41.064/41.064 NO 41.064/41.064 41.064/41.064 Problem 1: 41.064/41.064 41.064/41.064 41.064/41.064 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 41.064/41.064 Confluence Problem: 41.064/41.064 (VAR vNonEmpty x y z) 41.064/41.064 (REPLACEMENT-MAP 41.064/41.064 (a) 41.064/41.064 (f 1, 2) 41.064/41.064 (g 1, 2) 41.064/41.064 (k 1) 41.064/41.064 (b) 41.064/41.064 (fSNonEmpty) 41.064/41.064 (h 1) 41.064/41.064 ) 41.064/41.064 (RULES 41.064/41.064 a -> b 41.064/41.064 f(y,a) -> g(x,x) | h(z) ->* x, y ->* k(z), x ->* b 41.064/41.064 f(y,z) -> k(x) | z ->* h(a), y ->* k(a), a ->* x 41.064/41.064 g(x,x) -> a 41.064/41.064 k(x) -> a 41.064/41.064 k(x) -> b 41.064/41.064 k(x) -> h(x) 41.064/41.064 ) 41.064/41.064 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 41.064/41.064 41.064/41.064 41.064/41.064 Problem 1: 41.064/41.064 41.064/41.064 Inlining of Conditions Processor [STERN17]: 41.064/41.064 41.064/41.064 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 41.064/41.064 Confluence Problem: 41.064/41.064 (VAR vNonEmpty x y z) 41.064/41.064 (REPLACEMENT-MAP 41.064/41.064 (a) 41.064/41.064 (f 1, 2) 41.064/41.064 (g 1, 2) 41.064/41.064 (k 1) 41.064/41.064 (b) 41.064/41.064 (fSNonEmpty) 41.064/41.064 (h 1) 41.064/41.064 ) 41.064/41.064 (RULES 41.064/41.064 a -> b 41.064/41.064 f(y,a) -> g(h(z),h(z)) | h(z) ->* b, y ->* k(z) 41.064/41.064 f(y,z) -> k(a) | y ->* k(a), z ->* h(a) 41.064/41.064 g(x,x) -> a 41.064/41.064 k(x) -> a 41.064/41.064 k(x) -> b 41.064/41.064 k(x) -> h(x) 41.064/41.064 ) 41.064/41.064 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 41.064/41.064 41.064/41.064 41.064/41.064 Problem 1: 41.064/41.064 41.064/41.064 Clean CTRS Processor: 41.064/41.064 41.064/41.064 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 41.064/41.064 Confluence Problem: 41.064/41.064 (VAR x y z) 41.064/41.064 (REPLACEMENT-MAP 41.064/41.064 (a) 41.064/41.064 (f 1, 2) 41.064/41.064 (g 1, 2) 41.064/41.064 (k 1) 41.064/41.064 (b) 41.064/41.064 (fSNonEmpty) 41.064/41.064 (h 1) 41.064/41.064 ) 41.064/41.064 (RULES 41.064/41.064 a -> b 41.064/41.064 f(y,z) -> k(a) | y ->* k(a), z ->* h(a) 41.064/41.064 g(x,x) -> a 41.064/41.064 k(x) -> a 41.064/41.064 k(x) -> b 41.064/41.064 k(x) -> h(x) 41.064/41.064 ) 41.064/41.064 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 41.064/41.064 41.064/41.064 CRule InfChecker Info: 41.064/41.064 a -> b 41.064/41.064 Rule remains 41.064/41.064 Proof: 41.064/41.064 NO_CONDS 41.064/41.064 41.064/41.064 CRule InfChecker Info: 41.064/41.064 f(y,a) -> g(h(z),h(z)) | h(z) ->* b, y ->* k(z) 41.064/41.064 Rule deleted 41.064/41.064 Proof: 41.064/41.064 YES 41.064/41.064 41.064/41.064 Problem 1: 41.064/41.064 41.064/41.064 Infeasibility Problem: 41.064/41.064 [(VAR vNonEmpty x y z vNonEmpty y z) 41.064/41.064 (STRATEGY CONTEXTSENSITIVE 41.064/41.064 (a) 41.064/41.064 (f 1 2) 41.064/41.064 (g 1 2) 41.064/41.064 (k 1) 41.064/41.064 (b) 41.064/41.064 (fSNonEmpty) 41.064/41.064 (h 1) 41.064/41.064 ) 41.064/41.064 (RULES 41.064/41.064 a -> b 41.064/41.064 f(y,a) -> g(h(z),h(z)) | h(z) ->* b, y ->* k(z) 41.064/41.064 f(y,z) -> k(a) | y ->* k(a), z ->* h(a) 41.064/41.064 g(x,x) -> a 41.064/41.064 k(x) -> a 41.064/41.064 k(x) -> b 41.064/41.064 k(x) -> h(x) 41.064/41.064 )] 41.064/41.064 41.064/41.064 Infeasibility Conditions: 41.064/41.064 h(z) ->* b, y ->* k(z) 41.064/41.064 41.064/41.064 Problem 1: 41.064/41.064 41.064/41.064 Obtaining a model using Mace4: 41.064/41.064 41.064/41.064 -> Usable Rules: 41.064/41.064 a -> b 41.064/41.064 f(y,a) -> g(h(z),h(z)) | h(z) ->* b, y ->* k(z) 41.064/41.064 f(y,z) -> k(a) | y ->* k(a), z ->* h(a) 41.064/41.064 g(x,x) -> a 41.064/41.064 k(x) -> a 41.064/41.064 k(x) -> b 41.064/41.064 k(x) -> h(x) 41.064/41.064 41.064/41.064 -> Mace4 Output: 41.064/41.064 ============================== Mace4 ================================= 41.064/41.064 Mace4 (64) version 2009-11A, November 2009. 41.064/41.064 Process 81826 was started by ubuntu on ubuntu, 41.064/41.064 Wed Mar 9 10:29:15 2022 41.064/41.064 The command was "./mace4 -c -f /tmp/mace481815-2.in". 41.064/41.064 ============================== end of head =========================== 41.064/41.064 41.064/41.064 ============================== INPUT ================================= 41.064/41.064 41.064/41.064 % Reading from file /tmp/mace481815-2.in 41.064/41.064 41.064/41.064 assign(max_seconds,20). 41.064/41.064 41.064/41.064 formulas(assumptions). 41.064/41.064 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). 41.064/41.064 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). 41.064/41.064 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence). 41.064/41.064 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence). 41.064/41.064 ->(x1,y) -> ->(k(x1),k(y)) # label(congruence). 41.064/41.064 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence). 41.064/41.064 ->(a,b) # label(replacement). 41.064/41.064 ->*(h(x3),b) & ->*(x2,k(x3)) -> ->(f(x2,a),g(h(x3),h(x3))) # label(replacement). 41.064/41.064 ->*(x2,k(a)) & ->*(x3,h(a)) -> ->(f(x2,x3),k(a)) # label(replacement). 41.064/41.064 ->(g(x1,x1),a) # label(replacement). 41.064/41.064 ->(k(x1),a) # label(replacement). 41.064/41.064 ->(k(x1),b) # label(replacement). 41.064/41.064 ->(k(x1),h(x1)) # label(replacement). 41.064/41.064 ->*(x,x) # label(reflexivity). 41.064/41.064 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 41.064/41.064 end_of_list. 41.064/41.064 41.064/41.064 formulas(goals). 41.064/41.064 (exists x5 exists x6 (->*(h(x6),b) & ->*(x5,k(x6)))) # label(goal). 41.064/41.064 end_of_list. 41.064/41.064 41.064/41.064 ============================== end of input ========================== 41.064/41.064 41.064/41.064 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 41.064/41.064 41.064/41.064 % Formulas that are not ordinary clauses: 41.064/41.064 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 3 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 4 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 5 ->(x1,y) -> ->(k(x1),k(y)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 6 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 7 ->*(h(x3),b) & ->*(x2,k(x3)) -> ->(f(x2,a),g(h(x3),h(x3))) # label(replacement) # label(non_clause). [assumption]. 41.064/41.064 8 ->*(x2,k(a)) & ->*(x3,h(a)) -> ->(f(x2,x3),k(a)) # label(replacement) # label(non_clause). [assumption]. 41.064/41.064 9 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 41.064/41.064 10 (exists x5 exists x6 (->*(h(x6),b) & ->*(x5,k(x6)))) # label(goal) # label(non_clause) # label(goal). [goal]. 41.064/41.064 41.064/41.064 ============================== end of process non-clausal formulas === 41.064/41.064 41.064/41.064 ============================== CLAUSES FOR SEARCH ==================== 41.064/41.064 41.064/41.064 formulas(mace4_clauses). 41.064/41.064 -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). 41.064/41.064 -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). 41.064/41.064 -->(x,y) | ->(g(x,z),g(y,z)) # label(congruence). 41.064/41.064 -->(x,y) | ->(g(z,x),g(z,y)) # label(congruence). 41.064/41.064 -->(x,y) | ->(k(x),k(y)) # label(congruence). 41.064/41.064 -->(x,y) | ->(h(x),h(y)) # label(congruence). 41.064/41.064 ->(a,b) # label(replacement). 41.064/41.064 -->*(h(x),b) | -->*(y,k(x)) | ->(f(y,a),g(h(x),h(x))) # label(replacement). 41.064/41.064 -->*(x,k(a)) | -->*(y,h(a)) | ->(f(x,y),k(a)) # label(replacement). 41.064/41.064 ->(g(x,x),a) # label(replacement). 41.064/41.064 ->(k(x),a) # label(replacement). 41.064/41.064 ->(k(x),b) # label(replacement). 41.064/41.064 ->(k(x),h(x)) # label(replacement). 41.064/41.064 ->*(x,x) # label(reflexivity). 41.064/41.064 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 41.064/41.064 -->*(h(x),b) | -->*(y,k(x)) # label(goal). 41.064/41.064 end_of_list. 41.064/41.064 41.064/41.064 ============================== end of clauses for search ============= 41.064/41.064 41.064/41.064 % There are no natural numbers in the input. 41.064/41.064 41.064/41.064 ============================== DOMAIN SIZE 2 ========================= 41.064/41.064 41.064/41.064 ============================== MODEL ================================= 41.064/41.064 41.064/41.064 interpretation( 2, [number=1, seconds=0], [ 41.064/41.064 41.064/41.064 function(a, [ 0 ]), 41.064/41.064 41.064/41.064 function(b, [ 0 ]), 41.064/41.064 41.064/41.064 function(k(_), [ 0, 0 ]), 41.064/41.064 41.064/41.064 function(h(_), [ 1, 1 ]), 41.064/41.064 41.064/41.064 function(f(_,_), [ 41.064/41.064 0, 0, 41.064/41.064 0, 0 ]), 41.064/41.064 41.064/41.064 function(g(_,_), [ 41.064/41.064 0, 0, 41.064/41.064 0, 0 ]), 41.064/41.064 41.064/41.064 relation(->*(_,_), [ 41.064/41.064 1, 1, 41.064/41.064 0, 1 ]), 41.064/41.064 41.064/41.064 relation(->(_,_), [ 41.064/41.064 1, 1, 41.064/41.064 0, 1 ]) 41.064/41.064 ]). 41.064/41.064 41.064/41.064 ============================== end of model ========================== 41.064/41.064 41.064/41.064 ============================== STATISTICS ============================ 41.064/41.064 41.064/41.064 For domain size 2. 41.064/41.064 41.064/41.064 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 41.064/41.064 Ground clauses: seen=71, kept=67. 41.064/41.064 Selections=5, assignments=5, propagations=17, current_models=1. 41.064/41.064 Rewrite_terms=170, rewrite_bools=110, indexes=29. 41.064/41.064 Rules_from_neg_clauses=9, cross_offs=9. 41.064/41.064 41.064/41.064 ============================== end of statistics ===================== 41.064/41.064 41.064/41.064 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 41.064/41.064 41.064/41.064 Exiting with 1 model. 41.064/41.064 41.064/41.064 Process 81826 exit (max_models) Wed Mar 9 10:29:15 2022 41.064/41.064 The process finished Wed Mar 9 10:29:15 2022 41.064/41.064 41.064/41.064 41.064/41.064 Mace4 cooked interpretation: 41.064/41.064 41.064/41.064 41.064/41.064 41.064/41.064 The problem is infeasible. 41.064/41.064 41.064/41.064 41.064/41.064 CRule InfChecker Info: 41.064/41.064 f(y,z) -> k(a) | y ->* k(a), z ->* h(a) 41.064/41.064 Rule remains 41.064/41.064 Proof: 41.064/41.064 NO 41.064/41.064 41.064/41.064 Problem 1: 41.064/41.064 41.064/41.064 Infeasibility Problem: 41.064/41.064 [(VAR vNonEmpty x y z vNonEmpty y z) 41.064/41.064 (STRATEGY CONTEXTSENSITIVE 41.064/41.064 (a) 41.064/41.064 (f 1 2) 41.064/41.064 (g 1 2) 41.064/41.064 (k 1) 41.064/41.064 (b) 41.064/41.064 (fSNonEmpty) 41.064/41.064 (h 1) 41.064/41.064 ) 41.064/41.064 (RULES 41.064/41.064 a -> b 41.064/41.064 f(y,z) -> k(a) | y ->* k(a), z ->* h(a) 41.064/41.064 g(x,x) -> a 41.064/41.064 k(x) -> a 41.064/41.064 k(x) -> b 41.064/41.064 k(x) -> h(x) 41.064/41.064 )] 41.064/41.064 41.064/41.064 Infeasibility Conditions: 41.064/41.064 y ->* k(a), z ->* h(a) 41.064/41.064 41.064/41.064 Problem 1: 41.064/41.064 41.064/41.064 Obtaining a proof using Prover9: 41.064/41.064 41.064/41.064 -> Prover9 Output: 41.064/41.064 ============================== Prover9 =============================== 41.064/41.064 Prover9 (64) version 2009-11A, November 2009. 41.064/41.064 Process 81854 was started by ubuntu on ubuntu, 41.064/41.064 Wed Mar 9 10:29:35 2022 41.064/41.064 The command was "./prover9 -f /tmp/prover981845-0.in". 41.064/41.064 ============================== end of head =========================== 41.064/41.064 41.064/41.064 ============================== INPUT ================================= 41.064/41.064 41.064/41.064 % Reading from file /tmp/prover981845-0.in 41.064/41.064 41.064/41.064 assign(max_seconds,20). 41.064/41.064 41.064/41.064 formulas(assumptions). 41.064/41.064 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 41.064/41.064 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 41.064/41.064 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence). 41.064/41.064 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence). 41.064/41.064 ->_s0(x1,y) -> ->_s0(k(x1),k(y)) # label(congruence). 41.064/41.064 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 41.064/41.064 ->_s0(a,b) # label(replacement). 41.064/41.064 ->*_s0(x2,k(a)) & ->*_s0(x3,h(a)) -> ->_s0(f(x2,x3),k(a)) # label(replacement). 41.064/41.064 ->_s0(g(x1,x1),a) # label(replacement). 41.064/41.064 ->_s0(k(x1),a) # label(replacement). 41.064/41.064 ->_s0(k(x1),b) # label(replacement). 41.064/41.064 ->_s0(k(x1),h(x1)) # label(replacement). 41.064/41.064 ->*_s0(x,x) # label(reflexivity). 41.064/41.064 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 41.064/41.064 end_of_list. 41.064/41.064 41.064/41.064 formulas(goals). 41.064/41.064 (exists x5 exists x6 (->*_s0(x5,k(a)) & ->*_s0(x6,h(a)))) # label(goal). 41.064/41.064 end_of_list. 41.064/41.064 41.064/41.064 ============================== end of input ========================== 41.064/41.064 41.064/41.064 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 41.064/41.064 41.064/41.064 % Formulas that are not ordinary clauses: 41.064/41.064 1 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 2 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 3 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 4 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 5 ->_s0(x1,y) -> ->_s0(k(x1),k(y)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 6 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 7 ->*_s0(x2,k(a)) & ->*_s0(x3,h(a)) -> ->_s0(f(x2,x3),k(a)) # label(replacement) # label(non_clause). [assumption]. 41.064/41.064 8 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 41.064/41.064 9 (exists x5 exists x6 (->*_s0(x5,k(a)) & ->*_s0(x6,h(a)))) # label(goal) # label(non_clause) # label(goal). [goal]. 41.064/41.064 41.064/41.064 ============================== end of process non-clausal formulas === 41.064/41.064 41.064/41.064 ============================== PROCESS INITIAL CLAUSES =============== 41.064/41.064 41.064/41.064 % Clauses before input processing: 41.064/41.064 41.064/41.064 formulas(usable). 41.064/41.064 end_of_list. 41.064/41.064 41.064/41.064 formulas(sos). 41.064/41.064 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 41.064/41.064 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 41.064/41.064 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 41.064/41.064 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 41.064/41.064 -->_s0(x,y) | ->_s0(k(x),k(y)) # label(congruence). [clausify(5)]. 41.064/41.064 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(6)]. 41.064/41.064 ->_s0(a,b) # label(replacement). [assumption]. 41.064/41.064 -->*_s0(x,k(a)) | -->*_s0(y,h(a)) | ->_s0(f(x,y),k(a)) # label(replacement). [clausify(7)]. 41.064/41.064 ->_s0(g(x,x),a) # label(replacement). [assumption]. 41.064/41.064 ->_s0(k(x),a) # label(replacement). [assumption]. 41.064/41.064 ->_s0(k(x),b) # label(replacement). [assumption]. 41.064/41.064 ->_s0(k(x),h(x)) # label(replacement). [assumption]. 41.064/41.064 ->*_s0(x,x) # label(reflexivity). [assumption]. 41.064/41.064 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 41.064/41.064 -->*_s0(x,k(a)) | -->*_s0(y,h(a)) # label(goal). [deny(9)]. 41.064/41.064 end_of_list. 41.064/41.064 41.064/41.064 formulas(demodulators). 41.064/41.064 end_of_list. 41.064/41.064 41.064/41.064 ============================== PREDICATE ELIMINATION ================= 41.064/41.064 41.064/41.064 No predicates eliminated. 41.064/41.064 41.064/41.064 ============================== end predicate elimination ============= 41.064/41.064 41.064/41.064 Auto_denials: 41.064/41.064 % copying label goal to answer in negative clause 41.064/41.064 41.064/41.064 Term ordering decisions: 41.064/41.064 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 41.064/41.064 Function symbol precedence: function_order([ a, b, f, g, k, h ]). 41.064/41.064 After inverse_order: (no changes). 41.064/41.064 Unfolding symbols: (none). 41.064/41.064 41.064/41.064 Auto_inference settings: 41.064/41.064 % set(neg_binary_resolution). % (HNE depth_diff=-6) 41.064/41.064 % clear(ordered_res). % (HNE depth_diff=-6) 41.064/41.064 % set(ur_resolution). % (HNE depth_diff=-6) 41.064/41.064 % set(ur_resolution) -> set(pos_ur_resolution). 41.064/41.064 % set(ur_resolution) -> set(neg_ur_resolution). 41.064/41.064 41.064/41.064 Auto_process settings: 41.064/41.064 % set(unit_deletion). % (Horn set with negative nonunits) 41.064/41.064 41.064/41.064 kept: 10 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 41.064/41.064 kept: 11 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 41.064/41.064 kept: 12 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 41.064/41.064 kept: 13 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 41.064/41.064 kept: 14 -->_s0(x,y) | ->_s0(k(x),k(y)) # label(congruence). [clausify(5)]. 41.064/41.064 kept: 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(6)]. 41.064/41.064 kept: 16 ->_s0(a,b) # label(replacement). [assumption]. 41.064/41.064 kept: 17 -->*_s0(x,k(a)) | -->*_s0(y,h(a)) | ->_s0(f(x,y),k(a)) # label(replacement). [clausify(7)]. 41.064/41.064 kept: 18 ->_s0(g(x,x),a) # label(replacement). [assumption]. 41.064/41.064 kept: 19 ->_s0(k(x),a) # label(replacement). [assumption]. 41.064/41.064 kept: 20 ->_s0(k(x),b) # label(replacement). [assumption]. 41.064/41.064 kept: 21 ->_s0(k(x),h(x)) # label(replacement). [assumption]. 41.064/41.064 kept: 22 ->*_s0(x,x) # label(reflexivity). [assumption]. 41.064/41.064 kept: 23 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 41.064/41.064 kept: 24 -->*_s0(x,k(a)) | -->*_s0(y,h(a)) # label(goal) # answer(goal). [deny(9)]. 41.064/41.064 41.064/41.064 ============================== end of process initial clauses ======== 41.064/41.064 41.064/41.064 ============================== CLAUSES FOR SEARCH ==================== 41.064/41.064 41.064/41.064 % Clauses after input processing: 41.064/41.064 41.064/41.064 formulas(usable). 41.064/41.064 end_of_list. 41.064/41.064 41.064/41.064 formulas(sos). 41.064/41.064 10 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 41.064/41.064 11 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 41.064/41.064 12 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 41.064/41.064 13 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 41.064/41.064 14 -->_s0(x,y) | ->_s0(k(x),k(y)) # label(congruence). [clausify(5)]. 41.064/41.064 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(6)]. 41.064/41.064 16 ->_s0(a,b) # label(replacement). [assumption]. 41.064/41.064 18 ->_s0(g(x,x),a) # label(replacement). [assumption]. 41.064/41.064 19 ->_s0(k(x),a) # label(replacement). [assumption]. 41.064/41.064 20 ->_s0(k(x),b) # label(replacement). [assumption]. 41.064/41.064 21 ->_s0(k(x),h(x)) # label(replacement). [assumption]. 41.064/41.064 22 ->*_s0(x,x) # label(reflexivity). [assumption]. 41.064/41.064 23 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 41.064/41.064 24 -->*_s0(x,k(a)) | -->*_s0(y,h(a)) # label(goal) # answer(goal). [deny(9)]. 41.064/41.064 end_of_list. 41.064/41.064 41.064/41.064 formulas(demodulators). 41.064/41.064 end_of_list. 41.064/41.064 41.064/41.064 ============================== end of clauses for search ============= 41.064/41.064 41.064/41.064 ============================== SEARCH ================================ 41.064/41.064 41.064/41.064 % Starting search at 0.00 seconds. 41.064/41.064 41.064/41.064 given #1 (I,wt=10): 10 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 41.064/41.064 41.064/41.064 given #2 (I,wt=10): 11 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 41.064/41.064 41.064/41.064 given #3 (I,wt=10): 12 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 41.064/41.064 41.064/41.064 given #4 (I,wt=10): 13 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 41.064/41.064 41.064/41.064 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(k(x),k(y)) # label(congruence). [clausify(5)]. 41.064/41.064 41.064/41.064 given #6 (I,wt=8): 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(6)]. 41.064/41.064 41.064/41.064 given #7 (I,wt=3): 16 ->_s0(a,b) # label(replacement). [assumption]. 41.064/41.064 41.064/41.064 given #8 (I,wt=5): 18 ->_s0(g(x,x),a) # label(replacement). [assumption]. 41.064/41.064 41.064/41.064 given #9 (I,wt=4): 19 ->_s0(k(x),a) # label(replacement). [assumption]. 41.064/41.064 41.064/41.064 given #10 (I,wt=4): 20 ->_s0(k(x),b) # label(replacement). [assumption]. 41.064/41.064 41.064/41.064 given #11 (I,wt=5): 21 ->_s0(k(x),h(x)) # label(replacement). [assumption]. 41.064/41.064 41.064/41.064 given #12 (I,wt=3): 22 ->*_s0(x,x) # label(reflexivity). [assumption]. 41.064/41.064 41.064/41.064 given #13 (I,wt=9): 23 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 41.064/41.064 41.064/41.064 given #14 (I,wt=8): 24 -->*_s0(x,k(a)) | -->*_s0(y,h(a)) # label(goal) # answer(goal). [deny(9)]. 41.064/41.064 41.064/41.064 ============================== PROOF ================================= 41.064/41.064 41.064/41.064 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 41.064/41.064 % Length of proof is 9. 41.064/41.064 % Level of proof is 3. 41.064/41.064 % Maximum clause weight is 9.000. 41.064/41.064 % Given clauses 14. 41.064/41.064 41.064/41.064 8 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 41.064/41.064 9 (exists x5 exists x6 (->*_s0(x5,k(a)) & ->*_s0(x6,h(a)))) # label(goal) # label(non_clause) # label(goal). [goal]. 41.064/41.064 21 ->_s0(k(x),h(x)) # label(replacement). [assumption]. 41.064/41.064 22 ->*_s0(x,x) # label(reflexivity). [assumption]. 41.064/41.064 23 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 41.064/41.064 24 -->*_s0(x,k(a)) | -->*_s0(y,h(a)) # label(goal) # answer(goal). [deny(9)]. 41.064/41.064 55 ->*_s0(k(x),h(x)). [ur(23,a,21,a,b,22,a)]. 41.064/41.064 60 -->*_s0(x,h(a)) # answer(goal). [resolve(24,a,22,a)]. 41.064/41.064 61 $F # answer(goal). [resolve(60,a,55,a)]. 41.064/41.064 41.064/41.064 ============================== end of proof ========================== 41.064/41.064 41.064/41.064 ============================== STATISTICS ============================ 41.064/41.064 41.064/41.064 Given=14. Generated=52. Kept=51. proofs=1. 41.064/41.064 Usable=14. Sos=35. Demods=0. Limbo=0, Disabled=16. Hints=0. 41.064/41.064 Kept_by_rule=0, Deleted_by_rule=0. 41.064/41.064 Forward_subsumed=1. Back_subsumed=1. 41.064/41.064 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 41.064/41.064 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 41.064/41.064 Demod_attempts=0. Demod_rewrites=0. 41.064/41.064 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 41.064/41.064 Nonunit_fsub_feature_tests=3. Nonunit_bsub_feature_tests=16. 41.064/41.064 Megabytes=0.11. 41.064/41.064 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 41.064/41.064 41.064/41.064 ============================== end of statistics ===================== 41.064/41.064 41.064/41.064 ============================== end of search ========================= 41.064/41.064 41.064/41.064 THEOREM PROVED 41.064/41.064 41.064/41.064 Exiting with 1 proof. 41.064/41.064 41.064/41.064 Process 81854 exit (max_proofs) Wed Mar 9 10:29:35 2022 41.064/41.064 41.064/41.064 41.064/41.064 The problem is feasible. 41.064/41.064 41.064/41.064 41.064/41.064 CRule InfChecker Info: 41.064/41.064 g(x,x) -> a 41.064/41.064 Rule remains 41.064/41.064 Proof: 41.064/41.064 NO_CONDS 41.064/41.064 41.064/41.064 CRule InfChecker Info: 41.064/41.064 k(x) -> a 41.064/41.064 Rule remains 41.064/41.064 Proof: 41.064/41.064 NO_CONDS 41.064/41.064 41.064/41.064 CRule InfChecker Info: 41.064/41.064 k(x) -> b 41.064/41.064 Rule remains 41.064/41.064 Proof: 41.064/41.064 NO_CONDS 41.064/41.064 41.064/41.064 CRule InfChecker Info: 41.064/41.064 k(x) -> h(x) 41.064/41.064 Rule remains 41.064/41.064 Proof: 41.064/41.064 NO_CONDS 41.064/41.064 41.064/41.064 Problem 1: 41.064/41.064 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 41.064/41.064 Confluence Problem: 41.064/41.064 (VAR x y z) 41.064/41.064 (REPLACEMENT-MAP 41.064/41.064 (a) 41.064/41.064 (f 1, 2) 41.064/41.064 (g 1, 2) 41.064/41.064 (k 1) 41.064/41.064 (b) 41.064/41.064 (fSNonEmpty) 41.064/41.064 (h 1) 41.064/41.064 ) 41.064/41.064 (RULES 41.064/41.064 a -> b 41.064/41.064 f(y,z) -> k(a) | y ->* k(a), z ->* h(a) 41.064/41.064 g(x,x) -> a 41.064/41.064 k(x) -> a 41.064/41.064 k(x) -> b 41.064/41.064 k(x) -> h(x) 41.064/41.064 ) 41.064/41.064 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 41.064/41.064 41.064/41.064 Critical Pairs Processor: 41.064/41.064 -> Rules: 41.064/41.064 a -> b 41.064/41.064 f(y,z) -> k(a) | y ->* k(a), z ->* h(a) 41.064/41.064 g(x,x) -> a 41.064/41.064 k(x) -> a 41.064/41.064 k(x) -> b 41.064/41.064 k(x) -> h(x) 41.064/41.064 -> Vars: 41.064/41.064 "x", "y", "z" 41.064/41.064 41.064/41.064 -> Rlps: 41.064/41.064 crule: a -> b, id: 1, possubterms: a-> [] 41.064/41.064 crule: f(y,z) -> k(a) | y ->* k(a), z ->* h(a), id: 2, possubterms: f(y,z)-> [] 41.064/41.064 crule: g(x,x) -> a, id: 3, possubterms: g(x,x)-> [] 41.064/41.064 crule: k(x) -> a, id: 4, possubterms: k(x)-> [] 41.064/41.064 crule: k(x) -> b, id: 5, possubterms: k(x)-> [] 41.064/41.064 crule: k(x) -> h(x), id: 6, possubterms: k(x)-> [] 41.064/41.064 41.064/41.064 -> Unifications: 41.064/41.064 R5 unifies with R4 at p: [], l: k(x), lp: k(x), conds: {}, sig: {x -> x'}, l': k(x'), r: b, r': a 41.064/41.064 R6 unifies with R4 at p: [], l: k(x), lp: k(x), conds: {}, sig: {x -> x'}, l': k(x'), r: h(x), r': a 41.064/41.064 R6 unifies with R5 at p: [], l: k(x), lp: k(x), conds: {}, sig: {x -> x'}, l': k(x'), r: h(x), r': b 41.064/41.064 41.064/41.064 -> Critical pairs info: 41.064/41.064 => Not trivial, Overlay, NW2, N1 41.064/41.064 => Not trivial, Overlay, NW1, N2 41.064/41.064 => Not trivial, Overlay, NW1, N3 41.064/41.064 41.064/41.064 -> Problem conclusions: 41.064/41.064 Not left linear, Right linear, Not linear 41.064/41.064 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 41.064/41.064 CTRS Type: 1 41.064/41.064 Deterministic, Maybe strongly deterministic 41.064/41.064 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 41.064/41.064 Maybe right-stable CTRS, Overlay CTRS 41.064/41.064 Maybe normal CTRS, Maybe almost normal CTRS 41.064/41.064 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 41.064/41.064 Maybe level confluent 41.064/41.064 Maybe confluent 41.064/41.064 41.064/41.064 Problem 1: 41.064/41.064 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 41.064/41.064 Confluence Problem: 41.064/41.064 (VAR x y z x') 41.064/41.064 (REPLACEMENT-MAP 41.064/41.064 (a) 41.064/41.064 (f 1, 2) 41.064/41.064 (g 1, 2) 41.064/41.064 (k 1) 41.064/41.064 (b) 41.064/41.064 (fSNonEmpty) 41.064/41.064 (h 1) 41.064/41.064 ) 41.064/41.064 (RULES 41.064/41.064 a -> b 41.064/41.064 f(y,z) -> k(a) | y ->* k(a), z ->* h(a) 41.064/41.064 g(x,x) -> a 41.064/41.064 k(x) -> a 41.064/41.064 k(x) -> b 41.064/41.064 k(x) -> h(x) 41.064/41.064 ) 41.064/41.064 Critical Pairs: 41.064/41.064 => Not trivial, Overlay, NW2, N1 41.064/41.064 => Not trivial, Overlay, NW1, N2 41.064/41.064 => Not trivial, Overlay, NW1, N3 41.064/41.064 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 41.064/41.064 41.064/41.064 Conditional Critical Pairs Distributor Processor 41.064/41.064 41.064/41.064 Problem 1: 41.064/41.064 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 41.064/41.064 Confluence Problem: 41.064/41.064 (VAR x y z x') 41.064/41.064 (REPLACEMENT-MAP 41.064/41.064 (a) 41.064/41.064 (f 1, 2) 41.064/41.064 (g 1, 2) 41.064/41.064 (k 1) 41.064/41.064 (b) 41.064/41.064 (fSNonEmpty) 41.064/41.064 (h 1) 41.064/41.064 ) 41.064/41.064 (RULES 41.064/41.064 a -> b 41.064/41.064 f(y,z) -> k(a) | y ->* k(a), z ->* h(a) 41.064/41.064 g(x,x) -> a 41.064/41.064 k(x) -> a 41.064/41.064 k(x) -> b 41.064/41.064 k(x) -> h(x) 41.064/41.064 ) 41.064/41.064 Critical Pairs: 41.064/41.064 => Not trivial, Overlay, NW2, N1 41.064/41.064 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 41.064/41.064 41.064/41.064 Infeasible Convergence No Conditions CCP Processor: 41.064/41.064 Infeasible convergence? 41.064/41.064 YES 41.064/41.064 41.064/41.064 Problem 1: 41.064/41.064 41.064/41.064 Infeasibility Problem: 41.064/41.064 [(VAR vNonEmpty x y z x1 vNonEmpty z) 41.064/41.064 (STRATEGY CONTEXTSENSITIVE 41.064/41.064 (a) 41.064/41.064 (f 1 2) 41.064/41.064 (g 1 2) 41.064/41.064 (k 1) 41.064/41.064 (b) 41.064/41.064 (c_x1) 41.064/41.064 (fSNonEmpty) 41.064/41.064 (h 1) 41.064/41.064 ) 41.064/41.064 (RULES 41.064/41.064 a -> b 41.064/41.064 f(y,z) -> k(a) | y ->* k(a), z ->* h(a) 41.064/41.064 g(x,x) -> a 41.064/41.064 k(x) -> a 41.064/41.064 k(x) -> b 41.064/41.064 k(x) -> h(x) 41.064/41.064 )] 41.064/41.064 41.064/41.064 Infeasibility Conditions: 41.064/41.064 b ->* z, h(c_x1) ->* z 41.064/41.064 41.064/41.064 Problem 1: 41.064/41.064 41.064/41.064 Obtaining a model using Mace4: 41.064/41.064 41.064/41.064 -> Usable Rules: 41.064/41.064 Empty 41.064/41.064 41.064/41.064 -> Mace4 Output: 41.064/41.064 ============================== Mace4 ================================= 41.064/41.064 Mace4 (64) version 2009-11A, November 2009. 41.064/41.064 Process 81880 was started by ubuntu on ubuntu, 41.064/41.064 Wed Mar 9 10:29:35 2022 41.064/41.064 The command was "./mace4 -c -f /tmp/mace481867-2.in". 41.064/41.064 ============================== end of head =========================== 41.064/41.064 41.064/41.064 ============================== INPUT ================================= 41.064/41.064 41.064/41.064 % Reading from file /tmp/mace481867-2.in 41.064/41.064 41.064/41.064 assign(max_seconds,20). 41.064/41.064 41.064/41.064 formulas(assumptions). 41.064/41.064 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). 41.064/41.064 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). 41.064/41.064 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence). 41.064/41.064 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence). 41.064/41.064 ->(x1,y) -> ->(k(x1),k(y)) # label(congruence). 41.064/41.064 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence). 41.064/41.064 ->*(x,x) # label(reflexivity). 41.064/41.064 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 41.064/41.064 end_of_list. 41.064/41.064 41.064/41.064 formulas(goals). 41.064/41.064 (exists x6 (->*(b,x6) & ->*(h(c_x1),x6))) # label(goal). 41.064/41.064 end_of_list. 41.064/41.064 41.064/41.064 ============================== end of input ========================== 41.064/41.064 41.064/41.064 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 41.064/41.064 41.064/41.064 % Formulas that are not ordinary clauses: 41.064/41.064 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 3 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 4 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 5 ->(x1,y) -> ->(k(x1),k(y)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 6 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 41.064/41.064 7 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 41.064/41.064 8 (exists x6 (->*(b,x6) & ->*(h(c_x1),x6))) # label(goal) # label(non_clause) # label(goal). [goal]. 41.064/41.064 41.064/41.064 ============================== end of process non-clausal formulas === 41.064/41.064 41.064/41.064 ============================== CLAUSES FOR SEARCH ==================== 41.064/41.064 41.064/41.064 formulas(mace4_clauses). 41.064/41.064 -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). 41.064/41.064 -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). 41.064/41.064 -->(x,y) | ->(g(x,z),g(y,z)) # label(congruence). 41.064/41.064 -->(x,y) | ->(g(z,x),g(z,y)) # label(congruence). 41.064/41.064 -->(x,y) | ->(k(x),k(y)) # label(congruence). 41.064/41.064 -->(x,y) | ->(h(x),h(y)) # label(congruence). 41.064/41.064 ->*(x,x) # label(reflexivity). 41.064/41.064 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 41.064/41.064 -->*(b,x) | -->*(h(c_x1),x) # label(goal). 41.064/41.064 end_of_list. 41.064/41.064 41.064/41.064 ============================== end of clauses for search ============= 41.064/41.064 41.064/41.064 % There are no natural numbers in the input. 41.064/41.064 41.064/41.064 ============================== DOMAIN SIZE 2 ========================= 41.064/41.064 41.064/41.064 ============================== MODEL ================================= 41.064/41.064 41.064/41.064 interpretation( 2, [number=1, seconds=0], [ 41.064/41.064 41.064/41.064 function(b, [ 0 ]), 41.064/41.064 41.064/41.064 function(c_x1, [ 0 ]), 41.064/41.064 41.064/41.064 function(k(_), [ 0, 0 ]), 41.064/41.064 41.064/41.064 function(h(_), [ 1, 0 ]), 41.064/41.064 41.064/41.064 function(f(_,_), [ 41.064/41.064 0, 0, 41.064/41.064 0, 0 ]), 41.064/41.064 41.064/41.064 function(g(_,_), [ 41.064/41.064 0, 0, 41.064/41.064 0, 0 ]), 41.064/41.064 41.064/41.064 relation(->*(_,_), [ 41.064/41.064 1, 0, 41.064/41.064 0, 1 ]), 41.064/41.064 41.064/41.064 relation(->(_,_), [ 41.064/41.064 0, 0, 41.064/41.064 0, 0 ]) 41.064/41.064 ]). 41.064/41.064 41.064/41.064 ============================== end of model ========================== 41.064/41.064 41.064/41.064 ============================== STATISTICS ============================ 41.064/41.064 41.064/41.064 For domain size 2. 41.064/41.064 41.064/41.064 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 41.064/41.064 Ground clauses: seen=52, kept=48. 41.064/41.064 Selections=14, assignments=14, propagations=8, current_models=1. 41.064/41.064 Rewrite_terms=86, rewrite_bools=60, indexes=10. 41.064/41.064 Rules_from_neg_clauses=1, cross_offs=1. 41.064/41.064 41.064/41.064 ============================== end of statistics ===================== 41.064/41.064 41.064/41.064 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 41.064/41.064 41.064/41.064 Exiting with 1 model. 41.064/41.064 41.064/41.064 Process 81880 exit (max_models) Wed Mar 9 10:29:35 2022 41.064/41.064 The process finished Wed Mar 9 10:29:35 2022 41.064/41.064 41.064/41.064 41.064/41.064 Mace4 cooked interpretation: 41.064/41.064 41.064/41.064 41.064/41.064 41.064/41.064 The problem is infeasible. 41.064/41.064 41.064/41.064 41.064/41.064 The problem is not joinable. 41.064/41.064 45.22user 2.49system 0:41.64elapsed 114%CPU (0avgtext+0avgdata 510656maxresident)k 41.064/41.064 8inputs+0outputs (0major+201649minor)pagefaults 0swaps