NO

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x y z vNonEmpty x2 x1 x3)
(STRATEGY CONTEXTSENSITIVE
(pin 1)
(tc 1)
(a)
(b)
(c)
(fSNonEmpty)
(pout 1)
)
(RULES
pin(a) -> pout(b)
pin(b) -> pout(c)
tc(x) -> x
tc(x) -> y | pin(x) ->* pout(z), tc(z) ->* y
)
]

Infeasibility Conditions:
pin(x2) ->* pout(x1), tc(x1) ->* x3

Problem 1: 

Obtaining a proof using Prover9:

 -> Prover9 Output:
============================== Prover9 ===============================
Prover9 (64) version 2009-11A, November 2009.
Process 3294484 was started by sandbox on z009.star.cs.uiowa.edu,
Tue Jul 30 09:07:40 2024
The command was "./prover9 -f /tmp/prover93294475-0.in".
============================== end of head ===========================

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

% Reading from file /tmp/prover93294475-0.in

assign(max_seconds,20).

formulas(assumptions).
->_s0(x1,y) -> ->_s0(pin(x1),pin(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(tc(x1),tc(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(pout(x1),pout(y)) # label(congruence).
->_s0(pin(a),pout(b)) # label(replacement).
->_s0(pin(b),pout(c)) # label(replacement).
->_s0(tc(x1),x1) # label(replacement).
->*_s0(pin(x1),pout(x3)) & ->*_s0(tc(x3),x2) -> ->_s0(tc(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(pin(x5),pout(x6)) & ->*_s0(tc(x6),x7))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->_s0(x1,y) -> ->_s0(pin(x1),pin(y)) # label(congruence) # label(non_clause).  [assumption].
2 ->_s0(x1,y) -> ->_s0(tc(x1),tc(y)) # label(congruence) # label(non_clause).  [assumption].
3 ->_s0(x1,y) -> ->_s0(pout(x1),pout(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->*_s0(pin(x1),pout(x3)) & ->*_s0(tc(x3),x2) -> ->_s0(tc(x1),x2) # label(replacement) # label(non_clause).  [assumption].
5 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
6 (exists x5 exists x6 exists x7 (->*_s0(pin(x5),pout(x6)) & ->*_s0(tc(x6),x7))) # 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(pin(x),pin(y)) # label(congruence).  [clausify(1)].
-->_s0(x,y) | ->_s0(tc(x),tc(y)) # label(congruence).  [clausify(2)].
-->_s0(x,y) | ->_s0(pout(x),pout(y)) # label(congruence).  [clausify(3)].
->_s0(pin(a),pout(b)) # label(replacement).  [assumption].
->_s0(pin(b),pout(c)) # label(replacement).  [assumption].
->_s0(tc(x),x) # label(replacement).  [assumption].
-->*_s0(pin(x),pout(y)) | -->*_s0(tc(y),z) | ->_s0(tc(x),z) # label(replacement).  [clausify(4)].
->*_s0(x,x) # label(reflexivity).  [assumption].
-->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(5)].
-->*_s0(pin(x),pout(y)) | -->*_s0(tc(y),z) # label(goal).  [deny(6)].
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([ b, a, c, pin, tc, pout ]).
After inverse_order:  (no changes).
Unfolding symbols: (none).

Auto_inference settings:
  % set(neg_binary_resolution).  % (HNE depth_diff=-3)
  % clear(ordered_res).  % (HNE depth_diff=-3)
  % set(ur_resolution).  % (HNE depth_diff=-3)
    % 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:      7 -->_s0(x,y) | ->_s0(pin(x),pin(y)) # label(congruence).  [clausify(1)].
kept:      8 -->_s0(x,y) | ->_s0(tc(x),tc(y)) # label(congruence).  [clausify(2)].
kept:      9 -->_s0(x,y) | ->_s0(pout(x),pout(y)) # label(congruence).  [clausify(3)].
kept:      10 ->_s0(pin(a),pout(b)) # label(replacement).  [assumption].
kept:      11 ->_s0(pin(b),pout(c)) # label(replacement).  [assumption].
kept:      12 ->_s0(tc(x),x) # label(replacement).  [assumption].
kept:      13 -->*_s0(pin(x),pout(y)) | -->*_s0(tc(y),z) | ->_s0(tc(x),z) # label(replacement).  [clausify(4)].
kept:      14 ->*_s0(x,x) # label(reflexivity).  [assumption].
kept:      15 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(5)].
kept:      16 -->*_s0(pin(x),pout(y)) | -->*_s0(tc(y),z) # label(goal) # answer(goal).  [deny(6)].

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

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

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
7 -->_s0(x,y) | ->_s0(pin(x),pin(y)) # label(congruence).  [clausify(1)].
8 -->_s0(x,y) | ->_s0(tc(x),tc(y)) # label(congruence).  [clausify(2)].
9 -->_s0(x,y) | ->_s0(pout(x),pout(y)) # label(congruence).  [clausify(3)].
10 ->_s0(pin(a),pout(b)) # label(replacement).  [assumption].
11 ->_s0(pin(b),pout(c)) # label(replacement).  [assumption].
12 ->_s0(tc(x),x) # label(replacement).  [assumption].
14 ->*_s0(x,x) # label(reflexivity).  [assumption].
15 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(5)].
16 -->*_s0(pin(x),pout(y)) | -->*_s0(tc(y),z) # label(goal) # answer(goal).  [deny(6)].
end_of_list.

formulas(demodulators).
end_of_list.

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

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

% Starting search at 0.01 seconds.

given #1 (I,wt=8): 7 -->_s0(x,y) | ->_s0(pin(x),pin(y)) # label(congruence).  [clausify(1)].

given #2 (I,wt=8): 8 -->_s0(x,y) | ->_s0(tc(x),tc(y)) # label(congruence).  [clausify(2)].

given #3 (I,wt=8): 9 -->_s0(x,y) | ->_s0(pout(x),pout(y)) # label(congruence).  [clausify(3)].

given #4 (I,wt=5): 10 ->_s0(pin(a),pout(b)) # label(replacement).  [assumption].

given #5 (I,wt=5): 11 ->_s0(pin(b),pout(c)) # label(replacement).  [assumption].

given #6 (I,wt=4): 12 ->_s0(tc(x),x) # label(replacement).  [assumption].

given #7 (I,wt=3): 14 ->*_s0(x,x) # label(reflexivity).  [assumption].

given #8 (I,wt=9): 15 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(5)].

given #9 (I,wt=9): 16 -->*_s0(pin(x),pout(y)) | -->*_s0(tc(y),z) # label(goal) # answer(goal).  [deny(6)].

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

% Proof 1 at 0.01 (+ 0.00) seconds: goal.
% Length of proof is 9.
% Level of proof is 3.
% Maximum clause weight is 9.000.
% Given clauses 9.

5 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
6 (exists x5 exists x6 exists x7 (->*_s0(pin(x5),pout(x6)) & ->*_s0(tc(x6),x7))) # label(goal) # label(non_clause) # label(goal).  [goal].
10 ->_s0(pin(a),pout(b)) # label(replacement).  [assumption].
14 ->*_s0(x,x) # label(reflexivity).  [assumption].
15 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(5)].
16 -->*_s0(pin(x),pout(y)) | -->*_s0(tc(y),z) # label(goal) # answer(goal).  [deny(6)].
27 ->*_s0(pin(a),pout(b)).  [ur(15,a,10,a,b,14,a)].
30 -->*_s0(pin(x),pout(y)) # answer(goal).  [resolve(16,b,14,a)].
31 $F # answer(goal).  [resolve(30,a,27,a)].

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

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

Given=9. Generated=25. Kept=24. proofs=1.
Usable=9. Sos=11. Demods=0. Limbo=2, Disabled=11. Hints=0.
Kept_by_rule=0, Deleted_by_rule=0.
Forward_subsumed=1. 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=3. Nonunit_bsub_feature_tests=10.
Megabytes=0.08.
User_CPU=0.01, System_CPU=0.00, Wall_clock=0.

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

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

THEOREM PROVED

Exiting with 1 proof.

Process 3294484 exit (max_proofs) Tue Jul 30 09:07:40 2024


The problem is feasible.