YES(O(1), O(n^3)) 16.62/5.45 YES(O(1), O(n^3)) 16.62/5.47 16.62/5.47 16.62/5.47 16.62/5.47 16.62/5.47 16.62/5.47 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 16.62/5.47 16.62/5.47 16.62/5.47
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^3)).

0 CpxTRS
16.62/5.47 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
16.62/5.47 
↳2 CdtProblem
16.62/5.47 
↳3 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
16.62/5.47 
↳4 CdtProblem
16.62/5.47 
↳5 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
16.62/5.47 
↳6 CdtProblem
16.62/5.47 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
16.62/5.47 
↳8 CdtProblem
16.62/5.47 
↳9 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
16.62/5.47 
↳10 CdtProblem
16.62/5.47 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))))
16.62/5.47 
↳12 CdtProblem
16.62/5.47 
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID))
16.62/5.47 
↳14 BOUNDS(O(1), O(1))
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16.62/5.47

(0) Obligation:

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

le(0, y) → true 16.62/5.47
le(s(x), 0) → false 16.62/5.47
le(s(x), s(y)) → le(x, y) 16.62/5.47
minus(0, y) → 0 16.62/5.47
minus(s(x), y) → if_minus(le(s(x), y), s(x), y) 16.62/5.47
if_minus(true, s(x), y) → 0 16.62/5.47
if_minus(false, s(x), y) → s(minus(x, y)) 16.62/5.47
mod(0, y) → 0 16.62/5.47
mod(s(x), 0) → 0 16.62/5.47
mod(s(x), s(y)) → if_mod(le(y, x), s(x), s(y)) 16.62/5.47
if_mod(true, s(x), s(y)) → mod(minus(x, y), s(y)) 16.62/5.47
if_mod(false, s(x), s(y)) → s(x)

Rewrite Strategy: INNERMOST
16.62/5.47
16.62/5.47

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

Converted CpxTRS to CDT
16.62/5.47
16.62/5.47

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 16.62/5.47
le(s(z0), 0) → false 16.62/5.47
le(s(z0), s(z1)) → le(z0, z1) 16.62/5.47
minus(0, z0) → 0 16.62/5.47
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 16.62/5.47
if_minus(true, s(z0), z1) → 0 16.62/5.47
if_minus(false, s(z0), z1) → s(minus(z0, z1)) 16.62/5.48
mod(0, z0) → 0 16.62/5.48
mod(s(z0), 0) → 0 16.62/5.48
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1)) 16.62/5.48
if_mod(true, s(z0), s(z1)) → mod(minus(z0, z1), s(z1)) 16.62/5.48
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 16.62/5.48
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 16.62/5.48
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 16.62/5.48
MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 16.62/5.48
IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 16.62/5.48
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 16.62/5.48
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 16.62/5.48
MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 16.62/5.48
IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
K tuples:none
Defined Rule Symbols:

le, minus, if_minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, IF_MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c6, c9, c10

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16.62/5.48

(3) 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_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
We considered the (Usable) Rules:

minus(0, z0) → 0 16.62/5.48
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 16.62/5.49
le(s(z0), 0) → false 16.62/5.49
le(s(z0), s(z1)) → le(z0, z1) 16.62/5.49
le(0, z0) → true 16.62/5.49
if_minus(true, s(z0), z1) → 0 16.62/5.49
if_minus(false, s(z0), z1) → s(minus(z0, z1))
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 16.62/5.49
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 16.62/5.49
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 16.62/5.49
MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 16.62/5.49
IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 16.62/5.49

POL(0) = 0    16.62/5.49
POL(IF_MINUS(x1, x2, x3)) = 0    16.62/5.49
POL(IF_MOD(x1, x2, x3)) = [4]x2    16.62/5.49
POL(LE(x1, x2)) = 0    16.62/5.49
POL(MINUS(x1, x2)) = 0    16.62/5.49
POL(MOD(x1, x2)) = [4]x1    16.62/5.49
POL(c10(x1, x2)) = x1 + x2    16.62/5.49
POL(c2(x1)) = x1    16.62/5.49
POL(c4(x1, x2)) = x1 + x2    16.62/5.49
POL(c6(x1)) = x1    16.62/5.49
POL(c9(x1, x2)) = x1 + x2    16.62/5.49
POL(false) = 0    16.62/5.49
POL(if_minus(x1, x2, x3)) = x2    16.62/5.49
POL(le(x1, x2)) = 0    16.62/5.49
POL(minus(x1, x2)) = x1    16.62/5.49
POL(s(x1)) = [2] + x1    16.62/5.49
POL(true) = 0   
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16.62/5.49

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 16.62/5.49
le(s(z0), 0) → false 16.62/5.49
le(s(z0), s(z1)) → le(z0, z1) 16.62/5.49
minus(0, z0) → 0 16.62/5.49
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 16.62/5.49
if_minus(true, s(z0), z1) → 0 16.62/5.49
if_minus(false, s(z0), z1) → s(minus(z0, z1)) 16.62/5.49
mod(0, z0) → 0 16.62/5.49
mod(s(z0), 0) → 0 16.62/5.49
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1)) 16.62/5.49
if_mod(true, s(z0), s(z1)) → mod(minus(z0, z1), s(z1)) 16.62/5.49
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 16.62/5.49
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 16.62/5.49
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 16.62/5.49
MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 16.62/5.49
IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 16.62/5.49
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 16.62/5.49
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 16.62/5.49
MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
K tuples:

IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
Defined Rule Symbols:

le, minus, if_minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, IF_MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c6, c9, c10

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16.62/5.49

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

The following tuples could be moved from S to K by knowledge propagation:

MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 16.62/5.49
IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
16.62/5.49
16.62/5.49

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 16.62/5.49
le(s(z0), 0) → false 16.62/5.49
le(s(z0), s(z1)) → le(z0, z1) 16.62/5.49
minus(0, z0) → 0 16.62/5.49
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 16.62/5.49
if_minus(true, s(z0), z1) → 0 16.62/5.49
if_minus(false, s(z0), z1) → s(minus(z0, z1)) 16.62/5.49
mod(0, z0) → 0 16.62/5.49
mod(s(z0), 0) → 0 16.62/5.49
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1)) 16.62/5.49
if_mod(true, s(z0), s(z1)) → mod(minus(z0, z1), s(z1)) 16.62/5.49
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 16.62/5.49
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 16.62/5.49
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 16.62/5.49
MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 16.62/5.49
IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 16.62/5.49
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 16.62/5.49
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
K tuples:

IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 16.62/5.49
MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0))
Defined Rule Symbols:

le, minus, if_minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, IF_MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c6, c9, c10

16.62/5.49
16.62/5.49

(7) 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.

IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
We considered the (Usable) Rules:

minus(0, z0) → 0 16.62/5.49
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 16.62/5.49
le(s(z0), 0) → false 16.62/5.49
le(s(z0), s(z1)) → le(z0, z1) 16.62/5.49
le(0, z0) → true 16.62/5.49
if_minus(true, s(z0), z1) → 0 16.62/5.49
if_minus(false, s(z0), z1) → s(minus(z0, z1))
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 16.62/5.49
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 16.62/5.49
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 16.62/5.49
MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 16.62/5.49
IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 16.62/5.49

POL(0) = 0    16.62/5.49
POL(IF_MINUS(x1, x2, x3)) = x2    16.62/5.49
POL(IF_MOD(x1, x2, x3)) = [2]x22    16.62/5.49
POL(LE(x1, x2)) = 0    16.62/5.49
POL(MINUS(x1, x2)) = x1    16.62/5.49
POL(MOD(x1, x2)) = [2]x1 + [2]x12    16.62/5.49
POL(c10(x1, x2)) = x1 + x2    16.62/5.49
POL(c2(x1)) = x1    16.62/5.49
POL(c4(x1, x2)) = x1 + x2    16.62/5.49
POL(c6(x1)) = x1    16.62/5.49
POL(c9(x1, x2)) = x1 + x2    16.62/5.49
POL(false) = 0    16.62/5.49
POL(if_minus(x1, x2, x3)) = x2    16.62/5.49
POL(le(x1, x2)) = 0    16.62/5.49
POL(minus(x1, x2)) = x1    16.62/5.49
POL(s(x1)) = [2] + x1    16.62/5.49
POL(true) = 0   
16.62/5.49
16.62/5.49

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 16.62/5.49
le(s(z0), 0) → false 16.62/5.49
le(s(z0), s(z1)) → le(z0, z1) 16.62/5.49
minus(0, z0) → 0 16.62/5.49
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 16.62/5.49
if_minus(true, s(z0), z1) → 0 16.62/5.49
if_minus(false, s(z0), z1) → s(minus(z0, z1)) 16.62/5.49
mod(0, z0) → 0 16.62/5.49
mod(s(z0), 0) → 0 16.62/5.49
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1)) 16.62/5.49
if_mod(true, s(z0), s(z1)) → mod(minus(z0, z1), s(z1)) 16.62/5.49
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 16.62/5.49
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 16.62/5.49
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 16.62/5.49
MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 16.62/5.49
IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 16.62/5.49
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
K tuples:

IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 16.62/5.49
MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 16.62/5.49
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
Defined Rule Symbols:

le, minus, if_minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, IF_MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c6, c9, c10

16.62/5.49
16.62/5.49

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

The following tuples could be moved from S to K by knowledge propagation:

MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 16.62/5.49
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1))
16.62/5.49
16.62/5.49

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 16.62/5.49
le(s(z0), 0) → false 16.62/5.49
le(s(z0), s(z1)) → le(z0, z1) 16.62/5.49
minus(0, z0) → 0 16.62/5.49
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 16.62/5.49
if_minus(true, s(z0), z1) → 0 16.62/5.49
if_minus(false, s(z0), z1) → s(minus(z0, z1)) 16.62/5.49
mod(0, z0) → 0 16.62/5.49
mod(s(z0), 0) → 0 16.62/5.49
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1)) 16.62/5.49
if_mod(true, s(z0), s(z1)) → mod(minus(z0, z1), s(z1)) 16.62/5.49
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 16.62/5.49
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 16.62/5.49
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 16.62/5.49
MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 16.62/5.49
IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
S tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1))
K tuples:

IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 16.62/5.49
MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 16.62/5.49
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 16.62/5.49
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1))
Defined Rule Symbols:

le, minus, if_minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, IF_MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c6, c9, c10

16.62/5.49
16.62/5.49

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))) transformation)

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

LE(s(z0), s(z1)) → c2(LE(z0, z1))
We considered the (Usable) Rules:

minus(0, z0) → 0 16.62/5.49
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 16.62/5.49
le(s(z0), 0) → false 16.62/5.49
le(s(z0), s(z1)) → le(z0, z1) 16.62/5.49
le(0, z0) → true 16.62/5.49
if_minus(true, s(z0), z1) → 0 16.62/5.49
if_minus(false, s(z0), z1) → s(minus(z0, z1))
And the Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 16.62/5.49
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 16.62/5.49
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 16.62/5.49
MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 16.62/5.49
IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 16.62/5.49

POL(0) = 0    16.62/5.49
POL(IF_MINUS(x1, x2, x3)) = x22    16.62/5.49
POL(IF_MOD(x1, x2, x3)) = x2 + x2·x3 + x23    16.62/5.49
POL(LE(x1, x2)) = [1] + x1    16.62/5.49
POL(MINUS(x1, x2)) = [1] + x1 + x12    16.62/5.49
POL(MOD(x1, x2)) = x1 + x2 + x1·x2 + x13    16.62/5.49
POL(c10(x1, x2)) = x1 + x2    16.62/5.49
POL(c2(x1)) = x1    16.62/5.49
POL(c4(x1, x2)) = x1 + x2    16.62/5.49
POL(c6(x1)) = x1    16.62/5.49
POL(c9(x1, x2)) = x1 + x2    16.62/5.49
POL(false) = 0    16.62/5.49
POL(if_minus(x1, x2, x3)) = x2    16.62/5.49
POL(le(x1, x2)) = 0    16.62/5.49
POL(minus(x1, x2)) = x1    16.62/5.49
POL(s(x1)) = [1] + x1    16.62/5.49
POL(true) = 0   
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(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

le(0, z0) → true 16.62/5.49
le(s(z0), 0) → false 16.62/5.49
le(s(z0), s(z1)) → le(z0, z1) 16.62/5.49
minus(0, z0) → 0 16.62/5.49
minus(s(z0), z1) → if_minus(le(s(z0), z1), s(z0), z1) 16.62/5.49
if_minus(true, s(z0), z1) → 0 16.62/5.49
if_minus(false, s(z0), z1) → s(minus(z0, z1)) 16.62/5.49
mod(0, z0) → 0 16.62/5.49
mod(s(z0), 0) → 0 16.62/5.49
mod(s(z0), s(z1)) → if_mod(le(z1, z0), s(z0), s(z1)) 16.62/5.49
if_mod(true, s(z0), s(z1)) → mod(minus(z0, z1), s(z1)) 16.62/5.49
if_mod(false, s(z0), s(z1)) → s(z0)
Tuples:

LE(s(z0), s(z1)) → c2(LE(z0, z1)) 16.62/5.49
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 16.62/5.49
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 16.62/5.49
MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 16.62/5.49
IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1))
S tuples:none
K tuples:

IF_MOD(true, s(z0), s(z1)) → c10(MOD(minus(z0, z1), s(z1)), MINUS(z0, z1)) 16.62/5.49
MOD(s(z0), s(z1)) → c9(IF_MOD(le(z1, z0), s(z0), s(z1)), LE(z1, z0)) 16.62/5.49
IF_MINUS(false, s(z0), z1) → c6(MINUS(z0, z1)) 16.62/5.49
MINUS(s(z0), z1) → c4(IF_MINUS(le(s(z0), z1), s(z0), z1), LE(s(z0), z1)) 16.62/5.49
LE(s(z0), s(z1)) → c2(LE(z0, z1))
Defined Rule Symbols:

le, minus, if_minus, mod, if_mod

Defined Pair Symbols:

LE, MINUS, IF_MINUS, MOD, IF_MOD

Compound Symbols:

c2, c4, c6, c9, c10

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

The set S is empty
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(14) BOUNDS(O(1), O(1))

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16.91/5.56 EOF