NO
Problem 1:
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Confluence Problem:
(VAR vNonEmpty)
(REPLACEMENT-MAP
(a)
(b)
(c)
(f 1)
(fSNonEmpty)
(h 1, 2)
)
(RULES
a -> f(a)
b -> a
b -> c
b -> h(a,a)
b -> h(h(c,a),f(f(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)
(c)
(f 1)
(fSNonEmpty)
(h 1, 2)
)
(RULES
a -> f(a)
b -> a
b -> c
b -> h(a,a)
b -> h(h(c,a),f(f(b)))
)
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Huet Levy Ordered by Num of Vars and Symbs Procedure:
-> Rules:
a -> f(a)
b -> a
b -> c
b -> h(a,a)
b -> h(h(c,a),f(f(b)))
-> Vars:
-> Rlps:
(rule: a -> f(a), id: 1, possubterms: a->[])
(rule: b -> a, id: 2, possubterms: b->[])
(rule: b -> c, id: 3, possubterms: b->[])
(rule: b -> h(a,a), id: 4, possubterms: b->[])
(rule: b -> h(h(c,a),f(f(b))), id: 5, possubterms: b->[])
-> Unifications:
(R3 unifies with R2 at p: [], l: b, lp: b, sig: {}, l': b, r: c, r': a)
(R4 unifies with R2 at p: [], l: b, lp: b, sig: {}, l': b, r: h(a,a), r': a)
(R4 unifies with R3 at p: [], l: b, lp: b, sig: {}, l': b, r: h(a,a), r': c)
(R5 unifies with R2 at p: [], l: b, lp: b, sig: {}, l': b, r: h(h(c,a),f(f(b))), r': a)
(R5 unifies with R3 at p: [], l: b, lp: b, sig: {}, l': b, r: h(h(c,a),f(f(b))), r': c)
(R5 unifies with R4 at p: [], l: b, lp: b, sig: {}, l': b, r: h(h(c,a),f(f(b))), r': h(a,a))
-> Critical pairs info:
=> Not trivial, Overlay, Proper, NW0, N1
=> Not trivial, Overlay, Proper, NW0, N2
=> Not trivial, Overlay, Proper, NW0, N3
=> Not trivial, Overlay, Proper, NW0, N4
=> Not trivial, Overlay, Proper, NW0, N5
=> Not trivial, Overlay, Proper, NW0, N6
-> 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)
(c)
(f 1)
(fSNonEmpty)
(h 1, 2)
)
(RULES
a -> f(a)
b -> a
b -> c
b -> h(a,a)
b -> h(h(c,a),f(f(b)))
)
Critical Pairs:
=> Not trivial, Overlay, Proper, NW0, N1
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
No Convergence InfChecker Procedure:
Infeasible convergence?
YES
Problem 1:
Infeasibility Problem:
[(VAR vNonEmpty vNonEmpty z0)
(STRATEGY CONTEXTSENSITIVE
(a)
(b)
(c)
(f 1)
(fSNonEmpty)
(h 1 2)
)
(RULES
a -> f(a)
b -> a
b -> c
b -> h(a,a)
b -> h(h(c,a),f(f(b)))
)]
Infeasibility Conditions:
a ->* z0, c ->* z0
Problem 1:
Obtaining a model using Mace4:
-> Usable Rules:
a -> f(a)
-> Mace4 Output:
============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 2789777 was started by shintani on shintani-XPS-13-9310,
Fri Jun 9 17:19:40 2023
The command was "./mace4 -c -f /tmp/mace42789743-2.in".
============================== end of head ===========================
============================== INPUT =================================
% Reading from file /tmp/mace42789743-2.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,f(a)) # label(replacement).
->*(x,x) # label(reflexivity).
->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
end_of_list.
formulas(goals).
(exists x2 (->*(a,x2) & ->*(c,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 (->*(a,x2) & ->*(c,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,f(a)) # label(replacement).
->*(x,x) # label(reflexivity).
-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
-->*(a,x) | -->*(c,x) # label(goal).
end_of_list.
============================== end of clauses for search =============
% There are no natural numbers in the input.
============================== DOMAIN SIZE 2 =========================
============================== MODEL =================================
interpretation( 2, [number=1, seconds=0], [
function(a, [ 0 ]),
function(c, [ 1 ]),
function(f(_), [ 0, 0 ]),
function(h(_,_), [
0, 0,
0, 0 ]),
relation(->*(_,_), [
1, 0,
0, 1 ]),
relation(->(_,_), [
1, 0,
0, 0 ])
]).
============================== end of model ==========================
============================== STATISTICS ============================
For domain size 2.
Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds).
Ground clauses: seen=33, kept=29.
Selections=7, assignments=7, propagations=9, current_models=1.
Rewrite_terms=47, rewrite_bools=39, indexes=4.
Rules_from_neg_clauses=2, cross_offs=2.
============================== end of statistics =====================
User_CPU=0.00, System_CPU=0.00, Wall_clock=0.
Exiting with 1 model.
Process 2789777 exit (max_models) Fri Jun 9 17:19:40 2023
The process finished Fri Jun 9 17:19:40 2023
Mace4 cooked interpretation:
The problem is infeasible.
The problem is not confluent.