NO Problem 1: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty) (REPLACEMENT-MAP (a) (b) (c) (f 1) (fSNonEmpty) (h 1, 2) ) (RULES a -> f(a) b -> a b -> c b -> h(a,a) b -> h(h(c,a),f(f(b))) ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 1: Problem 1: Not CS-TRS Procedure: R is not a CS-TRS Problem 1: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty) (REPLACEMENT-MAP (a) (b) (c) (f 1) (fSNonEmpty) (h 1, 2) ) (RULES a -> f(a) b -> a b -> c b -> h(a,a) b -> h(h(c,a),f(f(b))) ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Huet Levy Ordered by Num of Vars and Symbs Procedure: -> Rules: a -> f(a) b -> a b -> c b -> h(a,a) b -> h(h(c,a),f(f(b))) -> Vars: -> Rlps: (rule: a -> f(a), id: 1, possubterms: a->[]) (rule: b -> a, id: 2, possubterms: b->[]) (rule: b -> c, id: 3, possubterms: b->[]) (rule: b -> h(a,a), id: 4, possubterms: b->[]) (rule: b -> h(h(c,a),f(f(b))), id: 5, possubterms: b->[]) -> Unifications: (R3 unifies with R2 at p: [], l: b, lp: b, sig: {}, l': b, r: c, r': a) (R4 unifies with R2 at p: [], l: b, lp: b, sig: {}, l': b, r: h(a,a), r': a) (R4 unifies with R3 at p: [], l: b, lp: b, sig: {}, l': b, r: h(a,a), r': c) (R5 unifies with R2 at p: [], l: b, lp: b, sig: {}, l': b, r: h(h(c,a),f(f(b))), r': a) (R5 unifies with R3 at p: [], l: b, lp: b, sig: {}, l': b, r: h(h(c,a),f(f(b))), r': c) (R5 unifies with R4 at p: [], l: b, lp: b, sig: {}, l': b, r: h(h(c,a),f(f(b))), r': h(a,a)) -> Critical pairs info: => Not trivial, Overlay, Proper, NW0, N1 => Not trivial, Overlay, Proper, NW0, N2 => Not trivial, Overlay, Proper, NW0, N3 => Not trivial, Overlay, Proper, NW0, N4 => Not trivial, Overlay, Proper, NW0, N5 => Not trivial, Overlay, Proper, NW0, N6 -> 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) (b) (c) (f 1) (fSNonEmpty) (h 1, 2) ) (RULES a -> f(a) b -> a b -> c b -> h(a,a) b -> h(h(c,a),f(f(b))) ) 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 (a) (b) (c) (f 1) (fSNonEmpty) (h 1 2) ) (RULES a -> f(a) b -> a b -> c b -> h(a,a) b -> h(h(c,a),f(f(b))) )] Infeasibility Conditions: a ->* z0, c ->* z0 Problem 1: Obtaining a model using Mace4: -> Usable Rules: a -> f(a) -> Mace4 Output: ============================== Mace4 ================================= Mace4 (64) version 2009-11A, November 2009. Process 2789777 was started by shintani on shintani-XPS-13-9310, Fri Jun 9 17:19:40 2023 The command was "./mace4 -c -f /tmp/mace42789743-2.in". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file /tmp/mace42789743-2.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,f(a)) # label(replacement). ->*(x,x) # label(reflexivity). ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). end_of_list. formulas(goals). (exists x2 (->*(a,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 (->*(a,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). ->(a,f(a)) # label(replacement). ->*(x,x) # label(reflexivity). -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). -->*(a,x) | -->*(c,x) # label(goal). end_of_list. ============================== end of clauses for search ============= % There are no natural numbers in the input. ============================== DOMAIN SIZE 2 ========================= ============================== MODEL ================================= interpretation( 2, [number=1, seconds=0], [ function(a, [ 0 ]), function(c, [ 1 ]), function(f(_), [ 0, 0 ]), function(h(_,_), [ 0, 0, 0, 0 ]), relation(->*(_,_), [ 1, 0, 0, 1 ]), relation(->(_,_), [ 1, 0, 0, 0 ]) ]). ============================== end of model ========================== ============================== STATISTICS ============================ For domain size 2. Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). Ground clauses: seen=33, kept=29. Selections=7, assignments=7, propagations=9, current_models=1. Rewrite_terms=47, rewrite_bools=39, indexes=4. Rules_from_neg_clauses=2, cross_offs=2. ============================== end of statistics ===================== User_CPU=0.00, System_CPU=0.00, Wall_clock=0. Exiting with 1 model. Process 2789777 exit (max_models) Fri Jun 9 17:19:40 2023 The process finished Fri Jun 9 17:19:40 2023 Mace4 cooked interpretation: The problem is infeasible. The problem is not confluent.