NO Problem 1: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty) (REPLACEMENT-MAP (a) (h 1, 2) (b) (c) (fSNonEmpty) ) (RULES a -> a a -> h(a,b) h(h(c,a),b) -> b ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 1: CleanTRS Procedure: R was updated by simple cleaning of the TRS ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty) (REPLACEMENT-MAP (a) (h 1, 2) (b) (c) (fSNonEmpty) ) (RULES a -> h(a,b) h(h(c,a),b) -> b ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 1: Not CS-TRS Procedure: R is not a CS-TRS Problem 1: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty) (REPLACEMENT-MAP (a) (h 1, 2) (b) (c) (fSNonEmpty) ) (RULES a -> h(a,b) h(h(c,a),b) -> b ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Huet Levy Ordered by Num of Vars and Symbs Procedure: -> Rules: a -> h(a,b) h(h(c,a),b) -> b -> Vars: -> Rlps: (rule: a -> h(a,b), id: 1, possubterms: a->[]) (rule: h(h(c,a),b) -> b, id: 2, possubterms: h(h(c,a),b)->[], h(c,a)->[1], c->[1, 1], a->[1, 2], b->[2]) -> Unifications: (R2 unifies with R1 at p: [1,2], l: h(h(c,a),b), lp: a, sig: {}, l': a, r: b, r': h(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) (h 1, 2) (b) (c) (fSNonEmpty) ) (RULES a -> h(a,b) h(h(c,a),b) -> b ) 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) (h 1 2) (b) (c) (fSNonEmpty) ) (RULES a -> h(a,b) h(h(c,a),b) -> b )] Infeasibility Conditions: h(h(c,h(a,b)),b) ->* z0, b ->* z0 Problem 1: Obtaining a model using Mace4: -> Usable Rules: a -> h(a,b) h(h(c,a),b) -> b -> Mace4 Output: ============================== Mace4 ================================= Mace4 (64) version 2009-11A, November 2009. Process 2791289 was started by shintani on shintani-XPS-13-9310, Fri Jun 9 17:31:22 2023 The command was "./mace4 -c -f /tmp/mace42791256-2.in". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file /tmp/mace42791256-2.in assign(max_seconds,10). formulas(assumptions). ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence). ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence). ->(a,h(a,b)) # label(replacement). ->(h(h(c,a),b),b) # label(replacement). ->*(x,x) # label(reflexivity). ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). end_of_list. formulas(goals). (exists x2 (->*(h(h(c,h(a,b)),b),x2) & ->*(b,x2))) # label(goal). end_of_list. ============================== end of input ========================== ============================== PROCESS NON-CLAUSAL FORMULAS ========== % Formulas that are not ordinary clauses: 1 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 2 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 3 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 4 (exists x2 (->*(h(h(c,h(a,b)),b),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) | ->(h(x,z),h(y,z)) # label(congruence). -->(x,y) | ->(h(z,x),h(z,y)) # label(congruence). ->(a,h(a,b)) # label(replacement). ->(h(h(c,a),b),b) # label(replacement). ->*(x,x) # label(reflexivity). -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). -->*(h(h(c,h(a,b)),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=30, kept=26. Selections=44, assignments=87, propagations=101, current_models=0. Rewrite_terms=673, rewrite_bools=442, indexes=162. Rules_from_neg_clauses=16, cross_offs=16. ============================== end of statistics ===================== ============================== DOMAIN SIZE 3 ========================= ============================== STATISTICS ============================ For domain size 3. Current CPU time: 0.00 seconds (total CPU time: 0.05 seconds). Ground clauses: seen=89, kept=80. Selections=18168, assignments=52936, propagations=27230, current_models=0. Rewrite_terms=337845, rewrite_bools=249226, indexes=30592. Rules_from_neg_clauses=2720, cross_offs=10693. ============================== end of statistics ===================== ============================== DOMAIN SIZE 4 ========================= ============================== MODEL ================================= interpretation( 4, [number=1, seconds=4], [ function(a, [ 0 ]), function(b, [ 1 ]), function(c, [ 0 ]), function(h(_,_), [ 0, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2 ]), relation(->*(_,_), [ 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1 ]), relation(->(_,_), [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1 ]) ]). ============================== end of model ========================== ============================== STATISTICS ============================ For domain size 4. Current CPU time: 0.00 seconds (total CPU time: 4.14 seconds). Ground clauses: seen=202, kept=186. Selections=2099808, assignments=8106237, propagations=1064481, current_models=1. Rewrite_terms=44752842, rewrite_bools=30933680, indexes=1302689. Rules_from_neg_clauses=77822, cross_offs=825362. ============================== end of statistics ===================== User_CPU=4.14, System_CPU=0.54, Wall_clock=5. Exiting with 1 model. Process 2791289 exit (max_models) Fri Jun 9 17:31:27 2023 The process finished Fri Jun 9 17:31:27 2023 Mace4 cooked interpretation: The problem is infeasible. The problem is not confluent.