NO

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x y l vNonEmpty x1 x2)
(STRATEGY CONTEXTSENSITIVE
(le 1 2)
(min 1)
(0)
(cons 1 2)
(fSNonEmpty)
(false)
(nil)
(s 1)
(true)
)
(RULES
le(0,s(x)) -> true
le(s(x),s(y)) -> le(x,y)
le(x,0) -> false
min(cons(x,nil)) -> x
min(cons(x,l)) -> min(l) | le(x,min(l)) ->* false
min(cons(x,l)) -> min(l) | min(l) ->* x
min(cons(x,l)) -> x | le(x,min(l)) ->* true
)
]

Infeasibility Conditions:
le(x1,min(x2)) ->* false, min(x2) ->* x1

Problem 1: 

Obtaining a proof using Prover9:

 -> Prover9 Output:
============================== Prover9 ===============================
Prover9 (64) version 2009-11A, November 2009.
Process 3262623 was started by sandbox2 on z024.star.cs.uiowa.edu,
Tue Jul 30 08:51:40 2024
The command was "./prover9 -f /tmp/prover93262616-0.in".
============================== end of head ===========================

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

% Reading from file /tmp/prover93262616-0.in

assign(max_seconds,20).

formulas(assumptions).
->_s0(x1,y) -> ->_s0(le(x1,x2),le(y,x2)) # label(congruence).
->_s0(x2,y) -> ->_s0(le(x1,x2),le(x1,y)) # label(congruence).
->_s0(x1,y) -> ->_s0(min(x1),min(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(cons(x1,x2),cons(y,x2)) # label(congruence).
->_s0(x2,y) -> ->_s0(cons(x1,x2),cons(x1,y)) # label(congruence).
->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence).
->_s0(le(0,s(x1)),true) # label(replacement).
->_s0(le(s(x1),s(x2)),le(x1,x2)) # label(replacement).
->_s0(le(x1,0),false) # label(replacement).
->_s0(min(cons(x1,nil)),x1) # label(replacement).
->*_s0(le(x1,min(x3)),false) -> ->_s0(min(cons(x1,x3)),min(x3)) # label(replacement).
->*_s0(min(x3),x1) -> ->_s0(min(cons(x1,x3)),min(x3)) # label(replacement).
->*_s0(le(x1,min(x3)),true) -> ->_s0(min(cons(x1,x3)),x1) # label(replacement).
->*_s0(x,x) # label(reflexivity).
->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x5 exists x6 (->*_s0(le(x5,min(x6)),false) & ->*_s0(min(x6),x5))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->_s0(x1,y) -> ->_s0(le(x1,x2),le(y,x2)) # label(congruence) # label(non_clause).  [assumption].
2 ->_s0(x2,y) -> ->_s0(le(x1,x2),le(x1,y)) # label(congruence) # label(non_clause).  [assumption].
3 ->_s0(x1,y) -> ->_s0(min(x1),min(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->_s0(x1,y) -> ->_s0(cons(x1,x2),cons(y,x2)) # label(congruence) # label(non_clause).  [assumption].
5 ->_s0(x2,y) -> ->_s0(cons(x1,x2),cons(x1,y)) # label(congruence) # label(non_clause).  [assumption].
6 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause).  [assumption].
7 ->*_s0(le(x1,min(x3)),false) -> ->_s0(min(cons(x1,x3)),min(x3)) # label(replacement) # label(non_clause).  [assumption].
8 ->*_s0(min(x3),x1) -> ->_s0(min(cons(x1,x3)),min(x3)) # label(replacement) # label(non_clause).  [assumption].
9 ->*_s0(le(x1,min(x3)),true) -> ->_s0(min(cons(x1,x3)),x1) # label(replacement) # label(non_clause).  [assumption].
10 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
11 (exists x5 exists x6 (->*_s0(le(x5,min(x6)),false) & ->*_s0(min(x6),x5))) # label(goal) # label(non_clause) # label(goal).  [goal].

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

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
-->_s0(x,y) | ->_s0(le(x,z),le(y,z)) # label(congruence).  [clausify(1)].
-->_s0(x,y) | ->_s0(le(z,x),le(z,y)) # label(congruence).  [clausify(2)].
-->_s0(x,y) | ->_s0(min(x),min(y)) # label(congruence).  [clausify(3)].
-->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence).  [clausify(4)].
-->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence).  [clausify(5)].
-->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(6)].
->_s0(le(0,s(x)),true) # label(replacement).  [assumption].
->_s0(le(s(x),s(y)),le(x,y)) # label(replacement).  [assumption].
->_s0(le(x,0),false) # label(replacement).  [assumption].
->_s0(min(cons(x,nil)),x) # label(replacement).  [assumption].
-->*_s0(le(x,min(y)),false) | ->_s0(min(cons(x,y)),min(y)) # label(replacement).  [clausify(7)].
-->*_s0(min(x),y) | ->_s0(min(cons(y,x)),min(x)) # label(replacement).  [clausify(8)].
-->*_s0(le(x,min(y)),true) | ->_s0(min(cons(x,y)),x) # label(replacement).  [clausify(9)].
->*_s0(x,x) # label(reflexivity).  [assumption].
-->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(10)].
-->*_s0(le(x,min(y)),false) | -->*_s0(min(y),x) # label(goal).  [deny(11)].
end_of_list.

formulas(demodulators).
end_of_list.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label goal to answer in negative clause

Term ordering decisions:
Predicate symbol precedence:  predicate_order([ ->_s0, ->*_s0 ]).
Function symbol precedence:  function_order([ true, 0, false, nil, le, cons, min, s ]).
After inverse_order:  (no changes).
Unfolding symbols: (none).

Auto_inference settings:
  % set(neg_binary_resolution).  % (HNE depth_diff=-7)
  % clear(ordered_res).  % (HNE depth_diff=-7)
  % set(ur_resolution).  % (HNE depth_diff=-7)
    % set(ur_resolution) -> set(pos_ur_resolution).
    % set(ur_resolution) -> set(neg_ur_resolution).

Auto_process settings:
  % set(unit_deletion).  % (Horn set with negative nonunits)

kept:      12 -->_s0(x,y) | ->_s0(le(x,z),le(y,z)) # label(congruence).  [clausify(1)].
kept:      13 -->_s0(x,y) | ->_s0(le(z,x),le(z,y)) # label(congruence).  [clausify(2)].
kept:      14 -->_s0(x,y) | ->_s0(min(x),min(y)) # label(congruence).  [clausify(3)].
kept:      15 -->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence).  [clausify(4)].
kept:      16 -->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence).  [clausify(5)].
kept:      17 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(6)].
kept:      18 ->_s0(le(0,s(x)),true) # label(replacement).  [assumption].
kept:      19 ->_s0(le(s(x),s(y)),le(x,y)) # label(replacement).  [assumption].
kept:      20 ->_s0(le(x,0),false) # label(replacement).  [assumption].
kept:      21 ->_s0(min(cons(x,nil)),x) # label(replacement).  [assumption].
kept:      22 -->*_s0(le(x,min(y)),false) | ->_s0(min(cons(x,y)),min(y)) # label(replacement).  [clausify(7)].
kept:      23 -->*_s0(min(x),y) | ->_s0(min(cons(y,x)),min(x)) # label(replacement).  [clausify(8)].
kept:      24 -->*_s0(le(x,min(y)),true) | ->_s0(min(cons(x,y)),x) # label(replacement).  [clausify(9)].
kept:      25 ->*_s0(x,x) # label(reflexivity).  [assumption].
kept:      26 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(10)].
kept:      27 -->*_s0(le(x,min(y)),false) | -->*_s0(min(y),x) # label(goal) # answer(goal).  [deny(11)].

============================== end of process initial clauses ========

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

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
12 -->_s0(x,y) | ->_s0(le(x,z),le(y,z)) # label(congruence).  [clausify(1)].
13 -->_s0(x,y) | ->_s0(le(z,x),le(z,y)) # label(congruence).  [clausify(2)].
14 -->_s0(x,y) | ->_s0(min(x),min(y)) # label(congruence).  [clausify(3)].
15 -->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence).  [clausify(4)].
16 -->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence).  [clausify(5)].
17 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(6)].
18 ->_s0(le(0,s(x)),true) # label(replacement).  [assumption].
19 ->_s0(le(s(x),s(y)),le(x,y)) # label(replacement).  [assumption].
20 ->_s0(le(x,0),false) # label(replacement).  [assumption].
21 ->_s0(min(cons(x,nil)),x) # label(replacement).  [assumption].
22 -->*_s0(le(x,min(y)),false) | ->_s0(min(cons(x,y)),min(y)) # label(replacement).  [clausify(7)].
23 -->*_s0(min(x),y) | ->_s0(min(cons(y,x)),min(x)) # label(replacement).  [clausify(8)].
24 -->*_s0(le(x,min(y)),true) | ->_s0(min(cons(x,y)),x) # label(replacement).  [clausify(9)].
25 ->*_s0(x,x) # label(reflexivity).  [assumption].
26 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(10)].
27 -->*_s0(le(x,min(y)),false) | -->*_s0(min(y),x) # label(goal) # answer(goal).  [deny(11)].
end_of_list.

formulas(demodulators).
end_of_list.

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

============================== SEARCH ================================

% Starting search at 0.01 seconds.

given #1 (I,wt=10): 12 -->_s0(x,y) | ->_s0(le(x,z),le(y,z)) # label(congruence).  [clausify(1)].

given #2 (I,wt=10): 13 -->_s0(x,y) | ->_s0(le(z,x),le(z,y)) # label(congruence).  [clausify(2)].

given #3 (I,wt=8): 14 -->_s0(x,y) | ->_s0(min(x),min(y)) # label(congruence).  [clausify(3)].

given #4 (I,wt=10): 15 -->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence).  [clausify(4)].

given #5 (I,wt=10): 16 -->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence).  [clausify(5)].

given #6 (I,wt=8): 17 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(6)].

given #7 (I,wt=6): 18 ->_s0(le(0,s(x)),true) # label(replacement).  [assumption].

given #8 (I,wt=9): 19 ->_s0(le(s(x),s(y)),le(x,y)) # label(replacement).  [assumption].

given #9 (I,wt=5): 20 ->_s0(le(x,0),false) # label(replacement).  [assumption].

given #10 (I,wt=6): 21 ->_s0(min(cons(x,nil)),x) # label(replacement).  [assumption].

given #11 (I,wt=13): 22 -->*_s0(le(x,min(y)),false) | ->_s0(min(cons(x,y)),min(y)) # label(replacement).  [clausify(7)].

given #12 (I,wt=11): 23 -->*_s0(min(x),y) | ->_s0(min(cons(y,x)),min(x)) # label(replacement).  [clausify(8)].

given #13 (I,wt=12): 24 -->*_s0(le(x,min(y)),true) | ->_s0(min(cons(x,y)),x) # label(replacement).  [clausify(9)].

given #14 (I,wt=3): 25 ->*_s0(x,x) # label(reflexivity).  [assumption].

given #15 (I,wt=9): 26 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(10)].

given #16 (I,wt=10): 27 -->*_s0(le(x,min(y)),false) | -->*_s0(min(y),x) # label(goal) # answer(goal).  [deny(11)].

given #17 (A,wt=8): 28 ->_s0(s(le(0,s(x))),s(true)).  [ur(17,a,18,a)].

given #18 (F,wt=7): 59 -->*_s0(le(min(x),min(x)),false) # answer(goal).  [resolve(27,b,25,a)].

given #19 (F,wt=7): 68 -->_s0(le(min(x),min(x)),false) # answer(goal).  [ur(26,b,25,a,c,59,a)].

given #20 (F,wt=10): 67 -->_s0(le(min(x),min(x)),y) | -->*_s0(y,false) # answer(goal).  [resolve(59,a,26,c)].

given #21 (F,wt=10): 69 -->*_s0(le(min(x),y),false) | -->_s0(min(x),y) # answer(goal).  [resolve(67,a,13,b)].

============================== PROOF =================================

% Proof 1 at 0.01 (+ 0.00) seconds: goal.
% Length of proof is 15.
% Level of proof is 6.
% Maximum clause weight is 10.000.
% Given clauses 21.

2 ->_s0(x2,y) -> ->_s0(le(x1,x2),le(x1,y)) # label(congruence) # label(non_clause).  [assumption].
10 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
11 (exists x5 exists x6 (->*_s0(le(x5,min(x6)),false) & ->*_s0(min(x6),x5))) # label(goal) # label(non_clause) # label(goal).  [goal].
13 -->_s0(x,y) | ->_s0(le(z,x),le(z,y)) # label(congruence).  [clausify(2)].
20 ->_s0(le(x,0),false) # label(replacement).  [assumption].
21 ->_s0(min(cons(x,nil)),x) # label(replacement).  [assumption].
25 ->*_s0(x,x) # label(reflexivity).  [assumption].
26 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(10)].
27 -->*_s0(le(x,min(y)),false) | -->*_s0(min(y),x) # label(goal) # answer(goal).  [deny(11)].
54 ->*_s0(le(x,0),false).  [ur(26,a,20,a,b,25,a)].
59 -->*_s0(le(min(x),min(x)),false) # answer(goal).  [resolve(27,b,25,a)].
67 -->_s0(le(min(x),min(x)),y) | -->*_s0(y,false) # answer(goal).  [resolve(59,a,26,c)].
69 -->*_s0(le(min(x),y),false) | -->_s0(min(x),y) # answer(goal).  [resolve(67,a,13,b)].
76 -->*_s0(le(min(cons(x,nil)),x),false) # answer(goal).  [resolve(69,b,21,a)].
77 $F # answer(goal).  [resolve(76,a,54,a)].

============================== end of proof ==========================

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

Given=21. Generated=68. Kept=65. proofs=1.
Usable=21. Sos=39. Demods=0. Limbo=4, Disabled=16. Hints=0.
Kept_by_rule=0, Deleted_by_rule=0.
Forward_subsumed=3. Back_subsumed=0.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0.
Demod_attempts=0. Demod_rewrites=0.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=19. Nonunit_bsub_feature_tests=43.
Megabytes=0.18.
User_CPU=0.01, System_CPU=0.00, Wall_clock=0.

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

============================== end of search =========================

THEOREM PROVED

Exiting with 1 proof.

Process 3262623 exit (max_proofs) Tue Jul 30 08:51:40 2024


The problem is feasible.