark(s(z0)) 234.18/71.91
s(ok(z0)) → ok(s(z0)) 234.18/71.91
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.91
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.91
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.91
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.91
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.91
proper(0) → ok(0) 234.18/71.91
proper(s(z0)) → s(proper(z0)) 234.18/71.91
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.91
proper(true) → ok(true) 234.18/71.91
proper(false) → ok(false) 234.18/71.91
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.91
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.91
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.91
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.91
top(mark(z0)) → top(proper(z0)) 234.18/71.91
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.91
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.91
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 234.18/71.91
S(mark(z0)) → c12(S(z0)) 234.18/71.91
S(ok(z0)) → c13(S(z0)) 234.18/71.91
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.91
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.91
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.91
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.91
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.91
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.91
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.91
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.91
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.91
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.91
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.91
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 234.18/71.91
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 234.18/71.91
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.91
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.91
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.91
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.91
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.91
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.91
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.91
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.91
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.91
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c10(DIV(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c10(DIV(mark(geq(z0, z1)), x1), ACTIVE(geq(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1))) 234.18/71.91
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1))) 234.18/71.91
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.91
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.91
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.91
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.91
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.91
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.91
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.91
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.91
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.91
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 234.18/71.91
S(mark(z0)) → c12(S(z0)) 234.18/71.91
S(ok(z0)) → c13(S(z0)) 234.18/71.91
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.91
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.91
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.91
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.91
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.91
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.91
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.91
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.91
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.91
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.91
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.91
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 234.18/71.91
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 234.18/71.91
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.91
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.91
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.91
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.91
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.91
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.91
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.91
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.91
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.91
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c10(DIV(mark(minus(z0, z1)), x1), ACTIVE(minus(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c10(DIV(mark(geq(z0, z1)), x1), ACTIVE(geq(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1))) 234.18/71.91
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1))) 234.18/71.91
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.91
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.91
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.91
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.91
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.91
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.91
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.91
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10

234.18/71.91
234.18/71.91

(17) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC
234.18/71.91
234.18/71.91

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.91
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.91
active(geq(z0, 0)) → mark(true) 234.18/71.91
active(geq(0, s(z0))) → mark(false) 234.18/71.91
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.91
active(div(0, s(z0))) → mark(0) 234.18/71.91
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.91
active(if(true, z0, z1)) → mark(z0) 234.18/71.91
active(if(false, z0, z1)) → mark(z1) 234.18/71.91
active(s(z0)) → s(active(z0)) 234.18/71.91
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.91
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.91
s(mark(z0)) → mark(s(z0)) 234.18/71.91
s(ok(z0)) → ok(s(z0)) 234.18/71.91
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.91
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.91
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.91
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.91
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.91
proper(0) → ok(0) 234.18/71.91
proper(s(z0)) → s(proper(z0)) 234.18/71.91
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.91
proper(true) → ok(true) 234.18/71.91
proper(false) → ok(false) 234.18/71.91
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.91
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.91
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.91
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.91
top(mark(z0)) → top(proper(z0)) 234.18/71.91
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.91
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.91
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 234.18/71.91
S(mark(z0)) → c12(S(z0)) 234.18/71.91
S(ok(z0)) → c13(S(z0)) 234.18/71.91
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.91
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.91
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.91
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.91
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.91
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.91
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.91
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.91
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.91
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.91
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.91
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 234.18/71.91
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 234.18/71.91
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.91
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.91
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.91
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.91
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.91
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.91
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.91
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.91
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.91
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1))) 234.18/71.91
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1))) 234.18/71.91
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.91
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.91
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.91
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.91
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.91
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.91
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.91
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.91
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.91
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.91
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.91
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.91
ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) 234.18/71.91
S(mark(z0)) → c12(S(z0)) 234.18/71.91
S(ok(z0)) → c13(S(z0)) 234.18/71.91
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.91
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.91
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.91
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.91
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.91
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.91
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.91
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.91
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.91
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.91
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.91
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.91
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.91
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 234.18/71.91
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 234.18/71.91
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.91
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.91
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.91
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.91
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.91
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.91
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.91
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.91
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.91
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.91
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1))) 234.18/71.91
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1))) 234.18/71.91
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.91
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.91
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.91
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.91
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.91
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.92
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.92
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.92
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.92
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.92
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.92
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1))))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c11, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3

234.18/71.92
234.18/71.92

(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(if(z0, z1, z2)) → c11(IF(active(z0), z1, z2), ACTIVE(z0)) by

ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(minus(0, z0))) 234.18/71.92
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c11(IF(mark(minus(z0, z1)), x1, x2), ACTIVE(minus(s(z0), s(z1)))) 234.18/71.92
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2), ACTIVE(geq(z0, 0))) 234.18/71.92
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(geq(0, s(z0)))) 234.18/71.92
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c11(IF(mark(geq(z0, z1)), x1, x2), ACTIVE(geq(s(z0), s(z1)))) 234.18/71.92
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(div(0, s(z0)))) 234.18/71.92
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.92
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(if(true, z0, z1))) 234.18/71.92
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2), ACTIVE(if(false, z0, z1))) 234.18/71.92
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.92
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.92
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
234.18/71.92
234.18/71.92

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.92
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.92
active(geq(z0, 0)) → mark(true) 234.18/71.92
active(geq(0, s(z0))) → mark(false) 234.18/71.92
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.92
active(div(0, s(z0))) → mark(0) 234.18/71.92
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.92
active(if(true, z0, z1)) → mark(z0) 234.18/71.92
active(if(false, z0, z1)) → mark(z1) 234.18/71.92
active(s(z0)) → s(active(z0)) 234.18/71.92
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.92
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.92
s(mark(z0)) → mark(s(z0)) 234.18/71.92
s(ok(z0)) → ok(s(z0)) 234.18/71.92
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.92
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.92
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.92
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.92
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.92
proper(0) → ok(0) 234.18/71.92
proper(s(z0)) → s(proper(z0)) 234.18/71.92
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.92
proper(true) → ok(true) 234.18/71.92
proper(false) → ok(false) 234.18/71.92
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.92
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.92
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.92
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.92
top(mark(z0)) → top(proper(z0)) 234.18/71.92
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.92
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.92
S(mark(z0)) → c12(S(z0)) 234.18/71.92
S(ok(z0)) → c13(S(z0)) 234.18/71.92
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.92
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.92
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.92
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.92
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.92
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.92
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.92
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.92
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.92
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.92
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.92
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.92
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.92
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.92
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.92
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.92
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.92
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 234.18/71.92
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 234.18/71.92
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.92
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.92
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.92
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.92
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.92
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.92
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.92
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.92
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.92
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.92
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.92
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1))) 234.18/71.92
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1))) 234.18/71.92
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.92
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.92
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.92
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.92
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.92
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.92
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.92
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.92
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.92
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.92
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.92
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.92
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(minus(0, z0))) 234.18/71.92
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c11(IF(mark(minus(z0, z1)), x1, x2), ACTIVE(minus(s(z0), s(z1)))) 234.18/71.92
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2), ACTIVE(geq(z0, 0))) 234.18/71.92
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(geq(0, s(z0)))) 234.18/71.92
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c11(IF(mark(geq(z0, z1)), x1, x2), ACTIVE(geq(s(z0), s(z1)))) 234.18/71.92
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(div(0, s(z0)))) 234.18/71.92
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.92
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(if(true, z0, z1))) 234.18/71.92
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2), ACTIVE(if(false, z0, z1))) 234.18/71.92
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.92
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.92
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.92
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.92
S(mark(z0)) → c12(S(z0)) 234.18/71.92
S(ok(z0)) → c13(S(z0)) 234.18/71.92
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.92
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.92
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.92
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.92
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.92
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.92
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.92
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.92
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.92
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.92
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.92
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.92
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.92
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.92
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.92
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.92
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.92
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 234.18/71.92
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 234.18/71.92
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.92
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.92
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.92
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.92
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.92
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.92
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.92
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.92
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.92
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.92
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.92
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1), ACTIVE(if(true, z0, z1))) 234.18/71.92
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1), ACTIVE(if(false, z0, z1))) 234.18/71.92
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.92
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.92
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.92
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.92
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.92
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.92
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.92
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.92
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.92
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.92
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.92
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.92
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(minus(0, z0))) 234.18/71.92
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c11(IF(mark(minus(z0, z1)), x1, x2), ACTIVE(minus(s(z0), s(z1)))) 234.18/71.92
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2), ACTIVE(geq(z0, 0))) 234.18/71.92
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2), ACTIVE(geq(0, s(z0)))) 234.18/71.92
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c11(IF(mark(geq(z0, z1)), x1, x2), ACTIVE(geq(s(z0), s(z1)))) 234.18/71.92
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2), ACTIVE(div(0, s(z0)))) 234.18/71.92
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.92
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2), ACTIVE(if(true, z0, z1))) 234.18/71.92
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2), ACTIVE(if(false, z0, z1))) 234.18/71.92
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.92
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.92
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2)))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11

234.18/71.92
234.18/71.92

(21) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 10 trailing tuple parts
234.18/71.92
234.18/71.92

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.92
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.92
active(geq(z0, 0)) → mark(true) 234.18/71.92
active(geq(0, s(z0))) → mark(false) 234.18/71.92
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.92
active(div(0, s(z0))) → mark(0) 234.18/71.92
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.92
active(if(true, z0, z1)) → mark(z0) 234.18/71.92
active(if(false, z0, z1)) → mark(z1) 234.18/71.92
active(s(z0)) → s(active(z0)) 234.18/71.92
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.92
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.92
s(mark(z0)) → mark(s(z0)) 234.18/71.92
s(ok(z0)) → ok(s(z0)) 234.18/71.92
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.92
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.92
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.92
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.92
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.92
proper(0) → ok(0) 234.18/71.92
proper(s(z0)) → s(proper(z0)) 234.18/71.92
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.92
proper(true) → ok(true) 234.18/71.92
proper(false) → ok(false) 234.18/71.92
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.92
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.92
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.92
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.92
top(mark(z0)) → top(proper(z0)) 234.18/71.92
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.92
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.92
S(mark(z0)) → c12(S(z0)) 234.18/71.92
S(ok(z0)) → c13(S(z0)) 234.18/71.92
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.92
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.92
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.92
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.92
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.92
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.92
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.92
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.92
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.92
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.92
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.92
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.92
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.92
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.92
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.92
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.92
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.92
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.92
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.92
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.92
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.92
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.92
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.92
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.92
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.92
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.92
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.92
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.92
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.92
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.92
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.92
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.92
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.92
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.92
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.92
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.92
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.92
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.92
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.92
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.92
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c11(IF(mark(minus(z0, z1)), x1, x2), ACTIVE(minus(s(z0), s(z1)))) 234.18/71.92
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c11(IF(mark(geq(z0, z1)), x1, x2), ACTIVE(geq(s(z0), s(z1)))) 234.18/71.92
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.92
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.92
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.92
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.92
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.92
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.92
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.92
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.92
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.92
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.92
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.92
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.92
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.92
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.92
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.92
S(mark(z0)) → c12(S(z0)) 234.18/71.92
S(ok(z0)) → c13(S(z0)) 234.18/71.92
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.92
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.92
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.92
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.92
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.92
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.92
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.92
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.92
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.92
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.92
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.92
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.92
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.92
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.92
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.92
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.92
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.92
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.92
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.92
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.92
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.92
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.92
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.92
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.92
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.92
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.92
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.92
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.92
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.92
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.98
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.98
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c11(IF(mark(minus(z0, z1)), x1, x2), ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c11(IF(mark(geq(z0, z1)), x1, x2), ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.98
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.98
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.98
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.98
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.98
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.98
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.98
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11

234.18/71.98
234.18/71.98

(23) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC
234.18/71.98
234.18/71.98

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.98
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.98
active(geq(z0, 0)) → mark(true) 234.18/71.98
active(geq(0, s(z0))) → mark(false) 234.18/71.98
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.98
active(div(0, s(z0))) → mark(0) 234.18/71.98
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.98
active(if(true, z0, z1)) → mark(z0) 234.18/71.98
active(if(false, z0, z1)) → mark(z1) 234.18/71.98
active(s(z0)) → s(active(z0)) 234.18/71.98
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.98
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.98
s(mark(z0)) → mark(s(z0)) 234.18/71.98
s(ok(z0)) → ok(s(z0)) 234.18/71.98
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.98
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.98
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.98
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.98
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.98
proper(0) → ok(0) 234.18/71.98
proper(s(z0)) → s(proper(z0)) 234.18/71.98
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.98
proper(true) → ok(true) 234.18/71.98
proper(false) → ok(false) 234.18/71.98
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.98
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.98
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.98
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.98
top(mark(z0)) → top(proper(z0)) 234.18/71.98
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.98
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.98
S(mark(z0)) → c12(S(z0)) 234.18/71.98
S(ok(z0)) → c13(S(z0)) 234.18/71.98
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.98
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.98
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.98
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.98
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.98
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.98
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.98
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.98
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.98
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.98
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.98
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.98
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.98
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.98
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.98
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.98
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.98
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.98
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.98
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.98
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.98
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.98
S(mark(z0)) → c12(S(z0)) 234.18/71.98
S(ok(z0)) → c13(S(z0)) 234.18/71.98
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.98
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.98
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.98
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.98
PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.98
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.98
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.98
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.98
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.98
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.98
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.98
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.98
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.98
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.98
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.98
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.98
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.98
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.98
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.98
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.98
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1))))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c18, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5

234.18/71.98
234.18/71.98

(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(minus(z0, z1)) → c18(MINUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 234.18/71.98
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.98
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 234.18/71.98
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 234.18/71.98
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.98
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 234.18/71.98
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.98
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 234.18/71.98
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 234.18/71.98
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
234.18/71.98
234.18/71.98

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.98
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.98
active(geq(z0, 0)) → mark(true) 234.18/71.98
active(geq(0, s(z0))) → mark(false) 234.18/71.98
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.98
active(div(0, s(z0))) → mark(0) 234.18/71.98
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.98
active(if(true, z0, z1)) → mark(z0) 234.18/71.98
active(if(false, z0, z1)) → mark(z1) 234.18/71.98
active(s(z0)) → s(active(z0)) 234.18/71.98
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.98
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.98
s(mark(z0)) → mark(s(z0)) 234.18/71.98
s(ok(z0)) → ok(s(z0)) 234.18/71.98
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.98
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.98
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.98
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.98
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.98
proper(0) → ok(0) 234.18/71.98
proper(s(z0)) → s(proper(z0)) 234.18/71.98
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.98
proper(true) → ok(true) 234.18/71.98
proper(false) → ok(false) 234.18/71.98
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.98
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.98
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.98
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.98
top(mark(z0)) → top(proper(z0)) 234.18/71.98
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.98
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.98
S(mark(z0)) → c12(S(z0)) 234.18/71.98
S(ok(z0)) → c13(S(z0)) 234.18/71.98
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.98
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.98
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.98
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.98
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.98
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.98
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.98
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.98
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.98
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.98
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.98
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.98
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.98
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.98
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.98
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.98
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.98
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.98
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.98
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.98
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 234.18/71.98
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.98
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 234.18/71.98
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 234.18/71.98
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.98
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 234.18/71.98
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.98
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 234.18/71.98
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 234.18/71.98
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.98
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.98
S(mark(z0)) → c12(S(z0)) 234.18/71.98
S(ok(z0)) → c13(S(z0)) 234.18/71.98
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.98
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.98
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.98
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.98
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.98
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.98
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.98
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.98
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.98
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.98
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.98
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.98
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.98
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.98
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.98
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.98
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.98
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.98
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.98
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.98
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 234.18/71.98
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.98
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 234.18/71.98
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 234.18/71.98
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.98
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 234.18/71.98
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.98
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 234.18/71.98
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 234.18/71.98
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18

234.18/71.98
234.18/71.98

(27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing tuple parts
234.18/71.98
234.18/71.98

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.98
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.98
active(geq(z0, 0)) → mark(true) 234.18/71.98
active(geq(0, s(z0))) → mark(false) 234.18/71.98
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.98
active(div(0, s(z0))) → mark(0) 234.18/71.98
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.98
active(if(true, z0, z1)) → mark(z0) 234.18/71.98
active(if(false, z0, z1)) → mark(z1) 234.18/71.98
active(s(z0)) → s(active(z0)) 234.18/71.98
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.98
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.98
s(mark(z0)) → mark(s(z0)) 234.18/71.98
s(ok(z0)) → ok(s(z0)) 234.18/71.98
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.98
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.98
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.98
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.98
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.98
proper(0) → ok(0) 234.18/71.98
proper(s(z0)) → s(proper(z0)) 234.18/71.98
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.98
proper(true) → ok(true) 234.18/71.98
proper(false) → ok(false) 234.18/71.98
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.98
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.98
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.98
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.98
top(mark(z0)) → top(proper(z0)) 234.18/71.98
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.98
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.98
S(mark(z0)) → c12(S(z0)) 234.18/71.98
S(ok(z0)) → c13(S(z0)) 234.18/71.98
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.98
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.98
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.98
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.98
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.98
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.98
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.98
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.98
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.98
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.98
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.98
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.98
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.98
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.98
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.98
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.98
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.98
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.98
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.98
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.98
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.98
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.98
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.98
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.98
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.98
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.98
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.98
S(mark(z0)) → c12(S(z0)) 234.18/71.98
S(ok(z0)) → c13(S(z0)) 234.18/71.98
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.98
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.98
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.98
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.98
PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) 234.18/71.98
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.98
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.98
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.98
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.98
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.98
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.98
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.98
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.98
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.98
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.98
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.98
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.98
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.98
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.98
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.98
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.98
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.98
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.98
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.98
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.98
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c20, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18

234.18/71.98
234.18/71.98

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(s(z0)) → c20(S(proper(z0)), PROPER(z0)) by

PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(s(0)) → c20(S(ok(0)), PROPER(0)) 234.18/71.98
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.98
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(s(true)) → c20(S(ok(true)), PROPER(true)) 234.18/71.98
PROPER(s(false)) → c20(S(ok(false)), PROPER(false)) 234.18/71.98
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.98
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
234.18/71.98
234.18/71.98

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.98
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.98
active(geq(z0, 0)) → mark(true) 234.18/71.98
active(geq(0, s(z0))) → mark(false) 234.18/71.98
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.98
active(div(0, s(z0))) → mark(0) 234.18/71.98
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.98
active(if(true, z0, z1)) → mark(z0) 234.18/71.98
active(if(false, z0, z1)) → mark(z1) 234.18/71.98
active(s(z0)) → s(active(z0)) 234.18/71.98
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.98
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.98
s(mark(z0)) → mark(s(z0)) 234.18/71.98
s(ok(z0)) → ok(s(z0)) 234.18/71.98
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.98
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.98
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.98
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.98
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.98
proper(0) → ok(0) 234.18/71.98
proper(s(z0)) → s(proper(z0)) 234.18/71.98
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.98
proper(true) → ok(true) 234.18/71.98
proper(false) → ok(false) 234.18/71.98
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.98
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.98
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.98
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.98
top(mark(z0)) → top(proper(z0)) 234.18/71.98
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.98
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.98
S(mark(z0)) → c12(S(z0)) 234.18/71.98
S(ok(z0)) → c13(S(z0)) 234.18/71.98
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.98
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.98
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.98
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.98
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.98
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.98
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.98
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.98
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.98
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.98
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.98
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.98
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.98
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.98
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.98
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.98
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.98
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.98
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.98
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.98
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.98
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.98
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.98
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.98
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(s(0)) → c20(S(ok(0)), PROPER(0)) 234.18/71.98
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.98
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(s(true)) → c20(S(ok(true)), PROPER(true)) 234.18/71.98
PROPER(s(false)) → c20(S(ok(false)), PROPER(false)) 234.18/71.98
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.98
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.98
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.98
S(mark(z0)) → c12(S(z0)) 234.18/71.98
S(ok(z0)) → c13(S(z0)) 234.18/71.98
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.98
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.98
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.98
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.98
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.98
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.98
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.98
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.98
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.98
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.98
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.98
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.98
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.98
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.98
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.98
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.98
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.98
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.98
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.98
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.98
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.98
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.98
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.98
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.98
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(s(0)) → c20(S(ok(0)), PROPER(0)) 234.18/71.98
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.98
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(s(true)) → c20(S(ok(true)), PROPER(true)) 234.18/71.98
PROPER(s(false)) → c20(S(ok(false)), PROPER(false)) 234.18/71.98
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.98
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20

234.18/71.98
234.18/71.98

(31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts
234.18/71.98
234.18/71.98

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.98
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.98
active(geq(z0, 0)) → mark(true) 234.18/71.98
active(geq(0, s(z0))) → mark(false) 234.18/71.98
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.98
active(div(0, s(z0))) → mark(0) 234.18/71.98
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.98
active(if(true, z0, z1)) → mark(z0) 234.18/71.98
active(if(false, z0, z1)) → mark(z1) 234.18/71.98
active(s(z0)) → s(active(z0)) 234.18/71.98
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.98
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.98
s(mark(z0)) → mark(s(z0)) 234.18/71.98
s(ok(z0)) → ok(s(z0)) 234.18/71.98
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.98
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.98
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.98
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.98
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.98
proper(0) → ok(0) 234.18/71.98
proper(s(z0)) → s(proper(z0)) 234.18/71.98
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.98
proper(true) → ok(true) 234.18/71.98
proper(false) → ok(false) 234.18/71.98
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.98
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.98
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.98
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.98
top(mark(z0)) → top(proper(z0)) 234.18/71.98
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.98
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.98
S(mark(z0)) → c12(S(z0)) 234.18/71.98
S(ok(z0)) → c13(S(z0)) 234.18/71.98
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.98
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.98
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.98
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.98
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.98
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.98
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.98
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.98
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.98
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.98
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.98
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.98
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.98
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.98
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.98
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.98
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.98
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.98
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.98
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.98
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.98
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.98
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.98
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.98
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.98
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.98
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(s(0)) → c20(S(ok(0))) 234.18/71.98
PROPER(s(true)) → c20(S(ok(true))) 234.18/71.98
PROPER(s(false)) → c20(S(ok(false)))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.98
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.98
S(mark(z0)) → c12(S(z0)) 234.18/71.98
S(ok(z0)) → c13(S(z0)) 234.18/71.98
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.98
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.98
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.98
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.98
PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.98
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.98
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.98
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.98
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.98
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.98
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.98
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.98
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.98
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.98
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.98
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.98
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.98
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.98
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.98
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.98
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.98
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.98
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.98
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.98
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.98
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.98
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(s(0)) → c20(S(ok(0))) 234.18/71.98
PROPER(s(true)) → c20(S(ok(true))) 234.18/71.98
PROPER(s(false)) → c20(S(ok(false)))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c21, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20

234.18/71.98
234.18/71.98

(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(geq(z0, z1)) → c21(GEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 234.18/71.98
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.98
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 234.18/71.98
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 234.18/71.98
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.98
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 234.18/71.98
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.98
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 234.18/71.98
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 234.18/71.98
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
234.18/71.98
234.18/71.98

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.98
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.98
active(geq(z0, 0)) → mark(true) 234.18/71.98
active(geq(0, s(z0))) → mark(false) 234.18/71.98
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.98
active(div(0, s(z0))) → mark(0) 234.18/71.98
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.98
active(if(true, z0, z1)) → mark(z0) 234.18/71.98
active(if(false, z0, z1)) → mark(z1) 234.18/71.98
active(s(z0)) → s(active(z0)) 234.18/71.98
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.98
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.98
s(mark(z0)) → mark(s(z0)) 234.18/71.98
s(ok(z0)) → ok(s(z0)) 234.18/71.98
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.98
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.98
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.98
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.98
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.98
proper(0) → ok(0) 234.18/71.98
proper(s(z0)) → s(proper(z0)) 234.18/71.98
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.98
proper(true) → ok(true) 234.18/71.98
proper(false) → ok(false) 234.18/71.98
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.98
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.98
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.98
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.98
top(mark(z0)) → top(proper(z0)) 234.18/71.98
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.98
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.98
S(mark(z0)) → c12(S(z0)) 234.18/71.98
S(ok(z0)) → c13(S(z0)) 234.18/71.98
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.98
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.98
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.98
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.98
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.98
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.98
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.98
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.98
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.98
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.98
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.98
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.98
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.98
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.98
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.98
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.98
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.98
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.98
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.98
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.98
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.98
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.98
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.98
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.98
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.98
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.98
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(s(0)) → c20(S(ok(0))) 234.18/71.98
PROPER(s(true)) → c20(S(ok(true))) 234.18/71.98
PROPER(s(false)) → c20(S(ok(false))) 234.18/71.98
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 234.18/71.98
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.98
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 234.18/71.98
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 234.18/71.98
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.98
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 234.18/71.98
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.98
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 234.18/71.98
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 234.18/71.98
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.98
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.98
S(mark(z0)) → c12(S(z0)) 234.18/71.98
S(ok(z0)) → c13(S(z0)) 234.18/71.98
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.98
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.98
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.98
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.98
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.98
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.98
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.98
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.98
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.98
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.98
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.98
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.98
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.98
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.98
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.98
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.98
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.98
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.98
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.98
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.98
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.98
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.98
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.98
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.98
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.98
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.98
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.98
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.98
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.98
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.98
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.98
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.98
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.98
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.98
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.98
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.98
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.98
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.98
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.98
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.98
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(s(0)) → c20(S(ok(0))) 234.18/71.98
PROPER(s(true)) → c20(S(ok(true))) 234.18/71.98
PROPER(s(false)) → c20(S(ok(false))) 234.18/71.98
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.98
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 234.18/71.98
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.98
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.98
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 234.18/71.98
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 234.18/71.98
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.98
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.98
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.98
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 234.18/71.98
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.98
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 234.18/71.99
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 234.18/71.99
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21

234.18/71.99
234.18/71.99

(35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing tuple parts
234.18/71.99
234.18/71.99

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.99
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.99
active(geq(z0, 0)) → mark(true) 234.18/71.99
active(geq(0, s(z0))) → mark(false) 234.18/71.99
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.99
active(div(0, s(z0))) → mark(0) 234.18/71.99
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.99
active(if(true, z0, z1)) → mark(z0) 234.18/71.99
active(if(false, z0, z1)) → mark(z1) 234.18/71.99
active(s(z0)) → s(active(z0)) 234.18/71.99
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.99
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.99
s(mark(z0)) → mark(s(z0)) 234.18/71.99
s(ok(z0)) → ok(s(z0)) 234.18/71.99
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.99
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.99
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.99
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.99
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.99
proper(0) → ok(0) 234.18/71.99
proper(s(z0)) → s(proper(z0)) 234.18/71.99
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.99
proper(true) → ok(true) 234.18/71.99
proper(false) → ok(false) 234.18/71.99
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.99
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.99
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.99
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.99
top(mark(z0)) → top(proper(z0)) 234.18/71.99
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.99
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.99
S(mark(z0)) → c12(S(z0)) 234.18/71.99
S(ok(z0)) → c13(S(z0)) 234.18/71.99
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.99
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.99
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.99
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.99
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.99
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.99
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.99
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.99
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.99
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.99
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.99
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.99
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.99
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.99
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.99
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.99
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.99
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.99
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.99
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.99
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.99
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.99
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(s(0)) → c20(S(ok(0))) 234.18/71.99
PROPER(s(true)) → c20(S(ok(true))) 234.18/71.99
PROPER(s(false)) → c20(S(ok(false))) 234.18/71.99
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.99
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.99
S(mark(z0)) → c12(S(z0)) 234.18/71.99
S(ok(z0)) → c13(S(z0)) 234.18/71.99
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.99
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.99
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.99
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.99
PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 234.18/71.99
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.99
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.99
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.99
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.99
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.99
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.99
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.99
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.99
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.99
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.99
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.99
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.99
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.99
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.99
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.99
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.99
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.99
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(s(0)) → c20(S(ok(0))) 234.18/71.99
PROPER(s(true)) → c20(S(ok(true))) 234.18/71.99
PROPER(s(false)) → c20(S(ok(false))) 234.18/71.99
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c24, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21

234.18/71.99
234.18/71.99

(37) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(div(z0, z1)) → c24(DIV(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 234.18/71.99
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 234.18/71.99
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 234.18/71.99
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 234.18/71.99
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 234.18/71.99
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 234.18/71.99
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
234.18/71.99
234.18/71.99

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.99
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.99
active(geq(z0, 0)) → mark(true) 234.18/71.99
active(geq(0, s(z0))) → mark(false) 234.18/71.99
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.99
active(div(0, s(z0))) → mark(0) 234.18/71.99
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.99
active(if(true, z0, z1)) → mark(z0) 234.18/71.99
active(if(false, z0, z1)) → mark(z1) 234.18/71.99
active(s(z0)) → s(active(z0)) 234.18/71.99
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.99
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.99
s(mark(z0)) → mark(s(z0)) 234.18/71.99
s(ok(z0)) → ok(s(z0)) 234.18/71.99
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.99
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.99
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.99
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.99
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.99
proper(0) → ok(0) 234.18/71.99
proper(s(z0)) → s(proper(z0)) 234.18/71.99
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.99
proper(true) → ok(true) 234.18/71.99
proper(false) → ok(false) 234.18/71.99
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.99
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.99
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.99
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.99
top(mark(z0)) → top(proper(z0)) 234.18/71.99
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.99
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.99
S(mark(z0)) → c12(S(z0)) 234.18/71.99
S(ok(z0)) → c13(S(z0)) 234.18/71.99
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.99
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.99
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.99
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.99
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.99
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.99
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.99
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.99
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.99
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.99
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.99
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.99
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.99
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.99
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.99
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.99
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.99
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.99
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.99
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.99
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.99
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(s(0)) → c20(S(ok(0))) 234.18/71.99
PROPER(s(true)) → c20(S(ok(true))) 234.18/71.99
PROPER(s(false)) → c20(S(ok(false))) 234.18/71.99
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 234.18/71.99
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 234.18/71.99
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 234.18/71.99
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 234.18/71.99
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 234.18/71.99
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 234.18/71.99
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.99
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.99
S(mark(z0)) → c12(S(z0)) 234.18/71.99
S(ok(z0)) → c13(S(z0)) 234.18/71.99
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.99
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.99
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.99
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.99
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.99
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.99
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.99
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.99
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.99
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.99
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.99
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.99
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.99
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.99
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.99
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.99
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.99
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.99
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.99
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.99
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.99
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(s(0)) → c20(S(ok(0))) 234.18/71.99
PROPER(s(true)) → c20(S(ok(true))) 234.18/71.99
PROPER(s(false)) → c20(S(ok(false))) 234.18/71.99
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 234.18/71.99
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 234.18/71.99
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 234.18/71.99
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 234.18/71.99
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 234.18/71.99
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 234.18/71.99
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24

234.18/71.99
234.18/71.99

(39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing tuple parts
234.18/71.99
234.18/71.99

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.99
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.99
active(geq(z0, 0)) → mark(true) 234.18/71.99
active(geq(0, s(z0))) → mark(false) 234.18/71.99
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.99
active(div(0, s(z0))) → mark(0) 234.18/71.99
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.99
active(if(true, z0, z1)) → mark(z0) 234.18/71.99
active(if(false, z0, z1)) → mark(z1) 234.18/71.99
active(s(z0)) → s(active(z0)) 234.18/71.99
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.99
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.99
s(mark(z0)) → mark(s(z0)) 234.18/71.99
s(ok(z0)) → ok(s(z0)) 234.18/71.99
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.99
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.99
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.99
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.99
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.99
proper(0) → ok(0) 234.18/71.99
proper(s(z0)) → s(proper(z0)) 234.18/71.99
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.99
proper(true) → ok(true) 234.18/71.99
proper(false) → ok(false) 234.18/71.99
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.99
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.99
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.99
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.99
top(mark(z0)) → top(proper(z0)) 234.18/71.99
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.99
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.99
S(mark(z0)) → c12(S(z0)) 234.18/71.99
S(ok(z0)) → c13(S(z0)) 234.18/71.99
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.99
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.99
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.99
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.99
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.99
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.99
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.99
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.99
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.99
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.99
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.99
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.99
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.99
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.99
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.99
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.99
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.99
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.99
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.99
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.99
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.99
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(s(0)) → c20(S(ok(0))) 234.18/71.99
PROPER(s(true)) → c20(S(ok(true))) 234.18/71.99
PROPER(s(false)) → c20(S(ok(false))) 234.18/71.99
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.99
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.99
S(mark(z0)) → c12(S(z0)) 234.18/71.99
S(ok(z0)) → c13(S(z0)) 234.18/71.99
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.99
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.99
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.99
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.99
PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 234.18/71.99
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.99
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.99
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.99
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.99
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.99
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.99
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.99
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.99
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.99
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.99
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.99
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.99
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.99
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.99
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.99
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.99
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(s(0)) → c20(S(ok(0))) 234.18/71.99
PROPER(s(true)) → c20(S(ok(true))) 234.18/71.99
PROPER(s(false)) → c20(S(ok(false))) 234.18/71.99
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, PROPER, MINUS, GEQ, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c25, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24

234.18/71.99
234.18/71.99

(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(if(z0, z1, z2)) → c25(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0)) 234.18/71.99
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.18/71.99
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true)) 234.18/71.99
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false)) 234.18/71.99
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.18/71.99
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2)) 234.18/71.99
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2)) 234.18/71.99
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2)) 234.18/71.99
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.18/71.99
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
234.18/71.99
234.18/71.99

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.99
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.99
active(geq(z0, 0)) → mark(true) 234.18/71.99
active(geq(0, s(z0))) → mark(false) 234.18/71.99
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.99
active(div(0, s(z0))) → mark(0) 234.18/71.99
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.99
active(if(true, z0, z1)) → mark(z0) 234.18/71.99
active(if(false, z0, z1)) → mark(z1) 234.18/71.99
active(s(z0)) → s(active(z0)) 234.18/71.99
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.99
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.99
s(mark(z0)) → mark(s(z0)) 234.18/71.99
s(ok(z0)) → ok(s(z0)) 234.18/71.99
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.99
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.99
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.99
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.99
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.99
proper(0) → ok(0) 234.18/71.99
proper(s(z0)) → s(proper(z0)) 234.18/71.99
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.99
proper(true) → ok(true) 234.18/71.99
proper(false) → ok(false) 234.18/71.99
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.99
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.99
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.99
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.99
top(mark(z0)) → top(proper(z0)) 234.18/71.99
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.99
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.99
S(mark(z0)) → c12(S(z0)) 234.18/71.99
S(ok(z0)) → c13(S(z0)) 234.18/71.99
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.99
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.99
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.99
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.99
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.99
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.99
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.99
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.99
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.99
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.99
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.99
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.99
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.99
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.99
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.99
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.99
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.99
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.99
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.99
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.99
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(s(0)) → c20(S(ok(0))) 234.18/71.99
PROPER(s(true)) → c20(S(ok(true))) 234.18/71.99
PROPER(s(false)) → c20(S(ok(false))) 234.18/71.99
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0)) 234.18/71.99
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.18/71.99
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true)) 234.18/71.99
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false)) 234.18/71.99
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.18/71.99
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2)) 234.18/71.99
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2)) 234.18/71.99
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2)) 234.18/71.99
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.18/71.99
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.99
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.99
S(mark(z0)) → c12(S(z0)) 234.18/71.99
S(ok(z0)) → c13(S(z0)) 234.18/71.99
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.99
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.99
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.99
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.99
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.99
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.99
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.99
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.99
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.99
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.99
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.99
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.99
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.99
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.99
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.99
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.99
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.99
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.99
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.99
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.99
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(s(0)) → c20(S(ok(0))) 234.18/71.99
PROPER(s(true)) → c20(S(ok(true))) 234.18/71.99
PROPER(s(false)) → c20(S(ok(false))) 234.18/71.99
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0)) 234.18/71.99
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.18/71.99
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true)) 234.18/71.99
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false)) 234.18/71.99
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.18/71.99
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2)) 234.18/71.99
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2)) 234.18/71.99
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2)) 234.18/71.99
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.18/71.99
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, TOP, PROPER

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25

234.18/71.99
234.18/71.99

(43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 9 trailing tuple parts
234.18/71.99
234.18/71.99

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.99
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.99
active(geq(z0, 0)) → mark(true) 234.18/71.99
active(geq(0, s(z0))) → mark(false) 234.18/71.99
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/71.99
active(div(0, s(z0))) → mark(0) 234.18/71.99
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/71.99
active(if(true, z0, z1)) → mark(z0) 234.18/71.99
active(if(false, z0, z1)) → mark(z1) 234.18/71.99
active(s(z0)) → s(active(z0)) 234.18/71.99
active(div(z0, z1)) → div(active(z0), z1) 234.18/71.99
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/71.99
s(mark(z0)) → mark(s(z0)) 234.18/71.99
s(ok(z0)) → ok(s(z0)) 234.18/71.99
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/71.99
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/71.99
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/71.99
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/71.99
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/71.99
proper(0) → ok(0) 234.18/71.99
proper(s(z0)) → s(proper(z0)) 234.18/71.99
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/71.99
proper(true) → ok(true) 234.18/71.99
proper(false) → ok(false) 234.18/71.99
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/71.99
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/71.99
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/71.99
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/71.99
top(mark(z0)) → top(proper(z0)) 234.18/71.99
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.99
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.99
S(mark(z0)) → c12(S(z0)) 234.18/71.99
S(ok(z0)) → c13(S(z0)) 234.18/71.99
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.99
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.99
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.99
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.99
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.99
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.99
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.99
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.99
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.99
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.99
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.99
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.99
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.99
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.99
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.99
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.99
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.99
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.99
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.99
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.99
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(s(0)) → c20(S(ok(0))) 234.18/71.99
PROPER(s(true)) → c20(S(ok(true))) 234.18/71.99
PROPER(s(false)) → c20(S(ok(false))) 234.18/71.99
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.18/71.99
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.18/71.99
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.18/71.99
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 234.18/71.99
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 234.18/71.99
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 234.18/71.99
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 234.18/71.99
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 234.18/71.99
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 234.18/71.99
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/71.99
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/71.99
S(mark(z0)) → c12(S(z0)) 234.18/71.99
S(ok(z0)) → c13(S(z0)) 234.18/71.99
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/71.99
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/71.99
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/71.99
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/71.99
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/71.99
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/71.99
TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) 234.18/71.99
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/71.99
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/71.99
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/71.99
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/71.99
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/71.99
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/71.99
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/71.99
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/71.99
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/71.99
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/71.99
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/71.99
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/71.99
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/71.99
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/71.99
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/71.99
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/71.99
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/71.99
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/71.99
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/71.99
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/71.99
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/71.99
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/71.99
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/71.99
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.99
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(s(0)) → c20(S(ok(0))) 234.18/71.99
PROPER(s(true)) → c20(S(ok(true))) 234.18/71.99
PROPER(s(false)) → c20(S(ok(false))) 234.18/71.99
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/71.99
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/71.99
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/71.99
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/71.99
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/71.99
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.18/71.99
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.18/71.99
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.18/71.99
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.18/71.99
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.18/71.99
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.18/71.99
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.18/71.99
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.18/71.99
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.18/71.99
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.18/71.99
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.18/71.99
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 234.18/71.99
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 234.18/71.99
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 234.18/71.99
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 234.18/71.99
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 234.18/71.99
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 234.18/71.99
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.18/71.99
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, TOP, PROPER

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c28, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25, c25

234.18/71.99
234.18/71.99

(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(mark(z0)) → c28(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/71.99
TOP(mark(0)) → c28(TOP(ok(0)), PROPER(0)) 234.18/71.99
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 234.18/71.99
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/71.99
TOP(mark(true)) → c28(TOP(ok(true)), PROPER(true)) 234.18/71.99
TOP(mark(false)) → c28(TOP(ok(false)), PROPER(false)) 234.18/71.99
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/71.99
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
234.18/71.99
234.18/71.99

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/71.99
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/71.99
active(geq(z0, 0)) → mark(true) 234.18/71.99
active(geq(0, s(z0))) → mark(false) 234.18/72.00
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/72.00
active(div(0, s(z0))) → mark(0) 234.18/72.00
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/72.00
active(if(true, z0, z1)) → mark(z0) 234.18/72.00
active(if(false, z0, z1)) → mark(z1) 234.18/72.00
active(s(z0)) → s(active(z0)) 234.18/72.00
active(div(z0, z1)) → div(active(z0), z1) 234.18/72.00
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/72.00
s(mark(z0)) → mark(s(z0)) 234.18/72.00
s(ok(z0)) → ok(s(z0)) 234.18/72.00
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/72.00
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/72.00
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/72.00
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/72.00
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/72.00
proper(0) → ok(0) 234.18/72.00
proper(s(z0)) → s(proper(z0)) 234.18/72.00
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/72.00
proper(true) → ok(true) 234.18/72.00
proper(false) → ok(false) 234.18/72.00
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/72.00
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/72.00
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/72.00
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/72.00
top(mark(z0)) → top(proper(z0)) 234.18/72.00
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/72.00
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/72.00
S(mark(z0)) → c12(S(z0)) 234.18/72.00
S(ok(z0)) → c13(S(z0)) 234.18/72.00
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/72.00
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/72.00
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/72.00
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/72.00
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/72.00
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/72.00
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/72.00
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/72.00
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/72.00
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/72.00
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/72.00
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/72.00
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/72.00
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/72.00
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/72.00
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/72.00
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/72.00
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/72.00
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/72.00
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/72.00
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/72.00
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/72.00
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/72.00
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/72.00
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/72.00
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/72.00
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/72.00
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/72.00
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/72.00
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/72.00
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/72.00
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/72.00
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/72.00
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/72.00
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/72.00
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/72.00
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/72.00
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/72.00
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/72.00
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/72.00
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/72.00
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/72.00
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/72.00
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/72.00
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/72.00
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/72.00
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/72.00
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/72.00
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/72.00
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/72.00
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/72.00
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/72.00
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/72.00
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/72.00
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/72.00
PROPER(s(0)) → c20(S(ok(0))) 234.18/72.00
PROPER(s(true)) → c20(S(ok(true))) 234.18/72.00
PROPER(s(false)) → c20(S(ok(false))) 234.18/72.00
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/72.00
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/72.00
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/72.00
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/72.00
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/72.00
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/72.00
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/72.00
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.18/72.00
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.18/72.00
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.18/72.00
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/72.00
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/72.00
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/72.00
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/72.00
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/72.00
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/72.00
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/72.00
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.18/72.00
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.18/72.00
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.18/72.00
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.18/72.00
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.18/72.00
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.18/72.00
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.18/72.00
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.18/72.00
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.18/72.00
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.18/72.00
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.18/72.00
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.18/72.00
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.18/72.00
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/72.00
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.18/72.00
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/72.00
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/72.00
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 234.18/72.00
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 234.18/72.00
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 234.18/72.00
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 234.18/72.00
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 234.18/72.00
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 234.18/72.00
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 234.18/72.00
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.18/72.00
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.18/72.00
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.18/72.00
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/72.00
TOP(mark(0)) → c28(TOP(ok(0)), PROPER(0)) 234.18/72.00
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 234.18/72.00
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/72.00
TOP(mark(true)) → c28(TOP(ok(true)), PROPER(true)) 234.18/72.00
TOP(mark(false)) → c28(TOP(ok(false)), PROPER(false)) 234.18/72.00
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/72.00
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/72.00
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/72.00
S(mark(z0)) → c12(S(z0)) 234.18/72.00
S(ok(z0)) → c13(S(z0)) 234.18/72.00
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/72.00
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/72.00
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/72.00
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/72.00
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/72.00
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/72.00
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/72.00
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/72.00
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/72.00
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/72.00
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/72.00
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/72.00
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/72.00
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/72.00
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/72.00
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/72.00
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/72.00
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/72.00
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/72.00
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/72.00
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/72.00
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/72.00
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/72.00
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/72.00
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/72.00
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/72.00
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/72.00
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/72.00
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/72.00
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/72.00
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/72.00
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/72.00
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/72.00
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/72.00
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/72.00
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/72.00
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/72.00
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/72.00
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/72.00
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/72.00
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/72.00
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/72.00
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/72.00
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/72.00
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/72.00
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/72.00
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/72.00
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/72.00
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/72.00
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/72.00
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/72.00
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/72.00
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/72.00
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/72.00
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.18/72.00
PROPER(s(0)) → c20(S(ok(0))) 234.18/72.00
PROPER(s(true)) → c20(S(ok(true))) 234.18/72.00
PROPER(s(false)) → c20(S(ok(false))) 234.18/72.00
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/72.00
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/72.00
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/72.00
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/72.00
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/72.00
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/72.00
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/72.00
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.18/72.00
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.18/72.00
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.18/72.00
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/72.00
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/72.00
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/72.00
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/72.00
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/72.00
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/72.00
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/72.00
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.18/72.00
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.18/72.00
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.18/72.00
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.18/72.00
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.18/72.00
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.18/72.00
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.18/72.00
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.18/72.00
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.18/72.00
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.18/72.00
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.18/72.00
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.18/72.00
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.18/72.00
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/72.00
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.18/72.00
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/72.00
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.18/72.00
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 234.18/72.00
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 234.18/72.00
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 234.18/72.00
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 234.18/72.00
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 234.18/72.00
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 234.18/72.00
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 234.18/72.00
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.18/72.00
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.18/72.00
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.18/72.00
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/72.00
TOP(mark(0)) → c28(TOP(ok(0)), PROPER(0)) 234.18/72.00
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 234.18/72.00
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/72.00
TOP(mark(true)) → c28(TOP(ok(true)), PROPER(true)) 234.18/72.00
TOP(mark(false)) → c28(TOP(ok(false)), PROPER(false)) 234.18/72.00
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/72.00
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, TOP, PROPER

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25, c25, c28

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(47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts
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(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.18/72.00
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.18/72.00
active(geq(z0, 0)) → mark(true) 234.18/72.00
active(geq(0, s(z0))) → mark(false) 234.18/72.00
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.18/72.00
active(div(0, s(z0))) → mark(0) 234.18/72.00
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.18/72.00
active(if(true, z0, z1)) → mark(z0) 234.18/72.00
active(if(false, z0, z1)) → mark(z1) 234.18/72.00
active(s(z0)) → s(active(z0)) 234.18/72.00
active(div(z0, z1)) → div(active(z0), z1) 234.18/72.00
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.18/72.00
s(mark(z0)) → mark(s(z0)) 234.18/72.00
s(ok(z0)) → ok(s(z0)) 234.18/72.00
div(mark(z0), z1) → mark(div(z0, z1)) 234.18/72.00
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.18/72.00
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.18/72.00
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.18/72.00
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.18/72.00
proper(0) → ok(0) 234.18/72.00
proper(s(z0)) → s(proper(z0)) 234.18/72.00
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.18/72.00
proper(true) → ok(true) 234.18/72.00
proper(false) → ok(false) 234.18/72.00
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.18/72.00
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.18/72.00
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.18/72.00
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.18/72.00
top(mark(z0)) → top(proper(z0)) 234.18/72.00
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.18/72.00
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.18/72.00
S(mark(z0)) → c12(S(z0)) 234.18/72.00
S(ok(z0)) → c13(S(z0)) 234.18/72.00
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.18/72.00
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.18/72.00
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.18/72.00
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.18/72.00
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.18/72.00
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.18/72.00
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.18/72.00
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.18/72.00
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.18/72.00
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.18/72.00
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.18/72.00
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.18/72.00
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.18/72.00
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.18/72.00
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.18/72.00
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.18/72.00
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.18/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.18/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.18/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.18/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.18/72.00
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.18/72.00
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.18/72.00
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.18/72.00
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.18/72.00
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.18/72.00
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.18/72.00
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.18/72.00
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.18/72.00
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.18/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.18/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.18/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.18/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.18/72.00
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.18/72.00
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.18/72.00
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.18/72.00
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.18/72.00
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.18/72.00
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.18/72.00
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.18/72.00
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.18/72.00
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/72.00
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.18/72.00
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.18/72.00
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.18/72.00
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.18/72.00
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.18/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.18/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.18/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.18/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.18/72.00
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.18/72.00
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.18/72.00
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.18/72.00
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.18/72.00
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.18/72.00
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.18/72.00
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.18/72.00
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.18/72.00
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.18/72.00
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.18/72.00
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.18/72.00
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.18/72.00
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.18/72.00
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.18/72.00
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.18/72.00
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.18/72.00
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(s(0)) → c20(S(ok(0))) 234.53/72.00
PROPER(s(true)) → c20(S(ok(true))) 234.53/72.00
PROPER(s(false)) → c20(S(ok(false))) 234.53/72.00
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.53/72.00
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.53/72.00
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.00
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 234.53/72.00
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.00
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.00
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.00
TOP(mark(0)) → c28(TOP(ok(0))) 234.53/72.00
TOP(mark(true)) → c28(TOP(ok(true))) 234.53/72.00
TOP(mark(false)) → c28(TOP(ok(false)))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.53/72.00
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.53/72.00
S(mark(z0)) → c12(S(z0)) 234.53/72.00
S(ok(z0)) → c13(S(z0)) 234.53/72.00
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.00
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.00
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.00
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.00
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.00
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.53/72.00
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.53/72.00
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.53/72.00
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.53/72.00
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.53/72.00
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.53/72.00
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.53/72.00
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.53/72.00
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.53/72.00
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.53/72.00
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.53/72.00
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.53/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.53/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.53/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.00
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.53/72.00
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.53/72.00
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.53/72.00
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.53/72.00
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.53/72.00
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.53/72.00
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.53/72.00
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.53/72.00
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.53/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.53/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.53/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.00
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.53/72.00
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.53/72.00
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.53/72.00
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.53/72.00
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.53/72.00
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.53/72.00
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.53/72.00
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.53/72.00
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.00
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.53/72.00
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.53/72.00
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.00
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.53/72.00
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.53/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.53/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.53/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.00
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.53/72.00
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.00
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(s(0)) → c20(S(ok(0))) 234.53/72.00
PROPER(s(true)) → c20(S(ok(true))) 234.53/72.00
PROPER(s(false)) → c20(S(ok(false))) 234.53/72.00
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.53/72.00
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.53/72.00
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.00
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 234.53/72.00
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.00
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.00
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.00
TOP(mark(0)) → c28(TOP(ok(0))) 234.53/72.00
TOP(mark(true)) → c28(TOP(ok(true))) 234.53/72.00
TOP(mark(false)) → c28(TOP(ok(false)))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, TOP, PROPER

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25, c25, c28, c28

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(49) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TOP(mark(0)) → c28(TOP(ok(0))) 234.53/72.00
TOP(mark(false)) → c28(TOP(ok(false)))
We considered the (Usable) Rules:

proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.00
proper(0) → ok(0) 234.53/72.00
proper(s(z0)) → s(proper(z0)) 234.53/72.00
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.00
proper(true) → ok(true) 234.53/72.00
proper(false) → ok(false) 234.53/72.00
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.00
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.00
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.00
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.00
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.00
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.00
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.00
s(mark(z0)) → mark(s(z0)) 234.53/72.00
s(ok(z0)) → ok(s(z0)) 234.53/72.00
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.00
active(minus(0, z0)) → mark(0) 234.53/72.00
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.00
active(geq(z0, 0)) → mark(true) 234.53/72.00
active(geq(0, s(z0))) → mark(false) 234.53/72.00
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.00
active(div(0, s(z0))) → mark(0) 234.53/72.00
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.00
active(if(true, z0, z1)) → mark(z0) 234.53/72.00
active(if(false, z0, z1)) → mark(z1) 234.53/72.00
active(s(z0)) → s(active(z0)) 234.53/72.00
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.00
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
And the Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.53/72.00
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.53/72.00
S(mark(z0)) → c12(S(z0)) 234.53/72.00
S(ok(z0)) → c13(S(z0)) 234.53/72.00
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.00
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.00
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.00
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.00
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.00
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.53/72.00
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.53/72.00
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.53/72.00
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.53/72.00
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.53/72.00
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.53/72.00
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.53/72.00
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.53/72.00
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.53/72.00
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.53/72.00
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.53/72.00
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.53/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.53/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.53/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.00
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.53/72.00
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.53/72.00
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.53/72.00
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.53/72.00
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.53/72.00
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.53/72.00
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.53/72.00
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.53/72.00
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.53/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.53/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.53/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.00
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.53/72.00
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.53/72.00
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.53/72.00
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.53/72.00
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.53/72.00
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.53/72.00
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.53/72.00
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.53/72.00
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.00
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.53/72.00
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.53/72.00
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.00
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.53/72.00
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.53/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.53/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.53/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.00
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.53/72.00
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.00
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(s(0)) → c20(S(ok(0))) 234.53/72.00
PROPER(s(true)) → c20(S(ok(true))) 234.53/72.00
PROPER(s(false)) → c20(S(ok(false))) 234.53/72.00
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.53/72.00
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.53/72.00
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.00
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 234.53/72.00
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.00
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.00
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.00
TOP(mark(0)) → c28(TOP(ok(0))) 234.53/72.00
TOP(mark(true)) → c28(TOP(ok(true))) 234.53/72.00
TOP(mark(false)) → c28(TOP(ok(false)))
The order we found is given by the following interpretation:
Polynomial interpretation : 234.53/72.00

POL(0) = 0    234.53/72.00
POL(ACTIVE(x1)) = 0    234.53/72.00
POL(DIV(x1, x2)) = 0    234.53/72.00
POL(GEQ(x1, x2)) = 0    234.53/72.00
POL(IF(x1, x2, x3)) = 0    234.53/72.00
POL(MINUS(x1, x2)) = 0    234.53/72.00
POL(PROPER(x1)) = 0    234.53/72.00
POL(S(x1)) = 0    234.53/72.00
POL(TOP(x1)) = [2]x1    234.53/72.00
POL(active(x1)) = x1    234.53/72.00
POL(c(x1)) = x1    234.53/72.00
POL(c1(x1)) = x1    234.53/72.00
POL(c10(x1)) = x1    234.53/72.00
POL(c10(x1, x2)) = x1 + x2    234.53/72.00
POL(c11(x1)) = x1    234.53/72.00
POL(c11(x1, x2)) = x1 + x2    234.53/72.00
POL(c12(x1)) = x1    234.53/72.00
POL(c13(x1)) = x1    234.53/72.00
POL(c14(x1)) = x1    234.53/72.00
POL(c15(x1)) = x1    234.53/72.00
POL(c16(x1)) = x1    234.53/72.00
POL(c17(x1)) = x1    234.53/72.00
POL(c18(x1, x2)) = x1 + x2    234.53/72.00
POL(c18(x1, x2, x3)) = x1 + x2 + x3    234.53/72.00
POL(c2(x1)) = x1    234.53/72.00
POL(c20(x1)) = x1    234.53/72.00
POL(c20(x1, x2)) = x1 + x2    234.53/72.00
POL(c21(x1, x2)) = x1 + x2    234.53/72.00
POL(c21(x1, x2, x3)) = x1 + x2 + x3    234.53/72.00
POL(c24(x1, x2)) = x1 + x2    234.53/72.00
POL(c24(x1, x2, x3)) = x1 + x2 + x3    234.53/72.00
POL(c25(x1, x2, x3)) = x1 + x2 + x3    234.53/72.00
POL(c25(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    234.53/72.00
POL(c26(x1)) = x1    234.53/72.00
POL(c27(x1)) = x1    234.53/72.00
POL(c28(x1)) = x1    234.53/72.00
POL(c28(x1, x2)) = x1 + x2    234.53/72.00
POL(c29(x1, x2)) = x1 + x2    234.53/72.00
POL(c3(x1)) = x1    234.53/72.00
POL(c4(x1)) = x1    234.53/72.00
POL(c5(x1)) = x1    234.53/72.00
POL(c9(x1)) = x1    234.53/72.00
POL(c9(x1, x2)) = x1 + x2    234.53/72.00
POL(div(x1, x2)) = [1]    234.53/72.00
POL(false) = 0    234.53/72.00
POL(geq(x1, x2)) = [1]    234.53/72.00
POL(if(x1, x2, x3)) = [1]    234.53/72.00
POL(mark(x1)) = [1]    234.53/72.00
POL(minus(x1, x2)) = [1]    234.53/72.00
POL(ok(x1)) = x1    234.53/72.00
POL(proper(x1)) = 0    234.53/72.00
POL(s(x1)) = [1]    234.53/72.00
POL(true) = [1]   
234.53/72.00
234.53/72.00

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.53/72.00
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.00
active(geq(z0, 0)) → mark(true) 234.53/72.00
active(geq(0, s(z0))) → mark(false) 234.53/72.00
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.00
active(div(0, s(z0))) → mark(0) 234.53/72.00
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.00
active(if(true, z0, z1)) → mark(z0) 234.53/72.00
active(if(false, z0, z1)) → mark(z1) 234.53/72.00
active(s(z0)) → s(active(z0)) 234.53/72.00
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.00
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.53/72.00
s(mark(z0)) → mark(s(z0)) 234.53/72.00
s(ok(z0)) → ok(s(z0)) 234.53/72.00
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.00
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.00
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.00
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.00
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.00
proper(0) → ok(0) 234.53/72.00
proper(s(z0)) → s(proper(z0)) 234.53/72.00
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.00
proper(true) → ok(true) 234.53/72.00
proper(false) → ok(false) 234.53/72.00
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.00
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.00
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.00
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.00
top(mark(z0)) → top(proper(z0)) 234.53/72.00
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.53/72.00
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.53/72.00
S(mark(z0)) → c12(S(z0)) 234.53/72.00
S(ok(z0)) → c13(S(z0)) 234.53/72.00
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.00
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.00
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.00
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.00
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.00
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.53/72.00
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.53/72.00
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.53/72.00
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.53/72.00
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.53/72.00
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.53/72.00
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.53/72.00
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.53/72.00
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.53/72.00
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.53/72.00
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.53/72.00
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.53/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.53/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.53/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.00
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.53/72.00
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.53/72.00
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.53/72.00
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.53/72.00
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.53/72.00
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.53/72.00
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.53/72.00
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.53/72.00
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.53/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.53/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.53/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.00
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.53/72.00
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.53/72.00
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.53/72.00
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.53/72.00
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.53/72.00
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.53/72.00
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.53/72.00
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.53/72.00
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.00
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.53/72.00
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.53/72.00
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.00
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.53/72.00
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.53/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.53/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.53/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.00
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.53/72.00
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.00
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(s(0)) → c20(S(ok(0))) 234.53/72.00
PROPER(s(true)) → c20(S(ok(true))) 234.53/72.00
PROPER(s(false)) → c20(S(ok(false))) 234.53/72.00
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.53/72.00
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.53/72.00
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.00
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 234.53/72.00
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.00
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.00
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.00
TOP(mark(0)) → c28(TOP(ok(0))) 234.53/72.00
TOP(mark(true)) → c28(TOP(ok(true))) 234.53/72.00
TOP(mark(false)) → c28(TOP(ok(false)))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.53/72.00
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.53/72.00
S(mark(z0)) → c12(S(z0)) 234.53/72.00
S(ok(z0)) → c13(S(z0)) 234.53/72.00
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.00
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.00
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.00
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.00
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.00
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.53/72.00
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.53/72.00
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.53/72.00
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.53/72.00
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.53/72.00
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.53/72.00
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.53/72.00
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.53/72.00
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.53/72.00
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.53/72.00
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.53/72.00
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.53/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.53/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.53/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.00
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.53/72.00
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.53/72.00
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.53/72.00
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.53/72.00
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.53/72.00
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.53/72.00
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.53/72.00
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.53/72.00
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.53/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.53/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.53/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.00
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.53/72.00
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.53/72.00
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.53/72.00
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.53/72.00
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.53/72.00
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.53/72.00
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.53/72.00
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.53/72.00
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.00
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.53/72.00
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.53/72.00
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.00
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.53/72.00
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.53/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.53/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.53/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.00
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.53/72.00
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.00
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(s(0)) → c20(S(ok(0))) 234.53/72.00
PROPER(s(true)) → c20(S(ok(true))) 234.53/72.00
PROPER(s(false)) → c20(S(ok(false))) 234.53/72.00
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.53/72.00
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.53/72.00
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.00
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 234.53/72.00
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.00
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.00
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.00
TOP(mark(true)) → c28(TOP(ok(true)))
K tuples:

TOP(mark(0)) → c28(TOP(ok(0))) 234.53/72.00
TOP(mark(false)) → c28(TOP(ok(false)))
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, TOP, PROPER

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25, c25, c28, c28

234.53/72.00
234.53/72.00

(51) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TOP(mark(true)) → c28(TOP(ok(true)))
We considered the (Usable) Rules:

proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.00
proper(0) → ok(0) 234.53/72.00
proper(s(z0)) → s(proper(z0)) 234.53/72.00
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.00
proper(true) → ok(true) 234.53/72.00
proper(false) → ok(false) 234.53/72.00
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.00
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.00
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.00
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.00
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.00
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.00
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.00
s(mark(z0)) → mark(s(z0)) 234.53/72.00
s(ok(z0)) → ok(s(z0)) 234.53/72.00
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.00
active(minus(0, z0)) → mark(0) 234.53/72.00
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.00
active(geq(z0, 0)) → mark(true) 234.53/72.00
active(geq(0, s(z0))) → mark(false) 234.53/72.00
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.00
active(div(0, s(z0))) → mark(0) 234.53/72.00
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.00
active(if(true, z0, z1)) → mark(z0) 234.53/72.00
active(if(false, z0, z1)) → mark(z1) 234.53/72.00
active(s(z0)) → s(active(z0)) 234.53/72.00
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.00
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
And the Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.53/72.00
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.53/72.00
S(mark(z0)) → c12(S(z0)) 234.53/72.00
S(ok(z0)) → c13(S(z0)) 234.53/72.00
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.00
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.00
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.00
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.00
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.00
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.53/72.00
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.53/72.00
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.53/72.00
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.53/72.00
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.53/72.00
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.53/72.00
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.53/72.00
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.53/72.00
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.53/72.00
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.53/72.00
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.53/72.00
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.53/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.53/72.00
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.53/72.00
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.00
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.53/72.00
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.53/72.00
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.53/72.00
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.53/72.00
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.53/72.00
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.53/72.00
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.53/72.00
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.53/72.00
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.53/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.53/72.00
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.53/72.00
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.00
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.53/72.00
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.53/72.00
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.53/72.00
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.53/72.00
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.53/72.00
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.53/72.00
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.53/72.00
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.53/72.00
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.00
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.53/72.00
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.53/72.00
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.00
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.53/72.00
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.53/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.53/72.00
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.53/72.00
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.00
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.53/72.00
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.00
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(s(0)) → c20(S(ok(0))) 234.53/72.00
PROPER(s(true)) → c20(S(ok(true))) 234.53/72.00
PROPER(s(false)) → c20(S(ok(false))) 234.53/72.00
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.00
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.00
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.00
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.00
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.00
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.53/72.00
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.53/72.00
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.53/72.00
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.53/72.00
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.53/72.00
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.53/72.00
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.53/72.00
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.53/72.00
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 234.53/72.00
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.00
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.00
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.00
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 234.53/72.00
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.00
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.00
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.00
TOP(mark(0)) → c28(TOP(ok(0))) 234.53/72.00
TOP(mark(true)) → c28(TOP(ok(true))) 234.53/72.00
TOP(mark(false)) → c28(TOP(ok(false)))
The order we found is given by the following interpretation:
Polynomial interpretation : 234.53/72.00

POL(0) = 0    234.53/72.00
POL(ACTIVE(x1)) = 0    234.53/72.00
POL(DIV(x1, x2)) = 0    234.53/72.00
POL(GEQ(x1, x2)) = 0    234.53/72.00
POL(IF(x1, x2, x3)) = 0    234.53/72.00
POL(MINUS(x1, x2)) = 0    234.53/72.00
POL(PROPER(x1)) = 0    234.53/72.00
POL(S(x1)) = 0    234.53/72.00
POL(TOP(x1)) = [2]x1    234.53/72.00
POL(active(x1)) = x1    234.53/72.00
POL(c(x1)) = x1    234.53/72.00
POL(c1(x1)) = x1    234.53/72.00
POL(c10(x1)) = x1    234.53/72.00
POL(c10(x1, x2)) = x1 + x2    234.53/72.00
POL(c11(x1)) = x1    234.53/72.00
POL(c11(x1, x2)) = x1 + x2    234.53/72.00
POL(c12(x1)) = x1    234.53/72.00
POL(c13(x1)) = x1    234.53/72.00
POL(c14(x1)) = x1    234.53/72.00
POL(c15(x1)) = x1    234.53/72.00
POL(c16(x1)) = x1    234.53/72.00
POL(c17(x1)) = x1    234.53/72.00
POL(c18(x1, x2)) = x1 + x2    234.53/72.00
POL(c18(x1, x2, x3)) = x1 + x2 + x3    234.53/72.00
POL(c2(x1)) = x1    234.53/72.00
POL(c20(x1)) = x1    234.53/72.00
POL(c20(x1, x2)) = x1 + x2    234.53/72.00
POL(c21(x1, x2)) = x1 + x2    234.53/72.00
POL(c21(x1, x2, x3)) = x1 + x2 + x3    234.53/72.00
POL(c24(x1, x2)) = x1 + x2    234.53/72.00
POL(c24(x1, x2, x3)) = x1 + x2 + x3    234.53/72.00
POL(c25(x1, x2, x3)) = x1 + x2 + x3    234.53/72.00
POL(c25(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    234.53/72.00
POL(c26(x1)) = x1    234.53/72.00
POL(c27(x1)) = x1    234.53/72.00
POL(c28(x1)) = x1    234.53/72.00
POL(c28(x1, x2)) = x1 + x2    234.53/72.00
POL(c29(x1, x2)) = x1 + x2    234.53/72.00
POL(c3(x1)) = x1    234.53/72.00
POL(c4(x1)) = x1    234.53/72.00
POL(c5(x1)) = x1    234.53/72.00
POL(c9(x1)) = x1    234.53/72.00
POL(c9(x1, x2)) = x1 + x2    234.53/72.00
POL(div(x1, x2)) = [2]    234.53/72.00
POL(false) = [1]    234.53/72.00
POL(geq(x1, x2)) = [2]    234.53/72.00
POL(if(x1, x2, x3)) = [2]    234.53/72.00
POL(mark(x1)) = [2]    234.53/72.00
POL(minus(x1, x2)) = [2]    234.53/72.00
POL(ok(x1)) = x1    234.53/72.00
POL(proper(x1)) = 0    234.53/72.00
POL(s(x1)) = [2]    234.53/72.00
POL(true) = 0   
234.53/72.00
234.53/72.00

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.53/72.00
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.00
active(geq(z0, 0)) → mark(true) 234.53/72.00
active(geq(0, s(z0))) → mark(false) 234.53/72.00
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.00
active(div(0, s(z0))) → mark(0) 234.53/72.00
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.00
active(if(true, z0, z1)) → mark(z0) 234.53/72.00
active(if(false, z0, z1)) → mark(z1) 234.53/72.00
active(s(z0)) → s(active(z0)) 234.53/72.00
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.00
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.53/72.00
s(mark(z0)) → mark(s(z0)) 234.53/72.00
s(ok(z0)) → ok(s(z0)) 234.53/72.00
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.00
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.00
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.00
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.00
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.00
proper(0) → ok(0) 234.53/72.00
proper(s(z0)) → s(proper(z0)) 234.53/72.00
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.00
proper(true) → ok(true) 234.53/72.00
proper(false) → ok(false) 234.53/72.00
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.00
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.00
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.00
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.00
top(mark(z0)) → top(proper(z0)) 234.53/72.00
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.53/72.00
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.53/72.00
S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.53/72.01
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.53/72.01
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.53/72.01
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.53/72.01
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.53/72.01
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.53/72.01
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.53/72.01
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.53/72.01
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.53/72.01
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.53/72.01
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.53/72.01
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.53/72.01
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.53/72.01
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.53/72.01
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.53/72.01
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.53/72.01
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.53/72.01
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.53/72.01
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.53/72.01
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.53/72.01
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.53/72.01
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.53/72.01
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.53/72.01
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.53/72.01
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.53/72.01
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.53/72.01
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.53/72.01
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.53/72.01
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.53/72.01
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.53/72.01
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.01
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.53/72.01
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.53/72.01
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.01
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.53/72.01
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.53/72.01
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.53/72.01
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.53/72.01
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.01
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.01
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.01
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.01
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.53/72.01
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.53/72.01
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.53/72.01
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.53/72.01
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.01
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(s(0)) → c20(S(ok(0))) 234.53/72.01
PROPER(s(true)) → c20(S(ok(true))) 234.53/72.01
PROPER(s(false)) → c20(S(ok(false))) 234.53/72.01
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.01
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.01
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.01
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.01
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.53/72.01
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.53/72.01
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.53/72.01
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.01
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.01
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.01
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.01
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.53/72.01
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.53/72.01
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.53/72.01
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.53/72.01
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.53/72.01
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.53/72.01
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 234.53/72.01
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 234.53/72.01
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 234.53/72.01
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.01
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.01
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.01
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.01
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.01
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 234.53/72.01
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.01
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.01
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.01
TOP(mark(0)) → c28(TOP(ok(0))) 234.53/72.01
TOP(mark(true)) → c28(TOP(ok(true))) 234.53/72.01
TOP(mark(false)) → c28(TOP(ok(false)))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.53/72.01
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.53/72.01
S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.53/72.01
TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) 234.53/72.01
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.53/72.01
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.53/72.01
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.53/72.01
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.53/72.01
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.53/72.01
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.53/72.01
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.53/72.01
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.53/72.01
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.53/72.01
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.53/72.01
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.53/72.01
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.53/72.01
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.53/72.01
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.53/72.01
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.53/72.01
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.53/72.01
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.53/72.01
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.53/72.01
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.53/72.01
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.53/72.01
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.53/72.01
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.53/72.01
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.53/72.01
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.53/72.01
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.53/72.01
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.53/72.01
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.53/72.01
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.53/72.01
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.01
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.53/72.01
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.53/72.01
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.01
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.53/72.01
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.53/72.01
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.53/72.01
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.53/72.01
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.01
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.01
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.01
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.01
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.53/72.01
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.53/72.01
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.53/72.01
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.53/72.01
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.01
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(s(0)) → c20(S(ok(0))) 234.53/72.01
PROPER(s(true)) → c20(S(ok(true))) 234.53/72.01
PROPER(s(false)) → c20(S(ok(false))) 234.53/72.01
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.01
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.01
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.01
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.01
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.53/72.01
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.53/72.01
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.53/72.01
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.01
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.01
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.01
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.01
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.53/72.01
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.53/72.01
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.53/72.01
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.53/72.01
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.53/72.01
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.53/72.01
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 234.53/72.01
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 234.53/72.01
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 234.53/72.01
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.01
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.01
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.01
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.01
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.01
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 234.53/72.01
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.01
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.01
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
K tuples:

TOP(mark(0)) → c28(TOP(ok(0))) 234.53/72.01
TOP(mark(false)) → c28(TOP(ok(false))) 234.53/72.01
TOP(mark(true)) → c28(TOP(ok(true)))
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, TOP, PROPER

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c29, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25, c25, c28, c28

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(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(ok(z0)) → c29(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(minus(0, z0))) → c29(TOP(mark(0)), ACTIVE(minus(0, z0))) 234.53/72.01
TOP(ok(minus(s(z0), s(z1)))) → c29(TOP(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
TOP(ok(geq(z0, 0))) → c29(TOP(mark(true)), ACTIVE(geq(z0, 0))) 234.53/72.01
TOP(ok(geq(0, s(z0)))) → c29(TOP(mark(false)), ACTIVE(geq(0, s(z0)))) 234.53/72.01
TOP(ok(geq(s(z0), s(z1)))) → c29(TOP(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
TOP(ok(div(0, s(z0)))) → c29(TOP(mark(0)), ACTIVE(div(0, s(z0)))) 234.53/72.01
TOP(ok(div(s(z0), s(z1)))) → c29(TOP(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
TOP(ok(if(true, z0, z1))) → c29(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 234.53/72.01
TOP(ok(if(false, z0, z1))) → c29(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 234.53/72.01
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0))) 234.53/72.01
TOP(ok(div(z0, z1))) → c29(TOP(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.53/72.01
TOP(ok(if(z0, z1, z2))) → c29(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
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(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.53/72.01
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.01
active(geq(z0, 0)) → mark(true) 234.53/72.01
active(geq(0, s(z0))) → mark(false) 234.53/72.01
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.01
active(div(0, s(z0))) → mark(0) 234.53/72.01
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.01
active(if(true, z0, z1)) → mark(z0) 234.53/72.01
active(if(false, z0, z1)) → mark(z1) 234.53/72.01
active(s(z0)) → s(active(z0)) 234.53/72.01
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.01
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.53/72.01
s(mark(z0)) → mark(s(z0)) 234.53/72.01
s(ok(z0)) → ok(s(z0)) 234.53/72.01
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.01
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.01
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.01
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.01
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.01
proper(0) → ok(0) 234.53/72.01
proper(s(z0)) → s(proper(z0)) 234.53/72.01
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.01
proper(true) → ok(true) 234.53/72.01
proper(false) → ok(false) 234.53/72.01
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.01
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.01
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.01
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.01
top(mark(z0)) → top(proper(z0)) 234.53/72.01
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.53/72.01
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.53/72.01
S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.53/72.01
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.53/72.01
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.53/72.01
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.53/72.01
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.53/72.01
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.53/72.01
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.53/72.01
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.53/72.01
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.53/72.01
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.53/72.01
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.53/72.01
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.53/72.01
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.53/72.01
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.53/72.01
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.53/72.01
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.53/72.01
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.53/72.01
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.53/72.01
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.53/72.01
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.53/72.01
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.53/72.01
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.53/72.01
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.53/72.01
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.53/72.01
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.53/72.01
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.53/72.01
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.53/72.01
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.53/72.01
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.53/72.01
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.01
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.53/72.01
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.53/72.01
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.01
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.53/72.01
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.53/72.01
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.53/72.01
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.53/72.01
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.01
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.01
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.01
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.01
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.53/72.01
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.53/72.01
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.53/72.01
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.53/72.01
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.01
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(s(0)) → c20(S(ok(0))) 234.53/72.01
PROPER(s(true)) → c20(S(ok(true))) 234.53/72.01
PROPER(s(false)) → c20(S(ok(false))) 234.53/72.01
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.01
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.01
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.01
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.01
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.53/72.01
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.53/72.01
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.53/72.01
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.01
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.01
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.01
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.01
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.53/72.01
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.53/72.01
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.53/72.01
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.53/72.01
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.53/72.01
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.53/72.01
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 234.53/72.01
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 234.53/72.01
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 234.53/72.01
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.01
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.01
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.01
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.01
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.01
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 234.53/72.01
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.01
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.01
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.01
TOP(mark(0)) → c28(TOP(ok(0))) 234.53/72.01
TOP(mark(true)) → c28(TOP(ok(true))) 234.53/72.01
TOP(mark(false)) → c28(TOP(ok(false))) 234.53/72.01
TOP(ok(minus(0, z0))) → c29(TOP(mark(0)), ACTIVE(minus(0, z0))) 234.53/72.01
TOP(ok(minus(s(z0), s(z1)))) → c29(TOP(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
TOP(ok(geq(z0, 0))) → c29(TOP(mark(true)), ACTIVE(geq(z0, 0))) 234.53/72.01
TOP(ok(geq(0, s(z0)))) → c29(TOP(mark(false)), ACTIVE(geq(0, s(z0)))) 234.53/72.01
TOP(ok(geq(s(z0), s(z1)))) → c29(TOP(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
TOP(ok(div(0, s(z0)))) → c29(TOP(mark(0)), ACTIVE(div(0, s(z0)))) 234.53/72.01
TOP(ok(div(s(z0), s(z1)))) → c29(TOP(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
TOP(ok(if(true, z0, z1))) → c29(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 234.53/72.01
TOP(ok(if(false, z0, z1))) → c29(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 234.53/72.01
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0))) 234.53/72.01
TOP(ok(div(z0, z1))) → c29(TOP(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.53/72.01
TOP(ok(if(z0, z1, z2))) → c29(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
S tuples:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.53/72.01
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.53/72.01
S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.53/72.01
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.53/72.01
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.53/72.01
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.53/72.01
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.53/72.01
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.53/72.01
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.53/72.01
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.53/72.01
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.53/72.01
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.53/72.01
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.53/72.01
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.53/72.01
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.53/72.01
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.53/72.01
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.53/72.01
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.53/72.01
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.53/72.01
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.53/72.01
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.53/72.01
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.53/72.01
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.53/72.01
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.53/72.01
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.53/72.01
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.53/72.01
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.53/72.01
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.53/72.01
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.53/72.01
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.53/72.01
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.53/72.01
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.01
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.53/72.01
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.53/72.01
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.01
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.53/72.01
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.53/72.01
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.53/72.01
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.53/72.01
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.01
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.01
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.01
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.01
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.53/72.01
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.53/72.01
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.53/72.01
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.53/72.01
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.01
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(s(0)) → c20(S(ok(0))) 234.53/72.01
PROPER(s(true)) → c20(S(ok(true))) 234.53/72.01
PROPER(s(false)) → c20(S(ok(false))) 234.53/72.01
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.01
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.01
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.01
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.01
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.53/72.01
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.53/72.01
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.53/72.01
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.01
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.01
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.01
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.01
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.53/72.01
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.53/72.01
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.53/72.01
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.53/72.01
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.53/72.01
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.53/72.01
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 234.53/72.01
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 234.53/72.01
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 234.53/72.01
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.01
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.01
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.01
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.01
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.01
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 234.53/72.01
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.01
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.01
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.01
TOP(ok(minus(0, z0))) → c29(TOP(mark(0)), ACTIVE(minus(0, z0))) 234.53/72.01
TOP(ok(minus(s(z0), s(z1)))) → c29(TOP(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
TOP(ok(geq(z0, 0))) → c29(TOP(mark(true)), ACTIVE(geq(z0, 0))) 234.53/72.01
TOP(ok(geq(0, s(z0)))) → c29(TOP(mark(false)), ACTIVE(geq(0, s(z0)))) 234.53/72.01
TOP(ok(geq(s(z0), s(z1)))) → c29(TOP(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
TOP(ok(div(0, s(z0)))) → c29(TOP(mark(0)), ACTIVE(div(0, s(z0)))) 234.53/72.01
TOP(ok(div(s(z0), s(z1)))) → c29(TOP(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
TOP(ok(if(true, z0, z1))) → c29(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 234.53/72.01
TOP(ok(if(false, z0, z1))) → c29(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 234.53/72.01
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0))) 234.53/72.01
TOP(ok(div(z0, z1))) → c29(TOP(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.53/72.01
TOP(ok(if(z0, z1, z2))) → c29(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
K tuples:

TOP(mark(0)) → c28(TOP(ok(0))) 234.53/72.01
TOP(mark(false)) → c28(TOP(ok(false))) 234.53/72.01
TOP(mark(true)) → c28(TOP(ok(true)))
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

ACTIVE, S, DIV, IF, MINUS, GEQ, PROPER, TOP

Compound Symbols:

c1, c4, c12, c13, c14, c15, c16, c17, c26, c27, c, c9, c9, c2, c10, c10, c3, c11, c11, c5, c18, c18, c20, c20, c21, c21, c24, c24, c25, c25, c28, c28, c29

234.53/72.01
234.53/72.01

(55) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

ACTIVE(minus(s(z0), s(z1))) → c1(MINUS(z0, z1)) 234.53/72.01
ACTIVE(geq(s(z0), s(z1))) → c4(GEQ(z0, z1)) 234.53/72.01
ACTIVE(div(s(z0), s(z1))) → c(GEQ(z0, z1)) 234.53/72.01
ACTIVE(div(s(z0), s(z1))) → c(MINUS(z0, z1)) 234.53/72.01
ACTIVE(div(s(z0), s(z1))) → c(S(z1)) 234.53/72.01
ACTIVE(s(div(s(z0), s(z1)))) → c9(S(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 234.53/72.01
ACTIVE(s(div(z0, z1))) → c9(S(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.53/72.01
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 234.53/72.01
ACTIVE(s(minus(0, z0))) → c9(S(mark(0))) 234.53/72.01
ACTIVE(s(geq(z0, 0))) → c9(S(mark(true))) 234.53/72.01
ACTIVE(s(geq(0, s(z0)))) → c9(S(mark(false))) 234.53/72.01
ACTIVE(s(minus(s(z0), s(z1)))) → c2(S(mark(minus(z0, z1)))) 234.53/72.01
ACTIVE(s(minus(s(z0), s(z1)))) → c2(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
ACTIVE(s(geq(s(z0), s(z1)))) → c2(S(mark(geq(z0, z1)))) 234.53/72.01
ACTIVE(s(geq(s(z0), s(z1)))) → c2(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
ACTIVE(div(div(s(z0), s(z1)), x1)) → c10(DIV(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
ACTIVE(div(s(z0), x1)) → c10(DIV(s(active(z0)), x1), ACTIVE(s(z0))) 234.53/72.01
ACTIVE(div(div(z0, z1), x1)) → c10(DIV(div(active(z0), z1), x1), ACTIVE(div(z0, z1))) 234.53/72.01
ACTIVE(div(if(z0, z1, z2), x1)) → c10(DIV(if(active(z0), z1, z2), x1), ACTIVE(if(z0, z1, z2))) 234.53/72.01
ACTIVE(s(div(0, s(z0)))) → c9(S(mark(0))) 234.53/72.01
ACTIVE(div(minus(0, z0), x1)) → c10(DIV(mark(0), x1)) 234.53/72.01
ACTIVE(div(geq(z0, 0), x1)) → c10(DIV(mark(true), x1)) 234.53/72.01
ACTIVE(div(geq(0, s(z0)), x1)) → c10(DIV(mark(false), x1)) 234.53/72.01
ACTIVE(div(div(0, s(z0)), x1)) → c10(DIV(mark(0), x1)) 234.53/72.01
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(DIV(mark(minus(z0, z1)), x1)) 234.53/72.01
ACTIVE(div(minus(s(z0), s(z1)), x1)) → c3(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(DIV(mark(geq(z0, z1)), x1)) 234.53/72.01
ACTIVE(div(geq(s(z0), s(z1)), x1)) → c3(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
ACTIVE(if(div(s(z0), s(z1)), x1, x2)) → c11(IF(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)), x1, x2), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
ACTIVE(if(s(z0), x1, x2)) → c11(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 234.53/72.01
ACTIVE(if(div(z0, z1), x1, x2)) → c11(IF(div(active(z0), z1), x1, x2), ACTIVE(div(z0, z1))) 234.53/72.01
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c11(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 234.53/72.01
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 234.53/72.01
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 234.53/72.01
ACTIVE(div(if(true, z0, z1), x1)) → c10(DIV(mark(z0), x1)) 234.53/72.01
ACTIVE(div(if(false, z0, z1), x1)) → c10(DIV(mark(z1), x1)) 234.53/72.01
ACTIVE(if(minus(0, z0), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.01
ACTIVE(if(geq(z0, 0), x1, x2)) → c11(IF(mark(true), x1, x2)) 234.53/72.01
ACTIVE(if(geq(0, s(z0)), x1, x2)) → c11(IF(mark(false), x1, x2)) 234.53/72.01
ACTIVE(if(div(0, s(z0)), x1, x2)) → c11(IF(mark(0), x1, x2)) 234.53/72.01
ACTIVE(if(if(true, z0, z1), x1, x2)) → c11(IF(mark(z0), x1, x2)) 234.53/72.01
ACTIVE(if(if(false, z0, z1), x1, x2)) → c11(IF(mark(z1), x1, x2)) 234.53/72.01
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(IF(mark(minus(z0, z1)), x1, x2)) 234.53/72.01
ACTIVE(if(minus(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(IF(mark(geq(z0, z1)), x1, x2)) 234.53/72.01
ACTIVE(if(geq(s(z0), s(z1)), x1, x2)) → c5(ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
PROPER(minus(x0, minus(z0, z1))) → c18(MINUS(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(minus(x0, s(z0))) → c18(MINUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.01
PROPER(minus(x0, geq(z0, z1))) → c18(MINUS(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(minus(x0, div(z0, z1))) → c18(MINUS(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.01
PROPER(minus(x0, if(z0, z1, z2))) → c18(MINUS(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(minus(minus(z0, z1), x1)) → c18(MINUS(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(minus(s(z0), x1)) → c18(MINUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.01
PROPER(minus(geq(z0, z1), x1)) → c18(MINUS(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(minus(div(z0, z1), x1)) → c18(MINUS(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(minus(if(z0, z1, z2), x1)) → c18(MINUS(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.01
PROPER(minus(x0, 0)) → c18(MINUS(proper(x0), ok(0)), PROPER(x0)) 234.53/72.01
PROPER(minus(x0, true)) → c18(MINUS(proper(x0), ok(true)), PROPER(x0)) 234.53/72.01
PROPER(minus(x0, false)) → c18(MINUS(proper(x0), ok(false)), PROPER(x0)) 234.53/72.01
PROPER(minus(0, x1)) → c18(MINUS(ok(0), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(minus(true, x1)) → c18(MINUS(ok(true), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(minus(false, x1)) → c18(MINUS(ok(false), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(s(minus(z0, z1))) → c20(S(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(s(s(z0))) → c20(S(s(proper(z0))), PROPER(s(z0))) 234.53/72.01
PROPER(s(geq(z0, z1))) → c20(S(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(s(div(z0, z1))) → c20(S(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.01
PROPER(s(if(z0, z1, z2))) → c20(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(s(0)) → c20(S(ok(0))) 234.53/72.01
PROPER(s(true)) → c20(S(ok(true))) 234.53/72.01
PROPER(s(false)) → c20(S(ok(false))) 234.53/72.01
PROPER(geq(x0, minus(z0, z1))) → c21(GEQ(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(geq(x0, s(z0))) → c21(GEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.01
PROPER(geq(x0, geq(z0, z1))) → c21(GEQ(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(geq(x0, div(z0, z1))) → c21(GEQ(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.01
PROPER(geq(x0, if(z0, z1, z2))) → c21(GEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(geq(minus(z0, z1), x1)) → c21(GEQ(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(geq(s(z0), x1)) → c21(GEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.01
PROPER(geq(geq(z0, z1), x1)) → c21(GEQ(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(geq(div(z0, z1), x1)) → c21(GEQ(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(geq(if(z0, z1, z2), x1)) → c21(GEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.01
PROPER(geq(x0, 0)) → c21(GEQ(proper(x0), ok(0)), PROPER(x0)) 234.53/72.01
PROPER(geq(x0, true)) → c21(GEQ(proper(x0), ok(true)), PROPER(x0)) 234.53/72.01
PROPER(geq(x0, false)) → c21(GEQ(proper(x0), ok(false)), PROPER(x0)) 234.53/72.01
PROPER(geq(0, x1)) → c21(GEQ(ok(0), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(geq(true, x1)) → c21(GEQ(ok(true), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(geq(false, x1)) → c21(GEQ(ok(false), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(div(x0, minus(z0, z1))) → c24(DIV(proper(x0), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(div(x0, s(z0))) → c24(DIV(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 234.53/72.01
PROPER(div(x0, geq(z0, z1))) → c24(DIV(proper(x0), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(div(x0, div(z0, z1))) → c24(DIV(proper(x0), div(proper(z0), proper(z1))), PROPER(x0), PROPER(div(z0, z1))) 234.53/72.01
PROPER(div(x0, if(z0, z1, z2))) → c24(DIV(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(div(minus(z0, z1), x1)) → c24(DIV(minus(proper(z0), proper(z1)), proper(x1)), PROPER(minus(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(div(s(z0), x1)) → c24(DIV(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 234.53/72.01
PROPER(div(geq(z0, z1), x1)) → c24(DIV(geq(proper(z0), proper(z1)), proper(x1)), PROPER(geq(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(div(div(z0, z1), x1)) → c24(DIV(div(proper(z0), proper(z1)), proper(x1)), PROPER(div(z0, z1)), PROPER(x1)) 234.53/72.01
PROPER(div(if(z0, z1, z2), x1)) → c24(DIV(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 234.53/72.01
PROPER(div(x0, 0)) → c24(DIV(proper(x0), ok(0)), PROPER(x0)) 234.53/72.01
PROPER(div(x0, true)) → c24(DIV(proper(x0), ok(true)), PROPER(x0)) 234.53/72.01
PROPER(div(x0, false)) → c24(DIV(proper(x0), ok(false)), PROPER(x0)) 234.53/72.01
PROPER(div(0, x1)) → c24(DIV(ok(0), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(div(true, x1)) → c24(DIV(ok(true), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(div(false, x1)) → c24(DIV(ok(false), proper(x1)), PROPER(x1)) 234.53/72.01
PROPER(if(x0, x1, minus(z0, z1))) → c25(IF(proper(x0), proper(x1), minus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(minus(z0, z1))) 234.53/72.01
PROPER(if(x0, x1, s(z0))) → c25(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 234.53/72.01
PROPER(if(x0, x1, geq(z0, z1))) → c25(IF(proper(x0), proper(x1), geq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(geq(z0, z1))) 234.53/72.01
PROPER(if(x0, x1, div(z0, z1))) → c25(IF(proper(x0), proper(x1), div(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(div(z0, z1))) 234.53/72.01
PROPER(if(x0, x1, if(z0, z1, z2))) → c25(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 234.53/72.01
PROPER(if(x0, minus(z0, z1), x2)) → c25(IF(proper(x0), minus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(minus(z0, z1)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, s(z0), x2)) → c25(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, geq(z0, z1), x2)) → c25(IF(proper(x0), geq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(geq(z0, z1)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, div(z0, z1), x2)) → c25(IF(proper(x0), div(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(div(z0, z1)), PROPER(x2)) 234.53/72.01
PROPER(if(x0, if(z0, z1, z2), x2)) → c25(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 234.53/72.01
PROPER(if(minus(z0, z1), x1, x2)) → c25(IF(minus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(minus(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(s(z0), x1, x2)) → c25(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(geq(z0, z1), x1, x2)) → c25(IF(geq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(geq(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(div(z0, z1), x1, x2)) → c25(IF(div(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(div(z0, z1)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(if(z0, z1, z2), x1, x2)) → c25(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(x0, x1, 0)) → c25(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 234.53/72.01
PROPER(if(x0, x1, true)) → c25(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 234.53/72.01
PROPER(if(x0, x1, false)) → c25(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 234.53/72.01
PROPER(if(x0, 0, x2)) → c25(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.01
PROPER(if(x0, true, x2)) → c25(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.01
PROPER(if(x0, false, x2)) → c25(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 234.53/72.01
PROPER(if(0, x1, x2)) → c25(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(true, x1, x2)) → c25(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.01
PROPER(if(false, x1, x2)) → c25(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 234.53/72.01
TOP(mark(minus(z0, z1))) → c28(TOP(minus(proper(z0), proper(z1))), PROPER(minus(z0, z1))) 234.53/72.01
TOP(mark(s(z0))) → c28(TOP(s(proper(z0))), PROPER(s(z0))) 234.53/72.01
TOP(mark(geq(z0, z1))) → c28(TOP(geq(proper(z0), proper(z1))), PROPER(geq(z0, z1))) 234.53/72.01
TOP(mark(div(z0, z1))) → c28(TOP(div(proper(z0), proper(z1))), PROPER(div(z0, z1))) 234.53/72.01
TOP(mark(if(z0, z1, z2))) → c28(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 234.53/72.01
TOP(ok(minus(0, z0))) → c29(TOP(mark(0)), ACTIVE(minus(0, z0))) 234.53/72.01
TOP(ok(minus(s(z0), s(z1)))) → c29(TOP(mark(minus(z0, z1))), ACTIVE(minus(s(z0), s(z1)))) 234.53/72.01
TOP(ok(geq(z0, 0))) → c29(TOP(mark(true)), ACTIVE(geq(z0, 0))) 234.53/72.01
TOP(ok(geq(0, s(z0)))) → c29(TOP(mark(false)), ACTIVE(geq(0, s(z0)))) 234.53/72.01
TOP(ok(geq(s(z0), s(z1)))) → c29(TOP(mark(geq(z0, z1))), ACTIVE(geq(s(z0), s(z1)))) 234.53/72.01
TOP(ok(div(0, s(z0)))) → c29(TOP(mark(0)), ACTIVE(div(0, s(z0)))) 234.53/72.01
TOP(ok(div(s(z0), s(z1)))) → c29(TOP(mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0))), ACTIVE(div(s(z0), s(z1)))) 234.53/72.01
TOP(ok(if(true, z0, z1))) → c29(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 234.53/72.01
TOP(ok(if(false, z0, z1))) → c29(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 234.53/72.01
TOP(ok(s(z0))) → c29(TOP(s(active(z0))), ACTIVE(s(z0))) 234.53/72.01
TOP(ok(div(z0, z1))) → c29(TOP(div(active(z0), z1)), ACTIVE(div(z0, z1))) 234.53/72.01
TOP(ok(if(z0, z1, z2))) → c29(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
234.53/72.01
234.53/72.01

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.53/72.01
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.01
active(geq(z0, 0)) → mark(true) 234.53/72.01
active(geq(0, s(z0))) → mark(false) 234.53/72.01
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.01
active(div(0, s(z0))) → mark(0) 234.53/72.01
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.01
active(if(true, z0, z1)) → mark(z0) 234.53/72.01
active(if(false, z0, z1)) → mark(z1) 234.53/72.01
active(s(z0)) → s(active(z0)) 234.53/72.01
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.01
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.53/72.01
s(mark(z0)) → mark(s(z0)) 234.53/72.01
s(ok(z0)) → ok(s(z0)) 234.53/72.01
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.01
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.01
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.01
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.01
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.01
proper(0) → ok(0) 234.53/72.01
proper(s(z0)) → s(proper(z0)) 234.53/72.01
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.01
proper(true) → ok(true) 234.53/72.01
proper(false) → ok(false) 234.53/72.01
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.01
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.01
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.01
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.01
top(mark(z0)) → top(proper(z0)) 234.53/72.01
top(ok(z0)) → top(active(z0))
Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.53/72.01
TOP(mark(0)) → c28(TOP(ok(0))) 234.53/72.01
TOP(mark(true)) → c28(TOP(ok(true))) 234.53/72.01
TOP(mark(false)) → c28(TOP(ok(false)))
S tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
K tuples:

TOP(mark(0)) → c28(TOP(ok(0))) 234.53/72.01
TOP(mark(false)) → c28(TOP(ok(false))) 234.53/72.01
TOP(mark(true)) → c28(TOP(ok(true)))
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

S, DIV, IF, MINUS, GEQ, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c26, c27, c28

234.53/72.01
234.53/72.01

(57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts
234.53/72.01
234.53/72.01

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.53/72.01
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.01
active(geq(z0, 0)) → mark(true) 234.53/72.01
active(geq(0, s(z0))) → mark(false) 234.53/72.01
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.01
active(div(0, s(z0))) → mark(0) 234.53/72.01
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.01
active(if(true, z0, z1)) → mark(z0) 234.53/72.01
active(if(false, z0, z1)) → mark(z1) 234.53/72.01
active(s(z0)) → s(active(z0)) 234.53/72.01
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.01
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.53/72.01
s(mark(z0)) → mark(s(z0)) 234.53/72.01
s(ok(z0)) → ok(s(z0)) 234.53/72.01
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.01
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.01
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.01
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.01
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.01
proper(0) → ok(0) 234.53/72.01
proper(s(z0)) → s(proper(z0)) 234.53/72.01
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.01
proper(true) → ok(true) 234.53/72.01
proper(false) → ok(false) 234.53/72.01
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.01
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.01
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.01
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.01
top(mark(z0)) → top(proper(z0)) 234.53/72.01
top(ok(z0)) → top(active(z0))
Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1)) 234.53/72.01
TOP(mark(0)) → c28 234.53/72.01
TOP(mark(true)) → c28 234.53/72.01
TOP(mark(false)) → c28
S tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
K tuples:

TOP(mark(0)) → c28 234.53/72.01
TOP(mark(true)) → c28 234.53/72.01
TOP(mark(false)) → c28
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

S, DIV, IF, MINUS, GEQ, TOP

Compound Symbols:

c12, c13, c14, c15, c16, c17, c26, c27, c28

234.53/72.01
234.53/72.01

(59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

TOP(mark(false)) → c28 234.53/72.01
TOP(mark(true)) → c28 234.53/72.01
TOP(mark(0)) → c28
234.53/72.01
234.53/72.01

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.53/72.01
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.01
active(geq(z0, 0)) → mark(true) 234.53/72.01
active(geq(0, s(z0))) → mark(false) 234.53/72.01
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.01
active(div(0, s(z0))) → mark(0) 234.53/72.01
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.01
active(if(true, z0, z1)) → mark(z0) 234.53/72.01
active(if(false, z0, z1)) → mark(z1) 234.53/72.01
active(s(z0)) → s(active(z0)) 234.53/72.01
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.01
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.53/72.01
s(mark(z0)) → mark(s(z0)) 234.53/72.01
s(ok(z0)) → ok(s(z0)) 234.53/72.01
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.01
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.01
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.01
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.01
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.01
proper(0) → ok(0) 234.53/72.01
proper(s(z0)) → s(proper(z0)) 234.53/72.01
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.01
proper(true) → ok(true) 234.53/72.01
proper(false) → ok(false) 234.53/72.01
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.01
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.01
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.01
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.01
top(mark(z0)) → top(proper(z0)) 234.53/72.01
top(ok(z0)) → top(active(z0))
Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
S tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
K tuples:none
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

S, DIV, IF, MINUS, GEQ

Compound Symbols:

c12, c13, c14, c15, c16, c17, c26, c27

234.53/72.01
234.53/72.01

(61) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 234.53/72.01

POL(DIV(x1, x2)) = 0    234.53/72.01
POL(GEQ(x1, x2)) = 0    234.53/72.01
POL(IF(x1, x2, x3)) = x3    234.53/72.01
POL(MINUS(x1, x2)) = 0    234.53/72.01
POL(S(x1)) = 0    234.53/72.01
POL(c12(x1)) = x1    234.53/72.01
POL(c13(x1)) = x1    234.53/72.01
POL(c14(x1)) = x1    234.53/72.01
POL(c15(x1)) = x1    234.53/72.01
POL(c16(x1)) = x1    234.53/72.01
POL(c17(x1)) = x1    234.53/72.01
POL(c26(x1)) = x1    234.53/72.01
POL(c27(x1)) = x1    234.53/72.01
POL(mark(x1)) = x1    234.53/72.01
POL(ok(x1)) = [1] + x1   
234.53/72.01
234.53/72.01

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.53/72.01
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.01
active(geq(z0, 0)) → mark(true) 234.53/72.01
active(geq(0, s(z0))) → mark(false) 234.53/72.01
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.01
active(div(0, s(z0))) → mark(0) 234.53/72.01
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.01
active(if(true, z0, z1)) → mark(z0) 234.53/72.01
active(if(false, z0, z1)) → mark(z1) 234.53/72.01
active(s(z0)) → s(active(z0)) 234.53/72.01
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.01
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.53/72.01
s(mark(z0)) → mark(s(z0)) 234.53/72.01
s(ok(z0)) → ok(s(z0)) 234.53/72.01
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.01
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.01
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.01
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.01
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.01
proper(0) → ok(0) 234.53/72.01
proper(s(z0)) → s(proper(z0)) 234.53/72.01
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.01
proper(true) → ok(true) 234.53/72.01
proper(false) → ok(false) 234.53/72.01
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.01
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.01
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.01
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.01
top(mark(z0)) → top(proper(z0)) 234.53/72.01
top(ok(z0)) → top(active(z0))
Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
S tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2))
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

S, DIV, IF, MINUS, GEQ

Compound Symbols:

c12, c13, c14, c15, c16, c17, c26, c27

234.53/72.01
234.53/72.01

(63) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

S(mark(z0)) → c12(S(z0))
We considered the (Usable) Rules:none
And the Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 234.53/72.01

POL(DIV(x1, x2)) = x2    234.53/72.01
POL(GEQ(x1, x2)) = x1 + x2    234.53/72.01
POL(IF(x1, x2, x3)) = [5]x2 + [5]x3    234.53/72.01
POL(MINUS(x1, x2)) = x1 + x2    234.53/72.01
POL(S(x1)) = [2]x1    234.53/72.01
POL(c12(x1)) = x1    234.53/72.01
POL(c13(x1)) = x1    234.53/72.01
POL(c14(x1)) = x1    234.53/72.01
POL(c15(x1)) = x1    234.53/72.01
POL(c16(x1)) = x1    234.53/72.01
POL(c17(x1)) = x1    234.53/72.01
POL(c26(x1)) = x1    234.53/72.01
POL(c27(x1)) = x1    234.53/72.01
POL(mark(x1)) = [1] + x1    234.53/72.01
POL(ok(x1)) = x1   
234.53/72.01
234.53/72.01

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.53/72.01
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.01
active(geq(z0, 0)) → mark(true) 234.53/72.01
active(geq(0, s(z0))) → mark(false) 234.53/72.01
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.01
active(div(0, s(z0))) → mark(0) 234.53/72.01
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.01
active(if(true, z0, z1)) → mark(z0) 234.53/72.01
active(if(false, z0, z1)) → mark(z1) 234.53/72.01
active(s(z0)) → s(active(z0)) 234.53/72.01
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.01
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.53/72.01
s(mark(z0)) → mark(s(z0)) 234.53/72.01
s(ok(z0)) → ok(s(z0)) 234.53/72.01
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.01
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.01
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.01
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.01
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.01
proper(0) → ok(0) 234.53/72.01
proper(s(z0)) → s(proper(z0)) 234.53/72.01
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.01
proper(true) → ok(true) 234.53/72.01
proper(false) → ok(false) 234.53/72.01
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.01
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.01
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.01
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.01
top(mark(z0)) → top(proper(z0)) 234.53/72.01
top(ok(z0)) → top(active(z0))
Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
S tuples:

S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
S(mark(z0)) → c12(S(z0))
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

S, DIV, IF, MINUS, GEQ

Compound Symbols:

c12, c13, c14, c15, c16, c17, c26, c27

234.53/72.01
234.53/72.01

(65) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 234.53/72.01

POL(DIV(x1, x2)) = [2]x1    234.53/72.01
POL(GEQ(x1, x2)) = 0    234.53/72.01
POL(IF(x1, x2, x3)) = x1 + [5]x2 + [5]x3    234.53/72.01
POL(MINUS(x1, x2)) = 0    234.53/72.01
POL(S(x1)) = 0    234.53/72.01
POL(c12(x1)) = x1    234.53/72.01
POL(c13(x1)) = x1    234.53/72.01
POL(c14(x1)) = x1    234.53/72.01
POL(c15(x1)) = x1    234.53/72.01
POL(c16(x1)) = x1    234.53/72.01
POL(c17(x1)) = x1    234.53/72.01
POL(c26(x1)) = x1    234.53/72.01
POL(c27(x1)) = x1    234.53/72.01
POL(mark(x1)) = x1    234.53/72.01
POL(ok(x1)) = [1] + x1   
234.53/72.01
234.53/72.01

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.53/72.01
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.01
active(geq(z0, 0)) → mark(true) 234.53/72.01
active(geq(0, s(z0))) → mark(false) 234.53/72.01
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.01
active(div(0, s(z0))) → mark(0) 234.53/72.01
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.01
active(if(true, z0, z1)) → mark(z0) 234.53/72.01
active(if(false, z0, z1)) → mark(z1) 234.53/72.01
active(s(z0)) → s(active(z0)) 234.53/72.01
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.01
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.53/72.01
s(mark(z0)) → mark(s(z0)) 234.53/72.01
s(ok(z0)) → ok(s(z0)) 234.53/72.01
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.01
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.01
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.01
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.01
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.01
proper(0) → ok(0) 234.53/72.01
proper(s(z0)) → s(proper(z0)) 234.53/72.01
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.01
proper(true) → ok(true) 234.53/72.01
proper(false) → ok(false) 234.53/72.01
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.01
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.01
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.01
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.01
top(mark(z0)) → top(proper(z0)) 234.53/72.01
top(ok(z0)) → top(active(z0))
Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
S tuples:

S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
S(mark(z0)) → c12(S(z0)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1))
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

S, DIV, IF, MINUS, GEQ

Compound Symbols:

c12, c13, c14, c15, c16, c17, c26, c27

234.53/72.01
234.53/72.01

(67) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

S(ok(z0)) → c13(S(z0))
We considered the (Usable) Rules:none
And the Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 234.53/72.01

POL(DIV(x1, x2)) = [5]x2    234.53/72.01
POL(GEQ(x1, x2)) = 0    234.53/72.01
POL(IF(x1, x2, x3)) = [5]x2 + [5]x3    234.53/72.01
POL(MINUS(x1, x2)) = 0    234.53/72.01
POL(S(x1)) = [2]x1    234.53/72.01
POL(c12(x1)) = x1    234.53/72.01
POL(c13(x1)) = x1    234.53/72.01
POL(c14(x1)) = x1    234.53/72.01
POL(c15(x1)) = x1    234.53/72.01
POL(c16(x1)) = x1    234.53/72.01
POL(c17(x1)) = x1    234.53/72.01
POL(c26(x1)) = x1    234.53/72.01
POL(c27(x1)) = x1    234.53/72.01
POL(mark(x1)) = x1    234.53/72.01
POL(ok(x1)) = [1] + x1   
234.53/72.01
234.53/72.01

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.53/72.01
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.01
active(geq(z0, 0)) → mark(true) 234.53/72.01
active(geq(0, s(z0))) → mark(false) 234.53/72.01
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.01
active(div(0, s(z0))) → mark(0) 234.53/72.01
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.01
active(if(true, z0, z1)) → mark(z0) 234.53/72.01
active(if(false, z0, z1)) → mark(z1) 234.53/72.01
active(s(z0)) → s(active(z0)) 234.53/72.01
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.01
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.53/72.01
s(mark(z0)) → mark(s(z0)) 234.53/72.01
s(ok(z0)) → ok(s(z0)) 234.53/72.01
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.01
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.01
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.01
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.01
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.01
proper(0) → ok(0) 234.53/72.01
proper(s(z0)) → s(proper(z0)) 234.53/72.01
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.01
proper(true) → ok(true) 234.53/72.01
proper(false) → ok(false) 234.53/72.01
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.01
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.01
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.01
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.01
top(mark(z0)) → top(proper(z0)) 234.53/72.01
top(ok(z0)) → top(active(z0))
Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
S tuples:

DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
S(mark(z0)) → c12(S(z0)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
S(ok(z0)) → c13(S(z0))
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

S, DIV, IF, MINUS, GEQ

Compound Symbols:

c12, c13, c14, c15, c16, c17, c26, c27

234.53/72.01
234.53/72.01

(69) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

DIV(mark(z0), z1) → c14(DIV(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 234.53/72.01

POL(DIV(x1, x2)) = [2]x1 + [3]x2    234.53/72.01
POL(GEQ(x1, x2)) = x1 + x2    234.53/72.01
POL(IF(x1, x2, x3)) = [3]x2 + [5]x3    234.53/72.01
POL(MINUS(x1, x2)) = 0    234.53/72.01
POL(S(x1)) = [5]x1    234.53/72.01
POL(c12(x1)) = x1    234.53/72.01
POL(c13(x1)) = x1    234.53/72.01
POL(c14(x1)) = x1    234.53/72.01
POL(c15(x1)) = x1    234.53/72.01
POL(c16(x1)) = x1    234.53/72.01
POL(c17(x1)) = x1    234.53/72.01
POL(c26(x1)) = x1    234.53/72.01
POL(c27(x1)) = x1    234.53/72.01
POL(mark(x1)) = [1] + x1    234.53/72.01
POL(ok(x1)) = x1   
234.53/72.01
234.53/72.01

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.53/72.01
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.01
active(geq(z0, 0)) → mark(true) 234.53/72.01
active(geq(0, s(z0))) → mark(false) 234.53/72.01
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.01
active(div(0, s(z0))) → mark(0) 234.53/72.01
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.01
active(if(true, z0, z1)) → mark(z0) 234.53/72.01
active(if(false, z0, z1)) → mark(z1) 234.53/72.01
active(s(z0)) → s(active(z0)) 234.53/72.01
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.01
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.53/72.01
s(mark(z0)) → mark(s(z0)) 234.53/72.01
s(ok(z0)) → ok(s(z0)) 234.53/72.01
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.01
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.01
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.01
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.01
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.01
proper(0) → ok(0) 234.53/72.01
proper(s(z0)) → s(proper(z0)) 234.53/72.01
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.01
proper(true) → ok(true) 234.53/72.01
proper(false) → ok(false) 234.53/72.01
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.01
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.01
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.01
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.01
top(mark(z0)) → top(proper(z0)) 234.53/72.01
top(ok(z0)) → top(active(z0))
Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
S tuples:

IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
S(mark(z0)) → c12(S(z0)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1))
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

S, DIV, IF, MINUS, GEQ

Compound Symbols:

c12, c13, c14, c15, c16, c17, c26, c27

234.53/72.01
234.53/72.01

(71) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 234.53/72.01

POL(DIV(x1, x2)) = [5]x1 + [5]x2    234.53/72.01
POL(GEQ(x1, x2)) = 0    234.53/72.01
POL(IF(x1, x2, x3)) = [4]x1 + [3]x2 + [5]x3    234.53/72.01
POL(MINUS(x1, x2)) = 0    234.53/72.01
POL(S(x1)) = [3]x1    234.53/72.01
POL(c12(x1)) = x1    234.53/72.01
POL(c13(x1)) = x1    234.53/72.01
POL(c14(x1)) = x1    234.53/72.01
POL(c15(x1)) = x1    234.53/72.01
POL(c16(x1)) = x1    234.53/72.01
POL(c17(x1)) = x1    234.53/72.01
POL(c26(x1)) = x1    234.53/72.01
POL(c27(x1)) = x1    234.53/72.01
POL(mark(x1)) = [1] + x1    234.53/72.01
POL(ok(x1)) = x1   
234.53/72.01
234.53/72.01

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.53/72.01
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.01
active(geq(z0, 0)) → mark(true) 234.53/72.01
active(geq(0, s(z0))) → mark(false) 234.53/72.01
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.01
active(div(0, s(z0))) → mark(0) 234.53/72.01
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.01
active(if(true, z0, z1)) → mark(z0) 234.53/72.01
active(if(false, z0, z1)) → mark(z1) 234.53/72.01
active(s(z0)) → s(active(z0)) 234.53/72.01
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.01
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.53/72.01
s(mark(z0)) → mark(s(z0)) 234.53/72.01
s(ok(z0)) → ok(s(z0)) 234.53/72.01
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.01
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.01
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.01
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.01
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.01
proper(0) → ok(0) 234.53/72.01
proper(s(z0)) → s(proper(z0)) 234.53/72.01
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.01
proper(true) → ok(true) 234.53/72.01
proper(false) → ok(false) 234.53/72.01
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.01
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.01
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.01
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.01
top(mark(z0)) → top(proper(z0)) 234.53/72.01
top(ok(z0)) → top(active(z0))
Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
S tuples:

MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
S(mark(z0)) → c12(S(z0)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2))
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

S, DIV, IF, MINUS, GEQ

Compound Symbols:

c12, c13, c14, c15, c16, c17, c26, c27

234.53/72.01
234.53/72.01

(73) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 234.53/72.01

POL(DIV(x1, x2)) = [3]x1 + [5]x2    234.53/72.01
POL(GEQ(x1, x2)) = 0    234.53/72.01
POL(IF(x1, x2, x3)) = [5]x1 + [3]x2 + [5]x3    234.53/72.01
POL(MINUS(x1, x2)) = x2    234.53/72.01
POL(S(x1)) = [3]x1    234.53/72.01
POL(c12(x1)) = x1    234.53/72.01
POL(c13(x1)) = x1    234.53/72.01
POL(c14(x1)) = x1    234.53/72.01
POL(c15(x1)) = x1    234.53/72.01
POL(c16(x1)) = x1    234.53/72.01
POL(c17(x1)) = x1    234.53/72.01
POL(c26(x1)) = x1    234.53/72.01
POL(c27(x1)) = x1    234.53/72.01
POL(mark(x1)) = x1    234.53/72.01
POL(ok(x1)) = [1] + x1   
234.53/72.01
234.53/72.01

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.53/72.01
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.01
active(geq(z0, 0)) → mark(true) 234.53/72.01
active(geq(0, s(z0))) → mark(false) 234.53/72.01
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.01
active(div(0, s(z0))) → mark(0) 234.53/72.01
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.01
active(if(true, z0, z1)) → mark(z0) 234.53/72.01
active(if(false, z0, z1)) → mark(z1) 234.53/72.01
active(s(z0)) → s(active(z0)) 234.53/72.01
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.01
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.53/72.01
s(mark(z0)) → mark(s(z0)) 234.53/72.01
s(ok(z0)) → ok(s(z0)) 234.53/72.01
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.01
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.01
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.01
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.01
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.01
proper(0) → ok(0) 234.53/72.01
proper(s(z0)) → s(proper(z0)) 234.53/72.01
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.01
proper(true) → ok(true) 234.53/72.01
proper(false) → ok(false) 234.53/72.01
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.01
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.01
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.01
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.01
top(mark(z0)) → top(proper(z0)) 234.53/72.01
top(ok(z0)) → top(active(z0))
Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
S tuples:

GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
S(mark(z0)) → c12(S(z0)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1))
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

S, DIV, IF, MINUS, GEQ

Compound Symbols:

c12, c13, c14, c15, c16, c17, c26, c27

234.53/72.01
234.53/72.01

(75) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 234.53/72.01

POL(DIV(x1, x2)) = [3]x1 + [3]x2    234.53/72.01
POL(GEQ(x1, x2)) = x1    234.53/72.01
POL(IF(x1, x2, x3)) = [3]x1 + [5]x2 + [3]x3    234.53/72.01
POL(MINUS(x1, x2)) = [5]x1 + [5]x2    234.53/72.01
POL(S(x1)) = [3]x1    234.53/72.01
POL(c12(x1)) = x1    234.53/72.01
POL(c13(x1)) = x1    234.53/72.01
POL(c14(x1)) = x1    234.53/72.01
POL(c15(x1)) = x1    234.53/72.01
POL(c16(x1)) = x1    234.53/72.01
POL(c17(x1)) = x1    234.53/72.01
POL(c26(x1)) = x1    234.53/72.01
POL(c27(x1)) = x1    234.53/72.01
POL(mark(x1)) = x1    234.53/72.01
POL(ok(x1)) = [2] + x1   
234.53/72.01
234.53/72.01

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(minus(0, z0)) → mark(0) 234.53/72.01
active(minus(s(z0), s(z1))) → mark(minus(z0, z1)) 234.53/72.01
active(geq(z0, 0)) → mark(true) 234.53/72.01
active(geq(0, s(z0))) → mark(false) 234.53/72.01
active(geq(s(z0), s(z1))) → mark(geq(z0, z1)) 234.53/72.01
active(div(0, s(z0))) → mark(0) 234.53/72.01
active(div(s(z0), s(z1))) → mark(if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0)) 234.53/72.01
active(if(true, z0, z1)) → mark(z0) 234.53/72.01
active(if(false, z0, z1)) → mark(z1) 234.53/72.01
active(s(z0)) → s(active(z0)) 234.53/72.01
active(div(z0, z1)) → div(active(z0), z1) 234.53/72.01
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 234.53/72.01
s(mark(z0)) → mark(s(z0)) 234.53/72.01
s(ok(z0)) → ok(s(z0)) 234.53/72.01
div(mark(z0), z1) → mark(div(z0, z1)) 234.53/72.01
div(ok(z0), ok(z1)) → ok(div(z0, z1)) 234.53/72.01
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 234.53/72.01
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 234.53/72.01
proper(minus(z0, z1)) → minus(proper(z0), proper(z1)) 234.53/72.01
proper(0) → ok(0) 234.53/72.01
proper(s(z0)) → s(proper(z0)) 234.53/72.01
proper(geq(z0, z1)) → geq(proper(z0), proper(z1)) 234.53/72.01
proper(true) → ok(true) 234.53/72.01
proper(false) → ok(false) 234.53/72.01
proper(div(z0, z1)) → div(proper(z0), proper(z1)) 234.53/72.01
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 234.53/72.01
minus(ok(z0), ok(z1)) → ok(minus(z0, z1)) 234.53/72.01
geq(ok(z0), ok(z1)) → ok(geq(z0, z1)) 234.53/72.01
top(mark(z0)) → top(proper(z0)) 234.53/72.01
top(ok(z0)) → top(active(z0))
Tuples:

S(mark(z0)) → c12(S(z0)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
S tuples:none
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c17(IF(z0, z1, z2)) 234.53/72.01
S(mark(z0)) → c12(S(z0)) 234.53/72.01
DIV(ok(z0), ok(z1)) → c15(DIV(z0, z1)) 234.53/72.01
S(ok(z0)) → c13(S(z0)) 234.53/72.01
DIV(mark(z0), z1) → c14(DIV(z0, z1)) 234.53/72.01
IF(mark(z0), z1, z2) → c16(IF(z0, z1, z2)) 234.53/72.01
MINUS(ok(z0), ok(z1)) → c26(MINUS(z0, z1)) 234.53/72.01
GEQ(ok(z0), ok(z1)) → c27(GEQ(z0, z1))
Defined Rule Symbols:

active, s, div, if, proper, minus, geq, top

Defined Pair Symbols:

S, DIV, IF, MINUS, GEQ

Compound Symbols:

c12, c13, c14, c15, c16, c17, c26, c27

234.53/72.01
234.53/72.01

(77) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty
234.53/72.01
234.53/72.01

(78) BOUNDS(O(1), O(1))

234.53/72.01
234.53/72.01
234.63/72.10 EOF