20.033/20.033 NO 20.033/20.033 20.033/20.033 Problem 1: 20.033/20.033 20.033/20.033 20.033/20.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.033/20.033 Confluence Problem: 20.033/20.033 (VAR vNonEmpty x y) 20.033/20.033 (REPLACEMENT-MAP 20.033/20.033 (a) 20.033/20.033 (b) 20.033/20.033 (e) 20.033/20.033 (f 1, 2) 20.033/20.033 (g 1) 20.033/20.033 (h 1) 20.033/20.033 (d) 20.033/20.033 (fSNonEmpty) 20.033/20.033 (s 1) 20.033/20.033 ) 20.033/20.033 (RULES 20.033/20.033 a -> d 20.033/20.033 b -> d 20.033/20.033 e -> e 20.033/20.033 f(x,y) -> g(x) | a ->* d 20.033/20.033 f(x,y) -> h(x) | b ->* d 20.033/20.033 g(s(x)) -> x 20.033/20.033 h(s(x)) -> x 20.033/20.033 ) 20.033/20.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.033/20.033 20.033/20.033 20.033/20.033 Problem 1: 20.033/20.033 20.033/20.033 Inlining of Conditions Processor [STERN17]: 20.033/20.033 20.033/20.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.033/20.033 Confluence Problem: 20.033/20.033 (VAR vNonEmpty x y) 20.033/20.033 (REPLACEMENT-MAP 20.033/20.033 (a) 20.033/20.033 (b) 20.033/20.033 (e) 20.033/20.033 (f 1, 2) 20.033/20.033 (g 1) 20.033/20.033 (h 1) 20.033/20.033 (d) 20.033/20.033 (fSNonEmpty) 20.033/20.033 (s 1) 20.033/20.033 ) 20.033/20.033 (RULES 20.033/20.033 a -> d 20.033/20.033 b -> d 20.033/20.033 e -> e 20.033/20.033 f(x,y) -> g(x) | a ->* d 20.033/20.033 f(x,y) -> h(x) | b ->* d 20.033/20.033 g(s(x)) -> x 20.033/20.033 h(s(x)) -> x 20.033/20.033 ) 20.033/20.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.033/20.033 20.033/20.033 20.033/20.033 Problem 1: 20.033/20.033 20.033/20.033 Clean CTRS Processor: 20.033/20.033 20.033/20.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.033/20.033 Confluence Problem: 20.033/20.033 (VAR x y) 20.033/20.033 (REPLACEMENT-MAP 20.033/20.033 (a) 20.033/20.033 (b) 20.033/20.033 (e) 20.033/20.033 (f 1, 2) 20.033/20.033 (g 1) 20.033/20.033 (h 1) 20.033/20.033 (d) 20.033/20.033 (fSNonEmpty) 20.033/20.033 (s 1) 20.033/20.033 ) 20.033/20.033 (RULES 20.033/20.033 a -> d 20.033/20.033 b -> d 20.033/20.033 f(x,y) -> g(x) | a ->* d 20.033/20.033 f(x,y) -> h(x) | b ->* d 20.033/20.033 g(s(x)) -> x 20.033/20.033 h(s(x)) -> x 20.033/20.033 ) 20.033/20.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.033/20.033 20.033/20.033 CRule InfChecker Info: 20.033/20.033 a -> d 20.033/20.033 Rule remains 20.033/20.033 Proof: 20.033/20.033 NO_CONDS 20.033/20.033 20.033/20.033 CRule InfChecker Info: 20.033/20.033 b -> d 20.033/20.033 Rule remains 20.033/20.033 Proof: 20.033/20.033 NO_CONDS 20.033/20.033 20.033/20.033 CRule InfChecker Info: 20.033/20.033 e -> e 20.033/20.033 Rule deleted 20.033/20.033 Proof: 20.033/20.033 SAME_LHS_AND_RHS 20.033/20.033 20.033/20.033 CRule InfChecker Info: 20.033/20.033 f(x,y) -> g(x) | a ->* d 20.033/20.033 Rule remains 20.033/20.033 Proof: 20.033/20.033 NO 20.033/20.033 20.033/20.033 Problem 1: 20.033/20.033 20.033/20.033 Infeasibility Problem: 20.033/20.033 [(VAR vNonEmpty x y vNonEmpty) 20.033/20.033 (STRATEGY CONTEXTSENSITIVE 20.033/20.033 (a) 20.033/20.033 (b) 20.033/20.033 (f 1 2) 20.033/20.033 (g 1) 20.033/20.033 (h 1) 20.033/20.033 (d) 20.033/20.033 (fSNonEmpty) 20.033/20.033 (s 1) 20.033/20.033 ) 20.033/20.033 (RULES 20.033/20.033 a -> d 20.033/20.033 b -> d 20.033/20.033 f(x,y) -> g(x) | a ->* d 20.033/20.033 f(x,y) -> h(x) | b ->* d 20.033/20.033 g(s(x)) -> x 20.033/20.033 h(s(x)) -> x 20.033/20.033 )] 20.033/20.033 20.033/20.033 Infeasibility Conditions: 20.033/20.033 a ->* d 20.033/20.033 20.033/20.033 Problem 1: 20.033/20.033 20.033/20.033 Obtaining a proof using Prover9: 20.033/20.033 20.033/20.033 -> Prover9 Output: 20.033/20.033 ============================== Prover9 =============================== 20.033/20.033 Prover9 (64) version 2009-11A, November 2009. 20.033/20.033 Process 72911 was started by ubuntu on ubuntu, 20.033/20.033 Wed Mar 9 10:04:48 2022 20.033/20.033 The command was "./prover9 -f /tmp/prover972902-0.in". 20.033/20.033 ============================== end of head =========================== 20.033/20.033 20.033/20.033 ============================== INPUT ================================= 20.033/20.033 20.033/20.033 % Reading from file /tmp/prover972902-0.in 20.033/20.033 20.033/20.033 assign(max_seconds,20). 20.033/20.033 20.033/20.033 formulas(assumptions). 20.033/20.033 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 20.033/20.033 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 20.033/20.033 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 20.033/20.033 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 20.033/20.033 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 20.033/20.033 ->_s0(a,d) # label(replacement). 20.033/20.033 ->*_s0(x,x) # label(reflexivity). 20.033/20.033 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(goals). 20.033/20.033 ->*_s0(a,d) # label(goal). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 ============================== end of input ========================== 20.033/20.033 20.033/20.033 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.033/20.033 20.033/20.033 % Formulas that are not ordinary clauses: 20.033/20.033 1 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 2 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 3 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 4 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 5 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 6 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.033/20.033 7 ->*_s0(a,d) # label(goal) # label(non_clause) # label(goal). [goal]. 20.033/20.033 20.033/20.033 ============================== end of process non-clausal formulas === 20.033/20.033 20.033/20.033 ============================== PROCESS INITIAL CLAUSES =============== 20.033/20.033 20.033/20.033 % Clauses before input processing: 20.033/20.033 20.033/20.033 formulas(usable). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(sos). 20.033/20.033 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.033/20.033 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.033/20.033 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.033/20.033 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.033/20.033 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 20.033/20.033 ->_s0(a,d) # label(replacement). [assumption]. 20.033/20.033 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.033/20.033 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(6)]. 20.033/20.033 -->*_s0(a,d) # label(goal). [deny(7)]. 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(demodulators). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 ============================== PREDICATE ELIMINATION ================= 20.033/20.033 20.033/20.033 No predicates eliminated. 20.033/20.033 20.033/20.033 ============================== end predicate elimination ============= 20.033/20.033 20.033/20.033 Auto_denials: 20.033/20.033 % copying label goal to answer in negative clause 20.033/20.033 20.033/20.033 Term ordering decisions: 20.033/20.033 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.033/20.033 Function symbol precedence: function_order([ a, d, f, g, h, s ]). 20.033/20.033 After inverse_order: (no changes). 20.033/20.033 Unfolding symbols: (none). 20.033/20.033 20.033/20.033 Auto_inference settings: 20.033/20.033 % set(neg_binary_resolution). % (HNE depth_diff=-5) 20.033/20.033 % clear(ordered_res). % (HNE depth_diff=-5) 20.033/20.033 % set(ur_resolution). % (HNE depth_diff=-5) 20.033/20.033 % set(ur_resolution) -> set(pos_ur_resolution). 20.033/20.033 % set(ur_resolution) -> set(neg_ur_resolution). 20.033/20.033 20.033/20.033 Auto_process settings: (no changes). 20.033/20.033 20.033/20.033 kept: 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.033/20.033 kept: 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.033/20.033 kept: 10 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.033/20.033 kept: 11 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.033/20.033 kept: 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 20.033/20.033 kept: 13 ->_s0(a,d) # label(replacement). [assumption]. 20.033/20.033 kept: 14 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.033/20.033 kept: 15 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(6)]. 20.033/20.033 kept: 16 -->*_s0(a,d) # label(goal) # answer(goal). [deny(7)]. 20.033/20.033 20.033/20.033 ============================== end of process initial clauses ======== 20.033/20.033 20.033/20.033 ============================== CLAUSES FOR SEARCH ==================== 20.033/20.033 20.033/20.033 % Clauses after input processing: 20.033/20.033 20.033/20.033 formulas(usable). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(sos). 20.033/20.033 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.033/20.033 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.033/20.033 10 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.033/20.033 11 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.033/20.033 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 20.033/20.033 13 ->_s0(a,d) # label(replacement). [assumption]. 20.033/20.033 14 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.033/20.033 15 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(6)]. 20.033/20.033 16 -->*_s0(a,d) # label(goal) # answer(goal). [deny(7)]. 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(demodulators). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 ============================== end of clauses for search ============= 20.033/20.033 20.033/20.033 ============================== SEARCH ================================ 20.033/20.033 20.033/20.033 % Starting search at 0.00 seconds. 20.033/20.033 20.033/20.033 given #1 (I,wt=10): 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.033/20.033 20.033/20.033 given #2 (I,wt=10): 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.033/20.033 20.033/20.033 given #3 (I,wt=8): 10 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.033/20.033 20.033/20.033 given #4 (I,wt=8): 11 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.033/20.033 20.033/20.033 given #5 (I,wt=8): 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 20.033/20.033 20.033/20.033 given #6 (I,wt=3): 13 ->_s0(a,d) # label(replacement). [assumption]. 20.033/20.033 20.033/20.033 given #7 (I,wt=3): 14 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.033/20.033 20.033/20.033 given #8 (I,wt=9): 15 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(6)]. 20.033/20.033 20.033/20.033 ============================== PROOF ================================= 20.033/20.033 20.033/20.033 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.033/20.033 % Length of proof is 8. 20.033/20.033 % Level of proof is 3. 20.033/20.033 % Maximum clause weight is 9.000. 20.033/20.033 % Given clauses 8. 20.033/20.033 20.033/20.033 6 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.033/20.033 7 ->*_s0(a,d) # label(goal) # label(non_clause) # label(goal). [goal]. 20.033/20.033 13 ->_s0(a,d) # label(replacement). [assumption]. 20.033/20.033 14 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.033/20.033 15 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(6)]. 20.033/20.033 16 -->*_s0(a,d) # label(goal) # answer(goal). [deny(7)]. 20.033/20.033 22 ->*_s0(a,d). [ur(15,a,13,a,b,14,a)]. 20.033/20.033 23 $F # answer(goal). [resolve(22,a,16,a)]. 20.033/20.033 20.033/20.033 ============================== end of proof ========================== 20.033/20.033 20.033/20.033 ============================== STATISTICS ============================ 20.033/20.033 20.033/20.033 Given=8. Generated=15. Kept=15. proofs=1. 20.033/20.033 Usable=8. Sos=6. Demods=0. Limbo=0, Disabled=9. Hints=0. 20.033/20.033 Kept_by_rule=0, Deleted_by_rule=0. 20.033/20.033 Forward_subsumed=0. Back_subsumed=0. 20.033/20.033 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.033/20.033 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.033/20.033 Demod_attempts=0. Demod_rewrites=0. 20.033/20.033 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.033/20.033 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=9. 20.033/20.033 Megabytes=0.06. 20.033/20.033 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.033/20.033 20.033/20.033 ============================== end of statistics ===================== 20.033/20.033 20.033/20.033 ============================== end of search ========================= 20.033/20.033 20.033/20.033 THEOREM PROVED 20.033/20.033 20.033/20.033 Exiting with 1 proof. 20.033/20.033 20.033/20.033 Process 72911 exit (max_proofs) Wed Mar 9 10:04:48 2022 20.033/20.033 20.033/20.033 20.033/20.033 The problem is feasible. 20.033/20.033 20.033/20.033 20.033/20.033 CRule InfChecker Info: 20.033/20.033 f(x,y) -> h(x) | b ->* d 20.033/20.033 Rule remains 20.033/20.033 Proof: 20.033/20.033 NO 20.033/20.033 20.033/20.033 Problem 1: 20.033/20.033 20.033/20.033 Infeasibility Problem: 20.033/20.033 [(VAR vNonEmpty x y vNonEmpty) 20.033/20.033 (STRATEGY CONTEXTSENSITIVE 20.033/20.033 (a) 20.033/20.033 (b) 20.033/20.033 (f 1 2) 20.033/20.033 (g 1) 20.033/20.033 (h 1) 20.033/20.033 (d) 20.033/20.033 (fSNonEmpty) 20.033/20.033 (s 1) 20.033/20.033 ) 20.033/20.033 (RULES 20.033/20.033 a -> d 20.033/20.033 b -> d 20.033/20.033 f(x,y) -> g(x) | a ->* d 20.033/20.033 f(x,y) -> h(x) | b ->* d 20.033/20.033 g(s(x)) -> x 20.033/20.033 h(s(x)) -> x 20.033/20.033 )] 20.033/20.033 20.033/20.033 Infeasibility Conditions: 20.033/20.033 b ->* d 20.033/20.033 20.033/20.033 Problem 1: 20.033/20.033 20.033/20.033 Obtaining a proof using Prover9: 20.033/20.033 20.033/20.033 -> Prover9 Output: 20.033/20.033 ============================== Prover9 =============================== 20.033/20.033 Prover9 (64) version 2009-11A, November 2009. 20.033/20.033 Process 72933 was started by ubuntu on ubuntu, 20.033/20.033 Wed Mar 9 10:04:48 2022 20.033/20.033 The command was "./prover9 -f /tmp/prover972924-0.in". 20.033/20.033 ============================== end of head =========================== 20.033/20.033 20.033/20.033 ============================== INPUT ================================= 20.033/20.033 20.033/20.033 % Reading from file /tmp/prover972924-0.in 20.033/20.033 20.033/20.033 assign(max_seconds,20). 20.033/20.033 20.033/20.033 formulas(assumptions). 20.033/20.033 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 20.033/20.033 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 20.033/20.033 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 20.033/20.033 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 20.033/20.033 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 20.033/20.033 ->_s0(b,d) # label(replacement). 20.033/20.033 ->*_s0(x,x) # label(reflexivity). 20.033/20.033 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(goals). 20.033/20.033 ->*_s0(b,d) # label(goal). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 ============================== end of input ========================== 20.033/20.033 20.033/20.033 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.033/20.033 20.033/20.033 % Formulas that are not ordinary clauses: 20.033/20.033 1 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 2 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 3 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 4 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 5 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 6 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.033/20.033 7 ->*_s0(b,d) # label(goal) # label(non_clause) # label(goal). [goal]. 20.033/20.033 20.033/20.033 ============================== end of process non-clausal formulas === 20.033/20.033 20.033/20.033 ============================== PROCESS INITIAL CLAUSES =============== 20.033/20.033 20.033/20.033 % Clauses before input processing: 20.033/20.033 20.033/20.033 formulas(usable). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(sos). 20.033/20.033 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.033/20.033 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.033/20.033 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.033/20.033 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.033/20.033 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 20.033/20.033 ->_s0(b,d) # label(replacement). [assumption]. 20.033/20.033 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.033/20.033 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(6)]. 20.033/20.033 -->*_s0(b,d) # label(goal). [deny(7)]. 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(demodulators). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 ============================== PREDICATE ELIMINATION ================= 20.033/20.033 20.033/20.033 No predicates eliminated. 20.033/20.033 20.033/20.033 ============================== end predicate elimination ============= 20.033/20.033 20.033/20.033 Auto_denials: 20.033/20.033 % copying label goal to answer in negative clause 20.033/20.033 20.033/20.033 Term ordering decisions: 20.033/20.033 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.033/20.033 Function symbol precedence: function_order([ b, d, f, g, h, s ]). 20.033/20.033 After inverse_order: (no changes). 20.033/20.033 Unfolding symbols: (none). 20.033/20.033 20.033/20.033 Auto_inference settings: 20.033/20.033 % set(neg_binary_resolution). % (HNE depth_diff=-5) 20.033/20.033 % clear(ordered_res). % (HNE depth_diff=-5) 20.033/20.033 % set(ur_resolution). % (HNE depth_diff=-5) 20.033/20.033 % set(ur_resolution) -> set(pos_ur_resolution). 20.033/20.033 % set(ur_resolution) -> set(neg_ur_resolution). 20.033/20.033 20.033/20.033 Auto_process settings: (no changes). 20.033/20.033 20.033/20.033 kept: 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.033/20.033 kept: 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.033/20.033 kept: 10 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.033/20.033 kept: 11 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.033/20.033 kept: 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 20.033/20.033 kept: 13 ->_s0(b,d) # label(replacement). [assumption]. 20.033/20.033 kept: 14 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.033/20.033 kept: 15 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(6)]. 20.033/20.033 kept: 16 -->*_s0(b,d) # label(goal) # answer(goal). [deny(7)]. 20.033/20.033 20.033/20.033 ============================== end of process initial clauses ======== 20.033/20.033 20.033/20.033 ============================== CLAUSES FOR SEARCH ==================== 20.033/20.033 20.033/20.033 % Clauses after input processing: 20.033/20.033 20.033/20.033 formulas(usable). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(sos). 20.033/20.033 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.033/20.033 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.033/20.033 10 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.033/20.033 11 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.033/20.033 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 20.033/20.033 13 ->_s0(b,d) # label(replacement). [assumption]. 20.033/20.033 14 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.033/20.033 15 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(6)]. 20.033/20.033 16 -->*_s0(b,d) # label(goal) # answer(goal). [deny(7)]. 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(demodulators). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 ============================== end of clauses for search ============= 20.033/20.033 20.033/20.033 ============================== SEARCH ================================ 20.033/20.033 20.033/20.033 % Starting search at 0.00 seconds. 20.033/20.033 20.033/20.033 given #1 (I,wt=10): 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.033/20.033 20.033/20.033 given #2 (I,wt=10): 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.033/20.033 20.033/20.033 given #3 (I,wt=8): 10 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.033/20.033 20.033/20.033 given #4 (I,wt=8): 11 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.033/20.033 20.033/20.033 given #5 (I,wt=8): 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 20.033/20.033 20.033/20.033 given #6 (I,wt=3): 13 ->_s0(b,d) # label(replacement). [assumption]. 20.033/20.033 20.033/20.033 given #7 (I,wt=3): 14 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.033/20.033 20.033/20.033 given #8 (I,wt=9): 15 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(6)]. 20.033/20.033 20.033/20.033 ============================== PROOF ================================= 20.033/20.033 20.033/20.033 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.033/20.033 % Length of proof is 8. 20.033/20.033 % Level of proof is 3. 20.033/20.033 % Maximum clause weight is 9.000. 20.033/20.033 % Given clauses 8. 20.033/20.033 20.033/20.033 6 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.033/20.033 7 ->*_s0(b,d) # label(goal) # label(non_clause) # label(goal). [goal]. 20.033/20.033 13 ->_s0(b,d) # label(replacement). [assumption]. 20.033/20.033 14 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.033/20.033 15 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(6)]. 20.033/20.033 16 -->*_s0(b,d) # label(goal) # answer(goal). [deny(7)]. 20.033/20.033 22 ->*_s0(b,d). [ur(15,a,13,a,b,14,a)]. 20.033/20.033 23 $F # answer(goal). [resolve(22,a,16,a)]. 20.033/20.033 20.033/20.033 ============================== end of proof ========================== 20.033/20.033 20.033/20.033 ============================== STATISTICS ============================ 20.033/20.033 20.033/20.033 Given=8. Generated=15. Kept=15. proofs=1. 20.033/20.033 Usable=8. Sos=6. Demods=0. Limbo=0, Disabled=9. Hints=0. 20.033/20.033 Kept_by_rule=0, Deleted_by_rule=0. 20.033/20.033 Forward_subsumed=0. Back_subsumed=0. 20.033/20.033 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.033/20.033 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.033/20.033 Demod_attempts=0. Demod_rewrites=0. 20.033/20.033 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.033/20.033 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=9. 20.033/20.033 Megabytes=0.06. 20.033/20.033 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.033/20.033 20.033/20.033 ============================== end of statistics ===================== 20.033/20.033 20.033/20.033 ============================== end of search ========================= 20.033/20.033 20.033/20.033 THEOREM PROVED 20.033/20.033 20.033/20.033 Exiting with 1 proof. 20.033/20.033 20.033/20.033 Process 72933 exit (max_proofs) Wed Mar 9 10:04:48 2022 20.033/20.033 20.033/20.033 20.033/20.033 The problem is feasible. 20.033/20.033 20.033/20.033 20.033/20.033 CRule InfChecker Info: 20.033/20.033 g(s(x)) -> x 20.033/20.033 Rule remains 20.033/20.033 Proof: 20.033/20.033 NO_CONDS 20.033/20.033 20.033/20.033 CRule InfChecker Info: 20.033/20.033 h(s(x)) -> x 20.033/20.033 Rule remains 20.033/20.033 Proof: 20.033/20.033 NO_CONDS 20.033/20.033 20.033/20.033 Problem 1: 20.033/20.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.033/20.033 Confluence Problem: 20.033/20.033 (VAR x y) 20.033/20.033 (REPLACEMENT-MAP 20.033/20.033 (a) 20.033/20.033 (b) 20.033/20.033 (e) 20.033/20.033 (f 1, 2) 20.033/20.033 (g 1) 20.033/20.033 (h 1) 20.033/20.033 (d) 20.033/20.033 (fSNonEmpty) 20.033/20.033 (s 1) 20.033/20.033 ) 20.033/20.033 (RULES 20.033/20.033 a -> d 20.033/20.033 b -> d 20.033/20.033 f(x,y) -> g(x) | a ->* d 20.033/20.033 f(x,y) -> h(x) | b ->* d 20.033/20.033 g(s(x)) -> x 20.033/20.033 h(s(x)) -> x 20.033/20.033 ) 20.033/20.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.033/20.033 20.033/20.033 Critical Pairs Processor: 20.033/20.033 -> Rules: 20.033/20.033 a -> d 20.033/20.033 b -> d 20.033/20.033 f(x,y) -> g(x) | a ->* d 20.033/20.033 f(x,y) -> h(x) | b ->* d 20.033/20.033 g(s(x)) -> x 20.033/20.033 h(s(x)) -> x 20.033/20.033 -> Vars: 20.033/20.033 "x", "y" 20.033/20.033 20.033/20.033 -> Rlps: 20.033/20.033 crule: a -> d, id: 1, possubterms: a-> [] 20.033/20.033 crule: b -> d, id: 2, possubterms: b-> [] 20.033/20.033 crule: f(x,y) -> g(x) | a ->* d, id: 3, possubterms: f(x,y)-> [] 20.033/20.033 crule: f(x,y) -> h(x) | b ->* d, id: 4, possubterms: f(x,y)-> [] 20.033/20.033 crule: g(s(x)) -> x, id: 5, possubterms: g(s(x))-> [], s(x)-> [1] 20.033/20.033 crule: h(s(x)) -> x, id: 6, possubterms: h(s(x))-> [], s(x)-> [1] 20.033/20.033 20.033/20.033 -> Unifications: 20.033/20.033 R4 unifies with R3 at p: [], l: f(x,y), lp: f(x,y), conds: {b ->* d, a ->* d}, sig: {x -> x',y -> y'}, l': f(x',y'), r: h(x), r': g(x') 20.033/20.033 20.033/20.033 -> Critical pairs info: 20.033/20.033 | b ->* d, a ->* d => Not trivial, Overlay, NW2, N1 20.033/20.033 20.033/20.033 -> Problem conclusions: 20.033/20.033 Left linear, Right linear, Linear 20.033/20.033 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.033/20.033 CTRS Type: 1 20.033/20.033 Deterministic, Strongly deterministic 20.033/20.033 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.033/20.033 Maybe right-stable CTRS, Overlay CTRS 20.033/20.033 Maybe normal CTRS, Maybe almost normal CTRS 20.033/20.033 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.033/20.033 Maybe level confluent 20.033/20.033 Maybe confluent 20.033/20.033 20.033/20.033 Problem 1: 20.033/20.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.033/20.033 Confluence Problem: 20.033/20.033 (VAR x y x') 20.033/20.033 (REPLACEMENT-MAP 20.033/20.033 (a) 20.033/20.033 (b) 20.033/20.033 (e) 20.033/20.033 (f 1, 2) 20.033/20.033 (g 1) 20.033/20.033 (h 1) 20.033/20.033 (d) 20.033/20.033 (fSNonEmpty) 20.033/20.033 (s 1) 20.033/20.033 ) 20.033/20.033 (RULES 20.033/20.033 a -> d 20.033/20.033 b -> d 20.033/20.033 f(x,y) -> g(x) | a ->* d 20.033/20.033 f(x,y) -> h(x) | b ->* d 20.033/20.033 g(s(x)) -> x 20.033/20.033 h(s(x)) -> x 20.033/20.033 ) 20.033/20.033 Critical Pairs: 20.033/20.033 | b ->* d, a ->* d => Not trivial, Overlay, NW2, N1 20.033/20.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.033/20.033 20.033/20.033 Conditional Critical Pairs Distributor Processor 20.033/20.033 20.033/20.033 Problem 1: 20.033/20.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.033/20.033 Confluence Problem: 20.033/20.033 (VAR x y x') 20.033/20.033 (REPLACEMENT-MAP 20.033/20.033 (a) 20.033/20.033 (b) 20.033/20.033 (e) 20.033/20.033 (f 1, 2) 20.033/20.033 (g 1) 20.033/20.033 (h 1) 20.033/20.033 (d) 20.033/20.033 (fSNonEmpty) 20.033/20.033 (s 1) 20.033/20.033 ) 20.033/20.033 (RULES 20.033/20.033 a -> d 20.033/20.033 b -> d 20.033/20.033 f(x,y) -> g(x) | a ->* d 20.033/20.033 f(x,y) -> h(x) | b ->* d 20.033/20.033 g(s(x)) -> x 20.033/20.033 h(s(x)) -> x 20.033/20.033 ) 20.033/20.033 Critical Pairs: 20.033/20.033 | b ->* d, a ->* d => Not trivial, Overlay, NW2, N1 20.033/20.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.033/20.033 20.033/20.033 Infeasible Convergence Constantized CCP Processor: 20.033/20.033 20.033/20.033 Infeasible constantized conditions? 20.033/20.033 NO 20.033/20.033 20.033/20.033 Problem 1: 20.033/20.033 20.033/20.033 Infeasibility Problem: 20.033/20.033 [(VAR vNonEmpty x y x1 vNonEmpty) 20.033/20.033 (STRATEGY CONTEXTSENSITIVE 20.033/20.033 (a) 20.033/20.033 (b) 20.033/20.033 (f 1 2) 20.033/20.033 (g 1) 20.033/20.033 (h 1) 20.033/20.033 (d) 20.033/20.033 (fSNonEmpty) 20.033/20.033 (s 1) 20.033/20.033 ) 20.033/20.033 (RULES 20.033/20.033 a -> d 20.033/20.033 b -> d 20.033/20.033 f(x,y) -> g(x) | a ->* d 20.033/20.033 f(x,y) -> h(x) | b ->* d 20.033/20.033 g(s(x)) -> x 20.033/20.033 h(s(x)) -> x 20.033/20.033 )] 20.033/20.033 20.033/20.033 Infeasibility Conditions: 20.033/20.033 b ->* d, a ->* d 20.033/20.033 20.033/20.033 Problem 1: 20.033/20.033 20.033/20.033 Obtaining a proof using Prover9: 20.033/20.033 20.033/20.033 -> Prover9 Output: 20.033/20.033 ============================== Prover9 =============================== 20.033/20.033 Prover9 (64) version 2009-11A, November 2009. 20.033/20.033 Process 72962 was started by ubuntu on ubuntu, 20.033/20.033 Wed Mar 9 10:04:49 2022 20.033/20.033 The command was "./prover9 -f /tmp/prover972947-0.in". 20.033/20.033 ============================== end of head =========================== 20.033/20.033 20.033/20.033 ============================== INPUT ================================= 20.033/20.033 20.033/20.033 % Reading from file /tmp/prover972947-0.in 20.033/20.033 20.033/20.033 assign(max_seconds,20). 20.033/20.033 20.033/20.033 formulas(assumptions). 20.033/20.033 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 20.033/20.033 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 20.033/20.033 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 20.033/20.033 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 20.033/20.033 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 20.033/20.033 ->_s0(a,d) # label(replacement). 20.033/20.033 ->_s0(b,d) # label(replacement). 20.033/20.033 ->*_s0(x,x) # label(reflexivity). 20.033/20.033 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(goals). 20.033/20.033 ->*_s0(b,d) & ->*_s0(a,d) # label(goal). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 ============================== end of input ========================== 20.033/20.033 20.033/20.033 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.033/20.033 20.033/20.033 % Formulas that are not ordinary clauses: 20.033/20.033 1 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 2 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 3 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 4 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 5 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 6 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.033/20.033 7 ->*_s0(b,d) & ->*_s0(a,d) # label(goal) # label(non_clause) # label(goal). [goal]. 20.033/20.033 20.033/20.033 ============================== end of process non-clausal formulas === 20.033/20.033 20.033/20.033 ============================== PROCESS INITIAL CLAUSES =============== 20.033/20.033 20.033/20.033 % Clauses before input processing: 20.033/20.033 20.033/20.033 formulas(usable). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(sos). 20.033/20.033 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.033/20.033 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.033/20.033 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.033/20.033 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.033/20.033 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 20.033/20.033 ->_s0(a,d) # label(replacement). [assumption]. 20.033/20.033 ->_s0(b,d) # label(replacement). [assumption]. 20.033/20.033 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.033/20.033 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(6)]. 20.033/20.033 -->*_s0(b,d) | -->*_s0(a,d) # label(goal). [deny(7)]. 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(demodulators). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 ============================== PREDICATE ELIMINATION ================= 20.033/20.033 20.033/20.033 No predicates eliminated. 20.033/20.033 20.033/20.033 ============================== end predicate elimination ============= 20.033/20.033 20.033/20.033 Auto_denials: 20.033/20.033 % copying label goal to answer in negative clause 20.033/20.033 20.033/20.033 Term ordering decisions: 20.033/20.033 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.033/20.033 Function symbol precedence: function_order([ d, a, b, f, g, h, s ]). 20.033/20.033 After inverse_order: (no changes). 20.033/20.033 Unfolding symbols: (none). 20.033/20.033 20.033/20.033 Auto_inference settings: 20.033/20.033 % set(neg_binary_resolution). % (HNE depth_diff=-5) 20.033/20.033 % clear(ordered_res). % (HNE depth_diff=-5) 20.033/20.033 % set(ur_resolution). % (HNE depth_diff=-5) 20.033/20.033 % set(ur_resolution) -> set(pos_ur_resolution). 20.033/20.033 % set(ur_resolution) -> set(neg_ur_resolution). 20.033/20.033 20.033/20.033 Auto_process settings: 20.033/20.033 % set(unit_deletion). % (Horn set with negative nonunits) 20.033/20.033 20.033/20.033 kept: 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.033/20.033 kept: 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.033/20.033 kept: 10 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.033/20.033 kept: 11 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.033/20.033 kept: 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 20.033/20.033 kept: 13 ->_s0(a,d) # label(replacement). [assumption]. 20.033/20.033 kept: 14 ->_s0(b,d) # label(replacement). [assumption]. 20.033/20.033 kept: 15 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.033/20.033 kept: 16 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(6)]. 20.033/20.033 kept: 17 -->*_s0(b,d) | -->*_s0(a,d) # label(goal) # answer(goal). [deny(7)]. 20.033/20.033 20.033/20.033 ============================== end of process initial clauses ======== 20.033/20.033 20.033/20.033 ============================== CLAUSES FOR SEARCH ==================== 20.033/20.033 20.033/20.033 % Clauses after input processing: 20.033/20.033 20.033/20.033 formulas(usable). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(sos). 20.033/20.033 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.033/20.033 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.033/20.033 10 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.033/20.033 11 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.033/20.033 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 20.033/20.033 13 ->_s0(a,d) # label(replacement). [assumption]. 20.033/20.033 14 ->_s0(b,d) # label(replacement). [assumption]. 20.033/20.033 15 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.033/20.033 16 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(6)]. 20.033/20.033 17 -->*_s0(b,d) | -->*_s0(a,d) # label(goal) # answer(goal). [deny(7)]. 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(demodulators). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 ============================== end of clauses for search ============= 20.033/20.033 20.033/20.033 ============================== SEARCH ================================ 20.033/20.033 20.033/20.033 % Starting search at 0.00 seconds. 20.033/20.033 20.033/20.033 given #1 (I,wt=10): 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.033/20.033 20.033/20.033 given #2 (I,wt=10): 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.033/20.033 20.033/20.033 given #3 (I,wt=8): 10 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.033/20.033 20.033/20.033 given #4 (I,wt=8): 11 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.033/20.033 20.033/20.033 given #5 (I,wt=8): 12 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 20.033/20.033 20.033/20.033 given #6 (I,wt=3): 13 ->_s0(a,d) # label(replacement). [assumption]. 20.033/20.033 20.033/20.033 given #7 (I,wt=3): 14 ->_s0(b,d) # label(replacement). [assumption]. 20.033/20.033 20.033/20.033 given #8 (I,wt=3): 15 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.033/20.033 20.033/20.033 given #9 (I,wt=9): 16 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(6)]. 20.033/20.033 20.033/20.033 ============================== PROOF ================================= 20.033/20.033 20.033/20.033 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.033/20.033 % Length of proof is 10. 20.033/20.033 % Level of proof is 3. 20.033/20.033 % Maximum clause weight is 9.000. 20.033/20.033 % Given clauses 9. 20.033/20.033 20.033/20.033 6 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.033/20.033 7 ->*_s0(b,d) & ->*_s0(a,d) # label(goal) # label(non_clause) # label(goal). [goal]. 20.033/20.033 13 ->_s0(a,d) # label(replacement). [assumption]. 20.033/20.033 14 ->_s0(b,d) # label(replacement). [assumption]. 20.033/20.033 15 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.033/20.033 16 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(6)]. 20.033/20.033 17 -->*_s0(b,d) | -->*_s0(a,d) # label(goal) # answer(goal). [deny(7)]. 20.033/20.033 28 ->*_s0(b,d). [ur(16,a,14,a,b,15,a)]. 20.033/20.033 29 ->*_s0(a,d). [ur(16,a,13,a,b,15,a)]. 20.033/20.033 30 $F # answer(goal). [back_unit_del(17),unit_del(a,28),unit_del(b,29)]. 20.033/20.033 20.033/20.033 ============================== end of proof ========================== 20.033/20.033 20.033/20.033 ============================== STATISTICS ============================ 20.033/20.033 20.033/20.033 Given=9. Generated=23. Kept=22. proofs=1. 20.033/20.033 Usable=9. Sos=10. Demods=0. Limbo=2, Disabled=11. Hints=0. 20.033/20.033 Kept_by_rule=0, Deleted_by_rule=0. 20.033/20.033 Forward_subsumed=0. Back_subsumed=0. 20.033/20.033 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.033/20.033 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=1. 20.033/20.033 Demod_attempts=0. Demod_rewrites=0. 20.033/20.033 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.033/20.033 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=10. 20.033/20.033 Megabytes=0.07. 20.033/20.033 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.033/20.033 20.033/20.033 ============================== end of statistics ===================== 20.033/20.033 20.033/20.033 ============================== end of search ========================= 20.033/20.033 20.033/20.033 THEOREM PROVED 20.033/20.033 20.033/20.033 Exiting with 1 proof. 20.033/20.033 20.033/20.033 Process 72962 exit (max_proofs) Wed Mar 9 10:04:49 2022 20.033/20.033 20.033/20.033 20.033/20.033 The problem is feasible. 20.033/20.033 20.033/20.033 Infeasible constantized CCP convergence? 20.033/20.033 YES 20.033/20.033 20.033/20.033 Problem 1: 20.033/20.033 20.033/20.033 Infeasibility Problem: 20.033/20.033 [(VAR vNonEmpty x y x1 vNonEmpty z) 20.033/20.033 (STRATEGY CONTEXTSENSITIVE 20.033/20.033 (a) 20.033/20.033 (b) 20.033/20.033 (f 1 2) 20.033/20.033 (g 1) 20.033/20.033 (h 1) 20.033/20.033 (c_x1) 20.033/20.033 (d) 20.033/20.033 (fSNonEmpty) 20.033/20.033 (s 1) 20.033/20.033 ) 20.033/20.033 (RULES 20.033/20.033 a -> d 20.033/20.033 b -> d 20.033/20.033 f(x,y) -> g(x) | a ->* d 20.033/20.033 f(x,y) -> h(x) | b ->* d 20.033/20.033 g(s(x)) -> x 20.033/20.033 h(s(x)) -> x 20.033/20.033 )] 20.033/20.033 20.033/20.033 Infeasibility Conditions: 20.033/20.033 g(c_x1) ->* z, h(c_x1) ->* z 20.033/20.033 20.033/20.033 Problem 1: 20.033/20.033 20.033/20.033 Obtaining a model using Mace4: 20.033/20.033 20.033/20.033 -> Usable Rules: 20.033/20.033 g(s(x)) -> x 20.033/20.033 h(s(x)) -> x 20.033/20.033 20.033/20.033 -> Mace4 Output: 20.033/20.033 ============================== Mace4 ================================= 20.033/20.033 Mace4 (64) version 2009-11A, November 2009. 20.033/20.033 Process 73005 was started by ubuntu on ubuntu, 20.033/20.033 Wed Mar 9 10:04:49 2022 20.033/20.033 The command was "./mace4 -c -f /tmp/mace472992-2.in". 20.033/20.033 ============================== end of head =========================== 20.033/20.033 20.033/20.033 ============================== INPUT ================================= 20.033/20.033 20.033/20.033 % Reading from file /tmp/mace472992-2.in 20.033/20.033 20.033/20.033 assign(max_seconds,20). 20.033/20.033 20.033/20.033 formulas(assumptions). 20.033/20.033 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). 20.033/20.033 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). 20.033/20.033 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence). 20.033/20.033 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence). 20.033/20.033 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 20.033/20.033 ->(g(s(x1)),x1) # label(replacement). 20.033/20.033 ->(h(s(x1)),x1) # label(replacement). 20.033/20.033 ->*(x,x) # label(reflexivity). 20.033/20.033 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 formulas(goals). 20.033/20.033 (exists x5 (->*(g(c_x1),x5) & ->*(h(c_x1),x5))) # label(goal). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 ============================== end of input ========================== 20.033/20.033 20.033/20.033 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.033/20.033 20.033/20.033 % Formulas that are not ordinary clauses: 20.033/20.033 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 3 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 4 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 5 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 20.033/20.033 6 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.033/20.033 7 (exists x5 (->*(g(c_x1),x5) & ->*(h(c_x1),x5))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.033/20.033 20.033/20.033 ============================== end of process non-clausal formulas === 20.033/20.033 20.033/20.033 ============================== CLAUSES FOR SEARCH ==================== 20.033/20.033 20.033/20.033 formulas(mace4_clauses). 20.033/20.033 -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). 20.033/20.033 -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). 20.033/20.033 -->(x,y) | ->(g(x),g(y)) # label(congruence). 20.033/20.033 -->(x,y) | ->(h(x),h(y)) # label(congruence). 20.033/20.033 -->(x,y) | ->(s(x),s(y)) # label(congruence). 20.033/20.033 ->(g(s(x)),x) # label(replacement). 20.033/20.033 ->(h(s(x)),x) # label(replacement). 20.033/20.033 ->*(x,x) # label(reflexivity). 20.033/20.033 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 20.033/20.033 -->*(g(c_x1),x) | -->*(h(c_x1),x) # label(goal). 20.033/20.033 end_of_list. 20.033/20.033 20.033/20.033 ============================== end of clauses for search ============= 20.033/20.033 20.033/20.033 % There are no natural numbers in the input. 20.033/20.033 20.033/20.033 ============================== DOMAIN SIZE 2 ========================= 20.033/20.033 20.033/20.033 ============================== STATISTICS ============================ 20.033/20.033 20.033/20.033 For domain size 2. 20.033/20.033 20.033/20.033 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.033/20.033 Ground clauses: seen=44, kept=40. 20.033/20.033 Selections=10, assignments=19, propagations=32, current_models=0. 20.033/20.033 Rewrite_terms=140, rewrite_bools=148, indexes=30. 20.033/20.033 Rules_from_neg_clauses=8, cross_offs=8. 20.033/20.033 20.033/20.033 ============================== end of statistics ===================== 20.033/20.033 20.033/20.033 ============================== DOMAIN SIZE 3 ========================= 20.033/20.033 20.033/20.033 ============================== MODEL ================================= 20.033/20.033 20.033/20.033 interpretation( 3, [number=1, seconds=0], [ 20.033/20.033 20.033/20.033 function(c_x1, [ 0 ]), 20.033/20.033 20.033/20.033 function(s(_), [ 1, 2, 2 ]), 20.033/20.033 20.033/20.033 function(g(_), [ 0, 0, 2 ]), 20.033/20.033 20.033/20.033 function(h(_), [ 1, 0, 2 ]), 20.033/20.033 20.033/20.033 function(f(_,_), [ 20.033/20.033 0, 0, 0, 20.033/20.033 0, 0, 0, 20.033/20.033 0, 0, 0 ]), 20.033/20.033 20.033/20.033 relation(->*(_,_), [ 20.033/20.033 1, 0, 0, 20.033/20.033 0, 1, 0, 20.033/20.033 1, 1, 1 ]), 20.033/20.033 20.033/20.033 relation(->(_,_), [ 20.033/20.033 1, 0, 0, 20.033/20.033 0, 1, 0, 20.033/20.033 1, 1, 1 ]) 20.033/20.033 ]). 20.033/20.033 20.033/20.033 ============================== end of model ========================== 20.033/20.033 20.033/20.033 ============================== STATISTICS ============================ 20.033/20.033 20.033/20.033 For domain size 3. 20.033/20.033 20.033/20.033 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.033/20.033 Ground clauses: seen=120, kept=111. 20.033/20.033 Selections=58, assignments=146, propagations=45, current_models=1. 20.033/20.033 Rewrite_terms=894, rewrite_bools=522, indexes=47. 20.033/20.033 Rules_from_neg_clauses=8, cross_offs=26. 20.033/20.033 20.033/20.033 ============================== end of statistics ===================== 20.033/20.033 20.033/20.033 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.033/20.033 20.033/20.033 Exiting with 1 model. 20.033/20.033 20.033/20.033 Process 73005 exit (max_models) Wed Mar 9 10:04:49 2022 20.033/20.033 The process finished Wed Mar 9 10:04:49 2022 20.033/20.033 20.033/20.033 20.033/20.033 Mace4 cooked interpretation: 20.033/20.033 20.033/20.033 20.033/20.033 20.033/20.033 The problem is infeasible. 20.033/20.033 20.033/20.033 20.033/20.033 The problem is not joinable. 20.033/20.033 15.02user 0.42system 0:20.33elapsed 76%CPU (0avgtext+0avgdata 64404maxresident)k 20.033/20.033 8inputs+0outputs (0major+34090minor)pagefaults 0swaps