Termination Proof using AProVE

Term Rewriting System R:
[t, n, x, a, b, c]
g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

FOLDB(t, s(n)) -> F(foldB(t, n), B)
FOLDB(t, s(n)) -> FOLDB(t, n)
FOLDC(t, s(n)) -> F(foldC(t, n), C)
FOLDC(t, s(n)) -> FOLDC(t, n)
F(t, x) -> F'(t, g(x))
F(t, x) -> G(x)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F'(triple(a, b, c), A) -> F''(foldB(triple(s(a), 0, c), b))
F'(triple(a, b, c), A) -> FOLDB(triple(s(a), 0, c), b)
F''(triple(a, b, c)) -> FOLDC(triple(a, b, 0), c)
FOLD(t, x, s(n)) -> F(fold(t, x, n), x)
FOLD(t, x, s(n)) -> FOLD(t, x, n)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

FOLDB(t, s(n)) -> FOLDB(t, n)
F'(triple(a, b, c), A) -> FOLDB(triple(s(a), 0, c), b)
FOLDC(t, s(n)) -> FOLDC(t, n)
FOLDC(t, s(n)) -> F(foldC(t, n), C)
F''(triple(a, b, c)) -> FOLDC(triple(a, b, 0), c)
F'(triple(a, b, c), A) -> F''(foldB(triple(s(a), 0, c), b))
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(t, x) -> F'(t, g(x))
FOLDB(t, s(n)) -> F(foldB(t, n), B)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(t, x) -> F'(t, g(x))

six new Dependency Pairs are created:

F(t, A) -> F'(t, A)
F(t, B) -> F'(t, A)
F(t, B) -> F'(t, B)
F(t, C) -> F'(t, A)
F(t, C) -> F'(t, B)
F(t, C) -> F'(t, C)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Narrowing Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

F(t, B) -> F'(t, B)
FOLDC(t, s(n)) -> FOLDC(t, n)
F(t, A) -> F'(t, A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(t, C) -> F'(t, B)
F'(triple(a, b, c), A) -> FOLDB(triple(s(a), 0, c), b)
F(t, C) -> F'(t, A)
FOLDC(t, s(n)) -> F(foldC(t, n), C)
F''(triple(a, b, c)) -> FOLDC(triple(a, b, 0), c)
F'(triple(a, b, c), A) -> F''(foldB(triple(s(a), 0, c), b))
F(t, B) -> F'(t, A)
FOLDB(t, s(n)) -> F(foldB(t, n), B)
FOLDB(t, s(n)) -> FOLDB(t, n)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F'(triple(a, b, c), A) -> F''(foldB(triple(s(a), 0, c), b))

two new Dependency Pairs are created:

F'(triple(a', 0, c'), A) -> F''(triple(s(a'), 0, c'))
F'(triple(a', s(n'), c'), A) -> F''(f(foldB(triple(s(a'), 0, c'), n'), B))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 4

                 ↳Narrowing Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

FOLDB(t, s(n)) -> FOLDB(t, n)
FOLDC(t, s(n)) -> FOLDC(t, n)
F(t, C) -> F'(t, B)
F'(triple(a', s(n'), c'), A) -> F''(f(foldB(triple(s(a'), 0, c'), n'), B))
F(t, C) -> F'(t, A)
FOLDC(t, s(n)) -> F(foldC(t, n), C)
F''(triple(a, b, c)) -> FOLDC(triple(a, b, 0), c)
F'(triple(a', 0, c'), A) -> F''(triple(s(a'), 0, c'))
F(t, B) -> F'(t, A)
FOLDB(t, s(n)) -> F(foldB(t, n), B)
F'(triple(a, b, c), A) -> FOLDB(triple(s(a), 0, c), b)
F(t, A) -> F'(t, A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(t, B) -> F'(t, B)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F'(triple(a', s(n'), c'), A) -> F''(f(foldB(triple(s(a'), 0, c'), n'), B))

three new Dependency Pairs are created:

F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 5

                 ↳Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

F(t, B) -> F'(t, B)
FOLDC(t, s(n)) -> FOLDC(t, n)
F(t, A) -> F'(t, A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(t, C) -> F'(t, B)
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(t, C) -> F'(t, A)
FOLDC(t, s(n)) -> F(foldC(t, n), C)
F''(triple(a, b, c)) -> FOLDC(triple(a, b, 0), c)
F'(triple(a', 0, c'), A) -> F''(triple(s(a'), 0, c'))
F'(triple(a, b, c), A) -> FOLDB(triple(s(a), 0, c), b)
F(t, B) -> F'(t, A)
FOLDB(t, s(n)) -> F(foldB(t, n), B)
FOLDB(t, s(n)) -> FOLDB(t, n)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

F(t, A) -> F'(t, A)

one new Dependency Pair is created:

F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 6

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

FOLDB(t, s(n)) -> FOLDB(t, n)
FOLDC(t, s(n)) -> FOLDC(t, n)
F(t, C) -> F'(t, B)
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(t, C) -> F'(t, A)
FOLDC(t, s(n)) -> F(foldC(t, n), C)
F''(triple(a, b, c)) -> FOLDC(triple(a, b, 0), c)
F'(triple(a', 0, c'), A) -> F''(triple(s(a'), 0, c'))
F(t, B) -> F'(t, A)
FOLDB(t, s(n)) -> F(foldB(t, n), B)
F'(triple(a, b, c), A) -> FOLDB(triple(s(a), 0, c), b)
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(t, B) -> F'(t, B)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

FOLDB(t, s(n)) -> FOLDB(t, n)

one new Dependency Pair is created:

FOLDB(t'', s(s(n''))) -> FOLDB(t'', s(n''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 7

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

F(t, C) -> F'(t, B)
FOLDB(t'', s(s(n''))) -> FOLDB(t'', s(n''))
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(t, B) -> F'(t, B)
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F''(triple(a, b, c)) -> FOLDC(triple(a, b, 0), c)
F'(triple(a', 0, c'), A) -> F''(triple(s(a'), 0, c'))
F(t, B) -> F'(t, A)
FOLDB(t, s(n)) -> F(foldB(t, n), B)
F'(triple(a, b, c), A) -> FOLDB(triple(s(a), 0, c), b)
F(t, C) -> F'(t, A)
FOLDC(t, s(n)) -> F(foldC(t, n), C)
FOLDC(t, s(n)) -> FOLDC(t, n)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

FOLDC(t, s(n)) -> FOLDC(t, n)

one new Dependency Pair is created:

FOLDC(t'', s(s(n''))) -> FOLDC(t'', s(n''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 8

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

FOLDB(t'', s(s(n''))) -> FOLDB(t'', s(n''))
F(t, B) -> F'(t, B)
FOLDC(t'', s(s(n''))) -> FOLDC(t'', s(n''))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(t, C) -> F'(t, A)
FOLDC(t, s(n)) -> F(foldC(t, n), C)
F''(triple(a, b, c)) -> FOLDC(triple(a, b, 0), c)
F'(triple(a', 0, c'), A) -> F''(triple(s(a'), 0, c'))
F(t, B) -> F'(t, A)
FOLDB(t, s(n)) -> F(foldB(t, n), B)
F'(triple(a, b, c), A) -> FOLDB(triple(s(a), 0, c), b)
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(t, C) -> F'(t, B)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

F'(triple(a, b, c), A) -> FOLDB(triple(s(a), 0, c), b)

two new Dependency Pairs are created:

F'(triple(a', s(n''), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(n''))
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 9

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

F(t, B) -> F'(t, B)
FOLDC(t'', s(s(n''))) -> FOLDC(t'', s(n''))
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(t, C) -> F'(t, B)
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F'(triple(a', s(n''), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(n''))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(t, C) -> F'(t, A)
FOLDC(t, s(n)) -> F(foldC(t, n), C)
F''(triple(a, b, c)) -> FOLDC(triple(a, b, 0), c)
F'(triple(a', 0, c'), A) -> F''(triple(s(a'), 0, c'))
F(t, B) -> F'(t, A)
FOLDB(t, s(n)) -> F(foldB(t, n), B)
FOLDB(t'', s(s(n''))) -> FOLDB(t'', s(n''))

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

F''(triple(a, b, c)) -> FOLDC(triple(a, b, 0), c)

two new Dependency Pairs are created:

F''(triple(a', b', s(n''))) -> FOLDC(triple(a', b', 0), s(n''))
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 10

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

F(t, C) -> F'(t, B)
FOLDB(t'', s(s(n''))) -> FOLDB(t'', s(n''))
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F(t, B) -> F'(t, A)
FOLDB(t, s(n)) -> F(foldB(t, n), B)
F'(triple(a', s(n''), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(n''))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
FOLDC(t'', s(s(n''))) -> FOLDC(t'', s(n''))
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(t, C) -> F'(t, A)
FOLDC(t, s(n)) -> F(foldC(t, n), C)
F''(triple(a', b', s(n''))) -> FOLDC(triple(a', b', 0), s(n''))
F'(triple(a', 0, c'), A) -> F''(triple(s(a'), 0, c'))
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(t, B) -> F'(t, B)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

F(t, B) -> F'(t, A)

six new Dependency Pairs are created:

F(triple(a''', 0, c'''), B) -> F'(triple(a''', 0, c'''), A)
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 11

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t'', s(s(n''))) -> FOLDB(t'', s(n''))
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
F(triple(a''', 0, c'''), B) -> F'(triple(a''', 0, c'''), A)
F(t, B) -> F'(t, B)
FOLDB(t, s(n)) -> F(foldB(t, n), B)
F'(triple(a', s(n''), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(n''))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
FOLDC(t'', s(s(n''))) -> FOLDC(t'', s(n''))
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(t, C) -> F'(t, A)
FOLDC(t, s(n)) -> F(foldC(t, n), C)
F''(triple(a', b', s(n''))) -> FOLDC(triple(a', b', 0), s(n''))
F'(triple(a', 0, c'), A) -> F''(triple(s(a'), 0, c'))
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(t, C) -> F'(t, B)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

F(t, B) -> F'(t, B)

one new Dependency Pair is created:

F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 12

                 ↳Narrowing Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

F(t, C) -> F'(t, B)
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t'', s(s(n''))) -> FOLDB(t'', s(n''))
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
F(triple(a''', 0, c'''), B) -> F'(triple(a''', 0, c'''), A)
FOLDB(t, s(n)) -> F(foldB(t, n), B)
F'(triple(a', s(n''), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(n''))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
FOLDC(t'', s(s(n''))) -> FOLDC(t'', s(n''))
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))
F'(triple(a', 0, c'), A) -> F''(triple(s(a'), 0, c'))
F(t, C) -> F'(t, A)
FOLDC(t, s(n)) -> F(foldC(t, n), C)
F''(triple(a', b', s(n''))) -> FOLDC(triple(a', b', 0), s(n''))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

FOLDB(t, s(n)) -> F(foldB(t, n), B)

two new Dependency Pairs are created:

FOLDB(t'', s(0)) -> F(t'', B)
FOLDB(t'', s(s(n''))) -> F(f(foldB(t'', n''), B), B)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 13

                 ↳Narrowing Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t'', s(s(n''))) -> F(f(foldB(t'', n''), B), B)
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
F(triple(a''', 0, c'''), B) -> F'(triple(a''', 0, c'''), A)
FOLDB(t'', s(0)) -> F(t'', B)
FOLDB(t'', s(s(n''))) -> FOLDB(t'', s(n''))
F'(triple(a', s(n''), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(n''))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
FOLDC(t'', s(s(n''))) -> FOLDC(t'', s(n''))
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(t, C) -> F'(t, A)
FOLDC(t, s(n)) -> F(foldC(t, n), C)
F''(triple(a', b', s(n''))) -> FOLDC(triple(a', b', 0), s(n''))
F'(triple(a', 0, c'), A) -> F''(triple(s(a'), 0, c'))
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(t, C) -> F'(t, B)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

FOLDB(t'', s(s(n''))) -> F(f(foldB(t'', n''), B), B)

three new Dependency Pairs are created:

FOLDB(t''', s(s(n'''))) -> F(f'(foldB(t''', n'''), g(B)), B)
FOLDB(t''', s(s(0))) -> F(f(t''', B), B)
FOLDB(t''', s(s(s(n')))) -> F(f(f(foldB(t''', n'), B), B), B)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 14

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

F(t, C) -> F'(t, B)
FOLDB(t''', s(s(s(n')))) -> F(f(f(foldB(t''', n'), B), B), B)
FOLDB(t''', s(s(0))) -> F(f(t''', B), B)
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t''', s(s(n'''))) -> F(f'(foldB(t''', n'''), g(B)), B)
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
F(triple(a''', 0, c'''), B) -> F'(triple(a''', 0, c'''), A)
FOLDB(t'', s(0)) -> F(t'', B)
FOLDB(t'', s(s(n''))) -> FOLDB(t'', s(n''))
F'(triple(a', s(n''), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(n''))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
FOLDC(t'', s(s(n''))) -> FOLDC(t'', s(n''))
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(t, C) -> F'(t, A)
FOLDC(t, s(n)) -> F(foldC(t, n), C)
F''(triple(a', b', s(n''))) -> FOLDC(triple(a', b', 0), s(n''))
F'(triple(a', 0, c'), A) -> F''(triple(s(a'), 0, c'))
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

F(t, C) -> F'(t, A)

six new Dependency Pairs are created:

F(triple(a''', 0, c'''), C) -> F'(triple(a''', 0, c'''), A)
F(triple(a'''', s(n''''), c''''), C) -> F'(triple(a'''', s(n''''), c''''), A)
F(triple(a'''', s(0), c''''), C) -> F'(triple(a'''', s(0), c''''), A)
F(triple(a'''', s(s(n'''')), c''''), C) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a''', s(n''''), c'''), C) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a''', s(s(n'''''')), c'''), C) -> F'(triple(a''', s(s(n'''''')), c'''), A)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 15

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

F(t, C) -> F'(t, B)

one new Dependency Pair is created:

F(triple(a'', b'', c''), C) -> F'(triple(a'', b'', c''), B)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 16

                 ↳Narrowing Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

F(triple(a'', b'', c''), C) -> F'(triple(a'', b'', c''), B)
F(triple(a''', s(n''''), c'''), C) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'''', s(s(n'''')), c''''), C) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), C) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t''', s(s(s(n')))) -> F(f(f(foldB(t''', n'), B), B), B)
FOLDB(t''', s(s(0))) -> F(f(t''', B), B)
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t''', s(s(n'''))) -> F(f'(foldB(t''', n'''), g(B)), B)
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
F(triple(a''', 0, c'''), B) -> F'(triple(a''', 0, c'''), A)
FOLDB(t'', s(0)) -> F(t'', B)
FOLDB(t'', s(s(n''))) -> FOLDB(t'', s(n''))
F'(triple(a', s(n''), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(n''))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F(triple(a'''', s(n''''), c''''), C) -> F'(triple(a'''', s(n''''), c''''), A)
FOLDC(t'', s(s(n''))) -> FOLDC(t'', s(n''))
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))
F'(triple(a', 0, c'), A) -> F''(triple(s(a'), 0, c'))
F(triple(a''', 0, c'''), C) -> F'(triple(a''', 0, c'''), A)
FOLDC(t, s(n)) -> F(foldC(t, n), C)
F''(triple(a', b', s(n''))) -> FOLDC(triple(a', b', 0), s(n''))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(triple(a''', s(s(n'''''')), c'''), C) -> F'(triple(a''', s(s(n'''''')), c'''), A)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

FOLDC(t, s(n)) -> F(foldC(t, n), C)

two new Dependency Pairs are created:

FOLDC(t'', s(0)) -> F(t'', C)
FOLDC(t'', s(s(n''))) -> F(f(foldC(t'', n''), C), C)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 17

                 ↳Narrowing Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

FOLDC(t'', s(s(n''))) -> F(f(foldC(t'', n''), C), C)

three new Dependency Pairs are created:

FOLDC(t''', s(s(n'''))) -> F(f'(foldC(t''', n'''), g(C)), C)
FOLDC(t''', s(s(0))) -> F(f(t''', C), C)
FOLDC(t''', s(s(s(n')))) -> F(f(f(foldC(t''', n'), C), C), C)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 18

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

FOLDC(t''', s(s(s(n')))) -> F(f(f(foldC(t''', n'), C), C), C)
FOLDC(t''', s(s(0))) -> F(f(t''', C), C)
F(triple(a'', b'', c''), C) -> F'(triple(a'', b'', c''), B)
F(triple(a''', s(n''''), c'''), C) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'''', s(s(n'''')), c''''), C) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), C) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t''', s(s(s(n')))) -> F(f(f(foldB(t''', n'), B), B), B)
FOLDB(t''', s(s(0))) -> F(f(t''', B), B)
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t''', s(s(n'''))) -> F(f'(foldB(t''', n'''), g(B)), B)
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
F(triple(a''', 0, c'''), B) -> F'(triple(a''', 0, c'''), A)
FOLDB(t'', s(0)) -> F(t'', B)
FOLDB(t'', s(s(n''))) -> FOLDB(t'', s(n''))
F'(triple(a', s(n''), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(n''))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F(triple(a'''', s(n''''), c''''), C) -> F'(triple(a'''', s(n''''), c''''), A)
FOLDC(t''', s(s(n'''))) -> F(f'(foldC(t''', n'''), g(C)), C)
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))
F'(triple(a', 0, c'), A) -> F''(triple(s(a'), 0, c'))
F(triple(a''', 0, c'''), C) -> F'(triple(a''', 0, c'''), A)
FOLDC(t'', s(0)) -> F(t'', C)
FOLDC(t'', s(s(n''))) -> FOLDC(t'', s(n''))
F''(triple(a', b', s(n''))) -> FOLDC(triple(a', b', 0), s(n''))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(triple(a''', s(s(n'''''')), c'''), C) -> F'(triple(a''', s(s(n'''''')), c'''), A)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

F'(triple(a', 0, c'), A) -> F''(triple(s(a'), 0, c'))

two new Dependency Pairs are created:

F'(triple(a''', 0, s(n'''')), A) -> F''(triple(s(a'''), 0, s(n'''')))
F'(triple(a''', 0, s(s(n''''''))), A) -> F''(triple(s(a'''), 0, s(s(n''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 19

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

FOLDC(t''', s(s(0))) -> F(f(t''', C), C)
F(triple(a'', b'', c''), C) -> F'(triple(a'', b'', c''), B)
F(triple(a''', s(s(n'''''')), c'''), C) -> F'(triple(a''', s(s(n'''''')), c'''), A)
F(triple(a''', s(n''''), c'''), C) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'''', s(s(n'''')), c''''), C) -> F'(triple(a'''', s(s(n'''')), c''''), A)
FOLDB(t''', s(s(s(n')))) -> F(f(f(foldB(t''', n'), B), B), B)
FOLDB(t''', s(s(0))) -> F(f(t''', B), B)
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t''', s(s(n'''))) -> F(f'(foldB(t''', n'''), g(B)), B)
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
F'(triple(a''', 0, s(s(n''''''))), A) -> F''(triple(s(a'''), 0, s(s(n''''''))))
F(triple(a''', 0, c'''), B) -> F'(triple(a''', 0, c'''), A)
FOLDB(t'', s(0)) -> F(t'', B)
FOLDB(t'', s(s(n''))) -> FOLDB(t'', s(n''))
F'(triple(a', s(n''), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(n''))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F(triple(a'''', s(0), c''''), C) -> F'(triple(a'''', s(0), c''''), A)
FOLDC(t''', s(s(n'''))) -> F(f'(foldC(t''', n'''), g(C)), C)
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(triple(a'''', s(n''''), c''''), C) -> F'(triple(a'''', s(n''''), c''''), A)
FOLDC(t'', s(0)) -> F(t'', C)
FOLDC(t'', s(s(n''))) -> FOLDC(t'', s(n''))
F''(triple(a', b', s(n''))) -> FOLDC(triple(a', b', 0), s(n''))
F'(triple(a''', 0, s(n'''')), A) -> F''(triple(s(a'''), 0, s(n'''')))
F(triple(a''', 0, c'''), C) -> F'(triple(a''', 0, c'''), A)
FOLDC(t''', s(s(s(n')))) -> F(f(f(foldC(t''', n'), C), C), C)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

FOLDB(t'', s(s(n''))) -> FOLDB(t'', s(n''))

five new Dependency Pairs are created:

FOLDB(t'''', s(s(s(n'''')))) -> FOLDB(t'''', s(s(n'''')))
FOLDB(t'''', s(s(0))) -> FOLDB(t'''', s(0))
FOLDB(t''', s(s(s(n''''')))) -> FOLDB(t''', s(s(n''''')))
FOLDB(t''', s(s(s(0)))) -> FOLDB(t''', s(s(0)))
FOLDB(t''', s(s(s(s(n''''))))) -> FOLDB(t''', s(s(s(n''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 20

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

FOLDC(t''', s(s(s(n')))) -> F(f(f(foldC(t''', n'), C), C), C)
F(triple(a'', b'', c''), C) -> F'(triple(a'', b'', c''), B)
F(triple(a''', s(s(n'''''')), c'''), C) -> F'(triple(a''', s(s(n'''''')), c'''), A)
F(triple(a''', s(n''''), c'''), C) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'''', s(s(n'''')), c''''), C) -> F'(triple(a'''', s(s(n'''')), c''''), A)
FOLDB(t''', s(s(s(s(n''''))))) -> FOLDB(t''', s(s(s(n''''))))
FOLDB(t''', s(s(s(0)))) -> FOLDB(t''', s(s(0)))
FOLDB(t''', s(s(s(n''''')))) -> FOLDB(t''', s(s(n''''')))
FOLDB(t'''', s(s(0))) -> FOLDB(t'''', s(0))
FOLDB(t'''', s(s(s(n'''')))) -> FOLDB(t'''', s(s(n'''')))
FOLDB(t''', s(s(s(n')))) -> F(f(f(foldB(t''', n'), B), B), B)
FOLDB(t''', s(s(0))) -> F(f(t''', B), B)
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t''', s(s(n'''))) -> F(f'(foldB(t''', n'''), g(B)), B)
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
F'(triple(a''', 0, s(s(n''''''))), A) -> F''(triple(s(a'''), 0, s(s(n''''''))))
F(triple(a''', 0, c'''), B) -> F'(triple(a''', 0, c'''), A)
FOLDB(t'', s(0)) -> F(t'', B)
F'(triple(a', s(n''), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(n''))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F(triple(a'''', s(0), c''''), C) -> F'(triple(a'''', s(0), c''''), A)
FOLDC(t''', s(s(n'''))) -> F(f'(foldC(t''', n'''), g(C)), C)
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(triple(a'''', s(n''''), c''''), C) -> F'(triple(a'''', s(n''''), c''''), A)
FOLDC(t'', s(0)) -> F(t'', C)
FOLDC(t'', s(s(n''))) -> FOLDC(t'', s(n''))
F''(triple(a', b', s(n''))) -> FOLDC(triple(a', b', 0), s(n''))
F'(triple(a''', 0, s(n'''')), A) -> F''(triple(s(a'''), 0, s(n'''')))
F(triple(a''', 0, c'''), C) -> F'(triple(a''', 0, c'''), A)
FOLDC(t''', s(s(0))) -> F(f(t''', C), C)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

FOLDC(t'', s(s(n''))) -> FOLDC(t'', s(n''))

five new Dependency Pairs are created:

FOLDC(t'''', s(s(s(n'''')))) -> FOLDC(t'''', s(s(n'''')))
FOLDC(t'''', s(s(0))) -> FOLDC(t'''', s(0))
FOLDC(t''', s(s(s(n''''')))) -> FOLDC(t''', s(s(n''''')))
FOLDC(t''', s(s(s(0)))) -> FOLDC(t''', s(s(0)))
FOLDC(t''', s(s(s(s(n''''))))) -> FOLDC(t''', s(s(s(n''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 21

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

FOLDC(t''', s(s(s(s(n''''))))) -> FOLDC(t''', s(s(s(n''''))))
FOLDC(t''', s(s(s(0)))) -> FOLDC(t''', s(s(0)))
FOLDC(t''', s(s(s(n''''')))) -> FOLDC(t''', s(s(n''''')))
FOLDC(t'''', s(s(0))) -> FOLDC(t'''', s(0))
FOLDC(t'''', s(s(s(n'''')))) -> FOLDC(t'''', s(s(n'''')))
FOLDC(t''', s(s(0))) -> F(f(t''', C), C)
F(triple(a'', b'', c''), C) -> F'(triple(a'', b'', c''), B)
F(triple(a''', s(s(n'''''')), c'''), C) -> F'(triple(a''', s(s(n'''''')), c'''), A)
F(triple(a''', s(n''''), c'''), C) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'''', s(s(n'''')), c''''), C) -> F'(triple(a'''', s(s(n'''')), c''''), A)
FOLDB(t''', s(s(s(s(n''''))))) -> FOLDB(t''', s(s(s(n''''))))
FOLDB(t''', s(s(s(0)))) -> FOLDB(t''', s(s(0)))
FOLDB(t''', s(s(s(n''''')))) -> FOLDB(t''', s(s(n''''')))
FOLDB(t'''', s(s(0))) -> FOLDB(t'''', s(0))
FOLDB(t'''', s(s(s(n'''')))) -> FOLDB(t'''', s(s(n'''')))
FOLDB(t''', s(s(s(n')))) -> F(f(f(foldB(t''', n'), B), B), B)
FOLDB(t''', s(s(0))) -> F(f(t''', B), B)
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t''', s(s(n'''))) -> F(f'(foldB(t''', n'''), g(B)), B)
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
F'(triple(a''', 0, s(s(n''''''))), A) -> F''(triple(s(a'''), 0, s(s(n''''''))))
F(triple(a''', 0, c'''), B) -> F'(triple(a''', 0, c'''), A)
FOLDB(t'', s(0)) -> F(t'', B)
F'(triple(a', s(n''), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(n''))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F(triple(a'''', s(0), c''''), C) -> F'(triple(a'''', s(0), c''''), A)
FOLDC(t''', s(s(n'''))) -> F(f'(foldC(t''', n'''), g(C)), C)
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(triple(a'''', s(n''''), c''''), C) -> F'(triple(a'''', s(n''''), c''''), A)
FOLDC(t'', s(0)) -> F(t'', C)
F''(triple(a', b', s(n''))) -> FOLDC(triple(a', b', 0), s(n''))
F'(triple(a''', 0, s(n'''')), A) -> F''(triple(s(a'''), 0, s(n'''')))
F(triple(a''', 0, c'''), C) -> F'(triple(a''', 0, c'''), A)
FOLDC(t''', s(s(s(n')))) -> F(f(f(foldC(t''', n'), C), C), C)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

F'(triple(a', s(n''), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(n''))

eight new Dependency Pairs are created:

F'(triple(a'', s(0), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(0))
F'(triple(a'', s(s(n''''')), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(n''''')))
F'(triple(a'', s(s(0)), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(0)))
F'(triple(a'', s(s(s(n''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''))))
F'(triple(a'', s(s(s(n''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''''))))
F'(triple(a'', s(s(s(n'''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n'''''''))))
F'(triple(a'', s(s(s(0))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(0))))
F'(triple(a'', s(s(s(s(n'''''')))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(s(n'''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 22

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

FOLDC(t''', s(s(s(0)))) -> FOLDC(t''', s(s(0)))
FOLDC(t''', s(s(s(n''''')))) -> FOLDC(t''', s(s(n''''')))
FOLDC(t'''', s(s(0))) -> FOLDC(t'''', s(0))
FOLDC(t'''', s(s(s(n'''')))) -> FOLDC(t'''', s(s(n'''')))
FOLDC(t''', s(s(s(n')))) -> F(f(f(foldC(t''', n'), C), C), C)
F(triple(a'', b'', c''), C) -> F'(triple(a'', b'', c''), B)
F(triple(a''', s(s(n'''''')), c'''), C) -> F'(triple(a''', s(s(n'''''')), c'''), A)
F(triple(a''', s(n''''), c'''), C) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'''', s(s(n'''')), c''''), C) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)
F'(triple(a'', s(s(s(s(n'''''')))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(s(n'''''')))))
F'(triple(a'', s(s(s(0))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(0))))
F'(triple(a'', s(s(s(n'''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n'''''''))))
FOLDB(t''', s(s(s(s(n''''))))) -> FOLDB(t''', s(s(s(n''''))))
FOLDB(t''', s(s(s(0)))) -> FOLDB(t''', s(s(0)))
FOLDB(t''', s(s(s(n''''')))) -> FOLDB(t''', s(s(n''''')))
FOLDB(t'''', s(s(s(n'''')))) -> FOLDB(t'''', s(s(n'''')))
F'(triple(a'', s(s(s(n''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''''))))
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)
FOLDB(t''', s(s(s(n')))) -> F(f(f(foldB(t''', n'), B), B), B)
F'(triple(a'', s(s(s(n''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''))))
FOLDB(t'''', s(s(0))) -> FOLDB(t'''', s(0))
F'(triple(a'', s(s(0)), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(0)))
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
FOLDB(t''', s(s(0))) -> F(f(t''', B), B)
F'(triple(a'', s(s(n''''')), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(n''''')))
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t''', s(s(n'''))) -> F(f'(foldB(t''', n'''), g(B)), B)
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
F'(triple(a''', 0, s(s(n''''''))), A) -> F''(triple(s(a'''), 0, s(s(n''''''))))
F(triple(a''', 0, c'''), B) -> F'(triple(a''', 0, c'''), A)
FOLDB(t'', s(0)) -> F(t'', B)
F'(triple(a'', s(0), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(0))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F(triple(a'''', s(0), c''''), C) -> F'(triple(a'''', s(0), c''''), A)
FOLDC(t''', s(s(0))) -> F(f(t''', C), C)
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(triple(a'''', s(n''''), c''''), C) -> F'(triple(a'''', s(n''''), c''''), A)
FOLDC(t'', s(0)) -> F(t'', C)
F''(triple(a', b', s(n''))) -> FOLDC(triple(a', b', 0), s(n''))
F'(triple(a''', 0, s(n'''')), A) -> F''(triple(s(a'''), 0, s(n'''')))
F(triple(a''', 0, c'''), C) -> F'(triple(a''', 0, c'''), A)
FOLDC(t''', s(s(n'''))) -> F(f'(foldC(t''', n'''), g(C)), C)
FOLDC(t''', s(s(s(s(n''''))))) -> FOLDC(t''', s(s(s(n''''))))

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

F''(triple(a', b', s(n''))) -> FOLDC(triple(a', b', 0), s(n''))

eight new Dependency Pairs are created:

F''(triple(a'', b'', s(0))) -> FOLDC(triple(a'', b'', 0), s(0))
F''(triple(a'', b'', s(s(n''''')))) -> FOLDC(triple(a'', b'', 0), s(s(n''''')))
F''(triple(a'', b'', s(s(0)))) -> FOLDC(triple(a'', b'', 0), s(s(0)))
F''(triple(a'', b'', s(s(s(n''''))))) -> FOLDC(triple(a'', b'', 0), s(s(s(n''''))))
F''(triple(a'', b'', s(s(s(n''''''))))) -> FOLDC(triple(a'', b'', 0), s(s(s(n''''''))))
F''(triple(a'', b'', s(s(s(n'''''''))))) -> FOLDC(triple(a'', b'', 0), s(s(s(n'''''''))))
F''(triple(a'', b'', s(s(s(0))))) -> FOLDC(triple(a'', b'', 0), s(s(s(0))))
F''(triple(a'', b'', s(s(s(s(n'''''')))))) -> FOLDC(triple(a'', b'', 0), s(s(s(s(n'''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 23

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

F(triple(a'', b'', c''), C) -> F'(triple(a'', b'', c''), B)
F(triple(a''', s(s(n'''''')), c'''), C) -> F'(triple(a''', s(s(n'''''')), c'''), A)
F(triple(a''', s(n''''), c'''), C) -> F'(triple(a''', s(n''''), c'''), A)
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)
F'(triple(a'', s(s(s(s(n'''''')))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(s(n'''''')))))
F'(triple(a'', s(s(s(0))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(0))))
F'(triple(a'', s(s(s(n'''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n'''''''))))
FOLDB(t''', s(s(s(s(n''''))))) -> FOLDB(t''', s(s(s(n''''))))
FOLDB(t''', s(s(s(0)))) -> FOLDB(t''', s(s(0)))
FOLDB(t''', s(s(s(n''''')))) -> FOLDB(t''', s(s(n''''')))
FOLDB(t'''', s(s(s(n'''')))) -> FOLDB(t'''', s(s(n'''')))
F'(triple(a'', s(s(s(n''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''''))))
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)
FOLDB(t''', s(s(s(n')))) -> F(f(f(foldB(t''', n'), B), B), B)
F'(triple(a'', s(s(s(n''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''))))
FOLDB(t'''', s(s(0))) -> FOLDB(t'''', s(0))
F'(triple(a'', s(s(0)), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(0)))
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
FOLDB(t''', s(s(0))) -> F(f(t''', B), B)
F'(triple(a'', s(s(n''''')), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(n''''')))
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t'', s(0)) -> F(t'', B)
F'(triple(a'', s(0), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(0))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
F'(triple(a''', 0, s(s(n''''''))), A) -> F''(triple(s(a'''), 0, s(s(n''''''))))
F(triple(a''', 0, c'''), B) -> F'(triple(a''', 0, c'''), A)
FOLDB(t''', s(s(n'''))) -> F(f'(foldB(t''', n'''), g(B)), B)
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F''(triple(a'', b'', s(s(s(s(n'''''')))))) -> FOLDC(triple(a'', b'', 0), s(s(s(s(n'''''')))))
F''(triple(a'', b'', s(s(s(0))))) -> FOLDC(triple(a'', b'', 0), s(s(s(0))))
F''(triple(a'', b'', s(s(s(n'''''''))))) -> FOLDC(triple(a'', b'', 0), s(s(s(n'''''''))))
F''(triple(a'', b'', s(s(s(n''''''))))) -> FOLDC(triple(a'', b'', 0), s(s(s(n''''''))))
FOLDC(t''', s(s(s(s(n''''))))) -> FOLDC(t''', s(s(s(n''''))))
FOLDC(t''', s(s(s(n''''')))) -> FOLDC(t''', s(s(n''''')))
FOLDC(t'''', s(s(s(n'''')))) -> FOLDC(t'''', s(s(n'''')))
F''(triple(a'', b'', s(s(s(n''''))))) -> FOLDC(triple(a'', b'', 0), s(s(s(n''''))))
FOLDC(t'''', s(s(0))) -> FOLDC(t'''', s(0))
F''(triple(a'', b'', s(s(0)))) -> FOLDC(triple(a'', b'', 0), s(s(0)))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F(triple(a'''', s(s(n'''')), c''''), C) -> F'(triple(a'''', s(s(n'''')), c''''), A)
FOLDC(t''', s(s(s(n')))) -> F(f(f(foldC(t''', n'), C), C), C)
F''(triple(a'', b'', s(s(n''''')))) -> FOLDC(triple(a'', b'', 0), s(s(n''''')))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F(triple(a'''', s(0), c''''), C) -> F'(triple(a'''', s(0), c''''), A)
FOLDC(t'', s(0)) -> F(t'', C)
F''(triple(a'', b'', s(0))) -> FOLDC(triple(a'', b'', 0), s(0))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(triple(a'''', s(n''''), c''''), C) -> F'(triple(a'''', s(n''''), c''''), A)
FOLDC(t''', s(s(0))) -> F(f(t''', C), C)
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))
F'(triple(a''', 0, s(n'''')), A) -> F''(triple(s(a'''), 0, s(n'''')))
F(triple(a''', 0, c'''), C) -> F'(triple(a''', 0, c'''), A)
FOLDC(t''', s(s(n'''))) -> F(f'(foldC(t''', n'''), g(C)), C)
FOLDC(t''', s(s(s(0)))) -> FOLDC(t''', s(s(0)))

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

FOLDB(t'', s(0)) -> F(t'', B)

seven new Dependency Pairs are created:

FOLDB(triple(a''''', 0, c'''''), s(0)) -> F(triple(a''''', 0, c'''''), B)
FOLDB(triple(a'''''', s(n''''''), c''''''), s(0)) -> F(triple(a'''''', s(n''''''), c''''''), B)
FOLDB(triple(a'''''', s(0), c''''''), s(0)) -> F(triple(a'''''', s(0), c''''''), B)
FOLDB(triple(a'''''', s(s(n'''''')), c''''''), s(0)) -> F(triple(a'''''', s(s(n'''''')), c''''''), B)
FOLDB(triple(a''''', s(n''''''), c'''''), s(0)) -> F(triple(a''''', s(n''''''), c'''''), B)
FOLDB(triple(a''''', s(s(n'''''''')), c'''''), s(0)) -> F(triple(a''''', s(s(n'''''''')), c'''''), B)
FOLDB(triple(a'''', b'''', c''''), s(0)) -> F(triple(a'''', b'''', c''''), B)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 24

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

F(triple(a''', s(s(n'''''')), c'''), C) -> F'(triple(a''', s(s(n'''''')), c'''), A)
F(triple(a''', s(n''''), c'''), C) -> F'(triple(a''', s(n''''), c'''), A)
FOLDB(triple(a'''', b'''', c''''), s(0)) -> F(triple(a'''', b'''', c''''), B)
FOLDB(triple(a''''', s(s(n'''''''')), c'''''), s(0)) -> F(triple(a''''', s(s(n'''''''')), c'''''), B)
FOLDB(triple(a''''', s(n''''''), c'''''), s(0)) -> F(triple(a''''', s(n''''''), c'''''), B)
FOLDB(triple(a'''''', s(s(n'''''')), c''''''), s(0)) -> F(triple(a'''''', s(s(n'''''')), c''''''), B)
FOLDB(triple(a'''''', s(0), c''''''), s(0)) -> F(triple(a'''''', s(0), c''''''), B)
F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)
F'(triple(a'', s(s(s(s(n'''''')))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(s(n'''''')))))
F'(triple(a'', s(s(s(0))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(0))))
F'(triple(a'', s(s(s(n'''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n'''''''))))
FOLDB(t''', s(s(s(s(n''''))))) -> FOLDB(t''', s(s(s(n''''))))
FOLDB(t''', s(s(s(0)))) -> FOLDB(t''', s(s(0)))
FOLDB(t''', s(s(s(n''''')))) -> FOLDB(t''', s(s(n''''')))
FOLDB(t'''', s(s(s(n'''')))) -> FOLDB(t'''', s(s(n'''')))
F'(triple(a'', s(s(s(n''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''''))))
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)
FOLDB(t''', s(s(s(n')))) -> F(f(f(foldB(t''', n'), B), B), B)
F'(triple(a'', s(s(s(n''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''))))
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
FOLDB(triple(a'''''', s(n''''''), c''''''), s(0)) -> F(triple(a'''''', s(n''''''), c''''''), B)
FOLDB(t'''', s(s(0))) -> FOLDB(t'''', s(0))
F'(triple(a'', s(s(0)), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(0)))
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t''', s(s(0))) -> F(f(t''', B), B)
F'(triple(a'', s(s(n''''')), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(n''''')))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
FOLDB(t''', s(s(n'''))) -> F(f'(foldB(t''', n'''), g(B)), B)
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F''(triple(a'', b'', s(s(s(s(n'''''')))))) -> FOLDC(triple(a'', b'', 0), s(s(s(s(n'''''')))))
F''(triple(a'', b'', s(s(s(0))))) -> FOLDC(triple(a'', b'', 0), s(s(s(0))))
F''(triple(a'', b'', s(s(s(n'''''''))))) -> FOLDC(triple(a'', b'', 0), s(s(s(n'''''''))))
FOLDC(t''', s(s(s(s(n''''))))) -> FOLDC(t''', s(s(s(n''''))))
FOLDC(t''', s(s(s(0)))) -> FOLDC(t''', s(s(0)))
FOLDC(t''', s(s(s(n''''')))) -> FOLDC(t''', s(s(n''''')))
FOLDC(t'''', s(s(s(n'''')))) -> FOLDC(t'''', s(s(n'''')))
F''(triple(a'', b'', s(s(s(n''''''))))) -> FOLDC(triple(a'', b'', 0), s(s(s(n''''''))))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F(triple(a'''', s(s(n'''')), c''''), C) -> F'(triple(a'''', s(s(n'''')), c''''), A)
FOLDC(t''', s(s(s(n')))) -> F(f(f(foldC(t''', n'), C), C), C)
F''(triple(a'', b'', s(s(s(n''''))))) -> FOLDC(triple(a'', b'', 0), s(s(s(n''''))))
FOLDC(t'''', s(s(0))) -> FOLDC(t'''', s(0))
F''(triple(a'', b'', s(s(0)))) -> FOLDC(triple(a'', b'', 0), s(s(0)))
F'(triple(a''', 0, s(s(n''''''))), A) -> F''(triple(s(a'''), 0, s(s(n''''''))))
F(triple(a''', 0, c'''), B) -> F'(triple(a''', 0, c'''), A)
FOLDB(triple(a''''', 0, c'''''), s(0)) -> F(triple(a''''', 0, c'''''), B)
F'(triple(a'', s(0), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(0))
F(triple(a'''', s(0), c''''), C) -> F'(triple(a'''', s(0), c''''), A)
FOLDC(t''', s(s(0))) -> F(f(t''', C), C)
F''(triple(a'', b'', s(s(n''''')))) -> FOLDC(triple(a'', b'', 0), s(s(n''''')))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F(triple(a'''', s(n''''), c''''), C) -> F'(triple(a'''', s(n''''), c''''), A)
FOLDC(t'', s(0)) -> F(t'', C)
F''(triple(a'', b'', s(0))) -> FOLDC(triple(a'', b'', 0), s(0))
F'(triple(a''', 0, s(n'''')), A) -> F''(triple(s(a'''), 0, s(n'''')))
F(triple(a''', 0, c'''), C) -> F'(triple(a''', 0, c'''), A)
FOLDC(t''', s(s(n'''))) -> F(f'(foldC(t''', n'''), g(C)), C)
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(triple(a'', b'', c''), C) -> F'(triple(a'', b'', c''), B)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

FOLDC(t'', s(0)) -> F(t'', C)

seven new Dependency Pairs are created:

FOLDC(triple(a''''', 0, c'''''), s(0)) -> F(triple(a''''', 0, c'''''), C)
FOLDC(triple(a'''''', s(n''''''), c''''''), s(0)) -> F(triple(a'''''', s(n''''''), c''''''), C)
FOLDC(triple(a'''''', s(0), c''''''), s(0)) -> F(triple(a'''''', s(0), c''''''), C)
FOLDC(triple(a'''''', s(s(n'''''')), c''''''), s(0)) -> F(triple(a'''''', s(s(n'''''')), c''''''), C)
FOLDC(triple(a''''', s(n''''''), c'''''), s(0)) -> F(triple(a''''', s(n''''''), c'''''), C)
FOLDC(triple(a''''', s(s(n'''''''')), c'''''), s(0)) -> F(triple(a''''', s(s(n'''''''')), c'''''), C)
FOLDC(triple(a'''', b'''', c''''), s(0)) -> F(triple(a'''', b'''', c''''), C)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 25

                 ↳Polynomial Ordering

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

FOLDC(triple(a'''', b'''', c''''), s(0)) -> F(triple(a'''', b'''', c''''), C)
FOLDC(triple(a''''', s(s(n'''''''')), c'''''), s(0)) -> F(triple(a''''', s(s(n'''''''')), c'''''), C)
FOLDC(triple(a''''', s(n''''''), c'''''), s(0)) -> F(triple(a''''', s(n''''''), c'''''), C)
FOLDC(triple(a'''''', s(s(n'''''')), c''''''), s(0)) -> F(triple(a'''''', s(s(n'''''')), c''''''), C)
FOLDC(triple(a'''''', s(0), c''''''), s(0)) -> F(triple(a'''''', s(0), c''''''), C)
F''(triple(a'', b'', s(s(s(s(n'''''')))))) -> FOLDC(triple(a'', b'', 0), s(s(s(s(n'''''')))))
F''(triple(a'', b'', s(s(s(0))))) -> FOLDC(triple(a'', b'', 0), s(s(s(0))))
F''(triple(a'', b'', s(s(s(n'''''''))))) -> FOLDC(triple(a'', b'', 0), s(s(s(n'''''''))))
F''(triple(a'', b'', s(s(s(n''''''))))) -> FOLDC(triple(a'', b'', 0), s(s(s(n''''''))))
FOLDC(t''', s(s(s(s(n''''))))) -> FOLDC(t''', s(s(s(n''''))))
FOLDC(t''', s(s(s(0)))) -> FOLDC(t''', s(s(0)))
FOLDC(t''', s(s(s(n''''')))) -> FOLDC(t''', s(s(n''''')))
FOLDC(t'''', s(s(s(n'''')))) -> FOLDC(t'''', s(s(n'''')))
F(triple(a''', s(n''''), c'''), C) -> F'(triple(a''', s(n''''), c'''), A)
FOLDB(triple(a'''', b'''', c''''), s(0)) -> F(triple(a'''', b'''', c''''), B)
FOLDB(triple(a''''', s(s(n'''''''')), c'''''), s(0)) -> F(triple(a''''', s(s(n'''''''')), c'''''), B)
FOLDB(triple(a''''', s(n''''''), c'''''), s(0)) -> F(triple(a''''', s(n''''''), c'''''), B)
FOLDB(triple(a'''''', s(s(n'''''')), c''''''), s(0)) -> F(triple(a'''''', s(s(n'''''')), c''''''), B)
FOLDB(triple(a'''''', s(0), c''''''), s(0)) -> F(triple(a'''''', s(0), c''''''), B)
F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)
F'(triple(a'', s(s(s(s(n'''''')))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(s(n'''''')))))
F'(triple(a'', s(s(s(0))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(0))))
F'(triple(a'', s(s(s(n'''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n'''''''))))
FOLDB(t''', s(s(s(s(n''''))))) -> FOLDB(t''', s(s(s(n''''))))
FOLDB(t''', s(s(s(0)))) -> FOLDB(t''', s(s(0)))
FOLDB(t''', s(s(s(n''''')))) -> FOLDB(t''', s(s(n''''')))
FOLDB(t'''', s(s(s(n'''')))) -> FOLDB(t'''', s(s(n'''')))
F'(triple(a'', s(s(s(n''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''''))))
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)
FOLDB(t''', s(s(s(n')))) -> F(f(f(foldB(t''', n'), B), B), B)
F'(triple(a'', s(s(s(n''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''))))
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
FOLDB(triple(a'''''', s(n''''''), c''''''), s(0)) -> F(triple(a'''''', s(n''''''), c''''''), B)
FOLDB(t'''', s(s(0))) -> FOLDB(t'''', s(0))
F'(triple(a'', s(s(0)), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(0)))
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t''', s(s(0))) -> F(f(t''', B), B)
F'(triple(a'', s(s(n''''')), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(n''''')))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
FOLDB(t''', s(s(n'''))) -> F(f'(foldB(t''', n'''), g(B)), B)
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F(triple(a'''', s(s(n'''')), c''''), C) -> F'(triple(a'''', s(s(n'''')), c''''), A)
FOLDC(t''', s(s(s(n')))) -> F(f(f(foldC(t''', n'), C), C), C)
F''(triple(a'', b'', s(s(s(n''''))))) -> FOLDC(triple(a'', b'', 0), s(s(s(n''''))))
F'(triple(a''', 0, s(s(n''''''))), A) -> F''(triple(s(a'''), 0, s(s(n''''''))))
F(triple(a''', 0, c'''), B) -> F'(triple(a''', 0, c'''), A)
FOLDB(triple(a''''', 0, c'''''), s(0)) -> F(triple(a''''', 0, c'''''), B)
F'(triple(a'', s(0), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(0))
F(triple(a'''', s(0), c''''), C) -> F'(triple(a'''', s(0), c''''), A)
FOLDC(triple(a'''''', s(n''''''), c''''''), s(0)) -> F(triple(a'''''', s(n''''''), c''''''), C)
FOLDC(t'''', s(s(0))) -> FOLDC(t'''', s(0))
F''(triple(a'', b'', s(s(0)))) -> FOLDC(triple(a'', b'', 0), s(s(0)))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F(triple(a'''', s(n''''), c''''), C) -> F'(triple(a'''', s(n''''), c''''), A)
FOLDC(t''', s(s(0))) -> F(f(t''', C), C)
F''(triple(a'', b'', s(s(n''''')))) -> FOLDC(triple(a'', b'', 0), s(s(n''''')))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(triple(a'', b'', c''), C) -> F'(triple(a'', b'', c''), B)
FOLDC(triple(a''''', 0, c'''''), s(0)) -> F(triple(a''''', 0, c'''''), C)
F''(triple(a'', b'', s(0))) -> FOLDC(triple(a'', b'', 0), s(0))
F'(triple(a''', 0, s(n'''')), A) -> F''(triple(s(a'''), 0, s(n'''')))
F(triple(a''', 0, c'''), C) -> F'(triple(a''', 0, c'''), A)
FOLDC(t''', s(s(n'''))) -> F(f'(foldC(t''', n'''), g(C)), C)
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(triple(a''', s(s(n'''''')), c'''), C) -> F'(triple(a''', s(s(n'''''')), c'''), A)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

FOLDC(triple(a'''', b'''', c''''), s(0)) -> F(triple(a'''', b'''', c''''), C)
FOLDC(triple(a''''', s(s(n'''''''')), c'''''), s(0)) -> F(triple(a''''', s(s(n'''''''')), c'''''), C)
FOLDC(triple(a''''', s(n''''''), c'''''), s(0)) -> F(triple(a''''', s(n''''''), c'''''), C)
FOLDC(triple(a'''''', s(s(n'''''')), c''''''), s(0)) -> F(triple(a'''''', s(s(n'''''')), c''''''), C)
FOLDC(triple(a'''''', s(0), c''''''), s(0)) -> F(triple(a'''''', s(0), c''''''), C)
FOLDC(t''', s(s(s(s(n''''))))) -> FOLDC(t''', s(s(s(n''''))))
FOLDC(t''', s(s(s(0)))) -> FOLDC(t''', s(s(0)))
FOLDC(t''', s(s(s(n''''')))) -> FOLDC(t''', s(s(n''''')))
FOLDC(t'''', s(s(s(n'''')))) -> FOLDC(t'''', s(s(n'''')))
FOLDC(t''', s(s(s(n')))) -> F(f(f(foldC(t''', n'), C), C), C)
FOLDC(triple(a'''''', s(n''''''), c''''''), s(0)) -> F(triple(a'''''', s(n''''''), c''''''), C)
FOLDC(t'''', s(s(0))) -> FOLDC(t'''', s(0))
FOLDC(t''', s(s(0))) -> F(f(t''', C), C)
FOLDC(triple(a''''', 0, c'''''), s(0)) -> F(triple(a''''', 0, c'''''), C)
FOLDC(t''', s(s(n'''))) -> F(f'(foldC(t''', n'''), g(C)), C)

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(foldB(x₁, x₂)) = x₁

POL(FOLDC(x₁, x₂)) = x₁ + x₂

POL(triple(x₁, x₂, x₃)) = x₃

POL(f'(x₁, x₂)) = x₁ + x₂

POL(F(x₁, x₂)) = x₁

POL(f(x₁, x₂)) = x₁ + x₂

POL(foldC(x₁, x₂)) = x₁ + x₂

POL(FOLDB(x₁, x₂)) = x₁

POL(F''(x₁)) = x₁

POL(0) = 0

POL(C) = 1

POL(B) = 0

POL(g(x₁)) = x₁

POL(F'(x₁, x₂)) = x₁

POL(s(x₁)) = 1 + x₁

POL(f''(x₁)) = x₁

POL(A) = 0

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 26

                 ↳Dependency Graph

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

F''(triple(a'', b'', s(s(s(s(n'''''')))))) -> FOLDC(triple(a'', b'', 0), s(s(s(s(n'''''')))))
F''(triple(a'', b'', s(s(s(0))))) -> FOLDC(triple(a'', b'', 0), s(s(s(0))))
F''(triple(a'', b'', s(s(s(n'''''''))))) -> FOLDC(triple(a'', b'', 0), s(s(s(n'''''''))))
F''(triple(a'', b'', s(s(s(n''''''))))) -> FOLDC(triple(a'', b'', 0), s(s(s(n''''''))))
F(triple(a''', s(n''''), c'''), C) -> F'(triple(a''', s(n''''), c'''), A)
FOLDB(triple(a'''', b'''', c''''), s(0)) -> F(triple(a'''', b'''', c''''), B)
FOLDB(triple(a''''', s(s(n'''''''')), c'''''), s(0)) -> F(triple(a''''', s(s(n'''''''')), c'''''), B)
FOLDB(triple(a''''', s(n''''''), c'''''), s(0)) -> F(triple(a''''', s(n''''''), c'''''), B)
FOLDB(triple(a'''''', s(s(n'''''')), c''''''), s(0)) -> F(triple(a'''''', s(s(n'''''')), c''''''), B)
FOLDB(triple(a'''''', s(0), c''''''), s(0)) -> F(triple(a'''''', s(0), c''''''), B)
F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)
F'(triple(a'', s(s(s(s(n'''''')))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(s(n'''''')))))
F'(triple(a'', s(s(s(0))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(0))))
F'(triple(a'', s(s(s(n'''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n'''''''))))
FOLDB(t''', s(s(s(s(n''''))))) -> FOLDB(t''', s(s(s(n''''))))
FOLDB(t''', s(s(s(0)))) -> FOLDB(t''', s(s(0)))
FOLDB(t''', s(s(s(n''''')))) -> FOLDB(t''', s(s(n''''')))
FOLDB(t'''', s(s(s(n'''')))) -> FOLDB(t'''', s(s(n'''')))
F'(triple(a'', s(s(s(n''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''''))))
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)
FOLDB(t''', s(s(s(n')))) -> F(f(f(foldB(t''', n'), B), B), B)
F'(triple(a'', s(s(s(n''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''))))
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
FOLDB(triple(a'''''', s(n''''''), c''''''), s(0)) -> F(triple(a'''''', s(n''''''), c''''''), B)
FOLDB(t'''', s(s(0))) -> FOLDB(t'''', s(0))
F'(triple(a'', s(s(0)), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(0)))
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t''', s(s(0))) -> F(f(t''', B), B)
F'(triple(a'', s(s(n''''')), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(n''''')))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
FOLDB(t''', s(s(n'''))) -> F(f'(foldB(t''', n'''), g(B)), B)
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))
F(triple(a'''', s(s(n'''')), c''''), C) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F''(triple(a'', b'', s(s(s(n''''))))) -> FOLDC(triple(a'', b'', 0), s(s(s(n''''))))
F'(triple(a''', 0, s(s(n''''''))), A) -> F''(triple(s(a'''), 0, s(s(n''''''))))
F(triple(a''', 0, c'''), B) -> F'(triple(a''', 0, c'''), A)
FOLDB(triple(a''''', 0, c'''''), s(0)) -> F(triple(a''''', 0, c'''''), B)
F'(triple(a'', s(0), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(0))
F(triple(a'''', s(0), c''''), C) -> F'(triple(a'''', s(0), c''''), A)
F''(triple(a'', b'', s(s(0)))) -> FOLDC(triple(a'', b'', 0), s(s(0)))
F'(triple(a'', s(s(n'')), c''), A) -> F''(f(f(foldB(triple(s(a''), 0, c''), n''), B), B))
F(triple(a'''', s(n''''), c''''), C) -> F'(triple(a'''', s(n''''), c''''), A)
F''(triple(a'', b'', s(s(n''''')))) -> FOLDC(triple(a'', b'', 0), s(s(n''''')))
F'(triple(a'', s(0), c''), A) -> F''(f(triple(s(a''), 0, c''), B))
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(triple(a'', b'', c''), C) -> F'(triple(a'', b'', c''), B)
F''(triple(a'', b'', s(0))) -> FOLDC(triple(a'', b'', 0), s(0))
F'(triple(a''', 0, s(n'''')), A) -> F''(triple(s(a'''), 0, s(n'''')))
F(triple(a''', 0, c'''), C) -> F'(triple(a''', 0, c'''), A)
F''(triple(a', b', s(s(n'''')))) -> FOLDC(triple(a', b', 0), s(s(n'''')))
F'(triple(a'', s(n''), c''), A) -> F''(f'(foldB(triple(s(a''), 0, c''), n''), g(B)))
F(triple(a''', s(s(n'''''')), c'''), C) -> F'(triple(a''', s(s(n'''''')), c'''), A)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 27

                 ↳Polynomial Ordering

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

FOLDB(triple(a'''', b'''', c''''), s(0)) -> F(triple(a'''', b'''', c''''), B)
FOLDB(triple(a''''', s(s(n'''''''')), c'''''), s(0)) -> F(triple(a''''', s(s(n'''''''')), c'''''), B)
FOLDB(triple(a''''', s(n''''''), c'''''), s(0)) -> F(triple(a''''', s(n''''''), c'''''), B)
FOLDB(triple(a'''''', s(s(n'''''')), c''''''), s(0)) -> F(triple(a'''''', s(s(n'''''')), c''''''), B)
FOLDB(triple(a'''''', s(0), c''''''), s(0)) -> F(triple(a'''''', s(0), c''''''), B)
F'(triple(a'', s(s(s(s(n'''''')))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(s(n'''''')))))
F'(triple(a'', s(s(s(0))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(0))))
F'(triple(a'', s(s(s(n'''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n'''''''))))
FOLDB(t''', s(s(s(s(n''''))))) -> FOLDB(t''', s(s(s(n''''))))
FOLDB(t''', s(s(s(0)))) -> FOLDB(t''', s(s(0)))
FOLDB(t''', s(s(s(n''''')))) -> FOLDB(t''', s(s(n''''')))
FOLDB(t'''', s(s(s(n'''')))) -> FOLDB(t'''', s(s(n'''')))
F'(triple(a'', s(s(s(n''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''''))))
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)
FOLDB(t''', s(s(s(n')))) -> F(f(f(foldB(t''', n'), B), B), B)
F'(triple(a'', s(s(s(n''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''))))
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
FOLDB(triple(a'''''', s(n''''''), c''''''), s(0)) -> F(triple(a'''''', s(n''''''), c''''''), B)
FOLDB(t'''', s(s(0))) -> FOLDB(t'''', s(0))
F'(triple(a'', s(s(0)), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(0)))
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
FOLDB(t''', s(s(0))) -> F(f(t''', B), B)
F'(triple(a'', s(s(n''''')), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(n''''')))
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)
FOLDB(triple(a''''', 0, c'''''), s(0)) -> F(triple(a''''', 0, c'''''), B)
F'(triple(a'', s(0), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(0))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
FOLDB(t''', s(s(n'''))) -> F(f'(foldB(t''', n'''), g(B)), B)
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

FOLDB(triple(a'''', b'''', c''''), s(0)) -> F(triple(a'''', b'''', c''''), B)
FOLDB(triple(a''''', s(s(n'''''''')), c'''''), s(0)) -> F(triple(a''''', s(s(n'''''''')), c'''''), B)
FOLDB(triple(a''''', s(n''''''), c'''''), s(0)) -> F(triple(a''''', s(n''''''), c'''''), B)
FOLDB(triple(a'''''', s(s(n'''''')), c''''''), s(0)) -> F(triple(a'''''', s(s(n'''''')), c''''''), B)
FOLDB(triple(a'''''', s(0), c''''''), s(0)) -> F(triple(a'''''', s(0), c''''''), B)
FOLDB(t''', s(s(s(n')))) -> F(f(f(foldB(t''', n'), B), B), B)
FOLDB(triple(a'''''', s(n''''''), c''''''), s(0)) -> F(triple(a'''''', s(n''''''), c''''''), B)
FOLDB(t''', s(s(0))) -> F(f(t''', B), B)
FOLDB(triple(a''''', 0, c'''''), s(0)) -> F(triple(a''''', 0, c'''''), B)
FOLDB(t''', s(s(n'''))) -> F(f'(foldB(t''', n'''), g(B)), B)

Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(foldB(x₁, x₂)) = x₁

POL(triple(x₁, x₂, x₃)) = x₂

POL(f'(x₁, x₂)) = x₁

POL(F(x₁, x₂)) = x₁

POL(f(x₁, x₂)) = x₁

POL(foldC(x₁, x₂)) = x₁

POL(FOLDB(x₁, x₂)) = 1 + x₁

POL(0) = 0

POL(C) = 0

POL(B) = 0

POL(g(x₁)) = 0

POL(F'(x₁, x₂)) = x₁

POL(s(x₁)) = 1

POL(f''(x₁)) = x₁

POL(A) = 0

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 28

                 ↳Dependency Graph

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

F'(triple(a'', s(s(s(s(n'''''')))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(s(n'''''')))))
F'(triple(a'', s(s(s(0))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(0))))
F'(triple(a'', s(s(s(n'''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n'''''''))))
FOLDB(t''', s(s(s(s(n''''))))) -> FOLDB(t''', s(s(s(n''''))))
FOLDB(t''', s(s(s(0)))) -> FOLDB(t''', s(s(0)))
FOLDB(t''', s(s(s(n''''')))) -> FOLDB(t''', s(s(n''''')))
FOLDB(t'''', s(s(s(n'''')))) -> FOLDB(t'''', s(s(n'''')))
F'(triple(a'', s(s(s(n''''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''''))))
F(triple(a''', s(s(n'''''')), c'''), B) -> F'(triple(a''', s(s(n'''''')), c'''), A)
F'(triple(a'', s(s(s(n''''))), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(s(n''''))))
F(triple(a''', s(n''''), c'''), B) -> F'(triple(a''', s(n''''), c'''), A)
FOLDB(t'''', s(s(0))) -> FOLDB(t'''', s(0))
F'(triple(a'', s(s(0)), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(0)))
F(triple(a'''', s(s(n'''')), c''''), B) -> F'(triple(a'''', s(s(n'''')), c''''), A)
F(triple(a'''', s(0), c''''), B) -> F'(triple(a'''', s(0), c''''), A)
F'(triple(a'', s(s(n''''')), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(s(n''''')))
F(triple(a'', b'', c''), A) -> F'(triple(a'', b'', c''), A)
F'(triple(a, b, c), B) -> F(triple(a, b, c), A)
F(triple(a'', b'', c''), B) -> F'(triple(a'', b'', c''), B)
F'(triple(a'', s(0), c''), A) -> FOLDB(triple(s(a''), 0, c''), s(0))
F(triple(a'''', s(n''''), c''''), B) -> F'(triple(a'''', s(n''''), c''''), A)
F'(triple(a', s(s(n'''')), c'), A) -> FOLDB(triple(s(a'), 0, c'), s(s(n'''')))

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 29

                 ↳Polynomial Ordering

       →DP Problem 2

         ↳FwdInst

Dependency Pairs:

FOLDB(t''', s(s(s(s(n''''))))) -> FOLDB(t''', s(s(s(n''''))))
FOLDB(t''', s(s(s(n''''')))) -> FOLDB(t''', s(s(n''''')))
FOLDB(t'''', s(s(s(n'''')))) -> FOLDB(t'''', s(s(n'''')))

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

The following dependency pairs can be strictly oriented:

FOLDB(t''', s(s(s(s(n''''))))) -> FOLDB(t''', s(s(s(n''''))))
FOLDB(t''', s(s(s(n''''')))) -> FOLDB(t''', s(s(n''''')))
FOLDB(t'''', s(s(s(n'''')))) -> FOLDB(t'''', s(s(n'''')))

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(s(x₁)) = 1 + x₁

POL(FOLDB(x₁, x₂)) = 1 + x₁ + x₂

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 3

             ↳Nar

...

               →DP Problem 30

                 ↳Dependency Graph

       →DP Problem 2

         ↳FwdInst

Dependency Pair:

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Forward Instantiation Transformation

Dependency Pair:

FOLD(t, x, s(n)) -> FOLD(t, x, n)

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

FOLD(t, x, s(n)) -> FOLD(t, x, n)

one new Dependency Pair is created:

FOLD(t'', x'', s(s(n''))) -> FOLD(t'', x'', s(n''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳FwdInst

           →DP Problem 31

             ↳Forward Instantiation Transformation

Dependency Pair:

FOLD(t'', x'', s(s(n''))) -> FOLD(t'', x'', s(n''))

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

FOLD(t'', x'', s(s(n''))) -> FOLD(t'', x'', s(n''))

one new Dependency Pair is created:

FOLD(t'''', x'''', s(s(s(n'''')))) -> FOLD(t'''', x'''', s(s(n'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳FwdInst

           →DP Problem 31

             ↳FwdInst

...

               →DP Problem 32

                 ↳Polynomial Ordering

Dependency Pair:

FOLD(t'''', x'''', s(s(s(n'''')))) -> FOLD(t'''', x'''', s(s(n'''')))

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

FOLD(t'''', x'''', s(s(s(n'''')))) -> FOLD(t'''', x'''', s(s(n'''')))

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(s(x₁)) = 1 + x₁

POL(FOLD(x₁, x₂, x₃)) = 1 + x₃

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳FwdInst

           →DP Problem 31

             ↳FwdInst

...

               →DP Problem 33

                 ↳Dependency Graph

Dependency Pair:

Rules:

g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0) -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0) -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(a, b, c), C) -> triple(a, b, s(c))
f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)
fold(t, x, 0) -> t
fold(t, x, s(n)) -> f(fold(t, x, n), x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:35 minutes

POL(foldB(x₁, x₂))	= x₁
POL(FOLDC(x₁, x₂))	= x₁ + x₂
POL(triple(x₁, x₂, x₃))	= x₃
POL(f'(x₁, x₂))	= x₁ + x₂
POL(F(x₁, x₂))	= x₁
POL(f(x₁, x₂))	= x₁ + x₂
POL(foldC(x₁, x₂))	= x₁ + x₂
POL(FOLDB(x₁, x₂))	= x₁
POL(F''(x₁))	= x₁
POL(0)	= 0
POL(C)	= 1
POL(B)	= 0
POL(g(x₁))	= x₁
POL(F'(x₁, x₂))	= x₁
POL(s(x₁))	= 1 + x₁
POL(f''(x₁))	= x₁
POL(A)	= 0

POL(s(x₁))	= 1 + x₁
POL(FOLD(x₁, x₂, x₃))	= 1 + x₃