YES

Problem 1: 


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(a)
(f 1, 2)
(b)
(fSNonEmpty)
(g 1)
)
(RULES
a -> b
f(a,a) -> g(f(a,a))
f(b,x) -> g(f(x,x))
f(x,b) -> g(f(x,x))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 

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

Problem 1: 
CSR Converter From Canonical u-Replacement Map Procedure [CSUR20]:

Original Replacement Map:
(a)
(f 1 2)
(b)
(fSNonEmpty)
(g 1)

New Replacement Map:
(a)
(f 1 2)
(b)
(fSNonEmpty)
(g)

New problem:

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(a)
(f 1, 2)
(b)
(fSNonEmpty)
(g)
)
(RULES
a -> b
f(a,a) -> g(f(a,a))
f(b,x) -> g(f(x,x))
f(x,b) -> g(f(x,x))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::


Problem 1: 
Termination Procedure:
Terminating?
YES

Problem 1: 

(VAR vu95NonEmpty vNonEmpty x)
(STRATEGY CONTEXTSENSITIVE
(a)
(f 1 2)
(b)
(fSNonEmpty)
(g)
)
(RULES
a -> b
f(a,a) -> g(f(a,a))
f(b,x) -> g(f(x,x))
f(x,b) -> g(f(x,x))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 Empty
-> Rules:
 a -> b
 f(a,a) -> g(f(a,a))
 f(b,x) -> g(f(x,x))
 f(x,b) -> g(f(x,x))
-> Unhiding Rules:
 Empty

Problem 1: 

Basic Processor:
-> Pairs:
 Empty
-> Rules:
 a -> b
 f(a,a) -> g(f(a,a))
 f(b,x) -> g(f(x,x))
 f(x,b) -> g(f(x,x))
-> Unhiding rules:
 Empty
-> Result:
 Set P is empty

The problem is finite.


Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty x)
(REPLACEMENT-MAP
(a)
(f 1, 2)
(b)
(fSNonEmpty)
(g)
)
(RULES
a -> b
f(a,a) -> g(f(a,a))
f(b,x) -> g(f(x,x))
f(x,b) -> g(f(x,x))
)
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet Levy NW Procedure:
-> Rules:
 a -> b
 f(a,a) -> g(f(a,a))
 f(b,x) -> g(f(x,x))
 f(x,b) -> g(f(x,x))
-> Vars:
 x, x
-> UVars:
 (UV-RuleId: 1, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 2, UV-LActive: [], UV-RActive: [], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 3, UV-LActive: [x], UV-RActive: [x], UV-LFrozen: [], UV-RFrozen: [])
 (UV-RuleId: 4, UV-LActive: [x], UV-RActive: [x], UV-LFrozen: [], UV-RFrozen: [])

-> Rlps:
 (rule: a -> b, id: 1, possubterms: a->[])
 (rule: f(a,a) -> g(f(a,a)), id: 2, possubterms: f(a,a)->[], a->[1], a->[2])
 (rule: f(b,x) -> g(f(x,x)), id: 3, possubterms: f(b,x)->[], b->[1])
 (rule: f(x,b) -> g(f(x,x)), id: 4, possubterms: f(x,b)->[], b->[2])

-> Unifications:
(R2 unifies with R1 at p: [1], l: f(a,a), lp: a, sig: {}, l': a, r: g(f(a,a)), r': b)
(R2 unifies with R1 at p: [2], l: f(a,a), lp: a, sig: {}, l': a, r: g(f(a,a)), r': b)
(R4 unifies with R3 at p: [], l: f(x,b), lp: f(x,b), sig: {x -> b,x' -> b}, l': f(b,x'), r: g(f(x,x)), r': g(f(x',x')))

-> Critical pairs info:
<g(f(b,b)),g(f(b,b))>   =>  Trivial, Overlay, Proper, NW0, N1
<f(b,a),g(f(a,a))>   =>  Not trivial, Not overlay, Proper, NW0, N2
<f(a,b),g(f(a,a))>   =>  Not trivial, Not overlay, Proper, NW0, N3

-> Problem conclusions:
 Left linear, Not right linear, Not linear
 Not weakly orthogonal, Not almost orthogonal, Not orthogonal
 Not Huet-Levy confluent, Not Newman confluent
 R is a CS-TRS, Left-homogeneous u-replacing variables



The problem is decomposed in 2 subproblems.

Problem 1.1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(a)
(f 1, 2)
(b)
(fSNonEmpty)
(g)
)
(RULES
a -> b
f(a,a) -> g(f(a,a))
f(b,x) -> g(f(x,x))
f(x,b) -> g(f(x,x))
)
Critical Pairs:
<f(b,a),g(f(a,a))>   =>  Not trivial, Not overlay, Proper, NW0, N2
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet InfChecker Procedure:
Infeasible convergence?
NO

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x vNonEmpty z0)
(STRATEGY CONTEXTSENSITIVE
(a)
(f 1 2)
(b)
(fSNonEmpty)
(g)
)
(RULES
a -> b
f(a,a) -> g(f(a,a))
f(b,x) -> g(f(x,x))
f(x,b) -> g(f(x,x))
)]
Infeasibility Conditions:
f(b,a) \->* z0, g(f(a,a)) \->* z0

Problem 1: 

Obtaining a proof using Prover9:

 -> Prover9 Output:
============================== Prover9 ===============================
Prover9 (64) version 2009-11A, November 2009.
Process 2639949 was started by shintani on shintani-XPS-13-9310,
Fri Jun  9 15:30:19 2023
The command was "./prover9 -f /tmp/prover92639939-0.in".
============================== end of head ===========================

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

% Reading from file /tmp/prover92639939-0.in

assign(max_seconds,20).

formulas(assumptions).
\->_s0(x1,y) -> \->_s0(f(x1,x2),f(y,x2)) # label(cscongruence).
\->_s0(x2,y) -> \->_s0(f(x1,x2),f(x1,y)) # label(cscongruence).
\->_s0(a,b) # label(csreplacement).
\->_s0(f(a,a),g(f(a,a))) # label(csreplacement).
\->_s0(f(b,x1),g(f(x1,x1))) # label(csreplacement).
\->_s0(f(x1,b),g(f(x1,x1))) # label(csreplacement).
\->*_s0(x,x) # label(csreflexivity).
\->_s0(x,y) & \->*_s0(y,z) -> \->*_s0(x,z) # label(cstransitivity).
end_of_list.

formulas(goals).
(exists x3 (\->*_s0(f(b,a),x3) & \->*_s0(g(f(a,a)),x3))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 \->_s0(x1,y) -> \->_s0(f(x1,x2),f(y,x2)) # label(cscongruence) # label(non_clause).  [assumption].
2 \->_s0(x2,y) -> \->_s0(f(x1,x2),f(x1,y)) # label(cscongruence) # label(non_clause).  [assumption].
3 \->_s0(x,y) & \->*_s0(y,z) -> \->*_s0(x,z) # label(cstransitivity) # label(non_clause).  [assumption].
4 (exists x3 (\->*_s0(f(b,a),x3) & \->*_s0(g(f(a,a)),x3))) # 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(f(x,z),f(y,z)) # label(cscongruence).  [clausify(1)].
-\->_s0(x,y) | \->_s0(f(z,x),f(z,y)) # label(cscongruence).  [clausify(2)].
\->_s0(a,b) # label(csreplacement).  [assumption].
\->_s0(f(a,a),g(f(a,a))) # label(csreplacement).  [assumption].
\->_s0(f(b,x),g(f(x,x))) # label(csreplacement).  [assumption].
\->_s0(f(x,b),g(f(x,x))) # label(csreplacement).  [assumption].
\->*_s0(x,x) # label(csreflexivity).  [assumption].
-\->_s0(x,y) | -\->*_s0(y,z) | \->*_s0(x,z) # label(cstransitivity).  [clausify(3)].
-\->*_s0(f(b,a),x) | -\->*_s0(g(f(a,a)),x) # label(goal).  [deny(4)].
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([ a, b, f, g ]).
After inverse_order:  (no changes).
Unfolding symbols: (none).

Auto_inference settings:
  % set(neg_binary_resolution).  % (HNE depth_diff=-2)
  % clear(ordered_res).  % (HNE depth_diff=-2)
  % set(ur_resolution).  % (HNE depth_diff=-2)
    % 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:      5 -\->_s0(x,y) | \->_s0(f(x,z),f(y,z)) # label(cscongruence).  [clausify(1)].
kept:      6 -\->_s0(x,y) | \->_s0(f(z,x),f(z,y)) # label(cscongruence).  [clausify(2)].
kept:      7 \->_s0(a,b) # label(csreplacement).  [assumption].
kept:      8 \->_s0(f(a,a),g(f(a,a))) # label(csreplacement).  [assumption].
kept:      9 \->_s0(f(b,x),g(f(x,x))) # label(csreplacement).  [assumption].
kept:      10 \->_s0(f(x,b),g(f(x,x))) # label(csreplacement).  [assumption].
kept:      11 \->*_s0(x,x) # label(csreflexivity).  [assumption].
kept:      12 -\->_s0(x,y) | -\->*_s0(y,z) | \->*_s0(x,z) # label(cstransitivity).  [clausify(3)].
kept:      13 -\->*_s0(f(b,a),x) | -\->*_s0(g(f(a,a)),x) # label(goal) # answer(goal).  [deny(4)].

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

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

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
5 -\->_s0(x,y) | \->_s0(f(x,z),f(y,z)) # label(cscongruence).  [clausify(1)].
6 -\->_s0(x,y) | \->_s0(f(z,x),f(z,y)) # label(cscongruence).  [clausify(2)].
7 \->_s0(a,b) # label(csreplacement).  [assumption].
8 \->_s0(f(a,a),g(f(a,a))) # label(csreplacement).  [assumption].
9 \->_s0(f(b,x),g(f(x,x))) # label(csreplacement).  [assumption].
10 \->_s0(f(x,b),g(f(x,x))) # label(csreplacement).  [assumption].
11 \->*_s0(x,x) # label(csreflexivity).  [assumption].
12 -\->_s0(x,y) | -\->*_s0(y,z) | \->*_s0(x,z) # label(cstransitivity).  [clausify(3)].
13 -\->*_s0(f(b,a),x) | -\->*_s0(g(f(a,a)),x) # label(goal) # answer(goal).  [deny(4)].
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): 5 -\->_s0(x,y) | \->_s0(f(x,z),f(y,z)) # label(cscongruence).  [clausify(1)].

given #2 (I,wt=10): 6 -\->_s0(x,y) | \->_s0(f(z,x),f(z,y)) # label(cscongruence).  [clausify(2)].

given #3 (I,wt=3): 7 \->_s0(a,b) # label(csreplacement).  [assumption].

given #4 (I,wt=8): 8 \->_s0(f(a,a),g(f(a,a))) # label(csreplacement).  [assumption].

given #5 (I,wt=8): 9 \->_s0(f(b,x),g(f(x,x))) # label(csreplacement).  [assumption].

given #6 (I,wt=8): 10 \->_s0(f(x,b),g(f(x,x))) # label(csreplacement).  [assumption].

given #7 (I,wt=3): 11 \->*_s0(x,x) # label(csreflexivity).  [assumption].

given #8 (I,wt=9): 12 -\->_s0(x,y) | -\->*_s0(y,z) | \->*_s0(x,z) # label(cstransitivity).  [clausify(3)].

given #9 (I,wt=11): 13 -\->*_s0(f(b,a),x) | -\->*_s0(g(f(a,a)),x) # label(goal) # answer(goal).  [deny(4)].

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

% Proof 1 at 0.00 (+ 0.00) seconds: goal.
% Length of proof is 8.
% Level of proof is 3.
% Maximum clause weight is 11.000.
% Given clauses 9.

3 \->_s0(x,y) & \->*_s0(y,z) -> \->*_s0(x,z) # label(cstransitivity) # label(non_clause).  [assumption].
4 (exists x3 (\->*_s0(f(b,a),x3) & \->*_s0(g(f(a,a)),x3))) # label(goal) # label(non_clause) # label(goal).  [goal].
9 \->_s0(f(b,x),g(f(x,x))) # label(csreplacement).  [assumption].
11 \->*_s0(x,x) # label(csreflexivity).  [assumption].
12 -\->_s0(x,y) | -\->*_s0(y,z) | \->*_s0(x,z) # label(cstransitivity).  [clausify(3)].
13 -\->*_s0(f(b,a),x) | -\->*_s0(g(f(a,a)),x) # label(goal) # answer(goal).  [deny(4)].
23 \->*_s0(f(b,x),g(f(x,x))).  [ur(12,a,9,a,b,11,a)].
29 $F # answer(goal).  [resolve(13,b,11,a),unit_del(a,23)].

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

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

Given=9. Generated=25. Kept=24. proofs=1.
Usable=9. Sos=12. Demods=0. Limbo=3, Disabled=9. Hints=0.
Kept_by_rule=0, Deleted_by_rule=0.
Forward_subsumed=0. 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=4. Nonunit_bsub_feature_tests=7.
Megabytes=0.08.
User_CPU=0.00, System_CPU=0.00, Wall_clock=0.

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

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

THEOREM PROVED

Exiting with 1 proof.

Process 2639949 exit (max_proofs) Fri Jun  9 15:30:19 2023


The problem is feasible.


The problem is confluent.

Problem 1.2: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR x)
(REPLACEMENT-MAP
(a)
(f 1, 2)
(b)
(fSNonEmpty)
(g)
)
(RULES
a -> b
f(a,a) -> g(f(a,a))
f(b,x) -> g(f(x,x))
f(x,b) -> g(f(x,x))
)
Critical Pairs:
<f(a,b),g(f(a,a))>   =>  Not trivial, Not overlay, Proper, NW0, N3
Terminating:"YES"
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

Huet InfChecker Procedure:
Infeasible convergence?
NO

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty x vNonEmpty z0)
(STRATEGY CONTEXTSENSITIVE
(a)
(f 1 2)
(b)
(fSNonEmpty)
(g)
)
(RULES
a -> b
f(a,a) -> g(f(a,a))
f(b,x) -> g(f(x,x))
f(x,b) -> g(f(x,x))
)]
Infeasibility Conditions:
f(a,b) \->* z0, g(f(a,a)) \->* z0

Problem 1: 

Obtaining a proof using Prover9:

 -> Prover9 Output:
============================== Prover9 ===============================
Prover9 (64) version 2009-11A, November 2009.
Process 2639977 was started by shintani on shintani-XPS-13-9310,
Fri Jun  9 15:30:19 2023
The command was "./prover9 -f /tmp/prover92639964-0.in".
============================== end of head ===========================

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

% Reading from file /tmp/prover92639964-0.in

assign(max_seconds,20).

formulas(assumptions).
\->_s0(x1,y) -> \->_s0(f(x1,x2),f(y,x2)) # label(cscongruence).
\->_s0(x2,y) -> \->_s0(f(x1,x2),f(x1,y)) # label(cscongruence).
\->_s0(a,b) # label(csreplacement).
\->_s0(f(a,a),g(f(a,a))) # label(csreplacement).
\->_s0(f(b,x1),g(f(x1,x1))) # label(csreplacement).
\->_s0(f(x1,b),g(f(x1,x1))) # label(csreplacement).
\->*_s0(x,x) # label(csreflexivity).
\->_s0(x,y) & \->*_s0(y,z) -> \->*_s0(x,z) # label(cstransitivity).
end_of_list.

formulas(goals).
(exists x3 (\->*_s0(f(a,b),x3) & \->*_s0(g(f(a,a)),x3))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 \->_s0(x1,y) -> \->_s0(f(x1,x2),f(y,x2)) # label(cscongruence) # label(non_clause).  [assumption].
2 \->_s0(x2,y) -> \->_s0(f(x1,x2),f(x1,y)) # label(cscongruence) # label(non_clause).  [assumption].
3 \->_s0(x,y) & \->*_s0(y,z) -> \->*_s0(x,z) # label(cstransitivity) # label(non_clause).  [assumption].
4 (exists x3 (\->*_s0(f(a,b),x3) & \->*_s0(g(f(a,a)),x3))) # 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(f(x,z),f(y,z)) # label(cscongruence).  [clausify(1)].
-\->_s0(x,y) | \->_s0(f(z,x),f(z,y)) # label(cscongruence).  [clausify(2)].
\->_s0(a,b) # label(csreplacement).  [assumption].
\->_s0(f(a,a),g(f(a,a))) # label(csreplacement).  [assumption].
\->_s0(f(b,x),g(f(x,x))) # label(csreplacement).  [assumption].
\->_s0(f(x,b),g(f(x,x))) # label(csreplacement).  [assumption].
\->*_s0(x,x) # label(csreflexivity).  [assumption].
-\->_s0(x,y) | -\->*_s0(y,z) | \->*_s0(x,z) # label(cstransitivity).  [clausify(3)].
-\->*_s0(f(a,b),x) | -\->*_s0(g(f(a,a)),x) # label(goal).  [deny(4)].
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([ a, b, f, g ]).
After inverse_order:  (no changes).
Unfolding symbols: (none).

Auto_inference settings:
  % set(neg_binary_resolution).  % (HNE depth_diff=-2)
  % clear(ordered_res).  % (HNE depth_diff=-2)
  % set(ur_resolution).  % (HNE depth_diff=-2)
    % 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:      5 -\->_s0(x,y) | \->_s0(f(x,z),f(y,z)) # label(cscongruence).  [clausify(1)].
kept:      6 -\->_s0(x,y) | \->_s0(f(z,x),f(z,y)) # label(cscongruence).  [clausify(2)].
kept:      7 \->_s0(a,b) # label(csreplacement).  [assumption].
kept:      8 \->_s0(f(a,a),g(f(a,a))) # label(csreplacement).  [assumption].
kept:      9 \->_s0(f(b,x),g(f(x,x))) # label(csreplacement).  [assumption].
kept:      10 \->_s0(f(x,b),g(f(x,x))) # label(csreplacement).  [assumption].
kept:      11 \->*_s0(x,x) # label(csreflexivity).  [assumption].
kept:      12 -\->_s0(x,y) | -\->*_s0(y,z) | \->*_s0(x,z) # label(cstransitivity).  [clausify(3)].
kept:      13 -\->*_s0(f(a,b),x) | -\->*_s0(g(f(a,a)),x) # label(goal) # answer(goal).  [deny(4)].

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

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

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
5 -\->_s0(x,y) | \->_s0(f(x,z),f(y,z)) # label(cscongruence).  [clausify(1)].
6 -\->_s0(x,y) | \->_s0(f(z,x),f(z,y)) # label(cscongruence).  [clausify(2)].
7 \->_s0(a,b) # label(csreplacement).  [assumption].
8 \->_s0(f(a,a),g(f(a,a))) # label(csreplacement).  [assumption].
9 \->_s0(f(b,x),g(f(x,x))) # label(csreplacement).  [assumption].
10 \->_s0(f(x,b),g(f(x,x))) # label(csreplacement).  [assumption].
11 \->*_s0(x,x) # label(csreflexivity).  [assumption].
12 -\->_s0(x,y) | -\->*_s0(y,z) | \->*_s0(x,z) # label(cstransitivity).  [clausify(3)].
13 -\->*_s0(f(a,b),x) | -\->*_s0(g(f(a,a)),x) # label(goal) # answer(goal).  [deny(4)].
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): 5 -\->_s0(x,y) | \->_s0(f(x,z),f(y,z)) # label(cscongruence).  [clausify(1)].

given #2 (I,wt=10): 6 -\->_s0(x,y) | \->_s0(f(z,x),f(z,y)) # label(cscongruence).  [clausify(2)].

given #3 (I,wt=3): 7 \->_s0(a,b) # label(csreplacement).  [assumption].

given #4 (I,wt=8): 8 \->_s0(f(a,a),g(f(a,a))) # label(csreplacement).  [assumption].

given #5 (I,wt=8): 9 \->_s0(f(b,x),g(f(x,x))) # label(csreplacement).  [assumption].

given #6 (I,wt=8): 10 \->_s0(f(x,b),g(f(x,x))) # label(csreplacement).  [assumption].

given #7 (I,wt=3): 11 \->*_s0(x,x) # label(csreflexivity).  [assumption].

given #8 (I,wt=9): 12 -\->_s0(x,y) | -\->*_s0(y,z) | \->*_s0(x,z) # label(cstransitivity).  [clausify(3)].

given #9 (I,wt=11): 13 -\->*_s0(f(a,b),x) | -\->*_s0(g(f(a,a)),x) # label(goal) # answer(goal).  [deny(4)].

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

% Proof 1 at 0.00 (+ 0.00) seconds: goal.
% Length of proof is 8.
% Level of proof is 3.
% Maximum clause weight is 11.000.
% Given clauses 9.

3 \->_s0(x,y) & \->*_s0(y,z) -> \->*_s0(x,z) # label(cstransitivity) # label(non_clause).  [assumption].
4 (exists x3 (\->*_s0(f(a,b),x3) & \->*_s0(g(f(a,a)),x3))) # label(goal) # label(non_clause) # label(goal).  [goal].
10 \->_s0(f(x,b),g(f(x,x))) # label(csreplacement).  [assumption].
11 \->*_s0(x,x) # label(csreflexivity).  [assumption].
12 -\->_s0(x,y) | -\->*_s0(y,z) | \->*_s0(x,z) # label(cstransitivity).  [clausify(3)].
13 -\->*_s0(f(a,b),x) | -\->*_s0(g(f(a,a)),x) # label(goal) # answer(goal).  [deny(4)].
22 \->*_s0(f(x,b),g(f(x,x))).  [ur(12,a,10,a,b,11,a)].
29 $F # answer(goal).  [resolve(13,b,11,a),unit_del(a,22)].

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

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

Given=9. Generated=25. Kept=24. proofs=1.
Usable=9. Sos=12. Demods=0. Limbo=3, Disabled=9. Hints=0.
Kept_by_rule=0, Deleted_by_rule=0.
Forward_subsumed=0. 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=4. Nonunit_bsub_feature_tests=7.
Megabytes=0.08.
User_CPU=0.00, System_CPU=0.00, Wall_clock=0.

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

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

THEOREM PROVED

Exiting with 1 proof.

Process 2639977 exit (max_proofs) Fri Jun  9 15:30:19 2023


The problem is feasible.


The problem is confluent.