YES

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(a 1)
(c)
(p 1)
(0)
(b)
(q 1)
)
(RULES
a(x) -> b | p(x) ->* 0
c -> c
p(q(x)) -> 0
)

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

Problem 1: 

Clean CTRS Procedure:

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(a 1)
(p 1)
(0)
(b)
(q 1)
)
(RULES
a(x) -> b | p(x) ->* 0
p(q(x)) -> 0
)

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

CRule InfChecker Info:
a(x) -> b | p(x) ->* 0
Rule remains
Proof:
NO

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x vNonEmpty x)
(STRATEGY CONTEXTSENSITIVE
(a 1)
(c)
(p 1)
(0)
(b)
(fSNonEmpty)
(q 1)
)
(RULES
a(x) -> b | p(x) ->* 0
c -> c
p(q(x)) -> 0
)
]

Infeasibility Conditions:
p(x) ->* 0

Problem 1: 

Obtaining a proof using Prover9:

 -> Prover9 Output:
============================== Prover9 ===============================
Prover9 (64) version 2009-11A, November 2009.
Process 2742459 was started by sandbox on z022.star.cs.uiowa.edu,
Thu Jun 27 11:11:08 2024
The command was "./prover9 -f /tmp/prover92742448-0.in".
============================== end of head ===========================

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

% Reading from file /tmp/prover92742448-0.in

assign(max_seconds,20).

formulas(assumptions).
->_s0(x1,y) -> ->_s0(a(x1),a(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(p(x1),p(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(q(x1),q(y)) # label(congruence).
->*_s0(p(x1),0) -> ->_s0(a(x1),b) # label(replacement).
->_s0(c,c) # label(replacement).
->_s0(p(q(x1)),0) # label(replacement).
->*_s0(x,x) # label(reflexivity).
->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x3 ->*_s0(p(x3),0)) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->_s0(x1,y) -> ->_s0(a(x1),a(y)) # label(congruence) # label(non_clause).  [assumption].
2 ->_s0(x1,y) -> ->_s0(p(x1),p(y)) # label(congruence) # label(non_clause).  [assumption].
3 ->_s0(x1,y) -> ->_s0(q(x1),q(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->*_s0(p(x1),0) -> ->_s0(a(x1),b) # label(replacement) # label(non_clause).  [assumption].
5 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
6 (exists x3 ->*_s0(p(x3),0)) # 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(a(x),a(y)) # label(congruence).  [clausify(1)].
-->_s0(x,y) | ->_s0(p(x),p(y)) # label(congruence).  [clausify(2)].
-->_s0(x,y) | ->_s0(q(x),q(y)) # label(congruence).  [clausify(3)].
-->*_s0(p(x),0) | ->_s0(a(x),b) # label(replacement).  [clausify(4)].
->_s0(c,c) # label(replacement).  [assumption].
->_s0(p(q(x)),0) # label(replacement).  [assumption].
->*_s0(x,x) # label(reflexivity).  [assumption].
-->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(5)].
-->*_s0(p(x),0) # 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([ c, 0, b, p, a, q ]).
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:  (no changes).

kept:      7 -->_s0(x,y) | ->_s0(a(x),a(y)) # label(congruence).  [clausify(1)].
kept:      8 -->_s0(x,y) | ->_s0(p(x),p(y)) # label(congruence).  [clausify(2)].
kept:      9 -->_s0(x,y) | ->_s0(q(x),q(y)) # label(congruence).  [clausify(3)].
kept:      10 -->*_s0(p(x),0) | ->_s0(a(x),b) # label(replacement).  [clausify(4)].
kept:      11 ->_s0(c,c) # label(replacement).  [assumption].
kept:      12 ->_s0(p(q(x)),0) # label(replacement).  [assumption].
kept:      13 ->*_s0(x,x) # label(reflexivity).  [assumption].
kept:      14 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(5)].
kept:      15 -->*_s0(p(x),0) # 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(a(x),a(y)) # label(congruence).  [clausify(1)].
8 -->_s0(x,y) | ->_s0(p(x),p(y)) # label(congruence).  [clausify(2)].
9 -->_s0(x,y) | ->_s0(q(x),q(y)) # label(congruence).  [clausify(3)].
11 ->_s0(c,c) # label(replacement).  [assumption].
12 ->_s0(p(q(x)),0) # label(replacement).  [assumption].
13 ->*_s0(x,x) # label(reflexivity).  [assumption].
14 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(5)].
15 -->*_s0(p(x),0) # 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(a(x),a(y)) # label(congruence).  [clausify(1)].

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

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

given #4 (I,wt=3): 11 ->_s0(c,c) # label(replacement).  [assumption].

given #5 (I,wt=5): 12 ->_s0(p(q(x)),0) # label(replacement).  [assumption].

given #6 (I,wt=3): 13 ->*_s0(x,x) # label(reflexivity).  [assumption].

given #7 (I,wt=9): 14 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(5)].

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

% Proof 1 at 0.01 (+ 0.00) seconds: goal.
% Length of proof is 8.
% Level of proof is 3.
% Maximum clause weight is 9.000.
% Given clauses 7.

5 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
6 (exists x3 ->*_s0(p(x3),0)) # label(goal) # label(non_clause) # label(goal).  [goal].
12 ->_s0(p(q(x)),0) # label(replacement).  [assumption].
13 ->*_s0(x,x) # label(reflexivity).  [assumption].
14 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(5)].
15 -->*_s0(p(x),0) # label(goal) # answer(goal).  [deny(6)].
22 ->*_s0(p(q(x)),0).  [ur(14,a,12,a,b,13,a)].
23 $F # answer(goal).  [resolve(22,a,15,a)].

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

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

Given=7. Generated=16. Kept=16. proofs=1.
Usable=7. Sos=7. Demods=0. Limbo=0, Disabled=10. Hints=0.
Kept_by_rule=0, Deleted_by_rule=0.
Forward_subsumed=0. 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=0. Nonunit_bsub_feature_tests=7.
Megabytes=0.06.
User_CPU=0.01, System_CPU=0.00, Wall_clock=0.

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

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

THEOREM PROVED

Exiting with 1 proof.

Process 2742459 exit (max_proofs) Thu Jun 27 11:11:08 2024


The problem is feasible.


CRule InfChecker Info:
c -> c
Rule deleted
Proof:
SAME_LHS_AND_RHS

CRule InfChecker Info:
p(q(x)) -> 0
Rule remains
Proof:
NO_CONDS

Problem 1: 

Problem 1: 

Problem 1: 

U_conf Transform CTRS Procedure:

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(a 1)
(p 1)
(0)
(b)
(q 1)
)
(RULES
a(x) -> b | p(x) ->* 0
p(q(x)) -> 0
)

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Resulting U_conf(R):
(VAR x)
(STRATEGY CONTEXTSENSITIVE
(a 1)
(p 1)
(0)
(b)
(q 1)
(U1 1 2)
)
(RULES
a(x) -> U1(p(x),x)
p(q(x)) -> 0
U1(0,x) -> b
)

UAlphas (debugging):
[{a(x) | p(x)}]

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(a 1)
(p 1)
(0)
(b)
(q 1)
(U1 1, 2)
)
(RULES
a(x) -> U1(p(x),x)
p(q(x)) -> 0
U1(0,x) -> b
)

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

Problem 1: 
Not CS-TRS Procedure:
R is not a CS-TRS

Problem 1: 
Linearity Procedure:
Linear?
NO

Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(a 1)
(p 1)
(0)
(b)
(U1 1, 2)
)
(RULES
a(x) -> U1(p(x),x)
p(q(x)) -> 0
U1(0,x) -> b
)

Linear:NO
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy Procedure:
-> Rules:
 a(x) -> U1(p(x),x)
 p(q(x)) -> 0
 U1(0,x) -> b
-> Vars:
 x, x, x

-> Rlps:
 (rule: a(x) -> U1(p(x),x), id: 1, possubterms: a(x)->[])
 (rule: p(q(x)) -> 0, id: 2, possubterms: p(q(x))->[], q(x)->[1])
 (rule: U1(0,x) -> b, id: 3, possubterms: U1(0,x)->[], 0->[1])

-> Unifications:


-> Critical pairs info:


-> Problem conclusions:
 Left linear, Not right linear, Not linear
 Weakly orthogonal, Almost orthogonal, Orthogonal, Not strongly orthogonal
 Huet-Levy confluent, Not Newman confluent
 R is a TRS 
 Confluent


The problem is confluent.