Term Rewriting System R:
[x, y]
f(0) -> 0
f(s(0)) -> s(0)
f(s(s(x))) -> p(h(g(x)))
f(s(s(x))) -> +(p(g(x)), q(g(x)))
g(0) -> pair(s(0), s(0))
g(s(x)) -> h(g(x))
g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))
h(x) -> pair(+(p(x), q(x)), p(x))
p(pair(x, y)) -> x
q(pair(x, y)) -> y
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(s(x))) -> P(h(g(x)))
F(s(s(x))) -> H(g(x))
F(s(s(x))) -> G(x)
F(s(s(x))) -> +'(p(g(x)), q(g(x)))
F(s(s(x))) -> P(g(x))
F(s(s(x))) -> Q(g(x))
G(s(x)) -> H(g(x))
G(s(x)) -> G(x)
G(s(x)) -> +'(p(g(x)), q(g(x)))
G(s(x)) -> P(g(x))
G(s(x)) -> Q(g(x))
H(x) -> +'(p(x), q(x))
H(x) -> P(x)
H(x) -> Q(x)
+'(x, s(y)) -> +'(x, y)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Usable Rules (Innermost)
       →DP Problem 2
         ↳UsableRules


Dependency Pair:

+'(x, s(y)) -> +'(x, y)


Rules:


f(0) -> 0
f(s(0)) -> s(0)
f(s(s(x))) -> p(h(g(x)))
f(s(s(x))) -> +(p(g(x)), q(g(x)))
g(0) -> pair(s(0), s(0))
g(s(x)) -> h(g(x))
g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))
h(x) -> pair(+(p(x), q(x)), p(x))
p(pair(x, y)) -> x
q(pair(x, y)) -> y
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




As we are in the innermost case, we can delete all 12 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 3
             ↳Size-Change Principle
       →DP Problem 2
         ↳UsableRules


Dependency Pair:

+'(x, s(y)) -> +'(x, y)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. +'(x, s(y)) -> +'(x, y)
and get the following Size-Change Graph(s):
{1} , {1}
1=1
2>2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1=1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
s(x1) -> s(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳Usable Rules (Innermost)


Dependency Pair:

G(s(x)) -> G(x)


Rules:


f(0) -> 0
f(s(0)) -> s(0)
f(s(s(x))) -> p(h(g(x)))
f(s(s(x))) -> +(p(g(x)), q(g(x)))
g(0) -> pair(s(0), s(0))
g(s(x)) -> h(g(x))
g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))
h(x) -> pair(+(p(x), q(x)), p(x))
p(pair(x, y)) -> x
q(pair(x, y)) -> y
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))


Strategy:

innermost




As we are in the innermost case, we can delete all 12 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
           →DP Problem 4
             ↳Size-Change Principle


Dependency Pair:

G(s(x)) -> G(x)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. G(s(x)) -> G(x)
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
s(x1) -> s(x1)

We obtain no new DP problems.

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