YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x y q n z vNonEmpty x1 x2 x4)
(STRATEGY CONTEXTSENSITIVE
(add 1 2)
(div 1 2)
(lte 1 2)
(minus 1 2)
(mod 1 2)
(mult 1 2)
(power 1 2)
(0)
(fSNonEmpty)
(false)
(s 1)
(true)
)
(RULES
add(0,x) -> x
add(s(x),y) -> s(add(x,y))
div(0,s(x)) -> 0
div(s(x),s(y)) -> 0 | lte(s(x),y) ->* true
div(s(x),s(y)) -> s(q) | lte(s(x),y) ->* false, div(minus(x,y),s(y)) ->* q
lte(0,y) -> true
lte(s(x),0) -> false
lte(s(x),s(y)) -> lte(x,y)
minus(0,s(y)) -> 0
minus(s(x),s(y)) -> minus(x,y)
minus(x,0) -> x
mod(0,y) -> 0
mod(x,0) -> x
mod(x,s(y)) -> mod(minus(x,s(y)),s(y)) | lte(s(y),x) ->* true
mod(x,s(y)) -> x | lte(s(y),x) ->* false
mult(0,y) -> 0
mult(s(x),y) -> add(mult(x,y),y)
power(x,0) -> s(0)
power(x,n) -> mult(mult(y,y),s(0)) | mod(n,s(s(0))) ->* 0, power(x,div(n,s(s(0)))) ->* y
power(x,n) -> mult(mult(y,y),x) | mod(n,s(s(0))) ->* s(z), power(x,div(n,s(s(0)))) ->* y
)
]

Infeasibility Conditions:
mod(0,s(s(0))) ->* s(x1), power(x2,div(0,s(s(0)))) ->* x4

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
add(0,x) -> x
add(s(x),y) -> s(add(x,y))
div(0,s(x)) -> 0
div(s(x),s(y)) -> 0 | lte(s(x),y) ->* true
div(s(x),s(y)) -> s(q) | lte(s(x),y) ->* false, div(minus(x,y),s(y)) ->* q
lte(0,y) -> true
lte(s(x),0) -> false
lte(s(x),s(y)) -> lte(x,y)
minus(0,s(y)) -> 0
minus(s(x),s(y)) -> minus(x,y)
minus(x,0) -> x
mod(0,y) -> 0
mod(x,0) -> x
mod(x,s(y)) -> mod(minus(x,s(y)),s(y)) | lte(s(y),x) ->* true
mod(x,s(y)) -> x | lte(s(y),x) ->* false
mult(0,y) -> 0
mult(s(x),y) -> add(mult(x,y),y)
power(x,0) -> s(0)
power(x,n) -> mult(mult(y,y),s(0)) | mod(n,s(s(0))) ->* 0, power(x,div(n,s(s(0)))) ->* y
power(x,n) -> mult(mult(y,y),x) | mod(n,s(s(0))) ->* s(z), power(x,div(n,s(s(0)))) ->* y

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3441252 was started by sandbox2 on z018.star.cs.uiowa.edu,
Tue Jul 30 09:59:55 2024
The command was "./mace4 -c -f /tmp/mace43441238-2.in".
============================== end of head ===========================

============================== INPUT =================================

% Reading from file /tmp/mace43441238-2.in

assign(max_seconds,100).

formulas(assumptions).
->(x1,y) -> ->(add(x1,x2),add(y,x2)) # label(congruence).
->(x2,y) -> ->(add(x1,x2),add(x1,y)) # label(congruence).
->(x1,y) -> ->(div(x1,x2),div(y,x2)) # label(congruence).
->(x2,y) -> ->(div(x1,x2),div(x1,y)) # label(congruence).
->(x1,y) -> ->(lte(x1,x2),lte(y,x2)) # label(congruence).
->(x2,y) -> ->(lte(x1,x2),lte(x1,y)) # label(congruence).
->(x1,y) -> ->(minus(x1,x2),minus(y,x2)) # label(congruence).
->(x2,y) -> ->(minus(x1,x2),minus(x1,y)) # label(congruence).
->(x1,y) -> ->(mod(x1,x2),mod(y,x2)) # label(congruence).
->(x2,y) -> ->(mod(x1,x2),mod(x1,y)) # label(congruence).
->(x1,y) -> ->(mult(x1,x2),mult(y,x2)) # label(congruence).
->(x2,y) -> ->(mult(x1,x2),mult(x1,y)) # label(congruence).
->(x1,y) -> ->(power(x1,x2),power(y,x2)) # label(congruence).
->(x2,y) -> ->(power(x1,x2),power(x1,y)) # label(congruence).
->(x1,y) -> ->(s(x1),s(y)) # label(congruence).
->(add(0,x1),x1) # label(replacement).
->(add(s(x1),x2),s(add(x1,x2))) # label(replacement).
->(div(0,s(x1)),0) # label(replacement).
->*(lte(s(x1),x2),true) -> ->(div(s(x1),s(x2)),0) # label(replacement).
->*(lte(s(x1),x2),false) & ->*(div(minus(x1,x2),s(x2)),x3) -> ->(div(s(x1),s(x2)),s(x3)) # label(replacement).
->(lte(0,x2),true) # label(replacement).
->(lte(s(x1),0),false) # label(replacement).
->(lte(s(x1),s(x2)),lte(x1,x2)) # label(replacement).
->(minus(0,s(x2)),0) # label(replacement).
->(minus(s(x1),s(x2)),minus(x1,x2)) # label(replacement).
->(minus(x1,0),x1) # label(replacement).
->(mod(0,x2),0) # label(replacement).
->(mod(x1,0),x1) # label(replacement).
->*(lte(s(x2),x1),true) -> ->(mod(x1,s(x2)),mod(minus(x1,s(x2)),s(x2))) # label(replacement).
->*(lte(s(x2),x1),false) -> ->(mod(x1,s(x2)),x1) # label(replacement).
->(mult(0,x2),0) # label(replacement).
->(mult(s(x1),x2),add(mult(x1,x2),x2)) # label(replacement).
->(power(x1,0),s(0)) # label(replacement).
->*(mod(x4,s(s(0))),0) & ->*(power(x1,div(x4,s(s(0)))),x2) -> ->(power(x1,x4),mult(mult(x2,x2),s(0))) # label(replacement).
->*(mod(x4,s(s(0))),s(x5)) & ->*(power(x1,div(x4,s(s(0)))),x2) -> ->(power(x1,x4),mult(mult(x2,x2),x1)) # label(replacement).
->*(x,x) # label(reflexivity).
->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x7 exists x8 exists x9 (->*(mod(0,s(s(0))),s(x7)) & ->*(power(x8,div(0,s(s(0)))),x9))) # label(goal).
end_of_list.

============================== end of input ==========================

============================== PROCESS NON-CLAUSAL FORMULAS ==========

% Formulas that are not ordinary clauses:
1 ->(x1,y) -> ->(add(x1,x2),add(y,x2)) # label(congruence) # label(non_clause).  [assumption].
2 ->(x2,y) -> ->(add(x1,x2),add(x1,y)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x1,y) -> ->(div(x1,x2),div(y,x2)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x2,y) -> ->(div(x1,x2),div(x1,y)) # label(congruence) # label(non_clause).  [assumption].
5 ->(x1,y) -> ->(lte(x1,x2),lte(y,x2)) # label(congruence) # label(non_clause).  [assumption].
6 ->(x2,y) -> ->(lte(x1,x2),lte(x1,y)) # label(congruence) # label(non_clause).  [assumption].
7 ->(x1,y) -> ->(minus(x1,x2),minus(y,x2)) # label(congruence) # label(non_clause).  [assumption].
8 ->(x2,y) -> ->(minus(x1,x2),minus(x1,y)) # label(congruence) # label(non_clause).  [assumption].
9 ->(x1,y) -> ->(mod(x1,x2),mod(y,x2)) # label(congruence) # label(non_clause).  [assumption].
10 ->(x2,y) -> ->(mod(x1,x2),mod(x1,y)) # label(congruence) # label(non_clause).  [assumption].
11 ->(x1,y) -> ->(mult(x1,x2),mult(y,x2)) # label(congruence) # label(non_clause).  [assumption].
12 ->(x2,y) -> ->(mult(x1,x2),mult(x1,y)) # label(congruence) # label(non_clause).  [assumption].
13 ->(x1,y) -> ->(power(x1,x2),power(y,x2)) # label(congruence) # label(non_clause).  [assumption].
14 ->(x2,y) -> ->(power(x1,x2),power(x1,y)) # label(congruence) # label(non_clause).  [assumption].
15 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause).  [assumption].
16 ->*(lte(s(x1),x2),true) -> ->(div(s(x1),s(x2)),0) # label(replacement) # label(non_clause).  [assumption].
17 ->*(lte(s(x1),x2),false) & ->*(div(minus(x1,x2),s(x2)),x3) -> ->(div(s(x1),s(x2)),s(x3)) # label(replacement) # label(non_clause).  [assumption].
18 ->*(lte(s(x2),x1),true) -> ->(mod(x1,s(x2)),mod(minus(x1,s(x2)),s(x2))) # label(replacement) # label(non_clause).  [assumption].
19 ->*(lte(s(x2),x1),false) -> ->(mod(x1,s(x2)),x1) # label(replacement) # label(non_clause).  [assumption].
20 ->*(mod(x4,s(s(0))),0) & ->*(power(x1,div(x4,s(s(0)))),x2) -> ->(power(x1,x4),mult(mult(x2,x2),s(0))) # label(replacement) # label(non_clause).  [assumption].
21 ->*(mod(x4,s(s(0))),s(x5)) & ->*(power(x1,div(x4,s(s(0)))),x2) -> ->(power(x1,x4),mult(mult(x2,x2),x1)) # label(replacement) # label(non_clause).  [assumption].
22 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
23 (exists x7 exists x8 exists x9 (->*(mod(0,s(s(0))),s(x7)) & ->*(power(x8,div(0,s(s(0)))),x9))) # label(goal) # label(non_clause) # label(goal).  [goal].

============================== end of process non-clausal formulas ===

============================== CLAUSES FOR SEARCH ====================

formulas(mace4_clauses).
-->(x,y) | ->(add(x,z),add(y,z)) # label(congruence).
-->(x,y) | ->(add(z,x),add(z,y)) # label(congruence).
-->(x,y) | ->(div(x,z),div(y,z)) # label(congruence).
-->(x,y) | ->(div(z,x),div(z,y)) # label(congruence).
-->(x,y) | ->(lte(x,z),lte(y,z)) # label(congruence).
-->(x,y) | ->(lte(z,x),lte(z,y)) # label(congruence).
-->(x,y) | ->(minus(x,z),minus(y,z)) # label(congruence).
-->(x,y) | ->(minus(z,x),minus(z,y)) # label(congruence).
-->(x,y) | ->(mod(x,z),mod(y,z)) # label(congruence).
-->(x,y) | ->(mod(z,x),mod(z,y)) # label(congruence).
-->(x,y) | ->(mult(x,z),mult(y,z)) # label(congruence).
-->(x,y) | ->(mult(z,x),mult(z,y)) # label(congruence).
-->(x,y) | ->(power(x,z),power(y,z)) # label(congruence).
-->(x,y) | ->(power(z,x),power(z,y)) # label(congruence).
-->(x,y) | ->(s(x),s(y)) # label(congruence).
->(add(0,x),x) # label(replacement).
->(add(s(x),y),s(add(x,y))) # label(replacement).
->(div(0,s(x)),0) # label(replacement).
-->*(lte(s(x),y),true) | ->(div(s(x),s(y)),0) # label(replacement).
-->*(lte(s(x),y),false) | -->*(div(minus(x,y),s(y)),z) | ->(div(s(x),s(y)),s(z)) # label(replacement).
->(lte(0,x),true) # label(replacement).
->(lte(s(x),0),false) # label(replacement).
->(lte(s(x),s(y)),lte(x,y)) # label(replacement).
->(minus(0,s(x)),0) # label(replacement).
->(minus(s(x),s(y)),minus(x,y)) # label(replacement).
->(minus(x,0),x) # label(replacement).
->(mod(0,x),0) # label(replacement).
->(mod(x,0),x) # label(replacement).
-->*(lte(s(x),y),true) | ->(mod(y,s(x)),mod(minus(y,s(x)),s(x))) # label(replacement).
-->*(lte(s(x),y),false) | ->(mod(y,s(x)),y) # label(replacement).
->(mult(0,x),0) # label(replacement).
->(mult(s(x),y),add(mult(x,y),y)) # label(replacement).
->(power(x,0),s(0)) # label(replacement).
-->*(mod(x,s(s(0))),0) | -->*(power(y,div(x,s(s(0)))),z) | ->(power(y,x),mult(mult(z,z),s(0))) # label(replacement).
-->*(mod(x,s(s(0))),s(y)) | -->*(power(z,div(x,s(s(0)))),u) | ->(power(z,x),mult(mult(u,u),z)) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(mod(0,s(s(0))),s(x)) | -->*(power(y,div(0,s(s(0)))),z) # label(goal).
end_of_list.

============================== end of clauses for search =============

% There are no natural numbers in the input.

============================== DOMAIN SIZE 2 =========================

============================== MODEL =================================

interpretation( 2, [number=1, seconds=0], [

        function(false, [ 0 ]),

        function(true, [ 0 ]),

        function(0, [ 0 ]),

        function(s(_), [ 1, 1 ]),

        function(add(_,_), [
			   0, 1,
			   1, 1 ]),

        function(div(_,_), [
			   0, 0,
			   0, 1 ]),

        function(lte(_,_), [
			   0, 0,
			   0, 0 ]),

        function(minus(_,_), [
			   0, 0,
			   1, 1 ]),

        function(mod(_,_), [
			   0, 0,
			   1, 1 ]),

        function(mult(_,_), [
			   0, 0,
			   0, 1 ]),

        function(power(_,_), [
			   1, 1,
			   1, 1 ]),

        relation(->*(_,_), [
			   1, 0,
			   1, 1 ]),

        relation(->(_,_), [
			   1, 0,
			   1, 1 ])
]).

============================== end of model ==========================

============================== STATISTICS ============================

For domain size 2.

Current CPU time: 0.00 seconds (total CPU time: 0.05 seconds).
Ground clauses: seen=214, kept=210.
Selections=7747, assignments=15477, propagations=7808, current_models=1.
Rewrite_terms=409082, rewrite_bools=267189, indexes=11391.
Rules_from_neg_clauses=7738, cross_offs=7738.

============================== end of statistics =====================

User_CPU=0.05, System_CPU=0.00, Wall_clock=0.

Exiting with 1 model.

Process 3441252 exit (max_models) Tue Jul 30 09:59:55 2024
The process finished Tue Jul 30 09:59:55 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 2

false = 0.

true = 0.

0 = 0.

s(0) = 1.
s(1) = 1.

add(0,0) = 0.
add(0,1) = 1.
add(1,0) = 1.
add(1,1) = 1.

div(0,0) = 0.
div(0,1) = 0.
div(1,0) = 0.
div(1,1) = 1.

lte(0,0) = 0.
lte(0,1) = 0.
lte(1,0) = 0.
lte(1,1) = 0.

minus(0,0) = 0.
minus(0,1) = 0.
minus(1,0) = 1.
minus(1,1) = 1.

mod(0,0) = 0.
mod(0,1) = 0.
mod(1,0) = 1.
mod(1,1) = 1.

mult(0,0) = 0.
mult(0,1) = 0.
mult(1,0) = 0.
mult(1,1) = 1.

power(0,0) = 1.
power(0,1) = 1.
power(1,0) = 1.
power(1,1) = 1.

  ->*(0,0).
- ->*(0,1).
  ->*(1,0).
  ->*(1,1).

  ->(0,0).
- ->(0,1).
  ->(1,0).
  ->(1,1).


The problem is infeasible.