The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).

(0) Obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).


The TRS R consists of the following rules:

f(0, 1, x) → f(h(x), h(x), x)
h(0) → 0
h(g(x, y)) → y

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

The duplicating contexts are:
f(0, 1, [])

The defined contexts are:
f([], x1, x2)
f(x0, [], x2)

[] just represents basic- or constructor-terms in the following defined contexts:
f([], x1, x2)
f(x0, [], x2)

As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.

(2) Obligation:

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


The TRS R consists of the following rules:

f(0, 1, x) → f(h(x), h(x), x)
h(0) → 0
h(g(x, y)) → y

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1

Tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0), H(z0), H(z0))
H(0) → c1
H(g(z0, z1)) → c2

S tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0), H(z0), H(z0))
H(0) → c1
H(g(z0, z1)) → c2

K tuples:none
Defined Rule Symbols:

f, h

Defined Pair Symbols:

F, H

Compound Symbols:

c, c1, c2

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

Removed 2 trailing nodes:

H(0) → c1
H(g(z0, z1)) → c2

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1

Tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0), H(z0), H(z0))

S tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0), H(z0), H(z0))

K tuples:none
Defined Rule Symbols:

f, h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 2 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1

Tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0))

S tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0))

K tuples:none
Defined Rule Symbols:

f, h

Defined Pair Symbols:

F

Compound Symbols:

c

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

f(0, 1, z0) → f(h(z0), h(z0), z0)

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1

Tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0))

S tuples:

F(0, 1, z0) → c(F(h(z0), h(z0), z0))

K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Use narrowing to replace F(0, 1, z0) → c(F(h(z0), h(z0), z0)) by

F(0, 1, 0) → c(F(h(0), 0, 0))
F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1

Tuples:

F(0, 1, 0) → c(F(h(0), 0, 0))
F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))

S tuples:

F(0, 1, 0) → c(F(h(0), 0, 0))
F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))

K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing nodes:

F(0, 1, 0) → c(F(h(0), 0, 0))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1

Tuples:

F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))

S tuples:

F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))

K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Use narrowing to replace F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1))) by

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1

Tuples:

F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

S tuples:

F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing nodes:

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1

Tuples:

F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))

S tuples:

F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))

K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Use narrowing to replace F(0, 1, 0) → c(F(0, h(0), 0)) by

F(0, 1, 0) → c(F(0, 0, 0))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1

Tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, 0) → c(F(0, 0, 0))

S tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, 0) → c(F(0, 0, 0))

K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing nodes:

F(0, 1, 0) → c(F(0, 0, 0))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(0) → 0
h(g(z0, z1)) → z1

Tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))

S tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))

K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

(23) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

h(0) → 0

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(g(z0, z1)) → z1

Tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))

S tuples:

F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))

K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Use narrowing to replace F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1))) by

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(g(z0, z1)) → z1

Tuples:

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

S tuples:

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:

F

Compound Symbols:

c

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

Removed 1 trailing nodes:

F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

h(g(z0, z1)) → z1

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

h

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(30) BOUNDS(1, 1)