20.031/20.031 YES 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 20.031/20.031 20.031/20.031 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.031/20.031 Confluence Problem: 20.031/20.031 (VAR vNonEmpty x) 20.031/20.031 (REPLACEMENT-MAP 20.031/20.031 (c) 20.031/20.031 (f 1, 2) 20.031/20.031 (g 1, 2) 20.031/20.031 (a) 20.031/20.031 (b) 20.031/20.031 (d) 20.031/20.031 (fSNonEmpty) 20.031/20.031 ) 20.031/20.031 (RULES 20.031/20.031 c -> c 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c | f(x,a) ->* a 20.031/20.031 g(x,a) -> d | f(x,b) ->* b 20.031/20.031 ) 20.031/20.031 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.031/20.031 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 20.031/20.031 Inlining of Conditions Processor [STERN17]: 20.031/20.031 20.031/20.031 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.031/20.031 Confluence Problem: 20.031/20.031 (VAR vNonEmpty x) 20.031/20.031 (REPLACEMENT-MAP 20.031/20.031 (c) 20.031/20.031 (f 1, 2) 20.031/20.031 (g 1, 2) 20.031/20.031 (a) 20.031/20.031 (b) 20.031/20.031 (d) 20.031/20.031 (fSNonEmpty) 20.031/20.031 ) 20.031/20.031 (RULES 20.031/20.031 c -> c 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c | f(x,a) ->* a 20.031/20.031 g(x,a) -> d | f(x,b) ->* b 20.031/20.031 ) 20.031/20.031 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.031/20.031 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 20.031/20.031 Clean CTRS Processor: 20.031/20.031 20.031/20.031 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.031/20.031 Confluence Problem: 20.031/20.031 (VAR x) 20.031/20.031 (REPLACEMENT-MAP 20.031/20.031 (c) 20.031/20.031 (f 1, 2) 20.031/20.031 (g 1, 2) 20.031/20.031 (a) 20.031/20.031 (b) 20.031/20.031 (d) 20.031/20.031 (fSNonEmpty) 20.031/20.031 ) 20.031/20.031 (RULES 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c | f(x,a) ->* a 20.031/20.031 g(x,a) -> d | f(x,b) ->* b 20.031/20.031 ) 20.031/20.031 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.031/20.031 20.031/20.031 CRule InfChecker Info: 20.031/20.031 c -> c 20.031/20.031 Rule deleted 20.031/20.031 Proof: 20.031/20.031 SAME_LHS_AND_RHS 20.031/20.031 20.031/20.031 CRule InfChecker Info: 20.031/20.031 f(a,x) -> a 20.031/20.031 Rule remains 20.031/20.031 Proof: 20.031/20.031 NO_CONDS 20.031/20.031 20.031/20.031 CRule InfChecker Info: 20.031/20.031 f(b,x) -> b 20.031/20.031 Rule remains 20.031/20.031 Proof: 20.031/20.031 NO_CONDS 20.031/20.031 20.031/20.031 CRule InfChecker Info: 20.031/20.031 g(a,x) -> c | f(x,a) ->* a 20.031/20.031 Rule remains 20.031/20.031 Proof: 20.031/20.031 NO 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 20.031/20.031 Infeasibility Problem: 20.031/20.031 [(VAR vNonEmpty x vNonEmpty x) 20.031/20.031 (STRATEGY CONTEXTSENSITIVE 20.031/20.031 (f 1 2) 20.031/20.031 (g 1 2) 20.031/20.031 (a) 20.031/20.031 (b) 20.031/20.031 (c) 20.031/20.031 (d) 20.031/20.031 (fSNonEmpty) 20.031/20.031 ) 20.031/20.031 (RULES 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c | f(x,a) ->* a 20.031/20.031 g(x,a) -> d | f(x,b) ->* b 20.031/20.031 )] 20.031/20.031 20.031/20.031 Infeasibility Conditions: 20.031/20.031 f(x,a) ->* a 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 20.031/20.031 Obtaining a proof using Prover9: 20.031/20.031 20.031/20.031 -> Prover9 Output: 20.031/20.031 ============================== Prover9 =============================== 20.031/20.031 Prover9 (64) version 2009-11A, November 2009. 20.031/20.031 Process 75264 was started by ubuntu on ubuntu, 20.031/20.031 Wed Mar 9 10:10:31 2022 20.031/20.031 The command was "./prover9 -f /tmp/prover975255-0.in". 20.031/20.031 ============================== end of head =========================== 20.031/20.031 20.031/20.031 ============================== INPUT ================================= 20.031/20.031 20.031/20.031 % Reading from file /tmp/prover975255-0.in 20.031/20.031 20.031/20.031 assign(max_seconds,20). 20.031/20.031 20.031/20.031 formulas(assumptions). 20.031/20.031 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 20.031/20.031 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 20.031/20.031 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence). 20.031/20.031 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence). 20.031/20.031 ->_s0(f(a,x1),a) # label(replacement). 20.031/20.031 ->_s0(f(b,x1),b) # label(replacement). 20.031/20.031 ->*_s0(f(x1,a),a) -> ->_s0(g(a,x1),c) # label(replacement). 20.031/20.031 ->*_s0(f(x1,b),b) -> ->_s0(g(x1,a),d) # label(replacement). 20.031/20.031 ->*_s0(x,x) # label(reflexivity). 20.031/20.031 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 formulas(goals). 20.031/20.031 (exists x3 ->*_s0(f(x3,a),a)) # label(goal). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 ============================== end of input ========================== 20.031/20.031 20.031/20.031 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.031/20.031 20.031/20.031 % Formulas that are not ordinary clauses: 20.031/20.031 1 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 2 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 3 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 4 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 5 ->*_s0(f(x1,a),a) -> ->_s0(g(a,x1),c) # label(replacement) # label(non_clause). [assumption]. 20.031/20.031 6 ->*_s0(f(x1,b),b) -> ->_s0(g(x1,a),d) # label(replacement) # label(non_clause). [assumption]. 20.031/20.031 7 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.031/20.031 8 (exists x3 ->*_s0(f(x3,a),a)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.031/20.031 20.031/20.031 ============================== end of process non-clausal formulas === 20.031/20.031 20.031/20.031 ============================== PROCESS INITIAL CLAUSES =============== 20.031/20.031 20.031/20.031 % Clauses before input processing: 20.031/20.031 20.031/20.031 formulas(usable). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 formulas(sos). 20.031/20.031 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.031/20.031 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.031/20.031 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 20.031/20.031 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 20.031/20.031 ->_s0(f(a,x),a) # label(replacement). [assumption]. 20.031/20.031 ->_s0(f(b,x),b) # label(replacement). [assumption]. 20.031/20.031 -->*_s0(f(x,a),a) | ->_s0(g(a,x),c) # label(replacement). [clausify(5)]. 20.031/20.031 -->*_s0(f(x,b),b) | ->_s0(g(x,a),d) # label(replacement). [clausify(6)]. 20.031/20.031 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.031/20.031 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 20.031/20.031 -->*_s0(f(x,a),a) # label(goal). [deny(8)]. 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 formulas(demodulators). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 ============================== PREDICATE ELIMINATION ================= 20.031/20.031 20.031/20.031 No predicates eliminated. 20.031/20.031 20.031/20.031 ============================== end predicate elimination ============= 20.031/20.031 20.031/20.031 Auto_denials: 20.031/20.031 % copying label goal to answer in negative clause 20.031/20.031 20.031/20.031 Term ordering decisions: 20.031/20.031 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.031/20.031 Function symbol precedence: function_order([ a, b, c, d, f, g ]). 20.031/20.031 After inverse_order: (no changes). 20.031/20.031 Unfolding symbols: (none). 20.031/20.031 20.031/20.031 Auto_inference settings: 20.031/20.031 % set(neg_binary_resolution). % (HNE depth_diff=-4) 20.031/20.031 % clear(ordered_res). % (HNE depth_diff=-4) 20.031/20.031 % set(ur_resolution). % (HNE depth_diff=-4) 20.031/20.031 % set(ur_resolution) -> set(pos_ur_resolution). 20.031/20.031 % set(ur_resolution) -> set(neg_ur_resolution). 20.031/20.031 20.031/20.031 Auto_process settings: (no changes). 20.031/20.031 20.031/20.031 kept: 9 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.031/20.031 kept: 10 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.031/20.031 kept: 11 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 20.031/20.031 kept: 12 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 20.031/20.031 kept: 13 ->_s0(f(a,x),a) # label(replacement). [assumption]. 20.031/20.031 kept: 14 ->_s0(f(b,x),b) # label(replacement). [assumption]. 20.031/20.031 kept: 15 -->*_s0(f(x,a),a) | ->_s0(g(a,x),c) # label(replacement). [clausify(5)]. 20.031/20.031 kept: 16 -->*_s0(f(x,b),b) | ->_s0(g(x,a),d) # label(replacement). [clausify(6)]. 20.031/20.031 kept: 17 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.031/20.031 kept: 18 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 20.031/20.031 kept: 19 -->*_s0(f(x,a),a) # label(goal) # answer(goal). [deny(8)]. 20.031/20.031 20.031/20.031 ============================== end of process initial clauses ======== 20.031/20.031 20.031/20.031 ============================== CLAUSES FOR SEARCH ==================== 20.031/20.031 20.031/20.031 % Clauses after input processing: 20.031/20.031 20.031/20.031 formulas(usable). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 formulas(sos). 20.031/20.031 9 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.031/20.031 10 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.031/20.031 11 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 20.031/20.031 12 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 20.031/20.031 13 ->_s0(f(a,x),a) # label(replacement). [assumption]. 20.031/20.031 14 ->_s0(f(b,x),b) # label(replacement). [assumption]. 20.031/20.031 16 -->*_s0(f(x,b),b) | ->_s0(g(x,a),d) # label(replacement). [clausify(6)]. 20.031/20.031 17 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.031/20.031 18 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 20.031/20.031 19 -->*_s0(f(x,a),a) # label(goal) # answer(goal). [deny(8)]. 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 formulas(demodulators). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 ============================== end of clauses for search ============= 20.031/20.031 20.031/20.031 ============================== SEARCH ================================ 20.031/20.031 20.031/20.031 % Starting search at 0.00 seconds. 20.031/20.031 20.031/20.031 given #1 (I,wt=10): 9 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.031/20.031 20.031/20.031 given #2 (I,wt=10): 10 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.031/20.031 20.031/20.031 given #3 (I,wt=10): 11 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 20.031/20.031 20.031/20.031 given #4 (I,wt=10): 12 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 20.031/20.031 20.031/20.031 given #5 (I,wt=5): 13 ->_s0(f(a,x),a) # label(replacement). [assumption]. 20.031/20.031 20.031/20.031 given #6 (I,wt=5): 14 ->_s0(f(b,x),b) # label(replacement). [assumption]. 20.031/20.031 20.031/20.031 given #7 (I,wt=10): 16 -->*_s0(f(x,b),b) | ->_s0(g(x,a),d) # label(replacement). [clausify(6)]. 20.031/20.031 20.031/20.031 given #8 (I,wt=3): 17 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.031/20.031 20.031/20.031 given #9 (I,wt=9): 18 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 20.031/20.031 20.031/20.031 ============================== PROOF ================================= 20.031/20.031 20.031/20.031 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.031/20.031 % Length of proof is 8. 20.031/20.031 % Level of proof is 3. 20.031/20.031 % Maximum clause weight is 9.000. 20.031/20.031 % Given clauses 9. 20.031/20.031 20.031/20.031 7 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.031/20.031 8 (exists x3 ->*_s0(f(x3,a),a)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.031/20.031 13 ->_s0(f(a,x),a) # label(replacement). [assumption]. 20.031/20.031 17 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.031/20.031 18 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 20.031/20.031 19 -->*_s0(f(x,a),a) # label(goal) # answer(goal). [deny(8)]. 20.031/20.031 29 ->*_s0(f(a,x),a). [ur(18,a,13,a,b,17,a)]. 20.031/20.031 30 $F # answer(goal). [resolve(29,a,19,a)]. 20.031/20.031 20.031/20.031 ============================== end of proof ========================== 20.031/20.031 20.031/20.031 ============================== STATISTICS ============================ 20.031/20.031 20.031/20.031 Given=9. Generated=21. Kept=21. proofs=1. 20.031/20.031 Usable=9. Sos=9. Demods=0. Limbo=1, Disabled=12. Hints=0. 20.031/20.031 Kept_by_rule=0, Deleted_by_rule=0. 20.031/20.031 Forward_subsumed=0. Back_subsumed=1. 20.031/20.031 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.031/20.031 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.031/20.031 Demod_attempts=0. Demod_rewrites=0. 20.031/20.031 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.031/20.031 Nonunit_fsub_feature_tests=2. Nonunit_bsub_feature_tests=13. 20.031/20.031 Megabytes=0.08. 20.031/20.031 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.031/20.031 20.031/20.031 ============================== end of statistics ===================== 20.031/20.031 20.031/20.031 ============================== end of search ========================= 20.031/20.031 20.031/20.031 THEOREM PROVED 20.031/20.031 20.031/20.031 Exiting with 1 proof. 20.031/20.031 20.031/20.031 Process 75264 exit (max_proofs) Wed Mar 9 10:10:31 2022 20.031/20.031 20.031/20.031 20.031/20.031 The problem is feasible. 20.031/20.031 20.031/20.031 20.031/20.031 CRule InfChecker Info: 20.031/20.031 g(x,a) -> d | f(x,b) ->* b 20.031/20.031 Rule remains 20.031/20.031 Proof: 20.031/20.031 NO 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 20.031/20.031 Infeasibility Problem: 20.031/20.031 [(VAR vNonEmpty x vNonEmpty x) 20.031/20.031 (STRATEGY CONTEXTSENSITIVE 20.031/20.031 (f 1 2) 20.031/20.031 (g 1 2) 20.031/20.031 (a) 20.031/20.031 (b) 20.031/20.031 (c) 20.031/20.031 (d) 20.031/20.031 (fSNonEmpty) 20.031/20.031 ) 20.031/20.031 (RULES 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c | f(x,a) ->* a 20.031/20.031 g(x,a) -> d | f(x,b) ->* b 20.031/20.031 )] 20.031/20.031 20.031/20.031 Infeasibility Conditions: 20.031/20.031 f(x,b) ->* b 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 20.031/20.031 Obtaining a proof using Prover9: 20.031/20.031 20.031/20.031 -> Prover9 Output: 20.031/20.031 ============================== Prover9 =============================== 20.031/20.031 Prover9 (64) version 2009-11A, November 2009. 20.031/20.031 Process 75286 was started by ubuntu on ubuntu, 20.031/20.031 Wed Mar 9 10:10:31 2022 20.031/20.031 The command was "./prover9 -f /tmp/prover975277-0.in". 20.031/20.031 ============================== end of head =========================== 20.031/20.031 20.031/20.031 ============================== INPUT ================================= 20.031/20.031 20.031/20.031 % Reading from file /tmp/prover975277-0.in 20.031/20.031 20.031/20.031 assign(max_seconds,20). 20.031/20.031 20.031/20.031 formulas(assumptions). 20.031/20.031 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 20.031/20.031 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 20.031/20.031 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence). 20.031/20.031 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence). 20.031/20.031 ->_s0(f(a,x1),a) # label(replacement). 20.031/20.031 ->_s0(f(b,x1),b) # label(replacement). 20.031/20.031 ->*_s0(f(x1,a),a) -> ->_s0(g(a,x1),c) # label(replacement). 20.031/20.031 ->*_s0(f(x1,b),b) -> ->_s0(g(x1,a),d) # label(replacement). 20.031/20.031 ->*_s0(x,x) # label(reflexivity). 20.031/20.031 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 formulas(goals). 20.031/20.031 (exists x3 ->*_s0(f(x3,b),b)) # label(goal). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 ============================== end of input ========================== 20.031/20.031 20.031/20.031 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.031/20.031 20.031/20.031 % Formulas that are not ordinary clauses: 20.031/20.031 1 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 2 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 3 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 4 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 5 ->*_s0(f(x1,a),a) -> ->_s0(g(a,x1),c) # label(replacement) # label(non_clause). [assumption]. 20.031/20.031 6 ->*_s0(f(x1,b),b) -> ->_s0(g(x1,a),d) # label(replacement) # label(non_clause). [assumption]. 20.031/20.031 7 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.031/20.031 8 (exists x3 ->*_s0(f(x3,b),b)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.031/20.031 20.031/20.031 ============================== end of process non-clausal formulas === 20.031/20.031 20.031/20.031 ============================== PROCESS INITIAL CLAUSES =============== 20.031/20.031 20.031/20.031 % Clauses before input processing: 20.031/20.031 20.031/20.031 formulas(usable). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 formulas(sos). 20.031/20.031 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.031/20.031 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.031/20.031 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 20.031/20.031 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 20.031/20.031 ->_s0(f(a,x),a) # label(replacement). [assumption]. 20.031/20.031 ->_s0(f(b,x),b) # label(replacement). [assumption]. 20.031/20.031 -->*_s0(f(x,a),a) | ->_s0(g(a,x),c) # label(replacement). [clausify(5)]. 20.031/20.031 -->*_s0(f(x,b),b) | ->_s0(g(x,a),d) # label(replacement). [clausify(6)]. 20.031/20.031 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.031/20.031 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 20.031/20.031 -->*_s0(f(x,b),b) # label(goal). [deny(8)]. 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 formulas(demodulators). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 ============================== PREDICATE ELIMINATION ================= 20.031/20.031 20.031/20.031 No predicates eliminated. 20.031/20.031 20.031/20.031 ============================== end predicate elimination ============= 20.031/20.031 20.031/20.031 Auto_denials: 20.031/20.031 % copying label goal to answer in negative clause 20.031/20.031 20.031/20.031 Term ordering decisions: 20.031/20.031 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.031/20.031 Function symbol precedence: function_order([ a, b, c, d, f, g ]). 20.031/20.031 After inverse_order: (no changes). 20.031/20.031 Unfolding symbols: (none). 20.031/20.031 20.031/20.031 Auto_inference settings: 20.031/20.031 % set(neg_binary_resolution). % (HNE depth_diff=-4) 20.031/20.031 % clear(ordered_res). % (HNE depth_diff=-4) 20.031/20.031 % set(ur_resolution). % (HNE depth_diff=-4) 20.031/20.031 % set(ur_resolution) -> set(pos_ur_resolution). 20.031/20.031 % set(ur_resolution) -> set(neg_ur_resolution). 20.031/20.031 20.031/20.031 Auto_process settings: (no changes). 20.031/20.031 20.031/20.031 kept: 9 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.031/20.031 kept: 10 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.031/20.031 kept: 11 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 20.031/20.031 kept: 12 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 20.031/20.031 kept: 13 ->_s0(f(a,x),a) # label(replacement). [assumption]. 20.031/20.031 kept: 14 ->_s0(f(b,x),b) # label(replacement). [assumption]. 20.031/20.031 kept: 15 -->*_s0(f(x,a),a) | ->_s0(g(a,x),c) # label(replacement). [clausify(5)]. 20.031/20.031 kept: 16 -->*_s0(f(x,b),b) | ->_s0(g(x,a),d) # label(replacement). [clausify(6)]. 20.031/20.031 kept: 17 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.031/20.031 kept: 18 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 20.031/20.031 kept: 19 -->*_s0(f(x,b),b) # label(goal) # answer(goal). [deny(8)]. 20.031/20.031 20.031/20.031 ============================== end of process initial clauses ======== 20.031/20.031 20.031/20.031 ============================== CLAUSES FOR SEARCH ==================== 20.031/20.031 20.031/20.031 % Clauses after input processing: 20.031/20.031 20.031/20.031 formulas(usable). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 formulas(sos). 20.031/20.031 9 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.031/20.031 10 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.031/20.031 11 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 20.031/20.031 12 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 20.031/20.031 13 ->_s0(f(a,x),a) # label(replacement). [assumption]. 20.031/20.031 14 ->_s0(f(b,x),b) # label(replacement). [assumption]. 20.031/20.031 15 -->*_s0(f(x,a),a) | ->_s0(g(a,x),c) # label(replacement). [clausify(5)]. 20.031/20.031 17 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.031/20.031 18 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 20.031/20.031 19 -->*_s0(f(x,b),b) # label(goal) # answer(goal). [deny(8)]. 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 formulas(demodulators). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 ============================== end of clauses for search ============= 20.031/20.031 20.031/20.031 ============================== SEARCH ================================ 20.031/20.031 20.031/20.031 % Starting search at 0.00 seconds. 20.031/20.031 20.031/20.031 given #1 (I,wt=10): 9 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.031/20.031 20.031/20.031 given #2 (I,wt=10): 10 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.031/20.031 20.031/20.031 given #3 (I,wt=10): 11 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 20.031/20.031 20.031/20.031 given #4 (I,wt=10): 12 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 20.031/20.031 20.031/20.031 given #5 (I,wt=5): 13 ->_s0(f(a,x),a) # label(replacement). [assumption]. 20.031/20.031 20.031/20.031 given #6 (I,wt=5): 14 ->_s0(f(b,x),b) # label(replacement). [assumption]. 20.031/20.031 20.031/20.031 given #7 (I,wt=10): 15 -->*_s0(f(x,a),a) | ->_s0(g(a,x),c) # label(replacement). [clausify(5)]. 20.031/20.031 20.031/20.031 given #8 (I,wt=3): 17 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.031/20.031 20.031/20.031 given #9 (I,wt=9): 18 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 20.031/20.031 20.031/20.031 ============================== PROOF ================================= 20.031/20.031 20.031/20.031 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.031/20.031 % Length of proof is 8. 20.031/20.031 % Level of proof is 3. 20.031/20.031 % Maximum clause weight is 9.000. 20.031/20.031 % Given clauses 9. 20.031/20.031 20.031/20.031 7 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.031/20.031 8 (exists x3 ->*_s0(f(x3,b),b)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.031/20.031 14 ->_s0(f(b,x),b) # label(replacement). [assumption]. 20.031/20.031 17 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.031/20.031 18 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 20.031/20.031 19 -->*_s0(f(x,b),b) # label(goal) # answer(goal). [deny(8)]. 20.031/20.031 28 ->*_s0(f(b,x),b). [ur(18,a,14,a,b,17,a)]. 20.031/20.031 29 $F # answer(goal). [resolve(28,a,19,a)]. 20.031/20.031 20.031/20.031 ============================== end of proof ========================== 20.031/20.031 20.031/20.031 ============================== STATISTICS ============================ 20.031/20.031 20.031/20.031 Given=9. Generated=20. Kept=20. proofs=1. 20.031/20.031 Usable=9. Sos=9. Demods=0. Limbo=0, Disabled=12. Hints=0. 20.031/20.031 Kept_by_rule=0, Deleted_by_rule=0. 20.031/20.031 Forward_subsumed=0. Back_subsumed=1. 20.031/20.031 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.031/20.031 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.031/20.031 Demod_attempts=0. Demod_rewrites=0. 20.031/20.031 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.031/20.031 Nonunit_fsub_feature_tests=2. Nonunit_bsub_feature_tests=13. 20.031/20.031 Megabytes=0.08. 20.031/20.031 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.031/20.031 20.031/20.031 ============================== end of statistics ===================== 20.031/20.031 20.031/20.031 ============================== end of search ========================= 20.031/20.031 20.031/20.031 THEOREM PROVED 20.031/20.031 20.031/20.031 Exiting with 1 proof. 20.031/20.031 20.031/20.031 Process 75286 exit (max_proofs) Wed Mar 9 10:10:31 2022 20.031/20.031 20.031/20.031 20.031/20.031 The problem is feasible. 20.031/20.031 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.031/20.031 Confluence Problem: 20.031/20.031 (VAR x) 20.031/20.031 (REPLACEMENT-MAP 20.031/20.031 (c) 20.031/20.031 (f 1, 2) 20.031/20.031 (g 1, 2) 20.031/20.031 (a) 20.031/20.031 (b) 20.031/20.031 (d) 20.031/20.031 (fSNonEmpty) 20.031/20.031 ) 20.031/20.031 (RULES 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c | f(x,a) ->* a 20.031/20.031 g(x,a) -> d | f(x,b) ->* b 20.031/20.031 ) 20.031/20.031 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.031/20.031 20.031/20.031 Critical Pairs Processor: 20.031/20.031 -> Rules: 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c | f(x,a) ->* a 20.031/20.031 g(x,a) -> d | f(x,b) ->* b 20.031/20.031 -> Vars: 20.031/20.031 "x" 20.031/20.031 20.031/20.031 -> Rlps: 20.031/20.031 crule: f(a,x) -> a, id: 1, possubterms: f(a,x)-> [], a-> [1] 20.031/20.031 crule: f(b,x) -> b, id: 2, possubterms: f(b,x)-> [], b-> [1] 20.031/20.031 crule: g(a,x) -> c | f(x,a) ->* a, id: 3, possubterms: g(a,x)-> [], a-> [1] 20.031/20.031 crule: g(x,a) -> d | f(x,b) ->* b, id: 4, possubterms: g(x,a)-> [], a-> [2] 20.031/20.031 20.031/20.031 -> Unifications: 20.031/20.031 R4 unifies with R3 at p: [], l: g(x,a), lp: g(x,a), conds: {f(x,b) ->* b, f(x',a) ->* a}, sig: {x -> a,x' -> a}, l': g(a,x'), r: d, r': c 20.031/20.031 20.031/20.031 -> Critical pairs info: 20.031/20.031 | f(a,b) ->* b, f(a,a) ->* a => Not trivial, Overlay, NW2, N1 20.031/20.031 20.031/20.031 -> Problem conclusions: 20.031/20.031 Left linear, Right linear, Linear 20.031/20.031 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.031/20.031 CTRS Type: 1 20.031/20.031 Deterministic, Strongly deterministic 20.031/20.031 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.031/20.031 Maybe right-stable CTRS, Overlay CTRS 20.031/20.031 Maybe normal CTRS, Maybe almost normal CTRS 20.031/20.031 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.031/20.031 Maybe level confluent 20.031/20.031 Maybe confluent 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 Clean Infeasible CCPs Processor: 20.031/20.031 Num of CPIs: 1 20.031/20.031 Timeout: 60 20.031/20.031 Timeout for each infeasibility problem: 60 s 20.031/20.031 | f(a,b) ->* b, f(a,a) ->* a => Not trivial, Overlay, NW2, N1 20.031/20.031 (CONDITIONTYPE ORIENTED) 20.031/20.031 (VAR x) 20.031/20.031 (RULES 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c | f(x,a) ->* a 20.031/20.031 g(x,a) -> d | f(x,b) ->* b) 20.031/20.031 (CONDITION f(a,b) ->* b, f(a,a) ->* a) 20.031/20.031 20.031/20.031 Proof: 20.031/20.031 YES 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 20.031/20.031 Infeasibility Problem: 20.031/20.031 [(VAR vNonEmpty x vNonEmpty) 20.031/20.031 (STRATEGY CONTEXTSENSITIVE 20.031/20.031 (f 1 2) 20.031/20.031 (g 1 2) 20.031/20.031 (a) 20.031/20.031 (b) 20.031/20.031 (c) 20.031/20.031 (d) 20.031/20.031 (fSNonEmpty) 20.031/20.031 ) 20.031/20.031 (RULES 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c | f(x,a) ->* a 20.031/20.031 g(x,a) -> d | f(x,b) ->* b 20.031/20.031 )] 20.031/20.031 20.031/20.031 Infeasibility Conditions: 20.031/20.031 f(a,b) ->* b, f(a,a) ->* a 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 20.031/20.031 Obtaining a model using Mace4: 20.031/20.031 20.031/20.031 -> Usable Rules: 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 20.031/20.031 -> Mace4 Output: 20.031/20.031 ============================== Mace4 ================================= 20.031/20.031 Mace4 (64) version 2009-11A, November 2009. 20.031/20.031 Process 75325 was started by ubuntu on ubuntu, 20.031/20.031 Wed Mar 9 10:10:31 2022 20.031/20.031 The command was "./mace4 -c -f /tmp/mace475303-2.in". 20.031/20.031 ============================== end of head =========================== 20.031/20.031 20.031/20.031 ============================== INPUT ================================= 20.031/20.031 20.031/20.031 % Reading from file /tmp/mace475303-2.in 20.031/20.031 20.031/20.031 assign(max_seconds,20). 20.031/20.031 20.031/20.031 formulas(assumptions). 20.031/20.031 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). 20.031/20.031 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). 20.031/20.031 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence). 20.031/20.031 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence). 20.031/20.031 ->(f(a,x1),a) # label(replacement). 20.031/20.031 ->(f(b,x1),b) # label(replacement). 20.031/20.031 ->*(x,x) # label(reflexivity). 20.031/20.031 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 formulas(goals). 20.031/20.031 ->*(f(a,b),b) & ->*(f(a,a),a) # label(goal). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 ============================== end of input ========================== 20.031/20.031 20.031/20.031 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.031/20.031 20.031/20.031 % Formulas that are not ordinary clauses: 20.031/20.031 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 3 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 4 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 5 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.031/20.031 6 ->*(f(a,b),b) & ->*(f(a,a),a) # label(goal) # label(non_clause) # label(goal). [goal]. 20.031/20.031 20.031/20.031 ============================== end of process non-clausal formulas === 20.031/20.031 20.031/20.031 ============================== CLAUSES FOR SEARCH ==================== 20.031/20.031 20.031/20.031 formulas(mace4_clauses). 20.031/20.031 -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). 20.031/20.031 -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). 20.031/20.031 -->(x,y) | ->(g(x,z),g(y,z)) # label(congruence). 20.031/20.031 -->(x,y) | ->(g(z,x),g(z,y)) # label(congruence). 20.031/20.031 ->(f(a,x),a) # label(replacement). 20.031/20.031 ->(f(b,x),b) # label(replacement). 20.031/20.031 ->*(x,x) # label(reflexivity). 20.031/20.031 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 20.031/20.031 -->*(f(a,b),b) | -->*(f(a,a),a) # label(goal). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 ============================== end of clauses for search ============= 20.031/20.031 20.031/20.031 % There are no natural numbers in the input. 20.031/20.031 20.031/20.031 ============================== DOMAIN SIZE 2 ========================= 20.031/20.031 20.031/20.031 ============================== MODEL ================================= 20.031/20.031 20.031/20.031 interpretation( 2, [number=1, seconds=0], [ 20.031/20.031 20.031/20.031 function(a, [ 0 ]), 20.031/20.031 20.031/20.031 function(b, [ 1 ]), 20.031/20.031 20.031/20.031 function(f(_,_), [ 20.031/20.031 0, 0, 20.031/20.031 1, 1 ]), 20.031/20.031 20.031/20.031 function(g(_,_), [ 20.031/20.031 0, 0, 20.031/20.031 0, 0 ]), 20.031/20.031 20.031/20.031 relation(->*(_,_), [ 20.031/20.031 1, 0, 20.031/20.031 0, 1 ]), 20.031/20.031 20.031/20.031 relation(->(_,_), [ 20.031/20.031 1, 0, 20.031/20.031 0, 1 ]) 20.031/20.031 ]). 20.031/20.031 20.031/20.031 ============================== end of model ========================== 20.031/20.031 20.031/20.031 ============================== STATISTICS ============================ 20.031/20.031 20.031/20.031 For domain size 2. 20.031/20.031 20.031/20.031 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.031/20.031 Ground clauses: seen=47, kept=43. 20.031/20.031 Selections=9, assignments=11, propagations=13, current_models=1. 20.031/20.031 Rewrite_terms=106, rewrite_bools=75, indexes=20. 20.031/20.031 Rules_from_neg_clauses=3, cross_offs=3. 20.031/20.031 20.031/20.031 ============================== end of statistics ===================== 20.031/20.031 20.031/20.031 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.031/20.031 20.031/20.031 Exiting with 1 model. 20.031/20.031 20.031/20.031 Process 75325 exit (max_models) Wed Mar 9 10:10:31 2022 20.031/20.031 The process finished Wed Mar 9 10:10:31 2022 20.031/20.031 20.031/20.031 20.031/20.031 Mace4 cooked interpretation: 20.031/20.031 20.031/20.031 20.031/20.031 20.031/20.031 The problem is infeasible. 20.031/20.031 20.031/20.031 20.031/20.031 20.031/20.031 20.031/20.031 -> Problem conclusions: 20.031/20.031 Left linear, Right linear, Linear 20.031/20.031 Weakly orthogonal, Almost orthogonal, Orthogonal 20.031/20.031 CTRS Type: 1 20.031/20.031 Deterministic, Strongly deterministic 20.031/20.031 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.031/20.031 Maybe right-stable CTRS, Overlay CTRS 20.031/20.031 Maybe normal CTRS, Maybe almost normal CTRS 20.031/20.031 Maybe terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.031/20.031 Maybe level confluent 20.031/20.031 Maybe confluent 20.031/20.031 20.031/20.031 Resulting CCPs: 20.031/20.031 No CCPs left 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 Underlying TRS Termination Processor: 20.031/20.031 20.031/20.031 Resulting Underlying TRS: 20.031/20.031 (VAR x) 20.031/20.031 (STRATEGY CONTEXTSENSITIVE 20.031/20.031 (c) 20.031/20.031 (f 1 2) 20.031/20.031 (g 1 2) 20.031/20.031 (a) 20.031/20.031 (b) 20.031/20.031 (d) 20.031/20.031 ) 20.031/20.031 (RULES 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c 20.031/20.031 g(x,a) -> d 20.031/20.031 ) 20.031/20.031 Underlying TRS terminating? 20.031/20.031 YES 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 20.031/20.031 (VAR vu95NonEmpty x) 20.031/20.031 (RULES 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c 20.031/20.031 g(x,a) -> d 20.031/20.031 ) 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 20.031/20.031 Dependency Pairs Processor: 20.031/20.031 -> Pairs: 20.031/20.031 Empty 20.031/20.031 -> Rules: 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c 20.031/20.031 g(x,a) -> d 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 20.031/20.031 SCC Processor: 20.031/20.031 -> Pairs: 20.031/20.031 Empty 20.031/20.031 -> Rules: 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c 20.031/20.031 g(x,a) -> d 20.031/20.031 ->Strongly Connected Components: 20.031/20.031 There is no strongly connected component 20.031/20.031 20.031/20.031 The problem is finite. 20.031/20.031 20.031/20.031 20.031/20.031 -> Problem conclusions: 20.031/20.031 Left linear, Right linear, Linear 20.031/20.031 Weakly orthogonal, Almost orthogonal, Orthogonal 20.031/20.031 CTRS Type: 1 20.031/20.031 Deterministic, Strongly deterministic 20.031/20.031 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.031/20.031 Maybe right-stable CTRS, Overlay CTRS 20.031/20.031 Maybe normal CTRS, Maybe almost normal CTRS 20.031/20.031 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.031/20.031 Maybe level confluent 20.031/20.031 Maybe confluent 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.031/20.031 Confluence Problem: 20.031/20.031 (VAR x) 20.031/20.031 (REPLACEMENT-MAP 20.031/20.031 (c) 20.031/20.031 (f 1, 2) 20.031/20.031 (g 1, 2) 20.031/20.031 (a) 20.031/20.031 (b) 20.031/20.031 (d) 20.031/20.031 (fSNonEmpty) 20.031/20.031 ) 20.031/20.031 (RULES 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c | f(x,a) ->* a 20.031/20.031 g(x,a) -> d | f(x,b) ->* b 20.031/20.031 ) 20.031/20.031 Critical Pairs: 20.031/20.031 20.031/20.031 Terminating:YES 20.031/20.031 20.031/20.031 -> Problem conclusions: 20.031/20.031 Left linear, Right linear, Linear 20.031/20.031 Weakly orthogonal, Almost orthogonal, Orthogonal 20.031/20.031 CTRS Type: 1 20.031/20.031 Deterministic, Strongly deterministic 20.031/20.031 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.031/20.031 Maybe right-stable CTRS, Overlay CTRS 20.031/20.031 Maybe normal CTRS, Maybe almost normal CTRS 20.031/20.031 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.031/20.031 Maybe level confluent 20.031/20.031 Maybe confluent 20.031/20.031 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.031/20.031 20.031/20.031 Right Stable Processor: 20.031/20.031 Right-stable CTRS 20.031/20.031 Justification: 20.031/20.031 20.031/20.031 -> Term right-stability conditions: 20.031/20.031 Term: a 20.031/20.031 Right-stable term 20.031/20.031 Linear constructor form 20.031/20.031 Don't know if term is a ground normal form (not needed) 20.031/20.031 Right-stability condition achieved 20.031/20.031 A call to InfChecker wasn't needed 20.031/20.031 20.031/20.031 20.031/20.031 -> Term right-stability conditions: 20.031/20.031 Term: b 20.031/20.031 Right-stable term 20.031/20.031 Linear constructor form 20.031/20.031 Don't know if term is a ground normal form (not needed) 20.031/20.031 Right-stability condition achieved 20.031/20.031 A call to InfChecker wasn't needed 20.031/20.031 20.031/20.031 -> Problem conclusions: 20.031/20.031 Left linear, Right linear, Linear 20.031/20.031 Weakly orthogonal, Almost orthogonal, Orthogonal 20.031/20.031 CTRS Type: 1 20.031/20.031 Deterministic, Strongly deterministic 20.031/20.031 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.031/20.031 Right-stable CTRS, Overlay CTRS 20.031/20.031 Maybe normal CTRS, Almost normal CTRS 20.031/20.031 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.031/20.031 Level confluent 20.031/20.031 Confluent 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.031/20.031 Confluence Problem: 20.031/20.031 (VAR x) 20.031/20.031 (REPLACEMENT-MAP 20.031/20.031 (c) 20.031/20.031 (f 1, 2) 20.031/20.031 (g 1, 2) 20.031/20.031 (a) 20.031/20.031 (b) 20.031/20.031 (d) 20.031/20.031 (fSNonEmpty) 20.031/20.031 ) 20.031/20.031 (RULES 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c | f(x,a) ->* a 20.031/20.031 g(x,a) -> d | f(x,b) ->* b 20.031/20.031 ) 20.031/20.031 Critical Pairs: 20.031/20.031 20.031/20.031 Terminating:YES 20.031/20.031 20.031/20.031 -> Problem conclusions: 20.031/20.031 Left linear, Right linear, Linear 20.031/20.031 Weakly orthogonal, Almost orthogonal, Orthogonal 20.031/20.031 CTRS Type: 1 20.031/20.031 Deterministic, Strongly deterministic 20.031/20.031 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.031/20.031 Right-stable CTRS, Overlay CTRS 20.031/20.031 Maybe normal CTRS, Almost normal CTRS 20.031/20.031 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.031/20.031 Level confluent 20.031/20.031 Confluent 20.031/20.031 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.031/20.031 20.031/20.031 Normal RConds Processor: 20.031/20.031 YES 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 20.031/20.031 Infeasibility Problem: 20.031/20.031 [(VAR vNonEmpty x vNonEmpty z z1) 20.031/20.031 (STRATEGY CONTEXTSENSITIVE 20.031/20.031 (f 1 2) 20.031/20.031 (g 1 2) 20.031/20.031 (a) 20.031/20.031 (b) 20.031/20.031 (c) 20.031/20.031 (d) 20.031/20.031 (fSNonEmpty) 20.031/20.031 ) 20.031/20.031 (RULES 20.031/20.031 f(a,x) -> a 20.031/20.031 f(b,x) -> b 20.031/20.031 g(a,x) -> c | f(x,a) ->* a 20.031/20.031 g(x,a) -> d | f(x,b) ->* b 20.031/20.031 )] 20.031/20.031 20.031/20.031 Infeasibility Conditions: 20.031/20.031 a -> z, b -> z1 20.031/20.031 20.031/20.031 Problem 1: 20.031/20.031 20.031/20.031 Obtaining a model using Mace4: 20.031/20.031 20.031/20.031 -> Usable Rules: 20.031/20.031 Empty 20.031/20.031 20.031/20.031 -> Mace4 Output: 20.031/20.031 ============================== Mace4 ================================= 20.031/20.031 Mace4 (64) version 2009-11A, November 2009. 20.031/20.031 Process 75396 was started by ubuntu on ubuntu, 20.031/20.031 Wed Mar 9 10:10:51 2022 20.031/20.031 The command was "./mace4 -c -f /tmp/mace475384-2.in". 20.031/20.031 ============================== end of head =========================== 20.031/20.031 20.031/20.031 ============================== INPUT ================================= 20.031/20.031 20.031/20.031 % Reading from file /tmp/mace475384-2.in 20.031/20.031 20.031/20.031 assign(max_seconds,13). 20.031/20.031 20.031/20.031 formulas(assumptions). 20.031/20.031 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). 20.031/20.031 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). 20.031/20.031 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence). 20.031/20.031 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 formulas(goals). 20.031/20.031 (exists x3 exists x4 (->(a,x3) & ->(b,x4))) # label(goal). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 ============================== end of input ========================== 20.031/20.031 20.031/20.031 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.031/20.031 20.031/20.031 % Formulas that are not ordinary clauses: 20.031/20.031 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 3 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 4 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.031/20.031 5 (exists x3 exists x4 (->(a,x3) & ->(b,x4))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.031/20.031 20.031/20.031 ============================== end of process non-clausal formulas === 20.031/20.031 20.031/20.031 ============================== CLAUSES FOR SEARCH ==================== 20.031/20.031 20.031/20.031 formulas(mace4_clauses). 20.031/20.031 -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). 20.031/20.031 -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). 20.031/20.031 -->(x,y) | ->(g(x,z),g(y,z)) # label(congruence). 20.031/20.031 -->(x,y) | ->(g(z,x),g(z,y)) # label(congruence). 20.031/20.031 -->(a,x) | -->(b,y) # label(goal). 20.031/20.031 end_of_list. 20.031/20.031 20.031/20.031 ============================== end of clauses for search ============= 20.031/20.031 20.031/20.031 % There are no natural numbers in the input. 20.031/20.031 20.031/20.031 ============================== DOMAIN SIZE 2 ========================= 20.031/20.031 20.031/20.031 ============================== MODEL ================================= 20.031/20.031 20.031/20.031 interpretation( 2, [number=1, seconds=0], [ 20.031/20.031 20.031/20.031 function(a, [ 0 ]), 20.031/20.031 20.031/20.031 function(b, [ 0 ]), 20.031/20.031 20.031/20.031 function(f(_,_), [ 20.031/20.031 0, 0, 20.031/20.031 0, 0 ]), 20.031/20.031 20.031/20.031 function(g(_,_), [ 20.031/20.031 0, 0, 20.031/20.031 0, 0 ]), 20.031/20.031 20.031/20.031 relation(->(_,_), [ 20.031/20.031 0, 0, 20.031/20.031 0, 0 ]) 20.031/20.031 ]). 20.031/20.031 20.031/20.031 ============================== end of model ========================== 20.031/20.031 20.031/20.031 ============================== STATISTICS ============================ 20.031/20.031 20.031/20.031 For domain size 2. 20.031/20.031 20.031/20.031 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.031/20.031 Ground clauses: seen=36, kept=36. 20.031/20.031 Selections=11, assignments=11, propagations=3, current_models=1. 20.031/20.031 Rewrite_terms=72, rewrite_bools=43, indexes=12. 20.031/20.031 Rules_from_neg_clauses=0, cross_offs=0. 20.031/20.031 20.031/20.031 ============================== end of statistics ===================== 20.031/20.031 20.031/20.031 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.031/20.031 20.031/20.031 Exiting with 1 model. 20.031/20.031 20.031/20.031 Process 75396 exit (max_models) Wed Mar 9 10:10:51 2022 20.031/20.031 The process finished Wed Mar 9 10:10:51 2022 20.031/20.031 20.031/20.031 20.031/20.031 Mace4 cooked interpretation: 20.031/20.031 20.031/20.031 20.031/20.031 20.031/20.031 The problem is infeasible. 20.031/20.031 20.031/20.031 20.031/20.031 -> Problem conclusions: 20.031/20.031 Left linear, Right linear, Linear 20.031/20.031 Weakly orthogonal, Almost orthogonal, Orthogonal 20.031/20.031 CTRS Type: 1 20.031/20.031 Deterministic, Strongly deterministic 20.031/20.031 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.031/20.031 Right-stable CTRS, Overlay CTRS 20.031/20.031 Normal CTRS, Almost normal CTRS 20.031/20.031 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.031/20.031 Level confluent 20.031/20.031 Confluent 20.031/20.031 20.031/20.031 The problem is joinable. 20.031/20.031 38.63user 1.54system 0:20.31elapsed 197%CPU (0avgtext+0avgdata 39980maxresident)k 20.031/20.031 8inputs+0outputs (0major+48843minor)pagefaults 0swaps