NO

Problem 1: 


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


Problem 1: 

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

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

Huet Levy Ordered by Num of Vars and Symbs Procedure:
-> Rules:
 a -> c
 b -> f(f(b))
 f(f(a)) -> b
 f(f(c)) -> h(a,h(h(c,c),b))
-> Vars:


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

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

-> Critical pairs info:
<f(f(c)),b>   =>  Not trivial, Not overlay, Proper, NW0, N1

-> Problem conclusions:
 Left linear, Right linear, Linear
 Not weakly orthogonal, Not almost orthogonal, Not orthogonal
 Not Huet-Levy confluent, Not Newman confluent
 R is a TRS 


Problem 1: 
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(REPLACEMENT-MAP
(a)
(b)
(f 1)
(c)
(fSNonEmpty)
(h 1, 2)
)
(RULES
a -> c
b -> f(f(b))
f(f(a)) -> b
f(f(c)) -> h(a,h(h(c,c),b))
)
Critical Pairs:
<f(f(c)),b>   =>  Not trivial, Not overlay, Proper, NW0, N1
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

No Convergence InfChecker Procedure:
Infeasible convergence?
YES

Problem 1: 

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

Problem 1: 

Obtaining a model using Mace4:

 -> Usable Rules:
a -> c
b -> f(f(b))
f(f(a)) -> b
f(f(c)) -> h(a,h(h(c,c),b))

 -> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 2792695 was started by shintani on shintani-XPS-13-9310,
Fri Jun  9 17:39:06 2023
The command was "./mace4 -c -f /tmp/mace42792678-2.in".
============================== end of head ===========================

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

% Reading from file /tmp/mace42792678-2.in

assign(max_seconds,10).

formulas(assumptions).
->(x1,y) -> ->(f(x1),f(y)) # label(congruence).
->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence).
->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence).
->(a,c) # label(replacement).
->(b,f(f(b))) # label(replacement).
->(f(f(a)),b) # label(replacement).
->(f(f(c)),h(a,h(h(c,c),b))) # label(replacement).
->*(x,x) # label(reflexivity).
->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x2 (->*(f(f(c)),x2) & ->*(b,x2))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause).  [assumption].
2 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause).  [assumption].
3 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause).  [assumption].
4 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
5 (exists x2 (->*(f(f(c)),x2) & ->*(b,x2))) # label(goal) # label(non_clause) # label(goal).  [goal].

============================== end of process non-clausal formulas ===

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

formulas(mace4_clauses).
-->(x,y) | ->(f(x),f(y)) # label(congruence).
-->(x,y) | ->(h(x,z),h(y,z)) # label(congruence).
-->(x,y) | ->(h(z,x),h(z,y)) # label(congruence).
->(a,c) # label(replacement).
->(b,f(f(b))) # label(replacement).
->(f(f(a)),b) # label(replacement).
->(f(f(c)),h(a,h(h(c,c),b))) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(f(f(c)),x) | -->*(b,x) # label(goal).
end_of_list.

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

% There are no natural numbers in the input.

============================== DOMAIN SIZE 2 =========================

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

For domain size 2.

Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds).
Ground clauses: seen=36, kept=32.
Selections=28, assignments=55, propagations=56, current_models=0.
Rewrite_terms=372, rewrite_bools=243, indexes=102.
Rules_from_neg_clauses=12, cross_offs=12.

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

============================== DOMAIN SIZE 3 =========================

============================== MODEL =================================

interpretation( 3, [number=1, seconds=0], [

        function(a, [ 0 ]),

        function(b, [ 1 ]),

        function(c, [ 2 ]),

        function(f(_), [ 0, 1, 2 ]),

        function(h(_,_), [
			   0, 2, 2,
			   2, 2, 2,
			   2, 2, 2 ]),

        relation(->*(_,_), [
			   1, 1, 1,
			   0, 1, 0,
			   0, 0, 1 ]),

        relation(->(_,_), [
			   0, 1, 1,
			   0, 1, 0,
			   0, 0, 1 ])
]).

============================== end of model ==========================

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

For domain size 3.

Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds).
Ground clauses: seen=100, kept=91.
Selections=186, assignments=527, propagations=815, current_models=1.
Rewrite_terms=4992, rewrite_bools=3982, indexes=1291.
Rules_from_neg_clauses=86, cross_offs=216.

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

User_CPU=0.00, System_CPU=0.00, Wall_clock=0.

Exiting with 1 model.

Process 2792695 exit (max_models) Fri Jun  9 17:39:06 2023
The process finished Fri Jun  9 17:39:06 2023


Mace4 cooked interpretation:



The problem is infeasible.


The problem is not confluent.