NO Problem 1: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty) (REPLACEMENT-MAP (a) (f 1, 2) (b) (fSNonEmpty) ) (RULES a -> f(a,b) f(a,b) -> f(b,a) ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 1: Problem 1: Not CS-TRS Procedure: R is not a CS-TRS Problem 1: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty) (REPLACEMENT-MAP (a) (f 1, 2) (b) (fSNonEmpty) ) (RULES a -> f(a,b) f(a,b) -> f(b,a) ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Huet Levy Ordered by Num of Vars and Symbs Procedure: -> Rules: a -> f(a,b) f(a,b) -> f(b,a) -> Vars: -> Rlps: (rule: a -> f(a,b), id: 1, possubterms: a->[]) (rule: f(a,b) -> f(b,a), id: 2, possubterms: f(a,b)->[], a->[1], b->[2]) -> Unifications: (R2 unifies with R1 at p: [1], l: f(a,b), lp: a, sig: {}, l': a, r: f(b,a), r': f(a,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: (REPLACEMENT-MAP (a) (f 1, 2) (b) (fSNonEmpty) ) (RULES a -> f(a,b) f(a,b) -> f(b,a) ) Critical Pairs: => 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) (f 1 2) (b) (fSNonEmpty) ) (RULES a -> f(a,b) f(a,b) -> f(b,a) )] Infeasibility Conditions: f(f(a,b),b) ->* z0, f(b,a) ->* z0 Problem 1: Obtaining a model using Mace4: -> Usable Rules: a -> f(a,b) f(a,b) -> f(b,a) -> Mace4 Output: ============================== Mace4 ================================= Mace4 (64) version 2009-11A, November 2009. Process 2692918 was started by shintani on shintani-XPS-13-9310, Fri Jun 9 15:43:23 2023 The command was "./mace4 -c -f /tmp/mace42692902-2.in". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file /tmp/mace42692902-2.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,f(a,b)) # label(replacement). ->(f(a,b),f(b,a)) # label(replacement). ->*(x,x) # label(reflexivity). ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). end_of_list. formulas(goals). (exists x2 (->*(f(f(a,b),b),x2) & ->*(f(b,a),x2))) # 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 x2 (->*(f(f(a,b),b),x2) & ->*(f(b,a),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,z),f(y,z)) # label(congruence). -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). ->(a,f(a,b)) # label(replacement). ->(f(a,b),f(b,a)) # label(replacement). ->*(x,x) # label(reflexivity). -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). -->*(f(f(a,b),b),x) | -->*(f(b,a),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=30, kept=26. Selections=11, assignments=21, propagations=42, current_models=0. Rewrite_terms=246, rewrite_bools=149, indexes=70. Rules_from_neg_clauses=12, cross_offs=12. ============================== 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=89, kept=80. Selections=32, assignments=89, propagations=310, current_models=0. Rewrite_terms=1616, rewrite_bools=1158, indexes=490. Rules_from_neg_clauses=41, cross_offs=105. ============================== end of statistics ===================== ============================== DOMAIN SIZE 4 ========================= ============================== STATISTICS ============================ For domain size 4. Current CPU time: 0.00 seconds (total CPU time: 0.01 seconds). Ground clauses: seen=202, kept=186. Selections=114, assignments=425, propagations=1842, current_models=0. Rewrite_terms=8274, rewrite_bools=10242, indexes=2587. Rules_from_neg_clauses=142, cross_offs=755. ============================== end of statistics ===================== ============================== DOMAIN SIZE 5 ========================= ============================== MODEL ================================= interpretation( 5, [number=1, seconds=0], [ function(a, [ 0 ]), function(b, [ 1 ]), function(f(_,_), [ 0, 2, 2, 2, 2, 3, 0, 3, 3, 3, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2 ]), relation(->*(_,_), [ 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1 ]), relation(->(_,_), [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1 ]) ]). ============================== end of model ========================== ============================== STATISTICS ============================ For domain size 5. Current CPU time: 0.00 seconds (total CPU time: 0.01 seconds). Ground clauses: seen=387, kept=362. Selections=59, assignments=180, propagations=492, current_models=1. Rewrite_terms=3116, rewrite_bools=3334, indexes=667. Rules_from_neg_clauses=46, cross_offs=405. ============================== end of statistics ===================== User_CPU=0.01, System_CPU=0.00, Wall_clock=0. Exiting with 1 model. Process 2692918 exit (max_models) Fri Jun 9 15:43:23 2023 The process finished Fri Jun 9 15:43:23 2023 Mace4 cooked interpretation: The problem is infeasible. The problem is not confluent.