NO Problem 1: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty x) (REPLACEMENT-MAP (a) (g 1) (b) (fSNonEmpty) ) (RULES a -> a g(g(a)) -> b g(g(x)) -> g(g(g(b))) ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 1: CleanTRS Procedure: R was updated by simple cleaning of the TRS ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty x) (REPLACEMENT-MAP (a) (g 1) (b) (fSNonEmpty) ) (RULES g(g(a)) -> b g(g(x)) -> g(g(g(b))) ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Problem 1: Not CS-TRS Procedure: R is not a CS-TRS Problem 1: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Confluence Problem: (VAR vNonEmpty x) (REPLACEMENT-MAP (a) (g 1) (b) (fSNonEmpty) ) (RULES g(g(a)) -> b g(g(x)) -> g(g(g(b))) ) ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Huet Levy Ordered by Num of Vars and Symbs Procedure: -> Rules: g(g(a)) -> b g(g(x)) -> g(g(g(b))) -> Vars: x -> Rlps: (rule: g(g(a)) -> b, id: 1, possubterms: g(g(a))->[], g(a)->[1], a->[1, 1]) (rule: g(g(x)) -> g(g(g(b))), id: 2, possubterms: g(g(x))->[], g(x)->[1]) -> Unifications: (R2 unifies with R1 at p: [], l: g(g(x)), lp: g(g(x)), sig: {x -> a}, l': g(g(a)), r: g(g(g(b))), r': b) (R2 unifies with R1 at p: [1], l: g(g(x)), lp: g(x), sig: {x -> g(a)}, l': g(g(a)), r: g(g(g(b))), r': b) (R2 unifies with R2 at p: [1], l: g(g(x)), lp: g(x), sig: {x -> g(x')}, l': g(g(x')), r: g(g(g(b))), r': g(g(g(b)))) -> Critical pairs info: => Not trivial, Overlay, Proper, NW0, N1 => Not trivial, Not overlay, Proper, NW0, N2 => Not trivial, Not overlay, Proper, NW0, N3 -> 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: (VAR x) (REPLACEMENT-MAP (a) (g 1) (b) (fSNonEmpty) ) (RULES g(g(a)) -> b g(g(x)) -> g(g(g(b))) ) Critical Pairs: => Not trivial, Overlay, Proper, NW0, N1 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: No Convergence InfChecker Procedure: Infeasible convergence? YES Problem 1: Infeasibility Problem: [(VAR vNonEmpty x vNonEmpty z0) (STRATEGY CONTEXTSENSITIVE (g 1) (a) (b) (fSNonEmpty) ) (RULES g(g(a)) -> b g(g(x)) -> g(g(g(b))) )] Infeasibility Conditions: b ->* z0, g(g(g(b))) ->* z0 Problem 1: Obtaining a model using Mace4: -> Usable Rules: g(g(a)) -> b g(g(x)) -> g(g(g(b))) -> Mace4 Output: ============================== Mace4 ================================= Mace4 (64) version 2009-11A, November 2009. Process 2793874 was started by shintani on shintani-XPS-13-9310, Fri Jun 9 17:47:37 2023 The command was "./mace4 -c -f /tmp/mace42793845-2.in". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file /tmp/mace42793845-2.in assign(max_seconds,10). formulas(assumptions). ->(x1,y) -> ->(g(x1),g(y)) # label(congruence). ->(g(g(a)),b) # label(replacement). ->(g(g(x1)),g(g(g(b)))) # label(replacement). ->*(x,x) # label(reflexivity). ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). end_of_list. formulas(goals). (exists x3 (->*(b,x3) & ->*(g(g(g(b))),x3))) # label(goal). end_of_list. ============================== end of input ========================== ============================== PROCESS NON-CLAUSAL FORMULAS ========== % Formulas that are not ordinary clauses: 1 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 2 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 3 (exists x3 (->*(b,x3) & ->*(g(g(g(b))),x3))) # label(goal) # label(non_clause) # label(goal). [goal]. ============================== end of process non-clausal formulas === ============================== CLAUSES FOR SEARCH ==================== formulas(mace4_clauses). -->(x,y) | ->(g(x),g(y)) # label(congruence). ->(g(g(a)),b) # label(replacement). ->(g(g(x)),g(g(g(b)))) # label(replacement). ->*(x,x) # label(reflexivity). -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). -->*(b,x) | -->*(g(g(g(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=19, kept=15. Selections=11, assignments=21, propagations=25, current_models=0. Rewrite_terms=185, rewrite_bools=75, indexes=40. Rules_from_neg_clauses=2, cross_offs=2. ============================== end of statistics ===================== ============================== DOMAIN SIZE 3 ========================= ============================== MODEL ================================= interpretation( 3, [number=1, seconds=0], [ function(a, [ 0 ]), function(b, [ 1 ]), function(g(_), [ 0, 2, 2 ]), relation(->*(_,_), [ 1, 1, 1, 0, 1, 0, 0, 0, 1 ]), relation(->(_,_), [ 0, 1, 1, 0, 0, 0, 0, 0, 1 ]) ]). ============================== end of model ========================== ============================== STATISTICS ============================ For domain size 3. Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). Ground clauses: seen=46, kept=37. Selections=13, assignments=24, propagations=51, current_models=1. Rewrite_terms=263, rewrite_bools=191, indexes=92. Rules_from_neg_clauses=2, cross_offs=7. ============================== end of statistics ===================== User_CPU=0.00, System_CPU=0.00, Wall_clock=0. Exiting with 1 model. Process 2793874 exit (max_models) Fri Jun 9 17:47:37 2023 The process finished Fri Jun 9 17:47:37 2023 Mace4 cooked interpretation: The problem is infeasible. The problem is not confluent.