NO Problem 1: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty) (REPLACEMENT-MAP (b) (f 1) (a) (c) (fSNonEmpty) (h 1, 2) ) (RULES b -> f(f(b)) b -> f(h(a,b)) f(a) -> b f(a) -> c ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 1: Problem 1: Not CS-TRS Procedure: R is not a CS-TRS Problem 1: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty) (REPLACEMENT-MAP (b) (f 1) (a) (c) (fSNonEmpty) (h 1, 2) ) (RULES b -> f(f(b)) b -> f(h(a,b)) f(a) -> b f(a) -> c ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Huet Levy Ordered by Num of Vars and Symbs Procedure: -> Rules: b -> f(f(b)) b -> f(h(a,b)) f(a) -> b f(a) -> c -> Vars: -> Rlps: (rule: b -> f(f(b)), id: 1, possubterms: b->[]) (rule: b -> f(h(a,b)), id: 2, possubterms: b->[]) (rule: f(a) -> b, id: 3, possubterms: f(a)->[], a->[1]) (rule: f(a) -> c, id: 4, possubterms: f(a)->[], a->[1]) -> Unifications: (R2 unifies with R1 at p: [], l: b, lp: b, sig: {}, l': b, r: f(h(a,b)), r': f(f(b))) (R4 unifies with R3 at p: [], l: f(a), lp: f(a), sig: {}, l': f(a), r: c, r': b) -> Critical pairs info: => Not trivial, Overlay, Proper, NW0, N1 => Not trivial, 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 (b) (f 1) (a) (c) (fSNonEmpty) (h 1, 2) ) (RULES b -> f(f(b)) b -> f(h(a,b)) f(a) -> b f(a) -> c ) 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 (b) (f 1) (a) (c) (fSNonEmpty) (h 1 2) ) (RULES b -> f(f(b)) b -> f(h(a,b)) f(a) -> b f(a) -> c )] Infeasibility Conditions: b ->* z0, c ->* z0 Problem 1: Obtaining a model using Mace4: -> Usable Rules: b -> f(f(b)) b -> f(h(a,b)) f(a) -> b f(a) -> c -> Mace4 Output: ============================== Mace4 ================================= Mace4 (64) version 2009-11A, November 2009. Process 2792863 was started by shintani on shintani-XPS-13-9310, Fri Jun 9 17:39:47 2023 The command was "./mace4 -c -f /tmp/mace42792835-2.in". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file /tmp/mace42792835-2.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). ->(b,f(f(b))) # label(replacement). ->(b,f(h(a,b))) # label(replacement). ->(f(a),b) # label(replacement). ->(f(a),c) # label(replacement). ->*(x,x) # label(reflexivity). ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). end_of_list. formulas(goals). (exists x2 (->*(b,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 (->*(b,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). ->(b,f(f(b))) # label(replacement). ->(b,f(h(a,b))) # label(replacement). ->(f(a),b) # label(replacement). ->(f(a),c) # label(replacement). ->*(x,x) # label(reflexivity). -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). -->*(b,x) | -->*(c,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=2, assignments=3, propagations=11, current_models=0. Rewrite_terms=20, rewrite_bools=26, indexes=11. Rules_from_neg_clauses=3, cross_offs=3. ============================== end of statistics ===================== ============================== DOMAIN SIZE 3 ========================= ============================== MODEL ================================= interpretation( 3, [number=1, seconds=0], [ function(b, [ 0 ]), function(a, [ 1 ]), function(c, [ 1 ]), function(f(_), [ 0, 2, 2 ]), function(h(_,_), [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ]), relation(->*(_,_), [ 1, 0, 0, 0, 1, 0, 1, 1, 1 ]), relation(->(_,_), [ 1, 0, 0, 0, 0, 0, 1, 1, 1 ]) ]). ============================== end of model ========================== ============================== STATISTICS ============================ For domain size 3. Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). Ground clauses: seen=100, kept=91. Selections=16, assignments=18, propagations=24, current_models=1. Rewrite_terms=167, rewrite_bools=155, indexes=24. Rules_from_neg_clauses=4, cross_offs=14. ============================== end of statistics ===================== User_CPU=0.00, System_CPU=0.00, Wall_clock=0. Exiting with 1 model. Process 2792863 exit (max_models) Fri Jun 9 17:39:47 2023 The process finished Fri Jun 9 17:39:47 2023 Mace4 cooked interpretation: The problem is infeasible. The problem is not confluent.