NO Problem 1: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty) (REPLACEMENT-MAP (a) (c) (f 1) (h 1, 2) (b) (fSNonEmpty) ) (RULES a -> f(a) c -> b f(h(b,f(c))) -> f(a) h(a,h(c,c)) -> f(h(b,f(c))) h(b,a) -> b ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 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) (h 1, 2) (b) (fSNonEmpty) ) (RULES a -> f(a) c -> b f(h(b,f(c))) -> f(a) h(a,h(c,c)) -> f(h(b,f(c))) h(b,a) -> b ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Huet Levy Ordered by Num of Vars and Symbs Procedure: -> Rules: a -> f(a) c -> b f(h(b,f(c))) -> f(a) h(a,h(c,c)) -> f(h(b,f(c))) h(b,a) -> b -> Vars: -> Rlps: (rule: a -> f(a), id: 1, possubterms: a->[]) (rule: c -> b, id: 2, possubterms: c->[]) (rule: f(h(b,f(c))) -> f(a), id: 3, possubterms: f(h(b,f(c)))->[], h(b,f(c))->[1], b->[1, 1], f(c)->[1, 2], c->[1, 2, 1]) (rule: h(a,h(c,c)) -> f(h(b,f(c))), id: 4, possubterms: h(a,h(c,c))->[], a->[1], h(c,c)->[2], c->[2, 1], c->[2, 2]) (rule: h(b,a) -> b, id: 5, possubterms: h(b,a)->[], b->[1], a->[2]) -> Unifications: (R3 unifies with R2 at p: [1,2,1], l: f(h(b,f(c))), lp: c, sig: {}, l': c, r: f(a), r': b) (R4 unifies with R1 at p: [1], l: h(a,h(c,c)), lp: a, sig: {}, l': a, r: f(h(b,f(c))), r': f(a)) (R4 unifies with R2 at p: [2,1], l: h(a,h(c,c)), lp: c, sig: {}, l': c, r: f(h(b,f(c))), r': b) (R4 unifies with R2 at p: [2,2], l: h(a,h(c,c)), lp: c, sig: {}, l': c, r: f(h(b,f(c))), r': b) (R5 unifies with R1 at p: [2], l: h(b,a), lp: a, sig: {}, l': a, r: b, r': f(a)) -> Critical pairs info: => Not trivial, Not overlay, Proper, NW0, N1 => Not trivial, Not overlay, Proper, NW0, N2 => Not trivial, Not overlay, Proper, NW0, N3 => Not trivial, Not overlay, Proper, NW0, N4 => Not trivial, Not overlay, Proper, NW0, N5 -> 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) (h 1, 2) (b) (fSNonEmpty) ) (RULES a -> f(a) c -> b f(h(b,f(c))) -> f(a) h(a,h(c,c)) -> f(h(b,f(c))) h(b,a) -> 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) (c) (f 1) (h 1 2) (b) (fSNonEmpty) ) (RULES a -> f(a) c -> b f(h(b,f(c))) -> f(a) h(a,h(c,c)) -> f(h(b,f(c))) h(b,a) -> b )] Infeasibility Conditions: f(h(b,f(b))) ->* z0, f(a) ->* z0 Problem 1: Obtaining a model using Mace4: -> Usable Rules: a -> f(a) c -> b f(h(b,f(c))) -> f(a) h(a,h(c,c)) -> f(h(b,f(c))) h(b,a) -> b -> Mace4 Output: ============================== Mace4 ================================= Mace4 (64) version 2009-11A, November 2009. Process 2791072 was started by shintani on shintani-XPS-13-9310, Fri Jun 9 17:30:42 2023 The command was "./mace4 -c -f /tmp/mace42791055-2.in". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file /tmp/mace42791055-2.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). ->(c,b) # label(replacement). ->(f(h(b,f(c))),f(a)) # label(replacement). ->(h(a,h(c,c)),f(h(b,f(c)))) # label(replacement). ->(h(b,a),b) # label(replacement). ->*(x,x) # label(reflexivity). ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). end_of_list. formulas(goals). (exists x2 (->*(f(h(b,f(b))),x2) & ->*(f(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(b,f(b))),x2) & ->*(f(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,f(a)) # label(replacement). ->(c,b) # label(replacement). ->(f(h(b,f(c))),f(a)) # label(replacement). ->(h(a,h(c,c)),f(h(b,f(c)))) # label(replacement). ->(h(b,a),b) # label(replacement). ->*(x,x) # label(reflexivity). -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). -->*(f(h(b,f(b))),x) | -->*(f(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=37, kept=33. Selections=135, assignments=269, propagations=109, current_models=0. Rewrite_terms=1768, rewrite_bools=1073, indexes=245. Rules_from_neg_clauses=9, cross_offs=9. ============================== end of statistics ===================== ============================== DOMAIN SIZE 3 ========================= ============================== MODEL ================================= interpretation( 3, [number=1, seconds=0], [ function(a, [ 0 ]), function(c, [ 1 ]), function(b, [ 2 ]), function(f(_), [ 0, 1, 2 ]), function(h(_,_), [ 0, 1, 0, 1, 1, 1, 1, 1, 2 ]), relation(->*(_,_), [ 1, 0, 0, 1, 1, 1, 0, 0, 1 ]), relation(->(_,_), [ 1, 0, 0, 1, 1, 1, 0, 0, 0 ]) ]). ============================== end of model ========================== ============================== STATISTICS ============================ For domain size 3. Current CPU time: 0.00 seconds (total CPU time: 0.13 seconds). Ground clauses: seen=101, kept=92. Selections=60551, assignments=178972, propagations=48766, current_models=1. Rewrite_terms=1176793, rewrite_bools=751472, indexes=57609. Rules_from_neg_clauses=3518, cross_offs=21214. ============================== end of statistics ===================== User_CPU=0.13, System_CPU=0.00, Wall_clock=0. Exiting with 1 model. Process 2791072 exit (max_models) Fri Jun 9 17:30:42 2023 The process finished Fri Jun 9 17:30:42 2023 Mace4 cooked interpretation: The problem is infeasible. The problem is not confluent.