NO Problem 1: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty) (REPLACEMENT-MAP (a) (c) (f 1) (b) (fSNonEmpty) (h 1, 2) ) (RULES a -> b c -> f(c) f(b) -> f(c) f(h(c,c)) -> a ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 1: Problem 1: Not CS-TRS Procedure: R is not a CS-TRS Problem 1: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty) (REPLACEMENT-MAP (a) (c) (f 1) (b) (fSNonEmpty) (h 1, 2) ) (RULES a -> b c -> f(c) f(b) -> f(c) f(h(c,c)) -> a ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Huet Levy Ordered by Num of Vars and Symbs Procedure: -> Rules: a -> b c -> f(c) f(b) -> f(c) f(h(c,c)) -> a -> Vars: -> Rlps: (rule: a -> b, id: 1, possubterms: a->[]) (rule: c -> f(c), id: 2, possubterms: c->[]) (rule: f(b) -> f(c), id: 3, possubterms: f(b)->[], b->[1]) (rule: f(h(c,c)) -> a, id: 4, possubterms: f(h(c,c))->[], h(c,c)->[1], c->[1, 1], c->[1, 2]) -> Unifications: (R4 unifies with R2 at p: [1,1], l: f(h(c,c)), lp: c, sig: {}, l': c, r: a, r': f(c)) (R4 unifies with R2 at p: [1,2], l: f(h(c,c)), lp: c, sig: {}, l': c, r: a, r': f(c)) -> Critical pairs info: => Not trivial, Not overlay, Proper, NW0, N1 => Not trivial, Not overlay, Proper, NW0, N2 -> 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) (c) (f 1) (b) (fSNonEmpty) (h 1, 2) ) (RULES a -> b c -> f(c) f(b) -> f(c) f(h(c,c)) -> 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) (c) (f 1) (b) (fSNonEmpty) (h 1 2) ) (RULES a -> b c -> f(c) f(b) -> f(c) f(h(c,c)) -> a )] Infeasibility Conditions: f(h(f(c),c)) ->* z0, a ->* z0 Problem 1: Obtaining a model using Mace4: -> Usable Rules: a -> b c -> f(c) f(b) -> f(c) f(h(c,c)) -> a -> Mace4 Output: ============================== Mace4 ================================= Mace4 (64) version 2009-11A, November 2009. Process 2792050 was started by shintani on shintani-XPS-13-9310, Fri Jun 9 17:35:15 2023 The command was "./mace4 -c -f /tmp/mace42792030-2.in". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file /tmp/mace42792030-2.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,b) # label(replacement). ->(c,f(c)) # label(replacement). ->(f(b),f(c)) # label(replacement). ->(f(h(c,c)),a) # label(replacement). ->*(x,x) # label(reflexivity). ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). end_of_list. formulas(goals). (exists x2 (->*(f(h(f(c),c)),x2) & ->*(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),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(h(f(c),c)),x2) & ->*(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),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,b) # label(replacement). ->(c,f(c)) # label(replacement). ->(f(b),f(c)) # label(replacement). ->(f(h(c,c)),a) # label(replacement). ->*(x,x) # label(reflexivity). -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). -->*(f(h(f(c),c)),x) | -->*(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=36, kept=32. Selections=134, assignments=267, propagations=146, current_models=0. Rewrite_terms=1510, rewrite_bools=1098, indexes=230. 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: 0.30 seconds). Ground clauses: seen=100, kept=91. Selections=142906, assignments=424547, propagations=51738, current_models=0. Rewrite_terms=2296269, rewrite_bools=1459179, indexes=45230. Rules_from_neg_clauses=5308, cross_offs=30212. ============================== end of statistics ===================== ============================== DOMAIN SIZE 4 ========================= ============================== MODEL ================================= interpretation( 4, [number=1, seconds=0], [ function(a, [ 0 ]), function(c, [ 0 ]), function(b, [ 0 ]), function(f(_), [ 1, 1, 2, 3 ]), function(h(_,_), [ 2, 2, 2, 0, 3, 3, 2, 0, 2, 2, 2, 0, 0, 0, 0, 0 ]), relation(->*(_,_), [ 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1 ]), relation(->(_,_), [ 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1 ]) ]). ============================== end of model ========================== ============================== STATISTICS ============================ For domain size 4. Current CPU time: 0.00 seconds (total CPU time: 0.30 seconds). Ground clauses: seen=220, kept=204. Selections=62, assignments=185, propagations=626, current_models=1. Rewrite_terms=2619, rewrite_bools=3167, indexes=737. Rules_from_neg_clauses=54, cross_offs=206. ============================== end of statistics ===================== User_CPU=0.30, System_CPU=0.02, Wall_clock=0. Exiting with 1 model. Process 2792050 exit (max_models) Fri Jun 9 17:35:15 2023 The process finished Fri Jun 9 17:35:15 2023 Mace4 cooked interpretation: The problem is infeasible. The problem is not confluent.