YES

Problem 1:

(VAR vu95NonEmpty x y)
(RULES
a -> c
a -> d
b -> c
b -> d
f(x) -> pair(x,y) | s(x) ->* t(y)
g(x,x) -> h(x,x)
s(c) -> t(k)
s(c) -> t(l)
)

Problem 1:

Well-founded Relation Processor:
-> Rules:
 a -> c
 a -> d
 b -> c
 b -> d
 f(x) -> pair(x,y) | s(x) ->* t(y)
 g(x,x) -> h(x,x)
 s(c) -> t(k)
 s(c) -> t(l)
->AGES Output:

Model Results

System:
mod InTheory is
sort S .
sort Bool .


op _->*_ : S S -> Bool [m = 2] .
op _->_ : S S -> Bool [m = 2] .
op a :  -> S .
op b :  -> S .
op f : S -> S .
op g : S S -> S .
op s : S -> S .
op c :  -> S .
op d :  -> S .
op fSNonEmpty :  -> S .
op h : S S -> S .
op k :  -> S .
op l :  -> S .
op pair : S S -> S .
op t : S -> S .
op sqsupset : S S -> Bool [wellfounded m = 1] .

endm


Property:
x ->R* x
x ->R y /\ y ->R* z => x ->R* z
x1 ->R y1 => f(x1) ->R f(y1)
x1 ->R y1 => g(x1,x2) ->R g(y1,x2)
x2 ->R y2 => g(x1,x2) ->R g(x1,y2)
x1 ->R y1 => s(x1) ->R s(y1)
x1 ->R y1 => h(x1,x2) ->R h(y1,x2)
x2 ->R y2 => h(x1,x2) ->R h(x1,y2)
x1 ->R y1 => pair(x1,x2) ->R pair(y1,x2)
x2 ->R y2 => pair(x1,x2) ->R pair(x1,y2)
x1 ->R y1 => t(x1) ->R t(y1)
a ->R c
a ->R d
b ->R c
b ->R d
s(x) ->R* t(y) => f(x) ->R pair(x,y)
g(x,x) ->R h(x,x)
s(c) ->R t(k)
s(c) ->R t(l)
x ->R y => sqsupset(x,y)

Results:


Domains:
S: |N \ {0}

Function Interpretations:
|[a]| = 3
|[b]| = 3
|[c]| = 2
|[d]| = 2
|[f(x_1_1:S)]| = 2 + 5.x_1_1:S
|[fSNonEmpty]| = 1
|[g(x_1_1:S,x_2_1:S)]| = 1 + 3.x_1_1:S + x_2_1:S
|[h(x_1_1:S,x_2_1:S)]| = x_1_1:S + 2.x_2_1:S
|[k]| = 1
|[l]| = 1
|[pair(x_1_1:S,x_2_1:S)]| = 3.x_1_1:S + x_2_1:S
|[s(x_1_1:S)]| = x_1_1:S
|[t(x_1_1:S)]| = x_1_1:S

Predicate Interpretations:
 x_1_1:S ->* x_2_1:S <=> ((x_1_1:S >= x_2_1:S) /\ (1 + x_1_1:S >= 0))
 x_1_1:S -> x_2_1:S <=> ((x_2_1:S >= 1) /\ (x_1_1:S >= 1 + x_2_1:S))
sqsupset(x_1_1:S,x_2_1:S) <=> (x_1_1:S >= 1 + x_2_1:S)

The problem is finite.
2.76user 0.19system 0:03.26elapsed 90%CPU (0avgtext+0avgdata 102244maxresident)k
25320inputs+176outputs (113major+26301minor)pagefaults 0swaps