YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty m n vNonEmpty)
(STRATEGY CONTEXTSENSITIVE
(ack 1 2)
(isNat 1)
(seven)
(O)
(fSNonEmpty)
(false)
(s 1)
(true)
)
(RULES
ack(O,n) -> s(n)
ack(s(m),O) -> ack(m,s(O))
ack(s(m),s(n)) -> ack(m,ack(s(m),n))
isNat(O) -> true
isNat(false) -> false
isNat(s(m)) -> isNat(m)
isNat(true) -> false
seven -> s(s(s(s(s(s(s(O)))))))
)
]
Infeasibility Conditions:
isNat(ack(seven,seven)) ->* false

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
ack(O,n) -> s(n)
ack(s(m),O) -> ack(m,s(O))
ack(s(m),s(n)) -> ack(m,ack(s(m),n))
isNat(O) -> true
isNat(false) -> false
isNat(s(m)) -> isNat(m)
isNat(true) -> false
seven -> s(s(s(s(s(s(s(O)))))))

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3206038 was started by sandbox on z029.star.cs.uiowa.edu,
Tue Jul 30 08:46:25 2024
The command was "./mace4 -c -f /tmp/mace43206027-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43206027-2.in

assign(max_seconds,100).

formulas(assumptions).
->(x1,y) -> ->(ack(x1,x2),ack(y,x2)) # label(congruence).
->(x2,y) -> ->(ack(x1,x2),ack(x1,y)) # label(congruence).
->(x1,y) -> ->(isNat(x1),isNat(y)) # label(congruence).
->(x1,y) -> ->(s(x1),s(y)) # label(congruence).
->(ack(O,x2),s(x2)) # label(replacement).
->(ack(s(x1),O),ack(x1,s(O))) # label(replacement).
->(ack(s(x1),s(x2)),ack(x1,ack(s(x1),x2))) # label(replacement).
->(isNat(O),true) # label(replacement).
->(isNat(false),false) # label(replacement).
->(isNat(s(x1)),isNat(x1)) # label(replacement).
->(isNat(true),false) # label(replacement).
->(seven,s(s(s(s(s(s(s(O)))))))) # label(replacement).
->*(x,x) # label(reflexivity).
->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
end_of_list.

formulas(goals).
->*(isNat(ack(seven,seven)),false) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->(x1,y) -> ->(ack(x1,x2),ack(y,x2)) # label(congruence) # label(non_clause).  [assumption].
2 ->(x2,y) -> ->(ack(x1,x2),ack(x1,y)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x1,y) -> ->(isNat(x1),isNat(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause).  [assumption].
5 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
6 ->*(isNat(ack(seven,seven)),false) # label(goal) # label(non_clause) # label(goal).  [goal].

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

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

formulas(mace4_clauses).
-->(x,y) | ->(ack(x,z),ack(y,z)) # label(congruence).
-->(x,y) | ->(ack(z,x),ack(z,y)) # label(congruence).
-->(x,y) | ->(isNat(x),isNat(y)) # label(congruence).
-->(x,y) | ->(s(x),s(y)) # label(congruence).
->(ack(O,x),s(x)) # label(replacement).
->(ack(s(x),O),ack(x,s(O))) # label(replacement).
->(ack(s(x),s(y)),ack(x,ack(s(x),y))) # label(replacement).
->(isNat(O),true) # label(replacement).
->(isNat(false),false) # label(replacement).
->(isNat(s(x)),isNat(x)) # label(replacement).
->(isNat(true),false) # label(replacement).
->(seven,s(s(s(s(s(s(s(O)))))))) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(isNat(ack(seven,seven)),false) # label(goal).
end_of_list.

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

% There are no natural numbers in the input.

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

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

For domain size 2.

Current CPU time: 0.00 seconds (total CPU time: 0.01 seconds).
Ground clauses: seen=49, kept=45.
Selections=65, assignments=129, propagations=153, current_models=0.
Rewrite_terms=1389, rewrite_bools=651, indexes=408.
Rules_from_neg_clauses=36, cross_offs=36.

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

============================== DOMAIN SIZE 3 =========================

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

interpretation( 3, [number=1, seconds=0], [

        function(seven, [ 0 ]),

        function(O, [ 0 ]),

        function(false, [ 1 ]),

        function(true, [ 2 ]),

        function(isNat(_), [ 2, 1, 1 ]),

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

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

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

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

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

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

For domain size 3.

Current CPU time: 0.00 seconds (total CPU time: 0.01 seconds).
Ground clauses: seen=125, kept=116.
Selections=102, assignments=270, propagations=565, current_models=1.
Rewrite_terms=3656, rewrite_bools=2448, indexes=1047.
Rules_from_neg_clauses=62, cross_offs=165.

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

User_CPU=0.01, System_CPU=0.00, Wall_clock=0.

Exiting with 1 model.

Process 3206038 exit (max_models) Tue Jul 30 08:46:25 2024
The process finished Tue Jul 30 08:46:25 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 3

seven = 0.

O = 0.

false = 1.

true = 2.

isNat(0) = 2.
isNat(1) = 1.
isNat(2) = 1.

s(0) = 0.
s(1) = 1.
s(2) = 1.

ack(0,0) = 0.
ack(0,1) = 1.
ack(0,2) = 1.
ack(1,0) = 0.
ack(1,1) = 0.
ack(1,2) = 0.
ack(2,0) = 0.
ack(2,1) = 0.
ack(2,2) = 0.

  ->*(0,0).
- ->*(0,1).
- ->*(0,2).
- ->*(1,0).
  ->*(1,1).
- ->*(1,2).
- ->*(2,0).
- ->*(2,1).
  ->*(2,2).

  ->(0,0).
- ->(0,1).
- ->(0,2).
- ->(1,0).
  ->(1,1).
- ->(1,2).
- ->(2,0).
- ->(2,1).
  ->(2,2).


The problem is infeasible.