YES(O(1), O(n^2)) 96.32/33.15 YES(O(1), O(n^2)) 96.32/33.19 96.32/33.19 96.32/33.19 96.32/33.19 96.32/33.19 96.32/33.19 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 96.32/33.19 96.32/33.19 96.32/33.19
96.32/33.19
96.32/33.19
96.32/33.19
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
96.32/33.19 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
96.32/33.19 
↳2 CdtProblem
96.32/33.19 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
96.32/33.19 
↳4 CdtProblem
96.32/33.19 
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
96.32/33.19 
↳6 CdtProblem
96.32/33.19 
↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
96.32/33.19 
↳8 CdtProblem
96.32/33.19 
↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
96.32/33.19 
↳10 CdtProblem
96.32/33.19 
↳11 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
96.32/33.19 
↳12 CdtProblem
96.32/33.19 
↳13 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
96.32/33.19 
↳14 CdtProblem
96.32/33.19 
↳15 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
96.32/33.19 
↳16 CdtProblem
96.32/33.19 
↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
96.32/33.19 
↳18 CdtProblem
96.32/33.19 
↳19 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
96.32/33.19 
↳20 CdtProblem
96.32/33.19 
↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
96.32/33.19 
↳22 CdtProblem
96.32/33.19 
↳23 CdtUnreachableProof (⇔)
96.32/33.19 
↳24 CdtProblem
96.32/33.19 
↳25 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
96.32/33.19 
↳26 CdtProblem
96.32/33.19 
↳27 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
96.32/33.19 
↳28 CdtProblem
96.32/33.19 
↳29 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
96.32/33.19 
↳30 CdtProblem
96.32/33.19 
↳31 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
96.32/33.19 
↳32 CdtProblem
96.32/33.19 
↳33 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
96.32/33.19 
↳34 CdtProblem
96.32/33.19 
↳35 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
96.32/33.19 
↳36 CdtProblem
96.32/33.19 
↳37 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
96.32/33.19 
↳38 CdtProblem
96.32/33.19 
↳39 SIsEmptyProof (BOTH BOUNDS(ID, ID))
96.32/33.19 
↳40 BOUNDS(O(1), O(1))
96.32/33.19
96.32/33.19 96.32/33.19
96.32/33.19
96.32/33.19

(0) Obligation:

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

active(f(X)) → mark(if(X, c, f(true))) 96.32/33.19
active(if(true, X, Y)) → mark(X) 96.32/33.19
active(if(false, X, Y)) → mark(Y) 96.32/33.19
active(f(X)) → f(active(X)) 96.32/33.19
active(if(X1, X2, X3)) → if(active(X1), X2, X3) 96.32/33.19
active(if(X1, X2, X3)) → if(X1, active(X2), X3) 96.32/33.19
f(mark(X)) → mark(f(X)) 96.32/33.19
if(mark(X1), X2, X3) → mark(if(X1, X2, X3)) 96.32/33.19
if(X1, mark(X2), X3) → mark(if(X1, X2, X3)) 96.32/33.19
proper(f(X)) → f(proper(X)) 96.32/33.19
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3)) 96.32/33.19
proper(c) → ok(c) 96.32/33.19
proper(true) → ok(true) 96.32/33.19
proper(false) → ok(false) 96.32/33.19
f(ok(X)) → ok(f(X)) 96.32/33.19
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3)) 96.32/33.19
top(mark(X)) → top(proper(X)) 96.32/33.19
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST
96.32/33.19
96.32/33.19

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

Converted CpxTRS to CDT
96.32/33.19
96.32/33.19

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.32/33.19
active(if(true, z0, z1)) → mark(z0) 96.32/33.19
active(if(false, z0, z1)) → mark(z1) 96.32/33.19
active(f(z0)) → f(active(z0)) 96.32/33.19
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.32/33.19
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.32/33.19
f(mark(z0)) → mark(f(z0)) 96.32/33.19
f(ok(z0)) → ok(f(z0)) 96.32/33.19
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.32/33.19
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.32/33.19
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.32/33.19
proper(f(z0)) → f(proper(z0)) 96.32/33.19
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.32/33.19
proper(c) → ok(c) 96.32/33.19
proper(true) → ok(true) 96.32/33.19
proper(false) → ok(false) 96.32/33.19
top(mark(z0)) → top(proper(z0)) 96.32/33.19
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c1(IF(z0, c, f(true)), F(true)) 96.32/33.19
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0)) 96.32/33.19
ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.25
F(mark(z0)) → c7(F(z0)) 96.59/33.25
F(ok(z0)) → c8(F(z0)) 96.59/33.25
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.25
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.25
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.25
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0)) 96.59/33.25
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.59/33.25
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0)) 96.59/33.25
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0)) → c1(IF(z0, c, f(true)), F(true)) 96.59/33.25
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.25
F(mark(z0)) → c7(F(z0)) 96.59/33.25
F(ok(z0)) → c8(F(z0)) 96.59/33.25
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.25
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.25
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.25
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0)) 96.59/33.25
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.59/33.25
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0)) 96.59/33.25
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c17, c18

96.59/33.25
96.59/33.25

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

Removed 1 trailing tuple parts
96.59/33.25
96.59/33.25

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.25
active(if(true, z0, z1)) → mark(z0) 96.59/33.25
active(if(false, z0, z1)) → mark(z1) 96.59/33.25
active(f(z0)) → f(active(z0)) 96.59/33.25
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.25
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.25
f(mark(z0)) → mark(f(z0)) 96.59/33.25
f(ok(z0)) → ok(f(z0)) 96.59/33.25
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.25
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.25
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.25
proper(f(z0)) → f(proper(z0)) 96.59/33.25
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.25
proper(c) → ok(c) 96.59/33.25
proper(true) → ok(true) 96.59/33.25
proper(false) → ok(false) 96.59/33.25
top(mark(z0)) → top(proper(z0)) 96.59/33.25
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.25
F(mark(z0)) → c7(F(z0)) 96.59/33.25
F(ok(z0)) → c8(F(z0)) 96.59/33.25
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.25
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.25
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.25
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0)) 96.59/33.25
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.59/33.25
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0)) 96.59/33.25
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
S tuples:

ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.25
F(mark(z0)) → c7(F(z0)) 96.59/33.25
F(ok(z0)) → c8(F(z0)) 96.59/33.25
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.25
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.25
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.25
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0)) 96.59/33.25
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.59/33.25
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0)) 96.59/33.25
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
K tuples:none
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c17, c18, c1

96.59/33.25
96.59/33.25

(5) 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.

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
We considered the (Usable) Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.25
active(if(true, z0, z1)) → mark(z0) 96.59/33.25
active(if(false, z0, z1)) → mark(z1) 96.59/33.25
active(f(z0)) → f(active(z0)) 96.59/33.25
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.25
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.25
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.25
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.25
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.25
f(mark(z0)) → mark(f(z0)) 96.59/33.25
f(ok(z0)) → ok(f(z0)) 96.59/33.25
proper(f(z0)) → f(proper(z0)) 96.59/33.25
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.25
proper(c) → ok(c) 96.59/33.25
proper(true) → ok(true) 96.59/33.25
proper(false) → ok(false)
And the Tuples:

ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.25
F(mark(z0)) → c7(F(z0)) 96.59/33.25
F(ok(z0)) → c8(F(z0)) 96.59/33.25
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.25
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.25
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.25
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0)) 96.59/33.25
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.59/33.25
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0)) 96.59/33.25
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
The order we found is given by the following interpretation:
Polynomial interpretation : 96.59/33.25

POL(ACTIVE(x1)) = 0    96.59/33.25
POL(F(x1)) = 0    96.59/33.25
POL(IF(x1, x2, x3)) = 0    96.59/33.25
POL(PROPER(x1)) = 0    96.59/33.25
POL(TOP(x1)) = [2]x1    96.59/33.25
POL(active(x1)) = x1    96.59/33.25
POL(c) = 0    96.59/33.25
POL(c1(x1)) = x1    96.59/33.25
POL(c10(x1)) = x1    96.59/33.25
POL(c11(x1)) = x1    96.59/33.25
POL(c12(x1, x2)) = x1 + x2    96.59/33.25
POL(c13(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    96.59/33.25
POL(c17(x1, x2)) = x1 + x2    96.59/33.25
POL(c18(x1, x2)) = x1 + x2    96.59/33.25
POL(c4(x1, x2)) = x1 + x2    96.59/33.25
POL(c5(x1, x2)) = x1 + x2    96.59/33.25
POL(c6(x1, x2)) = x1 + x2    96.59/33.25
POL(c7(x1)) = x1    96.59/33.25
POL(c8(x1)) = x1    96.59/33.25
POL(c9(x1)) = x1    96.59/33.25
POL(f(x1)) = [2] + [3]x1    96.59/33.25
POL(false) = [1]    96.59/33.25
POL(if(x1, x2, x3)) = [1] + x1 + x2 + x1·x3    96.59/33.25
POL(mark(x1)) = [1] + x1    96.59/33.25
POL(ok(x1)) = x1    96.59/33.25
POL(proper(x1)) = x1    96.59/33.25
POL(true) = 0   
96.59/33.25
96.59/33.25

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.25
active(if(true, z0, z1)) → mark(z0) 96.59/33.25
active(if(false, z0, z1)) → mark(z1) 96.59/33.25
active(f(z0)) → f(active(z0)) 96.59/33.25
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.25
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.25
f(mark(z0)) → mark(f(z0)) 96.59/33.25
f(ok(z0)) → ok(f(z0)) 96.59/33.25
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.25
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.25
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.25
proper(f(z0)) → f(proper(z0)) 96.59/33.25
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.25
proper(c) → ok(c) 96.59/33.25
proper(true) → ok(true) 96.59/33.25
proper(false) → ok(false) 96.59/33.25
top(mark(z0)) → top(proper(z0)) 96.59/33.25
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.25
F(mark(z0)) → c7(F(z0)) 96.59/33.25
F(ok(z0)) → c8(F(z0)) 96.59/33.25
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.25
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.25
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.25
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0)) 96.59/33.25
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.59/33.25
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0)) 96.59/33.25
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
S tuples:

ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.25
F(mark(z0)) → c7(F(z0)) 96.59/33.25
F(ok(z0)) → c8(F(z0)) 96.59/33.25
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.25
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.25
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.25
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0)) 96.59/33.25
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.59/33.25
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c17, c18, c1

96.59/33.25
96.59/33.25

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

Use narrowing to replace ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0)) by

ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.25
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
96.59/33.25
96.59/33.25

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.25
active(if(true, z0, z1)) → mark(z0) 96.59/33.25
active(if(false, z0, z1)) → mark(z1) 96.59/33.25
active(f(z0)) → f(active(z0)) 96.59/33.25
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.25
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.25
f(mark(z0)) → mark(f(z0)) 96.59/33.25
f(ok(z0)) → ok(f(z0)) 96.59/33.25
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.25
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.25
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.25
proper(f(z0)) → f(proper(z0)) 96.59/33.25
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.25
proper(c) → ok(c) 96.59/33.25
proper(true) → ok(true) 96.59/33.25
proper(false) → ok(false) 96.59/33.25
top(mark(z0)) → top(proper(z0)) 96.59/33.25
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.25
F(mark(z0)) → c7(F(z0)) 96.59/33.25
F(ok(z0)) → c8(F(z0)) 96.59/33.25
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.25
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.25
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.25
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0)) 96.59/33.25
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.59/33.25
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0)) 96.59/33.25
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.25
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.25
F(mark(z0)) → c7(F(z0)) 96.59/33.25
F(ok(z0)) → c8(F(z0)) 96.59/33.25
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.25
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.25
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.25
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0)) 96.59/33.25
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.59/33.25
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.25
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c17, c18, c1, c4

96.59/33.25
96.59/33.25

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

Use narrowing to replace PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0)) by

PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.25
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.25
PROPER(f(c)) → c12(F(ok(c)), PROPER(c)) 96.59/33.25
PROPER(f(true)) → c12(F(ok(true)), PROPER(true)) 96.59/33.25
PROPER(f(false)) → c12(F(ok(false)), PROPER(false))
96.59/33.25
96.59/33.25

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.25
active(if(true, z0, z1)) → mark(z0) 96.59/33.25
active(if(false, z0, z1)) → mark(z1) 96.59/33.25
active(f(z0)) → f(active(z0)) 96.59/33.25
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.25
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.25
f(mark(z0)) → mark(f(z0)) 96.59/33.25
f(ok(z0)) → ok(f(z0)) 96.59/33.25
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.25
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.25
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.25
proper(f(z0)) → f(proper(z0)) 96.59/33.25
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.25
proper(c) → ok(c) 96.59/33.25
proper(true) → ok(true) 96.59/33.25
proper(false) → ok(false) 96.59/33.25
top(mark(z0)) → top(proper(z0)) 96.59/33.25
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.25
F(mark(z0)) → c7(F(z0)) 96.59/33.25
F(ok(z0)) → c8(F(z0)) 96.59/33.25
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.25
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.25
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.25
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.59/33.25
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0)) 96.59/33.25
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.25
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.25
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.25
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.25
PROPER(f(c)) → c12(F(ok(c)), PROPER(c)) 96.59/33.25
PROPER(f(true)) → c12(F(ok(true)), PROPER(true)) 96.59/33.25
PROPER(f(false)) → c12(F(ok(false)), PROPER(false))
S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.25
F(mark(z0)) → c7(F(z0)) 96.59/33.25
F(ok(z0)) → c8(F(z0)) 96.59/33.25
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.25
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.25
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.25
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.59/33.25
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.25
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.25
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.25
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.25
PROPER(f(c)) → c12(F(ok(c)), PROPER(c)) 96.59/33.25
PROPER(f(true)) → c12(F(ok(true)), PROPER(true)) 96.59/33.25
PROPER(f(false)) → c12(F(ok(false)), PROPER(false))
K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c13, c17, c18, c1, c4, c12

96.59/33.25
96.59/33.25

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

Removed 3 trailing tuple parts
96.59/33.25
96.59/33.25

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.25
active(if(true, z0, z1)) → mark(z0) 96.59/33.25
active(if(false, z0, z1)) → mark(z1) 96.59/33.25
active(f(z0)) → f(active(z0)) 96.59/33.25
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.25
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.25
f(mark(z0)) → mark(f(z0)) 96.59/33.25
f(ok(z0)) → ok(f(z0)) 96.59/33.25
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.25
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.25
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.25
proper(f(z0)) → f(proper(z0)) 96.59/33.25
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.25
proper(c) → ok(c) 96.59/33.25
proper(true) → ok(true) 96.59/33.25
proper(false) → ok(false) 96.59/33.25
top(mark(z0)) → top(proper(z0)) 96.59/33.25
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.25
F(mark(z0)) → c7(F(z0)) 96.59/33.25
F(ok(z0)) → c8(F(z0)) 96.59/33.25
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.25
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.25
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.25
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.59/33.25
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0)) 96.59/33.25
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.25
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.25
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.25
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.25
PROPER(f(c)) → c12(F(ok(c))) 96.59/33.25
PROPER(f(true)) → c12(F(ok(true))) 96.59/33.25
PROPER(f(false)) → c12(F(ok(false)))
S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.25
F(mark(z0)) → c7(F(z0)) 96.59/33.25
F(ok(z0)) → c8(F(z0)) 96.59/33.25
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.25
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.25
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.25
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) 96.59/33.25
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.25
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.25
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.25
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.25
PROPER(f(c)) → c12(F(ok(c))) 96.59/33.25
PROPER(f(true)) → c12(F(ok(true))) 96.59/33.25
PROPER(f(false)) → c12(F(ok(false)))
K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c13, c17, c18, c1, c4, c12, c12

96.59/33.25
96.59/33.25

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

Use narrowing to replace PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0))) 96.59/33.25
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 96.59/33.25
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 96.59/33.25
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true)) 96.59/33.25
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false)) 96.59/33.25
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2)) 96.59/33.25
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 96.59/33.25
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 96.59/33.25
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2)) 96.59/33.25
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2)) 96.59/33.25
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2)) 96.59/33.25
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 96.59/33.25
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2)) 96.59/33.25
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2)) 96.59/33.25
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2))
96.59/33.25
96.59/33.25

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.25
active(if(true, z0, z1)) → mark(z0) 96.59/33.25
active(if(false, z0, z1)) → mark(z1) 96.59/33.25
active(f(z0)) → f(active(z0)) 96.59/33.25
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.25
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.25
f(mark(z0)) → mark(f(z0)) 96.59/33.25
f(ok(z0)) → ok(f(z0)) 96.59/33.25
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.25
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.25
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.25
proper(f(z0)) → f(proper(z0)) 96.59/33.25
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.25
proper(c) → ok(c) 96.59/33.25
proper(true) → ok(true) 96.59/33.25
proper(false) → ok(false) 96.59/33.25
top(mark(z0)) → top(proper(z0)) 96.59/33.25
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.25
F(mark(z0)) → c7(F(z0)) 96.59/33.25
F(ok(z0)) → c8(F(z0)) 96.59/33.25
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.25
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.25
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.25
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0)) 96.59/33.25
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.25
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.25
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.25
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.25
PROPER(f(c)) → c12(F(ok(c))) 96.59/33.25
PROPER(f(true)) → c12(F(ok(true))) 96.59/33.25
PROPER(f(false)) → c12(F(ok(false))) 96.59/33.25
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0))) 96.59/33.25
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 96.59/33.25
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 96.59/33.25
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true)) 96.59/33.25
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false)) 96.59/33.25
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2)) 96.59/33.25
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 96.59/33.25
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 96.59/33.25
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2)) 96.59/33.25
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2)) 96.59/33.25
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2)) 96.59/33.25
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 96.59/33.25
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2)) 96.59/33.25
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2)) 96.59/33.25
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.25
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.25
F(mark(z0)) → c7(F(z0)) 96.59/33.25
F(ok(z0)) → c8(F(z0)) 96.59/33.25
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.25
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.25
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.25
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.25
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.25
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.25
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.25
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.26
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.26
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.26
PROPER(f(c)) → c12(F(ok(c))) 96.59/33.26
PROPER(f(true)) → c12(F(ok(true))) 96.59/33.26
PROPER(f(false)) → c12(F(ok(false))) 96.59/33.26
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0))) 96.59/33.26
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 96.59/33.26
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c)) 96.59/33.26
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true)) 96.59/33.26
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false)) 96.59/33.26
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2)) 96.59/33.26
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 96.59/33.26
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2)) 96.59/33.26
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2)) 96.59/33.26
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2)) 96.59/33.26
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2))
K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, TOP, PROPER

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c17, c18, c1, c4, c12, c12, c13

96.59/33.26
96.59/33.26

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

Removed 9 trailing tuple parts
96.59/33.26
96.59/33.26

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.26
active(if(true, z0, z1)) → mark(z0) 96.59/33.26
active(if(false, z0, z1)) → mark(z1) 96.59/33.26
active(f(z0)) → f(active(z0)) 96.59/33.26
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.26
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.26
f(mark(z0)) → mark(f(z0)) 96.59/33.26
f(ok(z0)) → ok(f(z0)) 96.59/33.26
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.26
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.26
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.26
proper(f(z0)) → f(proper(z0)) 96.59/33.26
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.26
proper(c) → ok(c) 96.59/33.26
proper(true) → ok(true) 96.59/33.26
proper(false) → ok(false) 96.59/33.26
top(mark(z0)) → top(proper(z0)) 96.59/33.26
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.26
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.26
F(mark(z0)) → c7(F(z0)) 96.59/33.26
F(ok(z0)) → c8(F(z0)) 96.59/33.26
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.26
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.26
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.26
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0)) 96.59/33.26
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.26
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.26
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.26
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.26
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.26
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.26
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.26
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.26
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.26
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.26
PROPER(f(c)) → c12(F(ok(c))) 96.59/33.26
PROPER(f(true)) → c12(F(ok(true))) 96.59/33.26
PROPER(f(false)) → c12(F(ok(false))) 96.59/33.26
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0))) 96.59/33.26
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 96.59/33.26
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2)) 96.59/33.26
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 96.59/33.26
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.59/33.26
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 96.59/33.26
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 96.59/33.26
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.26
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.26
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.26
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.26
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.26
F(mark(z0)) → c7(F(z0)) 96.59/33.26
F(ok(z0)) → c8(F(z0)) 96.59/33.26
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.26
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.26
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.26
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.26
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.26
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.26
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.26
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.26
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.26
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.26
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.26
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.26
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.26
PROPER(f(c)) → c12(F(ok(c))) 96.59/33.26
PROPER(f(true)) → c12(F(ok(true))) 96.59/33.26
PROPER(f(false)) → c12(F(ok(false))) 96.59/33.26
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0))) 96.59/33.26
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 96.59/33.26
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2)) 96.59/33.26
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 96.59/33.26
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.59/33.26
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 96.59/33.26
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 96.59/33.26
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.26
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.26
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.26
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, TOP, PROPER

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c17, c18, c1, c4, c12, c12, c13, c13

96.59/33.26
96.59/33.26

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

Use narrowing to replace TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(f(z0))) → c17(TOP(f(proper(z0))), PROPER(f(z0))) 96.59/33.26
TOP(mark(if(z0, z1, z2))) → c17(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.26
TOP(mark(c)) → c17(TOP(ok(c)), PROPER(c)) 96.59/33.26
TOP(mark(true)) → c17(TOP(ok(true)), PROPER(true)) 96.59/33.26
TOP(mark(false)) → c17(TOP(ok(false)), PROPER(false))
96.59/33.26
96.59/33.26

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.26
active(if(true, z0, z1)) → mark(z0) 96.59/33.26
active(if(false, z0, z1)) → mark(z1) 96.59/33.26
active(f(z0)) → f(active(z0)) 96.59/33.26
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.26
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.26
f(mark(z0)) → mark(f(z0)) 96.59/33.26
f(ok(z0)) → ok(f(z0)) 96.59/33.26
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.26
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.26
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.26
proper(f(z0)) → f(proper(z0)) 96.59/33.26
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.26
proper(c) → ok(c) 96.59/33.26
proper(true) → ok(true) 96.59/33.26
proper(false) → ok(false) 96.59/33.26
top(mark(z0)) → top(proper(z0)) 96.59/33.26
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.26
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.26
F(mark(z0)) → c7(F(z0)) 96.59/33.26
F(ok(z0)) → c8(F(z0)) 96.59/33.26
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.26
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.26
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.26
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.26
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.26
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.26
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.26
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.26
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.26
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.26
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.26
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.26
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.26
PROPER(f(c)) → c12(F(ok(c))) 96.59/33.26
PROPER(f(true)) → c12(F(ok(true))) 96.59/33.26
PROPER(f(false)) → c12(F(ok(false))) 96.59/33.26
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0))) 96.59/33.26
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 96.59/33.26
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2)) 96.59/33.26
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 96.59/33.26
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.59/33.26
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 96.59/33.26
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 96.59/33.26
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.26
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.26
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.26
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.26
TOP(mark(f(z0))) → c17(TOP(f(proper(z0))), PROPER(f(z0))) 96.59/33.26
TOP(mark(if(z0, z1, z2))) → c17(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.26
TOP(mark(c)) → c17(TOP(ok(c)), PROPER(c)) 96.59/33.26
TOP(mark(true)) → c17(TOP(ok(true)), PROPER(true)) 96.59/33.26
TOP(mark(false)) → c17(TOP(ok(false)), PROPER(false))
S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.26
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.26
F(mark(z0)) → c7(F(z0)) 96.59/33.26
F(ok(z0)) → c8(F(z0)) 96.59/33.26
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.26
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.26
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.26
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.26
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.26
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.26
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.26
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.26
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.26
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.26
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.26
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.26
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.26
PROPER(f(c)) → c12(F(ok(c))) 96.59/33.26
PROPER(f(true)) → c12(F(ok(true))) 96.59/33.26
PROPER(f(false)) → c12(F(ok(false))) 96.59/33.26
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0))) 96.59/33.26
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 96.59/33.26
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2)) 96.59/33.26
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 96.59/33.26
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.59/33.26
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 96.59/33.26
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 96.59/33.26
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.26
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.26
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.26
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, TOP, PROPER

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c18, c1, c4, c12, c12, c13, c13, c17

96.59/33.26
96.59/33.26

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

Removed 3 trailing tuple parts
96.59/33.26
96.59/33.26

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.26
active(if(true, z0, z1)) → mark(z0) 96.59/33.26
active(if(false, z0, z1)) → mark(z1) 96.59/33.26
active(f(z0)) → f(active(z0)) 96.59/33.26
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.26
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.26
f(mark(z0)) → mark(f(z0)) 96.59/33.26
f(ok(z0)) → ok(f(z0)) 96.59/33.26
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.26
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.26
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.26
proper(f(z0)) → f(proper(z0)) 96.59/33.26
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.26
proper(c) → ok(c) 96.59/33.26
proper(true) → ok(true) 96.59/33.26
proper(false) → ok(false) 96.59/33.26
top(mark(z0)) → top(proper(z0)) 96.59/33.26
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.26
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.26
F(mark(z0)) → c7(F(z0)) 96.59/33.26
F(ok(z0)) → c8(F(z0)) 96.59/33.26
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.26
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.26
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.26
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.26
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.26
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.26
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.26
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.26
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.26
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.26
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.26
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.26
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.26
PROPER(f(c)) → c12(F(ok(c))) 96.59/33.26
PROPER(f(true)) → c12(F(ok(true))) 96.59/33.26
PROPER(f(false)) → c12(F(ok(false))) 96.59/33.26
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0))) 96.59/33.26
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 96.59/33.26
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2)) 96.59/33.26
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 96.59/33.26
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.59/33.26
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 96.59/33.26
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 96.59/33.26
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.26
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.26
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.26
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.26
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.27
TOP(mark(f(z0))) → c17(TOP(f(proper(z0))), PROPER(f(z0))) 96.59/33.27
TOP(mark(if(z0, z1, z2))) → c17(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.27
TOP(mark(c)) → c17(TOP(ok(c))) 96.59/33.27
TOP(mark(true)) → c17(TOP(ok(true))) 96.59/33.27
TOP(mark(false)) → c17(TOP(ok(false)))
S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.27
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.27
F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.27
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) 96.59/33.27
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.27
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.27
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.27
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.27
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.27
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.27
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.27
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.27
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.27
PROPER(f(c)) → c12(F(ok(c))) 96.59/33.27
PROPER(f(true)) → c12(F(ok(true))) 96.59/33.27
PROPER(f(false)) → c12(F(ok(false))) 96.59/33.27
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0))) 96.59/33.27
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 96.59/33.27
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2)) 96.59/33.27
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 96.59/33.27
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.59/33.27
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 96.59/33.27
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 96.59/33.27
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.27
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.27
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.27
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, TOP, PROPER

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c18, c1, c4, c12, c12, c13, c13, c17, c17

96.59/33.27
96.59/33.27

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

Use narrowing to replace TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(f(z0))) → c18(TOP(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.27
TOP(ok(if(true, z0, z1))) → c18(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.27
TOP(ok(if(false, z0, z1))) → c18(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.27
TOP(ok(f(z0))) → c18(TOP(f(active(z0))), ACTIVE(f(z0))) 96.59/33.27
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.27
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
96.59/33.27
96.59/33.27

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.27
active(if(true, z0, z1)) → mark(z0) 96.59/33.27
active(if(false, z0, z1)) → mark(z1) 96.59/33.27
active(f(z0)) → f(active(z0)) 96.59/33.27
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.27
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.27
f(mark(z0)) → mark(f(z0)) 96.59/33.27
f(ok(z0)) → ok(f(z0)) 96.59/33.27
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.27
proper(f(z0)) → f(proper(z0)) 96.59/33.27
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.27
proper(c) → ok(c) 96.59/33.27
proper(true) → ok(true) 96.59/33.27
proper(false) → ok(false) 96.59/33.27
top(mark(z0)) → top(proper(z0)) 96.59/33.27
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.27
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.27
F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.27
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.27
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.27
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.27
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.27
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.27
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.27
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.27
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.27
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.27
PROPER(f(c)) → c12(F(ok(c))) 96.59/33.27
PROPER(f(true)) → c12(F(ok(true))) 96.59/33.27
PROPER(f(false)) → c12(F(ok(false))) 96.59/33.27
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0))) 96.59/33.27
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 96.59/33.27
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2)) 96.59/33.27
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 96.59/33.27
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.59/33.27
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 96.59/33.27
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 96.59/33.27
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.27
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.27
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.27
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.27
TOP(mark(f(z0))) → c17(TOP(f(proper(z0))), PROPER(f(z0))) 96.59/33.27
TOP(mark(if(z0, z1, z2))) → c17(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.27
TOP(mark(c)) → c17(TOP(ok(c))) 96.59/33.27
TOP(mark(true)) → c17(TOP(ok(true))) 96.59/33.27
TOP(mark(false)) → c17(TOP(ok(false))) 96.59/33.27
TOP(ok(f(z0))) → c18(TOP(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.27
TOP(ok(if(true, z0, z1))) → c18(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.27
TOP(ok(if(false, z0, z1))) → c18(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.27
TOP(ok(f(z0))) → c18(TOP(f(active(z0))), ACTIVE(f(z0))) 96.59/33.27
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.27
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.27
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.27
F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.27
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.27
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.27
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.27
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.27
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.27
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.27
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.27
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.27
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.27
PROPER(f(c)) → c12(F(ok(c))) 96.59/33.27
PROPER(f(true)) → c12(F(ok(true))) 96.59/33.27
PROPER(f(false)) → c12(F(ok(false))) 96.59/33.27
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0))) 96.59/33.27
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 96.59/33.27
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2)) 96.59/33.27
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 96.59/33.27
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.59/33.27
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 96.59/33.27
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 96.59/33.27
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.27
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.27
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.27
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.27
TOP(ok(f(z0))) → c18(TOP(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.27
TOP(ok(if(true, z0, z1))) → c18(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.27
TOP(ok(if(false, z0, z1))) → c18(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.27
TOP(ok(f(z0))) → c18(TOP(f(active(z0))), ACTIVE(f(z0))) 96.59/33.27
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.27
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c1, c4, c12, c12, c13, c13, c17, c17, c18

96.59/33.27
96.59/33.27

(23) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0)) 96.59/33.27
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1)) 96.59/33.27
ACTIVE(f(z0)) → c1(IF(z0, c, f(true))) 96.59/33.27
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.27
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.27
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.27
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0))) 96.59/33.27
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.27
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.27
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0))) 96.59/33.27
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.27
PROPER(f(c)) → c12(F(ok(c))) 96.59/33.27
PROPER(f(true)) → c12(F(ok(true))) 96.59/33.27
PROPER(f(false)) → c12(F(ok(false))) 96.59/33.27
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0))) 96.59/33.27
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2))) 96.59/33.27
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2)) 96.59/33.27
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2)) 96.59/33.27
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1)) 96.59/33.27
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1)) 96.59/33.27
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1)) 96.59/33.27
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.27
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.27
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2)) 96.59/33.27
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.27
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2)) 96.59/33.27
TOP(mark(f(z0))) → c17(TOP(f(proper(z0))), PROPER(f(z0))) 96.59/33.27
TOP(mark(if(z0, z1, z2))) → c17(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2))) 96.59/33.27
TOP(ok(f(z0))) → c18(TOP(mark(if(z0, c, f(true)))), ACTIVE(f(z0))) 96.59/33.27
TOP(ok(if(true, z0, z1))) → c18(TOP(mark(z0)), ACTIVE(if(true, z0, z1))) 96.59/33.27
TOP(ok(if(false, z0, z1))) → c18(TOP(mark(z1)), ACTIVE(if(false, z0, z1))) 96.59/33.27
TOP(ok(f(z0))) → c18(TOP(f(active(z0))), ACTIVE(f(z0))) 96.59/33.27
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2))) 96.59/33.27
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
96.59/33.27
96.59/33.27

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.27
active(if(true, z0, z1)) → mark(z0) 96.59/33.27
active(if(false, z0, z1)) → mark(z1) 96.59/33.27
active(f(z0)) → f(active(z0)) 96.59/33.27
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.27
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.27
f(mark(z0)) → mark(f(z0)) 96.59/33.27
f(ok(z0)) → ok(f(z0)) 96.59/33.27
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.27
proper(f(z0)) → f(proper(z0)) 96.59/33.27
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.27
proper(c) → ok(c) 96.59/33.27
proper(true) → ok(true) 96.59/33.27
proper(false) → ok(false) 96.59/33.27
top(mark(z0)) → top(proper(z0)) 96.59/33.27
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.27
TOP(mark(c)) → c17(TOP(ok(c))) 96.59/33.27
TOP(mark(true)) → c17(TOP(ok(true))) 96.59/33.27
TOP(mark(false)) → c17(TOP(ok(false)))
S tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

F, IF, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c17

96.59/33.27
96.59/33.27

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

Removed 3 trailing tuple parts
96.59/33.27
96.59/33.27

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.27
active(if(true, z0, z1)) → mark(z0) 96.59/33.27
active(if(false, z0, z1)) → mark(z1) 96.59/33.27
active(f(z0)) → f(active(z0)) 96.59/33.27
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.27
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.27
f(mark(z0)) → mark(f(z0)) 96.59/33.27
f(ok(z0)) → ok(f(z0)) 96.59/33.27
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.27
proper(f(z0)) → f(proper(z0)) 96.59/33.27
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.27
proper(c) → ok(c) 96.59/33.27
proper(true) → ok(true) 96.59/33.27
proper(false) → ok(false) 96.59/33.27
top(mark(z0)) → top(proper(z0)) 96.59/33.27
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2)) 96.59/33.27
TOP(mark(c)) → c17 96.59/33.27
TOP(mark(true)) → c17 96.59/33.27
TOP(mark(false)) → c17
S tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

F, IF, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c17

96.59/33.27
96.59/33.27

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

Removed 3 trailing nodes:

TOP(mark(true)) → c17 96.59/33.27
TOP(mark(c)) → c17 96.59/33.27
TOP(mark(false)) → c17
96.59/33.27
96.59/33.27

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.27
active(if(true, z0, z1)) → mark(z0) 96.59/33.27
active(if(false, z0, z1)) → mark(z1) 96.59/33.27
active(f(z0)) → f(active(z0)) 96.59/33.27
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.27
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.27
f(mark(z0)) → mark(f(z0)) 96.59/33.27
f(ok(z0)) → ok(f(z0)) 96.59/33.27
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.27
proper(f(z0)) → f(proper(z0)) 96.59/33.27
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.27
proper(c) → ok(c) 96.59/33.27
proper(true) → ok(true) 96.59/33.27
proper(false) → ok(false) 96.59/33.27
top(mark(z0)) → top(proper(z0)) 96.59/33.27
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
S tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

F, IF

Compound Symbols:

c7, c8, c9, c10, c11

96.59/33.27
96.59/33.27

(29) 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) → c9(IF(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 96.59/33.27

POL(F(x1)) = 0    96.59/33.27
POL(IF(x1, x2, x3)) = x1    96.59/33.27
POL(c10(x1)) = x1    96.59/33.27
POL(c11(x1)) = x1    96.59/33.27
POL(c7(x1)) = x1    96.59/33.27
POL(c8(x1)) = x1    96.59/33.27
POL(c9(x1)) = x1    96.59/33.27
POL(mark(x1)) = [1] + x1    96.59/33.27
POL(ok(x1)) = x1   
96.59/33.27
96.59/33.27

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.27
active(if(true, z0, z1)) → mark(z0) 96.59/33.27
active(if(false, z0, z1)) → mark(z1) 96.59/33.27
active(f(z0)) → f(active(z0)) 96.59/33.27
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.27
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.27
f(mark(z0)) → mark(f(z0)) 96.59/33.27
f(ok(z0)) → ok(f(z0)) 96.59/33.27
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.27
proper(f(z0)) → f(proper(z0)) 96.59/33.27
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.27
proper(c) → ok(c) 96.59/33.27
proper(true) → ok(true) 96.59/33.27
proper(false) → ok(false) 96.59/33.27
top(mark(z0)) → top(proper(z0)) 96.59/33.27
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
S tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
K tuples:

IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

F, IF

Compound Symbols:

c7, c8, c9, c10, c11

96.59/33.27
96.59/33.27

(31) 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.

F(mark(z0)) → c7(F(z0))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 96.59/33.27

POL(F(x1)) = [2]x1    96.59/33.27
POL(IF(x1, x2, x3)) = x1    96.59/33.27
POL(c10(x1)) = x1    96.59/33.27
POL(c11(x1)) = x1    96.59/33.27
POL(c7(x1)) = x1    96.59/33.27
POL(c8(x1)) = x1    96.59/33.27
POL(c9(x1)) = x1    96.59/33.27
POL(mark(x1)) = [1] + x1    96.59/33.27
POL(ok(x1)) = x1   
96.59/33.27
96.59/33.27

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.27
active(if(true, z0, z1)) → mark(z0) 96.59/33.27
active(if(false, z0, z1)) → mark(z1) 96.59/33.27
active(f(z0)) → f(active(z0)) 96.59/33.27
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.27
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.27
f(mark(z0)) → mark(f(z0)) 96.59/33.27
f(ok(z0)) → ok(f(z0)) 96.59/33.27
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.27
proper(f(z0)) → f(proper(z0)) 96.59/33.27
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.27
proper(c) → ok(c) 96.59/33.27
proper(true) → ok(true) 96.59/33.27
proper(false) → ok(false) 96.59/33.27
top(mark(z0)) → top(proper(z0)) 96.59/33.27
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
S tuples:

F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
K tuples:

IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
F(mark(z0)) → c7(F(z0))
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

F, IF

Compound Symbols:

c7, c8, c9, c10, c11

96.59/33.27
96.59/33.27

(33) 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.

F(ok(z0)) → c8(F(z0))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 96.59/33.27

POL(F(x1)) = [2]x1    96.59/33.27
POL(IF(x1, x2, x3)) = 0    96.59/33.27
POL(c10(x1)) = x1    96.59/33.27
POL(c11(x1)) = x1    96.59/33.27
POL(c7(x1)) = x1    96.59/33.27
POL(c8(x1)) = x1    96.59/33.27
POL(c9(x1)) = x1    96.59/33.27
POL(mark(x1)) = [1] + x1    96.59/33.27
POL(ok(x1)) = [1] + x1   
96.59/33.27
96.59/33.27

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.27
active(if(true, z0, z1)) → mark(z0) 96.59/33.27
active(if(false, z0, z1)) → mark(z1) 96.59/33.27
active(f(z0)) → f(active(z0)) 96.59/33.27
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.27
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.27
f(mark(z0)) → mark(f(z0)) 96.59/33.27
f(ok(z0)) → ok(f(z0)) 96.59/33.27
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.27
proper(f(z0)) → f(proper(z0)) 96.59/33.27
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.27
proper(c) → ok(c) 96.59/33.27
proper(true) → ok(true) 96.59/33.27
proper(false) → ok(false) 96.59/33.27
top(mark(z0)) → top(proper(z0)) 96.59/33.27
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
S tuples:

IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
K tuples:

IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0))
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

F, IF

Compound Symbols:

c7, c8, c9, c10, c11

96.59/33.27
96.59/33.27

(35) 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(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 96.59/33.27

POL(F(x1)) = [3]x1    96.59/33.27
POL(IF(x1, x2, x3)) = [4]x2    96.59/33.27
POL(c10(x1)) = x1    96.59/33.27
POL(c11(x1)) = x1    96.59/33.27
POL(c7(x1)) = x1    96.59/33.27
POL(c8(x1)) = x1    96.59/33.27
POL(c9(x1)) = x1    96.59/33.27
POL(mark(x1)) = [4] + x1    96.59/33.27
POL(ok(x1)) = x1   
96.59/33.27
96.59/33.27

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.27
active(if(true, z0, z1)) → mark(z0) 96.59/33.27
active(if(false, z0, z1)) → mark(z1) 96.59/33.27
active(f(z0)) → f(active(z0)) 96.59/33.27
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.27
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.27
f(mark(z0)) → mark(f(z0)) 96.59/33.27
f(ok(z0)) → ok(f(z0)) 96.59/33.27
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.27
proper(f(z0)) → f(proper(z0)) 96.59/33.27
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.27
proper(c) → ok(c) 96.59/33.27
proper(true) → ok(true) 96.59/33.27
proper(false) → ok(false) 96.59/33.27
top(mark(z0)) → top(proper(z0)) 96.59/33.27
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
S tuples:

IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
K tuples:

IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

F, IF

Compound Symbols:

c7, c8, c9, c10, c11

96.59/33.27
96.59/33.27

(37) 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)) → c11(IF(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation : 96.59/33.27

POL(F(x1)) = [5]x1    96.59/33.27
POL(IF(x1, x2, x3)) = x3    96.59/33.27
POL(c10(x1)) = x1    96.59/33.27
POL(c11(x1)) = x1    96.59/33.27
POL(c7(x1)) = x1    96.59/33.27
POL(c8(x1)) = x1    96.59/33.27
POL(c9(x1)) = x1    96.59/33.27
POL(mark(x1)) = x1    96.59/33.27
POL(ok(x1)) = [2] + x1   
96.59/33.27
96.59/33.27

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0)) → mark(if(z0, c, f(true))) 96.59/33.27
active(if(true, z0, z1)) → mark(z0) 96.59/33.27
active(if(false, z0, z1)) → mark(z1) 96.59/33.27
active(f(z0)) → f(active(z0)) 96.59/33.27
active(if(z0, z1, z2)) → if(active(z0), z1, z2) 96.59/33.27
active(if(z0, z1, z2)) → if(z0, active(z1), z2) 96.59/33.27
f(mark(z0)) → mark(f(z0)) 96.59/33.27
f(ok(z0)) → ok(f(z0)) 96.59/33.27
if(mark(z0), z1, z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(z0, mark(z1), z2) → mark(if(z0, z1, z2)) 96.59/33.27
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2)) 96.59/33.27
proper(f(z0)) → f(proper(z0)) 96.59/33.27
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2)) 96.59/33.27
proper(c) → ok(c) 96.59/33.27
proper(true) → ok(true) 96.59/33.27
proper(false) → ok(false) 96.59/33.27
top(mark(z0)) → top(proper(z0)) 96.59/33.27
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
S tuples:none
K tuples:

IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2)) 96.59/33.27
F(mark(z0)) → c7(F(z0)) 96.59/33.27
F(ok(z0)) → c8(F(z0)) 96.59/33.27
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2)) 96.59/33.27
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

F, IF

Compound Symbols:

c7, c8, c9, c10, c11

96.59/33.27
96.59/33.27

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

The set S is empty
96.59/33.27
96.59/33.27

(40) BOUNDS(O(1), O(1))

96.59/33.27
96.59/33.27
96.96/33.34 EOF