0.009/0.009 YES 0.009/0.009 0.009/0.009 Problem 1: 0.009/0.009 0.009/0.009 0.009/0.009 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.009/0.009 Confluence Problem: 0.009/0.009 (VAR vNonEmpty x x' x'' y) 0.009/0.009 (REPLACEMENT-MAP 0.009/0.009 (f 1, 2) 0.009/0.009 (h 1, 2, 3) 0.009/0.009 (c) 0.009/0.009 (fSNonEmpty) 0.009/0.009 (g 1, 2) 0.009/0.009 ) 0.009/0.009 (RULES 0.009/0.009 f(x,x') -> g(y,y) | x ->* y, x' ->* y 0.009/0.009 h(x,x',x'') -> c | x ->* y, x' ->* y, x'' ->* y 0.009/0.009 ) 0.009/0.009 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.009/0.009 0.009/0.009 0.009/0.009 Problem 1: 0.009/0.009 0.009/0.009 Inlining of Conditions Processor [STERN17]: 0.009/0.009 0.009/0.009 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.009/0.009 Confluence Problem: 0.009/0.009 (VAR vNonEmpty x x' x'' y) 0.009/0.009 (REPLACEMENT-MAP 0.009/0.009 (f 1, 2) 0.009/0.009 (h 1, 2, 3) 0.009/0.009 (c) 0.009/0.009 (fSNonEmpty) 0.009/0.009 (g 1, 2) 0.009/0.009 ) 0.009/0.009 (RULES 0.009/0.009 f(x,x') -> g(y,y) | x ->* y, x' ->* y 0.009/0.009 h(x,x',x'') -> c | x ->* y, x' ->* y, x'' ->* y 0.009/0.009 ) 0.009/0.009 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.009/0.009 0.009/0.009 0.009/0.009 Problem 1: 0.009/0.009 0.009/0.009 Clean CTRS Processor: 0.009/0.009 0.009/0.009 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.009/0.009 Confluence Problem: 0.009/0.009 (VAR x x' x'' y) 0.009/0.009 (REPLACEMENT-MAP 0.009/0.009 (f 1, 2) 0.009/0.009 (h 1, 2, 3) 0.009/0.009 (c) 0.009/0.009 (fSNonEmpty) 0.009/0.009 (g 1, 2) 0.009/0.009 ) 0.009/0.009 (RULES 0.009/0.009 f(x,x') -> g(y,y) | x ->* y, x' ->* y 0.009/0.009 h(x,x',x'') -> c | x ->* y, x' ->* y, x'' ->* y 0.009/0.009 ) 0.009/0.009 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.009/0.009 0.009/0.009 CRule InfChecker Info: 0.009/0.009 f(x,x') -> g(y,y) | x ->* y, x' ->* y 0.009/0.009 Rule remains 0.009/0.009 Proof: 0.009/0.009 NO 0.009/0.009 0.009/0.009 Problem 1: 0.009/0.009 0.009/0.009 Infeasibility Problem: 0.009/0.009 [(VAR vNonEmpty x x' x'' y vNonEmpty x x' y) 0.009/0.009 (STRATEGY CONTEXTSENSITIVE 0.009/0.009 (f 1 2) 0.009/0.009 (h 1 2 3) 0.009/0.009 (c) 0.009/0.009 (fSNonEmpty) 0.009/0.009 (g 1 2) 0.009/0.009 ) 0.009/0.009 (RULES 0.009/0.009 f(x,x') -> g(y,y) | x ->* y, x' ->* y 0.009/0.009 h(x,x',x'') -> c | x ->* y, x' ->* y, x'' ->* y 0.009/0.009 )] 0.009/0.009 0.009/0.009 Infeasibility Conditions: 0.009/0.009 x ->* y, x' ->* y 0.009/0.009 0.009/0.009 Problem 1: 0.009/0.009 0.009/0.009 Obtaining a proof using Prover9: 0.009/0.009 0.009/0.009 -> Prover9 Output: 0.009/0.009 ============================== Prover9 =============================== 0.009/0.009 Prover9 (64) version 2009-11A, November 2009. 0.009/0.009 Process 20863 was started by ubuntu on ubuntu, 0.009/0.009 Wed Mar 9 09:31:21 2022 0.009/0.009 The command was "./prover9 -f /tmp/prover920854-0.in". 0.009/0.009 ============================== end of head =========================== 0.009/0.009 0.009/0.009 ============================== INPUT ================================= 0.009/0.009 0.009/0.009 % Reading from file /tmp/prover920854-0.in 0.009/0.009 0.009/0.009 assign(max_seconds,20). 0.009/0.009 0.009/0.009 formulas(assumptions). 0.009/0.009 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 0.009/0.009 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 0.009/0.009 ->_s0(x1,y) -> ->_s0(h(x1,x2,x3),h(y,x2,x3)) # label(congruence). 0.009/0.009 ->_s0(x2,y) -> ->_s0(h(x1,x2,x3),h(x1,y,x3)) # label(congruence). 0.009/0.009 ->_s0(x3,y) -> ->_s0(h(x1,x2,x3),h(x1,x2,y)) # label(congruence). 0.009/0.009 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence). 0.009/0.009 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence). 0.009/0.009 ->*_s0(x1,x4) & ->*_s0(x2,x4) -> ->_s0(f(x1,x2),g(x4,x4)) # label(replacement). 0.009/0.009 ->*_s0(x1,x4) & ->*_s0(x2,x4) & ->*_s0(x3,x4) -> ->_s0(h(x1,x2,x3),c) # label(replacement). 0.009/0.009 ->*_s0(x,x) # label(reflexivity). 0.009/0.009 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 formulas(goals). 0.009/0.009 (exists x6 exists x7 exists x8 (->*_s0(x6,x8) & ->*_s0(x7,x8))) # label(goal). 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 ============================== end of input ========================== 0.009/0.009 0.009/0.009 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.009/0.009 0.009/0.009 % Formulas that are not ordinary clauses: 0.009/0.009 1 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.009/0.009 2 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.009/0.009 3 ->_s0(x1,y) -> ->_s0(h(x1,x2,x3),h(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 0.009/0.009 4 ->_s0(x2,y) -> ->_s0(h(x1,x2,x3),h(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 0.009/0.009 5 ->_s0(x3,y) -> ->_s0(h(x1,x2,x3),h(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 0.009/0.009 6 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.009/0.009 7 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.009/0.009 8 ->*_s0(x1,x4) & ->*_s0(x2,x4) -> ->_s0(f(x1,x2),g(x4,x4)) # label(replacement) # label(non_clause). [assumption]. 0.009/0.009 9 ->*_s0(x1,x4) & ->*_s0(x2,x4) & ->*_s0(x3,x4) -> ->_s0(h(x1,x2,x3),c) # label(replacement) # label(non_clause). [assumption]. 0.009/0.009 10 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 0.009/0.009 11 (exists x6 exists x7 exists x8 (->*_s0(x6,x8) & ->*_s0(x7,x8))) # label(goal) # label(non_clause) # label(goal). [goal]. 0.009/0.009 0.009/0.009 ============================== end of process non-clausal formulas === 0.009/0.009 0.009/0.009 ============================== PROCESS INITIAL CLAUSES =============== 0.009/0.009 0.009/0.009 % Clauses before input processing: 0.009/0.009 0.009/0.009 formulas(usable). 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 formulas(sos). 0.009/0.009 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 0.009/0.009 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 0.009/0.009 -->_s0(x,y) | ->_s0(h(x,z,u),h(y,z,u)) # label(congruence). [clausify(3)]. 0.009/0.009 -->_s0(x,y) | ->_s0(h(z,x,u),h(z,y,u)) # label(congruence). [clausify(4)]. 0.009/0.009 -->_s0(x,y) | ->_s0(h(z,u,x),h(z,u,y)) # label(congruence). [clausify(5)]. 0.009/0.009 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(6)]. 0.009/0.009 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(7)]. 0.009/0.009 -->*_s0(x,y) | -->*_s0(z,y) | ->_s0(f(x,z),g(y,y)) # label(replacement). [clausify(8)]. 0.009/0.009 -->*_s0(x,y) | -->*_s0(z,y) | -->*_s0(u,y) | ->_s0(h(x,z,u),c) # label(replacement). [clausify(9)]. 0.009/0.009 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.009/0.009 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(10)]. 0.009/0.009 -->*_s0(x,y) | -->*_s0(z,y) # label(goal). [deny(11)]. 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 formulas(demodulators). 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 ============================== PREDICATE ELIMINATION ================= 0.009/0.009 0.009/0.009 No predicates eliminated. 0.009/0.009 0.009/0.009 ============================== end predicate elimination ============= 0.009/0.009 0.009/0.009 Auto_denials: 0.009/0.009 % copying label goal to answer in negative clause 0.009/0.009 0.009/0.009 Term ordering decisions: 0.009/0.009 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 0.009/0.009 Function symbol precedence: function_order([ c, f, g, h ]). 0.009/0.009 After inverse_order: (no changes). 0.009/0.009 Unfolding symbols: (none). 0.009/0.009 0.009/0.009 Auto_inference settings: 0.009/0.009 % set(neg_binary_resolution). % (HNE depth_diff=-9) 0.009/0.009 % clear(ordered_res). % (HNE depth_diff=-9) 0.009/0.009 % set(ur_resolution). % (HNE depth_diff=-9) 0.009/0.009 % set(ur_resolution) -> set(pos_ur_resolution). 0.009/0.009 % set(ur_resolution) -> set(neg_ur_resolution). 0.009/0.009 0.009/0.009 Auto_process settings: 0.009/0.009 % set(unit_deletion). % (Horn set with negative nonunits) 0.009/0.009 0.009/0.009 kept: 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 0.009/0.009 kept: 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 0.009/0.009 kept: 14 -->_s0(x,y) | ->_s0(h(x,z,u),h(y,z,u)) # label(congruence). [clausify(3)]. 0.009/0.009 kept: 15 -->_s0(x,y) | ->_s0(h(z,x,u),h(z,y,u)) # label(congruence). [clausify(4)]. 0.009/0.009 kept: 16 -->_s0(x,y) | ->_s0(h(z,u,x),h(z,u,y)) # label(congruence). [clausify(5)]. 0.009/0.009 kept: 17 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(6)]. 0.009/0.009 kept: 18 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(7)]. 0.009/0.009 kept: 19 -->*_s0(x,y) | -->*_s0(z,y) | ->_s0(f(x,z),g(y,y)) # label(replacement). [clausify(8)]. 0.009/0.009 kept: 20 -->*_s0(x,y) | -->*_s0(z,y) | -->*_s0(u,y) | ->_s0(h(x,z,u),c) # label(replacement). [clausify(9)]. 0.009/0.009 kept: 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.009/0.009 kept: 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(10)]. 0.009/0.009 kept: 23 -->*_s0(x,y) | -->*_s0(z,y) # label(goal) # answer(goal). [deny(11)]. 0.009/0.009 0.009/0.009 ============================== end of process initial clauses ======== 0.009/0.009 0.009/0.009 ============================== CLAUSES FOR SEARCH ==================== 0.009/0.009 0.009/0.009 % Clauses after input processing: 0.009/0.009 0.009/0.009 formulas(usable). 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 formulas(sos). 0.009/0.009 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 0.009/0.009 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 0.009/0.009 14 -->_s0(x,y) | ->_s0(h(x,z,u),h(y,z,u)) # label(congruence). [clausify(3)]. 0.009/0.009 15 -->_s0(x,y) | ->_s0(h(z,x,u),h(z,y,u)) # label(congruence). [clausify(4)]. 0.009/0.009 16 -->_s0(x,y) | ->_s0(h(z,u,x),h(z,u,y)) # label(congruence). [clausify(5)]. 0.009/0.009 17 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(6)]. 0.009/0.009 18 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(7)]. 0.009/0.009 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.009/0.009 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(10)]. 0.009/0.009 23 -->*_s0(x,y) | -->*_s0(z,y) # label(goal) # answer(goal). [deny(11)]. 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 formulas(demodulators). 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 ============================== end of clauses for search ============= 0.009/0.009 0.009/0.009 ============================== SEARCH ================================ 0.009/0.009 0.009/0.009 % Starting search at 0.00 seconds. 0.009/0.009 0.009/0.009 given #1 (I,wt=10): 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 0.009/0.009 0.009/0.009 given #2 (I,wt=10): 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 0.009/0.009 0.009/0.009 given #3 (I,wt=12): 14 -->_s0(x,y) | ->_s0(h(x,z,u),h(y,z,u)) # label(congruence). [clausify(3)]. 0.009/0.009 0.009/0.009 given #4 (I,wt=12): 15 -->_s0(x,y) | ->_s0(h(z,x,u),h(z,y,u)) # label(congruence). [clausify(4)]. 0.009/0.009 0.009/0.009 given #5 (I,wt=12): 16 -->_s0(x,y) | ->_s0(h(z,u,x),h(z,u,y)) # label(congruence). [clausify(5)]. 0.009/0.009 0.009/0.009 given #6 (I,wt=10): 17 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(6)]. 0.009/0.009 0.009/0.009 given #7 (I,wt=10): 18 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(7)]. 0.009/0.009 0.009/0.009 given #8 (I,wt=3): 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.009/0.009 0.009/0.009 given #9 (I,wt=9): 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(10)]. 0.009/0.009 0.009/0.009 given #10 (I,wt=6): 23 -->*_s0(x,y) | -->*_s0(z,y) # label(goal) # answer(goal). [deny(11)]. 0.009/0.009 0.009/0.009 ============================== PROOF ================================= 0.009/0.009 0.009/0.009 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 0.009/0.009 % Length of proof is 5. 0.009/0.009 % Level of proof is 3. 0.009/0.009 % Maximum clause weight is 6.000. 0.009/0.009 % Given clauses 10. 0.009/0.009 0.009/0.009 11 (exists x6 exists x7 exists x8 (->*_s0(x6,x8) & ->*_s0(x7,x8))) # label(goal) # label(non_clause) # label(goal). [goal]. 0.009/0.009 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.009/0.009 23 -->*_s0(x,y) | -->*_s0(z,y) # label(goal) # answer(goal). [deny(11)]. 0.009/0.009 24 -->*_s0(x,y) # answer(goal). [resolve(23,a,21,a)]. 0.009/0.009 25 $F # answer(goal). [resolve(24,a,21,a)]. 0.009/0.009 0.009/0.009 ============================== end of proof ========================== 0.009/0.009 0.009/0.009 ============================== STATISTICS ============================ 0.009/0.009 0.009/0.009 Given=10. Generated=14. Kept=13. proofs=1. 0.009/0.009 Usable=10. Sos=0. Demods=0. Limbo=0, Disabled=14. Hints=0. 0.009/0.009 Kept_by_rule=0, Deleted_by_rule=0. 0.009/0.009 Forward_subsumed=1. Back_subsumed=2. 0.009/0.009 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 0.009/0.009 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 0.009/0.009 Demod_attempts=0. Demod_rewrites=0. 0.009/0.009 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 0.009/0.009 Nonunit_fsub_feature_tests=6. Nonunit_bsub_feature_tests=24. 0.009/0.009 Megabytes=0.07. 0.009/0.009 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.009/0.009 0.009/0.009 ============================== end of statistics ===================== 0.009/0.009 0.009/0.009 ============================== end of search ========================= 0.009/0.009 0.009/0.009 THEOREM PROVED 0.009/0.009 0.009/0.009 Exiting with 1 proof. 0.009/0.009 0.009/0.009 Process 20863 exit (max_proofs) Wed Mar 9 09:31:21 2022 0.009/0.009 0.009/0.009 0.009/0.009 The problem is feasible. 0.009/0.009 0.009/0.009 0.009/0.009 CRule InfChecker Info: 0.009/0.009 h(x,x',x'') -> c | x ->* y, x' ->* y, x'' ->* y 0.009/0.009 Rule remains 0.009/0.009 Proof: 0.009/0.009 NO 0.009/0.009 0.009/0.009 Problem 1: 0.009/0.009 0.009/0.009 Infeasibility Problem: 0.009/0.009 [(VAR vNonEmpty x x' x'' y vNonEmpty x x' x'' y) 0.009/0.009 (STRATEGY CONTEXTSENSITIVE 0.009/0.009 (f 1 2) 0.009/0.009 (h 1 2 3) 0.009/0.009 (c) 0.009/0.009 (fSNonEmpty) 0.009/0.009 (g 1 2) 0.009/0.009 ) 0.009/0.009 (RULES 0.009/0.009 f(x,x') -> g(y,y) | x ->* y, x' ->* y 0.009/0.009 h(x,x',x'') -> c | x ->* y, x' ->* y, x'' ->* y 0.009/0.009 )] 0.009/0.009 0.009/0.009 Infeasibility Conditions: 0.009/0.009 x ->* y, x' ->* y, x'' ->* y 0.009/0.009 0.009/0.009 Problem 1: 0.009/0.009 0.009/0.009 Obtaining a proof using Prover9: 0.009/0.009 0.009/0.009 -> Prover9 Output: 0.009/0.009 ============================== Prover9 =============================== 0.009/0.009 Prover9 (64) version 2009-11A, November 2009. 0.009/0.009 Process 20885 was started by ubuntu on ubuntu, 0.009/0.009 Wed Mar 9 09:31:21 2022 0.009/0.009 The command was "./prover9 -f /tmp/prover920876-0.in". 0.009/0.009 ============================== end of head =========================== 0.009/0.009 0.009/0.009 ============================== INPUT ================================= 0.009/0.009 0.009/0.009 % Reading from file /tmp/prover920876-0.in 0.009/0.009 0.009/0.009 assign(max_seconds,20). 0.009/0.009 0.009/0.009 formulas(assumptions). 0.009/0.009 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 0.009/0.009 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 0.009/0.009 ->_s0(x1,y) -> ->_s0(h(x1,x2,x3),h(y,x2,x3)) # label(congruence). 0.009/0.009 ->_s0(x2,y) -> ->_s0(h(x1,x2,x3),h(x1,y,x3)) # label(congruence). 0.009/0.009 ->_s0(x3,y) -> ->_s0(h(x1,x2,x3),h(x1,x2,y)) # label(congruence). 0.009/0.009 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence). 0.009/0.009 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence). 0.009/0.009 ->*_s0(x1,x4) & ->*_s0(x2,x4) -> ->_s0(f(x1,x2),g(x4,x4)) # label(replacement). 0.009/0.009 ->*_s0(x1,x4) & ->*_s0(x2,x4) & ->*_s0(x3,x4) -> ->_s0(h(x1,x2,x3),c) # label(replacement). 0.009/0.009 ->*_s0(x,x) # label(reflexivity). 0.009/0.009 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 formulas(goals). 0.009/0.009 (exists x6 exists x7 exists x8 exists x9 (->*_s0(x6,x9) & ->*_s0(x7,x9) & ->*_s0(x8,x9))) # label(goal). 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 ============================== end of input ========================== 0.009/0.009 0.009/0.009 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.009/0.009 0.009/0.009 % Formulas that are not ordinary clauses: 0.009/0.009 1 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.009/0.009 2 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.009/0.009 3 ->_s0(x1,y) -> ->_s0(h(x1,x2,x3),h(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 0.009/0.009 4 ->_s0(x2,y) -> ->_s0(h(x1,x2,x3),h(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 0.009/0.009 5 ->_s0(x3,y) -> ->_s0(h(x1,x2,x3),h(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 0.009/0.009 6 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.009/0.009 7 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.009/0.009 8 ->*_s0(x1,x4) & ->*_s0(x2,x4) -> ->_s0(f(x1,x2),g(x4,x4)) # label(replacement) # label(non_clause). [assumption]. 0.009/0.009 9 ->*_s0(x1,x4) & ->*_s0(x2,x4) & ->*_s0(x3,x4) -> ->_s0(h(x1,x2,x3),c) # label(replacement) # label(non_clause). [assumption]. 0.009/0.009 10 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 0.009/0.009 11 (exists x6 exists x7 exists x8 exists x9 (->*_s0(x6,x9) & ->*_s0(x7,x9) & ->*_s0(x8,x9))) # label(goal) # label(non_clause) # label(goal). [goal]. 0.009/0.009 0.009/0.009 ============================== end of process non-clausal formulas === 0.009/0.009 0.009/0.009 ============================== PROCESS INITIAL CLAUSES =============== 0.009/0.009 0.009/0.009 % Clauses before input processing: 0.009/0.009 0.009/0.009 formulas(usable). 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 formulas(sos). 0.009/0.009 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 0.009/0.009 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 0.009/0.009 -->_s0(x,y) | ->_s0(h(x,z,u),h(y,z,u)) # label(congruence). [clausify(3)]. 0.009/0.009 -->_s0(x,y) | ->_s0(h(z,x,u),h(z,y,u)) # label(congruence). [clausify(4)]. 0.009/0.009 -->_s0(x,y) | ->_s0(h(z,u,x),h(z,u,y)) # label(congruence). [clausify(5)]. 0.009/0.009 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(6)]. 0.009/0.009 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(7)]. 0.009/0.009 -->*_s0(x,y) | -->*_s0(z,y) | ->_s0(f(x,z),g(y,y)) # label(replacement). [clausify(8)]. 0.009/0.009 -->*_s0(x,y) | -->*_s0(z,y) | -->*_s0(u,y) | ->_s0(h(x,z,u),c) # label(replacement). [clausify(9)]. 0.009/0.009 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.009/0.009 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(10)]. 0.009/0.009 -->*_s0(x,y) | -->*_s0(z,y) | -->*_s0(u,y) # label(goal). [deny(11)]. 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 formulas(demodulators). 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 ============================== PREDICATE ELIMINATION ================= 0.009/0.009 0.009/0.009 No predicates eliminated. 0.009/0.009 0.009/0.009 ============================== end predicate elimination ============= 0.009/0.009 0.009/0.009 Auto_denials: 0.009/0.009 % copying label goal to answer in negative clause 0.009/0.009 0.009/0.009 Term ordering decisions: 0.009/0.009 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 0.009/0.009 Function symbol precedence: function_order([ c, f, g, h ]). 0.009/0.009 After inverse_order: (no changes). 0.009/0.009 Unfolding symbols: (none). 0.009/0.009 0.009/0.009 Auto_inference settings: 0.009/0.009 % set(neg_binary_resolution). % (HNE depth_diff=-9) 0.009/0.009 % clear(ordered_res). % (HNE depth_diff=-9) 0.009/0.009 % set(ur_resolution). % (HNE depth_diff=-9) 0.009/0.009 % set(ur_resolution) -> set(pos_ur_resolution). 0.009/0.009 % set(ur_resolution) -> set(neg_ur_resolution). 0.009/0.009 0.009/0.009 Auto_process settings: 0.009/0.009 % set(unit_deletion). % (Horn set with negative nonunits) 0.009/0.009 0.009/0.009 kept: 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 0.009/0.009 kept: 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 0.009/0.009 kept: 14 -->_s0(x,y) | ->_s0(h(x,z,u),h(y,z,u)) # label(congruence). [clausify(3)]. 0.009/0.009 kept: 15 -->_s0(x,y) | ->_s0(h(z,x,u),h(z,y,u)) # label(congruence). [clausify(4)]. 0.009/0.009 kept: 16 -->_s0(x,y) | ->_s0(h(z,u,x),h(z,u,y)) # label(congruence). [clausify(5)]. 0.009/0.009 kept: 17 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(6)]. 0.009/0.009 kept: 18 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(7)]. 0.009/0.009 kept: 19 -->*_s0(x,y) | -->*_s0(z,y) | ->_s0(f(x,z),g(y,y)) # label(replacement). [clausify(8)]. 0.009/0.009 kept: 20 -->*_s0(x,y) | -->*_s0(z,y) | -->*_s0(u,y) | ->_s0(h(x,z,u),c) # label(replacement). [clausify(9)]. 0.009/0.009 kept: 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.009/0.009 kept: 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(10)]. 0.009/0.009 kept: 23 -->*_s0(x,y) | -->*_s0(z,y) | -->*_s0(u,y) # label(goal) # answer(goal). [deny(11)]. 0.009/0.009 0.009/0.009 ============================== end of process initial clauses ======== 0.009/0.009 0.009/0.009 ============================== CLAUSES FOR SEARCH ==================== 0.009/0.009 0.009/0.009 % Clauses after input processing: 0.009/0.009 0.009/0.009 formulas(usable). 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 formulas(sos). 0.009/0.009 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 0.009/0.009 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 0.009/0.009 14 -->_s0(x,y) | ->_s0(h(x,z,u),h(y,z,u)) # label(congruence). [clausify(3)]. 0.009/0.009 15 -->_s0(x,y) | ->_s0(h(z,x,u),h(z,y,u)) # label(congruence). [clausify(4)]. 0.009/0.009 16 -->_s0(x,y) | ->_s0(h(z,u,x),h(z,u,y)) # label(congruence). [clausify(5)]. 0.009/0.009 17 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(6)]. 0.009/0.009 18 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(7)]. 0.009/0.009 19 -->*_s0(x,y) | -->*_s0(z,y) | ->_s0(f(x,z),g(y,y)) # label(replacement). [clausify(8)]. 0.009/0.009 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.009/0.009 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(10)]. 0.009/0.009 23 -->*_s0(x,y) | -->*_s0(z,y) | -->*_s0(u,y) # label(goal) # answer(goal). [deny(11)]. 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 formulas(demodulators). 0.009/0.009 end_of_list. 0.009/0.009 0.009/0.009 ============================== end of clauses for search ============= 0.009/0.009 0.009/0.009 ============================== SEARCH ================================ 0.009/0.009 0.009/0.009 % Starting search at 0.00 seconds. 0.009/0.009 0.009/0.009 given #1 (I,wt=10): 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 0.009/0.009 0.009/0.009 given #2 (I,wt=10): 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 0.009/0.009 0.009/0.009 given #3 (I,wt=12): 14 -->_s0(x,y) | ->_s0(h(x,z,u),h(y,z,u)) # label(congruence). [clausify(3)]. 0.009/0.009 0.009/0.009 given #4 (I,wt=12): 15 -->_s0(x,y) | ->_s0(h(z,x,u),h(z,y,u)) # label(congruence). [clausify(4)]. 0.009/0.009 0.009/0.009 given #5 (I,wt=12): 16 -->_s0(x,y) | ->_s0(h(z,u,x),h(z,u,y)) # label(congruence). [clausify(5)]. 0.009/0.009 0.009/0.009 given #6 (I,wt=10): 17 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(6)]. 0.009/0.009 0.009/0.009 given #7 (I,wt=10): 18 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(7)]. 0.009/0.009 0.009/0.009 given #8 (I,wt=13): 19 -->*_s0(x,y) | -->*_s0(z,y) | ->_s0(f(x,z),g(y,y)) # label(replacement). [clausify(8)]. 0.009/0.009 0.009/0.009 given #9 (I,wt=3): 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.009/0.009 0.009/0.009 given #10 (I,wt=9): 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(10)]. 0.009/0.009 0.009/0.009 given #11 (I,wt=9): 23 -->*_s0(x,y) | -->*_s0(z,y) | -->*_s0(u,y) # label(goal) # answer(goal). [deny(11)]. 0.009/0.009 0.009/0.009 ============================== PROOF ================================= 0.009/0.009 0.009/0.009 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 0.009/0.009 % Length of proof is 5. 0.009/0.009 % Level of proof is 3. 0.009/0.009 % Maximum clause weight is 9.000. 0.009/0.009 % Given clauses 11. 0.009/0.009 0.009/0.009 11 (exists x6 exists x7 exists x8 exists x9 (->*_s0(x6,x9) & ->*_s0(x7,x9) & ->*_s0(x8,x9))) # label(goal) # label(non_clause) # label(goal). [goal]. 0.009/0.009 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.009/0.009 23 -->*_s0(x,y) | -->*_s0(z,y) | -->*_s0(u,y) # label(goal) # answer(goal). [deny(11)]. 0.009/0.009 26 -->*_s0(x,y) # answer(goal). [ur(23,b,21,a,c,21,a)]. 0.009/0.009 27 $F # answer(goal). [resolve(26,a,21,a)]. 0.009/0.009 0.009/0.009 ============================== end of proof ========================== 0.009/0.009 0.009/0.009 ============================== STATISTICS ============================ 0.009/0.009 0.009/0.009 Given=11. Generated=21. Kept=15. proofs=1. 0.009/0.009 Usable=11. Sos=1. Demods=0. Limbo=1, Disabled=13. Hints=0. 0.009/0.009 Kept_by_rule=0, Deleted_by_rule=0. 0.009/0.009 Forward_subsumed=6. Back_subsumed=1. 0.009/0.009 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 0.009/0.009 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 0.009/0.009 Demod_attempts=0. Demod_rewrites=0. 0.009/0.009 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 0.009/0.009 Nonunit_fsub_feature_tests=10. Nonunit_bsub_feature_tests=24. 0.009/0.009 Megabytes=0.08. 0.009/0.009 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.009/0.009 0.009/0.009 ============================== end of statistics ===================== 0.009/0.009 0.009/0.009 ============================== end of search ========================= 0.009/0.009 0.009/0.009 THEOREM PROVED 0.009/0.009 0.009/0.009 Exiting with 1 proof. 0.009/0.009 0.009/0.009 Process 20885 exit (max_proofs) Wed Mar 9 09:31:21 2022 0.009/0.009 0.009/0.009 0.009/0.009 The problem is feasible. 0.009/0.009 0.009/0.009 0.009/0.009 Problem 1: 0.009/0.009 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.009/0.009 Confluence Problem: 0.009/0.009 (VAR x x' x'' y) 0.009/0.009 (REPLACEMENT-MAP 0.009/0.009 (f 1, 2) 0.009/0.009 (h 1, 2, 3) 0.009/0.009 (c) 0.009/0.009 (fSNonEmpty) 0.009/0.009 (g 1, 2) 0.009/0.009 ) 0.009/0.009 (RULES 0.009/0.009 f(x,x') -> g(y,y) | x ->* y, x' ->* y 0.009/0.009 h(x,x',x'') -> c | x ->* y, x' ->* y, x'' ->* y 0.009/0.009 ) 0.009/0.009 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.009/0.009 0.009/0.009 Critical Pairs Processor: 0.009/0.009 -> Rules: 0.009/0.009 f(x,x') -> g(y,y) | x ->* y, x' ->* y 0.009/0.009 h(x,x',x'') -> c | x ->* y, x' ->* y, x'' ->* y 0.009/0.009 -> Vars: 0.009/0.009 "x", "x'", "x''", "y" 0.009/0.009 0.009/0.009 -> Rlps: 0.009/0.009 crule: f(x,x') -> g(y,y) | x ->* y, x' ->* y, id: 1, possubterms: f(x,x')-> [] 0.009/0.009 crule: h(x,x',x'') -> c | x ->* y, x' ->* y, x'' ->* y, id: 2, possubterms: h(x,x',x'')-> [] 0.009/0.009 0.009/0.009 -> Unifications: 0.009/0.009 0.009/0.009 0.009/0.009 -> Critical pairs info: 0.009/0.009 0.009/0.009 0.009/0.009 -> Problem conclusions: 0.009/0.009 Left linear, Not right linear, Not linear 0.009/0.009 Weakly orthogonal, Almost orthogonal, Orthogonal 0.009/0.009 CTRS Type: 3 0.009/0.009 Deterministic, Strongly deterministic 0.009/0.009 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.009/0.009 Maybe right-stable CTRS, Overlay CTRS 0.009/0.009 Maybe normal CTRS, Maybe almost normal CTRS 0.009/0.009 Maybe terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 0.009/0.009 Maybe level confluent 0.009/0.009 Maybe confluent 0.009/0.009 0.009/0.009 Problem 1: 0.009/0.009 Clean Infeasible CCPs Processor: 0.009/0.009 0.009/0.009 0.009/0.009 -> Problem conclusions: 0.009/0.009 Left linear, Not right linear, Not linear 0.009/0.009 Weakly orthogonal, Almost orthogonal, Orthogonal 0.009/0.009 CTRS Type: 3 0.009/0.009 Deterministic, Strongly deterministic 0.009/0.009 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.009/0.009 Maybe right-stable CTRS, Overlay CTRS 0.009/0.009 Maybe normal CTRS, Maybe almost normal CTRS 0.009/0.009 Maybe terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 0.009/0.009 Maybe level confluent 0.009/0.009 Maybe confluent 0.009/0.009 0.009/0.009 Resulting CCPs: 0.009/0.009 No CCPs left 0.009/0.009 0.009/0.009 Problem 1: 0.009/0.009 Operation Termination Processor: 0.009/0.009 Operational terminating? 0.009/0.009 YES 0.009/0.009 0.009/0.009 Problem 1: 0.009/0.009 0.009/0.009 (VAR vu95NonEmpty x xu39 xu39u39 y) 0.009/0.009 (RULES 0.009/0.009 f(x,xu39) -> g(y,y) | x ->* y, xu39 ->* y 0.009/0.009 h(x,xu39,xu39u39) -> c | x ->* y, xu39 ->* y, xu39u39 ->* y 0.009/0.009 ) 0.009/0.009 0.009/0.009 Problem 1: 0.009/0.009 Valid CTRS Processor: 0.009/0.009 -> Rules: 0.009/0.009 f(x,xu39) -> g(y,y) | x ->* y, xu39 ->* y 0.009/0.009 h(x,xu39,xu39u39) -> c | x ->* y, xu39 ->* y, xu39u39 ->* y 0.009/0.009 -> The system is a deterministic 3-CTRS. 0.009/0.009 0.009/0.009 Problem 1: 0.009/0.009 0.009/0.009 2D Dependency Pair Processor: 0.009/0.009 0.009/0.009 Conditional Termination Problem 1: 0.009/0.009 -> Pairs: 0.009/0.009 Empty 0.009/0.009 -> QPairs: 0.009/0.009 Empty 0.009/0.009 -> Rules: 0.009/0.009 f(x,xu39) -> g(y,y) | x ->* y, xu39 ->* y 0.009/0.009 h(x,xu39,xu39u39) -> c | x ->* y, xu39 ->* y, xu39u39 ->* y 0.009/0.009 0.009/0.009 Conditional Termination Problem 2: 0.009/0.009 -> Pairs: 0.009/0.009 Empty 0.009/0.009 -> QPairs: 0.009/0.009 Empty 0.009/0.009 -> Rules: 0.009/0.009 f(x,xu39) -> g(y,y) | x ->* y, xu39 ->* y 0.009/0.009 h(x,xu39,xu39u39) -> c | x ->* y, xu39 ->* y, xu39u39 ->* y 0.009/0.009 0.009/0.009 0.009/0.009 The problem is decomposed in 2 subproblems. 0.009/0.009 0.009/0.009 Problem 1.1: 0.009/0.009 0.009/0.009 SCC Processor: 0.009/0.009 -> Pairs: 0.009/0.009 Empty 0.009/0.009 -> QPairs: 0.009/0.009 Empty 0.009/0.009 -> Rules: 0.009/0.009 f(x,xu39) -> g(y,y) | x ->* y, xu39 ->* y 0.009/0.009 h(x,xu39,xu39u39) -> c | x ->* y, xu39 ->* y, xu39u39 ->* y 0.009/0.009 ->Strongly Connected Components: 0.009/0.009 There is no strongly connected component 0.009/0.009 0.009/0.009 The problem is finite. 0.009/0.009 0.009/0.009 Problem 1.2: 0.009/0.009 0.009/0.009 SCC Processor: 0.009/0.009 -> Pairs: 0.009/0.009 Empty 0.009/0.009 -> QPairs: 0.009/0.009 Empty 0.009/0.009 -> Rules: 0.009/0.009 f(x,xu39) -> g(y,y) | x ->* y, xu39 ->* y 0.009/0.009 h(x,xu39,xu39u39) -> c | x ->* y, xu39 ->* y, xu39u39 ->* y 0.009/0.009 ->Strongly Connected Components: 0.009/0.009 There is no strongly connected component 0.009/0.009 0.009/0.009 The problem is finite. 0.009/0.009 0.009/0.009 0.009/0.009 -> Problem conclusions: 0.009/0.009 Left linear, Not right linear, Not linear 0.009/0.009 Weakly orthogonal, Almost orthogonal, Orthogonal 0.009/0.009 CTRS Type: 3 0.009/0.009 Deterministic, Strongly deterministic 0.009/0.009 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.009/0.009 Maybe right-stable CTRS, Overlay CTRS 0.009/0.009 Maybe normal CTRS, Maybe almost normal CTRS 0.009/0.009 Terminating CTRS, Operational terminating CTRS, Joinable CCPs 0.009/0.009 Maybe level confluent 0.009/0.009 Confluent 0.009/0.009 0.009/0.009 The problem is joinable. 0.009/0.009 0.05user 0.02system 0:00.09elapsed 84%CPU (0avgtext+0avgdata 12428maxresident)k 0.009/0.009 8inputs+0outputs (0major+6823minor)pagefaults 0swaps