NO

Problem 1: 

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

Infeasibility Conditions:
les(x,x1) ->* true, les(x1,x2) ->* true

Problem 1: 

Obtaining a proof using Prover9:

 -> Prover9 Output:
============================== Prover9 ===============================
Prover9 (64) version 2009-11A, November 2009.
Process 3331822 was started by sandbox2 on z014.star.cs.uiowa.edu,
Tue Jul 30 09:20:10 2024
The command was "./prover9 -f /tmp/prover93331811-0.in".
============================== end of head ===========================

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

% Reading from file /tmp/prover93331811-0.in

assign(max_seconds,20).

formulas(assumptions).
->_s0(x1,y) -> ->_s0(dot(x1,x2),dot(y,x2)) # label(congruence).
->_s0(x2,y) -> ->_s0(dot(x1,x2),dot(x1,y)) # label(congruence).
->_s0(x1,y) -> ->_s0(les(x1,x2),les(y,x2)) # label(congruence).
->_s0(x2,y) -> ->_s0(les(x1,x2),les(x1,y)) # label(congruence).
->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence).
->*_s0(les(x1,x2),true) -> ->_s0(dot(x1,dot(x2,x3)),dot(x2,dot(x1,x3))) # label(replacement).
->_s0(les(0,0),false) # label(replacement).
->_s0(les(0,s(0)),true) # label(replacement).
->_s0(les(0,s(s(x1))),les(0,s(x1))) # label(replacement).
->_s0(les(s(0),0),false) # label(replacement).
->_s0(les(s(s(x1)),0),les(s(x1),0)) # label(replacement).
->_s0(les(s(x1),s(x2)),les(x1,x2)) # 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 exists x7 (->*_s0(les(x5,x6),true) & ->*_s0(les(x6,x7),true))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->_s0(x1,y) -> ->_s0(dot(x1,x2),dot(y,x2)) # label(congruence) # label(non_clause).  [assumption].
2 ->_s0(x2,y) -> ->_s0(dot(x1,x2),dot(x1,y)) # label(congruence) # label(non_clause).  [assumption].
3 ->_s0(x1,y) -> ->_s0(les(x1,x2),les(y,x2)) # label(congruence) # label(non_clause).  [assumption].
4 ->_s0(x2,y) -> ->_s0(les(x1,x2),les(x1,y)) # label(congruence) # label(non_clause).  [assumption].
5 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause).  [assumption].
6 ->*_s0(les(x1,x2),true) -> ->_s0(dot(x1,dot(x2,x3)),dot(x2,dot(x1,x3))) # label(replacement) # label(non_clause).  [assumption].
7 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
8 (exists x5 exists x6 exists x7 (->*_s0(les(x5,x6),true) & ->*_s0(les(x6,x7),true))) # 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(dot(x,z),dot(y,z)) # label(congruence).  [clausify(1)].
-->_s0(x,y) | ->_s0(dot(z,x),dot(z,y)) # label(congruence).  [clausify(2)].
-->_s0(x,y) | ->_s0(les(x,z),les(y,z)) # label(congruence).  [clausify(3)].
-->_s0(x,y) | ->_s0(les(z,x),les(z,y)) # label(congruence).  [clausify(4)].
-->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(5)].
-->*_s0(les(x,y),true) | ->_s0(dot(x,dot(y,z)),dot(y,dot(x,z))) # label(replacement).  [clausify(6)].
->_s0(les(0,0),false) # label(replacement).  [assumption].
->_s0(les(0,s(0)),true) # label(replacement).  [assumption].
->_s0(les(0,s(s(x))),les(0,s(x))) # label(replacement).  [assumption].
->_s0(les(s(0),0),false) # label(replacement).  [assumption].
->_s0(les(s(s(x)),0),les(s(x),0)) # label(replacement).  [assumption].
->_s0(les(s(x),s(y)),les(x,y)) # label(replacement).  [assumption].
->*_s0(x,x) # label(reflexivity).  [assumption].
-->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(7)].
-->*_s0(les(x,y),true) | -->*_s0(les(y,z),true) # label(goal).  [deny(8)].
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([ 0, false, true, les, dot, s ]).
After inverse_order:  (no changes).
Unfolding symbols: (none).

Auto_inference settings:
  % set(neg_binary_resolution).  % (HNE depth_diff=-6)
  % clear(ordered_res).  % (HNE depth_diff=-6)
  % set(ur_resolution).  % (HNE depth_diff=-6)
    % 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:      9 -->_s0(x,y) | ->_s0(dot(x,z),dot(y,z)) # label(congruence).  [clausify(1)].
kept:      10 -->_s0(x,y) | ->_s0(dot(z,x),dot(z,y)) # label(congruence).  [clausify(2)].
kept:      11 -->_s0(x,y) | ->_s0(les(x,z),les(y,z)) # label(congruence).  [clausify(3)].
kept:      12 -->_s0(x,y) | ->_s0(les(z,x),les(z,y)) # label(congruence).  [clausify(4)].
kept:      13 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(5)].
kept:      14 -->*_s0(les(x,y),true) | ->_s0(dot(x,dot(y,z)),dot(y,dot(x,z))) # label(replacement).  [clausify(6)].
kept:      15 ->_s0(les(0,0),false) # label(replacement).  [assumption].
kept:      16 ->_s0(les(0,s(0)),true) # label(replacement).  [assumption].
kept:      17 ->_s0(les(0,s(s(x))),les(0,s(x))) # label(replacement).  [assumption].
kept:      18 ->_s0(les(s(0),0),false) # label(replacement).  [assumption].
kept:      19 ->_s0(les(s(s(x)),0),les(s(x),0)) # label(replacement).  [assumption].
kept:      20 ->_s0(les(s(x),s(y)),les(x,y)) # label(replacement).  [assumption].
kept:      21 ->*_s0(x,x) # label(reflexivity).  [assumption].
kept:      22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(7)].
kept:      23 -->*_s0(les(x,y),true) | -->*_s0(les(y,z),true) # label(goal) # answer(goal).  [deny(8)].

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

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

% Clauses after input processing:

formulas(usable).
end_of_list.

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

formulas(demodulators).
end_of_list.

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

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

% Starting search at 0.00 seconds.

given #1 (I,wt=10): 9 -->_s0(x,y) | ->_s0(dot(x,z),dot(y,z)) # label(congruence).  [clausify(1)].

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

given #3 (I,wt=10): 11 -->_s0(x,y) | ->_s0(les(x,z),les(y,z)) # label(congruence).  [clausify(3)].

given #4 (I,wt=10): 12 -->_s0(x,y) | ->_s0(les(z,x),les(z,y)) # label(congruence).  [clausify(4)].

given #5 (I,wt=8): 13 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(5)].

given #6 (I,wt=16): 14 -->*_s0(les(x,y),true) | ->_s0(dot(x,dot(y,z)),dot(y,dot(x,z))) # label(replacement).  [clausify(6)].

given #7 (I,wt=5): 15 ->_s0(les(0,0),false) # label(replacement).  [assumption].

given #8 (I,wt=6): 16 ->_s0(les(0,s(0)),true) # label(replacement).  [assumption].

given #9 (I,wt=10): 17 ->_s0(les(0,s(s(x))),les(0,s(x))) # label(replacement).  [assumption].

given #10 (I,wt=6): 18 ->_s0(les(s(0),0),false) # label(replacement).  [assumption].

given #11 (I,wt=10): 19 ->_s0(les(s(s(x)),0),les(s(x),0)) # label(replacement).  [assumption].

given #12 (I,wt=9): 20 ->_s0(les(s(x),s(y)),les(x,y)) # label(replacement).  [assumption].

given #13 (I,wt=3): 21 ->*_s0(x,x) # label(reflexivity).  [assumption].

given #14 (I,wt=9): 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(7)].

given #15 (I,wt=10): 23 -->*_s0(les(x,y),true) | -->*_s0(les(y,z),true) # label(goal) # answer(goal).  [deny(8)].

given #16 (A,wt=7): 24 ->_s0(s(les(0,0)),s(false)).  [ur(13,a,15,a)].

given #17 (F,wt=13): 60 -->*_s0(les(x,y),true) | -->_s0(les(z,x),u) | -->*_s0(u,true) # answer(goal).  [resolve(23,a,22,c)].

given #18 (F,wt=6): 71 -->*_s0(les(s(0),x),true) # answer(goal).  [resolve(60,b,16,a),unit_del(b,21)].

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

% Proof 1 at 0.01 (+ 0.00) seconds: goal.
% Length of proof is 12.
% Level of proof is 5.
% Maximum clause weight is 13.000.
% Given clauses 18.

7 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
8 (exists x5 exists x6 exists x7 (->*_s0(les(x5,x6),true) & ->*_s0(les(x6,x7),true))) # label(goal) # label(non_clause) # label(goal).  [goal].
16 ->_s0(les(0,s(0)),true) # label(replacement).  [assumption].
20 ->_s0(les(s(x),s(y)),les(x,y)) # label(replacement).  [assumption].
21 ->*_s0(x,x) # label(reflexivity).  [assumption].
22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(7)].
23 -->*_s0(les(x,y),true) | -->*_s0(les(y,z),true) # label(goal) # answer(goal).  [deny(8)].
58 ->*_s0(les(0,s(0)),true).  [ur(22,a,16,a,b,21,a)].
60 -->*_s0(les(x,y),true) | -->_s0(les(z,x),u) | -->*_s0(u,true) # answer(goal).  [resolve(23,a,22,c)].
71 -->*_s0(les(s(0),x),true) # answer(goal).  [resolve(60,b,16,a),unit_del(b,21)].
77 -->*_s0(les(0,x),true) # answer(goal).  [ur(22,a,20,a,c,71,a)].
78 $F # answer(goal).  [resolve(77,a,58,a)].

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

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

Given=18. Generated=74. Kept=69. proofs=1.
Usable=18. Sos=48. Demods=0. Limbo=2, Disabled=15. Hints=0.
Kept_by_rule=0, Deleted_by_rule=0.
Forward_subsumed=5. 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=22. Nonunit_bsub_feature_tests=34.
Megabytes=0.17.
User_CPU=0.01, System_CPU=0.00, Wall_clock=0.

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

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

THEOREM PROVED

Exiting with 1 proof.

Process 3331822 exit (max_proofs) Tue Jul 30 09:20:10 2024


The problem is feasible.