Runtime Complexity (innermost) proof of /scratch/tmpnFXJRP/z118.xml

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

0 CpxTRS

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

↳2 CdtProblem

↳3 CdtUnreachableProof (⇔)

↳4 CdtProblem

↳5 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)

↳6 CdtProblem

↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

↳8 CdtProblem

↳9 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))

↳10 CdtProblem

↳11 CdtKnowledgeProof (⇔)

↳12 CdtProblem

↳13 CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))

↳14 CdtProblem

↳15 CdtUnreachableProof (⇔)

↳16 CdtProblem

↳17 CdtMatrixRedPairProof (UPPER BOUND (ADD(O(n^2))))

↳18 CdtProblem

↳19 SIsEmptyProof (⇔)

↳20 BOUNDS(O(1), O(1))

(0) Obligation:

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

a(x1) → g(d(x1))
b(b(b(x1))) → c(d(c(x1)))
b(b(x1)) → a(g(g(x1)))
c(d(x1)) → g(g(x1))
g(g(g(x1))) → b(b(x1))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → g(d(z0))
b(b(b(z0))) → c(d(c(z0)))
b(b(z0)) → a(g(g(z0)))
c(d(z0)) → g(g(z0))
g(g(g(z0))) → b(b(z0))

Tuples:

A(z0) → c1(G(d(z0)))
B(b(b(z0))) → c2(C(d(c(z0))), C(z0))
B(b(z0)) → c3(A(g(g(z0))), G(g(z0)), G(z0))
C(d(z0)) → c4(G(g(z0)), G(z0))
G(g(g(z0))) → c5(B(b(z0)), B(z0))

S tuples:

A(z0) → c1(G(d(z0)))
B(b(b(z0))) → c2(C(d(c(z0))), C(z0))
B(b(z0)) → c3(A(g(g(z0))), G(g(z0)), G(z0))
C(d(z0)) → c4(G(g(z0)), G(z0))
G(g(g(z0))) → c5(B(b(z0)), B(z0))

K tuples:none
Defined Rule Symbols:

a, b, c, g

Defined Pair Symbols:

A, B, C, G

Compound Symbols:

c1, c2, c3, c4, c5

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

B(b(b(z0))) → c2(C(d(c(z0))), C(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → g(d(z0))
b(b(b(z0))) → c(d(c(z0)))
b(b(z0)) → a(g(g(z0)))
c(d(z0)) → g(g(z0))
g(g(g(z0))) → b(b(z0))

Tuples:

A(z0) → c1(G(d(z0)))
C(d(z0)) → c4(G(g(z0)), G(z0))
G(g(g(z0))) → c5(B(b(z0)), B(z0))
B(b(z0)) → c3(A(g(g(z0))), G(g(z0)), G(z0))

S tuples:

A(z0) → c1(G(d(z0)))
B(b(z0)) → c3(A(g(g(z0))), G(g(z0)), G(z0))
C(d(z0)) → c4(G(g(z0)), G(z0))
G(g(g(z0))) → c5(B(b(z0)), B(z0))

K tuples:none
Defined Rule Symbols:

a, b, c, g

Defined Pair Symbols:

A, C, G, B

Compound Symbols:

c1, c4, c5, c3

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 4 dangling nodes:

A(z0) → c1(G(d(z0)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → g(d(z0))
b(b(b(z0))) → c(d(c(z0)))
b(b(z0)) → a(g(g(z0)))
c(d(z0)) → g(g(z0))
g(g(g(z0))) → b(b(z0))

Tuples:

C(d(z0)) → c4(G(g(z0)), G(z0))
G(g(g(z0))) → c5(B(b(z0)), B(z0))
B(b(z0)) → c3(A(g(g(z0))), G(g(z0)), G(z0))

S tuples:

B(b(z0)) → c3(A(g(g(z0))), G(g(z0)), G(z0))
C(d(z0)) → c4(G(g(z0)), G(z0))
G(g(g(z0))) → c5(B(b(z0)), B(z0))

K tuples:none
Defined Rule Symbols:

a, b, c, g

Defined Pair Symbols:

C, G, B

Compound Symbols:

c4, c5, c3

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

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → g(d(z0))
b(b(b(z0))) → c(d(c(z0)))
b(b(z0)) → a(g(g(z0)))
c(d(z0)) → g(g(z0))
g(g(g(z0))) → b(b(z0))

Tuples:

G(g(g(z0))) → c5(B(b(z0)), B(z0))
B(b(z0)) → c3(A(g(g(z0))), G(g(z0)), G(z0))
C(d(z0)) → c1(G(g(z0)))
C(d(z0)) → c1(G(z0))

S tuples:

B(b(z0)) → c3(A(g(g(z0))), G(g(z0)), G(z0))
G(g(g(z0))) → c5(B(b(z0)), B(z0))
C(d(z0)) → c1(G(g(z0)))
C(d(z0)) → c1(G(z0))

K tuples:none
Defined Rule Symbols:

a, b, c, g

Defined Pair Symbols:

G, B, C

Compound Symbols:

c5, c3, c1

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

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → g(d(z0))
b(b(b(z0))) → c(d(c(z0)))
b(b(z0)) → a(g(g(z0)))
c(d(z0)) → g(g(z0))
g(g(g(z0))) → b(b(z0))

Tuples:

G(g(g(z0))) → c5(B(b(z0)), B(z0))
C(d(z0)) → c1(G(g(z0)))
C(d(z0)) → c1(G(z0))
B(b(z0)) → c3(G(g(z0)), G(z0))

S tuples:

G(g(g(z0))) → c5(B(b(z0)), B(z0))
C(d(z0)) → c1(G(g(z0)))
C(d(z0)) → c1(G(z0))
B(b(z0)) → c3(G(g(z0)), G(z0))

K tuples:none
Defined Rule Symbols:

a, b, c, g

Defined Pair Symbols:

G, C, B

Compound Symbols:

c5, c1, c3

(11) CdtKnowledgeProof (EQUIVALENT transformation)

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

C(d(z0)) → c1(G(g(z0)))
C(d(z0)) → c1(G(z0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → g(d(z0))
b(b(b(z0))) → c(d(c(z0)))
b(b(z0)) → a(g(g(z0)))
c(d(z0)) → g(g(z0))
g(g(g(z0))) → b(b(z0))

Tuples:

G(g(g(z0))) → c5(B(b(z0)), B(z0))
C(d(z0)) → c1(G(g(z0)))
C(d(z0)) → c1(G(z0))
B(b(z0)) → c3(G(g(z0)), G(z0))

S tuples:

G(g(g(z0))) → c5(B(b(z0)), B(z0))
B(b(z0)) → c3(G(g(z0)), G(z0))

K tuples:

C(d(z0)) → c1(G(g(z0)))
C(d(z0)) → c1(G(z0))

Defined Rule Symbols:

a, b, c, g

Defined Pair Symbols:

G, C, B

Compound Symbols:

c5, c1, c3

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

Use forward instantiation to replace C(d(z0)) → c1(G(z0)) by

C(d(g(g(y0)))) → c1(G(g(g(y0))))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → g(d(z0))
b(b(b(z0))) → c(d(c(z0)))
b(b(z0)) → a(g(g(z0)))
c(d(z0)) → g(g(z0))
g(g(g(z0))) → b(b(z0))

Tuples:

G(g(g(z0))) → c5(B(b(z0)), B(z0))
C(d(z0)) → c1(G(g(z0)))
B(b(z0)) → c3(G(g(z0)), G(z0))
C(d(g(g(y0)))) → c1(G(g(g(y0))))

S tuples:

G(g(g(z0))) → c5(B(b(z0)), B(z0))
B(b(z0)) → c3(G(g(z0)), G(z0))

K tuples:

C(d(z0)) → c1(G(g(z0)))
C(d(g(g(y0)))) → c1(G(g(g(y0))))

Defined Rule Symbols:

a, b, c, g

Defined Pair Symbols:

G, C, B

Compound Symbols:

c5, c1, c3

(15) CdtUnreachableProof (EQUIVALENT transformation)

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

C(d(g(g(y0)))) → c1(G(g(g(y0))))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → g(d(z0))
b(b(b(z0))) → c(d(c(z0)))
b(b(z0)) → a(g(g(z0)))
c(d(z0)) → g(g(z0))
g(g(g(z0))) → b(b(z0))

Tuples:

C(d(z0)) → c1(G(g(z0)))
G(g(g(z0))) → c5(B(b(z0)), B(z0))
B(b(z0)) → c3(G(g(z0)), G(z0))

S tuples:

G(g(g(z0))) → c5(B(b(z0)), B(z0))
B(b(z0)) → c3(G(g(z0)), G(z0))

K tuples:

C(d(z0)) → c1(G(g(z0)))

Defined Rule Symbols:

a, b, c, g

Defined Pair Symbols:

C, G, B

Compound Symbols:

c1, c5, c3

(17) CdtMatrixRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

G(g(g(z0))) → c5(B(b(z0)), B(z0))
B(b(z0)) → c3(G(g(z0)), G(z0))

We considered the (Usable) Rules:

g(g(g(z0))) → b(b(z0))
b(b(z0)) → a(g(g(z0)))
a(z0) → g(d(z0))

And the Tuples:

C(d(z0)) → c1(G(g(z0)))
G(g(g(z0))) → c5(B(b(z0)), B(z0))
B(b(z0)) → c3(G(g(z0)), G(z0))

The order we found is given by the following interpretation:
Matrix interpretation [MATRO]:
Non-tuple symbols:

M( c(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( a(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( g(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	4	/

x₁

M( b(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	3	/

x₁

M( d(x₁) ) =

/	1	\
\	0	/

/	1	4	\
\	0	0	/

x₁

Tuple symbols:

M( C(x₁) ) =

/	3	\
\	1	/

/	1	1	\
\	0	0	/

x₁

M( c(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( c1(x₁) ) =

/	3	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( c5(x₁, x₂) ) =

/	0	\
\	0	/

/	1	2	\
\	0	0	/

x₁

/	1	2	\
\	0	0	/

x₂

M( B(x₁) ) =

/	0	\
\	0	/

/	0	2	\
\	0	1	/

x₁

M( a(x₁) ) =

/	0	\
\	4	/

/	0	0	\
\	0	0	/

x₁

M( g(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	4	/

x₁

M( c3(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	4	\
\	0	0	/

x₂

M( G(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( b(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	3	/

x₁

Matrix type:
We used a basic matrix type which is not further parametrizeable.

As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

a(z0) → g(d(z0))
b(b(b(z0))) → c(d(c(z0)))
b(b(z0)) → a(g(g(z0)))
c(d(z0)) → g(g(z0))
g(g(g(z0))) → b(b(z0))

Tuples:

C(d(z0)) → c1(G(g(z0)))
G(g(g(z0))) → c5(B(b(z0)), B(z0))
B(b(z0)) → c3(G(g(z0)), G(z0))

S tuples:none
K tuples:

C(d(z0)) → c1(G(g(z0)))
G(g(g(z0))) → c5(B(b(z0)), B(z0))
B(b(z0)) → c3(G(g(z0)), G(z0))

Defined Rule Symbols:

a, b, c, g

Defined Pair Symbols:

C, G, B

Compound Symbols:

c1, c5, c3

(19) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

(3) CdtUnreachableProof (EQUIVALENT transformation)

(4) Obligation:

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

(11) CdtKnowledgeProof (EQUIVALENT transformation)

(12) Obligation:

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

(14) Obligation:

(15) CdtUnreachableProof (EQUIVALENT transformation)

(16) Obligation:

(17) CdtMatrixRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

(18) Obligation:

(19) SIsEmptyProof (EQUIVALENT transformation)

(20) BOUNDS(O(1), O(1))