YES(O(1), O(n^2)) 212.95/68.51 YES(O(1), O(n^2)) 212.95/68.56 212.95/68.56 212.95/68.56 212.95/68.56 212.95/68.56 212.95/68.56 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 212.95/68.56 212.95/68.56 212.95/68.56
212.95/68.56 213.31/68.65 213.31/68.65
213.31/68.65
213.31/68.65

(0) Obligation:

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

active(p(0)) → mark(0) 213.31/68.65
active(p(s(X))) → mark(X) 213.31/68.65
active(leq(0, Y)) → mark(true) 213.31/68.65
active(leq(s(X), 0)) → mark(false) 213.31/68.65
active(leq(s(X), s(Y))) → mark(leq(X, Y)) 213.31/68.65
active(if(true, X, Y)) → mark(X) 213.31/68.65
active(if(false, X, Y)) → mark(Y) 213.31/68.65
active(diff(X, Y)) → mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) 213.31/68.65
active(p(X)) → p(active(X)) 213.31/68.65
active(s(X)) → s(active(X)) 213.31/68.65
active(leq(X1, X2)) → leq(active(X1), X2) 213.31/68.65
active(leq(X1, X2)) → leq(X1, active(X2)) 213.31/68.65
active(if(X1, X2, X3)) → if(active(X1), X2, X3) 213.31/68.65
active(diff(X1, X2)) → diff(active(X1), X2) 213.31/68.65
active(diff(X1, X2)) → diff(X1, active(X2)) 213.31/68.65
p(mark(X)) → mark(p(X)) 213.31/68.65
s(mark(X)) → mark(s(X)) 213.31/68.65
leq(mark(X1), X2) → mark(leq(X1, X2)) 213.31/68.65
leq(X1, mark(X2)) → mark(leq(X1, X2)) 213.31/68.65
if(mark(X1), X2, X3) → mark(if(X1, X2, X3)) 213.31/68.65
diff(mark(X1), X2) → mark(diff(X1, X2)) 213.31/68.65
diff(X1, mark(X2)) → mark(diff(X1, X2)) 213.31/68.65
proper(p(X)) → p(proper(X)) 213.31/68.65
proper(0) → ok(0) 213.31/68.65
proper(s(X)) → s(proper(X)) 213.31/68.65
proper(leq(X1, X2)) → leq(proper(X1), proper(X2)) 213.31/68.65
proper(true) → ok(true) 213.31/68.65
proper(false) → ok(false) 213.31/68.65
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3)) 213.31/68.65
proper(diff(X1, X2)) → diff(proper(X1), proper(X2)) 213.31/68.65
p(ok(X)) → ok(p(X)) 213.31/68.65
s(ok(X)) → ok(s(X)) 213.31/68.65
leq(ok(X1), ok(X2)) → ok(leq(X1, X2)) 213.31/68.65
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3)) 213.31/68.65
diff(ok(X1), ok(X2)) → ok(diff(X1, X2)) 213.31/68.65
top(mark(X)) → top(proper(X)) 213.31/68.65
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST
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213.31/68.65

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
213.31/68.65
213.31/68.65

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.31/68.65
active(p(s(z0))) → mark(z0) 213.31/68.65
active(leq(0, z0)) → mark(true) 213.31/68.65
active(leq(s(z0), 0)) → mark(false) 213.31/68.65
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.31/68.65
active(if(true, z0, z1)) → mark(z0) 213.31/68.65
active(if(false, z0, z1)) → mark(z1) 213.31/68.65
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.31/68.65
active(p(z0)) → p(active(z0)) 213.31/68.65
active(s(z0)) → s(active(z0)) 213.31/68.65
active(leq(z0, z1)) → leq(active(z0), z1) 213.31/68.65
active(leq(z0, z1)) → leq(z0, active(z1)) 213.31/68.65
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.31/68.65
active(diff(z0, z1)) → diff(active(z0), z1) 213.31/68.65
active(diff(z0, z1)) → diff(z0, active(z1)) 213.31/68.65
p(mark(z0)) → mark(p(z0)) 213.31/68.65
p(ok(z0)) → ok(p(z0)) 213.31/68.65
s(mark(z0)) → mark(s(z0)) 213.31/68.65
s(ok(z0)) → ok(s(z0)) 213.31/68.65
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.31/68.65
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.31/68.65
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.31/68.65
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.31/68.65
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.31/68.65
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.31/68.65
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.31/68.65
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.31/68.65
proper(p(z0)) → p(proper(z0)) 213.31/68.65
proper(0) → ok(0) 213.31/68.65
proper(s(z0)) → s(proper(z0)) 213.31/68.65
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.31/68.65
proper(true) → ok(true) 213.31/68.65
proper(false) → ok(false) 213.31/68.65
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.31/68.65
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.31/68.65
top(mark(z0)) → top(proper(z0)) 213.31/68.65
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.31/68.65
ACTIVE(diff(z0, z1)) → c7(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), LEQ(z0, z1), S(diff(p(z0), z1)), DIFF(p(z0), z1), P(z0)) 213.31/68.65
ACTIVE(p(z0)) → c8(P(active(z0)), ACTIVE(z0)) 213.31/68.65
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 213.31/68.65
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.31/68.65
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.31/68.65
ACTIVE(if(z0, z1, z2)) → c12(IF(active(z0), z1, z2), ACTIVE(z0)) 213.31/68.65
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.31/68.65
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.31/68.65
P(mark(z0)) → c15(P(z0)) 213.31/68.65
P(ok(z0)) → c16(P(z0)) 213.31/68.65
S(mark(z0)) → c17(S(z0)) 213.31/68.65
S(ok(z0)) → c18(S(z0)) 213.31/68.65
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.31/68.65
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.31/68.65
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.31/68.65
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.31/68.65
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.31/68.65
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.31/68.65
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.31/68.65
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.31/68.65
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.31/68.65
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.31/68.65
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.31/68.65
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.31/68.65
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.31/68.65
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.31/68.65
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.31/68.65
ACTIVE(diff(z0, z1)) → c7(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), LEQ(z0, z1), S(diff(p(z0), z1)), DIFF(p(z0), z1), P(z0)) 213.31/68.65
ACTIVE(p(z0)) → c8(P(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.76
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.76
ACTIVE(if(z0, z1, z2)) → c12(IF(active(z0), z1, z2), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.76
P(mark(z0)) → c15(P(z0)) 213.71/68.76
P(ok(z0)) → c16(P(z0)) 213.71/68.76
S(mark(z0)) → c17(S(z0)) 213.71/68.76
S(ok(z0)) → c18(S(z0)) 213.71/68.76
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.76
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.76
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.76
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.76
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.76
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.76
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.76
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.76
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.76
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.76
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.76
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.76
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.76
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.76
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c29, c30, c33, c34, c35, c36

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213.71/68.76

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

Removed 1 trailing tuple parts
213.71/68.76
213.71/68.76

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.76
active(p(s(z0))) → mark(z0) 213.71/68.76
active(leq(0, z0)) → mark(true) 213.71/68.76
active(leq(s(z0), 0)) → mark(false) 213.71/68.76
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.76
active(if(true, z0, z1)) → mark(z0) 213.71/68.76
active(if(false, z0, z1)) → mark(z1) 213.71/68.76
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.76
active(p(z0)) → p(active(z0)) 213.71/68.76
active(s(z0)) → s(active(z0)) 213.71/68.76
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.76
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.76
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.76
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.76
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.76
p(mark(z0)) → mark(p(z0)) 213.71/68.76
p(ok(z0)) → ok(p(z0)) 213.71/68.76
s(mark(z0)) → mark(s(z0)) 213.71/68.76
s(ok(z0)) → ok(s(z0)) 213.71/68.76
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.76
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.76
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.76
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.76
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.76
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.76
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.76
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.76
proper(p(z0)) → p(proper(z0)) 213.71/68.76
proper(0) → ok(0) 213.71/68.76
proper(s(z0)) → s(proper(z0)) 213.71/68.76
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.76
proper(true) → ok(true) 213.71/68.76
proper(false) → ok(false) 213.71/68.76
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.76
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.76
top(mark(z0)) → top(proper(z0)) 213.71/68.76
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.76
ACTIVE(p(z0)) → c8(P(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.76
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.76
ACTIVE(if(z0, z1, z2)) → c12(IF(active(z0), z1, z2), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.76
P(mark(z0)) → c15(P(z0)) 213.71/68.76
P(ok(z0)) → c16(P(z0)) 213.71/68.76
S(mark(z0)) → c17(S(z0)) 213.71/68.76
S(ok(z0)) → c18(S(z0)) 213.71/68.76
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.76
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.76
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.76
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.76
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.76
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.76
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.76
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.76
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.76
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.76
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.76
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.76
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.76
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.76
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c7(LEQ(z0, z1), S(diff(p(z0), z1)), DIFF(p(z0), z1), P(z0))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.76
ACTIVE(p(z0)) → c8(P(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.76
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.76
ACTIVE(if(z0, z1, z2)) → c12(IF(active(z0), z1, z2), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.76
P(mark(z0)) → c15(P(z0)) 213.71/68.76
P(ok(z0)) → c16(P(z0)) 213.71/68.76
S(mark(z0)) → c17(S(z0)) 213.71/68.76
S(ok(z0)) → c18(S(z0)) 213.71/68.76
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.76
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.76
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.76
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.76
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.76
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.76
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.76
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.76
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.76
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.76
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.76
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.76
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.76
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.76
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c7(LEQ(z0, z1), S(diff(p(z0), z1)), DIFF(p(z0), z1), P(z0))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c29, c30, c33, c34, c35, c36, c7

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213.71/68.76

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

Split RHS of tuples not part of any SCC
213.71/68.76
213.71/68.76

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.76
active(p(s(z0))) → mark(z0) 213.71/68.76
active(leq(0, z0)) → mark(true) 213.71/68.76
active(leq(s(z0), 0)) → mark(false) 213.71/68.76
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.76
active(if(true, z0, z1)) → mark(z0) 213.71/68.76
active(if(false, z0, z1)) → mark(z1) 213.71/68.76
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.76
active(p(z0)) → p(active(z0)) 213.71/68.76
active(s(z0)) → s(active(z0)) 213.71/68.76
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.76
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.76
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.76
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.76
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.76
p(mark(z0)) → mark(p(z0)) 213.71/68.76
p(ok(z0)) → ok(p(z0)) 213.71/68.76
s(mark(z0)) → mark(s(z0)) 213.71/68.76
s(ok(z0)) → ok(s(z0)) 213.71/68.76
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.76
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.76
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.76
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.76
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.76
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.76
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.76
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.76
proper(p(z0)) → p(proper(z0)) 213.71/68.76
proper(0) → ok(0) 213.71/68.76
proper(s(z0)) → s(proper(z0)) 213.71/68.76
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.76
proper(true) → ok(true) 213.71/68.76
proper(false) → ok(false) 213.71/68.76
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.76
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.76
top(mark(z0)) → top(proper(z0)) 213.71/68.76
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.76
ACTIVE(p(z0)) → c8(P(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.76
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.76
ACTIVE(if(z0, z1, z2)) → c12(IF(active(z0), z1, z2), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.76
P(mark(z0)) → c15(P(z0)) 213.71/68.76
P(ok(z0)) → c16(P(z0)) 213.71/68.76
S(mark(z0)) → c17(S(z0)) 213.71/68.76
S(ok(z0)) → c18(S(z0)) 213.71/68.76
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.76
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.76
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.76
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.76
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.76
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.76
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.76
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.76
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.76
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.76
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.76
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.76
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.76
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.76
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.76
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c(P(z0))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.76
ACTIVE(p(z0)) → c8(P(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.76
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.76
ACTIVE(if(z0, z1, z2)) → c12(IF(active(z0), z1, z2), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.76
P(mark(z0)) → c15(P(z0)) 213.71/68.76
P(ok(z0)) → c16(P(z0)) 213.71/68.76
S(mark(z0)) → c17(S(z0)) 213.71/68.76
S(ok(z0)) → c18(S(z0)) 213.71/68.76
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.76
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.76
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.76
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.76
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.76
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.76
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.76
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.76
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.76
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.76
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.76
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.76
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.76
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.76
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.76
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c(P(z0))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c29, c30, c33, c34, c35, c36, c

213.71/68.76
213.71/68.76

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

Use narrowing to replace ACTIVE(p(z0)) → c8(P(active(z0)), ACTIVE(z0)) by

ACTIVE(p(p(0))) → c8(P(mark(0)), ACTIVE(p(0))) 213.71/68.76
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.76
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.76
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.76
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.76
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.76
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.76
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.76
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.76
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.76
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.76
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.76
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.76
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.76
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1)))
213.71/68.76
213.71/68.76

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.76
active(p(s(z0))) → mark(z0) 213.71/68.76
active(leq(0, z0)) → mark(true) 213.71/68.76
active(leq(s(z0), 0)) → mark(false) 213.71/68.76
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.76
active(if(true, z0, z1)) → mark(z0) 213.71/68.76
active(if(false, z0, z1)) → mark(z1) 213.71/68.76
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.76
active(p(z0)) → p(active(z0)) 213.71/68.76
active(s(z0)) → s(active(z0)) 213.71/68.76
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.76
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.76
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.76
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.76
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.76
p(mark(z0)) → mark(p(z0)) 213.71/68.76
p(ok(z0)) → ok(p(z0)) 213.71/68.76
s(mark(z0)) → mark(s(z0)) 213.71/68.76
s(ok(z0)) → ok(s(z0)) 213.71/68.76
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.76
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.76
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.76
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.76
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.76
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.76
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.76
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.76
proper(p(z0)) → p(proper(z0)) 213.71/68.76
proper(0) → ok(0) 213.71/68.76
proper(s(z0)) → s(proper(z0)) 213.71/68.76
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.76
proper(true) → ok(true) 213.71/68.76
proper(false) → ok(false) 213.71/68.76
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.76
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.76
top(mark(z0)) → top(proper(z0)) 213.71/68.76
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.76
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.76
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.76
ACTIVE(if(z0, z1, z2)) → c12(IF(active(z0), z1, z2), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.76
P(mark(z0)) → c15(P(z0)) 213.71/68.76
P(ok(z0)) → c16(P(z0)) 213.71/68.76
S(mark(z0)) → c17(S(z0)) 213.71/68.76
S(ok(z0)) → c18(S(z0)) 213.71/68.76
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.76
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.76
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.76
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.76
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.76
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.76
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.76
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.76
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.76
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.76
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.76
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.76
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.76
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.76
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.76
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.76
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.76
ACTIVE(p(p(0))) → c8(P(mark(0)), ACTIVE(p(0))) 213.71/68.76
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.76
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.76
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.76
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.76
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.76
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.76
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.76
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.76
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.76
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.76
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.76
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1)))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.77
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
ACTIVE(if(z0, z1, z2)) → c12(IF(active(z0), z1, z2), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
P(mark(z0)) → c15(P(z0)) 213.71/68.77
P(ok(z0)) → c16(P(z0)) 213.71/68.77
S(mark(z0)) → c17(S(z0)) 213.71/68.77
S(ok(z0)) → c18(S(z0)) 213.71/68.77
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.77
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.77
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.77
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.77
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.77
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.77
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.77
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.77
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.77
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.77
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.77
ACTIVE(p(p(0))) → c8(P(mark(0)), ACTIVE(p(0))) 213.71/68.77
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c29, c30, c33, c34, c35, c36, c, c8

213.71/68.77
213.71/68.77

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

Removed 1 trailing tuple parts
213.71/68.77
213.71/68.77

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.77
active(p(s(z0))) → mark(z0) 213.71/68.77
active(leq(0, z0)) → mark(true) 213.71/68.77
active(leq(s(z0), 0)) → mark(false) 213.71/68.77
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.77
active(if(true, z0, z1)) → mark(z0) 213.71/68.77
active(if(false, z0, z1)) → mark(z1) 213.71/68.77
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.77
active(p(z0)) → p(active(z0)) 213.71/68.77
active(s(z0)) → s(active(z0)) 213.71/68.77
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.77
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.77
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.77
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.77
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.77
p(mark(z0)) → mark(p(z0)) 213.71/68.77
p(ok(z0)) → ok(p(z0)) 213.71/68.77
s(mark(z0)) → mark(s(z0)) 213.71/68.77
s(ok(z0)) → ok(s(z0)) 213.71/68.77
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.77
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.77
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.77
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.77
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.77
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.77
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.77
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.77
proper(p(z0)) → p(proper(z0)) 213.71/68.77
proper(0) → ok(0) 213.71/68.77
proper(s(z0)) → s(proper(z0)) 213.71/68.77
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.77
proper(true) → ok(true) 213.71/68.77
proper(false) → ok(false) 213.71/68.77
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.77
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.77
top(mark(z0)) → top(proper(z0)) 213.71/68.77
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.77
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
ACTIVE(if(z0, z1, z2)) → c12(IF(active(z0), z1, z2), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
P(mark(z0)) → c15(P(z0)) 213.71/68.77
P(ok(z0)) → c16(P(z0)) 213.71/68.77
S(mark(z0)) → c17(S(z0)) 213.71/68.77
S(ok(z0)) → c18(S(z0)) 213.71/68.77
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.77
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.77
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.77
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.77
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.77
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.77
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.77
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.77
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.77
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.77
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.77
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(0))) → c8(P(mark(0)))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.77
ACTIVE(s(z0)) → c9(S(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
ACTIVE(if(z0, z1, z2)) → c12(IF(active(z0), z1, z2), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
P(mark(z0)) → c15(P(z0)) 213.71/68.77
P(ok(z0)) → c16(P(z0)) 213.71/68.77
S(mark(z0)) → c17(S(z0)) 213.71/68.77
S(ok(z0)) → c18(S(z0)) 213.71/68.77
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.77
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.77
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.77
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.77
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.77
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.77
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.77
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.77
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.77
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.77
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.77
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(0))) → c8(P(mark(0)))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c29, c30, c33, c34, c35, c36, c, c8, c8

213.71/68.77
213.71/68.77

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

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

ACTIVE(s(p(0))) → c9(S(mark(0)), ACTIVE(p(0))) 213.71/68.77
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1)))
213.71/68.77
213.71/68.77

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.77
active(p(s(z0))) → mark(z0) 213.71/68.77
active(leq(0, z0)) → mark(true) 213.71/68.77
active(leq(s(z0), 0)) → mark(false) 213.71/68.77
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.77
active(if(true, z0, z1)) → mark(z0) 213.71/68.77
active(if(false, z0, z1)) → mark(z1) 213.71/68.77
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.77
active(p(z0)) → p(active(z0)) 213.71/68.77
active(s(z0)) → s(active(z0)) 213.71/68.77
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.77
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.77
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.77
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.77
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.77
p(mark(z0)) → mark(p(z0)) 213.71/68.77
p(ok(z0)) → ok(p(z0)) 213.71/68.77
s(mark(z0)) → mark(s(z0)) 213.71/68.77
s(ok(z0)) → ok(s(z0)) 213.71/68.77
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.77
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.77
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.77
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.77
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.77
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.77
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.77
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.77
proper(p(z0)) → p(proper(z0)) 213.71/68.77
proper(0) → ok(0) 213.71/68.77
proper(s(z0)) → s(proper(z0)) 213.71/68.77
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.77
proper(true) → ok(true) 213.71/68.77
proper(false) → ok(false) 213.71/68.77
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.77
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.77
top(mark(z0)) → top(proper(z0)) 213.71/68.77
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
ACTIVE(if(z0, z1, z2)) → c12(IF(active(z0), z1, z2), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
P(mark(z0)) → c15(P(z0)) 213.71/68.77
P(ok(z0)) → c16(P(z0)) 213.71/68.77
S(mark(z0)) → c17(S(z0)) 213.71/68.77
S(ok(z0)) → c18(S(z0)) 213.71/68.77
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.77
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.77
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.77
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.77
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.77
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.77
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.77
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.77
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.77
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.77
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.77
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.77
ACTIVE(s(p(0))) → c9(S(mark(0)), ACTIVE(p(0))) 213.71/68.77
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1)))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
ACTIVE(if(z0, z1, z2)) → c12(IF(active(z0), z1, z2), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
P(mark(z0)) → c15(P(z0)) 213.71/68.77
P(ok(z0)) → c16(P(z0)) 213.71/68.77
S(mark(z0)) → c17(S(z0)) 213.71/68.77
S(ok(z0)) → c18(S(z0)) 213.71/68.77
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.77
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.77
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.77
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.77
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.77
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.77
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.77
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.77
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.77
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.77
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.77
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.77
ACTIVE(s(p(0))) → c9(S(mark(0)), ACTIVE(p(0))) 213.71/68.77
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c29, c30, c33, c34, c35, c36, c, c8, c8, c9

213.71/68.77
213.71/68.77

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

Removed 1 trailing tuple parts
213.71/68.77
213.71/68.77

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.77
active(p(s(z0))) → mark(z0) 213.71/68.77
active(leq(0, z0)) → mark(true) 213.71/68.77
active(leq(s(z0), 0)) → mark(false) 213.71/68.77
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.77
active(if(true, z0, z1)) → mark(z0) 213.71/68.77
active(if(false, z0, z1)) → mark(z1) 213.71/68.77
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.77
active(p(z0)) → p(active(z0)) 213.71/68.77
active(s(z0)) → s(active(z0)) 213.71/68.77
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.77
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.77
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.77
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.77
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.77
p(mark(z0)) → mark(p(z0)) 213.71/68.77
p(ok(z0)) → ok(p(z0)) 213.71/68.77
s(mark(z0)) → mark(s(z0)) 213.71/68.77
s(ok(z0)) → ok(s(z0)) 213.71/68.77
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.77
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.77
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.77
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.77
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.77
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.77
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.77
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.77
proper(p(z0)) → p(proper(z0)) 213.71/68.77
proper(0) → ok(0) 213.71/68.77
proper(s(z0)) → s(proper(z0)) 213.71/68.77
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.77
proper(true) → ok(true) 213.71/68.77
proper(false) → ok(false) 213.71/68.77
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.77
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.77
top(mark(z0)) → top(proper(z0)) 213.71/68.77
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
ACTIVE(if(z0, z1, z2)) → c12(IF(active(z0), z1, z2), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
P(mark(z0)) → c15(P(z0)) 213.71/68.77
P(ok(z0)) → c16(P(z0)) 213.71/68.77
S(mark(z0)) → c17(S(z0)) 213.71/68.77
S(ok(z0)) → c18(S(z0)) 213.71/68.77
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.77
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.77
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.77
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.77
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.77
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.77
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.77
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.77
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.77
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.77
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.77
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.77
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(0))) → c9(S(mark(0)))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
ACTIVE(if(z0, z1, z2)) → c12(IF(active(z0), z1, z2), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
P(mark(z0)) → c15(P(z0)) 213.71/68.77
P(ok(z0)) → c16(P(z0)) 213.71/68.77
S(mark(z0)) → c17(S(z0)) 213.71/68.77
S(ok(z0)) → c18(S(z0)) 213.71/68.77
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.77
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.77
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.77
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.77
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.77
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.77
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.77
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.77
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.77
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.77
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.77
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.77
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(0))) → c9(S(mark(0)))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c29, c30, c33, c34, c35, c36, c, c8, c8, c9, c9

213.71/68.77
213.71/68.77

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

Use narrowing to replace ACTIVE(if(z0, z1, z2)) → c12(IF(active(z0), z1, z2), ACTIVE(z0)) by

ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2), ACTIVE(p(0))) 213.71/68.77
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1)))
213.71/68.77
213.71/68.77

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.77
active(p(s(z0))) → mark(z0) 213.71/68.77
active(leq(0, z0)) → mark(true) 213.71/68.77
active(leq(s(z0), 0)) → mark(false) 213.71/68.77
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.77
active(if(true, z0, z1)) → mark(z0) 213.71/68.77
active(if(false, z0, z1)) → mark(z1) 213.71/68.77
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.77
active(p(z0)) → p(active(z0)) 213.71/68.77
active(s(z0)) → s(active(z0)) 213.71/68.77
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.77
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.77
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.77
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.77
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.77
p(mark(z0)) → mark(p(z0)) 213.71/68.77
p(ok(z0)) → ok(p(z0)) 213.71/68.77
s(mark(z0)) → mark(s(z0)) 213.71/68.77
s(ok(z0)) → ok(s(z0)) 213.71/68.77
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.77
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.77
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.77
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.77
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.77
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.77
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.77
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.77
proper(p(z0)) → p(proper(z0)) 213.71/68.77
proper(0) → ok(0) 213.71/68.77
proper(s(z0)) → s(proper(z0)) 213.71/68.77
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.77
proper(true) → ok(true) 213.71/68.77
proper(false) → ok(false) 213.71/68.77
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.77
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.77
top(mark(z0)) → top(proper(z0)) 213.71/68.77
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
P(mark(z0)) → c15(P(z0)) 213.71/68.77
P(ok(z0)) → c16(P(z0)) 213.71/68.77
S(mark(z0)) → c17(S(z0)) 213.71/68.77
S(ok(z0)) → c18(S(z0)) 213.71/68.77
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.77
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.77
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.77
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.77
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.77
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.77
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.77
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.77
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.77
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.77
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.77
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.77
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.77
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2), ACTIVE(p(0))) 213.71/68.77
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1)))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
P(mark(z0)) → c15(P(z0)) 213.71/68.77
P(ok(z0)) → c16(P(z0)) 213.71/68.77
S(mark(z0)) → c17(S(z0)) 213.71/68.77
S(ok(z0)) → c18(S(z0)) 213.71/68.77
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.77
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.77
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.77
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.77
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.77
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.77
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.77
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.77
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.77
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.77
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.77
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.77
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0)), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1)), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.77
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2), ACTIVE(p(0))) 213.71/68.77
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(if(true, z0, z1))) 213.71/68.77
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2), ACTIVE(if(false, z0, z1))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c29, c30, c33, c34, c35, c36, c, c8, c8, c9, c9, c12

213.71/68.77
213.71/68.77

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

Removed 7 trailing tuple parts
213.71/68.77
213.71/68.77

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.77
active(p(s(z0))) → mark(z0) 213.71/68.77
active(leq(0, z0)) → mark(true) 213.71/68.77
active(leq(s(z0), 0)) → mark(false) 213.71/68.77
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.77
active(if(true, z0, z1)) → mark(z0) 213.71/68.77
active(if(false, z0, z1)) → mark(z1) 213.71/68.77
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.77
active(p(z0)) → p(active(z0)) 213.71/68.77
active(s(z0)) → s(active(z0)) 213.71/68.77
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.77
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.77
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.77
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.77
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.77
p(mark(z0)) → mark(p(z0)) 213.71/68.77
p(ok(z0)) → ok(p(z0)) 213.71/68.77
s(mark(z0)) → mark(s(z0)) 213.71/68.77
s(ok(z0)) → ok(s(z0)) 213.71/68.77
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.77
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.77
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.77
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.77
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.77
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.77
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.77
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.77
proper(p(z0)) → p(proper(z0)) 213.71/68.77
proper(0) → ok(0) 213.71/68.77
proper(s(z0)) → s(proper(z0)) 213.71/68.77
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.77
proper(true) → ok(true) 213.71/68.77
proper(false) → ok(false) 213.71/68.77
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.77
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.77
top(mark(z0)) → top(proper(z0)) 213.71/68.77
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
P(mark(z0)) → c15(P(z0)) 213.71/68.77
P(ok(z0)) → c16(P(z0)) 213.71/68.77
S(mark(z0)) → c17(S(z0)) 213.71/68.77
S(ok(z0)) → c18(S(z0)) 213.71/68.77
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.77
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.77
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.77
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.77
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.77
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.77
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.77
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.77
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.77
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.77
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.77
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.77
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.77
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.77
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.77
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.77
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.77
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.77
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.77
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
P(mark(z0)) → c15(P(z0)) 213.71/68.77
P(ok(z0)) → c16(P(z0)) 213.71/68.77
S(mark(z0)) → c17(S(z0)) 213.71/68.77
S(ok(z0)) → c18(S(z0)) 213.71/68.77
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.77
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.77
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.77
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.77
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.77
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.77
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.77
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.77
PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.77
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.77
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.77
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.77
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.77
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.77
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.77
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.77
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.77
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.77
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.77
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c29, c30, c33, c34, c35, c36, c, c8, c8, c9, c9, c12, c12

213.71/68.77
213.71/68.77

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

Use narrowing to replace PROPER(p(z0)) → c27(P(proper(z0)), PROPER(z0)) by

PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.77
PROPER(p(0)) → c27(P(ok(0)), PROPER(0)) 213.71/68.77
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.77
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.77
PROPER(p(true)) → c27(P(ok(true)), PROPER(true)) 213.71/68.77
PROPER(p(false)) → c27(P(ok(false)), PROPER(false)) 213.71/68.77
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.77
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1)))
213.71/68.77
213.71/68.77

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.77
active(p(s(z0))) → mark(z0) 213.71/68.77
active(leq(0, z0)) → mark(true) 213.71/68.77
active(leq(s(z0), 0)) → mark(false) 213.71/68.77
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.77
active(if(true, z0, z1)) → mark(z0) 213.71/68.77
active(if(false, z0, z1)) → mark(z1) 213.71/68.77
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.77
active(p(z0)) → p(active(z0)) 213.71/68.77
active(s(z0)) → s(active(z0)) 213.71/68.77
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.77
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.77
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.77
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.77
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.77
p(mark(z0)) → mark(p(z0)) 213.71/68.77
p(ok(z0)) → ok(p(z0)) 213.71/68.77
s(mark(z0)) → mark(s(z0)) 213.71/68.77
s(ok(z0)) → ok(s(z0)) 213.71/68.77
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.77
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.77
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.77
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.77
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.77
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.77
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.77
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.77
proper(p(z0)) → p(proper(z0)) 213.71/68.77
proper(0) → ok(0) 213.71/68.77
proper(s(z0)) → s(proper(z0)) 213.71/68.77
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.77
proper(true) → ok(true) 213.71/68.77
proper(false) → ok(false) 213.71/68.77
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.77
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.77
top(mark(z0)) → top(proper(z0)) 213.71/68.77
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
P(mark(z0)) → c15(P(z0)) 213.71/68.77
P(ok(z0)) → c16(P(z0)) 213.71/68.77
S(mark(z0)) → c17(S(z0)) 213.71/68.77
S(ok(z0)) → c18(S(z0)) 213.71/68.77
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.77
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.77
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.77
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.77
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.77
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.77
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.77
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.77
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.77
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.77
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.77
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.77
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.77
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.77
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.77
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.77
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.77
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.77
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.77
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.77
PROPER(p(0)) → c27(P(ok(0)), PROPER(0)) 213.71/68.77
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.77
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.77
PROPER(p(true)) → c27(P(ok(true)), PROPER(true)) 213.71/68.77
PROPER(p(false)) → c27(P(ok(false)), PROPER(false)) 213.71/68.77
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.77
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1)))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
P(mark(z0)) → c15(P(z0)) 213.71/68.77
P(ok(z0)) → c16(P(z0)) 213.71/68.77
S(mark(z0)) → c17(S(z0)) 213.71/68.77
S(ok(z0)) → c18(S(z0)) 213.71/68.77
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.77
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.77
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.77
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.77
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.77
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.77
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.77
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.77
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.77
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.77
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.77
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.77
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.77
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.77
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.77
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.77
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.77
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.77
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.77
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.77
PROPER(p(0)) → c27(P(ok(0)), PROPER(0)) 213.71/68.77
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.77
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.77
PROPER(p(true)) → c27(P(ok(true)), PROPER(true)) 213.71/68.77
PROPER(p(false)) → c27(P(ok(false)), PROPER(false)) 213.71/68.77
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.77
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c29, c30, c33, c34, c35, c36, c, c8, c8, c9, c9, c12, c12, c27

213.71/68.77
213.71/68.77

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

Removed 3 trailing tuple parts
213.71/68.77
213.71/68.77

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.77
active(p(s(z0))) → mark(z0) 213.71/68.77
active(leq(0, z0)) → mark(true) 213.71/68.77
active(leq(s(z0), 0)) → mark(false) 213.71/68.77
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.77
active(if(true, z0, z1)) → mark(z0) 213.71/68.77
active(if(false, z0, z1)) → mark(z1) 213.71/68.77
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.77
active(p(z0)) → p(active(z0)) 213.71/68.77
active(s(z0)) → s(active(z0)) 213.71/68.77
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.77
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.77
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.77
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.77
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.77
p(mark(z0)) → mark(p(z0)) 213.71/68.77
p(ok(z0)) → ok(p(z0)) 213.71/68.77
s(mark(z0)) → mark(s(z0)) 213.71/68.77
s(ok(z0)) → ok(s(z0)) 213.71/68.77
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.77
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.77
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.77
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.77
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.77
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.77
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.77
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.77
proper(p(z0)) → p(proper(z0)) 213.71/68.77
proper(0) → ok(0) 213.71/68.77
proper(s(z0)) → s(proper(z0)) 213.71/68.77
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.77
proper(true) → ok(true) 213.71/68.77
proper(false) → ok(false) 213.71/68.77
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.77
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.77
top(mark(z0)) → top(proper(z0)) 213.71/68.77
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
P(mark(z0)) → c15(P(z0)) 213.71/68.77
P(ok(z0)) → c16(P(z0)) 213.71/68.77
S(mark(z0)) → c17(S(z0)) 213.71/68.77
S(ok(z0)) → c18(S(z0)) 213.71/68.77
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.77
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.77
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.77
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.77
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.77
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.77
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.77
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.77
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.77
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.77
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.77
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.77
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.77
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.77
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.77
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.77
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.77
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.77
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.77
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.77
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.77
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.77
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.77
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.77
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.77
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.77
PROPER(p(false)) → c27(P(ok(false)))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.77
P(mark(z0)) → c15(P(z0)) 213.71/68.77
P(ok(z0)) → c16(P(z0)) 213.71/68.77
S(mark(z0)) → c17(S(z0)) 213.71/68.77
S(ok(z0)) → c18(S(z0)) 213.71/68.77
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.77
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.77
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.77
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.77
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.77
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.77
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.77
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.77
PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) 213.71/68.77
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.77
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.77
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.77
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.77
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.77
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.77
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.77
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.77
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.77
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.77
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.77
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.77
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.77
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.77
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.77
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.77
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.77
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.77
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.77
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.77
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.77
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.77
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.77
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.77
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.77
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.77
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.77
PROPER(p(false)) → c27(P(ok(false)))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c29, c30, c33, c34, c35, c36, c, c8, c8, c9, c9, c12, c12, c27, c27

213.71/68.77
213.71/68.77

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

Use narrowing to replace PROPER(s(z0)) → c29(S(proper(z0)), PROPER(z0)) by

PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.77
PROPER(s(0)) → c29(S(ok(0)), PROPER(0)) 213.71/68.77
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.77
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.77
PROPER(s(true)) → c29(S(ok(true)), PROPER(true)) 213.71/68.77
PROPER(s(false)) → c29(S(ok(false)), PROPER(false)) 213.71/68.77
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.77
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1)))
213.71/68.77
213.71/68.77

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.77
active(p(s(z0))) → mark(z0) 213.71/68.77
active(leq(0, z0)) → mark(true) 213.71/68.77
active(leq(s(z0), 0)) → mark(false) 213.71/68.77
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.77
active(if(true, z0, z1)) → mark(z0) 213.71/68.77
active(if(false, z0, z1)) → mark(z1) 213.71/68.77
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.77
active(p(z0)) → p(active(z0)) 213.71/68.77
active(s(z0)) → s(active(z0)) 213.71/68.77
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.77
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.77
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.77
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.77
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.77
p(mark(z0)) → mark(p(z0)) 213.71/68.77
p(ok(z0)) → ok(p(z0)) 213.71/68.77
s(mark(z0)) → mark(s(z0)) 213.71/68.78
s(ok(z0)) → ok(s(z0)) 213.71/68.78
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.78
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.78
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.78
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.78
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.78
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.78
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.78
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.78
proper(p(z0)) → p(proper(z0)) 213.71/68.78
proper(0) → ok(0) 213.71/68.78
proper(s(z0)) → s(proper(z0)) 213.71/68.78
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.78
proper(true) → ok(true) 213.71/68.78
proper(false) → ok(false) 213.71/68.78
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.78
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.78
top(mark(z0)) → top(proper(z0)) 213.71/68.78
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
P(mark(z0)) → c15(P(z0)) 213.71/68.78
P(ok(z0)) → c16(P(z0)) 213.71/68.78
S(mark(z0)) → c17(S(z0)) 213.71/68.78
S(ok(z0)) → c18(S(z0)) 213.71/68.78
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.78
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.78
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.78
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.78
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.78
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.78
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.78
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.78
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.78
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.78
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.78
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.78
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.78
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.78
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.78
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.78
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.78
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.78
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.78
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.78
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.78
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.78
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.78
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(s(0)) → c29(S(ok(0)), PROPER(0)) 213.71/68.78
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(s(true)) → c29(S(ok(true)), PROPER(true)) 213.71/68.78
PROPER(s(false)) → c29(S(ok(false)), PROPER(false)) 213.71/68.78
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1)))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
P(mark(z0)) → c15(P(z0)) 213.71/68.78
P(ok(z0)) → c16(P(z0)) 213.71/68.78
S(mark(z0)) → c17(S(z0)) 213.71/68.78
S(ok(z0)) → c18(S(z0)) 213.71/68.78
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.78
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.78
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.78
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.78
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.78
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.78
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.78
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.78
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.78
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.78
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.78
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.78
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.78
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.78
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.78
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.78
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.78
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.78
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.78
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.78
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.78
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.78
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.78
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(s(0)) → c29(S(ok(0)), PROPER(0)) 213.71/68.78
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(s(true)) → c29(S(ok(true)), PROPER(true)) 213.71/68.78
PROPER(s(false)) → c29(S(ok(false)), PROPER(false)) 213.71/68.78
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c33, c34, c35, c36, c, c8, c8, c9, c9, c12, c12, c27, c27, c29

213.71/68.78
213.71/68.78

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

Removed 3 trailing tuple parts
213.71/68.78
213.71/68.78

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.78
active(p(s(z0))) → mark(z0) 213.71/68.78
active(leq(0, z0)) → mark(true) 213.71/68.78
active(leq(s(z0), 0)) → mark(false) 213.71/68.78
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.78
active(if(true, z0, z1)) → mark(z0) 213.71/68.78
active(if(false, z0, z1)) → mark(z1) 213.71/68.78
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.78
active(p(z0)) → p(active(z0)) 213.71/68.78
active(s(z0)) → s(active(z0)) 213.71/68.78
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.78
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.78
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.78
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.78
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.78
p(mark(z0)) → mark(p(z0)) 213.71/68.78
p(ok(z0)) → ok(p(z0)) 213.71/68.78
s(mark(z0)) → mark(s(z0)) 213.71/68.78
s(ok(z0)) → ok(s(z0)) 213.71/68.78
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.78
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.78
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.78
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.78
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.78
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.78
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.78
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.78
proper(p(z0)) → p(proper(z0)) 213.71/68.78
proper(0) → ok(0) 213.71/68.78
proper(s(z0)) → s(proper(z0)) 213.71/68.78
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.78
proper(true) → ok(true) 213.71/68.78
proper(false) → ok(false) 213.71/68.78
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.78
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.78
top(mark(z0)) → top(proper(z0)) 213.71/68.78
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
P(mark(z0)) → c15(P(z0)) 213.71/68.78
P(ok(z0)) → c16(P(z0)) 213.71/68.78
S(mark(z0)) → c17(S(z0)) 213.71/68.78
S(ok(z0)) → c18(S(z0)) 213.71/68.78
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.78
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.78
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.78
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.78
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.78
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.78
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.78
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.78
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.78
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.78
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.78
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.78
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.78
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.78
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.78
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.78
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.78
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.78
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.78
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.78
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.78
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.78
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.78
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.78
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.78
PROPER(s(false)) → c29(S(ok(false)))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
P(mark(z0)) → c15(P(z0)) 213.71/68.78
P(ok(z0)) → c16(P(z0)) 213.71/68.78
S(mark(z0)) → c17(S(z0)) 213.71/68.78
S(ok(z0)) → c18(S(z0)) 213.71/68.78
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.78
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.78
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.78
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.78
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.78
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.78
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.78
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.78
PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.78
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.78
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.78
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.78
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.78
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.78
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.78
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.78
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.78
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.78
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.78
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.78
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.78
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.78
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.78
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.78
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.78
PROPER(s(false)) → c29(S(ok(false)))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c33, c34, c35, c36, c, c8, c8, c9, c9, c12, c12, c27, c27, c29, c29

213.71/68.78
213.71/68.78

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

Use narrowing to replace PROPER(leq(z0, z1)) → c30(LEQ(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.78
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 213.71/68.78
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.78
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 213.71/68.78
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 213.71/68.78
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 213.71/68.78
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.78
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 213.71/68.78
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 213.71/68.78
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.78
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1))
213.71/68.78
213.71/68.78

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.78
active(p(s(z0))) → mark(z0) 213.71/68.78
active(leq(0, z0)) → mark(true) 213.71/68.78
active(leq(s(z0), 0)) → mark(false) 213.71/68.78
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.78
active(if(true, z0, z1)) → mark(z0) 213.71/68.78
active(if(false, z0, z1)) → mark(z1) 213.71/68.78
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.78
active(p(z0)) → p(active(z0)) 213.71/68.78
active(s(z0)) → s(active(z0)) 213.71/68.78
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.78
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.78
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.78
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.78
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.78
p(mark(z0)) → mark(p(z0)) 213.71/68.78
p(ok(z0)) → ok(p(z0)) 213.71/68.78
s(mark(z0)) → mark(s(z0)) 213.71/68.78
s(ok(z0)) → ok(s(z0)) 213.71/68.78
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.78
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.78
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.78
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.78
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.78
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.78
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.78
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.78
proper(p(z0)) → p(proper(z0)) 213.71/68.78
proper(0) → ok(0) 213.71/68.78
proper(s(z0)) → s(proper(z0)) 213.71/68.78
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.78
proper(true) → ok(true) 213.71/68.78
proper(false) → ok(false) 213.71/68.78
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.78
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.78
top(mark(z0)) → top(proper(z0)) 213.71/68.78
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
P(mark(z0)) → c15(P(z0)) 213.71/68.78
P(ok(z0)) → c16(P(z0)) 213.71/68.78
S(mark(z0)) → c17(S(z0)) 213.71/68.78
S(ok(z0)) → c18(S(z0)) 213.71/68.78
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.78
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.78
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.78
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.78
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.78
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.78
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.78
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.78
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.78
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.78
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.78
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.78
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.78
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.78
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.78
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.78
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.78
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.78
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.78
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.78
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.78
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.78
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.78
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.78
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.78
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.78
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.78
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 213.71/68.78
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.78
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 213.71/68.78
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 213.71/68.78
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 213.71/68.78
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.78
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 213.71/68.78
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 213.71/68.78
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.78
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
P(mark(z0)) → c15(P(z0)) 213.71/68.78
P(ok(z0)) → c16(P(z0)) 213.71/68.78
S(mark(z0)) → c17(S(z0)) 213.71/68.78
S(ok(z0)) → c18(S(z0)) 213.71/68.78
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.78
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.78
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.78
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.78
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.78
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.78
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.78
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.78
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.78
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.78
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.78
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.78
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.78
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.78
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.78
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.78
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.78
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.78
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.78
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.78
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.78
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.78
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.78
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.78
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.78
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.78
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.78
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 213.71/68.78
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.78
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 213.71/68.78
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 213.71/68.78
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 213.71/68.78
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.78
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 213.71/68.78
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 213.71/68.78
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.78
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c33, c34, c35, c36, c, c8, c8, c9, c9, c12, c12, c27, c27, c29, c29, c30

213.71/68.78
213.71/68.78

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

Removed 6 trailing tuple parts
213.71/68.78
213.71/68.78

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.78
active(p(s(z0))) → mark(z0) 213.71/68.78
active(leq(0, z0)) → mark(true) 213.71/68.78
active(leq(s(z0), 0)) → mark(false) 213.71/68.78
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.78
active(if(true, z0, z1)) → mark(z0) 213.71/68.78
active(if(false, z0, z1)) → mark(z1) 213.71/68.78
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.78
active(p(z0)) → p(active(z0)) 213.71/68.78
active(s(z0)) → s(active(z0)) 213.71/68.78
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.78
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.78
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.78
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.78
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.78
p(mark(z0)) → mark(p(z0)) 213.71/68.78
p(ok(z0)) → ok(p(z0)) 213.71/68.78
s(mark(z0)) → mark(s(z0)) 213.71/68.78
s(ok(z0)) → ok(s(z0)) 213.71/68.78
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.78
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.78
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.78
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.78
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.78
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.78
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.78
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.78
proper(p(z0)) → p(proper(z0)) 213.71/68.78
proper(0) → ok(0) 213.71/68.78
proper(s(z0)) → s(proper(z0)) 213.71/68.78
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.78
proper(true) → ok(true) 213.71/68.78
proper(false) → ok(false) 213.71/68.78
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.78
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.78
top(mark(z0)) → top(proper(z0)) 213.71/68.78
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
P(mark(z0)) → c15(P(z0)) 213.71/68.78
P(ok(z0)) → c16(P(z0)) 213.71/68.78
S(mark(z0)) → c17(S(z0)) 213.71/68.78
S(ok(z0)) → c18(S(z0)) 213.71/68.78
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.78
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.78
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.78
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.78
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.78
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.78
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.78
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.78
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.78
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.78
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.78
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.78
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.78
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.78
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.78
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.78
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.78
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.78
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.78
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.78
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.78
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.78
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.78
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.78
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.78
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.78
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.78
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.78
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.78
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.78
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.78
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.78
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.78
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.78
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.78
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.78
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
P(mark(z0)) → c15(P(z0)) 213.71/68.78
P(ok(z0)) → c16(P(z0)) 213.71/68.78
S(mark(z0)) → c17(S(z0)) 213.71/68.78
S(ok(z0)) → c18(S(z0)) 213.71/68.78
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.78
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.78
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.78
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.78
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.78
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.78
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.78
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.78
PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 213.71/68.78
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.78
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.78
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.78
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.78
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.78
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.78
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.78
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.78
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.78
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.78
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.78
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.78
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.78
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.78
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.78
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.78
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.78
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.78
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.78
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.78
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.78
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.78
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.78
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.78
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.78
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.78
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.78
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c33, c34, c35, c36, c, c8, c8, c9, c9, c12, c12, c27, c27, c29, c29, c30, c30

213.71/68.78
213.71/68.78

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

Use narrowing to replace PROPER(if(z0, z1, z2)) → c33(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 213.71/68.78
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0)) 213.71/68.78
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 213.71/68.78
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true)) 213.71/68.78
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false)) 213.71/68.78
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2)) 213.71/68.78
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2)) 213.71/68.78
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2)) 213.71/68.78
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 213.71/68.78
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2))
213.71/68.78
213.71/68.78

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.78
active(p(s(z0))) → mark(z0) 213.71/68.78
active(leq(0, z0)) → mark(true) 213.71/68.78
active(leq(s(z0), 0)) → mark(false) 213.71/68.78
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.78
active(if(true, z0, z1)) → mark(z0) 213.71/68.78
active(if(false, z0, z1)) → mark(z1) 213.71/68.78
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.78
active(p(z0)) → p(active(z0)) 213.71/68.78
active(s(z0)) → s(active(z0)) 213.71/68.78
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.78
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.78
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.78
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.78
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.78
p(mark(z0)) → mark(p(z0)) 213.71/68.78
p(ok(z0)) → ok(p(z0)) 213.71/68.78
s(mark(z0)) → mark(s(z0)) 213.71/68.78
s(ok(z0)) → ok(s(z0)) 213.71/68.78
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.78
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.78
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.78
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.78
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.78
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.78
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.78
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.78
proper(p(z0)) → p(proper(z0)) 213.71/68.78
proper(0) → ok(0) 213.71/68.78
proper(s(z0)) → s(proper(z0)) 213.71/68.78
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.78
proper(true) → ok(true) 213.71/68.78
proper(false) → ok(false) 213.71/68.78
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.78
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.78
top(mark(z0)) → top(proper(z0)) 213.71/68.78
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
P(mark(z0)) → c15(P(z0)) 213.71/68.78
P(ok(z0)) → c16(P(z0)) 213.71/68.78
S(mark(z0)) → c17(S(z0)) 213.71/68.78
S(ok(z0)) → c18(S(z0)) 213.71/68.78
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.78
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.78
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.78
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.78
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.78
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.78
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.78
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.78
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.78
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.78
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.78
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.78
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.78
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.78
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.78
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.78
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.78
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.78
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.78
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.78
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.78
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.78
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.78
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.78
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.78
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.78
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.78
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.78
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.78
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.78
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.78
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.78
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.78
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.78
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.78
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 213.71/68.78
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 213.71/68.78
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0)) 213.71/68.78
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 213.71/68.78
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true)) 213.71/68.78
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false)) 213.71/68.78
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2)) 213.71/68.78
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2)) 213.71/68.78
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2)) 213.71/68.78
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 213.71/68.78
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
P(mark(z0)) → c15(P(z0)) 213.71/68.78
P(ok(z0)) → c16(P(z0)) 213.71/68.78
S(mark(z0)) → c17(S(z0)) 213.71/68.78
S(ok(z0)) → c18(S(z0)) 213.71/68.78
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.78
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.78
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.78
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.78
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.78
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.78
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.78
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.78
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.78
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.78
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.78
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.78
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.78
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.78
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.78
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.78
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.78
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.78
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.78
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.78
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.78
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.78
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.78
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.78
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.78
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.78
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.78
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.78
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.78
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.78
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.78
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.78
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.78
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.78
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.78
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 213.71/68.78
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 213.71/68.78
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0)) 213.71/68.78
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 213.71/68.78
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true)) 213.71/68.78
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false)) 213.71/68.78
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2)) 213.71/68.78
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2)) 213.71/68.78
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2)) 213.71/68.78
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 213.71/68.78
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c34, c35, c36, c, c8, c8, c9, c9, c12, c12, c27, c27, c29, c29, c30, c30, c33

213.71/68.78
213.71/68.78

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

Removed 9 trailing tuple parts
213.71/68.78
213.71/68.78

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.78
active(p(s(z0))) → mark(z0) 213.71/68.78
active(leq(0, z0)) → mark(true) 213.71/68.78
active(leq(s(z0), 0)) → mark(false) 213.71/68.78
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.78
active(if(true, z0, z1)) → mark(z0) 213.71/68.78
active(if(false, z0, z1)) → mark(z1) 213.71/68.78
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.78
active(p(z0)) → p(active(z0)) 213.71/68.78
active(s(z0)) → s(active(z0)) 213.71/68.78
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.78
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.78
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.78
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.78
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.78
p(mark(z0)) → mark(p(z0)) 213.71/68.78
p(ok(z0)) → ok(p(z0)) 213.71/68.78
s(mark(z0)) → mark(s(z0)) 213.71/68.78
s(ok(z0)) → ok(s(z0)) 213.71/68.78
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.78
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.78
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.78
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.78
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.78
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.78
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.78
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.78
proper(p(z0)) → p(proper(z0)) 213.71/68.78
proper(0) → ok(0) 213.71/68.78
proper(s(z0)) → s(proper(z0)) 213.71/68.78
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.78
proper(true) → ok(true) 213.71/68.78
proper(false) → ok(false) 213.71/68.78
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.78
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.78
top(mark(z0)) → top(proper(z0)) 213.71/68.78
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
P(mark(z0)) → c15(P(z0)) 213.71/68.78
P(ok(z0)) → c16(P(z0)) 213.71/68.78
S(mark(z0)) → c17(S(z0)) 213.71/68.78
S(ok(z0)) → c18(S(z0)) 213.71/68.78
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.78
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.78
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.78
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.78
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.78
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.78
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.78
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.78
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.78
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.78
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.78
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.78
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.78
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.78
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.78
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.78
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.78
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.78
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.78
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.78
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.78
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.78
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.78
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.78
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.78
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.78
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.78
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.78
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.78
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.78
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.78
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.78
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.78
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.78
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.78
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.78
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.78
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.78
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 213.71/68.78
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 213.71/68.78
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 213.71/68.78
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 213.71/68.78
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 213.71/68.78
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 213.71/68.78
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 213.71/68.78
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 213.71/68.78
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 213.71/68.78
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 213.71/68.78
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 213.71/68.78
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.78
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.78
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.78
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.78
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.78
P(mark(z0)) → c15(P(z0)) 213.71/68.78
P(ok(z0)) → c16(P(z0)) 213.71/68.78
S(mark(z0)) → c17(S(z0)) 213.71/68.78
S(ok(z0)) → c18(S(z0)) 213.71/68.78
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.78
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.78
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.78
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.78
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.78
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.78
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.78
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.78
PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 213.71/68.78
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.78
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.78
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.78
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.78
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.78
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.78
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.78
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.78
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.78
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.78
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.78
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.79
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.79
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.79
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.79
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.79
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.79
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.79
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.79
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.79
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.79
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.79
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.79
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 213.71/68.79
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 213.71/68.79
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c34, c35, c36, c, c8, c8, c9, c9, c12, c12, c27, c27, c29, c29, c30, c30, c33, c33

213.71/68.79
213.71/68.79

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

Use narrowing to replace PROPER(diff(z0, z1)) → c34(DIFF(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(diff(x0, p(z0))) → c34(DIFF(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(diff(x0, 0)) → c34(DIFF(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 213.71/68.79
PROPER(diff(x0, s(z0))) → c34(DIFF(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(diff(x0, leq(z0, z1))) → c34(DIFF(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(diff(x0, true)) → c34(DIFF(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 213.71/68.79
PROPER(diff(x0, false)) → c34(DIFF(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 213.71/68.79
PROPER(diff(x0, if(z0, z1, z2))) → c34(DIFF(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(diff(x0, diff(z0, z1))) → c34(DIFF(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(diff(p(z0), x1)) → c34(DIFF(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(0, x1)) → c34(DIFF(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 213.71/68.79
PROPER(diff(s(z0), x1)) → c34(DIFF(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(leq(z0, z1), x1)) → c34(DIFF(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(diff(true, x1)) → c34(DIFF(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 213.71/68.79
PROPER(diff(false, x1)) → c34(DIFF(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 213.71/68.79
PROPER(diff(if(z0, z1, z2), x1)) → c34(DIFF(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(diff(diff(z0, z1), x1)) → c34(DIFF(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1))
213.71/68.79
213.71/68.79

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.79
active(p(s(z0))) → mark(z0) 213.71/68.79
active(leq(0, z0)) → mark(true) 213.71/68.79
active(leq(s(z0), 0)) → mark(false) 213.71/68.79
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.79
active(if(true, z0, z1)) → mark(z0) 213.71/68.79
active(if(false, z0, z1)) → mark(z1) 213.71/68.79
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.79
active(p(z0)) → p(active(z0)) 213.71/68.79
active(s(z0)) → s(active(z0)) 213.71/68.79
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.79
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.79
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.79
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.79
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.79
p(mark(z0)) → mark(p(z0)) 213.71/68.79
p(ok(z0)) → ok(p(z0)) 213.71/68.79
s(mark(z0)) → mark(s(z0)) 213.71/68.79
s(ok(z0)) → ok(s(z0)) 213.71/68.79
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.79
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.79
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.79
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.79
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.79
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.79
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.79
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.79
proper(p(z0)) → p(proper(z0)) 213.71/68.79
proper(0) → ok(0) 213.71/68.79
proper(s(z0)) → s(proper(z0)) 213.71/68.79
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.79
proper(true) → ok(true) 213.71/68.79
proper(false) → ok(false) 213.71/68.79
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.79
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.79
top(mark(z0)) → top(proper(z0)) 213.71/68.79
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
P(mark(z0)) → c15(P(z0)) 213.71/68.79
P(ok(z0)) → c16(P(z0)) 213.71/68.79
S(mark(z0)) → c17(S(z0)) 213.71/68.79
S(ok(z0)) → c18(S(z0)) 213.71/68.79
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.79
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.79
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.79
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.79
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.79
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.79
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.79
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.79
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.79
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.79
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.79
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.79
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.79
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.79
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.79
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.79
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.79
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.79
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.79
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.79
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.79
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.79
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.79
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.79
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 213.71/68.79
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 213.71/68.79
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(diff(x0, p(z0))) → c34(DIFF(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(diff(x0, 0)) → c34(DIFF(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 213.71/68.79
PROPER(diff(x0, s(z0))) → c34(DIFF(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(diff(x0, leq(z0, z1))) → c34(DIFF(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(diff(x0, true)) → c34(DIFF(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 213.71/68.79
PROPER(diff(x0, false)) → c34(DIFF(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 213.71/68.79
PROPER(diff(x0, if(z0, z1, z2))) → c34(DIFF(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(diff(x0, diff(z0, z1))) → c34(DIFF(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(diff(p(z0), x1)) → c34(DIFF(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(0, x1)) → c34(DIFF(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 213.71/68.79
PROPER(diff(s(z0), x1)) → c34(DIFF(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(leq(z0, z1), x1)) → c34(DIFF(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(diff(true, x1)) → c34(DIFF(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 213.71/68.79
PROPER(diff(false, x1)) → c34(DIFF(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 213.71/68.79
PROPER(diff(if(z0, z1, z2), x1)) → c34(DIFF(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(diff(diff(z0, z1), x1)) → c34(DIFF(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
P(mark(z0)) → c15(P(z0)) 213.71/68.79
P(ok(z0)) → c16(P(z0)) 213.71/68.79
S(mark(z0)) → c17(S(z0)) 213.71/68.79
S(ok(z0)) → c18(S(z0)) 213.71/68.79
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.79
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.79
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.79
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.79
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.79
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.79
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.79
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.79
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.79
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.79
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.79
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.79
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.79
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.79
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.79
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.79
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.79
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.79
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.79
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.79
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.79
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.79
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.79
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.79
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 213.71/68.79
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 213.71/68.79
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(diff(x0, p(z0))) → c34(DIFF(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(diff(x0, 0)) → c34(DIFF(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 213.71/68.79
PROPER(diff(x0, s(z0))) → c34(DIFF(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(diff(x0, leq(z0, z1))) → c34(DIFF(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(diff(x0, true)) → c34(DIFF(proper(x0), ok(true)), PROPER(x0), PROPER(true)) 213.71/68.79
PROPER(diff(x0, false)) → c34(DIFF(proper(x0), ok(false)), PROPER(x0), PROPER(false)) 213.71/68.79
PROPER(diff(x0, if(z0, z1, z2))) → c34(DIFF(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(diff(x0, diff(z0, z1))) → c34(DIFF(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(diff(p(z0), x1)) → c34(DIFF(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(0, x1)) → c34(DIFF(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 213.71/68.79
PROPER(diff(s(z0), x1)) → c34(DIFF(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(leq(z0, z1), x1)) → c34(DIFF(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(diff(true, x1)) → c34(DIFF(ok(true), proper(x1)), PROPER(true), PROPER(x1)) 213.71/68.79
PROPER(diff(false, x1)) → c34(DIFF(ok(false), proper(x1)), PROPER(false), PROPER(x1)) 213.71/68.79
PROPER(diff(if(z0, z1, z2), x1)) → c34(DIFF(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(diff(diff(z0, z1), x1)) → c34(DIFF(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, TOP, PROPER

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c35, c36, c, c8, c8, c9, c9, c12, c12, c27, c27, c29, c29, c30, c30, c33, c33, c34

213.71/68.79
213.71/68.79

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

Removed 6 trailing tuple parts
213.71/68.79
213.71/68.79

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.79
active(p(s(z0))) → mark(z0) 213.71/68.79
active(leq(0, z0)) → mark(true) 213.71/68.79
active(leq(s(z0), 0)) → mark(false) 213.71/68.79
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.79
active(if(true, z0, z1)) → mark(z0) 213.71/68.79
active(if(false, z0, z1)) → mark(z1) 213.71/68.79
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.79
active(p(z0)) → p(active(z0)) 213.71/68.79
active(s(z0)) → s(active(z0)) 213.71/68.79
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.79
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.79
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.79
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.79
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.79
p(mark(z0)) → mark(p(z0)) 213.71/68.79
p(ok(z0)) → ok(p(z0)) 213.71/68.79
s(mark(z0)) → mark(s(z0)) 213.71/68.79
s(ok(z0)) → ok(s(z0)) 213.71/68.79
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.79
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.79
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.79
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.79
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.79
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.79
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.79
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.79
proper(p(z0)) → p(proper(z0)) 213.71/68.79
proper(0) → ok(0) 213.71/68.79
proper(s(z0)) → s(proper(z0)) 213.71/68.79
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.79
proper(true) → ok(true) 213.71/68.79
proper(false) → ok(false) 213.71/68.79
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.79
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.79
top(mark(z0)) → top(proper(z0)) 213.71/68.79
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
P(mark(z0)) → c15(P(z0)) 213.71/68.79
P(ok(z0)) → c16(P(z0)) 213.71/68.79
S(mark(z0)) → c17(S(z0)) 213.71/68.79
S(ok(z0)) → c18(S(z0)) 213.71/68.79
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.79
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.79
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.79
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.79
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.79
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.79
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.79
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.79
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.79
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.79
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.79
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.79
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.79
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.79
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.79
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.79
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.79
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.79
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.79
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.79
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.79
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.79
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.79
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.79
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 213.71/68.79
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 213.71/68.79
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(diff(x0, p(z0))) → c34(DIFF(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(diff(x0, s(z0))) → c34(DIFF(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(diff(x0, leq(z0, z1))) → c34(DIFF(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(diff(x0, if(z0, z1, z2))) → c34(DIFF(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(diff(x0, diff(z0, z1))) → c34(DIFF(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(diff(p(z0), x1)) → c34(DIFF(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(s(z0), x1)) → c34(DIFF(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(leq(z0, z1), x1)) → c34(DIFF(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(diff(if(z0, z1, z2), x1)) → c34(DIFF(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(diff(diff(z0, z1), x1)) → c34(DIFF(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(diff(x0, 0)) → c34(DIFF(proper(x0), ok(0)), PROPER(x0)) 213.71/68.79
PROPER(diff(x0, true)) → c34(DIFF(proper(x0), ok(true)), PROPER(x0)) 213.71/68.79
PROPER(diff(x0, false)) → c34(DIFF(proper(x0), ok(false)), PROPER(x0)) 213.71/68.79
PROPER(diff(0, x1)) → c34(DIFF(ok(0), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(diff(true, x1)) → c34(DIFF(ok(true), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(diff(false, x1)) → c34(DIFF(ok(false), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
P(mark(z0)) → c15(P(z0)) 213.71/68.79
P(ok(z0)) → c16(P(z0)) 213.71/68.79
S(mark(z0)) → c17(S(z0)) 213.71/68.79
S(ok(z0)) → c18(S(z0)) 213.71/68.79
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.79
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.79
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.79
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.79
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.79
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.79
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.79
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.79
TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) 213.71/68.79
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.79
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.79
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.79
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.79
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.79
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.79
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.79
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.79
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.79
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.79
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.79
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.79
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.79
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.79
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.79
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 213.71/68.79
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 213.71/68.79
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(diff(x0, p(z0))) → c34(DIFF(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(diff(x0, s(z0))) → c34(DIFF(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(diff(x0, leq(z0, z1))) → c34(DIFF(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(diff(x0, if(z0, z1, z2))) → c34(DIFF(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(diff(x0, diff(z0, z1))) → c34(DIFF(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(diff(p(z0), x1)) → c34(DIFF(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(s(z0), x1)) → c34(DIFF(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(leq(z0, z1), x1)) → c34(DIFF(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(diff(if(z0, z1, z2), x1)) → c34(DIFF(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(diff(diff(z0, z1), x1)) → c34(DIFF(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(diff(x0, 0)) → c34(DIFF(proper(x0), ok(0)), PROPER(x0)) 213.71/68.79
PROPER(diff(x0, true)) → c34(DIFF(proper(x0), ok(true)), PROPER(x0)) 213.71/68.79
PROPER(diff(x0, false)) → c34(DIFF(proper(x0), ok(false)), PROPER(x0)) 213.71/68.79
PROPER(diff(0, x1)) → c34(DIFF(ok(0), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(diff(true, x1)) → c34(DIFF(ok(true), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(diff(false, x1)) → c34(DIFF(ok(false), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, TOP, PROPER

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c35, c36, c, c8, c8, c9, c9, c12, c12, c27, c27, c29, c29, c30, c30, c33, c33, c34, c34

213.71/68.79
213.71/68.79

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

Use narrowing to replace TOP(mark(z0)) → c35(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(p(z0))) → c35(TOP(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
TOP(mark(0)) → c35(TOP(ok(0)), PROPER(0)) 213.71/68.79
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
TOP(mark(leq(z0, z1))) → c35(TOP(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
TOP(mark(true)) → c35(TOP(ok(true)), PROPER(true)) 213.71/68.79
TOP(mark(false)) → c35(TOP(ok(false)), PROPER(false)) 213.71/68.79
TOP(mark(if(z0, z1, z2))) → c35(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
TOP(mark(diff(z0, z1))) → c35(TOP(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1)))
213.71/68.79
213.71/68.79

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.79
active(p(s(z0))) → mark(z0) 213.71/68.79
active(leq(0, z0)) → mark(true) 213.71/68.79
active(leq(s(z0), 0)) → mark(false) 213.71/68.79
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.79
active(if(true, z0, z1)) → mark(z0) 213.71/68.79
active(if(false, z0, z1)) → mark(z1) 213.71/68.79
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.79
active(p(z0)) → p(active(z0)) 213.71/68.79
active(s(z0)) → s(active(z0)) 213.71/68.79
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.79
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.79
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.79
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.79
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.79
p(mark(z0)) → mark(p(z0)) 213.71/68.79
p(ok(z0)) → ok(p(z0)) 213.71/68.79
s(mark(z0)) → mark(s(z0)) 213.71/68.79
s(ok(z0)) → ok(s(z0)) 213.71/68.79
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.79
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.79
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.79
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.79
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.79
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.79
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.79
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.79
proper(p(z0)) → p(proper(z0)) 213.71/68.79
proper(0) → ok(0) 213.71/68.79
proper(s(z0)) → s(proper(z0)) 213.71/68.79
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.79
proper(true) → ok(true) 213.71/68.79
proper(false) → ok(false) 213.71/68.79
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.79
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.79
top(mark(z0)) → top(proper(z0)) 213.71/68.79
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
P(mark(z0)) → c15(P(z0)) 213.71/68.79
P(ok(z0)) → c16(P(z0)) 213.71/68.79
S(mark(z0)) → c17(S(z0)) 213.71/68.79
S(ok(z0)) → c18(S(z0)) 213.71/68.79
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.79
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.79
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.79
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.79
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.79
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.79
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.79
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.79
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.79
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.79
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.79
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.79
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.79
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.79
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.79
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.79
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.79
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.79
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.79
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.79
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.79
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.79
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.79
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 213.71/68.79
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 213.71/68.79
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(diff(x0, p(z0))) → c34(DIFF(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(diff(x0, s(z0))) → c34(DIFF(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(diff(x0, leq(z0, z1))) → c34(DIFF(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(diff(x0, if(z0, z1, z2))) → c34(DIFF(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(diff(x0, diff(z0, z1))) → c34(DIFF(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(diff(p(z0), x1)) → c34(DIFF(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(s(z0), x1)) → c34(DIFF(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(leq(z0, z1), x1)) → c34(DIFF(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(diff(if(z0, z1, z2), x1)) → c34(DIFF(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(diff(diff(z0, z1), x1)) → c34(DIFF(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(diff(x0, 0)) → c34(DIFF(proper(x0), ok(0)), PROPER(x0)) 213.71/68.79
PROPER(diff(x0, true)) → c34(DIFF(proper(x0), ok(true)), PROPER(x0)) 213.71/68.79
PROPER(diff(x0, false)) → c34(DIFF(proper(x0), ok(false)), PROPER(x0)) 213.71/68.79
PROPER(diff(0, x1)) → c34(DIFF(ok(0), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(diff(true, x1)) → c34(DIFF(ok(true), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(diff(false, x1)) → c34(DIFF(ok(false), proper(x1)), PROPER(x1)) 213.71/68.79
TOP(mark(p(z0))) → c35(TOP(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
TOP(mark(0)) → c35(TOP(ok(0)), PROPER(0)) 213.71/68.79
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
TOP(mark(leq(z0, z1))) → c35(TOP(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
TOP(mark(true)) → c35(TOP(ok(true)), PROPER(true)) 213.71/68.79
TOP(mark(false)) → c35(TOP(ok(false)), PROPER(false)) 213.71/68.79
TOP(mark(if(z0, z1, z2))) → c35(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
TOP(mark(diff(z0, z1))) → c35(TOP(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1)))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
P(mark(z0)) → c15(P(z0)) 213.71/68.79
P(ok(z0)) → c16(P(z0)) 213.71/68.79
S(mark(z0)) → c17(S(z0)) 213.71/68.79
S(ok(z0)) → c18(S(z0)) 213.71/68.79
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.79
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.79
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.79
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.79
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.79
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.79
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.79
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.79
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.79
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.79
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.79
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.79
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.79
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.79
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.79
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.79
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.79
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.79
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.79
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.79
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.79
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.79
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.79
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 213.71/68.79
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 213.71/68.79
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(diff(x0, p(z0))) → c34(DIFF(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(diff(x0, s(z0))) → c34(DIFF(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(diff(x0, leq(z0, z1))) → c34(DIFF(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(diff(x0, if(z0, z1, z2))) → c34(DIFF(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(diff(x0, diff(z0, z1))) → c34(DIFF(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(diff(p(z0), x1)) → c34(DIFF(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(s(z0), x1)) → c34(DIFF(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(leq(z0, z1), x1)) → c34(DIFF(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(diff(if(z0, z1, z2), x1)) → c34(DIFF(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(diff(diff(z0, z1), x1)) → c34(DIFF(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(diff(x0, 0)) → c34(DIFF(proper(x0), ok(0)), PROPER(x0)) 213.71/68.79
PROPER(diff(x0, true)) → c34(DIFF(proper(x0), ok(true)), PROPER(x0)) 213.71/68.79
PROPER(diff(x0, false)) → c34(DIFF(proper(x0), ok(false)), PROPER(x0)) 213.71/68.79
PROPER(diff(0, x1)) → c34(DIFF(ok(0), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(diff(true, x1)) → c34(DIFF(ok(true), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(diff(false, x1)) → c34(DIFF(ok(false), proper(x1)), PROPER(x1)) 213.71/68.79
TOP(mark(p(z0))) → c35(TOP(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
TOP(mark(0)) → c35(TOP(ok(0)), PROPER(0)) 213.71/68.79
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
TOP(mark(leq(z0, z1))) → c35(TOP(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
TOP(mark(true)) → c35(TOP(ok(true)), PROPER(true)) 213.71/68.79
TOP(mark(false)) → c35(TOP(ok(false)), PROPER(false)) 213.71/68.79
TOP(mark(if(z0, z1, z2))) → c35(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
TOP(mark(diff(z0, z1))) → c35(TOP(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, TOP, PROPER

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c36, c, c8, c8, c9, c9, c12, c12, c27, c27, c29, c29, c30, c30, c33, c33, c34, c34, c35

213.71/68.79
213.71/68.79

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

Removed 3 trailing tuple parts
213.71/68.79
213.71/68.79

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.79
active(p(s(z0))) → mark(z0) 213.71/68.79
active(leq(0, z0)) → mark(true) 213.71/68.79
active(leq(s(z0), 0)) → mark(false) 213.71/68.79
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.79
active(if(true, z0, z1)) → mark(z0) 213.71/68.79
active(if(false, z0, z1)) → mark(z1) 213.71/68.79
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.79
active(p(z0)) → p(active(z0)) 213.71/68.79
active(s(z0)) → s(active(z0)) 213.71/68.79
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.79
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.79
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.79
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.79
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.79
p(mark(z0)) → mark(p(z0)) 213.71/68.79
p(ok(z0)) → ok(p(z0)) 213.71/68.79
s(mark(z0)) → mark(s(z0)) 213.71/68.79
s(ok(z0)) → ok(s(z0)) 213.71/68.79
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.79
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.79
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.79
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.79
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.79
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.79
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.79
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.79
proper(p(z0)) → p(proper(z0)) 213.71/68.79
proper(0) → ok(0) 213.71/68.79
proper(s(z0)) → s(proper(z0)) 213.71/68.79
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.79
proper(true) → ok(true) 213.71/68.79
proper(false) → ok(false) 213.71/68.79
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.79
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.79
top(mark(z0)) → top(proper(z0)) 213.71/68.79
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
P(mark(z0)) → c15(P(z0)) 213.71/68.79
P(ok(z0)) → c16(P(z0)) 213.71/68.79
S(mark(z0)) → c17(S(z0)) 213.71/68.79
S(ok(z0)) → c18(S(z0)) 213.71/68.79
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.79
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.79
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.79
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.79
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.79
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.79
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.79
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.79
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.79
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.79
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.79
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.79
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.79
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.79
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.79
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.79
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.79
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.79
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.79
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.79
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.79
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.79
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.79
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 213.71/68.79
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 213.71/68.79
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(diff(x0, p(z0))) → c34(DIFF(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(diff(x0, s(z0))) → c34(DIFF(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(diff(x0, leq(z0, z1))) → c34(DIFF(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(diff(x0, if(z0, z1, z2))) → c34(DIFF(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(diff(x0, diff(z0, z1))) → c34(DIFF(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(diff(p(z0), x1)) → c34(DIFF(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(s(z0), x1)) → c34(DIFF(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(leq(z0, z1), x1)) → c34(DIFF(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(diff(if(z0, z1, z2), x1)) → c34(DIFF(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(diff(diff(z0, z1), x1)) → c34(DIFF(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(diff(x0, 0)) → c34(DIFF(proper(x0), ok(0)), PROPER(x0)) 213.71/68.79
PROPER(diff(x0, true)) → c34(DIFF(proper(x0), ok(true)), PROPER(x0)) 213.71/68.79
PROPER(diff(x0, false)) → c34(DIFF(proper(x0), ok(false)), PROPER(x0)) 213.71/68.79
PROPER(diff(0, x1)) → c34(DIFF(ok(0), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(diff(true, x1)) → c34(DIFF(ok(true), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(diff(false, x1)) → c34(DIFF(ok(false), proper(x1)), PROPER(x1)) 213.71/68.79
TOP(mark(p(z0))) → c35(TOP(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
TOP(mark(leq(z0, z1))) → c35(TOP(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
TOP(mark(if(z0, z1, z2))) → c35(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
TOP(mark(diff(z0, z1))) → c35(TOP(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
TOP(mark(0)) → c35(TOP(ok(0))) 213.71/68.79
TOP(mark(true)) → c35(TOP(ok(true))) 213.71/68.79
TOP(mark(false)) → c35(TOP(ok(false)))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
P(mark(z0)) → c15(P(z0)) 213.71/68.79
P(ok(z0)) → c16(P(z0)) 213.71/68.79
S(mark(z0)) → c17(S(z0)) 213.71/68.79
S(ok(z0)) → c18(S(z0)) 213.71/68.79
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.79
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.79
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.79
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.79
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.79
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.79
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.79
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.79
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.79
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.79
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.79
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.79
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.79
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.79
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.79
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.79
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.79
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.79
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.79
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.79
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.79
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.79
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.79
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.79
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.79
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.79
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 213.71/68.79
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 213.71/68.79
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 213.71/68.79
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 213.71/68.79
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 213.71/68.79
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.79
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.79
PROPER(diff(x0, p(z0))) → c34(DIFF(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.79
PROPER(diff(x0, s(z0))) → c34(DIFF(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.79
PROPER(diff(x0, leq(z0, z1))) → c34(DIFF(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.79
PROPER(diff(x0, if(z0, z1, z2))) → c34(DIFF(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.79
PROPER(diff(x0, diff(z0, z1))) → c34(DIFF(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.79
PROPER(diff(p(z0), x1)) → c34(DIFF(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(s(z0), x1)) → c34(DIFF(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.79
PROPER(diff(leq(z0, z1), x1)) → c34(DIFF(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(diff(if(z0, z1, z2), x1)) → c34(DIFF(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.79
PROPER(diff(diff(z0, z1), x1)) → c34(DIFF(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.79
PROPER(diff(x0, 0)) → c34(DIFF(proper(x0), ok(0)), PROPER(x0)) 213.71/68.79
PROPER(diff(x0, true)) → c34(DIFF(proper(x0), ok(true)), PROPER(x0)) 213.71/68.79
PROPER(diff(x0, false)) → c34(DIFF(proper(x0), ok(false)), PROPER(x0)) 213.71/68.79
PROPER(diff(0, x1)) → c34(DIFF(ok(0), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(diff(true, x1)) → c34(DIFF(ok(true), proper(x1)), PROPER(x1)) 213.71/68.79
PROPER(diff(false, x1)) → c34(DIFF(ok(false), proper(x1)), PROPER(x1)) 213.71/68.79
TOP(mark(p(z0))) → c35(TOP(p(proper(z0))), PROPER(p(z0))) 213.71/68.79
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0))) 213.71/68.79
TOP(mark(leq(z0, z1))) → c35(TOP(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.79
TOP(mark(if(z0, z1, z2))) → c35(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.79
TOP(mark(diff(z0, z1))) → c35(TOP(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.79
TOP(mark(0)) → c35(TOP(ok(0))) 213.71/68.79
TOP(mark(true)) → c35(TOP(ok(true))) 213.71/68.79
TOP(mark(false)) → c35(TOP(ok(false)))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, TOP, PROPER

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c36, c, c8, c8, c9, c9, c12, c12, c27, c27, c29, c29, c30, c30, c33, c33, c34, c34, c35, c35

213.71/68.79
213.71/68.79

(43) 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)) → c35(TOP(ok(0))) 213.71/68.79
TOP(mark(true)) → c35(TOP(ok(true))) 213.71/68.79
TOP(mark(false)) → c35(TOP(ok(false)))
We considered the (Usable) Rules:

proper(p(z0)) → p(proper(z0)) 213.71/68.79
proper(0) → ok(0) 213.71/68.79
proper(s(z0)) → s(proper(z0)) 213.71/68.79
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.79
proper(true) → ok(true) 213.71/68.79
proper(false) → ok(false) 213.71/68.79
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.79
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.79
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.79
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.79
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.79
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.79
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.79
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.79
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.79
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.79
s(mark(z0)) → mark(s(z0)) 213.71/68.79
s(ok(z0)) → ok(s(z0)) 213.71/68.79
p(mark(z0)) → mark(p(z0)) 213.71/68.79
p(ok(z0)) → ok(p(z0)) 213.71/68.79
active(p(0)) → mark(0) 213.71/68.79
active(p(s(z0))) → mark(z0) 213.71/68.79
active(leq(0, z0)) → mark(true) 213.71/68.79
active(leq(s(z0), 0)) → mark(false) 213.71/68.79
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.79
active(if(true, z0, z1)) → mark(z0) 213.71/68.79
active(if(false, z0, z1)) → mark(z1) 213.71/68.79
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.79
active(p(z0)) → p(active(z0)) 213.71/68.79
active(s(z0)) → s(active(z0)) 213.71/68.79
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.79
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.79
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.79
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.79
active(diff(z0, z1)) → diff(z0, active(z1))
And the Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.79
P(mark(z0)) → c15(P(z0)) 213.71/68.79
P(ok(z0)) → c16(P(z0)) 213.71/68.79
S(mark(z0)) → c17(S(z0)) 213.71/68.79
S(ok(z0)) → c18(S(z0)) 213.71/68.79
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.79
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.79
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.79
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.79
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.79
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.79
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.79
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.79
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.79
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.79
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.79
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.79
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.79
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.79
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.79
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.79
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.79
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.79
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.80
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.80
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.80
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.80
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.80
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.80
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.80
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.80
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.80
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.80
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.80
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.80
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.80
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.80
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.80
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.80
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.80
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.80
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.80
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.80
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.80
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.80
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.80
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.80
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.80
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.80
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.80
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.80
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.80
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.80
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.80
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.80
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.80
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.80
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.80
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.80
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.80
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.80
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.80
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.80
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.80
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.80
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.80
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.80
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.80
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.80
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.80
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.80
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.80
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.80
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.80
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.80
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 213.71/68.80
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 213.71/68.80
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 213.71/68.80
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 213.71/68.80
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 213.71/68.80
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 213.71/68.80
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 213.71/68.80
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 213.71/68.80
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 213.71/68.80
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 213.71/68.80
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 213.71/68.80
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 213.71/68.80
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 213.71/68.80
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 213.71/68.80
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.80
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.80
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.80
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(diff(x0, p(z0))) → c34(DIFF(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.80
PROPER(diff(x0, s(z0))) → c34(DIFF(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.80
PROPER(diff(x0, leq(z0, z1))) → c34(DIFF(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.80
PROPER(diff(x0, if(z0, z1, z2))) → c34(DIFF(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.80
PROPER(diff(x0, diff(z0, z1))) → c34(DIFF(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.80
PROPER(diff(p(z0), x1)) → c34(DIFF(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.80
PROPER(diff(s(z0), x1)) → c34(DIFF(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.80
PROPER(diff(leq(z0, z1), x1)) → c34(DIFF(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.80
PROPER(diff(if(z0, z1, z2), x1)) → c34(DIFF(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.80
PROPER(diff(diff(z0, z1), x1)) → c34(DIFF(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.80
PROPER(diff(x0, 0)) → c34(DIFF(proper(x0), ok(0)), PROPER(x0)) 213.71/68.80
PROPER(diff(x0, true)) → c34(DIFF(proper(x0), ok(true)), PROPER(x0)) 213.71/68.80
PROPER(diff(x0, false)) → c34(DIFF(proper(x0), ok(false)), PROPER(x0)) 213.71/68.80
PROPER(diff(0, x1)) → c34(DIFF(ok(0), proper(x1)), PROPER(x1)) 213.71/68.80
PROPER(diff(true, x1)) → c34(DIFF(ok(true), proper(x1)), PROPER(x1)) 213.71/68.80
PROPER(diff(false, x1)) → c34(DIFF(ok(false), proper(x1)), PROPER(x1)) 213.71/68.80
TOP(mark(p(z0))) → c35(TOP(p(proper(z0))), PROPER(p(z0))) 213.71/68.80
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0))) 213.71/68.80
TOP(mark(leq(z0, z1))) → c35(TOP(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.80
TOP(mark(if(z0, z1, z2))) → c35(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.80
TOP(mark(diff(z0, z1))) → c35(TOP(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.80
TOP(mark(0)) → c35(TOP(ok(0))) 213.71/68.80
TOP(mark(true)) → c35(TOP(ok(true))) 213.71/68.80
TOP(mark(false)) → c35(TOP(ok(false)))
The order we found is given by the following interpretation:
Polynomial interpretation : 213.71/68.80

POL(0) = 0    213.71/68.80
POL(ACTIVE(x1)) = 0    213.71/68.80
POL(DIFF(x1, x2)) = 0    213.71/68.80
POL(IF(x1, x2, x3)) = 0    213.71/68.80
POL(LEQ(x1, x2)) = 0    213.71/68.80
POL(P(x1)) = 0    213.71/68.80
POL(PROPER(x1)) = 0    213.71/68.80
POL(S(x1)) = 0    213.71/68.80
POL(TOP(x1)) = [2]x1    213.71/68.80
POL(active(x1)) = x1    213.71/68.80
POL(c(x1)) = x1    213.71/68.80
POL(c10(x1, x2)) = x1 + x2    213.71/68.80
POL(c11(x1, x2)) = x1 + x2    213.71/68.80
POL(c12(x1)) = x1    213.71/68.80
POL(c12(x1, x2)) = x1 + x2    213.71/68.80
POL(c13(x1, x2)) = x1 + x2    213.71/68.80
POL(c14(x1, x2)) = x1 + x2    213.71/68.80
POL(c15(x1)) = x1    213.71/68.80
POL(c16(x1)) = x1    213.71/68.80
POL(c17(x1)) = x1    213.71/68.80
POL(c18(x1)) = x1    213.71/68.80
POL(c19(x1)) = x1    213.71/68.80
POL(c20(x1)) = x1    213.71/68.80
POL(c21(x1)) = x1    213.71/68.80
POL(c22(x1)) = x1    213.71/68.80
POL(c23(x1)) = x1    213.71/68.80
POL(c24(x1)) = x1    213.71/68.80
POL(c25(x1)) = x1    213.71/68.80
POL(c26(x1)) = x1    213.71/68.80
POL(c27(x1)) = x1    213.71/68.80
POL(c27(x1, x2)) = x1 + x2    213.71/68.80
POL(c29(x1)) = x1    213.71/68.80
POL(c29(x1, x2)) = x1 + x2    213.71/68.80
POL(c30(x1, x2)) = x1 + x2    213.71/68.80
POL(c30(x1, x2, x3)) = x1 + x2 + x3    213.71/68.80
POL(c33(x1, x2, x3)) = x1 + x2 + x3    213.71/68.80
POL(c33(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    213.71/68.80
POL(c34(x1, x2)) = x1 + x2    213.71/68.80
POL(c34(x1, x2, x3)) = x1 + x2 + x3    213.71/68.80
POL(c35(x1)) = x1    213.71/68.80
POL(c35(x1, x2)) = x1 + x2    213.71/68.80
POL(c36(x1, x2)) = x1 + x2    213.71/68.80
POL(c4(x1)) = x1    213.71/68.80
POL(c8(x1)) = x1    213.71/68.80
POL(c8(x1, x2)) = x1 + x2    213.71/68.80
POL(c9(x1)) = x1    213.71/68.80
POL(c9(x1, x2)) = x1 + x2    213.71/68.80
POL(diff(x1, x2)) = [4]    213.71/68.80
POL(false) = 0    213.71/68.80
POL(if(x1, x2, x3)) = [4]    213.71/68.80
POL(leq(x1, x2)) = [4]    213.71/68.80
POL(mark(x1)) = [4]    213.71/68.80
POL(ok(x1)) = x1    213.71/68.80
POL(p(x1)) = [4]    213.71/68.80
POL(proper(x1)) = [2] + [4]x1    213.71/68.80
POL(s(x1)) = [4]    213.71/68.80
POL(true) = 0   
213.71/68.80
213.71/68.80

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 213.71/68.80
active(p(s(z0))) → mark(z0) 213.71/68.80
active(leq(0, z0)) → mark(true) 213.71/68.80
active(leq(s(z0), 0)) → mark(false) 213.71/68.80
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 213.71/68.80
active(if(true, z0, z1)) → mark(z0) 213.71/68.80
active(if(false, z0, z1)) → mark(z1) 213.71/68.80
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 213.71/68.80
active(p(z0)) → p(active(z0)) 213.71/68.80
active(s(z0)) → s(active(z0)) 213.71/68.80
active(leq(z0, z1)) → leq(active(z0), z1) 213.71/68.80
active(leq(z0, z1)) → leq(z0, active(z1)) 213.71/68.80
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 213.71/68.80
active(diff(z0, z1)) → diff(active(z0), z1) 213.71/68.80
active(diff(z0, z1)) → diff(z0, active(z1)) 213.71/68.80
p(mark(z0)) → mark(p(z0)) 213.71/68.80
p(ok(z0)) → ok(p(z0)) 213.71/68.80
s(mark(z0)) → mark(s(z0)) 213.71/68.80
s(ok(z0)) → ok(s(z0)) 213.71/68.80
leq(mark(z0), z1) → mark(leq(z0, z1)) 213.71/68.80
leq(z0, mark(z1)) → mark(leq(z0, z1)) 213.71/68.80
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 213.71/68.80
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 213.71/68.80
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 213.71/68.80
diff(mark(z0), z1) → mark(diff(z0, z1)) 213.71/68.80
diff(z0, mark(z1)) → mark(diff(z0, z1)) 213.71/68.80
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 213.71/68.80
proper(p(z0)) → p(proper(z0)) 213.71/68.80
proper(0) → ok(0) 213.71/68.80
proper(s(z0)) → s(proper(z0)) 213.71/68.80
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 213.71/68.80
proper(true) → ok(true) 213.71/68.80
proper(false) → ok(false) 213.71/68.80
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 213.71/68.80
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 213.71/68.80
top(mark(z0)) → top(proper(z0)) 213.71/68.80
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.80
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.80
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.80
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.80
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.80
P(mark(z0)) → c15(P(z0)) 213.71/68.80
P(ok(z0)) → c16(P(z0)) 213.71/68.80
S(mark(z0)) → c17(S(z0)) 213.71/68.80
S(ok(z0)) → c18(S(z0)) 213.71/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 213.71/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 213.71/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 213.71/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 213.71/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 213.71/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 213.71/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 213.71/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 213.71/68.80
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 213.71/68.80
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 213.71/68.80
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 213.71/68.80
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 213.71/68.80
ACTIVE(diff(z0, z1)) → c(P(z0)) 213.71/68.80
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.80
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.80
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.80
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.80
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.80
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 213.71/68.80
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 213.71/68.80
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.80
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.80
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.80
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.80
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.80
ACTIVE(p(p(0))) → c8(P(mark(0))) 213.71/68.80
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 213.71/68.80
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 213.71/68.80
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 213.71/68.80
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 213.71/68.80
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 213.71/68.80
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 213.71/68.80
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 213.71/68.80
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 213.71/68.80
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 213.71/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 213.71/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 213.71/68.80
ACTIVE(s(p(0))) → c9(S(mark(0))) 213.71/68.80
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 213.71/68.80
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 213.71/68.80
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 213.71/68.80
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 213.71/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.80
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 213.71/68.80
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 213.71/68.80
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.80
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 213.71/68.80
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 213.71/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 213.71/68.80
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 213.71/68.80
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 213.71/68.80
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 213.71/68.80
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 213.71/68.80
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 213.71/68.80
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 213.71/68.80
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 213.71/68.80
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 213.71/68.80
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 213.71/68.80
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.80
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.80
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.80
PROPER(p(0)) → c27(P(ok(0))) 213.71/68.80
PROPER(p(true)) → c27(P(ok(true))) 213.71/68.80
PROPER(p(false)) → c27(P(ok(false))) 213.71/68.80
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 213.71/68.80
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 213.71/68.80
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.80
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.80
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.80
PROPER(s(0)) → c29(S(ok(0))) 213.71/68.80
PROPER(s(true)) → c29(S(ok(true))) 213.71/68.80
PROPER(s(false)) → c29(S(ok(false))) 213.71/68.80
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.80
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.80
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.80
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.80
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.80
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.80
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.80
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.80
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.80
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.80
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 213.71/68.80
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 213.71/68.80
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 213.71/68.80
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 213.71/68.80
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 213.71/68.80
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 213.71/68.80
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 213.71/68.80
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 213.71/68.80
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 213.71/68.80
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 213.71/68.80
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 213.71/68.80
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 213.71/68.80
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 213.71/68.80
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 213.71/68.80
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 213.71/68.80
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 213.71/68.80
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 213.71/68.80
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 213.71/68.80
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 213.71/68.80
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.80
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.80
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 213.71/68.80
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 213.71/68.80
PROPER(diff(x0, p(z0))) → c34(DIFF(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 213.71/68.80
PROPER(diff(x0, s(z0))) → c34(DIFF(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 213.71/68.80
PROPER(diff(x0, leq(z0, z1))) → c34(DIFF(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 213.71/68.80
PROPER(diff(x0, if(z0, z1, z2))) → c34(DIFF(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 213.71/68.80
PROPER(diff(x0, diff(z0, z1))) → c34(DIFF(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 213.71/68.80
PROPER(diff(p(z0), x1)) → c34(DIFF(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 213.71/68.80
PROPER(diff(s(z0), x1)) → c34(DIFF(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 213.71/68.80
PROPER(diff(leq(z0, z1), x1)) → c34(DIFF(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 213.71/68.80
PROPER(diff(if(z0, z1, z2), x1)) → c34(DIFF(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 213.71/68.80
PROPER(diff(diff(z0, z1), x1)) → c34(DIFF(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 213.71/68.80
PROPER(diff(x0, 0)) → c34(DIFF(proper(x0), ok(0)), PROPER(x0)) 213.71/68.80
PROPER(diff(x0, true)) → c34(DIFF(proper(x0), ok(true)), PROPER(x0)) 213.71/68.80
PROPER(diff(x0, false)) → c34(DIFF(proper(x0), ok(false)), PROPER(x0)) 213.71/68.80
PROPER(diff(0, x1)) → c34(DIFF(ok(0), proper(x1)), PROPER(x1)) 213.71/68.80
PROPER(diff(true, x1)) → c34(DIFF(ok(true), proper(x1)), PROPER(x1)) 213.71/68.80
PROPER(diff(false, x1)) → c34(DIFF(ok(false), proper(x1)), PROPER(x1)) 213.71/68.80
TOP(mark(p(z0))) → c35(TOP(p(proper(z0))), PROPER(p(z0))) 213.71/68.80
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0))) 213.71/68.80
TOP(mark(leq(z0, z1))) → c35(TOP(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 213.71/68.80
TOP(mark(if(z0, z1, z2))) → c35(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 213.71/68.80
TOP(mark(diff(z0, z1))) → c35(TOP(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 213.71/68.80
TOP(mark(0)) → c35(TOP(ok(0))) 213.71/68.80
TOP(mark(true)) → c35(TOP(ok(true))) 213.71/68.80
TOP(mark(false)) → c35(TOP(ok(false)))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 213.71/68.80
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 213.71/68.80
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 213.71/68.80
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 213.71/68.80
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 213.71/68.80
P(mark(z0)) → c15(P(z0)) 213.71/68.80
P(ok(z0)) → c16(P(z0)) 213.71/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 214.02/68.80
TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(P(z0)) 214.02/68.80
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 214.02/68.80
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 214.02/68.80
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 214.02/68.80
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 214.02/68.80
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 214.02/68.80
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 214.02/68.80
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(p(0))) → c8(P(mark(0))) 214.02/68.80
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 214.02/68.80
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 214.02/68.80
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 214.02/68.80
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 214.02/68.80
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 214.02/68.80
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 214.02/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(s(p(0))) → c9(S(mark(0))) 214.02/68.80
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 214.02/68.80
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 214.02/68.80
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 214.02/68.80
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 214.02/68.80
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 214.02/68.80
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 214.02/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 214.02/68.80
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 214.02/68.80
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 214.02/68.80
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 214.02/68.80
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 214.02/68.80
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 214.02/68.80
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 214.02/68.80
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 214.02/68.80
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 214.02/68.80
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(p(0)) → c27(P(ok(0))) 214.02/68.80
PROPER(p(true)) → c27(P(ok(true))) 214.02/68.80
PROPER(p(false)) → c27(P(ok(false))) 214.02/68.80
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 214.02/68.80
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 214.02/68.80
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(s(0)) → c29(S(ok(0))) 214.02/68.80
PROPER(s(true)) → c29(S(ok(true))) 214.02/68.80
PROPER(s(false)) → c29(S(ok(false))) 214.02/68.80
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 214.02/68.80
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 214.02/68.80
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 214.02/68.80
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 214.02/68.80
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 214.02/68.80
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 214.02/68.80
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 214.02/68.80
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 214.02/68.80
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 214.02/68.80
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 214.02/68.80
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 214.02/68.80
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 214.02/68.80
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 214.02/68.80
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 214.02/68.80
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 214.02/68.80
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 214.02/68.80
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 214.02/68.80
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(diff(x0, p(z0))) → c34(DIFF(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 214.02/68.80
PROPER(diff(x0, s(z0))) → c34(DIFF(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 214.02/68.80
PROPER(diff(x0, leq(z0, z1))) → c34(DIFF(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(diff(x0, if(z0, z1, z2))) → c34(DIFF(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(diff(x0, diff(z0, z1))) → c34(DIFF(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(diff(p(z0), x1)) → c34(DIFF(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 214.02/68.80
PROPER(diff(s(z0), x1)) → c34(DIFF(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 214.02/68.80
PROPER(diff(leq(z0, z1), x1)) → c34(DIFF(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(diff(if(z0, z1, z2), x1)) → c34(DIFF(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 214.02/68.80
PROPER(diff(diff(z0, z1), x1)) → c34(DIFF(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(diff(x0, 0)) → c34(DIFF(proper(x0), ok(0)), PROPER(x0)) 214.02/68.80
PROPER(diff(x0, true)) → c34(DIFF(proper(x0), ok(true)), PROPER(x0)) 214.02/68.80
PROPER(diff(x0, false)) → c34(DIFF(proper(x0), ok(false)), PROPER(x0)) 214.02/68.80
PROPER(diff(0, x1)) → c34(DIFF(ok(0), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(diff(true, x1)) → c34(DIFF(ok(true), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(diff(false, x1)) → c34(DIFF(ok(false), proper(x1)), PROPER(x1)) 214.02/68.80
TOP(mark(p(z0))) → c35(TOP(p(proper(z0))), PROPER(p(z0))) 214.02/68.80
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0))) 214.02/68.80
TOP(mark(leq(z0, z1))) → c35(TOP(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 214.02/68.80
TOP(mark(if(z0, z1, z2))) → c35(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 214.02/68.80
TOP(mark(diff(z0, z1))) → c35(TOP(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1)))
K tuples:

TOP(mark(0)) → c35(TOP(ok(0))) 214.02/68.80
TOP(mark(true)) → c35(TOP(ok(true))) 214.02/68.80
TOP(mark(false)) → c35(TOP(ok(false)))
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, TOP, PROPER

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c36, c, c8, c8, c9, c9, c12, c12, c27, c27, c29, c29, c30, c30, c33, c33, c34, c34, c35, c35

214.02/68.80
214.02/68.80

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

Use narrowing to replace TOP(ok(z0)) → c36(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(p(0))) → c36(TOP(mark(0)), ACTIVE(p(0))) 214.02/68.80
TOP(ok(p(s(z0)))) → c36(TOP(mark(z0)), ACTIVE(p(s(z0)))) 214.02/68.80
TOP(ok(leq(0, z0))) → c36(TOP(mark(true)), ACTIVE(leq(0, z0))) 214.02/68.80
TOP(ok(leq(s(z0), 0))) → c36(TOP(mark(false)), ACTIVE(leq(s(z0), 0))) 214.02/68.80
TOP(ok(leq(s(z0), s(z1)))) → c36(TOP(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
TOP(ok(if(true, z0, z1))) → c36(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 214.02/68.80
TOP(ok(if(false, z0, z1))) → c36(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 214.02/68.80
TOP(ok(diff(z0, z1))) → c36(TOP(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 214.02/68.80
TOP(ok(p(z0))) → c36(TOP(p(active(z0))), ACTIVE(p(z0))) 214.02/68.80
TOP(ok(s(z0))) → c36(TOP(s(active(z0))), ACTIVE(s(z0))) 214.02/68.80
TOP(ok(leq(z0, z1))) → c36(TOP(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 214.02/68.80
TOP(ok(leq(z0, z1))) → c36(TOP(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 214.02/68.80
TOP(ok(if(z0, z1, z2))) → c36(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 214.02/68.80
TOP(ok(diff(z0, z1))) → c36(TOP(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 214.02/68.80
TOP(ok(diff(z0, z1))) → c36(TOP(diff(z0, active(z1))), ACTIVE(diff(z0, z1)))
214.02/68.80
214.02/68.80

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.80
active(p(s(z0))) → mark(z0) 214.02/68.80
active(leq(0, z0)) → mark(true) 214.02/68.80
active(leq(s(z0), 0)) → mark(false) 214.02/68.80
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.80
active(if(true, z0, z1)) → mark(z0) 214.02/68.80
active(if(false, z0, z1)) → mark(z1) 214.02/68.80
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.80
active(p(z0)) → p(active(z0)) 214.02/68.80
active(s(z0)) → s(active(z0)) 214.02/68.80
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.80
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.80
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.80
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.80
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.80
p(mark(z0)) → mark(p(z0)) 214.02/68.80
p(ok(z0)) → ok(p(z0)) 214.02/68.80
s(mark(z0)) → mark(s(z0)) 214.02/68.80
s(ok(z0)) → ok(s(z0)) 214.02/68.80
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.80
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.80
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.80
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.80
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.80
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.80
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.80
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.80
proper(p(z0)) → p(proper(z0)) 214.02/68.80
proper(0) → ok(0) 214.02/68.80
proper(s(z0)) → s(proper(z0)) 214.02/68.80
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.80
proper(true) → ok(true) 214.02/68.80
proper(false) → ok(false) 214.02/68.80
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.80
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.80
top(mark(z0)) → top(proper(z0)) 214.02/68.80
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 214.02/68.80
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 214.02/68.80
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 214.02/68.80
P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(P(z0)) 214.02/68.80
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 214.02/68.80
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 214.02/68.80
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 214.02/68.80
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 214.02/68.80
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 214.02/68.80
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 214.02/68.80
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(p(0))) → c8(P(mark(0))) 214.02/68.80
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 214.02/68.80
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 214.02/68.80
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 214.02/68.80
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 214.02/68.80
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 214.02/68.80
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 214.02/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(s(p(0))) → c9(S(mark(0))) 214.02/68.80
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 214.02/68.80
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 214.02/68.80
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 214.02/68.80
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 214.02/68.80
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 214.02/68.80
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 214.02/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 214.02/68.80
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 214.02/68.80
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 214.02/68.80
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 214.02/68.80
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 214.02/68.80
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 214.02/68.80
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 214.02/68.80
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 214.02/68.80
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 214.02/68.80
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(p(0)) → c27(P(ok(0))) 214.02/68.80
PROPER(p(true)) → c27(P(ok(true))) 214.02/68.80
PROPER(p(false)) → c27(P(ok(false))) 214.02/68.80
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 214.02/68.80
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 214.02/68.80
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(s(0)) → c29(S(ok(0))) 214.02/68.80
PROPER(s(true)) → c29(S(ok(true))) 214.02/68.80
PROPER(s(false)) → c29(S(ok(false))) 214.02/68.80
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 214.02/68.80
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 214.02/68.80
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 214.02/68.80
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 214.02/68.80
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 214.02/68.80
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 214.02/68.80
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 214.02/68.80
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 214.02/68.80
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 214.02/68.80
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 214.02/68.80
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 214.02/68.80
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 214.02/68.80
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 214.02/68.80
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 214.02/68.80
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 214.02/68.80
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 214.02/68.80
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 214.02/68.80
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(diff(x0, p(z0))) → c34(DIFF(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 214.02/68.80
PROPER(diff(x0, s(z0))) → c34(DIFF(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 214.02/68.80
PROPER(diff(x0, leq(z0, z1))) → c34(DIFF(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(diff(x0, if(z0, z1, z2))) → c34(DIFF(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(diff(x0, diff(z0, z1))) → c34(DIFF(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(diff(p(z0), x1)) → c34(DIFF(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 214.02/68.80
PROPER(diff(s(z0), x1)) → c34(DIFF(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 214.02/68.80
PROPER(diff(leq(z0, z1), x1)) → c34(DIFF(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(diff(if(z0, z1, z2), x1)) → c34(DIFF(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 214.02/68.80
PROPER(diff(diff(z0, z1), x1)) → c34(DIFF(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(diff(x0, 0)) → c34(DIFF(proper(x0), ok(0)), PROPER(x0)) 214.02/68.80
PROPER(diff(x0, true)) → c34(DIFF(proper(x0), ok(true)), PROPER(x0)) 214.02/68.80
PROPER(diff(x0, false)) → c34(DIFF(proper(x0), ok(false)), PROPER(x0)) 214.02/68.80
PROPER(diff(0, x1)) → c34(DIFF(ok(0), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(diff(true, x1)) → c34(DIFF(ok(true), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(diff(false, x1)) → c34(DIFF(ok(false), proper(x1)), PROPER(x1)) 214.02/68.80
TOP(mark(p(z0))) → c35(TOP(p(proper(z0))), PROPER(p(z0))) 214.02/68.80
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0))) 214.02/68.80
TOP(mark(leq(z0, z1))) → c35(TOP(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 214.02/68.80
TOP(mark(if(z0, z1, z2))) → c35(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 214.02/68.80
TOP(mark(diff(z0, z1))) → c35(TOP(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 214.02/68.80
TOP(mark(0)) → c35(TOP(ok(0))) 214.02/68.80
TOP(mark(true)) → c35(TOP(ok(true))) 214.02/68.80
TOP(mark(false)) → c35(TOP(ok(false))) 214.02/68.80
TOP(ok(p(0))) → c36(TOP(mark(0)), ACTIVE(p(0))) 214.02/68.80
TOP(ok(p(s(z0)))) → c36(TOP(mark(z0)), ACTIVE(p(s(z0)))) 214.02/68.80
TOP(ok(leq(0, z0))) → c36(TOP(mark(true)), ACTIVE(leq(0, z0))) 214.02/68.80
TOP(ok(leq(s(z0), 0))) → c36(TOP(mark(false)), ACTIVE(leq(s(z0), 0))) 214.02/68.80
TOP(ok(leq(s(z0), s(z1)))) → c36(TOP(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
TOP(ok(if(true, z0, z1))) → c36(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 214.02/68.80
TOP(ok(if(false, z0, z1))) → c36(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 214.02/68.80
TOP(ok(diff(z0, z1))) → c36(TOP(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 214.02/68.80
TOP(ok(p(z0))) → c36(TOP(p(active(z0))), ACTIVE(p(z0))) 214.02/68.80
TOP(ok(s(z0))) → c36(TOP(s(active(z0))), ACTIVE(s(z0))) 214.02/68.80
TOP(ok(leq(z0, z1))) → c36(TOP(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 214.02/68.80
TOP(ok(leq(z0, z1))) → c36(TOP(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 214.02/68.80
TOP(ok(if(z0, z1, z2))) → c36(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 214.02/68.80
TOP(ok(diff(z0, z1))) → c36(TOP(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 214.02/68.80
TOP(ok(diff(z0, z1))) → c36(TOP(diff(z0, active(z1))), ACTIVE(diff(z0, z1)))
S tuples:

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 214.02/68.80
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 214.02/68.80
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 214.02/68.80
P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(P(z0)) 214.02/68.80
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 214.02/68.80
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 214.02/68.80
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 214.02/68.80
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 214.02/68.80
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 214.02/68.80
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 214.02/68.80
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(p(0))) → c8(P(mark(0))) 214.02/68.80
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 214.02/68.80
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 214.02/68.80
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 214.02/68.80
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 214.02/68.80
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 214.02/68.80
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 214.02/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(s(p(0))) → c9(S(mark(0))) 214.02/68.80
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 214.02/68.80
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 214.02/68.80
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 214.02/68.80
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 214.02/68.80
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 214.02/68.80
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 214.02/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 214.02/68.80
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 214.02/68.80
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 214.02/68.80
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 214.02/68.80
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 214.02/68.80
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 214.02/68.80
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 214.02/68.80
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 214.02/68.80
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 214.02/68.80
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(p(0)) → c27(P(ok(0))) 214.02/68.80
PROPER(p(true)) → c27(P(ok(true))) 214.02/68.80
PROPER(p(false)) → c27(P(ok(false))) 214.02/68.80
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 214.02/68.80
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 214.02/68.80
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(s(0)) → c29(S(ok(0))) 214.02/68.80
PROPER(s(true)) → c29(S(ok(true))) 214.02/68.80
PROPER(s(false)) → c29(S(ok(false))) 214.02/68.80
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 214.02/68.80
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 214.02/68.80
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 214.02/68.80
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 214.02/68.80
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 214.02/68.80
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 214.02/68.80
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 214.02/68.80
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 214.02/68.80
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 214.02/68.80
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 214.02/68.80
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 214.02/68.80
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 214.02/68.80
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 214.02/68.80
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 214.02/68.80
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 214.02/68.80
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 214.02/68.80
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 214.02/68.80
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(diff(x0, p(z0))) → c34(DIFF(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 214.02/68.80
PROPER(diff(x0, s(z0))) → c34(DIFF(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 214.02/68.80
PROPER(diff(x0, leq(z0, z1))) → c34(DIFF(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(diff(x0, if(z0, z1, z2))) → c34(DIFF(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(diff(x0, diff(z0, z1))) → c34(DIFF(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(diff(p(z0), x1)) → c34(DIFF(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 214.02/68.80
PROPER(diff(s(z0), x1)) → c34(DIFF(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 214.02/68.80
PROPER(diff(leq(z0, z1), x1)) → c34(DIFF(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(diff(if(z0, z1, z2), x1)) → c34(DIFF(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 214.02/68.80
PROPER(diff(diff(z0, z1), x1)) → c34(DIFF(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(diff(x0, 0)) → c34(DIFF(proper(x0), ok(0)), PROPER(x0)) 214.02/68.80
PROPER(diff(x0, true)) → c34(DIFF(proper(x0), ok(true)), PROPER(x0)) 214.02/68.80
PROPER(diff(x0, false)) → c34(DIFF(proper(x0), ok(false)), PROPER(x0)) 214.02/68.80
PROPER(diff(0, x1)) → c34(DIFF(ok(0), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(diff(true, x1)) → c34(DIFF(ok(true), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(diff(false, x1)) → c34(DIFF(ok(false), proper(x1)), PROPER(x1)) 214.02/68.80
TOP(mark(p(z0))) → c35(TOP(p(proper(z0))), PROPER(p(z0))) 214.02/68.80
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0))) 214.02/68.80
TOP(mark(leq(z0, z1))) → c35(TOP(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 214.02/68.80
TOP(mark(if(z0, z1, z2))) → c35(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 214.02/68.80
TOP(mark(diff(z0, z1))) → c35(TOP(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 214.02/68.80
TOP(ok(p(0))) → c36(TOP(mark(0)), ACTIVE(p(0))) 214.02/68.80
TOP(ok(p(s(z0)))) → c36(TOP(mark(z0)), ACTIVE(p(s(z0)))) 214.02/68.80
TOP(ok(leq(0, z0))) → c36(TOP(mark(true)), ACTIVE(leq(0, z0))) 214.02/68.80
TOP(ok(leq(s(z0), 0))) → c36(TOP(mark(false)), ACTIVE(leq(s(z0), 0))) 214.02/68.80
TOP(ok(leq(s(z0), s(z1)))) → c36(TOP(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
TOP(ok(if(true, z0, z1))) → c36(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 214.02/68.80
TOP(ok(if(false, z0, z1))) → c36(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 214.02/68.80
TOP(ok(diff(z0, z1))) → c36(TOP(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 214.02/68.80
TOP(ok(p(z0))) → c36(TOP(p(active(z0))), ACTIVE(p(z0))) 214.02/68.80
TOP(ok(s(z0))) → c36(TOP(s(active(z0))), ACTIVE(s(z0))) 214.02/68.80
TOP(ok(leq(z0, z1))) → c36(TOP(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 214.02/68.80
TOP(ok(leq(z0, z1))) → c36(TOP(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 214.02/68.80
TOP(ok(if(z0, z1, z2))) → c36(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 214.02/68.80
TOP(ok(diff(z0, z1))) → c36(TOP(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 214.02/68.80
TOP(ok(diff(z0, z1))) → c36(TOP(diff(z0, active(z1))), ACTIVE(diff(z0, z1)))
K tuples:

TOP(mark(0)) → c35(TOP(ok(0))) 214.02/68.80
TOP(mark(true)) → c35(TOP(ok(true))) 214.02/68.80
TOP(mark(false)) → c35(TOP(ok(false)))
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

ACTIVE, P, S, LEQ, IF, DIFF, PROPER, TOP

Compound Symbols:

c4, c10, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c, c8, c8, c9, c9, c12, c12, c27, c27, c29, c29, c30, c30, c33, c33, c34, c34, c35, c35, c36

214.02/68.80
214.02/68.80

(47) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(leq(s(z0), s(z1))) → c4(LEQ(z0, z1)) 214.02/68.80
ACTIVE(leq(z0, z1)) → c10(LEQ(active(z0), z1), ACTIVE(z0)) 214.02/68.80
ACTIVE(leq(z0, z1)) → c11(LEQ(z0, active(z1)), ACTIVE(z1)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c13(DIFF(active(z0), z1), ACTIVE(z0)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c14(DIFF(z0, active(z1)), ACTIVE(z1)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(LEQ(z0, z1)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(S(diff(p(z0), z1))) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(DIFF(p(z0), z1)) 214.02/68.80
ACTIVE(diff(z0, z1)) → c(P(z0)) 214.02/68.80
ACTIVE(p(p(s(z0)))) → c8(P(mark(z0)), ACTIVE(p(s(z0)))) 214.02/68.80
ACTIVE(p(leq(0, z0))) → c8(P(mark(true)), ACTIVE(leq(0, z0))) 214.02/68.80
ACTIVE(p(leq(s(z0), 0))) → c8(P(mark(false)), ACTIVE(leq(s(z0), 0))) 214.02/68.80
ACTIVE(p(leq(s(z0), s(z1)))) → c8(P(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
ACTIVE(p(diff(z0, z1))) → c8(P(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(p(z0))) → c8(P(p(active(z0))), ACTIVE(p(z0))) 214.02/68.80
ACTIVE(p(s(z0))) → c8(P(s(active(z0))), ACTIVE(s(z0))) 214.02/68.80
ACTIVE(p(leq(z0, z1))) → c8(P(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(p(leq(z0, z1))) → c8(P(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(p(if(z0, z1, z2))) → c8(P(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 214.02/68.80
ACTIVE(p(diff(z0, z1))) → c8(P(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(diff(z0, z1))) → c8(P(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(p(0))) → c8(P(mark(0))) 214.02/68.80
ACTIVE(s(p(s(z0)))) → c9(S(mark(z0)), ACTIVE(p(s(z0)))) 214.02/68.80
ACTIVE(s(leq(0, z0))) → c9(S(mark(true)), ACTIVE(leq(0, z0))) 214.02/68.80
ACTIVE(s(leq(s(z0), 0))) → c9(S(mark(false)), ACTIVE(leq(s(z0), 0))) 214.02/68.80
ACTIVE(s(leq(s(z0), s(z1)))) → c9(S(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(s(p(z0))) → c9(S(p(active(z0))), ACTIVE(p(z0))) 214.02/68.80
ACTIVE(s(s(z0))) → c9(S(s(active(z0))), ACTIVE(s(z0))) 214.02/68.80
ACTIVE(s(leq(z0, z1))) → c9(S(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(s(leq(z0, z1))) → c9(S(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(s(if(z0, z1, z2))) → c9(S(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 214.02/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(s(diff(z0, z1))) → c9(S(diff(z0, active(z1))), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(s(p(0))) → c9(S(mark(0))) 214.02/68.80
ACTIVE(if(p(s(z0)), x1, x2)) → c12(IF(mark(z0), x1, x2), ACTIVE(p(s(z0)))) 214.02/68.80
ACTIVE(if(leq(0, z0), x1, x2)) → c12(IF(mark(true), x1, x2), ACTIVE(leq(0, z0))) 214.02/68.80
ACTIVE(if(leq(s(z0), 0), x1, x2)) → c12(IF(mark(false), x1, x2), ACTIVE(leq(s(z0), 0))) 214.02/68.80
ACTIVE(if(leq(s(z0), s(z1)), x1, x2)) → c12(IF(mark(leq(z0, z1)), x1, x2), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))), x1, x2), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(if(p(z0), x1, x2)) → c12(IF(p(active(z0)), x1, x2), ACTIVE(p(z0))) 214.02/68.80
ACTIVE(if(s(z0), x1, x2)) → c12(IF(s(active(z0)), x1, x2), ACTIVE(s(z0))) 214.02/68.80
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(active(z0), z1), x1, x2), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(if(leq(z0, z1), x1, x2)) → c12(IF(leq(z0, active(z1)), x1, x2), ACTIVE(leq(z0, z1))) 214.02/68.80
ACTIVE(if(if(z0, z1, z2), x1, x2)) → c12(IF(if(active(z0), z1, z2), x1, x2), ACTIVE(if(z0, z1, z2))) 214.02/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(active(z0), z1), x1, x2), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(if(diff(z0, z1), x1, x2)) → c12(IF(diff(z0, active(z1)), x1, x2), ACTIVE(diff(z0, z1))) 214.02/68.80
ACTIVE(p(if(true, z0, z1))) → c8(P(mark(z0))) 214.02/68.80
ACTIVE(p(if(false, z0, z1))) → c8(P(mark(z1))) 214.02/68.80
ACTIVE(s(if(true, z0, z1))) → c9(S(mark(z0))) 214.02/68.80
ACTIVE(s(if(false, z0, z1))) → c9(S(mark(z1))) 214.02/68.80
ACTIVE(if(p(0), x1, x2)) → c12(IF(mark(0), x1, x2)) 214.02/68.80
ACTIVE(if(if(true, z0, z1), x1, x2)) → c12(IF(mark(z0), x1, x2)) 214.02/68.80
ACTIVE(if(if(false, z0, z1), x1, x2)) → c12(IF(mark(z1), x1, x2)) 214.02/68.80
PROPER(p(p(z0))) → c27(P(p(proper(z0))), PROPER(p(z0))) 214.02/68.80
PROPER(p(s(z0))) → c27(P(s(proper(z0))), PROPER(s(z0))) 214.02/68.80
PROPER(p(leq(z0, z1))) → c27(P(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(p(if(z0, z1, z2))) → c27(P(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(p(diff(z0, z1))) → c27(P(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(p(0)) → c27(P(ok(0))) 214.02/68.80
PROPER(p(true)) → c27(P(ok(true))) 214.02/68.80
PROPER(p(false)) → c27(P(ok(false))) 214.02/68.80
PROPER(s(p(z0))) → c29(S(p(proper(z0))), PROPER(p(z0))) 214.02/68.80
PROPER(s(s(z0))) → c29(S(s(proper(z0))), PROPER(s(z0))) 214.02/68.80
PROPER(s(leq(z0, z1))) → c29(S(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(s(if(z0, z1, z2))) → c29(S(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(s(diff(z0, z1))) → c29(S(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(s(0)) → c29(S(ok(0))) 214.02/68.80
PROPER(s(true)) → c29(S(ok(true))) 214.02/68.80
PROPER(s(false)) → c29(S(ok(false))) 214.02/68.80
PROPER(leq(x0, p(z0))) → c30(LEQ(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 214.02/68.80
PROPER(leq(x0, s(z0))) → c30(LEQ(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 214.02/68.80
PROPER(leq(x0, leq(z0, z1))) → c30(LEQ(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(leq(x0, if(z0, z1, z2))) → c30(LEQ(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(leq(x0, diff(z0, z1))) → c30(LEQ(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(leq(p(z0), x1)) → c30(LEQ(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 214.02/68.80
PROPER(leq(s(z0), x1)) → c30(LEQ(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 214.02/68.80
PROPER(leq(leq(z0, z1), x1)) → c30(LEQ(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(leq(if(z0, z1, z2), x1)) → c30(LEQ(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 214.02/68.80
PROPER(leq(diff(z0, z1), x1)) → c30(LEQ(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(leq(x0, 0)) → c30(LEQ(proper(x0), ok(0)), PROPER(x0)) 214.02/68.80
PROPER(leq(x0, true)) → c30(LEQ(proper(x0), ok(true)), PROPER(x0)) 214.02/68.80
PROPER(leq(x0, false)) → c30(LEQ(proper(x0), ok(false)), PROPER(x0)) 214.02/68.80
PROPER(leq(0, x1)) → c30(LEQ(ok(0), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(leq(true, x1)) → c30(LEQ(ok(true), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(leq(false, x1)) → c30(LEQ(ok(false), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(if(x0, x1, p(z0))) → c33(IF(proper(x0), proper(x1), p(proper(z0))), PROPER(x0), PROPER(x1), PROPER(p(z0))) 214.02/68.80
PROPER(if(x0, x1, s(z0))) → c33(IF(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0))) 214.02/68.80
PROPER(if(x0, x1, leq(z0, z1))) → c33(IF(proper(x0), proper(x1), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(if(x0, x1, if(z0, z1, z2))) → c33(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(if(x0, x1, diff(z0, z1))) → c33(IF(proper(x0), proper(x1), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(if(x0, p(z0), x2)) → c33(IF(proper(x0), p(proper(z0)), proper(x2)), PROPER(x0), PROPER(p(z0)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, s(z0), x2)) → c33(IF(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, leq(z0, z1), x2)) → c33(IF(proper(x0), leq(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(leq(z0, z1)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, if(z0, z1, z2), x2)) → c33(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 214.02/68.80
PROPER(if(x0, diff(z0, z1), x2)) → c33(IF(proper(x0), diff(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(diff(z0, z1)), PROPER(x2)) 214.02/68.80
PROPER(if(p(z0), x1, x2)) → c33(IF(p(proper(z0)), proper(x1), proper(x2)), PROPER(p(z0)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(s(z0), x1, x2)) → c33(IF(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(leq(z0, z1), x1, x2)) → c33(IF(leq(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(leq(z0, z1)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(if(z0, z1, z2), x1, x2)) → c33(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(diff(z0, z1), x1, x2)) → c33(IF(diff(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(diff(z0, z1)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(x0, x1, 0)) → c33(IF(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1)) 214.02/68.80
PROPER(if(x0, x1, true)) → c33(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 214.02/68.80
PROPER(if(x0, x1, false)) → c33(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 214.02/68.80
PROPER(if(x0, 0, x2)) → c33(IF(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2)) 214.02/68.80
PROPER(if(x0, true, x2)) → c33(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 214.02/68.80
PROPER(if(x0, false, x2)) → c33(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 214.02/68.80
PROPER(if(0, x1, x2)) → c33(IF(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(true, x1, x2)) → c33(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(if(false, x1, x2)) → c33(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 214.02/68.80
PROPER(diff(x0, p(z0))) → c34(DIFF(proper(x0), p(proper(z0))), PROPER(x0), PROPER(p(z0))) 214.02/68.80
PROPER(diff(x0, s(z0))) → c34(DIFF(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 214.02/68.80
PROPER(diff(x0, leq(z0, z1))) → c34(DIFF(proper(x0), leq(proper(z0), proper(z1))), PROPER(x0), PROPER(leq(z0, z1))) 214.02/68.80
PROPER(diff(x0, if(z0, z1, z2))) → c34(DIFF(proper(x0), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(if(z0, z1, z2))) 214.02/68.80
PROPER(diff(x0, diff(z0, z1))) → c34(DIFF(proper(x0), diff(proper(z0), proper(z1))), PROPER(x0), PROPER(diff(z0, z1))) 214.02/68.80
PROPER(diff(p(z0), x1)) → c34(DIFF(p(proper(z0)), proper(x1)), PROPER(p(z0)), PROPER(x1)) 214.02/68.80
PROPER(diff(s(z0), x1)) → c34(DIFF(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 214.02/68.80
PROPER(diff(leq(z0, z1), x1)) → c34(DIFF(leq(proper(z0), proper(z1)), proper(x1)), PROPER(leq(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(diff(if(z0, z1, z2), x1)) → c34(DIFF(if(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(if(z0, z1, z2)), PROPER(x1)) 214.02/68.80
PROPER(diff(diff(z0, z1), x1)) → c34(DIFF(diff(proper(z0), proper(z1)), proper(x1)), PROPER(diff(z0, z1)), PROPER(x1)) 214.02/68.80
PROPER(diff(x0, 0)) → c34(DIFF(proper(x0), ok(0)), PROPER(x0)) 214.02/68.80
PROPER(diff(x0, true)) → c34(DIFF(proper(x0), ok(true)), PROPER(x0)) 214.02/68.80
PROPER(diff(x0, false)) → c34(DIFF(proper(x0), ok(false)), PROPER(x0)) 214.02/68.80
PROPER(diff(0, x1)) → c34(DIFF(ok(0), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(diff(true, x1)) → c34(DIFF(ok(true), proper(x1)), PROPER(x1)) 214.02/68.80
PROPER(diff(false, x1)) → c34(DIFF(ok(false), proper(x1)), PROPER(x1)) 214.02/68.80
TOP(mark(p(z0))) → c35(TOP(p(proper(z0))), PROPER(p(z0))) 214.02/68.80
TOP(mark(s(z0))) → c35(TOP(s(proper(z0))), PROPER(s(z0))) 214.02/68.80
TOP(mark(leq(z0, z1))) → c35(TOP(leq(proper(z0), proper(z1))), PROPER(leq(z0, z1))) 214.02/68.80
TOP(mark(if(z0, z1, z2))) → c35(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 214.02/68.80
TOP(mark(diff(z0, z1))) → c35(TOP(diff(proper(z0), proper(z1))), PROPER(diff(z0, z1))) 214.02/68.80
TOP(ok(p(0))) → c36(TOP(mark(0)), ACTIVE(p(0))) 214.02/68.80
TOP(ok(p(s(z0)))) → c36(TOP(mark(z0)), ACTIVE(p(s(z0)))) 214.02/68.80
TOP(ok(leq(0, z0))) → c36(TOP(mark(true)), ACTIVE(leq(0, z0))) 214.02/68.80
TOP(ok(leq(s(z0), 0))) → c36(TOP(mark(false)), ACTIVE(leq(s(z0), 0))) 214.02/68.80
TOP(ok(leq(s(z0), s(z1)))) → c36(TOP(mark(leq(z0, z1))), ACTIVE(leq(s(z0), s(z1)))) 214.02/68.80
TOP(ok(if(true, z0, z1))) → c36(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 214.02/68.80
TOP(ok(if(false, z0, z1))) → c36(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 214.02/68.80
TOP(ok(diff(z0, z1))) → c36(TOP(mark(if(leq(z0, z1), 0, s(diff(p(z0), z1))))), ACTIVE(diff(z0, z1))) 214.02/68.80
TOP(ok(p(z0))) → c36(TOP(p(active(z0))), ACTIVE(p(z0))) 214.02/68.80
TOP(ok(s(z0))) → c36(TOP(s(active(z0))), ACTIVE(s(z0))) 214.02/68.80
TOP(ok(leq(z0, z1))) → c36(TOP(leq(active(z0), z1)), ACTIVE(leq(z0, z1))) 214.02/68.80
TOP(ok(leq(z0, z1))) → c36(TOP(leq(z0, active(z1))), ACTIVE(leq(z0, z1))) 214.02/68.80
TOP(ok(if(z0, z1, z2))) → c36(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 214.02/68.80
TOP(ok(diff(z0, z1))) → c36(TOP(diff(active(z0), z1)), ACTIVE(diff(z0, z1))) 214.02/68.80
TOP(ok(diff(z0, z1))) → c36(TOP(diff(z0, active(z1))), ACTIVE(diff(z0, z1)))
214.02/68.80
214.02/68.80

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.80
active(p(s(z0))) → mark(z0) 214.02/68.80
active(leq(0, z0)) → mark(true) 214.02/68.80
active(leq(s(z0), 0)) → mark(false) 214.02/68.80
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.80
active(if(true, z0, z1)) → mark(z0) 214.02/68.80
active(if(false, z0, z1)) → mark(z1) 214.02/68.80
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.80
active(p(z0)) → p(active(z0)) 214.02/68.80
active(s(z0)) → s(active(z0)) 214.02/68.80
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.80
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.80
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.80
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.80
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.80
p(mark(z0)) → mark(p(z0)) 214.02/68.80
p(ok(z0)) → ok(p(z0)) 214.02/68.80
s(mark(z0)) → mark(s(z0)) 214.02/68.80
s(ok(z0)) → ok(s(z0)) 214.02/68.80
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.80
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.80
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.80
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.80
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.80
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.80
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.80
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.80
proper(p(z0)) → p(proper(z0)) 214.02/68.80
proper(0) → ok(0) 214.02/68.80
proper(s(z0)) → s(proper(z0)) 214.02/68.80
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.80
proper(true) → ok(true) 214.02/68.80
proper(false) → ok(false) 214.02/68.80
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.80
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.80
top(mark(z0)) → top(proper(z0)) 214.02/68.80
top(ok(z0)) → top(active(z0))
Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 214.02/68.80
TOP(mark(0)) → c35(TOP(ok(0))) 214.02/68.80
TOP(mark(true)) → c35(TOP(ok(true))) 214.02/68.80
TOP(mark(false)) → c35(TOP(ok(false)))
S tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
K tuples:

TOP(mark(0)) → c35(TOP(ok(0))) 214.02/68.80
TOP(mark(true)) → c35(TOP(ok(true))) 214.02/68.80
TOP(mark(false)) → c35(TOP(ok(false)))
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

P, S, LEQ, IF, DIFF, TOP

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c35

214.02/68.80
214.02/68.80

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

Removed 3 trailing tuple parts
214.02/68.80
214.02/68.80

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.80
active(p(s(z0))) → mark(z0) 214.02/68.80
active(leq(0, z0)) → mark(true) 214.02/68.80
active(leq(s(z0), 0)) → mark(false) 214.02/68.80
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.80
active(if(true, z0, z1)) → mark(z0) 214.02/68.80
active(if(false, z0, z1)) → mark(z1) 214.02/68.80
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.80
active(p(z0)) → p(active(z0)) 214.02/68.80
active(s(z0)) → s(active(z0)) 214.02/68.80
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.80
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.80
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.80
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.80
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.80
p(mark(z0)) → mark(p(z0)) 214.02/68.80
p(ok(z0)) → ok(p(z0)) 214.02/68.80
s(mark(z0)) → mark(s(z0)) 214.02/68.80
s(ok(z0)) → ok(s(z0)) 214.02/68.80
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.80
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.80
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.80
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.80
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.80
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.80
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.80
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.80
proper(p(z0)) → p(proper(z0)) 214.02/68.80
proper(0) → ok(0) 214.02/68.80
proper(s(z0)) → s(proper(z0)) 214.02/68.80
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.80
proper(true) → ok(true) 214.02/68.80
proper(false) → ok(false) 214.02/68.80
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.80
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.80
top(mark(z0)) → top(proper(z0)) 214.02/68.80
top(ok(z0)) → top(active(z0))
Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1)) 214.02/68.80
TOP(mark(0)) → c35 214.02/68.80
TOP(mark(true)) → c35 214.02/68.80
TOP(mark(false)) → c35
S tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
K tuples:

TOP(mark(0)) → c35 214.02/68.80
TOP(mark(true)) → c35 214.02/68.80
TOP(mark(false)) → c35
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

P, S, LEQ, IF, DIFF, TOP

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c35

214.02/68.80
214.02/68.80

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

Removed 3 trailing nodes:

TOP(mark(true)) → c35 214.02/68.80
TOP(mark(false)) → c35 214.02/68.80
TOP(mark(0)) → c35
214.02/68.80
214.02/68.80

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.80
active(p(s(z0))) → mark(z0) 214.02/68.80
active(leq(0, z0)) → mark(true) 214.02/68.80
active(leq(s(z0), 0)) → mark(false) 214.02/68.80
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.80
active(if(true, z0, z1)) → mark(z0) 214.02/68.80
active(if(false, z0, z1)) → mark(z1) 214.02/68.80
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.80
active(p(z0)) → p(active(z0)) 214.02/68.80
active(s(z0)) → s(active(z0)) 214.02/68.80
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.80
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.80
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.80
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.80
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.80
p(mark(z0)) → mark(p(z0)) 214.02/68.80
p(ok(z0)) → ok(p(z0)) 214.02/68.80
s(mark(z0)) → mark(s(z0)) 214.02/68.80
s(ok(z0)) → ok(s(z0)) 214.02/68.80
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.80
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.80
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.80
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.80
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.80
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.80
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.80
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.80
proper(p(z0)) → p(proper(z0)) 214.02/68.80
proper(0) → ok(0) 214.02/68.80
proper(s(z0)) → s(proper(z0)) 214.02/68.80
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.80
proper(true) → ok(true) 214.02/68.80
proper(false) → ok(false) 214.02/68.80
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.80
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.80
top(mark(z0)) → top(proper(z0)) 214.02/68.80
top(ok(z0)) → top(active(z0))
Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
S tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
K tuples:none
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

P, S, LEQ, IF, DIFF

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26

214.02/68.80
214.02/68.80

(53) 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)) → c23(IF(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 214.02/68.80

POL(DIFF(x1, x2)) = 0    214.02/68.80
POL(IF(x1, x2, x3)) = [2]x1    214.02/68.80
POL(LEQ(x1, x2)) = 0    214.02/68.80
POL(P(x1)) = 0    214.02/68.80
POL(S(x1)) = 0    214.02/68.80
POL(c15(x1)) = x1    214.02/68.80
POL(c16(x1)) = x1    214.02/68.80
POL(c17(x1)) = x1    214.02/68.80
POL(c18(x1)) = x1    214.02/68.80
POL(c19(x1)) = x1    214.02/68.80
POL(c20(x1)) = x1    214.02/68.80
POL(c21(x1)) = x1    214.02/68.80
POL(c22(x1)) = x1    214.02/68.80
POL(c23(x1)) = x1    214.02/68.80
POL(c24(x1)) = x1    214.02/68.80
POL(c25(x1)) = x1    214.02/68.80
POL(c26(x1)) = x1    214.02/68.80
POL(mark(x1)) = x1    214.02/68.80
POL(ok(x1)) = [1] + x1   
214.02/68.80
214.02/68.80

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.80
active(p(s(z0))) → mark(z0) 214.02/68.80
active(leq(0, z0)) → mark(true) 214.02/68.80
active(leq(s(z0), 0)) → mark(false) 214.02/68.80
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.80
active(if(true, z0, z1)) → mark(z0) 214.02/68.80
active(if(false, z0, z1)) → mark(z1) 214.02/68.80
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.80
active(p(z0)) → p(active(z0)) 214.02/68.80
active(s(z0)) → s(active(z0)) 214.02/68.80
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.80
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.80
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.80
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.80
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.80
p(mark(z0)) → mark(p(z0)) 214.02/68.80
p(ok(z0)) → ok(p(z0)) 214.02/68.80
s(mark(z0)) → mark(s(z0)) 214.02/68.80
s(ok(z0)) → ok(s(z0)) 214.02/68.80
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.80
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.80
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.80
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.80
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.80
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.80
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.80
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.80
proper(p(z0)) → p(proper(z0)) 214.02/68.80
proper(0) → ok(0) 214.02/68.80
proper(s(z0)) → s(proper(z0)) 214.02/68.80
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.80
proper(true) → ok(true) 214.02/68.80
proper(false) → ok(false) 214.02/68.80
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.80
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.80
top(mark(z0)) → top(proper(z0)) 214.02/68.80
top(ok(z0)) → top(active(z0))
Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
S tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2))
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

P, S, LEQ, IF, DIFF

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26

214.02/68.80
214.02/68.80

(55) 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.

LEQ(mark(z0), z1) → c19(LEQ(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 214.02/68.80

POL(DIFF(x1, x2)) = 0    214.02/68.80
POL(IF(x1, x2, x3)) = [5]x2 + [5]x3    214.02/68.80
POL(LEQ(x1, x2)) = x1    214.02/68.80
POL(P(x1)) = 0    214.02/68.80
POL(S(x1)) = 0    214.02/68.80
POL(c15(x1)) = x1    214.02/68.80
POL(c16(x1)) = x1    214.02/68.80
POL(c17(x1)) = x1    214.02/68.80
POL(c18(x1)) = x1    214.02/68.80
POL(c19(x1)) = x1    214.02/68.80
POL(c20(x1)) = x1    214.02/68.80
POL(c21(x1)) = x1    214.02/68.80
POL(c22(x1)) = x1    214.02/68.80
POL(c23(x1)) = x1    214.02/68.80
POL(c24(x1)) = x1    214.02/68.80
POL(c25(x1)) = x1    214.02/68.80
POL(c26(x1)) = x1    214.02/68.80
POL(mark(x1)) = [1] + x1    214.02/68.80
POL(ok(x1)) = x1   
214.02/68.80
214.02/68.80

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.80
active(p(s(z0))) → mark(z0) 214.02/68.80
active(leq(0, z0)) → mark(true) 214.02/68.80
active(leq(s(z0), 0)) → mark(false) 214.02/68.80
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.80
active(if(true, z0, z1)) → mark(z0) 214.02/68.80
active(if(false, z0, z1)) → mark(z1) 214.02/68.80
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.80
active(p(z0)) → p(active(z0)) 214.02/68.80
active(s(z0)) → s(active(z0)) 214.02/68.80
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.80
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.80
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.80
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.80
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.80
p(mark(z0)) → mark(p(z0)) 214.02/68.80
p(ok(z0)) → ok(p(z0)) 214.02/68.80
s(mark(z0)) → mark(s(z0)) 214.02/68.80
s(ok(z0)) → ok(s(z0)) 214.02/68.80
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.80
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.80
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.80
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.80
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.80
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.80
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.80
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.80
proper(p(z0)) → p(proper(z0)) 214.02/68.80
proper(0) → ok(0) 214.02/68.80
proper(s(z0)) → s(proper(z0)) 214.02/68.80
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.80
proper(true) → ok(true) 214.02/68.80
proper(false) → ok(false) 214.02/68.80
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.80
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.80
top(mark(z0)) → top(proper(z0)) 214.02/68.80
top(ok(z0)) → top(active(z0))
Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
S tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1))
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

P, S, LEQ, IF, DIFF

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26

214.02/68.80
214.02/68.80

(57) 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) → c22(IF(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 214.02/68.80

POL(DIFF(x1, x2)) = 0    214.02/68.80
POL(IF(x1, x2, x3)) = x1 + [5]x2 + [5]x3    214.02/68.80
POL(LEQ(x1, x2)) = 0    214.02/68.80
POL(P(x1)) = 0    214.02/68.80
POL(S(x1)) = 0    214.02/68.80
POL(c15(x1)) = x1    214.02/68.80
POL(c16(x1)) = x1    214.02/68.80
POL(c17(x1)) = x1    214.02/68.80
POL(c18(x1)) = x1    214.02/68.80
POL(c19(x1)) = x1    214.02/68.80
POL(c20(x1)) = x1    214.02/68.80
POL(c21(x1)) = x1    214.02/68.80
POL(c22(x1)) = x1    214.02/68.80
POL(c23(x1)) = x1    214.02/68.80
POL(c24(x1)) = x1    214.02/68.80
POL(c25(x1)) = x1    214.02/68.80
POL(c26(x1)) = x1    214.02/68.80
POL(mark(x1)) = [1] + x1    214.02/68.80
POL(ok(x1)) = x1   
214.02/68.80
214.02/68.80

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.80
active(p(s(z0))) → mark(z0) 214.02/68.80
active(leq(0, z0)) → mark(true) 214.02/68.80
active(leq(s(z0), 0)) → mark(false) 214.02/68.80
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.80
active(if(true, z0, z1)) → mark(z0) 214.02/68.80
active(if(false, z0, z1)) → mark(z1) 214.02/68.80
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.80
active(p(z0)) → p(active(z0)) 214.02/68.80
active(s(z0)) → s(active(z0)) 214.02/68.80
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.80
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.80
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.80
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.80
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.80
p(mark(z0)) → mark(p(z0)) 214.02/68.80
p(ok(z0)) → ok(p(z0)) 214.02/68.80
s(mark(z0)) → mark(s(z0)) 214.02/68.80
s(ok(z0)) → ok(s(z0)) 214.02/68.80
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.80
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.80
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.80
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.80
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.80
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.80
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.80
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.80
proper(p(z0)) → p(proper(z0)) 214.02/68.80
proper(0) → ok(0) 214.02/68.80
proper(s(z0)) → s(proper(z0)) 214.02/68.80
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.80
proper(true) → ok(true) 214.02/68.80
proper(false) → ok(false) 214.02/68.80
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.80
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.80
top(mark(z0)) → top(proper(z0)) 214.02/68.80
top(ok(z0)) → top(active(z0))
Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
S tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2))
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

P, S, LEQ, IF, DIFF

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26

214.02/68.80
214.02/68.80

(59) 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.

DIFF(mark(z0), z1) → c24(DIFF(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 214.02/68.80

POL(DIFF(x1, x2)) = x1    214.02/68.80
POL(IF(x1, x2, x3)) = [5]x1 + [5]x2 + [5]x3    214.02/68.80
POL(LEQ(x1, x2)) = 0    214.02/68.80
POL(P(x1)) = 0    214.02/68.80
POL(S(x1)) = 0    214.02/68.80
POL(c15(x1)) = x1    214.02/68.80
POL(c16(x1)) = x1    214.02/68.80
POL(c17(x1)) = x1    214.02/68.80
POL(c18(x1)) = x1    214.02/68.80
POL(c19(x1)) = x1    214.02/68.80
POL(c20(x1)) = x1    214.02/68.80
POL(c21(x1)) = x1    214.02/68.80
POL(c22(x1)) = x1    214.02/68.80
POL(c23(x1)) = x1    214.02/68.80
POL(c24(x1)) = x1    214.02/68.80
POL(c25(x1)) = x1    214.02/68.80
POL(c26(x1)) = x1    214.02/68.80
POL(mark(x1)) = [1] + x1    214.02/68.80
POL(ok(x1)) = x1   
214.02/68.80
214.02/68.80

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.80
active(p(s(z0))) → mark(z0) 214.02/68.80
active(leq(0, z0)) → mark(true) 214.02/68.80
active(leq(s(z0), 0)) → mark(false) 214.02/68.80
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.80
active(if(true, z0, z1)) → mark(z0) 214.02/68.80
active(if(false, z0, z1)) → mark(z1) 214.02/68.80
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.80
active(p(z0)) → p(active(z0)) 214.02/68.80
active(s(z0)) → s(active(z0)) 214.02/68.80
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.80
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.80
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.80
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.80
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.80
p(mark(z0)) → mark(p(z0)) 214.02/68.80
p(ok(z0)) → ok(p(z0)) 214.02/68.80
s(mark(z0)) → mark(s(z0)) 214.02/68.80
s(ok(z0)) → ok(s(z0)) 214.02/68.80
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.80
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.80
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.80
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.80
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.80
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.80
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.80
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.80
proper(p(z0)) → p(proper(z0)) 214.02/68.80
proper(0) → ok(0) 214.02/68.80
proper(s(z0)) → s(proper(z0)) 214.02/68.80
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.80
proper(true) → ok(true) 214.02/68.80
proper(false) → ok(false) 214.02/68.80
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.80
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.80
top(mark(z0)) → top(proper(z0)) 214.02/68.80
top(ok(z0)) → top(active(z0))
Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
S tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1))
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

P, S, LEQ, IF, DIFF

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26

214.02/68.80
214.02/68.80

(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.

P(mark(z0)) → c15(P(z0))
We considered the (Usable) Rules:none
And the Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.80
P(ok(z0)) → c16(P(z0)) 214.02/68.80
S(mark(z0)) → c17(S(z0)) 214.02/68.80
S(ok(z0)) → c18(S(z0)) 214.02/68.80
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.80
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.80
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.80
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.80
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.80
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.80
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.80
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 214.02/68.80

POL(DIFF(x1, x2)) = 0    214.02/68.80
POL(IF(x1, x2, x3)) = [3]x1 + [3]x2 + [5]x3    214.02/68.80
POL(LEQ(x1, x2)) = 0    214.02/68.80
POL(P(x1)) = [2]x1    214.02/68.80
POL(S(x1)) = 0    214.02/68.80
POL(c15(x1)) = x1    214.02/68.80
POL(c16(x1)) = x1    214.02/68.80
POL(c17(x1)) = x1    214.02/68.80
POL(c18(x1)) = x1    214.02/68.80
POL(c19(x1)) = x1    214.02/68.80
POL(c20(x1)) = x1    214.02/68.80
POL(c21(x1)) = x1    214.02/68.81
POL(c22(x1)) = x1    214.02/68.81
POL(c23(x1)) = x1    214.02/68.81
POL(c24(x1)) = x1    214.02/68.81
POL(c25(x1)) = x1    214.02/68.81
POL(c26(x1)) = x1    214.02/68.81
POL(mark(x1)) = [1] + x1    214.02/68.81
POL(ok(x1)) = x1   
214.02/68.81
214.02/68.81

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.81
active(p(s(z0))) → mark(z0) 214.02/68.81
active(leq(0, z0)) → mark(true) 214.02/68.81
active(leq(s(z0), 0)) → mark(false) 214.02/68.81
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.81
active(if(true, z0, z1)) → mark(z0) 214.02/68.81
active(if(false, z0, z1)) → mark(z1) 214.02/68.81
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.81
active(p(z0)) → p(active(z0)) 214.02/68.81
active(s(z0)) → s(active(z0)) 214.02/68.81
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.81
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.81
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.81
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.81
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.81
p(mark(z0)) → mark(p(z0)) 214.02/68.81
p(ok(z0)) → ok(p(z0)) 214.02/68.81
s(mark(z0)) → mark(s(z0)) 214.02/68.81
s(ok(z0)) → ok(s(z0)) 214.02/68.81
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.81
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.81
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.81
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.81
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.81
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.81
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.81
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.81
proper(p(z0)) → p(proper(z0)) 214.02/68.81
proper(0) → ok(0) 214.02/68.81
proper(s(z0)) → s(proper(z0)) 214.02/68.81
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.81
proper(true) → ok(true) 214.02/68.81
proper(false) → ok(false) 214.02/68.81
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.81
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.81
top(mark(z0)) → top(proper(z0)) 214.02/68.81
top(ok(z0)) → top(active(z0))
Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
S tuples:

P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
P(mark(z0)) → c15(P(z0))
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

P, S, LEQ, IF, DIFF

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26

214.02/68.81
214.02/68.81

(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.

P(ok(z0)) → c16(P(z0))
We considered the (Usable) Rules:none
And the Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 214.02/68.81

POL(DIFF(x1, x2)) = 0    214.02/68.81
POL(IF(x1, x2, x3)) = [5]x1 + [3]x2 + [5]x3    214.02/68.81
POL(LEQ(x1, x2)) = 0    214.02/68.81
POL(P(x1)) = [2]x1    214.02/68.81
POL(S(x1)) = 0    214.02/68.81
POL(c15(x1)) = x1    214.02/68.81
POL(c16(x1)) = x1    214.02/68.81
POL(c17(x1)) = x1    214.02/68.81
POL(c18(x1)) = x1    214.02/68.81
POL(c19(x1)) = x1    214.02/68.81
POL(c20(x1)) = x1    214.02/68.81
POL(c21(x1)) = x1    214.02/68.81
POL(c22(x1)) = x1    214.02/68.81
POL(c23(x1)) = x1    214.02/68.81
POL(c24(x1)) = x1    214.02/68.81
POL(c25(x1)) = x1    214.02/68.81
POL(c26(x1)) = x1    214.02/68.81
POL(mark(x1)) = x1    214.02/68.81
POL(ok(x1)) = [1] + x1   
214.02/68.81
214.02/68.81

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.81
active(p(s(z0))) → mark(z0) 214.02/68.81
active(leq(0, z0)) → mark(true) 214.02/68.81
active(leq(s(z0), 0)) → mark(false) 214.02/68.81
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.81
active(if(true, z0, z1)) → mark(z0) 214.02/68.81
active(if(false, z0, z1)) → mark(z1) 214.02/68.81
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.81
active(p(z0)) → p(active(z0)) 214.02/68.81
active(s(z0)) → s(active(z0)) 214.02/68.81
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.81
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.81
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.81
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.81
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.81
p(mark(z0)) → mark(p(z0)) 214.02/68.81
p(ok(z0)) → ok(p(z0)) 214.02/68.81
s(mark(z0)) → mark(s(z0)) 214.02/68.81
s(ok(z0)) → ok(s(z0)) 214.02/68.81
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.81
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.81
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.81
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.81
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.81
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.81
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.81
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.81
proper(p(z0)) → p(proper(z0)) 214.02/68.81
proper(0) → ok(0) 214.02/68.81
proper(s(z0)) → s(proper(z0)) 214.02/68.81
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.81
proper(true) → ok(true) 214.02/68.81
proper(false) → ok(false) 214.02/68.81
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.81
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.81
top(mark(z0)) → top(proper(z0)) 214.02/68.81
top(ok(z0)) → top(active(z0))
Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
S tuples:

S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0))
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

P, S, LEQ, IF, DIFF

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26

214.02/68.81
214.02/68.81

(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.

LEQ(z0, mark(z1)) → c20(LEQ(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 214.02/68.81

POL(DIFF(x1, x2)) = 0    214.02/68.81
POL(IF(x1, x2, x3)) = [3]x1 + [5]x2 + [5]x3    214.02/68.81
POL(LEQ(x1, x2)) = [2]x2    214.02/68.81
POL(P(x1)) = [3]x1    214.02/68.81
POL(S(x1)) = 0    214.02/68.81
POL(c15(x1)) = x1    214.02/68.81
POL(c16(x1)) = x1    214.02/68.81
POL(c17(x1)) = x1    214.02/68.81
POL(c18(x1)) = x1    214.02/68.81
POL(c19(x1)) = x1    214.02/68.81
POL(c20(x1)) = x1    214.02/68.81
POL(c21(x1)) = x1    214.02/68.81
POL(c22(x1)) = x1    214.02/68.81
POL(c23(x1)) = x1    214.02/68.81
POL(c24(x1)) = x1    214.02/68.81
POL(c25(x1)) = x1    214.02/68.81
POL(c26(x1)) = x1    214.02/68.81
POL(mark(x1)) = [1] + x1    214.02/68.81
POL(ok(x1)) = x1   
214.02/68.81
214.02/68.81

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.81
active(p(s(z0))) → mark(z0) 214.02/68.81
active(leq(0, z0)) → mark(true) 214.02/68.81
active(leq(s(z0), 0)) → mark(false) 214.02/68.81
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.81
active(if(true, z0, z1)) → mark(z0) 214.02/68.81
active(if(false, z0, z1)) → mark(z1) 214.02/68.81
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.81
active(p(z0)) → p(active(z0)) 214.02/68.81
active(s(z0)) → s(active(z0)) 214.02/68.81
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.81
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.81
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.81
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.81
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.81
p(mark(z0)) → mark(p(z0)) 214.02/68.81
p(ok(z0)) → ok(p(z0)) 214.02/68.81
s(mark(z0)) → mark(s(z0)) 214.02/68.81
s(ok(z0)) → ok(s(z0)) 214.02/68.81
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.81
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.81
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.81
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.81
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.81
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.81
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.81
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.81
proper(p(z0)) → p(proper(z0)) 214.02/68.81
proper(0) → ok(0) 214.02/68.81
proper(s(z0)) → s(proper(z0)) 214.02/68.81
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.81
proper(true) → ok(true) 214.02/68.81
proper(false) → ok(false) 214.02/68.81
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.81
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.81
top(mark(z0)) → top(proper(z0)) 214.02/68.81
top(ok(z0)) → top(active(z0))
Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
S tuples:

S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1))
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

P, S, LEQ, IF, DIFF

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26

214.02/68.81
214.02/68.81

(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.

LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 214.02/68.81

POL(DIFF(x1, x2)) = 0    214.02/68.81
POL(IF(x1, x2, x3)) = [3]x1 + [5]x2 + [5]x3    214.02/68.81
POL(LEQ(x1, x2)) = x1    214.02/68.81
POL(P(x1)) = [3]x1    214.02/68.81
POL(S(x1)) = 0    214.02/68.81
POL(c15(x1)) = x1    214.02/68.81
POL(c16(x1)) = x1    214.02/68.81
POL(c17(x1)) = x1    214.02/68.81
POL(c18(x1)) = x1    214.02/68.81
POL(c19(x1)) = x1    214.02/68.81
POL(c20(x1)) = x1    214.02/68.81
POL(c21(x1)) = x1    214.02/68.81
POL(c22(x1)) = x1    214.02/68.81
POL(c23(x1)) = x1    214.02/68.81
POL(c24(x1)) = x1    214.02/68.81
POL(c25(x1)) = x1    214.02/68.81
POL(c26(x1)) = x1    214.02/68.81
POL(mark(x1)) = x1    214.02/68.81
POL(ok(x1)) = [1] + x1   
214.02/68.81
214.02/68.81

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.81
active(p(s(z0))) → mark(z0) 214.02/68.81
active(leq(0, z0)) → mark(true) 214.02/68.81
active(leq(s(z0), 0)) → mark(false) 214.02/68.81
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.81
active(if(true, z0, z1)) → mark(z0) 214.02/68.81
active(if(false, z0, z1)) → mark(z1) 214.02/68.81
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.81
active(p(z0)) → p(active(z0)) 214.02/68.81
active(s(z0)) → s(active(z0)) 214.02/68.81
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.81
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.81
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.81
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.81
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.81
p(mark(z0)) → mark(p(z0)) 214.02/68.81
p(ok(z0)) → ok(p(z0)) 214.02/68.81
s(mark(z0)) → mark(s(z0)) 214.02/68.81
s(ok(z0)) → ok(s(z0)) 214.02/68.81
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.81
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.81
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.81
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.81
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.81
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.81
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.81
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.81
proper(p(z0)) → p(proper(z0)) 214.02/68.81
proper(0) → ok(0) 214.02/68.81
proper(s(z0)) → s(proper(z0)) 214.02/68.81
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.81
proper(true) → ok(true) 214.02/68.81
proper(false) → ok(false) 214.02/68.81
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.81
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.81
top(mark(z0)) → top(proper(z0)) 214.02/68.81
top(ok(z0)) → top(active(z0))
Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
S tuples:

S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1))
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

P, S, LEQ, IF, DIFF

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26

214.02/68.81
214.02/68.81

(69) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

DIFF(z0, mark(z1)) → c25(DIFF(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 214.02/68.81

POL(DIFF(x1, x2)) = [2]x22    214.02/68.81
POL(IF(x1, x2, x3)) = [3]x32 + [3]x2·x3 + [3]x1·x3 + [3]x12 + [3]x1·x2 + [3]x22    214.02/68.81
POL(LEQ(x1, x2)) = x1 + x2 + [3]x22 + [3]x1·x2 + [3]x12    214.02/68.81
POL(P(x1)) = [3]x1 + [3]x12    214.02/68.81
POL(S(x1)) = 0    214.02/68.81
POL(c15(x1)) = x1    214.02/68.81
POL(c16(x1)) = x1    214.02/68.81
POL(c17(x1)) = x1    214.02/68.81
POL(c18(x1)) = x1    214.02/68.81
POL(c19(x1)) = x1    214.02/68.81
POL(c20(x1)) = x1    214.02/68.81
POL(c21(x1)) = x1    214.02/68.81
POL(c22(x1)) = x1    214.02/68.81
POL(c23(x1)) = x1    214.02/68.81
POL(c24(x1)) = x1    214.02/68.81
POL(c25(x1)) = x1    214.02/68.81
POL(c26(x1)) = x1    214.02/68.81
POL(mark(x1)) = [1] + x1    214.02/68.81
POL(ok(x1)) = x1   
214.02/68.81
214.02/68.81

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.81
active(p(s(z0))) → mark(z0) 214.02/68.81
active(leq(0, z0)) → mark(true) 214.02/68.81
active(leq(s(z0), 0)) → mark(false) 214.02/68.81
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.81
active(if(true, z0, z1)) → mark(z0) 214.02/68.81
active(if(false, z0, z1)) → mark(z1) 214.02/68.81
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.81
active(p(z0)) → p(active(z0)) 214.02/68.81
active(s(z0)) → s(active(z0)) 214.02/68.81
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.81
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.81
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.81
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.81
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.81
p(mark(z0)) → mark(p(z0)) 214.02/68.81
p(ok(z0)) → ok(p(z0)) 214.02/68.81
s(mark(z0)) → mark(s(z0)) 214.02/68.81
s(ok(z0)) → ok(s(z0)) 214.02/68.81
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.81
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.81
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.81
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.81
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.81
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.81
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.81
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.81
proper(p(z0)) → p(proper(z0)) 214.02/68.81
proper(0) → ok(0) 214.02/68.81
proper(s(z0)) → s(proper(z0)) 214.02/68.81
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.81
proper(true) → ok(true) 214.02/68.81
proper(false) → ok(false) 214.02/68.81
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.81
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.81
top(mark(z0)) → top(proper(z0)) 214.02/68.81
top(ok(z0)) → top(active(z0))
Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
S tuples:

S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1))
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

P, S, LEQ, IF, DIFF

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26

214.02/68.81
214.02/68.81

(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.

S(mark(z0)) → c17(S(z0))
We considered the (Usable) Rules:none
And the Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 214.02/68.81

POL(DIFF(x1, x2)) = 0    214.02/68.81
POL(IF(x1, x2, x3)) = [3]x1 + [3]x2 + [3]x3    214.02/68.81
POL(LEQ(x1, x2)) = [5]x1 + [3]x2    214.02/68.81
POL(P(x1)) = [3]x1    214.02/68.81
POL(S(x1)) = [4]x1    214.02/68.81
POL(c15(x1)) = x1    214.02/68.81
POL(c16(x1)) = x1    214.02/68.81
POL(c17(x1)) = x1    214.02/68.81
POL(c18(x1)) = x1    214.02/68.81
POL(c19(x1)) = x1    214.02/68.81
POL(c20(x1)) = x1    214.02/68.81
POL(c21(x1)) = x1    214.02/68.81
POL(c22(x1)) = x1    214.02/68.81
POL(c23(x1)) = x1    214.02/68.81
POL(c24(x1)) = x1    214.02/68.81
POL(c25(x1)) = x1    214.02/68.81
POL(c26(x1)) = x1    214.02/68.81
POL(mark(x1)) = [4] + x1    214.02/68.81
POL(ok(x1)) = x1   
214.02/68.81
214.02/68.81

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.81
active(p(s(z0))) → mark(z0) 214.02/68.81
active(leq(0, z0)) → mark(true) 214.02/68.81
active(leq(s(z0), 0)) → mark(false) 214.02/68.81
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.81
active(if(true, z0, z1)) → mark(z0) 214.02/68.81
active(if(false, z0, z1)) → mark(z1) 214.02/68.81
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.81
active(p(z0)) → p(active(z0)) 214.02/68.81
active(s(z0)) → s(active(z0)) 214.02/68.81
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.81
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.81
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.81
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.81
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.81
p(mark(z0)) → mark(p(z0)) 214.02/68.81
p(ok(z0)) → ok(p(z0)) 214.02/68.81
s(mark(z0)) → mark(s(z0)) 214.02/68.81
s(ok(z0)) → ok(s(z0)) 214.02/68.81
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.81
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.81
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.81
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.81
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.81
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.81
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.81
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.81
proper(p(z0)) → p(proper(z0)) 214.02/68.81
proper(0) → ok(0) 214.02/68.81
proper(s(z0)) → s(proper(z0)) 214.02/68.81
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.81
proper(true) → ok(true) 214.02/68.81
proper(false) → ok(false) 214.02/68.81
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.81
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.81
top(mark(z0)) → top(proper(z0)) 214.02/68.81
top(ok(z0)) → top(active(z0))
Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
S tuples:

S(ok(z0)) → c18(S(z0)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
S(mark(z0)) → c17(S(z0))
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

P, S, LEQ, IF, DIFF

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26

214.02/68.81
214.02/68.81

(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.

S(ok(z0)) → c18(S(z0))
We considered the (Usable) Rules:none
And the Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 214.02/68.81

POL(DIFF(x1, x2)) = 0    214.02/68.81
POL(IF(x1, x2, x3)) = [5]x1 + [3]x2 + [5]x3    214.02/68.81
POL(LEQ(x1, x2)) = [3]x1 + [5]x2    214.02/68.81
POL(P(x1)) = [5]x1    214.02/68.81
POL(S(x1)) = [2]x1    214.02/68.81
POL(c15(x1)) = x1    214.02/68.81
POL(c16(x1)) = x1    214.02/68.81
POL(c17(x1)) = x1    214.02/68.81
POL(c18(x1)) = x1    214.02/68.81
POL(c19(x1)) = x1    214.02/68.81
POL(c20(x1)) = x1    214.02/68.81
POL(c21(x1)) = x1    214.02/68.81
POL(c22(x1)) = x1    214.02/68.81
POL(c23(x1)) = x1    214.02/68.81
POL(c24(x1)) = x1    214.02/68.81
POL(c25(x1)) = x1    214.02/68.81
POL(c26(x1)) = x1    214.02/68.81
POL(mark(x1)) = x1    214.02/68.81
POL(ok(x1)) = [1] + x1   
214.02/68.81
214.02/68.81

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.81
active(p(s(z0))) → mark(z0) 214.02/68.81
active(leq(0, z0)) → mark(true) 214.02/68.81
active(leq(s(z0), 0)) → mark(false) 214.02/68.81
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.81
active(if(true, z0, z1)) → mark(z0) 214.02/68.81
active(if(false, z0, z1)) → mark(z1) 214.02/68.81
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.81
active(p(z0)) → p(active(z0)) 214.02/68.81
active(s(z0)) → s(active(z0)) 214.02/68.81
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.81
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.81
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.81
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.81
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.81
p(mark(z0)) → mark(p(z0)) 214.02/68.81
p(ok(z0)) → ok(p(z0)) 214.02/68.81
s(mark(z0)) → mark(s(z0)) 214.02/68.81
s(ok(z0)) → ok(s(z0)) 214.02/68.81
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.81
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.81
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.81
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.81
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.81
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.81
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.81
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.81
proper(p(z0)) → p(proper(z0)) 214.02/68.81
proper(0) → ok(0) 214.02/68.81
proper(s(z0)) → s(proper(z0)) 214.02/68.81
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.81
proper(true) → ok(true) 214.02/68.81
proper(false) → ok(false) 214.02/68.81
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.81
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.81
top(mark(z0)) → top(proper(z0)) 214.02/68.81
top(ok(z0)) → top(active(z0))
Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
S tuples:

DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0))
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

P, S, LEQ, IF, DIFF

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26

214.02/68.81
214.02/68.81

(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.

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

P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 214.02/68.81

POL(DIFF(x1, x2)) = x2    214.02/68.81
POL(IF(x1, x2, x3)) = [5]x1 + [5]x2 + [5]x3    214.02/68.81
POL(LEQ(x1, x2)) = [3]x1 + [3]x2    214.02/68.81
POL(P(x1)) = [5]x1    214.02/68.81
POL(S(x1)) = [3]x1    214.02/68.81
POL(c15(x1)) = x1    214.02/68.81
POL(c16(x1)) = x1    214.02/68.81
POL(c17(x1)) = x1    214.02/68.81
POL(c18(x1)) = x1    214.02/68.81
POL(c19(x1)) = x1    214.02/68.81
POL(c20(x1)) = x1    214.02/68.81
POL(c21(x1)) = x1    214.02/68.81
POL(c22(x1)) = x1    214.02/68.81
POL(c23(x1)) = x1    214.02/68.81
POL(c24(x1)) = x1    214.02/68.81
POL(c25(x1)) = x1    214.02/68.81
POL(c26(x1)) = x1    214.02/68.81
POL(mark(x1)) = x1    214.02/68.81
POL(ok(x1)) = [1] + x1   
214.02/68.81
214.02/68.81

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(p(0)) → mark(0) 214.02/68.81
active(p(s(z0))) → mark(z0) 214.02/68.81
active(leq(0, z0)) → mark(true) 214.02/68.81
active(leq(s(z0), 0)) → mark(false) 214.02/68.81
active(leq(s(z0), s(z1))) → mark(leq(z0, z1)) 214.02/68.81
active(if(true, z0, z1)) → mark(z0) 214.02/68.81
active(if(false, z0, z1)) → mark(z1) 214.02/68.81
active(diff(z0, z1)) → mark(if(leq(z0, z1), 0, s(diff(p(z0), z1)))) 214.02/68.81
active(p(z0)) → p(active(z0)) 214.02/68.81
active(s(z0)) → s(active(z0)) 214.02/68.81
active(leq(z0, z1)) → leq(active(z0), z1) 214.02/68.81
active(leq(z0, z1)) → leq(z0, active(z1)) 214.02/68.81
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 214.02/68.81
active(diff(z0, z1)) → diff(active(z0), z1) 214.02/68.81
active(diff(z0, z1)) → diff(z0, active(z1)) 214.02/68.81
p(mark(z0)) → mark(p(z0)) 214.02/68.81
p(ok(z0)) → ok(p(z0)) 214.02/68.81
s(mark(z0)) → mark(s(z0)) 214.02/68.81
s(ok(z0)) → ok(s(z0)) 214.02/68.81
leq(mark(z0), z1) → mark(leq(z0, z1)) 214.02/68.81
leq(z0, mark(z1)) → mark(leq(z0, z1)) 214.02/68.81
leq(ok(z0), ok(z1)) → ok(leq(z0, z1)) 214.02/68.81
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 214.02/68.81
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 214.02/68.81
diff(mark(z0), z1) → mark(diff(z0, z1)) 214.02/68.81
diff(z0, mark(z1)) → mark(diff(z0, z1)) 214.02/68.81
diff(ok(z0), ok(z1)) → ok(diff(z0, z1)) 214.02/68.81
proper(p(z0)) → p(proper(z0)) 214.02/68.81
proper(0) → ok(0) 214.02/68.81
proper(s(z0)) → s(proper(z0)) 214.02/68.81
proper(leq(z0, z1)) → leq(proper(z0), proper(z1)) 214.02/68.81
proper(true) → ok(true) 214.02/68.81
proper(false) → ok(false) 214.02/68.81
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 214.02/68.81
proper(diff(z0, z1)) → diff(proper(z0), proper(z1)) 214.02/68.81
top(mark(z0)) → top(proper(z0)) 214.02/68.81
top(ok(z0)) → top(active(z0))
Tuples:

P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
S tuples:none
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c23(IF(z0, z1, z2)) 214.02/68.81
LEQ(mark(z0), z1) → c19(LEQ(z0, z1)) 214.02/68.81
IF(mark(z0), z1, z2) → c22(IF(z0, z1, z2)) 214.02/68.81
DIFF(mark(z0), z1) → c24(DIFF(z0, z1)) 214.02/68.81
P(mark(z0)) → c15(P(z0)) 214.02/68.81
P(ok(z0)) → c16(P(z0)) 214.02/68.81
LEQ(z0, mark(z1)) → c20(LEQ(z0, z1)) 214.02/68.81
LEQ(ok(z0), ok(z1)) → c21(LEQ(z0, z1)) 214.02/68.81
DIFF(z0, mark(z1)) → c25(DIFF(z0, z1)) 214.02/68.81
S(mark(z0)) → c17(S(z0)) 214.02/68.81
S(ok(z0)) → c18(S(z0)) 214.02/68.81
DIFF(ok(z0), ok(z1)) → c26(DIFF(z0, z1))
Defined Rule Symbols:

active, p, s, leq, if, diff, proper, top

Defined Pair Symbols:

P, S, LEQ, IF, DIFF

Compound Symbols:

c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26

214.02/68.81
214.02/68.81

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

The set S is empty
214.02/68.81
214.02/68.81

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

214.02/68.81
214.02/68.81
214.02/68.89 EOF