22.003/22.003 NO 22.003/22.003 22.003/22.003 Problem 1: 22.003/22.003 22.003/22.003 22.003/22.003 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 22.003/22.003 Confluence Problem: 22.003/22.003 (VAR vNonEmpty x) 22.003/22.003 (REPLACEMENT-MAP 22.003/22.003 (a) 22.003/22.003 (b) 22.003/22.003 (c) 22.003/22.003 (d) 22.003/22.003 (f 1) 22.003/22.003 (g 1, 2) 22.003/22.003 (h 1) 22.003/22.003 (A) 22.003/22.003 (B) 22.003/22.003 (C) 22.003/22.003 (e) 22.003/22.003 (fSNonEmpty) 22.003/22.003 (i 1, 2) 22.003/22.003 (k) 22.003/22.003 ) 22.003/22.003 (RULES 22.003/22.003 a -> c 22.003/22.003 a -> d 22.003/22.003 b -> c 22.003/22.003 b -> d 22.003/22.003 c -> e 22.003/22.003 c -> k 22.003/22.003 d -> k 22.003/22.003 f(x) -> x | x ->* e 22.003/22.003 g(x,x) -> C | A ->* B 22.003/22.003 h(x) -> i(x,x) 22.003/22.003 ) 22.003/22.003 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 22.003/22.003 22.003/22.003 22.003/22.003 Problem 1: 22.003/22.003 22.003/22.003 Inlining of Conditions Processor [STERN17]: 22.003/22.003 22.003/22.003 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 22.003/22.003 Confluence Problem: 22.003/22.003 (VAR vNonEmpty x) 22.003/22.003 (REPLACEMENT-MAP 22.003/22.003 (a) 22.003/22.003 (b) 22.003/22.003 (c) 22.003/22.003 (d) 22.003/22.003 (f 1) 22.003/22.003 (g 1, 2) 22.003/22.003 (h 1) 22.003/22.003 (A) 22.003/22.003 (B) 22.003/22.003 (C) 22.003/22.003 (e) 22.003/22.003 (fSNonEmpty) 22.003/22.003 (i 1, 2) 22.003/22.003 (k) 22.003/22.003 ) 22.003/22.003 (RULES 22.003/22.003 a -> c 22.003/22.003 a -> d 22.003/22.003 b -> c 22.003/22.003 b -> d 22.003/22.003 c -> e 22.003/22.003 c -> k 22.003/22.003 d -> k 22.003/22.003 f(x) -> x | x ->* e 22.003/22.003 g(x,x) -> C | A ->* B 22.003/22.003 h(x) -> i(x,x) 22.003/22.003 ) 22.003/22.003 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 22.003/22.003 22.003/22.003 22.003/22.003 Problem 1: 22.003/22.003 22.003/22.003 Clean CTRS Processor: 22.003/22.003 22.003/22.003 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 22.003/22.003 Confluence Problem: 22.003/22.003 (VAR x) 22.003/22.003 (REPLACEMENT-MAP 22.003/22.003 (a) 22.003/22.003 (b) 22.003/22.003 (c) 22.003/22.003 (d) 22.003/22.003 (f 1) 22.003/22.003 (g 1, 2) 22.003/22.003 (h 1) 22.003/22.003 (A) 22.003/22.003 (B) 22.003/22.003 (C) 22.003/22.003 (e) 22.003/22.003 (fSNonEmpty) 22.003/22.003 (i 1, 2) 22.003/22.003 (k) 22.003/22.003 ) 22.003/22.003 (RULES 22.003/22.003 a -> c 22.003/22.003 a -> d 22.003/22.003 b -> c 22.003/22.003 b -> d 22.003/22.003 c -> e 22.003/22.003 c -> k 22.003/22.003 d -> k 22.003/22.003 f(x) -> x | x ->* e 22.003/22.003 h(x) -> i(x,x) 22.003/22.003 ) 22.003/22.003 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 22.003/22.003 22.003/22.003 CRule InfChecker Info: 22.003/22.003 a -> c 22.003/22.003 Rule remains 22.003/22.003 Proof: 22.003/22.003 NO_CONDS 22.003/22.003 22.003/22.003 CRule InfChecker Info: 22.003/22.003 a -> d 22.003/22.003 Rule remains 22.003/22.003 Proof: 22.003/22.003 NO_CONDS 22.003/22.003 22.003/22.003 CRule InfChecker Info: 22.003/22.003 b -> c 22.003/22.003 Rule remains 22.003/22.003 Proof: 22.003/22.003 NO_CONDS 22.003/22.003 22.003/22.003 CRule InfChecker Info: 22.003/22.003 b -> d 22.003/22.003 Rule remains 22.003/22.003 Proof: 22.003/22.003 NO_CONDS 22.003/22.003 22.003/22.003 CRule InfChecker Info: 22.003/22.003 c -> e 22.003/22.003 Rule remains 22.003/22.003 Proof: 22.003/22.003 NO_CONDS 22.003/22.003 22.003/22.003 CRule InfChecker Info: 22.003/22.003 c -> k 22.003/22.003 Rule remains 22.003/22.003 Proof: 22.003/22.003 NO_CONDS 22.003/22.003 22.003/22.003 CRule InfChecker Info: 22.003/22.003 d -> k 22.003/22.003 Rule remains 22.003/22.003 Proof: 22.003/22.003 NO_CONDS 22.003/22.003 22.003/22.003 CRule InfChecker Info: 22.003/22.003 f(x) -> x | x ->* e 22.003/22.003 Rule remains 22.003/22.003 Proof: 22.003/22.003 NO 22.003/22.003 22.003/22.003 Problem 1: 22.003/22.003 22.003/22.003 Infeasibility Problem: 22.003/22.003 [(VAR vNonEmpty x vNonEmpty x) 22.003/22.003 (STRATEGY CONTEXTSENSITIVE 22.003/22.003 (a) 22.003/22.003 (b) 22.003/22.003 (c) 22.003/22.003 (d) 22.003/22.003 (f 1) 22.003/22.003 (g 1 2) 22.003/22.003 (h 1) 22.003/22.003 (A) 22.003/22.003 (B) 22.003/22.003 (C) 22.003/22.003 (e) 22.003/22.003 (fSNonEmpty) 22.003/22.003 (i 1 2) 22.003/22.003 (k) 22.003/22.003 ) 22.003/22.003 (RULES 22.003/22.003 a -> c 22.003/22.003 a -> d 22.003/22.003 b -> c 22.003/22.003 b -> d 22.003/22.003 c -> e 22.003/22.003 c -> k 22.003/22.003 d -> k 22.003/22.003 f(x) -> x | x ->* e 22.003/22.003 g(x,x) -> C | A ->* B 22.003/22.003 h(x) -> i(x,x) 22.003/22.003 )] 22.003/22.003 22.003/22.003 Infeasibility Conditions: 22.003/22.003 x ->* e 22.003/22.003 22.003/22.003 Problem 1: 22.003/22.003 22.003/22.003 Obtaining a proof using Prover9: 22.003/22.003 22.003/22.003 -> Prover9 Output: 22.003/22.003 ============================== Prover9 =============================== 22.003/22.003 Prover9 (64) version 2009-11A, November 2009. 22.003/22.003 Process 61624 was started by ubuntu on ubuntu, 22.003/22.003 Wed Mar 9 09:48:56 2022 22.003/22.003 The command was "./prover9 -f /tmp/prover961615-0.in". 22.003/22.003 ============================== end of head =========================== 22.003/22.003 22.003/22.003 ============================== INPUT ================================= 22.003/22.003 22.003/22.003 % Reading from file /tmp/prover961615-0.in 22.003/22.003 22.003/22.003 assign(max_seconds,20). 22.003/22.003 22.003/22.003 formulas(assumptions). 22.003/22.003 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 22.003/22.003 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence). 22.003/22.003 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence). 22.003/22.003 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 22.003/22.003 ->_s0(x1,y) -> ->_s0(i(x1,x2),i(y,x2)) # label(congruence). 22.003/22.003 ->_s0(x2,y) -> ->_s0(i(x1,x2),i(x1,y)) # label(congruence). 22.003/22.003 ->_s0(a,c) # label(replacement). 22.003/22.003 ->_s0(a,d) # label(replacement). 22.003/22.003 ->_s0(b,c) # label(replacement). 22.003/22.003 ->_s0(b,d) # label(replacement). 22.003/22.003 ->_s0(c,e) # label(replacement). 22.003/22.003 ->_s0(c,k) # label(replacement). 22.003/22.003 ->_s0(d,k) # label(replacement). 22.003/22.003 ->*_s0(x1,e) -> ->_s0(f(x1),x1) # label(replacement). 22.003/22.003 ->*_s0(A,B) -> ->_s0(g(x1,x1),C) # label(replacement). 22.003/22.003 ->_s0(h(x1),i(x1,x1)) # label(replacement). 22.003/22.003 ->*_s0(x,x) # label(reflexivity). 22.003/22.003 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 22.003/22.003 end_of_list. 22.003/22.003 22.003/22.003 formulas(goals). 22.003/22.003 (exists x3 ->*_s0(x3,e)) # label(goal). 22.003/22.003 end_of_list. 22.003/22.003 22.003/22.003 ============================== end of input ========================== 22.003/22.003 22.003/22.003 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 22.003/22.003 22.003/22.003 % Formulas that are not ordinary clauses: 22.003/22.003 1 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 2 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 3 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 4 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 5 ->_s0(x1,y) -> ->_s0(i(x1,x2),i(y,x2)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 6 ->_s0(x2,y) -> ->_s0(i(x1,x2),i(x1,y)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 7 ->*_s0(x1,e) -> ->_s0(f(x1),x1) # label(replacement) # label(non_clause). [assumption]. 22.003/22.003 8 ->*_s0(A,B) -> ->_s0(g(x1,x1),C) # label(replacement) # label(non_clause). [assumption]. 22.003/22.003 9 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 22.003/22.003 10 (exists x3 ->*_s0(x3,e)) # label(goal) # label(non_clause) # label(goal). [goal]. 22.003/22.003 22.003/22.003 ============================== end of process non-clausal formulas === 22.003/22.003 22.003/22.003 ============================== PROCESS INITIAL CLAUSES =============== 22.003/22.003 22.003/22.003 % Clauses before input processing: 22.003/22.003 22.003/22.003 formulas(usable). 22.003/22.003 end_of_list. 22.003/22.003 22.003/22.003 formulas(sos). 22.003/22.003 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 22.003/22.003 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(2)]. 22.003/22.003 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(3)]. 22.003/22.003 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 22.003/22.003 -->_s0(x,y) | ->_s0(i(x,z),i(y,z)) # label(congruence). [clausify(5)]. 22.003/22.003 -->_s0(x,y) | ->_s0(i(z,x),i(z,y)) # label(congruence). [clausify(6)]. 22.003/22.003 ->_s0(a,c) # label(replacement). [assumption]. 22.003/22.003 ->_s0(a,d) # label(replacement). [assumption]. 22.003/22.003 ->_s0(b,c) # label(replacement). [assumption]. 22.003/22.003 ->_s0(b,d) # label(replacement). [assumption]. 22.003/22.003 ->_s0(c,e) # label(replacement). [assumption]. 22.003/22.003 ->_s0(c,k) # label(replacement). [assumption]. 22.003/22.003 ->_s0(d,k) # label(replacement). [assumption]. 22.003/22.003 -->*_s0(x,e) | ->_s0(f(x),x) # label(replacement). [clausify(7)]. 22.003/22.003 -->*_s0(A,B) | ->_s0(g(x,x),C) # label(replacement). [clausify(8)]. 22.003/22.003 ->_s0(h(x),i(x,x)) # label(replacement). [assumption]. 22.003/22.003 ->*_s0(x,x) # label(reflexivity). [assumption]. 22.003/22.003 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 22.003/22.003 -->*_s0(x,e) # label(goal). [deny(10)]. 22.003/22.003 end_of_list. 22.003/22.003 22.003/22.003 formulas(demodulators). 22.003/22.003 end_of_list. 22.003/22.003 22.003/22.003 ============================== PREDICATE ELIMINATION ================= 22.003/22.003 22.003/22.003 No predicates eliminated. 22.003/22.003 22.003/22.003 ============================== end predicate elimination ============= 22.003/22.003 22.003/22.003 Auto_denials: 22.003/22.003 % copying label goal to answer in negative clause 22.003/22.003 22.003/22.003 Term ordering decisions: 22.003/22.003 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 22.003/22.003 Function symbol precedence: function_order([ c, d, e, k, a, b, B, C, A, i, g, f, h ]). 22.003/22.003 After inverse_order: (no changes). 22.003/22.003 Unfolding symbols: (none). 22.003/22.003 22.003/22.003 Auto_inference settings: 22.003/22.003 % set(neg_binary_resolution). % (HNE depth_diff=-8) 22.003/22.003 % clear(ordered_res). % (HNE depth_diff=-8) 22.003/22.003 % set(ur_resolution). % (HNE depth_diff=-8) 22.003/22.003 % set(ur_resolution) -> set(pos_ur_resolution). 22.003/22.003 % set(ur_resolution) -> set(neg_ur_resolution). 22.003/22.003 22.003/22.003 Auto_process settings: (no changes). 22.003/22.003 22.003/22.003 kept: 11 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 22.003/22.003 kept: 12 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(2)]. 22.003/22.003 kept: 13 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(3)]. 22.003/22.003 kept: 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 22.003/22.003 kept: 15 -->_s0(x,y) | ->_s0(i(x,z),i(y,z)) # label(congruence). [clausify(5)]. 22.003/22.003 kept: 16 -->_s0(x,y) | ->_s0(i(z,x),i(z,y)) # label(congruence). [clausify(6)]. 22.003/22.003 kept: 17 ->_s0(a,c) # label(replacement). [assumption]. 22.003/22.003 kept: 18 ->_s0(a,d) # label(replacement). [assumption]. 22.003/22.003 kept: 19 ->_s0(b,c) # label(replacement). [assumption]. 22.003/22.003 kept: 20 ->_s0(b,d) # label(replacement). [assumption]. 22.003/22.003 kept: 21 ->_s0(c,e) # label(replacement). [assumption]. 22.003/22.003 kept: 22 ->_s0(c,k) # label(replacement). [assumption]. 22.003/22.003 kept: 23 ->_s0(d,k) # label(replacement). [assumption]. 22.003/22.003 kept: 24 -->*_s0(x,e) | ->_s0(f(x),x) # label(replacement). [clausify(7)]. 22.003/22.003 kept: 25 -->*_s0(A,B) | ->_s0(g(x,x),C) # label(replacement). [clausify(8)]. 22.003/22.003 kept: 26 ->_s0(h(x),i(x,x)) # label(replacement). [assumption]. 22.003/22.003 kept: 27 ->*_s0(x,x) # label(reflexivity). [assumption]. 22.003/22.003 kept: 28 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 22.003/22.003 kept: 29 -->*_s0(x,e) # label(goal) # answer(goal). [deny(10)]. 22.003/22.003 22.003/22.003 ============================== PROOF ================================= 22.003/22.003 22.003/22.003 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 22.003/22.003 % Length of proof is 4. 22.003/22.003 % Level of proof is 2. 22.003/22.003 % Maximum clause weight is 3.000. 22.003/22.003 % Given clauses 0. 22.003/22.003 22.003/22.003 10 (exists x3 ->*_s0(x3,e)) # label(goal) # label(non_clause) # label(goal). [goal]. 22.003/22.003 27 ->*_s0(x,x) # label(reflexivity). [assumption]. 22.003/22.003 29 -->*_s0(x,e) # label(goal) # answer(goal). [deny(10)]. 22.003/22.003 30 $F # answer(goal). [resolve(29,a,27,a)]. 22.003/22.003 22.003/22.003 ============================== end of proof ========================== 22.003/22.003 22.003/22.003 ============================== STATISTICS ============================ 22.003/22.003 22.003/22.003 Given=0. Generated=19. Kept=19. proofs=1. 22.003/22.003 Usable=0. Sos=0. Demods=0. Limbo=18, Disabled=19. Hints=0. 22.003/22.003 Kept_by_rule=0, Deleted_by_rule=0. 22.003/22.003 Forward_subsumed=0. Back_subsumed=0. 22.003/22.003 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 22.003/22.003 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 22.003/22.003 Demod_attempts=0. Demod_rewrites=0. 22.003/22.003 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 22.003/22.003 Nonunit_fsub_feature_tests=2. Nonunit_bsub_feature_tests=0. 22.003/22.003 Megabytes=0.07. 22.003/22.003 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 22.003/22.003 22.003/22.003 ============================== end of statistics ===================== 22.003/22.003 22.003/22.003 ============================== end of search ========================= 22.003/22.003 22.003/22.003 THEOREM PROVED 22.003/22.003 22.003/22.003 Exiting with 1 proof. 22.003/22.003 22.003/22.003 Process 61624 exit (max_proofs) Wed Mar 9 09:48:56 2022 22.003/22.003 22.003/22.003 22.003/22.003 The problem is feasible. 22.003/22.003 22.003/22.003 22.003/22.003 CRule InfChecker Info: 22.003/22.003 g(x,x) -> C | A ->* B 22.003/22.003 Rule deleted 22.003/22.003 Proof: 22.003/22.003 YES 22.003/22.003 22.003/22.003 Problem 1: 22.003/22.003 22.003/22.003 Infeasibility Problem: 22.003/22.003 [(VAR vNonEmpty x vNonEmpty) 22.003/22.003 (STRATEGY CONTEXTSENSITIVE 22.003/22.003 (a) 22.003/22.003 (b) 22.003/22.003 (c) 22.003/22.003 (d) 22.003/22.003 (f 1) 22.003/22.003 (g 1 2) 22.003/22.003 (h 1) 22.003/22.003 (A) 22.003/22.003 (B) 22.003/22.003 (C) 22.003/22.003 (e) 22.003/22.003 (fSNonEmpty) 22.003/22.003 (i 1 2) 22.003/22.003 (k) 22.003/22.003 ) 22.003/22.003 (RULES 22.003/22.003 a -> c 22.003/22.003 a -> d 22.003/22.003 b -> c 22.003/22.003 b -> d 22.003/22.003 c -> e 22.003/22.003 c -> k 22.003/22.003 d -> k 22.003/22.003 f(x) -> x | x ->* e 22.003/22.003 g(x,x) -> C | A ->* B 22.003/22.003 h(x) -> i(x,x) 22.003/22.003 )] 22.003/22.003 22.003/22.003 Infeasibility Conditions: 22.003/22.003 A ->* B 22.003/22.003 22.003/22.003 Problem 1: 22.003/22.003 22.003/22.003 Obtaining a model using Mace4: 22.003/22.003 22.003/22.003 -> Usable Rules: 22.003/22.003 Empty 22.003/22.003 22.003/22.003 -> Mace4 Output: 22.003/22.003 ============================== Mace4 ================================= 22.003/22.003 Mace4 (64) version 2009-11A, November 2009. 22.003/22.003 Process 61652 was started by ubuntu on ubuntu, 22.003/22.003 Wed Mar 9 09:48:56 2022 22.003/22.003 The command was "./mace4 -c -f /tmp/mace461639-2.in". 22.003/22.003 ============================== end of head =========================== 22.003/22.003 22.003/22.003 ============================== INPUT ================================= 22.003/22.003 22.003/22.003 % Reading from file /tmp/mace461639-2.in 22.003/22.003 22.003/22.003 assign(max_seconds,20). 22.003/22.003 22.003/22.003 formulas(assumptions). 22.003/22.003 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 22.003/22.003 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence). 22.003/22.003 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence). 22.003/22.003 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence). 22.003/22.003 ->(x1,y) -> ->(i(x1,x2),i(y,x2)) # label(congruence). 22.003/22.003 ->(x2,y) -> ->(i(x1,x2),i(x1,y)) # label(congruence). 22.003/22.003 ->*(x,x) # label(reflexivity). 22.003/22.003 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 22.003/22.003 end_of_list. 22.003/22.003 22.003/22.003 formulas(goals). 22.003/22.003 ->*(A,B) # label(goal). 22.003/22.003 end_of_list. 22.003/22.003 22.003/22.003 ============================== end of input ========================== 22.003/22.003 22.003/22.003 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 22.003/22.003 22.003/22.003 % Formulas that are not ordinary clauses: 22.003/22.003 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 2 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 3 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 4 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 5 ->(x1,y) -> ->(i(x1,x2),i(y,x2)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 6 ->(x2,y) -> ->(i(x1,x2),i(x1,y)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 7 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 22.003/22.003 8 ->*(A,B) # label(goal) # label(non_clause) # label(goal). [goal]. 22.003/22.003 22.003/22.003 ============================== end of process non-clausal formulas === 22.003/22.003 22.003/22.003 ============================== CLAUSES FOR SEARCH ==================== 22.003/22.003 22.003/22.003 formulas(mace4_clauses). 22.003/22.003 -->(x,y) | ->(f(x),f(y)) # label(congruence). 22.003/22.003 -->(x,y) | ->(g(x,z),g(y,z)) # label(congruence). 22.003/22.003 -->(x,y) | ->(g(z,x),g(z,y)) # label(congruence). 22.003/22.003 -->(x,y) | ->(h(x),h(y)) # label(congruence). 22.003/22.003 -->(x,y) | ->(i(x,z),i(y,z)) # label(congruence). 22.003/22.003 -->(x,y) | ->(i(z,x),i(z,y)) # label(congruence). 22.003/22.003 ->*(x,x) # label(reflexivity). 22.003/22.003 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 22.003/22.003 -->*(A,B) # label(goal). 22.003/22.003 end_of_list. 22.003/22.003 22.003/22.003 ============================== end of clauses for search ============= 22.003/22.003 22.003/22.003 % There are no natural numbers in the input. 22.003/22.003 22.003/22.003 ============================== DOMAIN SIZE 2 ========================= 22.003/22.003 22.003/22.003 ============================== MODEL ================================= 22.003/22.003 22.003/22.003 interpretation( 2, [number=1, seconds=0], [ 22.003/22.003 22.003/22.003 function(B, [ 0 ]), 22.003/22.003 22.003/22.003 function(A, [ 1 ]), 22.003/22.003 22.003/22.003 function(f(_), [ 0, 0 ]), 22.003/22.003 22.003/22.003 function(h(_), [ 0, 0 ]), 22.003/22.003 22.003/22.003 function(i(_,_), [ 22.003/22.003 0, 0, 22.003/22.003 0, 0 ]), 22.003/22.003 22.003/22.003 function(g(_,_), [ 22.003/22.003 0, 0, 22.003/22.003 0, 0 ]), 22.003/22.003 22.003/22.003 relation(->*(_,_), [ 22.003/22.003 1, 0, 22.003/22.003 0, 1 ]), 22.003/22.003 22.003/22.003 relation(->(_,_), [ 22.003/22.003 0, 0, 22.003/22.003 0, 0 ]) 22.003/22.003 ]). 22.003/22.003 22.003/22.003 ============================== end of model ========================== 22.003/22.003 22.003/22.003 ============================== STATISTICS ============================ 22.003/22.003 22.003/22.003 For domain size 2. 22.003/22.003 22.003/22.003 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 22.003/22.003 Ground clauses: seen=51, kept=47. 22.003/22.003 Selections=15, assignments=15, propagations=7, current_models=1. 22.003/22.003 Rewrite_terms=82, rewrite_bools=57, indexes=7. 22.003/22.003 Rules_from_neg_clauses=1, cross_offs=1. 22.003/22.003 22.003/22.003 ============================== end of statistics ===================== 22.003/22.003 22.003/22.003 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 22.003/22.003 22.003/22.003 Exiting with 1 model. 22.003/22.003 22.003/22.003 Process 61652 exit (max_models) Wed Mar 9 09:48:56 2022 22.003/22.003 The process finished Wed Mar 9 09:48:56 2022 22.003/22.003 22.003/22.003 22.003/22.003 Mace4 cooked interpretation: 22.003/22.003 22.003/22.003 22.003/22.003 22.003/22.003 The problem is infeasible. 22.003/22.003 22.003/22.003 22.003/22.003 CRule InfChecker Info: 22.003/22.003 h(x) -> i(x,x) 22.003/22.003 Rule remains 22.003/22.003 Proof: 22.003/22.003 NO_CONDS 22.003/22.003 22.003/22.003 Problem 1: 22.003/22.003 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 22.003/22.003 Confluence Problem: 22.003/22.003 (VAR x) 22.003/22.003 (REPLACEMENT-MAP 22.003/22.003 (a) 22.003/22.003 (b) 22.003/22.003 (c) 22.003/22.003 (d) 22.003/22.003 (f 1) 22.003/22.003 (g 1, 2) 22.003/22.003 (h 1) 22.003/22.003 (A) 22.003/22.003 (B) 22.003/22.003 (C) 22.003/22.003 (e) 22.003/22.003 (fSNonEmpty) 22.003/22.003 (i 1, 2) 22.003/22.003 (k) 22.003/22.003 ) 22.003/22.003 (RULES 22.003/22.003 a -> c 22.003/22.003 a -> d 22.003/22.003 b -> c 22.003/22.003 b -> d 22.003/22.003 c -> e 22.003/22.003 c -> k 22.003/22.003 d -> k 22.003/22.003 f(x) -> x | x ->* e 22.003/22.003 h(x) -> i(x,x) 22.003/22.003 ) 22.003/22.003 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 22.003/22.003 22.003/22.003 Critical Pairs Processor: 22.003/22.003 -> Rules: 22.003/22.003 a -> c 22.003/22.003 a -> d 22.003/22.003 b -> c 22.003/22.003 b -> d 22.003/22.003 c -> e 22.003/22.003 c -> k 22.003/22.003 d -> k 22.003/22.003 f(x) -> x | x ->* e 22.003/22.003 h(x) -> i(x,x) 22.003/22.003 -> Vars: 22.003/22.003 "x" 22.003/22.003 22.003/22.003 -> Rlps: 22.003/22.003 crule: a -> c, id: 1, possubterms: a-> [] 22.003/22.003 crule: a -> d, id: 2, possubterms: a-> [] 22.003/22.003 crule: b -> c, id: 3, possubterms: b-> [] 22.003/22.003 crule: b -> d, id: 4, possubterms: b-> [] 22.003/22.003 crule: c -> e, id: 5, possubterms: c-> [] 22.003/22.003 crule: c -> k, id: 6, possubterms: c-> [] 22.003/22.003 crule: d -> k, id: 7, possubterms: d-> [] 22.003/22.003 crule: f(x) -> x | x ->* e, id: 8, possubterms: f(x)-> [] 22.003/22.003 crule: h(x) -> i(x,x), id: 9, possubterms: h(x)-> [] 22.003/22.003 22.003/22.003 -> Unifications: 22.003/22.003 R2 unifies with R1 at p: [], l: a, lp: a, conds: {}, sig: {}, l': a, r: d, r': c 22.003/22.003 R4 unifies with R3 at p: [], l: b, lp: b, conds: {}, sig: {}, l': b, r: d, r': c 22.003/22.003 R6 unifies with R5 at p: [], l: c, lp: c, conds: {}, sig: {}, l': c, r: k, r': e 22.003/22.003 22.003/22.003 -> Critical pairs info: 22.003/22.003 => Not trivial, Overlay, NW2, N1 22.003/22.003 => Not trivial, Overlay, NW0, N2 22.003/22.003 22.003/22.003 -> Problem conclusions: 22.003/22.003 Left linear, Not right linear, Not linear 22.003/22.003 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 22.003/22.003 CTRS Type: 1 22.003/22.003 Deterministic, Strongly deterministic 22.003/22.003 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 22.003/22.003 Maybe right-stable CTRS, Overlay CTRS 22.003/22.003 Maybe normal CTRS, Maybe almost normal CTRS 22.003/22.003 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 22.003/22.003 Maybe level confluent 22.003/22.003 Maybe confluent 22.003/22.003 22.003/22.003 Problem 1: 22.003/22.003 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 22.003/22.003 Confluence Problem: 22.003/22.003 (VAR x) 22.003/22.003 (REPLACEMENT-MAP 22.003/22.003 (a) 22.003/22.003 (b) 22.003/22.003 (c) 22.003/22.003 (d) 22.003/22.003 (f 1) 22.003/22.003 (g 1, 2) 22.003/22.003 (h 1) 22.003/22.003 (A) 22.003/22.003 (B) 22.003/22.003 (C) 22.003/22.003 (e) 22.003/22.003 (fSNonEmpty) 22.003/22.003 (i 1, 2) 22.003/22.003 (k) 22.003/22.003 ) 22.003/22.003 (RULES 22.003/22.003 a -> c 22.003/22.003 a -> d 22.003/22.003 b -> c 22.003/22.003 b -> d 22.003/22.003 c -> e 22.003/22.003 c -> k 22.003/22.003 d -> k 22.003/22.003 f(x) -> x | x ->* e 22.003/22.003 h(x) -> i(x,x) 22.003/22.003 ) 22.003/22.003 Critical Pairs: 22.003/22.003 => Not trivial, Overlay, NW2, N1 22.003/22.003 => Not trivial, Overlay, NW0, N2 22.003/22.003 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 22.003/22.003 22.003/22.003 Conditional Critical Pairs Distributor Processor 22.003/22.003 22.003/22.003 Problem 1: 22.003/22.003 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 22.003/22.003 Confluence Problem: 22.003/22.003 (VAR x) 22.003/22.003 (REPLACEMENT-MAP 22.003/22.003 (a) 22.003/22.003 (b) 22.003/22.003 (c) 22.003/22.003 (d) 22.003/22.003 (f 1) 22.003/22.003 (g 1, 2) 22.003/22.003 (h 1) 22.003/22.003 (A) 22.003/22.003 (B) 22.003/22.003 (C) 22.003/22.003 (e) 22.003/22.003 (fSNonEmpty) 22.003/22.003 (i 1, 2) 22.003/22.003 (k) 22.003/22.003 ) 22.003/22.003 (RULES 22.003/22.003 a -> c 22.003/22.003 a -> d 22.003/22.003 b -> c 22.003/22.003 b -> d 22.003/22.003 c -> e 22.003/22.003 c -> k 22.003/22.003 d -> k 22.003/22.003 f(x) -> x | x ->* e 22.003/22.003 h(x) -> i(x,x) 22.003/22.003 ) 22.003/22.003 Critical Pairs: 22.003/22.003 => Not trivial, Overlay, NW2, N1 22.003/22.003 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 22.003/22.003 22.003/22.003 InfChecker Different Ground and Normal No Conditions CCP Terms Processor: 22.003/22.003 Proof: 22.003/22.003 YES 22.003/22.003 22.003/22.003 Problem 1: 22.003/22.003 22.003/22.003 Infeasibility Problem: 22.003/22.003 [(VAR vNonEmpty x vNonEmpty y) 22.003/22.003 (STRATEGY CONTEXTSENSITIVE 22.003/22.003 (a) 22.003/22.003 (b) 22.003/22.003 (c) 22.003/22.003 (d) 22.003/22.003 (f 1) 22.003/22.003 (h 1) 22.003/22.003 (e) 22.003/22.003 (fSNonEmpty) 22.003/22.003 (i 1 2) 22.003/22.003 (k) 22.003/22.003 ) 22.003/22.003 (RULES 22.003/22.003 a -> c 22.003/22.003 a -> d 22.003/22.003 b -> c 22.003/22.003 b -> d 22.003/22.003 c -> e 22.003/22.003 c -> k 22.003/22.003 d -> k 22.003/22.003 f(x) -> x | x ->* e 22.003/22.003 h(x) -> i(x,x) 22.003/22.003 )] 22.003/22.003 22.003/22.003 Infeasibility Conditions: 22.003/22.003 e -> y 22.003/22.003 22.003/22.003 Problem 1: 22.003/22.003 22.003/22.003 Obtaining a model using Mace4: 22.003/22.003 22.003/22.003 -> Usable Rules: 22.003/22.003 Empty 22.003/22.003 22.003/22.003 -> Mace4 Output: 22.003/22.003 ============================== Mace4 ================================= 22.003/22.003 Mace4 (64) version 2009-11A, November 2009. 22.003/22.003 Process 61697 was started by ubuntu on ubuntu, 22.003/22.003 Wed Mar 9 09:49:18 2022 22.003/22.003 The command was "./mace4 -c -f /tmp/mace461674-2.in". 22.003/22.003 ============================== end of head =========================== 22.003/22.003 22.003/22.003 ============================== INPUT ================================= 22.003/22.003 22.003/22.003 % Reading from file /tmp/mace461674-2.in 22.003/22.003 22.003/22.003 assign(max_seconds,6). 22.003/22.003 22.003/22.003 formulas(assumptions). 22.003/22.003 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 22.003/22.003 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence). 22.003/22.003 ->(x1,y) -> ->(i(x1,x2),i(y,x2)) # label(congruence). 22.003/22.003 ->(x2,y) -> ->(i(x1,x2),i(x1,y)) # label(congruence). 22.003/22.003 end_of_list. 22.003/22.003 22.003/22.003 formulas(goals). 22.003/22.003 (exists x3 ->(e,x3)) # label(goal). 22.003/22.003 end_of_list. 22.003/22.003 22.003/22.003 ============================== end of input ========================== 22.003/22.003 22.003/22.003 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 22.003/22.003 22.003/22.003 % Formulas that are not ordinary clauses: 22.003/22.003 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 2 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 3 ->(x1,y) -> ->(i(x1,x2),i(y,x2)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 4 ->(x2,y) -> ->(i(x1,x2),i(x1,y)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 5 (exists x3 ->(e,x3)) # label(goal) # label(non_clause) # label(goal). [goal]. 22.003/22.003 22.003/22.003 ============================== end of process non-clausal formulas === 22.003/22.003 22.003/22.003 ============================== CLAUSES FOR SEARCH ==================== 22.003/22.003 22.003/22.003 formulas(mace4_clauses). 22.003/22.003 -->(x,y) | ->(f(x),f(y)) # label(congruence). 22.003/22.003 -->(x,y) | ->(h(x),h(y)) # label(congruence). 22.003/22.003 -->(x,y) | ->(i(x,z),i(y,z)) # label(congruence). 22.003/22.003 -->(x,y) | ->(i(z,x),i(z,y)) # label(congruence). 22.003/22.003 -->(e,x) # label(goal). 22.003/22.003 end_of_list. 22.003/22.003 22.003/22.003 ============================== end of clauses for search ============= 22.003/22.003 22.003/22.003 % There are no natural numbers in the input. 22.003/22.003 22.003/22.003 ============================== DOMAIN SIZE 2 ========================= 22.003/22.003 22.003/22.003 ============================== MODEL ================================= 22.003/22.003 22.003/22.003 interpretation( 2, [number=1, seconds=0], [ 22.003/22.003 22.003/22.003 function(e, [ 0 ]), 22.003/22.003 22.003/22.003 function(f(_), [ 0, 0 ]), 22.003/22.003 22.003/22.003 function(h(_), [ 0, 0 ]), 22.003/22.003 22.003/22.003 function(i(_,_), [ 22.003/22.003 0, 0, 22.003/22.003 0, 0 ]), 22.003/22.003 22.003/22.003 relation(->(_,_), [ 22.003/22.003 0, 0, 22.003/22.003 0, 0 ]) 22.003/22.003 ]). 22.003/22.003 22.003/22.003 ============================== end of model ========================== 22.003/22.003 22.003/22.003 ============================== STATISTICS ============================ 22.003/22.003 22.003/22.003 For domain size 2. 22.003/22.003 22.003/22.003 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 22.003/22.003 Ground clauses: seen=26, kept=26. 22.003/22.003 Selections=9, assignments=9, propagations=4, current_models=1. 22.003/22.003 Rewrite_terms=50, rewrite_bools=28, indexes=2. 22.003/22.003 Rules_from_neg_clauses=0, cross_offs=0. 22.003/22.003 22.003/22.003 ============================== end of statistics ===================== 22.003/22.003 22.003/22.003 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 22.003/22.003 22.003/22.003 Exiting with 1 model. 22.003/22.003 22.003/22.003 Process 61697 exit (max_models) Wed Mar 9 09:49:18 2022 22.003/22.003 The process finished Wed Mar 9 09:49:18 2022 22.003/22.003 22.003/22.003 22.003/22.003 Mace4 cooked interpretation: 22.003/22.003 22.003/22.003 22.003/22.003 22.003/22.003 The problem is infeasible. 22.003/22.003 YES 22.003/22.003 22.003/22.003 Problem 1: 22.003/22.003 22.003/22.003 Infeasibility Problem: 22.003/22.003 [(VAR vNonEmpty x vNonEmpty x) 22.003/22.003 (STRATEGY CONTEXTSENSITIVE 22.003/22.003 (a) 22.003/22.003 (b) 22.003/22.003 (c) 22.003/22.003 (d) 22.003/22.003 (f 1) 22.003/22.003 (h 1) 22.003/22.003 (e) 22.003/22.003 (fSNonEmpty) 22.003/22.003 (i 1 2) 22.003/22.003 (k) 22.003/22.003 ) 22.003/22.003 (RULES 22.003/22.003 a -> c 22.003/22.003 a -> d 22.003/22.003 b -> c 22.003/22.003 b -> d 22.003/22.003 c -> e 22.003/22.003 c -> k 22.003/22.003 d -> k 22.003/22.003 f(x) -> x | x ->* e 22.003/22.003 h(x) -> i(x,x) 22.003/22.003 )] 22.003/22.003 22.003/22.003 Infeasibility Conditions: 22.003/22.003 k -> x 22.003/22.003 22.003/22.003 Problem 1: 22.003/22.003 22.003/22.003 Obtaining a model using Mace4: 22.003/22.003 22.003/22.003 -> Usable Rules: 22.003/22.003 Empty 22.003/22.003 22.003/22.003 -> Mace4 Output: 22.003/22.003 ============================== Mace4 ================================= 22.003/22.003 Mace4 (64) version 2009-11A, November 2009. 22.003/22.003 Process 61735 was started by ubuntu on ubuntu, 22.003/22.003 Wed Mar 9 09:49:18 2022 22.003/22.003 The command was "./mace4 -c -f /tmp/mace461722-2.in". 22.003/22.003 ============================== end of head =========================== 22.003/22.003 22.003/22.003 ============================== INPUT ================================= 22.003/22.003 22.003/22.003 % Reading from file /tmp/mace461722-2.in 22.003/22.003 22.003/22.003 assign(max_seconds,6). 22.003/22.003 22.003/22.003 formulas(assumptions). 22.003/22.003 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 22.003/22.003 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence). 22.003/22.003 ->(x1,y) -> ->(i(x1,x2),i(y,x2)) # label(congruence). 22.003/22.003 ->(x2,y) -> ->(i(x1,x2),i(x1,y)) # label(congruence). 22.003/22.003 end_of_list. 22.003/22.003 22.003/22.003 formulas(goals). 22.003/22.003 (exists x3 ->(k,x3)) # label(goal). 22.003/22.003 end_of_list. 22.003/22.003 22.003/22.003 ============================== end of input ========================== 22.003/22.003 22.003/22.003 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 22.003/22.003 22.003/22.003 % Formulas that are not ordinary clauses: 22.003/22.003 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 2 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 3 ->(x1,y) -> ->(i(x1,x2),i(y,x2)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 4 ->(x2,y) -> ->(i(x1,x2),i(x1,y)) # label(congruence) # label(non_clause). [assumption]. 22.003/22.003 5 (exists x3 ->(k,x3)) # label(goal) # label(non_clause) # label(goal). [goal]. 22.003/22.003 22.003/22.003 ============================== end of process non-clausal formulas === 22.003/22.003 22.003/22.003 ============================== CLAUSES FOR SEARCH ==================== 22.003/22.003 22.003/22.003 formulas(mace4_clauses). 22.003/22.003 -->(x,y) | ->(f(x),f(y)) # label(congruence). 22.003/22.003 -->(x,y) | ->(h(x),h(y)) # label(congruence). 22.003/22.003 -->(x,y) | ->(i(x,z),i(y,z)) # label(congruence). 22.003/22.003 -->(x,y) | ->(i(z,x),i(z,y)) # label(congruence). 22.003/22.003 -->(k,x) # label(goal). 22.003/22.003 end_of_list. 22.003/22.003 22.003/22.003 ============================== end of clauses for search ============= 22.003/22.003 22.003/22.003 % There are no natural numbers in the input. 22.003/22.003 22.003/22.003 ============================== DOMAIN SIZE 2 ========================= 22.003/22.003 22.003/22.003 ============================== MODEL ================================= 22.003/22.003 22.003/22.003 interpretation( 2, [number=1, seconds=0], [ 22.003/22.003 22.003/22.003 function(k, [ 0 ]), 22.003/22.003 22.003/22.003 function(f(_), [ 0, 0 ]), 22.003/22.003 22.003/22.003 function(h(_), [ 0, 0 ]), 22.003/22.003 22.003/22.003 function(i(_,_), [ 22.003/22.003 0, 0, 22.003/22.003 0, 0 ]), 22.003/22.003 22.003/22.003 relation(->(_,_), [ 22.003/22.003 0, 0, 22.003/22.003 0, 0 ]) 22.003/22.003 ]). 22.003/22.003 22.003/22.003 ============================== end of model ========================== 22.003/22.003 22.003/22.003 ============================== STATISTICS ============================ 22.003/22.003 22.003/22.003 For domain size 2. 22.003/22.003 22.003/22.003 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 22.003/22.003 Ground clauses: seen=26, kept=26. 22.003/22.003 Selections=9, assignments=9, propagations=4, current_models=1. 22.003/22.003 Rewrite_terms=50, rewrite_bools=28, indexes=2. 22.003/22.003 Rules_from_neg_clauses=0, cross_offs=0. 22.003/22.003 22.003/22.003 ============================== end of statistics ===================== 22.003/22.003 22.003/22.003 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 22.003/22.003 22.003/22.003 Exiting with 1 model. 22.003/22.003 22.003/22.003 Process 61735 exit (max_models) Wed Mar 9 09:49:18 2022 22.003/22.003 The process finished Wed Mar 9 09:49:18 2022 22.003/22.003 22.003/22.003 22.003/22.003 Mace4 cooked interpretation: 22.003/22.003 22.003/22.003 22.003/22.003 22.003/22.003 The problem is infeasible. 22.003/22.003 22.003/22.003 22.003/22.003 The problem is not joinable. 22.003/22.003 23.73user 0.96system 0:22.03elapsed 112%CPU (0avgtext+0avgdata 856048maxresident)k 22.003/22.003 8inputs+0outputs (0major+265775minor)pagefaults 0swaps