YES

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x y xs vNonEmpty x3 x1)
(STRATEGY CONTEXTSENSITIVE
(le 1 2)
(min 1)
(0)
(cons 1 2)
(fSNonEmpty)
(false)
(nil)
(s 1)
(true)
)
(RULES
le(0,s(y)) -> true
le(s(x),s(y)) -> le(x,y)
le(x,0) -> false
min(cons(x,nil)) -> x
min(cons(x,xs)) -> x | min(xs) ->* y, le(x,y) ->* true
min(cons(x,xs)) -> y | min(xs) ->* y, le(x,y) ->* false
)
]

Infeasibility Conditions:
min(nil) ->* x3, le(x1,x3) ->* false

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
le(0,s(y)) -> true
le(s(x),s(y)) -> le(x,y)
le(x,0) -> false
min(cons(x,nil)) -> x
min(cons(x,xs)) -> x | min(xs) ->* y, le(x,y) ->* true
min(cons(x,xs)) -> y | min(xs) ->* y, le(x,y) ->* false

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 3449932 was started by sandbox on z024.star.cs.uiowa.edu,
Tue Jul 30 09:54:40 2024
The command was "./mace4 -c -f /tmp/mace43449921-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace43449921-2.in

assign(max_seconds,100).

formulas(assumptions).
->(x1,y) -> ->(le(x1,x2),le(y,x2)) # label(congruence).
->(x2,y) -> ->(le(x1,x2),le(x1,y)) # label(congruence).
->(x1,y) -> ->(min(x1),min(y)) # label(congruence).
->(x1,y) -> ->(cons(x1,x2),cons(y,x2)) # label(congruence).
->(x2,y) -> ->(cons(x1,x2),cons(x1,y)) # label(congruence).
->(x1,y) -> ->(s(x1),s(y)) # label(congruence).
->(le(0,s(x2)),true) # label(replacement).
->(le(s(x1),s(x2)),le(x1,x2)) # label(replacement).
->(le(x1,0),false) # label(replacement).
->(min(cons(x1,nil)),x1) # label(replacement).
->*(min(x3),x2) & ->*(le(x1,x2),true) -> ->(min(cons(x1,x3)),x1) # label(replacement).
->*(min(x3),x2) & ->*(le(x1,x2),false) -> ->(min(cons(x1,x3)),x2) # label(replacement).
->*(x,x) # label(reflexivity).
->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x5 exists x6 (->*(min(nil),x5) & ->*(le(x6,x5),false))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->(x1,y) -> ->(le(x1,x2),le(y,x2)) # label(congruence) # label(non_clause).  [assumption].
2 ->(x2,y) -> ->(le(x1,x2),le(x1,y)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x1,y) -> ->(min(x1),min(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x1,y) -> ->(cons(x1,x2),cons(y,x2)) # label(congruence) # label(non_clause).  [assumption].
5 ->(x2,y) -> ->(cons(x1,x2),cons(x1,y)) # label(congruence) # label(non_clause).  [assumption].
6 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause).  [assumption].
7 ->*(min(x3),x2) & ->*(le(x1,x2),true) -> ->(min(cons(x1,x3)),x1) # label(replacement) # label(non_clause).  [assumption].
8 ->*(min(x3),x2) & ->*(le(x1,x2),false) -> ->(min(cons(x1,x3)),x2) # label(replacement) # label(non_clause).  [assumption].
9 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
10 (exists x5 exists x6 (->*(min(nil),x5) & ->*(le(x6,x5),false))) # label(goal) # label(non_clause) # label(goal).  [goal].

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

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

formulas(mace4_clauses).
-->(x,y) | ->(le(x,z),le(y,z)) # label(congruence).
-->(x,y) | ->(le(z,x),le(z,y)) # label(congruence).
-->(x,y) | ->(min(x),min(y)) # label(congruence).
-->(x,y) | ->(cons(x,z),cons(y,z)) # label(congruence).
-->(x,y) | ->(cons(z,x),cons(z,y)) # label(congruence).
-->(x,y) | ->(s(x),s(y)) # label(congruence).
->(le(0,s(x)),true) # label(replacement).
->(le(s(x),s(y)),le(x,y)) # label(replacement).
->(le(x,0),false) # label(replacement).
->(min(cons(x,nil)),x) # label(replacement).
-->*(min(x),y) | -->*(le(z,y),true) | ->(min(cons(z,x)),z) # label(replacement).
-->*(min(x),y) | -->*(le(z,y),false) | ->(min(cons(z,x)),y) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(min(nil),x) | -->*(le(y,x),false) # 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(true, [ 0 ]),

        function(0, [ 0 ]),

        function(false, [ 0 ]),

        function(nil, [ 1 ]),

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

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

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

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

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

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

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

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

For domain size 2.

Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds).
Ground clauses: seen=80, kept=76.
Selections=17, assignments=22, propagations=24, current_models=1.
Rewrite_terms=383, rewrite_bools=231, indexes=72.
Rules_from_neg_clauses=9, cross_offs=9.

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

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

Exiting with 1 model.

Process 3449932 exit (max_models) Tue Jul 30 09:54:40 2024
The process finished Tue Jul 30 09:54:40 2024


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 2

true = 0.

0 = 0.

false = 0.

nil = 1.

min(0) = 0.
min(1) = 1.

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

le(0,0) = 0.
le(0,1) = 1.
le(1,0) = 0.
le(1,1) = 1.

cons(0,0) = 0.
cons(0,1) = 0.
cons(1,0) = 0.
cons(1,1) = 0.

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

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


The problem is infeasible.