Term Rewriting System R:
[X, Z, Y]
h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

H(X, Z) -> F(X, s(X), Z)
F(X, Y, g(X, Y)) -> H(0, g(X, Y))
G(X, s(Y)) -> G(X, Y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pair:

G(X, s(Y)) -> G(X, Y)


Rules:


h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)


Strategy:

innermost




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

G(X, s(Y)) -> G(X, Y)
one new Dependency Pair is created:

G(X'', s(s(Y''))) -> G(X'', s(Y''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Forward Instantiation Transformation


Dependency Pair:

G(X'', s(s(Y''))) -> G(X'', s(Y''))


Rules:


h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)


Strategy:

innermost




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

G(X'', s(s(Y''))) -> G(X'', s(Y''))
one new Dependency Pair is created:

G(X'''', s(s(s(Y'''')))) -> G(X'''', s(s(Y'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Argument Filtering and Ordering


Dependency Pair:

G(X'''', s(s(s(Y'''')))) -> G(X'''', s(s(Y'''')))


Rules:


h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)


Strategy:

innermost




The following dependency pair can be strictly oriented:

G(X'''', s(s(s(Y'''')))) -> G(X'''', s(s(Y'''')))


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
G(x1, x2) -> G(x1, x2)
s(x1) -> s(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳Dependency Graph


Dependency Pair:


Rules:


h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes