NO

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x y z vNonEmpty x y z)
(STRATEGY CONTEXTSENSITIVE
(and 1 2)
(f 1)
(implies 1 2)
(not 1)
(or 1 2)
(0)
(1)
(fSNonEmpty)
)
(RULES
and(not(x),x) -> 0
and(0,x) -> 0
and(1,x) -> x
and(x,not(x)) -> 0
and(x,0) -> 0
and(x,1) -> x
f(x) -> f(0) | implies(x,0) ->* y, implies(x,y) ->* z, implies(z,0) ->* 1
implies(x,y) -> 0 | x ->* 1, y ->* 0
implies(x,y) -> 1 | not(x) ->* 1
implies(x,y) -> 1 | y ->* 1
not(0) -> 1
not(1) -> 0
or(not(x),x) -> 1
or(0,x) -> x
or(1,x) -> 1
or(x,not(x)) -> 1
or(x,0) -> x
or(x,1) -> 1
)
]

Infeasibility Conditions:
implies(x,0) ->* y, implies(x,y) ->* z, implies(z,0) ->* 1

Problem 1: 

Obtaining a proof using Prover9:

 -> Prover9 Output:
============================== Prover9 ===============================
Prover9 (64) version 2009-11A, November 2009.
Process 3450203 was started by sandbox2 on z028.star.cs.uiowa.edu,
Tue Jul 30 09:55:40 2024
The command was "./prover9 -f /tmp/prover93450196-0.in".
============================== end of head ===========================

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

% Reading from file /tmp/prover93450196-0.in

assign(max_seconds,20).

formulas(assumptions).
->_s0(x1,y) -> ->_s0(and(x1,x2),and(y,x2)) # label(congruence).
->_s0(x2,y) -> ->_s0(and(x1,x2),and(x1,y)) # label(congruence).
->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(implies(x1,x2),implies(y,x2)) # label(congruence).
->_s0(x2,y) -> ->_s0(implies(x1,x2),implies(x1,y)) # label(congruence).
->_s0(x1,y) -> ->_s0(not(x1),not(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(or(x1,x2),or(y,x2)) # label(congruence).
->_s0(x2,y) -> ->_s0(or(x1,x2),or(x1,y)) # label(congruence).
->_s0(and(not(x1),x1),0) # label(replacement).
->_s0(and(0,x1),0) # label(replacement).
->_s0(and(1,x1),x1) # label(replacement).
->_s0(and(x1,not(x1)),0) # label(replacement).
->_s0(and(x1,0),0) # label(replacement).
->_s0(and(x1,1),x1) # label(replacement).
->*_s0(implies(x1,0),x2) & ->*_s0(implies(x1,x2),x3) & ->*_s0(implies(x3,0),1) -> ->_s0(f(x1),f(0)) # label(replacement).
->*_s0(x1,1) & ->*_s0(x2,0) -> ->_s0(implies(x1,x2),0) # label(replacement).
->*_s0(not(x1),1) -> ->_s0(implies(x1,x2),1) # label(replacement).
->*_s0(x2,1) -> ->_s0(implies(x1,x2),1) # label(replacement).
->_s0(not(0),1) # label(replacement).
->_s0(not(1),0) # label(replacement).
->_s0(or(not(x1),x1),1) # label(replacement).
->_s0(or(0,x1),x1) # label(replacement).
->_s0(or(1,x1),1) # label(replacement).
->_s0(or(x1,not(x1)),1) # label(replacement).
->_s0(or(x1,0),x1) # label(replacement).
->_s0(or(x1,1),1) # 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(implies(x5,0),x6) & ->*_s0(implies(x5,x6),x7) & ->*_s0(implies(x7,0),1))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->_s0(x1,y) -> ->_s0(and(x1,x2),and(y,x2)) # label(congruence) # label(non_clause).  [assumption].
2 ->_s0(x2,y) -> ->_s0(and(x1,x2),and(x1,y)) # label(congruence) # label(non_clause).  [assumption].
3 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->_s0(x1,y) -> ->_s0(implies(x1,x2),implies(y,x2)) # label(congruence) # label(non_clause).  [assumption].
5 ->_s0(x2,y) -> ->_s0(implies(x1,x2),implies(x1,y)) # label(congruence) # label(non_clause).  [assumption].
6 ->_s0(x1,y) -> ->_s0(not(x1),not(y)) # label(congruence) # label(non_clause).  [assumption].
7 ->_s0(x1,y) -> ->_s0(or(x1,x2),or(y,x2)) # label(congruence) # label(non_clause).  [assumption].
8 ->_s0(x2,y) -> ->_s0(or(x1,x2),or(x1,y)) # label(congruence) # label(non_clause).  [assumption].
9 ->*_s0(implies(x1,0),x2) & ->*_s0(implies(x1,x2),x3) & ->*_s0(implies(x3,0),1) -> ->_s0(f(x1),f(0)) # label(replacement) # label(non_clause).  [assumption].
10 ->*_s0(x1,1) & ->*_s0(x2,0) -> ->_s0(implies(x1,x2),0) # label(replacement) # label(non_clause).  [assumption].
11 ->*_s0(not(x1),1) -> ->_s0(implies(x1,x2),1) # label(replacement) # label(non_clause).  [assumption].
12 ->*_s0(x2,1) -> ->_s0(implies(x1,x2),1) # label(replacement) # label(non_clause).  [assumption].
13 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
14 (exists x5 exists x6 exists x7 (->*_s0(implies(x5,0),x6) & ->*_s0(implies(x5,x6),x7) & ->*_s0(implies(x7,0),1))) # 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(and(x,z),and(y,z)) # label(congruence).  [clausify(1)].
-->_s0(x,y) | ->_s0(and(z,x),and(z,y)) # label(congruence).  [clausify(2)].
-->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence).  [clausify(3)].
-->_s0(x,y) | ->_s0(implies(x,z),implies(y,z)) # label(congruence).  [clausify(4)].
-->_s0(x,y) | ->_s0(implies(z,x),implies(z,y)) # label(congruence).  [clausify(5)].
-->_s0(x,y) | ->_s0(not(x),not(y)) # label(congruence).  [clausify(6)].
-->_s0(x,y) | ->_s0(or(x,z),or(y,z)) # label(congruence).  [clausify(7)].
-->_s0(x,y) | ->_s0(or(z,x),or(z,y)) # label(congruence).  [clausify(8)].
->_s0(and(not(x),x),0) # label(replacement).  [assumption].
->_s0(and(0,x),0) # label(replacement).  [assumption].
->_s0(and(1,x),x) # label(replacement).  [assumption].
->_s0(and(x,not(x)),0) # label(replacement).  [assumption].
->_s0(and(x,0),0) # label(replacement).  [assumption].
->_s0(and(x,1),x) # label(replacement).  [assumption].
-->*_s0(implies(x,0),y) | -->*_s0(implies(x,y),z) | -->*_s0(implies(z,0),1) | ->_s0(f(x),f(0)) # label(replacement).  [clausify(9)].
-->*_s0(x,1) | -->*_s0(y,0) | ->_s0(implies(x,y),0) # label(replacement).  [clausify(10)].
-->*_s0(not(x),1) | ->_s0(implies(x,y),1) # label(replacement).  [clausify(11)].
-->*_s0(x,1) | ->_s0(implies(y,x),1) # label(replacement).  [clausify(12)].
->_s0(not(0),1) # label(replacement).  [assumption].
->_s0(not(1),0) # label(replacement).  [assumption].
->_s0(or(not(x),x),1) # label(replacement).  [assumption].
->_s0(or(0,x),x) # label(replacement).  [assumption].
->_s0(or(1,x),1) # label(replacement).  [assumption].
->_s0(or(x,not(x)),1) # label(replacement).  [assumption].
->_s0(or(x,0),x) # label(replacement).  [assumption].
->_s0(or(x,1),1) # label(replacement).  [assumption].
->*_s0(x,x) # label(reflexivity).  [assumption].
-->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(13)].
-->*_s0(implies(x,0),y) | -->*_s0(implies(x,y),z) | -->*_s0(implies(z,0),1) # label(goal).  [deny(14)].
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([ 1, 0, and, implies, or, not, f ]).
After inverse_order:  (no changes).
Unfolding symbols: (none).

Auto_inference settings:
  % set(neg_binary_resolution).  % (HNE depth_diff=-10)
  % clear(ordered_res).  % (HNE depth_diff=-10)
  % set(ur_resolution).  % (HNE depth_diff=-10)
    % 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:      15 -->_s0(x,y) | ->_s0(and(x,z),and(y,z)) # label(congruence).  [clausify(1)].
kept:      16 -->_s0(x,y) | ->_s0(and(z,x),and(z,y)) # label(congruence).  [clausify(2)].
kept:      17 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence).  [clausify(3)].
kept:      18 -->_s0(x,y) | ->_s0(implies(x,z),implies(y,z)) # label(congruence).  [clausify(4)].
kept:      19 -->_s0(x,y) | ->_s0(implies(z,x),implies(z,y)) # label(congruence).  [clausify(5)].
kept:      20 -->_s0(x,y) | ->_s0(not(x),not(y)) # label(congruence).  [clausify(6)].
kept:      21 -->_s0(x,y) | ->_s0(or(x,z),or(y,z)) # label(congruence).  [clausify(7)].
kept:      22 -->_s0(x,y) | ->_s0(or(z,x),or(z,y)) # label(congruence).  [clausify(8)].
kept:      23 ->_s0(and(not(x),x),0) # label(replacement).  [assumption].
kept:      24 ->_s0(and(0,x),0) # label(replacement).  [assumption].
kept:      25 ->_s0(and(1,x),x) # label(replacement).  [assumption].
kept:      26 ->_s0(and(x,not(x)),0) # label(replacement).  [assumption].
kept:      27 ->_s0(and(x,0),0) # label(replacement).  [assumption].
kept:      28 ->_s0(and(x,1),x) # label(replacement).  [assumption].
kept:      29 -->*_s0(implies(x,0),y) | -->*_s0(implies(x,y),z) | -->*_s0(implies(z,0),1) | ->_s0(f(x),f(0)) # label(replacement).  [clausify(9)].
kept:      30 -->*_s0(x,1) | -->*_s0(y,0) | ->_s0(implies(x,y),0) # label(replacement).  [clausify(10)].
kept:      31 -->*_s0(not(x),1) | ->_s0(implies(x,y),1) # label(replacement).  [clausify(11)].
kept:      32 -->*_s0(x,1) | ->_s0(implies(y,x),1) # label(replacement).  [clausify(12)].
kept:      33 ->_s0(not(0),1) # label(replacement).  [assumption].
kept:      34 ->_s0(not(1),0) # label(replacement).  [assumption].
kept:      35 ->_s0(or(not(x),x),1) # label(replacement).  [assumption].
kept:      36 ->_s0(or(0,x),x) # label(replacement).  [assumption].
kept:      37 ->_s0(or(1,x),1) # label(replacement).  [assumption].
kept:      38 ->_s0(or(x,not(x)),1) # label(replacement).  [assumption].
kept:      39 ->_s0(or(x,0),x) # label(replacement).  [assumption].
kept:      40 ->_s0(or(x,1),1) # label(replacement).  [assumption].
kept:      41 ->*_s0(x,x) # label(reflexivity).  [assumption].
kept:      42 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(13)].
kept:      43 -->*_s0(implies(x,0),y) | -->*_s0(implies(x,y),z) | -->*_s0(implies(z,0),1) # label(goal) # answer(goal).  [deny(14)].

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

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

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
15 -->_s0(x,y) | ->_s0(and(x,z),and(y,z)) # label(congruence).  [clausify(1)].
16 -->_s0(x,y) | ->_s0(and(z,x),and(z,y)) # label(congruence).  [clausify(2)].
17 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence).  [clausify(3)].
18 -->_s0(x,y) | ->_s0(implies(x,z),implies(y,z)) # label(congruence).  [clausify(4)].
19 -->_s0(x,y) | ->_s0(implies(z,x),implies(z,y)) # label(congruence).  [clausify(5)].
20 -->_s0(x,y) | ->_s0(not(x),not(y)) # label(congruence).  [clausify(6)].
21 -->_s0(x,y) | ->_s0(or(x,z),or(y,z)) # label(congruence).  [clausify(7)].
22 -->_s0(x,y) | ->_s0(or(z,x),or(z,y)) # label(congruence).  [clausify(8)].
23 ->_s0(and(not(x),x),0) # label(replacement).  [assumption].
24 ->_s0(and(0,x),0) # label(replacement).  [assumption].
25 ->_s0(and(1,x),x) # label(replacement).  [assumption].
26 ->_s0(and(x,not(x)),0) # label(replacement).  [assumption].
27 ->_s0(and(x,0),0) # label(replacement).  [assumption].
28 ->_s0(and(x,1),x) # label(replacement).  [assumption].
30 -->*_s0(x,1) | -->*_s0(y,0) | ->_s0(implies(x,y),0) # label(replacement).  [clausify(10)].
31 -->*_s0(not(x),1) | ->_s0(implies(x,y),1) # label(replacement).  [clausify(11)].
32 -->*_s0(x,1) | ->_s0(implies(y,x),1) # label(replacement).  [clausify(12)].
33 ->_s0(not(0),1) # label(replacement).  [assumption].
34 ->_s0(not(1),0) # label(replacement).  [assumption].
35 ->_s0(or(not(x),x),1) # label(replacement).  [assumption].
36 ->_s0(or(0,x),x) # label(replacement).  [assumption].
37 ->_s0(or(1,x),1) # label(replacement).  [assumption].
38 ->_s0(or(x,not(x)),1) # label(replacement).  [assumption].
39 ->_s0(or(x,0),x) # label(replacement).  [assumption].
40 ->_s0(or(x,1),1) # label(replacement).  [assumption].
41 ->*_s0(x,x) # label(reflexivity).  [assumption].
42 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(13)].
43 -->*_s0(implies(x,0),y) | -->*_s0(implies(x,y),z) | -->*_s0(implies(z,0),1) # label(goal) # answer(goal).  [deny(14)].
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): 15 -->_s0(x,y) | ->_s0(and(x,z),and(y,z)) # label(congruence).  [clausify(1)].

given #2 (I,wt=10): 16 -->_s0(x,y) | ->_s0(and(z,x),and(z,y)) # label(congruence).  [clausify(2)].

given #3 (I,wt=8): 17 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence).  [clausify(3)].

given #4 (I,wt=10): 18 -->_s0(x,y) | ->_s0(implies(x,z),implies(y,z)) # label(congruence).  [clausify(4)].

given #5 (I,wt=10): 19 -->_s0(x,y) | ->_s0(implies(z,x),implies(z,y)) # label(congruence).  [clausify(5)].

given #6 (I,wt=8): 20 -->_s0(x,y) | ->_s0(not(x),not(y)) # label(congruence).  [clausify(6)].

given #7 (I,wt=10): 21 -->_s0(x,y) | ->_s0(or(x,z),or(y,z)) # label(congruence).  [clausify(7)].

given #8 (I,wt=10): 22 -->_s0(x,y) | ->_s0(or(z,x),or(z,y)) # label(congruence).  [clausify(8)].

given #9 (I,wt=6): 23 ->_s0(and(not(x),x),0) # label(replacement).  [assumption].

given #10 (I,wt=5): 24 ->_s0(and(0,x),0) # label(replacement).  [assumption].

given #11 (I,wt=5): 25 ->_s0(and(1,x),x) # label(replacement).  [assumption].

given #12 (I,wt=6): 26 ->_s0(and(x,not(x)),0) # label(replacement).  [assumption].

given #13 (I,wt=5): 27 ->_s0(and(x,0),0) # label(replacement).  [assumption].

given #14 (I,wt=5): 28 ->_s0(and(x,1),x) # label(replacement).  [assumption].

given #15 (I,wt=11): 30 -->*_s0(x,1) | -->*_s0(y,0) | ->_s0(implies(x,y),0) # label(replacement).  [clausify(10)].

given #16 (I,wt=9): 31 -->*_s0(not(x),1) | ->_s0(implies(x,y),1) # label(replacement).  [clausify(11)].

given #17 (I,wt=8): 32 -->*_s0(x,1) | ->_s0(implies(y,x),1) # label(replacement).  [clausify(12)].

given #18 (I,wt=4): 33 ->_s0(not(0),1) # label(replacement).  [assumption].

given #19 (I,wt=4): 34 ->_s0(not(1),0) # label(replacement).  [assumption].

given #20 (I,wt=6): 35 ->_s0(or(not(x),x),1) # label(replacement).  [assumption].

given #21 (I,wt=5): 36 ->_s0(or(0,x),x) # label(replacement).  [assumption].

given #22 (I,wt=5): 37 ->_s0(or(1,x),1) # label(replacement).  [assumption].

given #23 (I,wt=6): 38 ->_s0(or(x,not(x)),1) # label(replacement).  [assumption].

given #24 (I,wt=5): 39 ->_s0(or(x,0),x) # label(replacement).  [assumption].

given #25 (I,wt=5): 40 ->_s0(or(x,1),1) # label(replacement).  [assumption].

given #26 (I,wt=3): 41 ->*_s0(x,x) # label(reflexivity).  [assumption].

given #27 (I,wt=9): 42 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(13)].

given #28 (I,wt=15): 43 -->*_s0(implies(x,0),y) | -->*_s0(implies(x,y),z) | -->*_s0(implies(z,0),1) # label(goal) # answer(goal).  [deny(14)].

given #29 (A,wt=10): 44 ->_s0(or(x,and(not(y),y)),or(x,0)).  [ur(22,a,23,a)].

given #30 (F,wt=9): 177 -->*_s0(implies(implies(x,implies(x,0)),0),1) # answer(goal).  [ur(43,a,41,a,b,41,a)].

given #31 (F,wt=9): 188 -->_s0(implies(implies(x,implies(x,0)),0),1) # answer(goal).  [ur(42,b,41,a,c,177,a)].

given #32 (F,wt=3): 189 -->*_s0(0,1) # answer(goal).  [resolve(188,a,32,b)].

given #33 (F,wt=3): 192 -->_s0(0,1) # answer(goal).  [ur(42,b,41,a,c,189,a)].

given #34 (T,wt=4): 164 ->*_s0(not(1),0).  [ur(42,a,34,a,b,41,a)].

given #35 (T,wt=4): 165 ->*_s0(not(0),1).  [ur(42,a,33,a,b,41,a)].

given #36 (T,wt=5): 156 ->_s0(implies(x,1),1).  [ur(32,a,41,a)].

given #37 (T,wt=5): 157 ->_s0(implies(1,0),0).  [ur(30,a,41,a,b,41,a)].

given #38 (A,wt=10): 45 ->_s0(or(and(not(x),x),y),or(0,y)).  [ur(21,a,23,a)].

given #39 (F,wt=4): 202 -->_s0(0,not(0)) # answer(goal).  [ur(42,b,165,a,c,189,a)].

given #40 (F,wt=6): 191 -->_s0(0,x) | -->*_s0(x,1) # answer(goal).  [resolve(189,a,42,c)].

given #41 (F,wt=8): 190 -->*_s0(not(implies(x,implies(x,0))),1) # answer(goal).  [resolve(188,a,31,b)].

given #42 (F,wt=8): 234 -->_s0(not(implies(x,implies(x,0))),1) # answer(goal).  [ur(42,b,41,a,c,190,a)].

given #43 (T,wt=5): 158 ->*_s0(or(x,1),1).  [ur(42,a,40,a,b,41,a)].

given #44 (T,wt=5): 159 ->*_s0(or(x,0),x).  [ur(42,a,39,a,b,41,a)].

given #45 (T,wt=5): 161 ->*_s0(or(1,x),1).  [ur(42,a,37,a,b,41,a)].

given #46 (T,wt=5): 162 ->*_s0(or(0,x),x).  [ur(42,a,36,a,b,41,a)].

given #47 (A,wt=8): 46 ->_s0(not(and(not(x),x)),not(0)).  [ur(20,a,23,a)].

given #48 (F,wt=5): 235 -->_s0(0,or(x,1)) # answer(goal).  [resolve(158,a,191,b)].

given #49 (F,wt=5): 261 -->_s0(0,or(1,x)) # answer(goal).  [resolve(161,a,191,b)].

given #50 (F,wt=9): 231 -->_s0(0,x) | -->_s0(x,y) | -->*_s0(y,1) # answer(goal).  [resolve(191,b,42,c)].

given #51 (F,wt=5): 287 -->_s0(0,implies(x,1)) # answer(goal).  [resolve(231,b,156,a),unit_del(b,41)].

given #52 (T,wt=5): 166 ->*_s0(and(x,1),x).  [ur(42,a,28,a,b,41,a)].

given #53 (T,wt=5): 167 ->*_s0(and(x,0),0).  [ur(42,a,27,a,b,41,a)].

given #54 (T,wt=5): 169 ->*_s0(and(1,x),x).  [ur(42,a,25,a,b,41,a)].

given #55 (T,wt=5): 170 ->*_s0(and(0,x),0).  [ur(42,a,24,a,b,41,a)].

given #56 (A,wt=10): 47 ->_s0(implies(x,and(not(y),y)),implies(x,0)).  [ur(19,a,23,a)].

given #57 (F,wt=5): 323 -->_s0(0,and(1,1)) # answer(goal).  [ur(231,b,28,a,c,41,a)].

given #58 (F,wt=6): 292 -->_s0(0,or(x,not(x))) # answer(goal).  [resolve(231,b,38,a),unit_del(b,41)].

given #59 (F,wt=6): 294 -->_s0(0,or(not(x),x)) # answer(goal).  [resolve(231,b,35,a),unit_del(b,41)].

given #60 (F,wt=6): 312 -->_s0(0,x) | -->_s0(x,1) # answer(goal).  [resolve(231,c,41,a)].

given #61 (T,wt=5): 199 ->_s0(implies(0,x),1).  [ur(31,a,165,a)].

given #62 (T,wt=5): 204 ->*_s0(implies(x,1),1).  [ur(42,a,156,a,b,41,a)].

given #63 (T,wt=5): 213 ->*_s0(implies(1,0),0).  [ur(42,a,157,a,b,41,a)].

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

% Proof 1 at 0.02 (+ 0.00) seconds: goal.
% Length of proof is 16.
% Level of proof is 5.
% Maximum clause weight is 15.000.
% Given clauses 63.

10 ->*_s0(x1,1) & ->*_s0(x2,0) -> ->_s0(implies(x1,x2),0) # label(replacement) # label(non_clause).  [assumption].
11 ->*_s0(not(x1),1) -> ->_s0(implies(x1,x2),1) # label(replacement) # label(non_clause).  [assumption].
13 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
14 (exists x5 exists x6 exists x7 (->*_s0(implies(x5,0),x6) & ->*_s0(implies(x5,x6),x7) & ->*_s0(implies(x7,0),1))) # label(goal) # label(non_clause) # label(goal).  [goal].
30 -->*_s0(x,1) | -->*_s0(y,0) | ->_s0(implies(x,y),0) # label(replacement).  [clausify(10)].
31 -->*_s0(not(x),1) | ->_s0(implies(x,y),1) # label(replacement).  [clausify(11)].
33 ->_s0(not(0),1) # label(replacement).  [assumption].
41 ->*_s0(x,x) # label(reflexivity).  [assumption].
42 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(13)].
43 -->*_s0(implies(x,0),y) | -->*_s0(implies(x,y),z) | -->*_s0(implies(z,0),1) # label(goal) # answer(goal).  [deny(14)].
157 ->_s0(implies(1,0),0).  [ur(30,a,41,a,b,41,a)].
165 ->*_s0(not(0),1).  [ur(42,a,33,a,b,41,a)].
199 ->_s0(implies(0,x),1).  [ur(31,a,165,a)].
213 ->*_s0(implies(1,0),0).  [ur(42,a,157,a,b,41,a)].
384 ->*_s0(implies(0,x),1).  [ur(42,a,199,a,b,41,a)].
412 $F # answer(goal).  [resolve(213,a,43,b),unit_del(a,213),unit_del(b,384)].

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

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

Given=63. Generated=531. Kept=397. proofs=1.
Usable=63. Sos=311. Demods=0. Limbo=0, Disabled=52. Hints=0.
Kept_by_rule=0, Deleted_by_rule=0.
Forward_subsumed=133. Back_subsumed=23.
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=60. Nonunit_bsub_feature_tests=136.
Megabytes=0.59.
User_CPU=0.02, System_CPU=0.00, Wall_clock=0.

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

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

THEOREM PROVED

Exiting with 1 proof.

Process 3450203 exit (max_proofs) Tue Jul 30 09:55:40 2024


The problem is feasible.