Runtime Complexity (innermost) proof of /tmp/tmpdsH137/MYNAT_nosorts-noand

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 14 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳4 CdtProblem

↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 CdtProblem

↳7 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 216 ms)

↳8 CdtProblem

↳9 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)

↳10 CdtProblem

↳11 CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)), 733 ms)

↳12 CdtProblem

↳13 CdtKnowledgeProof (⇔, 0 ms)

↳14 BOUNDS(1, 1)

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

The TRS R consists of the following rules:

U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)
x(z0, 0) → 0
x(z0, s(z1)) → U21(tt, z1, z0)
activate(z0) → z0

Tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
PLUS(z0, 0) → c4
PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, 0) → c6
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
ACTIVATE(z0) → c8

S tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
PLUS(z0, 0) → c4
PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, 0) → c6
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
ACTIVATE(z0) → c8

K tuples:none
Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

U11', U12', U21', U22', PLUS, X, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8

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

Removed 3 trailing nodes:

X(z0, 0) → c6
ACTIVATE(z0) → c8
PLUS(z0, 0) → c4

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)
x(z0, 0) → 0
x(z0, s(z1)) → U21(tt, z1, z0)
activate(z0) → z0

Tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))

S tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))

K tuples:none
Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

U11', U12', U21', U22', PLUS, X

Compound Symbols:

c, c1, c2, c3, c5, c7

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

Removed 9 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)
x(z0, 0) → 0
x(z0, s(z1)) → U21(tt, z1, z0)
activate(z0) → z0

Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))

S tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))

K tuples:none
Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

PLUS, X, U11', U12', U21', U22'

Compound Symbols:

c5, c7, c, c1, c2, c3

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

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

X(z0, s(z1)) → c7(U21'(tt, z1, z0))

We considered the (Usable) Rules:

activate(z0) → z0

And the Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(PLUS(x₁, x₂)) = 0
POL(U11(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U21(x₁, x₂, x₃)) = 0
POL(U21'(x₁, x₂, x₃)) = x₂
POL(U22(x₁, x₂, x₃)) = 0
POL(U22'(x₁, x₂, x₃)) = x₂
POL(X(x₁, x₂)) = x₂
POL(activate(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(plus(x₁, x₂)) = 0
POL(s(x₁)) = [1] + x₁
POL(tt) = 0
POL(x(x₁, x₂)) = 0

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)
x(z0, 0) → 0
x(z0, s(z1)) → U21(tt, z1, z0)
activate(z0) → z0

Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))

S tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))

K tuples:

X(z0, s(z1)) → c7(U21'(tt, z1, z0))

Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

PLUS, X, U11', U12', U21', U22'

Compound Symbols:

c5, c7, c, c1, c2, c3

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

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

U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)
x(z0, 0) → 0
x(z0, s(z1)) → U21(tt, z1, z0)
activate(z0) → z0

Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))

S tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))

K tuples:

X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))

Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

PLUS, X, U11', U12', U21', U22'

Compound Symbols:

c5, c7, c, c1, c2, c3

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))

We considered the (Usable) Rules:

activate(z0) → z0

And the Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]
POL(PLUS(x₁, x₂)) = x₂
POL(U11(x₁, x₂, x₃)) = [2]x₁ + [2]x₂ + x₃ + [2]x₃² + [2]x₁·x₂ + [2]x₂²
POL(U11'(x₁, x₂, x₃)) = [2]x₁ + x₂ + x₁·x₃ + [2]x₁·x₂
POL(U12(x₁, x₂, x₃)) = [2] + [2]x₁ + x₃ + x₂·x₃ + x₁·x₃ + [2]x₁² + [2]x₁·x₂ + [2]x₂²
POL(U12'(x₁, x₂, x₃)) = [2]x₁ + x₂
POL(U21(x₁, x₂, x₃)) = [1] + [2]x₂ + x₃ + [2]x₃² + x₂·x₃ + [2]x₁·x₃ + x₁² + [2]x₁·x₂ + [2]x₂²
POL(U21'(x₁, x₂, x₃)) = [2]x₁ + [2]x₃ + x₂·x₃ + [2]x₁·x₃ + x₁² + x₁·x₂
POL(U22(x₁, x₂, x₃)) = [1] + x₁ + x₃ + x₃² + [2]x₁·x₃ + [2]x₁² + [2]x₂²
POL(U22'(x₁, x₂, x₃)) = [2]x₁ + x₃ + x₂·x₃
POL(X(x₁, x₂)) = x₁·x₂
POL(activate(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁, x₂)) = x₁ + x₂
POL(c5(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(plus(x₁, x₂)) = [2] + x₁²
POL(s(x₁)) = [2] + x₁
POL(tt) = 0
POL(x(x₁, x₂)) = 0

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1))
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1))
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0) → z0
plus(z0, s(z1)) → U11(tt, z1, z0)
x(z0, 0) → 0
x(z0, s(z1)) → U21(tt, z1, z0)
activate(z0) → z0

Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))

S tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))

K tuples:

X(z0, s(z1)) → c7(U21'(tt, z1, z0))
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))

Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

PLUS, X, U11', U12', U21', U22'

Compound Symbols:

c5, c7, c, c1, c2, c3

(13) CdtKnowledgeProof (EQUIVALENT transformation)

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

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0))

Now S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

(8) Obligation:

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

(10) Obligation:

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

(12) Obligation:

(13) CdtKnowledgeProof (EQUIVALENT transformation)

(14) BOUNDS(1, 1)