NO Problem 1: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty x) (REPLACEMENT-MAP (a) (b) (f 1, 2) (fSNonEmpty) ) (RULES a -> b b -> f(a,a) f(x,a) -> b ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 1: Problem 1: Not CS-TRS Procedure: R is not a CS-TRS Problem 1: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty x) (REPLACEMENT-MAP (a) (b) (f 1, 2) (fSNonEmpty) ) (RULES a -> b b -> f(a,a) f(x,a) -> b ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Huet Levy Ordered by Num of Vars and Symbs Procedure: -> Rules: a -> b b -> f(a,a) f(x,a) -> b -> Vars: x -> Rlps: (rule: a -> b, id: 1, possubterms: a->[]) (rule: b -> f(a,a), id: 2, possubterms: b->[]) (rule: f(x,a) -> b, id: 3, possubterms: f(x,a)->[], a->[2]) -> Unifications: (R3 unifies with R1 at p: [2], l: f(x,a), lp: a, sig: {}, l': a, r: b, r': b) -> Critical pairs info: => 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: (VAR x) (REPLACEMENT-MAP (a) (b) (f 1, 2) (fSNonEmpty) ) (RULES a -> b b -> f(a,a) f(x,a) -> b ) Critical Pairs: => Not trivial, Not overlay, Proper, NW0, N1 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: No Convergence InfChecker Procedure: Infeasible convergence? YES Problem 1: Infeasibility Problem: [(VAR vNonEmpty x vNonEmpty z0) (STRATEGY CONTEXTSENSITIVE (a) (b) (f 1 2) (c_x) (fSNonEmpty) ) (RULES a -> b b -> f(a,a) f(x,a) -> b )] Infeasibility Conditions: f(c_x,b) ->* z0, b ->* z0 Problem 1: Obtaining a model using Mace4: -> Usable Rules: a -> b b -> f(a,a) f(x,a) -> b -> Mace4 Output: ============================== Mace4 ================================= Mace4 (64) version 2009-11A, November 2009. Process 2778478 was started by shintani on shintani-XPS-13-9310, Fri Jun 9 16:47:54 2023 The command was "./mace4 -c -f /tmp/mace42778456-2.in". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file /tmp/mace42778456-2.in assign(max_seconds,10). formulas(assumptions). ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). ->(a,b) # label(replacement). ->(b,f(a,a)) # label(replacement). ->(f(x1,a),b) # label(replacement). ->*(x,x) # label(reflexivity). ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). end_of_list. formulas(goals). (exists x3 (->*(f(c_x,b),x3) & ->*(b,x3))) # label(goal). end_of_list. ============================== end of input ========================== ============================== PROCESS NON-CLAUSAL FORMULAS ========== % Formulas that are not ordinary clauses: 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 3 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 4 (exists x3 (->*(f(c_x,b),x3) & ->*(b,x3))) # label(goal) # label(non_clause) # label(goal). [goal]. ============================== end of process non-clausal formulas === ============================== CLAUSES FOR SEARCH ==================== formulas(mace4_clauses). -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). ->(a,b) # label(replacement). ->(b,f(a,a)) # label(replacement). ->(f(x,a),b) # label(replacement). ->*(x,x) # label(reflexivity). -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). -->*(f(c_x,b),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=32, kept=28. Selections=4, assignments=7, propagations=18, current_models=0. Rewrite_terms=52, rewrite_bools=37, indexes=22. Rules_from_neg_clauses=4, cross_offs=4. ============================== end of statistics ===================== ============================== DOMAIN SIZE 3 ========================= ============================== STATISTICS ============================ For domain size 3. Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). Ground clauses: seen=92, kept=83. Selections=6, assignments=13, propagations=48, current_models=0. Rewrite_terms=181, rewrite_bools=181, indexes=61. Rules_from_neg_clauses=8, cross_offs=21. ============================== end of statistics ===================== ============================== DOMAIN SIZE 4 ========================= ============================== MODEL ================================= interpretation( 4, [number=1, seconds=0], [ function(a, [ 0 ]), function(b, [ 1 ]), function(c_x, [ 2 ]), function(f(_,_), [ 1, 1, 0, 0, 1, 1, 1, 1, 3, 2, 2, 2, 3, 3, 3, 3 ]), relation(->*(_,_), [ 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1 ]), relation(->(_,_), [ 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1 ]) ]). ============================== end of model ========================== ============================== STATISTICS ============================ For domain size 4. Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). Ground clauses: seen=206, kept=190. Selections=21, assignments=46, propagations=108, current_models=1. Rewrite_terms=608, rewrite_bools=649, indexes=118. Rules_from_neg_clauses=14, cross_offs=74. ============================== end of statistics ===================== User_CPU=0.00, System_CPU=0.00, Wall_clock=0. Exiting with 1 model. Process 2778478 exit (max_models) Fri Jun 9 16:47:54 2023 The process finished Fri Jun 9 16:47:54 2023 Mace4 cooked interpretation: The problem is infeasible. The problem is not confluent.