NO

Problem 1: 

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

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

Problem 1: 

Obtaining a proof using Prover9:

 -> Prover9 Output:
============================== Prover9 ===============================
Prover9 (64) version 2009-11A, November 2009.
Process 3344595 was started by sandbox on z020.star.cs.uiowa.edu,
Tue Jul 30 09:31:10 2024
The command was "./prover9 -f /tmp/prover93344585-0.in".
============================== end of head ===========================

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

% Reading from file /tmp/prover93344585-0.in

assign(max_seconds,20).

formulas(assumptions).
->_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(lt(x1,x2),lt(y,x2)) # label(congruence).
->_s0(x2,y) -> ->_s0(lt(x1,x2),lt(x1,y)) # label(congruence).
->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence).
->*_s0(lt(x1,x2),true) -> ->_s0(cons(x1,cons(x2,x3)),cons(x2,cons(x1,x3))) # label(replacement).
->_s0(lt(0,s(x2)),true) # label(replacement).
->_s0(lt(s(x1),s(x2)),lt(x1,x2)) # label(replacement).
->_s0(lt(x1,0),false) # 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(lt(x5,x6),true) & ->*_s0(lt(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(cons(x1,x2),cons(y,x2)) # label(congruence) # label(non_clause).  [assumption].
2 ->_s0(x2,y) -> ->_s0(cons(x1,x2),cons(x1,y)) # label(congruence) # label(non_clause).  [assumption].
3 ->_s0(x1,y) -> ->_s0(lt(x1,x2),lt(y,x2)) # label(congruence) # label(non_clause).  [assumption].
4 ->_s0(x2,y) -> ->_s0(lt(x1,x2),lt(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(lt(x1,x2),true) -> ->_s0(cons(x1,cons(x2,x3)),cons(x2,cons(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(lt(x5,x6),true) & ->*_s0(lt(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(cons(x,z),cons(y,z)) # label(congruence).  [clausify(1)].
-->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence).  [clausify(2)].
-->_s0(x,y) | ->_s0(lt(x,z),lt(y,z)) # label(congruence).  [clausify(3)].
-->_s0(x,y) | ->_s0(lt(z,x),lt(z,y)) # label(congruence).  [clausify(4)].
-->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(5)].
-->*_s0(lt(x,y),true) | ->_s0(cons(x,cons(y,z)),cons(y,cons(x,z))) # label(replacement).  [clausify(6)].
->_s0(lt(0,s(x)),true) # label(replacement).  [assumption].
->_s0(lt(s(x),s(y)),lt(x,y)) # label(replacement).  [assumption].
->_s0(lt(x,0),false) # label(replacement).  [assumption].
->*_s0(x,x) # label(reflexivity).  [assumption].
-->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(7)].
-->*_s0(lt(x,y),true) | -->*_s0(lt(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, true, false, lt, cons, 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(cons(x,z),cons(y,z)) # label(congruence).  [clausify(1)].
kept:      10 -->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence).  [clausify(2)].
kept:      11 -->_s0(x,y) | ->_s0(lt(x,z),lt(y,z)) # label(congruence).  [clausify(3)].
kept:      12 -->_s0(x,y) | ->_s0(lt(z,x),lt(z,y)) # label(congruence).  [clausify(4)].
kept:      13 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(5)].
kept:      14 -->*_s0(lt(x,y),true) | ->_s0(cons(x,cons(y,z)),cons(y,cons(x,z))) # label(replacement).  [clausify(6)].
kept:      15 ->_s0(lt(0,s(x)),true) # label(replacement).  [assumption].
kept:      16 ->_s0(lt(s(x),s(y)),lt(x,y)) # label(replacement).  [assumption].
kept:      17 ->_s0(lt(x,0),false) # label(replacement).  [assumption].
kept:      18 ->*_s0(x,x) # label(reflexivity).  [assumption].
kept:      19 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(7)].
kept:      20 -->*_s0(lt(x,y),true) | -->*_s0(lt(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(cons(x,z),cons(y,z)) # label(congruence).  [clausify(1)].
10 -->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence).  [clausify(2)].
11 -->_s0(x,y) | ->_s0(lt(x,z),lt(y,z)) # label(congruence).  [clausify(3)].
12 -->_s0(x,y) | ->_s0(lt(z,x),lt(z,y)) # label(congruence).  [clausify(4)].
13 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(5)].
14 -->*_s0(lt(x,y),true) | ->_s0(cons(x,cons(y,z)),cons(y,cons(x,z))) # label(replacement).  [clausify(6)].
15 ->_s0(lt(0,s(x)),true) # label(replacement).  [assumption].
16 ->_s0(lt(s(x),s(y)),lt(x,y)) # label(replacement).  [assumption].
17 ->_s0(lt(x,0),false) # label(replacement).  [assumption].
18 ->*_s0(x,x) # label(reflexivity).  [assumption].
19 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(7)].
20 -->*_s0(lt(x,y),true) | -->*_s0(lt(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(cons(x,z),cons(y,z)) # label(congruence).  [clausify(1)].

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

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

given #4 (I,wt=10): 12 -->_s0(x,y) | ->_s0(lt(z,x),lt(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(lt(x,y),true) | ->_s0(cons(x,cons(y,z)),cons(y,cons(x,z))) # label(replacement).  [clausify(6)].

given #7 (I,wt=6): 15 ->_s0(lt(0,s(x)),true) # label(replacement).  [assumption].

given #8 (I,wt=9): 16 ->_s0(lt(s(x),s(y)),lt(x,y)) # label(replacement).  [assumption].

given #9 (I,wt=5): 17 ->_s0(lt(x,0),false) # label(replacement).  [assumption].

given #10 (I,wt=3): 18 ->*_s0(x,x) # label(reflexivity).  [assumption].

given #11 (I,wt=9): 19 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(7)].

given #12 (I,wt=10): 20 -->*_s0(lt(x,y),true) | -->*_s0(lt(y,z),true) # label(goal) # answer(goal).  [deny(8)].

given #13 (A,wt=8): 21 ->_s0(s(lt(0,s(x))),s(true)).  [ur(13,a,15,a)].

given #14 (F,wt=13): 39 -->*_s0(lt(x,y),true) | -->_s0(lt(z,x),u) | -->*_s0(u,true) # answer(goal).  [resolve(20,a,19,c)].

given #15 (F,wt=6): 50 -->*_s0(lt(s(x),y),true) # answer(goal).  [resolve(39,b,15,a),unit_del(b,18)].

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

% Proof 1 at 0.00 (+ 0.01) seconds: goal.
% Length of proof is 12.
% Level of proof is 5.
% Maximum clause weight is 13.000.
% Given clauses 15.

7 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
8 (exists x5 exists x6 exists x7 (->*_s0(lt(x5,x6),true) & ->*_s0(lt(x6,x7),true))) # label(goal) # label(non_clause) # label(goal).  [goal].
15 ->_s0(lt(0,s(x)),true) # label(replacement).  [assumption].
16 ->_s0(lt(s(x),s(y)),lt(x,y)) # label(replacement).  [assumption].
18 ->*_s0(x,x) # label(reflexivity).  [assumption].
19 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(7)].
20 -->*_s0(lt(x,y),true) | -->*_s0(lt(y,z),true) # label(goal) # answer(goal).  [deny(8)].
38 ->*_s0(lt(0,s(x)),true).  [ur(19,a,15,a,b,18,a)].
39 -->*_s0(lt(x,y),true) | -->_s0(lt(z,x),u) | -->*_s0(u,true) # answer(goal).  [resolve(20,a,19,c)].
50 -->*_s0(lt(s(x),y),true) # answer(goal).  [resolve(39,b,15,a),unit_del(b,18)].
57 -->*_s0(lt(x,y),true) # answer(goal).  [ur(19,a,16,a,c,50,a)].
58 $F # answer(goal).  [resolve(57,a,38,a)].

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

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

Given=15. Generated=51. Kept=49. proofs=1.
Usable=15. Sos=29. Demods=0. Limbo=3, Disabled=13. Hints=0.
Kept_by_rule=0, Deleted_by_rule=0.
Forward_subsumed=2. Back_subsumed=1.
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=18. Nonunit_bsub_feature_tests=35.
Megabytes=0.14.
User_CPU=0.00, System_CPU=0.01, Wall_clock=0.

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

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

THEOREM PROVED

Exiting with 1 proof.

Process 3344595 exit (max_proofs) Tue Jul 30 09:31:10 2024


The problem is feasible.