The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
    ↳3 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳4 CdtProblem
    ↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 CdtProblem
    ↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳8 CdtProblem
    ↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳10 CdtProblem
    ↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 7 ms)
    ↳12 CdtProblem
    ↳13 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳14 CdtProblem
    ↳15 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳16 CdtProblem
    ↳17 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 12 ms)
    ↳18 CdtProblem
    ↳19 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳20 CdtProblem
↳21 CpxTrsMatchBoundsTAProof (⇔, 14 ms)
↳22 BOUNDS(O(1), O(n^1))

(0) Obligation:

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

+(*(x, y), *(x, z)) → *(x, +(y, z))
+(+(x, y), z) → +(x, +(y, z))
+(*(x, y), +(*(x, z), u)) → +(*(x, +(y, z)), u)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2))
+(+(z0, z1), z2) → +(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(*(z0, +(z1, z2)), u), +'(z1, z2))
S tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(*(z0, +(z1, z2)), u), +'(z1, z2))
K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2

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

Use narrowing to replace +'(*(z0, z1), +(*(z0, z2), u)) → c2(+'(*(z0, +(z1, z2)), u), +'(z1, z2)) by

+'(*(x0, *(z0, z1)), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, *(z0, +(z1, z2))), u), +'(*(z0, z1), *(z0, z2)))
+'(*(x0, +(z0, z1)), +(*(x0, z2), u)) → c2(+'(*(x0, +(z0, +(z1, z2))), u), +'(+(z0, z1), z2))
+'(*(x0, *(z0, z1)), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(*(z0, +(z1, z2)), u)), u), +'(*(z0, z1), +(*(z0, z2), u)))
+'(*(x0, x1), +(*(x0, x2), u)) → c2

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2))
+(+(z0, z1), z2) → +(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(z0, z1)), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, *(z0, +(z1, z2))), u), +'(*(z0, z1), *(z0, z2)))
+'(*(x0, +(z0, z1)), +(*(x0, z2), u)) → c2(+'(*(x0, +(z0, +(z1, z2))), u), +'(+(z0, z1), z2))
+'(*(x0, *(z0, z1)), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(*(z0, +(z1, z2)), u)), u), +'(*(z0, z1), +(*(z0, z2), u)))
+'(*(x0, x1), +(*(x0, x2), u)) → c2
S tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(z0, z1)), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, *(z0, +(z1, z2))), u), +'(*(z0, z1), *(z0, z2)))
+'(*(x0, +(z0, z1)), +(*(x0, z2), u)) → c2(+'(*(x0, +(z0, +(z1, z2))), u), +'(+(z0, z1), z2))
+'(*(x0, *(z0, z1)), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(*(z0, +(z1, z2)), u)), u), +'(*(z0, z1), +(*(z0, z2), u)))
+'(*(x0, x1), +(*(x0, x2), u)) → c2
K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2, c2

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

Removed 1 trailing nodes:

+'(*(x0, x1), +(*(x0, x2), u)) → c2

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2))
+(+(z0, z1), z2) → +(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(z0, z1)), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, *(z0, +(z1, z2))), u), +'(*(z0, z1), *(z0, z2)))
+'(*(x0, +(z0, z1)), +(*(x0, z2), u)) → c2(+'(*(x0, +(z0, +(z1, z2))), u), +'(+(z0, z1), z2))
+'(*(x0, *(z0, z1)), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(*(z0, +(z1, z2)), u)), u), +'(*(z0, z1), +(*(z0, z2), u)))
S tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(z0, z1)), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, *(z0, +(z1, z2))), u), +'(*(z0, z1), *(z0, z2)))
+'(*(x0, +(z0, z1)), +(*(x0, z2), u)) → c2(+'(*(x0, +(z0, +(z1, z2))), u), +'(+(z0, z1), z2))
+'(*(x0, *(z0, z1)), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(*(z0, +(z1, z2)), u)), u), +'(*(z0, z1), +(*(z0, z2), u)))
K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2

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

Use narrowing to replace +'(*(x0, *(z0, z1)), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, *(z0, +(z1, z2))), u), +'(*(z0, z1), *(z0, z2))) by

+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, *(z0, z2))), u)) → c2(+'(*(x0, *(x1, *(z0, +(z1, z2)))), u), +'(*(x1, *(z0, z1)), *(x1, *(z0, z2))))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, *(x1, z2)), u)) → c2(+'(*(x0, *(x1, +(z0, +(z1, z2)))), u), +'(*(x1, +(z0, z1)), *(x1, z2)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, +(*(z0, z2), u))), u)) → c2(+'(*(x0, *(x1, +(*(z0, +(z1, z2)), u))), u), +'(*(x1, *(z0, z1)), *(x1, +(*(z0, z2), u))))
+'(*(x0, *(x1, x2)), +(*(x0, *(x1, x3)), u)) → c2(+'(*(x1, x2), *(x1, x3)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2))
+(+(z0, z1), z2) → +(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, +(z0, z1)), +(*(x0, z2), u)) → c2(+'(*(x0, +(z0, +(z1, z2))), u), +'(+(z0, z1), z2))
+'(*(x0, *(z0, z1)), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(*(z0, +(z1, z2)), u)), u), +'(*(z0, z1), +(*(z0, z2), u)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, *(z0, z2))), u)) → c2(+'(*(x0, *(x1, *(z0, +(z1, z2)))), u), +'(*(x1, *(z0, z1)), *(x1, *(z0, z2))))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, *(x1, z2)), u)) → c2(+'(*(x0, *(x1, +(z0, +(z1, z2)))), u), +'(*(x1, +(z0, z1)), *(x1, z2)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, +(*(z0, z2), u))), u)) → c2(+'(*(x0, *(x1, +(*(z0, +(z1, z2)), u))), u), +'(*(x1, *(z0, z1)), *(x1, +(*(z0, z2), u))))
+'(*(x0, *(x1, x2)), +(*(x0, *(x1, x3)), u)) → c2(+'(*(x1, x2), *(x1, x3)))
S tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, +(z0, z1)), +(*(x0, z2), u)) → c2(+'(*(x0, +(z0, +(z1, z2))), u), +'(+(z0, z1), z2))
+'(*(x0, *(z0, z1)), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(*(z0, +(z1, z2)), u)), u), +'(*(z0, z1), +(*(z0, z2), u)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, *(z0, z2))), u)) → c2(+'(*(x0, *(x1, *(z0, +(z1, z2)))), u), +'(*(x1, *(z0, z1)), *(x1, *(z0, z2))))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, *(x1, z2)), u)) → c2(+'(*(x0, *(x1, +(z0, +(z1, z2)))), u), +'(*(x1, +(z0, z1)), *(x1, z2)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, +(*(z0, z2), u))), u)) → c2(+'(*(x0, *(x1, +(*(z0, +(z1, z2)), u))), u), +'(*(x1, *(z0, z1)), *(x1, +(*(z0, z2), u))))
+'(*(x0, *(x1, x2)), +(*(x0, *(x1, x3)), u)) → c2(+'(*(x1, x2), *(x1, x3)))
K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2, c2

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

Use narrowing to replace +'(*(x0, +(z0, z1)), +(*(x0, z2), u)) → c2(+'(*(x0, +(z0, +(z1, z2))), u), +'(+(z0, z1), z2)) by

+'(*(x0, +(x1, *(z0, z1))), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, +(x1, *(z0, +(z1, z2)))), u), +'(+(x1, *(z0, z1)), *(z0, z2)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(x1, +(*(z0, +(z1, z2)), u))), u), +'(+(x1, *(z0, z1)), +(*(z0, z2), u)))
+'(*(x0, +(x1, +(z0, z1))), +(*(x0, z2), u)) → c2(+'(*(x0, +(x1, +(z0, +(z1, z2)))), u), +'(+(x1, +(z0, z1)), z2))
+'(*(x0, +(x1, x2)), +(*(x0, x3), u)) → c2(+'(+(x1, x2), x3))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2))
+(+(z0, z1), z2) → +(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(z0, z1)), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(*(z0, +(z1, z2)), u)), u), +'(*(z0, z1), +(*(z0, z2), u)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, *(z0, z2))), u)) → c2(+'(*(x0, *(x1, *(z0, +(z1, z2)))), u), +'(*(x1, *(z0, z1)), *(x1, *(z0, z2))))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, *(x1, z2)), u)) → c2(+'(*(x0, *(x1, +(z0, +(z1, z2)))), u), +'(*(x1, +(z0, z1)), *(x1, z2)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, +(*(z0, z2), u))), u)) → c2(+'(*(x0, *(x1, +(*(z0, +(z1, z2)), u))), u), +'(*(x1, *(z0, z1)), *(x1, +(*(z0, z2), u))))
+'(*(x0, *(x1, x2)), +(*(x0, *(x1, x3)), u)) → c2(+'(*(x1, x2), *(x1, x3)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, +(x1, *(z0, +(z1, z2)))), u), +'(+(x1, *(z0, z1)), *(z0, z2)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(x1, +(*(z0, +(z1, z2)), u))), u), +'(+(x1, *(z0, z1)), +(*(z0, z2), u)))
+'(*(x0, +(x1, +(z0, z1))), +(*(x0, z2), u)) → c2(+'(*(x0, +(x1, +(z0, +(z1, z2)))), u), +'(+(x1, +(z0, z1)), z2))
+'(*(x0, +(x1, x2)), +(*(x0, x3), u)) → c2(+'(+(x1, x2), x3))
S tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(z0, z1)), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(*(z0, +(z1, z2)), u)), u), +'(*(z0, z1), +(*(z0, z2), u)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, *(z0, z2))), u)) → c2(+'(*(x0, *(x1, *(z0, +(z1, z2)))), u), +'(*(x1, *(z0, z1)), *(x1, *(z0, z2))))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, *(x1, z2)), u)) → c2(+'(*(x0, *(x1, +(z0, +(z1, z2)))), u), +'(*(x1, +(z0, z1)), *(x1, z2)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, +(*(z0, z2), u))), u)) → c2(+'(*(x0, *(x1, +(*(z0, +(z1, z2)), u))), u), +'(*(x1, *(z0, z1)), *(x1, +(*(z0, z2), u))))
+'(*(x0, *(x1, x2)), +(*(x0, *(x1, x3)), u)) → c2(+'(*(x1, x2), *(x1, x3)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, +(x1, *(z0, +(z1, z2)))), u), +'(+(x1, *(z0, z1)), *(z0, z2)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(x1, +(*(z0, +(z1, z2)), u))), u), +'(+(x1, *(z0, z1)), +(*(z0, z2), u)))
+'(*(x0, +(x1, +(z0, z1))), +(*(x0, z2), u)) → c2(+'(*(x0, +(x1, +(z0, +(z1, z2)))), u), +'(+(x1, +(z0, z1)), z2))
+'(*(x0, +(x1, x2)), +(*(x0, x3), u)) → c2(+'(+(x1, x2), x3))
K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2, c2

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

Use narrowing to replace +'(*(x0, *(z0, z1)), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(*(z0, +(z1, z2)), u)), u), +'(*(z0, z1), +(*(z0, z2), u))) by

+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, *(z0, z2)), u)), u)) → c2(+'(*(x0, +(*(x1, *(z0, +(z1, z2))), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, *(z0, z2)), u)))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, +(*(x1, z2), u)), u)) → c2(+'(*(x0, +(*(x1, +(z0, +(z1, z2))), u)), u), +'(*(x1, +(z0, z1)), +(*(x1, z2), u)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, +(*(z0, z2), u)), u)), u)) → c2(+'(*(x0, +(*(x1, +(*(z0, +(z1, z2)), u)), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, +(*(z0, z2), u)), u)))
+'(*(x0, *(x1, x2)), +(*(x0, +(*(x1, x3), u)), u)) → c2

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2))
+(+(z0, z1), z2) → +(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, *(z0, z2))), u)) → c2(+'(*(x0, *(x1, *(z0, +(z1, z2)))), u), +'(*(x1, *(z0, z1)), *(x1, *(z0, z2))))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, *(x1, z2)), u)) → c2(+'(*(x0, *(x1, +(z0, +(z1, z2)))), u), +'(*(x1, +(z0, z1)), *(x1, z2)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, +(*(z0, z2), u))), u)) → c2(+'(*(x0, *(x1, +(*(z0, +(z1, z2)), u))), u), +'(*(x1, *(z0, z1)), *(x1, +(*(z0, z2), u))))
+'(*(x0, *(x1, x2)), +(*(x0, *(x1, x3)), u)) → c2(+'(*(x1, x2), *(x1, x3)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, +(x1, *(z0, +(z1, z2)))), u), +'(+(x1, *(z0, z1)), *(z0, z2)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(x1, +(*(z0, +(z1, z2)), u))), u), +'(+(x1, *(z0, z1)), +(*(z0, z2), u)))
+'(*(x0, +(x1, +(z0, z1))), +(*(x0, z2), u)) → c2(+'(*(x0, +(x1, +(z0, +(z1, z2)))), u), +'(+(x1, +(z0, z1)), z2))
+'(*(x0, +(x1, x2)), +(*(x0, x3), u)) → c2(+'(+(x1, x2), x3))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, *(z0, z2)), u)), u)) → c2(+'(*(x0, +(*(x1, *(z0, +(z1, z2))), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, *(z0, z2)), u)))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, +(*(x1, z2), u)), u)) → c2(+'(*(x0, +(*(x1, +(z0, +(z1, z2))), u)), u), +'(*(x1, +(z0, z1)), +(*(x1, z2), u)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, +(*(z0, z2), u)), u)), u)) → c2(+'(*(x0, +(*(x1, +(*(z0, +(z1, z2)), u)), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, +(*(z0, z2), u)), u)))
+'(*(x0, *(x1, x2)), +(*(x0, +(*(x1, x3), u)), u)) → c2
S tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, *(z0, z2))), u)) → c2(+'(*(x0, *(x1, *(z0, +(z1, z2)))), u), +'(*(x1, *(z0, z1)), *(x1, *(z0, z2))))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, *(x1, z2)), u)) → c2(+'(*(x0, *(x1, +(z0, +(z1, z2)))), u), +'(*(x1, +(z0, z1)), *(x1, z2)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, +(*(z0, z2), u))), u)) → c2(+'(*(x0, *(x1, +(*(z0, +(z1, z2)), u))), u), +'(*(x1, *(z0, z1)), *(x1, +(*(z0, z2), u))))
+'(*(x0, *(x1, x2)), +(*(x0, *(x1, x3)), u)) → c2(+'(*(x1, x2), *(x1, x3)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, +(x1, *(z0, +(z1, z2)))), u), +'(+(x1, *(z0, z1)), *(z0, z2)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(x1, +(*(z0, +(z1, z2)), u))), u), +'(+(x1, *(z0, z1)), +(*(z0, z2), u)))
+'(*(x0, +(x1, +(z0, z1))), +(*(x0, z2), u)) → c2(+'(*(x0, +(x1, +(z0, +(z1, z2)))), u), +'(+(x1, +(z0, z1)), z2))
+'(*(x0, +(x1, x2)), +(*(x0, x3), u)) → c2(+'(+(x1, x2), x3))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, *(z0, z2)), u)), u)) → c2(+'(*(x0, +(*(x1, *(z0, +(z1, z2))), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, *(z0, z2)), u)))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, +(*(x1, z2), u)), u)) → c2(+'(*(x0, +(*(x1, +(z0, +(z1, z2))), u)), u), +'(*(x1, +(z0, z1)), +(*(x1, z2), u)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, +(*(z0, z2), u)), u)), u)) → c2(+'(*(x0, +(*(x1, +(*(z0, +(z1, z2)), u)), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, +(*(z0, z2), u)), u)))
+'(*(x0, *(x1, x2)), +(*(x0, +(*(x1, x3), u)), u)) → c2
K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2, c2, c2

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

Removed 1 trailing nodes:

+'(*(x0, *(x1, x2)), +(*(x0, +(*(x1, x3), u)), u)) → c2

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2))
+(+(z0, z1), z2) → +(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
Tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, *(z0, z2))), u)) → c2(+'(*(x0, *(x1, *(z0, +(z1, z2)))), u), +'(*(x1, *(z0, z1)), *(x1, *(z0, z2))))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, *(x1, z2)), u)) → c2(+'(*(x0, *(x1, +(z0, +(z1, z2)))), u), +'(*(x1, +(z0, z1)), *(x1, z2)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, +(*(z0, z2), u))), u)) → c2(+'(*(x0, *(x1, +(*(z0, +(z1, z2)), u))), u), +'(*(x1, *(z0, z1)), *(x1, +(*(z0, z2), u))))
+'(*(x0, *(x1, x2)), +(*(x0, *(x1, x3)), u)) → c2(+'(*(x1, x2), *(x1, x3)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, +(x1, *(z0, +(z1, z2)))), u), +'(+(x1, *(z0, z1)), *(z0, z2)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(x1, +(*(z0, +(z1, z2)), u))), u), +'(+(x1, *(z0, z1)), +(*(z0, z2), u)))
+'(*(x0, +(x1, +(z0, z1))), +(*(x0, z2), u)) → c2(+'(*(x0, +(x1, +(z0, +(z1, z2)))), u), +'(+(x1, +(z0, z1)), z2))
+'(*(x0, +(x1, x2)), +(*(x0, x3), u)) → c2(+'(+(x1, x2), x3))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, *(z0, z2)), u)), u)) → c2(+'(*(x0, +(*(x1, *(z0, +(z1, z2))), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, *(z0, z2)), u)))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, +(*(x1, z2), u)), u)) → c2(+'(*(x0, +(*(x1, +(z0, +(z1, z2))), u)), u), +'(*(x1, +(z0, z1)), +(*(x1, z2), u)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, +(*(z0, z2), u)), u)), u)) → c2(+'(*(x0, +(*(x1, +(*(z0, +(z1, z2)), u)), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, +(*(z0, z2), u)), u)))
S tuples:

+'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2))
+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, *(z0, z2))), u)) → c2(+'(*(x0, *(x1, *(z0, +(z1, z2)))), u), +'(*(x1, *(z0, z1)), *(x1, *(z0, z2))))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, *(x1, z2)), u)) → c2(+'(*(x0, *(x1, +(z0, +(z1, z2)))), u), +'(*(x1, +(z0, z1)), *(x1, z2)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, +(*(z0, z2), u))), u)) → c2(+'(*(x0, *(x1, +(*(z0, +(z1, z2)), u))), u), +'(*(x1, *(z0, z1)), *(x1, +(*(z0, z2), u))))
+'(*(x0, *(x1, x2)), +(*(x0, *(x1, x3)), u)) → c2(+'(*(x1, x2), *(x1, x3)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, +(x1, *(z0, +(z1, z2)))), u), +'(+(x1, *(z0, z1)), *(z0, z2)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(x1, +(*(z0, +(z1, z2)), u))), u), +'(+(x1, *(z0, z1)), +(*(z0, z2), u)))
+'(*(x0, +(x1, +(z0, z1))), +(*(x0, z2), u)) → c2(+'(*(x0, +(x1, +(z0, +(z1, z2)))), u), +'(+(x1, +(z0, z1)), z2))
+'(*(x0, +(x1, x2)), +(*(x0, x3), u)) → c2(+'(+(x1, x2), x3))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, *(z0, z2)), u)), u)) → c2(+'(*(x0, +(*(x1, *(z0, +(z1, z2))), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, *(z0, z2)), u)))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, +(*(x1, z2), u)), u)) → c2(+'(*(x0, +(*(x1, +(z0, +(z1, z2))), u)), u), +'(*(x1, +(z0, z1)), +(*(x1, z2), u)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, +(*(z0, z2), u)), u)), u)) → c2(+'(*(x0, +(*(x1, +(*(z0, +(z1, z2)), u)), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, +(*(z0, z2), u)), u)))
K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2, c2

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

Use forward instantiation to replace +'(*(z0, z1), *(z0, z2)) → c(+'(z1, z2)) by

+'(*(z0, *(y0, y1)), *(z0, *(y2, y3))) → c(+'(*(y0, y1), *(y2, y3)))
+'(*(z0, +(y0, y1)), *(z0, z2)) → c(+'(+(y0, y1), z2))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, *(y6, y7))), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, *(y5, *(y6, y7))), u)))
+'(*(z0, *(y0, *(y1, +(y2, y3)))), *(z0, +(*(y4, *(y5, y6)), u))) → c(+'(*(y0, *(y1, +(y2, y3))), +(*(y4, *(y5, y6)), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, +(*(y6, y7), u))), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, *(y5, +(*(y6, y7), u))), u)))
+'(*(z0, *(y0, *(y1, y2))), *(z0, +(*(y3, *(y4, y5)), u))) → c(+'(*(y0, *(y1, y2)), +(*(y3, *(y4, y5)), u)))
+'(*(z0, *(y0, +(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, y6)), u))) → c(+'(*(y0, +(y1, *(y2, y3))), +(*(y4, *(y5, y6)), u)))
+'(*(z0, *(y0, +(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, y6), u)), u))) → c(+'(*(y0, +(y1, *(y2, y3))), +(*(y4, +(*(y5, y6), u)), u)))
+'(*(z0, *(y0, +(y1, +(y2, y3)))), *(z0, +(*(y4, y5), u))) → c(+'(*(y0, +(y1, +(y2, y3))), +(*(y4, y5), u)))
+'(*(z0, *(y0, +(y1, y2))), *(z0, +(*(y3, y4), u))) → c(+'(*(y0, +(y1, y2)), +(*(y3, y4), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, *(y6, y7)), u)), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, +(*(y5, *(y6, y7)), u)), u)))
+'(*(z0, *(y0, *(y1, +(y2, y3)))), *(z0, +(*(y4, +(*(y5, y6), u)), u))) → c(+'(*(y0, *(y1, +(y2, y3))), +(*(y4, +(*(y5, y6), u)), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, +(*(y6, y7), u)), u)), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, +(*(y5, +(*(y6, y7), u)), u)), u)))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2))
+(+(z0, z1), z2) → +(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
Tuples:

+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, *(z0, z2))), u)) → c2(+'(*(x0, *(x1, *(z0, +(z1, z2)))), u), +'(*(x1, *(z0, z1)), *(x1, *(z0, z2))))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, *(x1, z2)), u)) → c2(+'(*(x0, *(x1, +(z0, +(z1, z2)))), u), +'(*(x1, +(z0, z1)), *(x1, z2)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, +(*(z0, z2), u))), u)) → c2(+'(*(x0, *(x1, +(*(z0, +(z1, z2)), u))), u), +'(*(x1, *(z0, z1)), *(x1, +(*(z0, z2), u))))
+'(*(x0, *(x1, x2)), +(*(x0, *(x1, x3)), u)) → c2(+'(*(x1, x2), *(x1, x3)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, +(x1, *(z0, +(z1, z2)))), u), +'(+(x1, *(z0, z1)), *(z0, z2)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(x1, +(*(z0, +(z1, z2)), u))), u), +'(+(x1, *(z0, z1)), +(*(z0, z2), u)))
+'(*(x0, +(x1, +(z0, z1))), +(*(x0, z2), u)) → c2(+'(*(x0, +(x1, +(z0, +(z1, z2)))), u), +'(+(x1, +(z0, z1)), z2))
+'(*(x0, +(x1, x2)), +(*(x0, x3), u)) → c2(+'(+(x1, x2), x3))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, *(z0, z2)), u)), u)) → c2(+'(*(x0, +(*(x1, *(z0, +(z1, z2))), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, *(z0, z2)), u)))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, +(*(x1, z2), u)), u)) → c2(+'(*(x0, +(*(x1, +(z0, +(z1, z2))), u)), u), +'(*(x1, +(z0, z1)), +(*(x1, z2), u)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, +(*(z0, z2), u)), u)), u)) → c2(+'(*(x0, +(*(x1, +(*(z0, +(z1, z2)), u)), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, +(*(z0, z2), u)), u)))
+'(*(z0, *(y0, y1)), *(z0, *(y2, y3))) → c(+'(*(y0, y1), *(y2, y3)))
+'(*(z0, +(y0, y1)), *(z0, z2)) → c(+'(+(y0, y1), z2))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, *(y6, y7))), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, *(y5, *(y6, y7))), u)))
+'(*(z0, *(y0, *(y1, +(y2, y3)))), *(z0, +(*(y4, *(y5, y6)), u))) → c(+'(*(y0, *(y1, +(y2, y3))), +(*(y4, *(y5, y6)), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, +(*(y6, y7), u))), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, *(y5, +(*(y6, y7), u))), u)))
+'(*(z0, *(y0, *(y1, y2))), *(z0, +(*(y3, *(y4, y5)), u))) → c(+'(*(y0, *(y1, y2)), +(*(y3, *(y4, y5)), u)))
+'(*(z0, *(y0, +(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, y6)), u))) → c(+'(*(y0, +(y1, *(y2, y3))), +(*(y4, *(y5, y6)), u)))
+'(*(z0, *(y0, +(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, y6), u)), u))) → c(+'(*(y0, +(y1, *(y2, y3))), +(*(y4, +(*(y5, y6), u)), u)))
+'(*(z0, *(y0, +(y1, +(y2, y3)))), *(z0, +(*(y4, y5), u))) → c(+'(*(y0, +(y1, +(y2, y3))), +(*(y4, y5), u)))
+'(*(z0, *(y0, +(y1, y2))), *(z0, +(*(y3, y4), u))) → c(+'(*(y0, +(y1, y2)), +(*(y3, y4), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, *(y6, y7)), u)), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, +(*(y5, *(y6, y7)), u)), u)))
+'(*(z0, *(y0, *(y1, +(y2, y3)))), *(z0, +(*(y4, +(*(y5, y6), u)), u))) → c(+'(*(y0, *(y1, +(y2, y3))), +(*(y4, +(*(y5, y6), u)), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, +(*(y6, y7), u)), u)), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, +(*(y5, +(*(y6, y7), u)), u)), u)))
S tuples:

+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, *(z0, z2))), u)) → c2(+'(*(x0, *(x1, *(z0, +(z1, z2)))), u), +'(*(x1, *(z0, z1)), *(x1, *(z0, z2))))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, *(x1, z2)), u)) → c2(+'(*(x0, *(x1, +(z0, +(z1, z2)))), u), +'(*(x1, +(z0, z1)), *(x1, z2)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, +(*(z0, z2), u))), u)) → c2(+'(*(x0, *(x1, +(*(z0, +(z1, z2)), u))), u), +'(*(x1, *(z0, z1)), *(x1, +(*(z0, z2), u))))
+'(*(x0, *(x1, x2)), +(*(x0, *(x1, x3)), u)) → c2(+'(*(x1, x2), *(x1, x3)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, +(x1, *(z0, +(z1, z2)))), u), +'(+(x1, *(z0, z1)), *(z0, z2)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(x1, +(*(z0, +(z1, z2)), u))), u), +'(+(x1, *(z0, z1)), +(*(z0, z2), u)))
+'(*(x0, +(x1, +(z0, z1))), +(*(x0, z2), u)) → c2(+'(*(x0, +(x1, +(z0, +(z1, z2)))), u), +'(+(x1, +(z0, z1)), z2))
+'(*(x0, +(x1, x2)), +(*(x0, x3), u)) → c2(+'(+(x1, x2), x3))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, *(z0, z2)), u)), u)) → c2(+'(*(x0, +(*(x1, *(z0, +(z1, z2))), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, *(z0, z2)), u)))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, +(*(x1, z2), u)), u)) → c2(+'(*(x0, +(*(x1, +(z0, +(z1, z2))), u)), u), +'(*(x1, +(z0, z1)), +(*(x1, z2), u)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, +(*(z0, z2), u)), u)), u)) → c2(+'(*(x0, +(*(x1, +(*(z0, +(z1, z2)), u)), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, +(*(z0, z2), u)), u)))
+'(*(z0, *(y0, y1)), *(z0, *(y2, y3))) → c(+'(*(y0, y1), *(y2, y3)))
+'(*(z0, +(y0, y1)), *(z0, z2)) → c(+'(+(y0, y1), z2))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, *(y6, y7))), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, *(y5, *(y6, y7))), u)))
+'(*(z0, *(y0, *(y1, +(y2, y3)))), *(z0, +(*(y4, *(y5, y6)), u))) → c(+'(*(y0, *(y1, +(y2, y3))), +(*(y4, *(y5, y6)), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, +(*(y6, y7), u))), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, *(y5, +(*(y6, y7), u))), u)))
+'(*(z0, *(y0, *(y1, y2))), *(z0, +(*(y3, *(y4, y5)), u))) → c(+'(*(y0, *(y1, y2)), +(*(y3, *(y4, y5)), u)))
+'(*(z0, *(y0, +(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, y6)), u))) → c(+'(*(y0, +(y1, *(y2, y3))), +(*(y4, *(y5, y6)), u)))
+'(*(z0, *(y0, +(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, y6), u)), u))) → c(+'(*(y0, +(y1, *(y2, y3))), +(*(y4, +(*(y5, y6), u)), u)))
+'(*(z0, *(y0, +(y1, +(y2, y3)))), *(z0, +(*(y4, y5), u))) → c(+'(*(y0, +(y1, +(y2, y3))), +(*(y4, y5), u)))
+'(*(z0, *(y0, +(y1, y2))), *(z0, +(*(y3, y4), u))) → c(+'(*(y0, +(y1, y2)), +(*(y3, y4), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, *(y6, y7)), u)), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, +(*(y5, *(y6, y7)), u)), u)))
+'(*(z0, *(y0, *(y1, +(y2, y3)))), *(z0, +(*(y4, +(*(y5, y6), u)), u))) → c(+'(*(y0, *(y1, +(y2, y3))), +(*(y4, +(*(y5, y6), u)), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, +(*(y6, y7), u)), u)), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, +(*(y5, +(*(y6, y7), u)), u)), u)))
K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c1, c2, c2, c

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

Use forward instantiation to replace +'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, +(*(z0, z2), u))), u)) → c2(+'(*(x0, *(x1, +(*(z0, +(z1, z2)), u))), u), +'(*(x1, *(z0, z1)), *(x1, +(*(z0, z2), u)))) by

+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, *(y8, y9))), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), *(y7, *(y8, y9)))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, *(y7, *(y8, y9))), u))))
+'(*(z0, *(z1, *(z2, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, y8)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, +(y3, y4)), *(y7, y8))), u))), u), +'(*(z1, *(z2, *(y2, +(y3, y4)))), *(z1, +(*(z2, *(y7, y8)), u))))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, +(*(y8, y9), u))), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), *(y7, +(*(y8, y9), u)))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, *(y7, +(*(y8, y9), u))), u))))
+'(*(z0, *(z1, *(z2, *(y2, y3)))), +(*(z0, *(z1, +(*(z2, *(y6, y7)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, y3), *(y6, y7))), u))), u), +'(*(z1, *(z2, *(y2, y3))), *(z1, +(*(z2, *(y6, y7)), u))))
+'(*(z0, *(z1, *(z2, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, y8)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, *(y3, y4)), *(y7, y8))), u))), u), +'(*(z1, *(z2, +(y2, *(y3, y4)))), *(z1, +(*(z2, *(y7, y8)), u))))
+'(*(z0, *(z1, *(z2, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, *(y3, y4)), +(*(y7, y8), u))), u))), u), +'(*(z1, *(z2, +(y2, *(y3, y4)))), *(z1, +(*(z2, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(z2, +(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(z2, z4), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, +(y3, y4)), z4)), u))), u), +'(*(z1, *(z2, +(y2, +(y3, y4)))), *(z1, +(*(z2, z4), u))))
+'(*(z0, *(z1, *(z2, +(y2, y3)))), +(*(z0, *(z1, +(*(z2, z4), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, y3), z4)), u))), u), +'(*(z1, *(z2, +(y2, y3))), *(z1, +(*(z2, z4), u))))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, *(y8, y9)), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), +(*(y7, *(y8, y9)), u))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, +(*(y7, *(y8, y9)), u)), u))))
+'(*(z0, *(z1, *(z2, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, +(y3, y4)), +(*(y7, y8), u))), u))), u), +'(*(z1, *(z2, *(y2, +(y3, y4)))), *(z1, +(*(z2, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, +(*(y8, y9), u)), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), +(*(y7, +(*(y8, y9), u)), u))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, +(*(y7, +(*(y8, y9), u)), u)), u))))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2))
+(+(z0, z1), z2) → +(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
Tuples:

+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, *(z0, z2))), u)) → c2(+'(*(x0, *(x1, *(z0, +(z1, z2)))), u), +'(*(x1, *(z0, z1)), *(x1, *(z0, z2))))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, *(x1, z2)), u)) → c2(+'(*(x0, *(x1, +(z0, +(z1, z2)))), u), +'(*(x1, +(z0, z1)), *(x1, z2)))
+'(*(x0, *(x1, x2)), +(*(x0, *(x1, x3)), u)) → c2(+'(*(x1, x2), *(x1, x3)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, +(x1, *(z0, +(z1, z2)))), u), +'(+(x1, *(z0, z1)), *(z0, z2)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(x1, +(*(z0, +(z1, z2)), u))), u), +'(+(x1, *(z0, z1)), +(*(z0, z2), u)))
+'(*(x0, +(x1, +(z0, z1))), +(*(x0, z2), u)) → c2(+'(*(x0, +(x1, +(z0, +(z1, z2)))), u), +'(+(x1, +(z0, z1)), z2))
+'(*(x0, +(x1, x2)), +(*(x0, x3), u)) → c2(+'(+(x1, x2), x3))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, *(z0, z2)), u)), u)) → c2(+'(*(x0, +(*(x1, *(z0, +(z1, z2))), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, *(z0, z2)), u)))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, +(*(x1, z2), u)), u)) → c2(+'(*(x0, +(*(x1, +(z0, +(z1, z2))), u)), u), +'(*(x1, +(z0, z1)), +(*(x1, z2), u)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, +(*(z0, z2), u)), u)), u)) → c2(+'(*(x0, +(*(x1, +(*(z0, +(z1, z2)), u)), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, +(*(z0, z2), u)), u)))
+'(*(z0, *(y0, y1)), *(z0, *(y2, y3))) → c(+'(*(y0, y1), *(y2, y3)))
+'(*(z0, +(y0, y1)), *(z0, z2)) → c(+'(+(y0, y1), z2))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, *(y6, y7))), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, *(y5, *(y6, y7))), u)))
+'(*(z0, *(y0, *(y1, +(y2, y3)))), *(z0, +(*(y4, *(y5, y6)), u))) → c(+'(*(y0, *(y1, +(y2, y3))), +(*(y4, *(y5, y6)), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, +(*(y6, y7), u))), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, *(y5, +(*(y6, y7), u))), u)))
+'(*(z0, *(y0, *(y1, y2))), *(z0, +(*(y3, *(y4, y5)), u))) → c(+'(*(y0, *(y1, y2)), +(*(y3, *(y4, y5)), u)))
+'(*(z0, *(y0, +(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, y6)), u))) → c(+'(*(y0, +(y1, *(y2, y3))), +(*(y4, *(y5, y6)), u)))
+'(*(z0, *(y0, +(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, y6), u)), u))) → c(+'(*(y0, +(y1, *(y2, y3))), +(*(y4, +(*(y5, y6), u)), u)))
+'(*(z0, *(y0, +(y1, +(y2, y3)))), *(z0, +(*(y4, y5), u))) → c(+'(*(y0, +(y1, +(y2, y3))), +(*(y4, y5), u)))
+'(*(z0, *(y0, +(y1, y2))), *(z0, +(*(y3, y4), u))) → c(+'(*(y0, +(y1, y2)), +(*(y3, y4), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, *(y6, y7)), u)), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, +(*(y5, *(y6, y7)), u)), u)))
+'(*(z0, *(y0, *(y1, +(y2, y3)))), *(z0, +(*(y4, +(*(y5, y6), u)), u))) → c(+'(*(y0, *(y1, +(y2, y3))), +(*(y4, +(*(y5, y6), u)), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, +(*(y6, y7), u)), u)), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, +(*(y5, +(*(y6, y7), u)), u)), u)))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, *(y8, y9))), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), *(y7, *(y8, y9)))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, *(y7, *(y8, y9))), u))))
+'(*(z0, *(z1, *(z2, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, y8)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, +(y3, y4)), *(y7, y8))), u))), u), +'(*(z1, *(z2, *(y2, +(y3, y4)))), *(z1, +(*(z2, *(y7, y8)), u))))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, +(*(y8, y9), u))), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), *(y7, +(*(y8, y9), u)))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, *(y7, +(*(y8, y9), u))), u))))
+'(*(z0, *(z1, *(z2, *(y2, y3)))), +(*(z0, *(z1, +(*(z2, *(y6, y7)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, y3), *(y6, y7))), u))), u), +'(*(z1, *(z2, *(y2, y3))), *(z1, +(*(z2, *(y6, y7)), u))))
+'(*(z0, *(z1, *(z2, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, y8)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, *(y3, y4)), *(y7, y8))), u))), u), +'(*(z1, *(z2, +(y2, *(y3, y4)))), *(z1, +(*(z2, *(y7, y8)), u))))
+'(*(z0, *(z1, *(z2, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, *(y3, y4)), +(*(y7, y8), u))), u))), u), +'(*(z1, *(z2, +(y2, *(y3, y4)))), *(z1, +(*(z2, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(z2, +(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(z2, z4), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, +(y3, y4)), z4)), u))), u), +'(*(z1, *(z2, +(y2, +(y3, y4)))), *(z1, +(*(z2, z4), u))))
+'(*(z0, *(z1, *(z2, +(y2, y3)))), +(*(z0, *(z1, +(*(z2, z4), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, y3), z4)), u))), u), +'(*(z1, *(z2, +(y2, y3))), *(z1, +(*(z2, z4), u))))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, *(y8, y9)), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), +(*(y7, *(y8, y9)), u))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, +(*(y7, *(y8, y9)), u)), u))))
+'(*(z0, *(z1, *(z2, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, +(y3, y4)), +(*(y7, y8), u))), u))), u), +'(*(z1, *(z2, *(y2, +(y3, y4)))), *(z1, +(*(z2, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, +(*(y8, y9), u)), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), +(*(y7, +(*(y8, y9), u)), u))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, +(*(y7, +(*(y8, y9), u)), u)), u))))
S tuples:

+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, *(z0, z2))), u)) → c2(+'(*(x0, *(x1, *(z0, +(z1, z2)))), u), +'(*(x1, *(z0, z1)), *(x1, *(z0, z2))))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, *(x1, z2)), u)) → c2(+'(*(x0, *(x1, +(z0, +(z1, z2)))), u), +'(*(x1, +(z0, z1)), *(x1, z2)))
+'(*(x0, *(x1, x2)), +(*(x0, *(x1, x3)), u)) → c2(+'(*(x1, x2), *(x1, x3)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, +(x1, *(z0, +(z1, z2)))), u), +'(+(x1, *(z0, z1)), *(z0, z2)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(x1, +(*(z0, +(z1, z2)), u))), u), +'(+(x1, *(z0, z1)), +(*(z0, z2), u)))
+'(*(x0, +(x1, +(z0, z1))), +(*(x0, z2), u)) → c2(+'(*(x0, +(x1, +(z0, +(z1, z2)))), u), +'(+(x1, +(z0, z1)), z2))
+'(*(x0, +(x1, x2)), +(*(x0, x3), u)) → c2(+'(+(x1, x2), x3))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, *(z0, z2)), u)), u)) → c2(+'(*(x0, +(*(x1, *(z0, +(z1, z2))), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, *(z0, z2)), u)))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, +(*(x1, z2), u)), u)) → c2(+'(*(x0, +(*(x1, +(z0, +(z1, z2))), u)), u), +'(*(x1, +(z0, z1)), +(*(x1, z2), u)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, +(*(z0, z2), u)), u)), u)) → c2(+'(*(x0, +(*(x1, +(*(z0, +(z1, z2)), u)), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, +(*(z0, z2), u)), u)))
+'(*(z0, *(y0, y1)), *(z0, *(y2, y3))) → c(+'(*(y0, y1), *(y2, y3)))
+'(*(z0, +(y0, y1)), *(z0, z2)) → c(+'(+(y0, y1), z2))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, *(y6, y7))), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, *(y5, *(y6, y7))), u)))
+'(*(z0, *(y0, *(y1, +(y2, y3)))), *(z0, +(*(y4, *(y5, y6)), u))) → c(+'(*(y0, *(y1, +(y2, y3))), +(*(y4, *(y5, y6)), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, +(*(y6, y7), u))), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, *(y5, +(*(y6, y7), u))), u)))
+'(*(z0, *(y0, *(y1, y2))), *(z0, +(*(y3, *(y4, y5)), u))) → c(+'(*(y0, *(y1, y2)), +(*(y3, *(y4, y5)), u)))
+'(*(z0, *(y0, +(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, y6)), u))) → c(+'(*(y0, +(y1, *(y2, y3))), +(*(y4, *(y5, y6)), u)))
+'(*(z0, *(y0, +(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, y6), u)), u))) → c(+'(*(y0, +(y1, *(y2, y3))), +(*(y4, +(*(y5, y6), u)), u)))
+'(*(z0, *(y0, +(y1, +(y2, y3)))), *(z0, +(*(y4, y5), u))) → c(+'(*(y0, +(y1, +(y2, y3))), +(*(y4, y5), u)))
+'(*(z0, *(y0, +(y1, y2))), *(z0, +(*(y3, y4), u))) → c(+'(*(y0, +(y1, y2)), +(*(y3, y4), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, *(y6, y7)), u)), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, +(*(y5, *(y6, y7)), u)), u)))
+'(*(z0, *(y0, *(y1, +(y2, y3)))), *(z0, +(*(y4, +(*(y5, y6), u)), u))) → c(+'(*(y0, *(y1, +(y2, y3))), +(*(y4, +(*(y5, y6), u)), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, +(*(y6, y7), u)), u)), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, +(*(y5, +(*(y6, y7), u)), u)), u)))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, *(y8, y9))), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), *(y7, *(y8, y9)))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, *(y7, *(y8, y9))), u))))
+'(*(z0, *(z1, *(z2, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, y8)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, +(y3, y4)), *(y7, y8))), u))), u), +'(*(z1, *(z2, *(y2, +(y3, y4)))), *(z1, +(*(z2, *(y7, y8)), u))))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, +(*(y8, y9), u))), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), *(y7, +(*(y8, y9), u)))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, *(y7, +(*(y8, y9), u))), u))))
+'(*(z0, *(z1, *(z2, *(y2, y3)))), +(*(z0, *(z1, +(*(z2, *(y6, y7)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, y3), *(y6, y7))), u))), u), +'(*(z1, *(z2, *(y2, y3))), *(z1, +(*(z2, *(y6, y7)), u))))
+'(*(z0, *(z1, *(z2, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, y8)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, *(y3, y4)), *(y7, y8))), u))), u), +'(*(z1, *(z2, +(y2, *(y3, y4)))), *(z1, +(*(z2, *(y7, y8)), u))))
+'(*(z0, *(z1, *(z2, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, *(y3, y4)), +(*(y7, y8), u))), u))), u), +'(*(z1, *(z2, +(y2, *(y3, y4)))), *(z1, +(*(z2, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(z2, +(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(z2, z4), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, +(y3, y4)), z4)), u))), u), +'(*(z1, *(z2, +(y2, +(y3, y4)))), *(z1, +(*(z2, z4), u))))
+'(*(z0, *(z1, *(z2, +(y2, y3)))), +(*(z0, *(z1, +(*(z2, z4), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, y3), z4)), u))), u), +'(*(z1, *(z2, +(y2, y3))), *(z1, +(*(z2, z4), u))))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, *(y8, y9)), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), +(*(y7, *(y8, y9)), u))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, +(*(y7, *(y8, y9)), u)), u))))
+'(*(z0, *(z1, *(z2, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, +(y3, y4)), +(*(y7, y8), u))), u))), u), +'(*(z1, *(z2, *(y2, +(y3, y4)))), *(z1, +(*(z2, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, +(*(y8, y9), u)), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), +(*(y7, +(*(y8, y9), u)), u))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, +(*(y7, +(*(y8, y9), u)), u)), u))))
K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c1, c2, c2, c

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

Use forward instantiation to replace +'(*(x0, *(x1, x2)), +(*(x0, *(x1, x3)), u)) → c2(+'(*(x1, x2), *(x1, x3))) by

+'(*(z0, *(z1, *(y1, y2))), +(*(z0, *(z1, *(y4, y5))), u)) → c2(+'(*(z1, *(y1, y2)), *(z1, *(y4, y5))))
+'(*(z0, *(z1, +(y1, y2))), +(*(z0, *(z1, z3)), u)) → c2(+'(*(z1, +(y1, y2)), *(z1, z3)))
+'(*(z0, *(z1, *(y1, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, *(y7, *(y8, y9))), u))), u)) → c2(+'(*(z1, *(y1, *(y2, *(y3, y4)))), *(z1, +(*(y6, *(y7, *(y8, y9))), u))))
+'(*(z0, *(z1, *(y1, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(y6, *(y7, y8)), u))), u)) → c2(+'(*(z1, *(y1, *(y2, +(y3, y4)))), *(z1, +(*(y6, *(y7, y8)), u))))
+'(*(z0, *(z1, *(y1, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, *(y7, +(*(y8, y9), u))), u))), u)) → c2(+'(*(z1, *(y1, *(y2, *(y3, y4)))), *(z1, +(*(y6, *(y7, +(*(y8, y9), u))), u))))
+'(*(z0, *(z1, *(y1, *(y2, y3)))), +(*(z0, *(z1, +(*(y5, *(y6, y7)), u))), u)) → c2(+'(*(z1, *(y1, *(y2, y3))), *(z1, +(*(y5, *(y6, y7)), u))))
+'(*(z0, *(z1, *(y1, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, *(y7, y8)), u))), u)) → c2(+'(*(z1, *(y1, +(y2, *(y3, y4)))), *(z1, +(*(y6, *(y7, y8)), u))))
+'(*(z0, *(z1, *(y1, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z1, *(y1, +(y2, *(y3, y4)))), *(z1, +(*(y6, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(y1, +(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(y6, y7), u))), u)) → c2(+'(*(z1, *(y1, +(y2, +(y3, y4)))), *(z1, +(*(y6, y7), u))))
+'(*(z0, *(z1, *(y1, +(y2, y3)))), +(*(z0, *(z1, +(*(y5, y6), u))), u)) → c2(+'(*(z1, *(y1, +(y2, y3))), *(z1, +(*(y5, y6), u))))
+'(*(z0, *(z1, *(y1, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, +(*(y7, *(y8, y9)), u)), u))), u)) → c2(+'(*(z1, *(y1, *(y2, *(y3, y4)))), *(z1, +(*(y6, +(*(y7, *(y8, y9)), u)), u))))
+'(*(z0, *(z1, *(y1, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(y6, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z1, *(y1, *(y2, +(y3, y4)))), *(z1, +(*(y6, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(y1, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, +(*(y7, +(*(y8, y9), u)), u)), u))), u)) → c2(+'(*(z1, *(y1, *(y2, *(y3, y4)))), *(z1, +(*(y6, +(*(y7, +(*(y8, y9), u)), u)), u))))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(*(z0, z1), *(z0, z2)) → *(z0, +(z1, z2))
+(+(z0, z1), z2) → +(z0, +(z1, z2))
+(*(z0, z1), +(*(z0, z2), u)) → +(*(z0, +(z1, z2)), u)
Tuples:

+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, *(z0, z2))), u)) → c2(+'(*(x0, *(x1, *(z0, +(z1, z2)))), u), +'(*(x1, *(z0, z1)), *(x1, *(z0, z2))))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, *(x1, z2)), u)) → c2(+'(*(x0, *(x1, +(z0, +(z1, z2)))), u), +'(*(x1, +(z0, z1)), *(x1, z2)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, +(x1, *(z0, +(z1, z2)))), u), +'(+(x1, *(z0, z1)), *(z0, z2)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(x1, +(*(z0, +(z1, z2)), u))), u), +'(+(x1, *(z0, z1)), +(*(z0, z2), u)))
+'(*(x0, +(x1, +(z0, z1))), +(*(x0, z2), u)) → c2(+'(*(x0, +(x1, +(z0, +(z1, z2)))), u), +'(+(x1, +(z0, z1)), z2))
+'(*(x0, +(x1, x2)), +(*(x0, x3), u)) → c2(+'(+(x1, x2), x3))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, *(z0, z2)), u)), u)) → c2(+'(*(x0, +(*(x1, *(z0, +(z1, z2))), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, *(z0, z2)), u)))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, +(*(x1, z2), u)), u)) → c2(+'(*(x0, +(*(x1, +(z0, +(z1, z2))), u)), u), +'(*(x1, +(z0, z1)), +(*(x1, z2), u)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, +(*(z0, z2), u)), u)), u)) → c2(+'(*(x0, +(*(x1, +(*(z0, +(z1, z2)), u)), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, +(*(z0, z2), u)), u)))
+'(*(z0, *(y0, y1)), *(z0, *(y2, y3))) → c(+'(*(y0, y1), *(y2, y3)))
+'(*(z0, +(y0, y1)), *(z0, z2)) → c(+'(+(y0, y1), z2))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, *(y6, y7))), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, *(y5, *(y6, y7))), u)))
+'(*(z0, *(y0, *(y1, +(y2, y3)))), *(z0, +(*(y4, *(y5, y6)), u))) → c(+'(*(y0, *(y1, +(y2, y3))), +(*(y4, *(y5, y6)), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, +(*(y6, y7), u))), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, *(y5, +(*(y6, y7), u))), u)))
+'(*(z0, *(y0, *(y1, y2))), *(z0, +(*(y3, *(y4, y5)), u))) → c(+'(*(y0, *(y1, y2)), +(*(y3, *(y4, y5)), u)))
+'(*(z0, *(y0, +(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, y6)), u))) → c(+'(*(y0, +(y1, *(y2, y3))), +(*(y4, *(y5, y6)), u)))
+'(*(z0, *(y0, +(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, y6), u)), u))) → c(+'(*(y0, +(y1, *(y2, y3))), +(*(y4, +(*(y5, y6), u)), u)))
+'(*(z0, *(y0, +(y1, +(y2, y3)))), *(z0, +(*(y4, y5), u))) → c(+'(*(y0, +(y1, +(y2, y3))), +(*(y4, y5), u)))
+'(*(z0, *(y0, +(y1, y2))), *(z0, +(*(y3, y4), u))) → c(+'(*(y0, +(y1, y2)), +(*(y3, y4), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, *(y6, y7)), u)), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, +(*(y5, *(y6, y7)), u)), u)))
+'(*(z0, *(y0, *(y1, +(y2, y3)))), *(z0, +(*(y4, +(*(y5, y6), u)), u))) → c(+'(*(y0, *(y1, +(y2, y3))), +(*(y4, +(*(y5, y6), u)), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, +(*(y6, y7), u)), u)), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, +(*(y5, +(*(y6, y7), u)), u)), u)))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, *(y8, y9))), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), *(y7, *(y8, y9)))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, *(y7, *(y8, y9))), u))))
+'(*(z0, *(z1, *(z2, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, y8)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, +(y3, y4)), *(y7, y8))), u))), u), +'(*(z1, *(z2, *(y2, +(y3, y4)))), *(z1, +(*(z2, *(y7, y8)), u))))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, +(*(y8, y9), u))), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), *(y7, +(*(y8, y9), u)))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, *(y7, +(*(y8, y9), u))), u))))
+'(*(z0, *(z1, *(z2, *(y2, y3)))), +(*(z0, *(z1, +(*(z2, *(y6, y7)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, y3), *(y6, y7))), u))), u), +'(*(z1, *(z2, *(y2, y3))), *(z1, +(*(z2, *(y6, y7)), u))))
+'(*(z0, *(z1, *(z2, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, y8)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, *(y3, y4)), *(y7, y8))), u))), u), +'(*(z1, *(z2, +(y2, *(y3, y4)))), *(z1, +(*(z2, *(y7, y8)), u))))
+'(*(z0, *(z1, *(z2, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, *(y3, y4)), +(*(y7, y8), u))), u))), u), +'(*(z1, *(z2, +(y2, *(y3, y4)))), *(z1, +(*(z2, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(z2, +(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(z2, z4), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, +(y3, y4)), z4)), u))), u), +'(*(z1, *(z2, +(y2, +(y3, y4)))), *(z1, +(*(z2, z4), u))))
+'(*(z0, *(z1, *(z2, +(y2, y3)))), +(*(z0, *(z1, +(*(z2, z4), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, y3), z4)), u))), u), +'(*(z1, *(z2, +(y2, y3))), *(z1, +(*(z2, z4), u))))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, *(y8, y9)), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), +(*(y7, *(y8, y9)), u))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, +(*(y7, *(y8, y9)), u)), u))))
+'(*(z0, *(z1, *(z2, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, +(y3, y4)), +(*(y7, y8), u))), u))), u), +'(*(z1, *(z2, *(y2, +(y3, y4)))), *(z1, +(*(z2, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, +(*(y8, y9), u)), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), +(*(y7, +(*(y8, y9), u)), u))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, +(*(y7, +(*(y8, y9), u)), u)), u))))
+'(*(z0, *(z1, *(y1, y2))), +(*(z0, *(z1, *(y4, y5))), u)) → c2(+'(*(z1, *(y1, y2)), *(z1, *(y4, y5))))
+'(*(z0, *(z1, +(y1, y2))), +(*(z0, *(z1, z3)), u)) → c2(+'(*(z1, +(y1, y2)), *(z1, z3)))
+'(*(z0, *(z1, *(y1, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, *(y7, *(y8, y9))), u))), u)) → c2(+'(*(z1, *(y1, *(y2, *(y3, y4)))), *(z1, +(*(y6, *(y7, *(y8, y9))), u))))
+'(*(z0, *(z1, *(y1, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(y6, *(y7, y8)), u))), u)) → c2(+'(*(z1, *(y1, *(y2, +(y3, y4)))), *(z1, +(*(y6, *(y7, y8)), u))))
+'(*(z0, *(z1, *(y1, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, *(y7, +(*(y8, y9), u))), u))), u)) → c2(+'(*(z1, *(y1, *(y2, *(y3, y4)))), *(z1, +(*(y6, *(y7, +(*(y8, y9), u))), u))))
+'(*(z0, *(z1, *(y1, *(y2, y3)))), +(*(z0, *(z1, +(*(y5, *(y6, y7)), u))), u)) → c2(+'(*(z1, *(y1, *(y2, y3))), *(z1, +(*(y5, *(y6, y7)), u))))
+'(*(z0, *(z1, *(y1, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, *(y7, y8)), u))), u)) → c2(+'(*(z1, *(y1, +(y2, *(y3, y4)))), *(z1, +(*(y6, *(y7, y8)), u))))
+'(*(z0, *(z1, *(y1, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z1, *(y1, +(y2, *(y3, y4)))), *(z1, +(*(y6, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(y1, +(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(y6, y7), u))), u)) → c2(+'(*(z1, *(y1, +(y2, +(y3, y4)))), *(z1, +(*(y6, y7), u))))
+'(*(z0, *(z1, *(y1, +(y2, y3)))), +(*(z0, *(z1, +(*(y5, y6), u))), u)) → c2(+'(*(z1, *(y1, +(y2, y3))), *(z1, +(*(y5, y6), u))))
+'(*(z0, *(z1, *(y1, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, +(*(y7, *(y8, y9)), u)), u))), u)) → c2(+'(*(z1, *(y1, *(y2, *(y3, y4)))), *(z1, +(*(y6, +(*(y7, *(y8, y9)), u)), u))))
+'(*(z0, *(z1, *(y1, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(y6, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z1, *(y1, *(y2, +(y3, y4)))), *(z1, +(*(y6, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(y1, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, +(*(y7, +(*(y8, y9), u)), u)), u))), u)) → c2(+'(*(z1, *(y1, *(y2, *(y3, y4)))), *(z1, +(*(y6, +(*(y7, +(*(y8, y9), u)), u)), u))))
S tuples:

+'(+(z0, z1), z2) → c1(+'(z0, +(z1, z2)), +'(z1, z2))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, *(x1, *(z0, z2))), u)) → c2(+'(*(x0, *(x1, *(z0, +(z1, z2)))), u), +'(*(x1, *(z0, z1)), *(x1, *(z0, z2))))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, *(x1, z2)), u)) → c2(+'(*(x0, *(x1, +(z0, +(z1, z2)))), u), +'(*(x1, +(z0, z1)), *(x1, z2)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, *(z0, z2)), u)) → c2(+'(*(x0, +(x1, *(z0, +(z1, z2)))), u), +'(+(x1, *(z0, z1)), *(z0, z2)))
+'(*(x0, +(x1, *(z0, z1))), +(*(x0, +(*(z0, z2), u)), u)) → c2(+'(*(x0, +(x1, +(*(z0, +(z1, z2)), u))), u), +'(+(x1, *(z0, z1)), +(*(z0, z2), u)))
+'(*(x0, +(x1, +(z0, z1))), +(*(x0, z2), u)) → c2(+'(*(x0, +(x1, +(z0, +(z1, z2)))), u), +'(+(x1, +(z0, z1)), z2))
+'(*(x0, +(x1, x2)), +(*(x0, x3), u)) → c2(+'(+(x1, x2), x3))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, *(z0, z2)), u)), u)) → c2(+'(*(x0, +(*(x1, *(z0, +(z1, z2))), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, *(z0, z2)), u)))
+'(*(x0, *(x1, +(z0, z1))), +(*(x0, +(*(x1, z2), u)), u)) → c2(+'(*(x0, +(*(x1, +(z0, +(z1, z2))), u)), u), +'(*(x1, +(z0, z1)), +(*(x1, z2), u)))
+'(*(x0, *(x1, *(z0, z1))), +(*(x0, +(*(x1, +(*(z0, z2), u)), u)), u)) → c2(+'(*(x0, +(*(x1, +(*(z0, +(z1, z2)), u)), u)), u), +'(*(x1, *(z0, z1)), +(*(x1, +(*(z0, z2), u)), u)))
+'(*(z0, *(y0, y1)), *(z0, *(y2, y3))) → c(+'(*(y0, y1), *(y2, y3)))
+'(*(z0, +(y0, y1)), *(z0, z2)) → c(+'(+(y0, y1), z2))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, *(y6, y7))), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, *(y5, *(y6, y7))), u)))
+'(*(z0, *(y0, *(y1, +(y2, y3)))), *(z0, +(*(y4, *(y5, y6)), u))) → c(+'(*(y0, *(y1, +(y2, y3))), +(*(y4, *(y5, y6)), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, +(*(y6, y7), u))), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, *(y5, +(*(y6, y7), u))), u)))
+'(*(z0, *(y0, *(y1, y2))), *(z0, +(*(y3, *(y4, y5)), u))) → c(+'(*(y0, *(y1, y2)), +(*(y3, *(y4, y5)), u)))
+'(*(z0, *(y0, +(y1, *(y2, y3)))), *(z0, +(*(y4, *(y5, y6)), u))) → c(+'(*(y0, +(y1, *(y2, y3))), +(*(y4, *(y5, y6)), u)))
+'(*(z0, *(y0, +(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, y6), u)), u))) → c(+'(*(y0, +(y1, *(y2, y3))), +(*(y4, +(*(y5, y6), u)), u)))
+'(*(z0, *(y0, +(y1, +(y2, y3)))), *(z0, +(*(y4, y5), u))) → c(+'(*(y0, +(y1, +(y2, y3))), +(*(y4, y5), u)))
+'(*(z0, *(y0, +(y1, y2))), *(z0, +(*(y3, y4), u))) → c(+'(*(y0, +(y1, y2)), +(*(y3, y4), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, *(y6, y7)), u)), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, +(*(y5, *(y6, y7)), u)), u)))
+'(*(z0, *(y0, *(y1, +(y2, y3)))), *(z0, +(*(y4, +(*(y5, y6), u)), u))) → c(+'(*(y0, *(y1, +(y2, y3))), +(*(y4, +(*(y5, y6), u)), u)))
+'(*(z0, *(y0, *(y1, *(y2, y3)))), *(z0, +(*(y4, +(*(y5, +(*(y6, y7), u)), u)), u))) → c(+'(*(y0, *(y1, *(y2, y3))), +(*(y4, +(*(y5, +(*(y6, y7), u)), u)), u)))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, *(y8, y9))), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), *(y7, *(y8, y9)))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, *(y7, *(y8, y9))), u))))
+'(*(z0, *(z1, *(z2, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, y8)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, +(y3, y4)), *(y7, y8))), u))), u), +'(*(z1, *(z2, *(y2, +(y3, y4)))), *(z1, +(*(z2, *(y7, y8)), u))))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, +(*(y8, y9), u))), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), *(y7, +(*(y8, y9), u)))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, *(y7, +(*(y8, y9), u))), u))))
+'(*(z0, *(z1, *(z2, *(y2, y3)))), +(*(z0, *(z1, +(*(z2, *(y6, y7)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, y3), *(y6, y7))), u))), u), +'(*(z1, *(z2, *(y2, y3))), *(z1, +(*(z2, *(y6, y7)), u))))
+'(*(z0, *(z1, *(z2, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, *(y7, y8)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, *(y3, y4)), *(y7, y8))), u))), u), +'(*(z1, *(z2, +(y2, *(y3, y4)))), *(z1, +(*(z2, *(y7, y8)), u))))
+'(*(z0, *(z1, *(z2, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, *(y3, y4)), +(*(y7, y8), u))), u))), u), +'(*(z1, *(z2, +(y2, *(y3, y4)))), *(z1, +(*(z2, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(z2, +(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(z2, z4), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, +(y3, y4)), z4)), u))), u), +'(*(z1, *(z2, +(y2, +(y3, y4)))), *(z1, +(*(z2, z4), u))))
+'(*(z0, *(z1, *(z2, +(y2, y3)))), +(*(z0, *(z1, +(*(z2, z4), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(+(y2, y3), z4)), u))), u), +'(*(z1, *(z2, +(y2, y3))), *(z1, +(*(z2, z4), u))))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, *(y8, y9)), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), +(*(y7, *(y8, y9)), u))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, +(*(y7, *(y8, y9)), u)), u))))
+'(*(z0, *(z1, *(z2, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, +(y3, y4)), +(*(y7, y8), u))), u))), u), +'(*(z1, *(z2, *(y2, +(y3, y4)))), *(z1, +(*(z2, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(z2, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(z2, +(*(y7, +(*(y8, y9), u)), u)), u))), u)) → c2(+'(*(z0, *(z1, +(*(z2, +(*(y2, *(y3, y4)), +(*(y7, +(*(y8, y9), u)), u))), u))), u), +'(*(z1, *(z2, *(y2, *(y3, y4)))), *(z1, +(*(z2, +(*(y7, +(*(y8, y9), u)), u)), u))))
+'(*(z0, *(z1, *(y1, y2))), +(*(z0, *(z1, *(y4, y5))), u)) → c2(+'(*(z1, *(y1, y2)), *(z1, *(y4, y5))))
+'(*(z0, *(z1, +(y1, y2))), +(*(z0, *(z1, z3)), u)) → c2(+'(*(z1, +(y1, y2)), *(z1, z3)))
+'(*(z0, *(z1, *(y1, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, *(y7, *(y8, y9))), u))), u)) → c2(+'(*(z1, *(y1, *(y2, *(y3, y4)))), *(z1, +(*(y6, *(y7, *(y8, y9))), u))))
+'(*(z0, *(z1, *(y1, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(y6, *(y7, y8)), u))), u)) → c2(+'(*(z1, *(y1, *(y2, +(y3, y4)))), *(z1, +(*(y6, *(y7, y8)), u))))
+'(*(z0, *(z1, *(y1, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, *(y7, +(*(y8, y9), u))), u))), u)) → c2(+'(*(z1, *(y1, *(y2, *(y3, y4)))), *(z1, +(*(y6, *(y7, +(*(y8, y9), u))), u))))
+'(*(z0, *(z1, *(y1, *(y2, y3)))), +(*(z0, *(z1, +(*(y5, *(y6, y7)), u))), u)) → c2(+'(*(z1, *(y1, *(y2, y3))), *(z1, +(*(y5, *(y6, y7)), u))))
+'(*(z0, *(z1, *(y1, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, *(y7, y8)), u))), u)) → c2(+'(*(z1, *(y1, +(y2, *(y3, y4)))), *(z1, +(*(y6, *(y7, y8)), u))))
+'(*(z0, *(z1, *(y1, +(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z1, *(y1, +(y2, *(y3, y4)))), *(z1, +(*(y6, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(y1, +(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(y6, y7), u))), u)) → c2(+'(*(z1, *(y1, +(y2, +(y3, y4)))), *(z1, +(*(y6, y7), u))))
+'(*(z0, *(z1, *(y1, +(y2, y3)))), +(*(z0, *(z1, +(*(y5, y6), u))), u)) → c2(+'(*(z1, *(y1, +(y2, y3))), *(z1, +(*(y5, y6), u))))
+'(*(z0, *(z1, *(y1, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, +(*(y7, *(y8, y9)), u)), u))), u)) → c2(+'(*(z1, *(y1, *(y2, *(y3, y4)))), *(z1, +(*(y6, +(*(y7, *(y8, y9)), u)), u))))
+'(*(z0, *(z1, *(y1, *(y2, +(y3, y4))))), +(*(z0, *(z1, +(*(y6, +(*(y7, y8), u)), u))), u)) → c2(+'(*(z1, *(y1, *(y2, +(y3, y4)))), *(z1, +(*(y6, +(*(y7, y8), u)), u))))
+'(*(z0, *(z1, *(y1, *(y2, *(y3, y4))))), +(*(z0, *(z1, +(*(y6, +(*(y7, +(*(y8, y9), u)), u)), u))), u)) → c2(+'(*(z1, *(y1, *(y2, *(y3, y4)))), *(z1, +(*(y6, +(*(y7, +(*(y8, y9), u)), u)), u))))
K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c1, c2, c2, c

(21) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match(-raise)-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1.

The compatible tree automaton used to show the Match(-raise)-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 3]
transitions:
*0(0, 0) → 0
u0() → 0
+1(0, 0) → 2
*1(0, 2) → 1
+1(0, 0) → 1
+1(0, 0) → 3
*1(0, 3) → 1
*1(0, 2) → 3
*1(0, 3) → 3

(22) BOUNDS(O(1), O(n^1))