25.033/25.033 NO 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 Confluence Problem: 25.033/25.033 (VAR vNonEmpty:S x:S y:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (e) 25.033/25.033 (f 1 2) 25.033/25.033 (g 1) 25.033/25.033 (h 1) 25.033/25.033 (d) 25.033/25.033 (fSNonEmpty) 25.033/25.033 (s 1) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 a -> d 25.033/25.033 b -> d 25.033/25.033 e -> e 25.033/25.033 f(x:S,y:S) -> g(x:S) | a ->* d 25.033/25.033 f(x:S,y:S) -> h(x:S) | b ->* d 25.033/25.033 g(s(x:S)) -> x:S 25.033/25.033 h(s(x:S)) -> x:S 25.033/25.033 ) 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Inlining of Conditions Processor [STERN17]: 25.033/25.033 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 Confluence Problem: 25.033/25.033 (VAR vNonEmpty:S x:S y:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (e) 25.033/25.033 (f 1 2) 25.033/25.033 (g 1) 25.033/25.033 (h 1) 25.033/25.033 (d) 25.033/25.033 (fSNonEmpty) 25.033/25.033 (s 1) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 a -> d 25.033/25.033 b -> d 25.033/25.033 e -> e 25.033/25.033 f(x:S,y:S) -> g(x:S) | a ->* d 25.033/25.033 f(x:S,y:S) -> h(x:S) | b ->* d 25.033/25.033 g(s(x:S)) -> x:S 25.033/25.033 h(s(x:S)) -> x:S 25.033/25.033 ) 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Clean CTRS Processor: 25.033/25.033 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 Confluence Problem: 25.033/25.033 (VAR x:S y:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (e) 25.033/25.033 (f 1 2) 25.033/25.033 (g 1) 25.033/25.033 (h 1) 25.033/25.033 (d) 25.033/25.033 (fSNonEmpty) 25.033/25.033 (s 1) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 a -> d 25.033/25.033 b -> d 25.033/25.033 f(x:S,y:S) -> g(x:S) | a ->* d 25.033/25.033 f(x:S,y:S) -> h(x:S) | b ->* d 25.033/25.033 g(s(x:S)) -> x:S 25.033/25.033 h(s(x:S)) -> x:S 25.033/25.033 ) 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 25.033/25.033 CRule InfChecker Info: 25.033/25.033 a -> d 25.033/25.033 Rule remains 25.033/25.033 Proof: 25.033/25.033 NO_CONDS 25.033/25.033 25.033/25.033 CRule InfChecker Info: 25.033/25.033 b -> d 25.033/25.033 Rule remains 25.033/25.033 Proof: 25.033/25.033 NO_CONDS 25.033/25.033 25.033/25.033 CRule InfChecker Info: 25.033/25.033 e -> e 25.033/25.033 Rule deleted 25.033/25.033 Proof: 25.033/25.033 SAME_LHS_AND_RHS 25.033/25.033 25.033/25.033 CRule InfChecker Info: 25.033/25.033 f(x:S,y:S) -> g(x:S) | a ->* d 25.033/25.033 Rule remains 25.033/25.033 Proof: 25.033/25.033 NO 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Infeasibility Problem: 25.033/25.033 [(VAR vNonEmpty:S x:S:S y:S:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (f 1 2) 25.033/25.033 (g 1) 25.033/25.033 (h 1) 25.033/25.033 (d) 25.033/25.033 (fSNonEmpty) 25.033/25.033 (s 1) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 a -> d 25.033/25.033 b -> d 25.033/25.033 f(x:S:S,y:S:S) -> g(x:S:S) | a ->* d 25.033/25.033 f(x:S:S,y:S:S) -> h(x:S:S) | b ->* d 25.033/25.033 g(s(x:S:S)) -> x:S:S 25.033/25.033 h(s(x:S:S)) -> x:S:S 25.033/25.033 )] 25.033/25.033 25.033/25.033 Infeasibility Conditions: 25.033/25.033 a ->* d 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Obtaining a proof using Prover9: 25.033/25.033 25.033/25.033 -> Prover9 Output: 25.033/25.033 ============================== Prover9 =============================== 25.033/25.033 Prover9 (64) version 2009-11A, November 2009. 25.033/25.033 Process 115680 was started by ubuntu on ubuntu, 25.033/25.033 Fri Jul 16 15:24:21 2021 25.033/25.033 The command was "./prover9 -f /tmp/prover9115671-0.in". 25.033/25.033 ============================== end of head =========================== 25.033/25.033 25.033/25.033 ============================== INPUT ================================= 25.033/25.033 25.033/25.033 % Reading from file /tmp/prover9115671-0.in 25.033/25.033 25.033/25.033 assign(max_seconds,20). 25.033/25.033 25.033/25.033 formulas(assumptions). 25.033/25.033 ->*_s0(x,x) # label(reflexivity). 25.033/25.033 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.033/25.033 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 25.033/25.033 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 25.033/25.033 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 25.033/25.033 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 25.033/25.033 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 25.033/25.033 ->_s0(a,d) # label(replacement). 25.033/25.033 ->_s0(b,d) # label(replacement). 25.033/25.033 ->*_s0(a,d) -> ->_s0(f(x1,x2),g(x1)) # label(replacement). 25.033/25.033 ->*_s0(b,d) -> ->_s0(f(x1,x2),h(x1)) # label(replacement). 25.033/25.033 ->_s0(g(s(x1)),x1) # label(replacement). 25.033/25.033 ->_s0(h(s(x1)),x1) # label(replacement). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(goals). 25.033/25.033 ->*_s0(a,d) # label(goal). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== end of input ========================== 25.033/25.033 25.033/25.033 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.033/25.033 25.033/25.033 % Formulas that are not ordinary clauses: 25.033/25.033 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.033/25.033 2 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 3 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 4 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 5 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 6 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 7 ->*_s0(a,d) -> ->_s0(f(x1,x2),g(x1)) # label(replacement) # label(non_clause). [assumption]. 25.033/25.033 8 ->*_s0(b,d) -> ->_s0(f(x1,x2),h(x1)) # label(replacement) # label(non_clause). [assumption]. 25.033/25.033 9 ->*_s0(a,d) # label(goal) # label(non_clause) # label(goal). [goal]. 25.033/25.033 25.033/25.033 ============================== end of process non-clausal formulas === 25.033/25.033 25.033/25.033 ============================== PROCESS INITIAL CLAUSES =============== 25.033/25.033 25.033/25.033 % Clauses before input processing: 25.033/25.033 25.033/25.033 formulas(usable). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(sos). 25.033/25.033 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.033/25.033 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.033/25.033 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.033/25.033 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.033/25.033 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.033/25.033 ->_s0(a,d) # label(replacement). [assumption]. 25.033/25.033 ->_s0(b,d) # label(replacement). [assumption]. 25.033/25.033 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.033/25.033 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.033/25.033 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 -->*_s0(a,d) # label(goal). [deny(9)]. 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(demodulators). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== PREDICATE ELIMINATION ================= 25.033/25.033 25.033/25.033 No predicates eliminated. 25.033/25.033 25.033/25.033 ============================== end predicate elimination ============= 25.033/25.033 25.033/25.033 Auto_denials: 25.033/25.033 % copying label goal to answer in negative clause 25.033/25.033 25.033/25.033 Term ordering decisions: 25.033/25.033 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.033/25.033 Function symbol precedence: function_order([ d, a, b, f, g, h, s ]). 25.033/25.033 After inverse_order: (no changes). 25.033/25.033 Unfolding symbols: (none). 25.033/25.033 25.033/25.033 Auto_inference settings: 25.033/25.033 % set(neg_binary_resolution). % (HNE depth_diff=-7) 25.033/25.033 % clear(ordered_res). % (HNE depth_diff=-7) 25.033/25.033 % set(ur_resolution). % (HNE depth_diff=-7) 25.033/25.033 % set(ur_resolution) -> set(pos_ur_resolution). 25.033/25.033 % set(ur_resolution) -> set(neg_ur_resolution). 25.033/25.033 25.033/25.033 Auto_process settings: (no changes). 25.033/25.033 25.033/25.033 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 kept: 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.033/25.033 kept: 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.033/25.033 kept: 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.033/25.033 kept: 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.033/25.033 kept: 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.033/25.033 kept: 17 ->_s0(a,d) # label(replacement). [assumption]. 25.033/25.033 kept: 18 ->_s0(b,d) # label(replacement). [assumption]. 25.033/25.033 kept: 19 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.033/25.033 kept: 20 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.033/25.033 kept: 21 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 kept: 22 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 kept: 23 -->*_s0(a,d) # label(goal) # answer(goal). [deny(9)]. 25.033/25.033 25.033/25.033 ============================== end of process initial clauses ======== 25.033/25.033 25.033/25.033 ============================== CLAUSES FOR SEARCH ==================== 25.033/25.033 25.033/25.033 % Clauses after input processing: 25.033/25.033 25.033/25.033 formulas(usable). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(sos). 25.033/25.033 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.033/25.033 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.033/25.033 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.033/25.033 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.033/25.033 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.033/25.033 17 ->_s0(a,d) # label(replacement). [assumption]. 25.033/25.033 18 ->_s0(b,d) # label(replacement). [assumption]. 25.033/25.033 20 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.033/25.033 21 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 22 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 23 -->*_s0(a,d) # label(goal) # answer(goal). [deny(9)]. 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(demodulators). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== end of clauses for search ============= 25.033/25.033 25.033/25.033 ============================== SEARCH ================================ 25.033/25.033 25.033/25.033 % Starting search at 0.00 seconds. 25.033/25.033 25.033/25.033 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 25.033/25.033 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 25.033/25.033 given #3 (I,wt=10): 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.033/25.033 25.033/25.033 given #4 (I,wt=10): 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.033/25.033 25.033/25.033 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.033/25.033 25.033/25.033 given #6 (I,wt=8): 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.033/25.033 25.033/25.033 given #7 (I,wt=8): 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.033/25.033 25.033/25.033 given #8 (I,wt=3): 17 ->_s0(a,d) # label(replacement). [assumption]. 25.033/25.033 25.033/25.033 ============================== PROOF ================================= 25.033/25.033 25.033/25.033 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.033/25.033 % Length of proof is 8. 25.033/25.033 % Level of proof is 3. 25.033/25.033 % Maximum clause weight is 9.000. 25.033/25.033 % Given clauses 8. 25.033/25.033 25.033/25.033 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.033/25.033 9 ->*_s0(a,d) # label(goal) # label(non_clause) # label(goal). [goal]. 25.033/25.033 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 17 ->_s0(a,d) # label(replacement). [assumption]. 25.033/25.033 23 -->*_s0(a,d) # label(goal) # answer(goal). [deny(9)]. 25.033/25.033 29 ->*_s0(a,d). [ur(11,a,17,a,b,10,a)]. 25.033/25.033 30 $F # answer(goal). [resolve(29,a,23,a)]. 25.033/25.033 25.033/25.033 ============================== end of proof ========================== 25.033/25.033 25.033/25.033 ============================== STATISTICS ============================ 25.033/25.033 25.033/25.033 Given=8. Generated=20. Kept=20. proofs=1. 25.033/25.033 Usable=8. Sos=5. Demods=0. Limbo=5, Disabled=15. Hints=0. 25.033/25.033 Kept_by_rule=0, Deleted_by_rule=0. 25.033/25.033 Forward_subsumed=0. Back_subsumed=1. 25.033/25.033 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.033/25.033 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.033/25.033 Demod_attempts=0. Demod_rewrites=0. 25.033/25.033 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.033/25.033 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=12. 25.033/25.033 Megabytes=0.07. 25.033/25.033 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.033/25.033 25.033/25.033 ============================== end of statistics ===================== 25.033/25.033 25.033/25.033 ============================== end of search ========================= 25.033/25.033 25.033/25.033 THEOREM PROVED 25.033/25.033 25.033/25.033 Exiting with 1 proof. 25.033/25.033 25.033/25.033 Process 115680 exit (max_proofs) Fri Jul 16 15:24:21 2021 25.033/25.033 25.033/25.033 25.033/25.033 The problem is feasible. 25.033/25.033 25.033/25.033 25.033/25.033 CRule InfChecker Info: 25.033/25.033 f(x:S,y:S) -> h(x:S) | b ->* d 25.033/25.033 Rule remains 25.033/25.033 Proof: 25.033/25.033 NO 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Infeasibility Problem: 25.033/25.033 [(VAR vNonEmpty:S x:S:S y:S:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (f 1 2) 25.033/25.033 (g 1) 25.033/25.033 (h 1) 25.033/25.033 (d) 25.033/25.033 (fSNonEmpty) 25.033/25.033 (s 1) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 a -> d 25.033/25.033 b -> d 25.033/25.033 f(x:S:S,y:S:S) -> g(x:S:S) | a ->* d 25.033/25.033 f(x:S:S,y:S:S) -> h(x:S:S) | b ->* d 25.033/25.033 g(s(x:S:S)) -> x:S:S 25.033/25.033 h(s(x:S:S)) -> x:S:S 25.033/25.033 )] 25.033/25.033 25.033/25.033 Infeasibility Conditions: 25.033/25.033 b ->* d 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Obtaining a proof using Prover9: 25.033/25.033 25.033/25.033 -> Prover9 Output: 25.033/25.033 ============================== Prover9 =============================== 25.033/25.033 Prover9 (64) version 2009-11A, November 2009. 25.033/25.033 Process 115702 was started by ubuntu on ubuntu, 25.033/25.033 Fri Jul 16 15:24:21 2021 25.033/25.033 The command was "./prover9 -f /tmp/prover9115693-0.in". 25.033/25.033 ============================== end of head =========================== 25.033/25.033 25.033/25.033 ============================== INPUT ================================= 25.033/25.033 25.033/25.033 % Reading from file /tmp/prover9115693-0.in 25.033/25.033 25.033/25.033 assign(max_seconds,20). 25.033/25.033 25.033/25.033 formulas(assumptions). 25.033/25.033 ->*_s0(x,x) # label(reflexivity). 25.033/25.033 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.033/25.033 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 25.033/25.033 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 25.033/25.033 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 25.033/25.033 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 25.033/25.033 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 25.033/25.033 ->_s0(a,d) # label(replacement). 25.033/25.033 ->_s0(b,d) # label(replacement). 25.033/25.033 ->*_s0(a,d) -> ->_s0(f(x1,x2),g(x1)) # label(replacement). 25.033/25.033 ->*_s0(b,d) -> ->_s0(f(x1,x2),h(x1)) # label(replacement). 25.033/25.033 ->_s0(g(s(x1)),x1) # label(replacement). 25.033/25.033 ->_s0(h(s(x1)),x1) # label(replacement). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(goals). 25.033/25.033 ->*_s0(b,d) # label(goal). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== end of input ========================== 25.033/25.033 25.033/25.033 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.033/25.033 25.033/25.033 % Formulas that are not ordinary clauses: 25.033/25.033 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.033/25.033 2 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 3 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 4 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 5 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 6 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 7 ->*_s0(a,d) -> ->_s0(f(x1,x2),g(x1)) # label(replacement) # label(non_clause). [assumption]. 25.033/25.033 8 ->*_s0(b,d) -> ->_s0(f(x1,x2),h(x1)) # label(replacement) # label(non_clause). [assumption]. 25.033/25.033 9 ->*_s0(b,d) # label(goal) # label(non_clause) # label(goal). [goal]. 25.033/25.033 25.033/25.033 ============================== end of process non-clausal formulas === 25.033/25.033 25.033/25.033 ============================== PROCESS INITIAL CLAUSES =============== 25.033/25.033 25.033/25.033 % Clauses before input processing: 25.033/25.033 25.033/25.033 formulas(usable). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(sos). 25.033/25.033 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.033/25.033 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.033/25.033 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.033/25.033 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.033/25.033 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.033/25.033 ->_s0(a,d) # label(replacement). [assumption]. 25.033/25.033 ->_s0(b,d) # label(replacement). [assumption]. 25.033/25.033 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.033/25.033 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.033/25.033 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 -->*_s0(b,d) # label(goal). [deny(9)]. 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(demodulators). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== PREDICATE ELIMINATION ================= 25.033/25.033 25.033/25.033 No predicates eliminated. 25.033/25.033 25.033/25.033 ============================== end predicate elimination ============= 25.033/25.033 25.033/25.033 Auto_denials: 25.033/25.033 % copying label goal to answer in negative clause 25.033/25.033 25.033/25.033 Term ordering decisions: 25.033/25.033 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.033/25.033 Function symbol precedence: function_order([ d, a, b, f, g, h, s ]). 25.033/25.033 After inverse_order: (no changes). 25.033/25.033 Unfolding symbols: (none). 25.033/25.033 25.033/25.033 Auto_inference settings: 25.033/25.033 % set(neg_binary_resolution). % (HNE depth_diff=-7) 25.033/25.033 % clear(ordered_res). % (HNE depth_diff=-7) 25.033/25.033 % set(ur_resolution). % (HNE depth_diff=-7) 25.033/25.033 % set(ur_resolution) -> set(pos_ur_resolution). 25.033/25.033 % set(ur_resolution) -> set(neg_ur_resolution). 25.033/25.033 25.033/25.033 Auto_process settings: (no changes). 25.033/25.033 25.033/25.033 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 kept: 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.033/25.033 kept: 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.033/25.033 kept: 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.033/25.033 kept: 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.033/25.033 kept: 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.033/25.033 kept: 17 ->_s0(a,d) # label(replacement). [assumption]. 25.033/25.033 kept: 18 ->_s0(b,d) # label(replacement). [assumption]. 25.033/25.033 kept: 19 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.033/25.033 kept: 20 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.033/25.033 kept: 21 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 kept: 22 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 kept: 23 -->*_s0(b,d) # label(goal) # answer(goal). [deny(9)]. 25.033/25.033 25.033/25.033 ============================== end of process initial clauses ======== 25.033/25.033 25.033/25.033 ============================== CLAUSES FOR SEARCH ==================== 25.033/25.033 25.033/25.033 % Clauses after input processing: 25.033/25.033 25.033/25.033 formulas(usable). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(sos). 25.033/25.033 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.033/25.033 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.033/25.033 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.033/25.033 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.033/25.033 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.033/25.033 17 ->_s0(a,d) # label(replacement). [assumption]. 25.033/25.033 18 ->_s0(b,d) # label(replacement). [assumption]. 25.033/25.033 19 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.033/25.033 21 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 22 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 23 -->*_s0(b,d) # label(goal) # answer(goal). [deny(9)]. 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(demodulators). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== end of clauses for search ============= 25.033/25.033 25.033/25.033 ============================== SEARCH ================================ 25.033/25.033 25.033/25.033 % Starting search at 0.00 seconds. 25.033/25.033 25.033/25.033 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 25.033/25.033 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 25.033/25.033 given #3 (I,wt=10): 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.033/25.033 25.033/25.033 given #4 (I,wt=10): 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.033/25.033 25.033/25.033 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.033/25.033 25.033/25.033 given #6 (I,wt=8): 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.033/25.033 25.033/25.033 given #7 (I,wt=8): 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.033/25.033 25.033/25.033 given #8 (I,wt=3): 17 ->_s0(a,d) # label(replacement). [assumption]. 25.033/25.033 25.033/25.033 given #9 (I,wt=3): 18 ->_s0(b,d) # label(replacement). [assumption]. 25.033/25.033 25.033/25.033 ============================== PROOF ================================= 25.033/25.033 25.033/25.033 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.033/25.033 % Length of proof is 8. 25.033/25.033 % Level of proof is 3. 25.033/25.033 % Maximum clause weight is 9.000. 25.033/25.033 % Given clauses 9. 25.033/25.033 25.033/25.033 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.033/25.033 9 ->*_s0(b,d) # label(goal) # label(non_clause) # label(goal). [goal]. 25.033/25.033 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 18 ->_s0(b,d) # label(replacement). [assumption]. 25.033/25.033 23 -->*_s0(b,d) # label(goal) # answer(goal). [deny(9)]. 25.033/25.033 35 ->*_s0(b,d). [ur(11,a,18,a,b,10,a)]. 25.033/25.033 36 $F # answer(goal). [resolve(35,a,23,a)]. 25.033/25.033 25.033/25.033 ============================== end of proof ========================== 25.033/25.033 25.033/25.033 ============================== STATISTICS ============================ 25.033/25.033 25.033/25.033 Given=9. Generated=26. Kept=26. proofs=1. 25.033/25.033 Usable=9. Sos=10. Demods=0. Limbo=5, Disabled=15. Hints=0. 25.033/25.033 Kept_by_rule=0, Deleted_by_rule=0. 25.033/25.033 Forward_subsumed=0. Back_subsumed=1. 25.033/25.033 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.033/25.033 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.033/25.033 Demod_attempts=0. Demod_rewrites=0. 25.033/25.033 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.033/25.033 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=12. 25.033/25.033 Megabytes=0.08. 25.033/25.033 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.033/25.033 25.033/25.033 ============================== end of statistics ===================== 25.033/25.033 25.033/25.033 ============================== end of search ========================= 25.033/25.033 25.033/25.033 THEOREM PROVED 25.033/25.033 25.033/25.033 Exiting with 1 proof. 25.033/25.033 25.033/25.033 Process 115702 exit (max_proofs) Fri Jul 16 15:24:21 2021 25.033/25.033 25.033/25.033 25.033/25.033 The problem is feasible. 25.033/25.033 25.033/25.033 25.033/25.033 CRule InfChecker Info: 25.033/25.033 g(s(x:S)) -> x:S 25.033/25.033 Rule remains 25.033/25.033 Proof: 25.033/25.033 NO_CONDS 25.033/25.033 25.033/25.033 CRule InfChecker Info: 25.033/25.033 h(s(x:S)) -> x:S 25.033/25.033 Rule remains 25.033/25.033 Proof: 25.033/25.033 NO_CONDS 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 Confluence Problem: 25.033/25.033 (VAR x:S y:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (e) 25.033/25.033 (f 1 2) 25.033/25.033 (g 1) 25.033/25.033 (h 1) 25.033/25.033 (d) 25.033/25.033 (fSNonEmpty) 25.033/25.033 (s 1) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 a -> d 25.033/25.033 b -> d 25.033/25.033 f(x:S,y:S) -> g(x:S) | a ->* d 25.033/25.033 f(x:S,y:S) -> h(x:S) | b ->* d 25.033/25.033 g(s(x:S)) -> x:S 25.033/25.033 h(s(x:S)) -> x:S 25.033/25.033 ) 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 25.033/25.033 Critical Pairs Processor: 25.033/25.033 -> Rules: 25.033/25.033 a -> d 25.033/25.033 b -> d 25.033/25.033 f(x:S,y:S) -> g(x:S) | a ->* d 25.033/25.033 f(x:S,y:S) -> h(x:S) | b ->* d 25.033/25.033 g(s(x:S)) -> x:S 25.033/25.033 h(s(x:S)) -> x:S 25.033/25.033 -> Vars: 25.033/25.033 "x", "y", "x", "y", "x", "x" 25.033/25.033 -> FVars: 25.033/25.033 "x3", "x4", "x5", "x6", "x7", "x8" 25.033/25.033 -> PVars: 25.033/25.033 "x": ["x3", "x5", "x7", "x8"], "y": ["x4", "x6"] 25.033/25.033 25.033/25.033 -> Rlps: 25.033/25.033 crule: a -> d, id: 1, possubterms: a-> [] 25.033/25.033 crule: b -> d, id: 2, possubterms: b-> [] 25.033/25.033 crule: f(x3:S,x4:S) -> g(x3:S) | a ->* d, id: 3, possubterms: f(x3:S,x4:S)-> [] 25.033/25.033 crule: f(x5:S,x6:S) -> h(x5:S) | b ->* d, id: 4, possubterms: f(x5:S,x6:S)-> [] 25.033/25.033 crule: g(s(x7:S)) -> x7:S, id: 5, possubterms: g(s(x7:S))-> [], s(x7:S)-> [1] 25.033/25.033 crule: h(s(x8:S)) -> x8:S, id: 6, possubterms: h(s(x8:S))-> [], s(x8:S)-> [1] 25.033/25.033 25.033/25.033 -> Unifications: 25.033/25.033 R4 unifies with R3 at p: [], l: f(x5:S,x6:S), lp: f(x5:S,x6:S), conds: {b ->* d, a ->* d}, sig: {x5:S -> x:S,x6:S -> y:S}, l': f(x:S,y:S), r: h(x5:S), r': g(x:S) 25.033/25.033 25.033/25.033 -> Critical pairs info: 25.033/25.033 | b ->* d, a ->* d => Not trivial, Overlay, N1 25.033/25.033 25.033/25.033 -> Problem conclusions: 25.033/25.033 Left linear, Right linear, Linear 25.033/25.033 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 25.033/25.033 CTRS Type: 1 25.033/25.033 Deterministic, Strongly deterministic 25.033/25.033 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.033/25.033 Maybe right-stable CTRS, Overlay CTRS 25.033/25.033 Maybe normal CTRS, Maybe almost normal CTRS 25.033/25.033 Maybe terminating CTRS, Maybe joinable CCPs 25.033/25.033 Maybe level confluent 25.033/25.033 Maybe confluent 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 Confluence Problem: 25.033/25.033 (VAR x:S y:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (e) 25.033/25.033 (f 1 2) 25.033/25.033 (g 1) 25.033/25.033 (h 1) 25.033/25.033 (d) 25.033/25.033 (fSNonEmpty) 25.033/25.033 (s 1) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 a -> d 25.033/25.033 b -> d 25.033/25.033 f(x:S,y:S) -> g(x:S) | a ->* d 25.033/25.033 f(x:S,y:S) -> h(x:S) | b ->* d 25.033/25.033 g(s(x:S)) -> x:S 25.033/25.033 h(s(x:S)) -> x:S 25.033/25.033 ) 25.033/25.033 Critical Pairs: 25.033/25.033 | b ->* d, a ->* d => Not trivial, Overlay, N1 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 25.033/25.033 Conditional Critical Pairs Distributor Processor 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 Confluence Problem: 25.033/25.033 (VAR x:S y:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (e) 25.033/25.033 (f 1 2) 25.033/25.033 (g 1) 25.033/25.033 (h 1) 25.033/25.033 (d) 25.033/25.033 (fSNonEmpty) 25.033/25.033 (s 1) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 a -> d 25.033/25.033 b -> d 25.033/25.033 f(x:S,y:S) -> g(x:S) | a ->* d 25.033/25.033 f(x:S,y:S) -> h(x:S) | b ->* d 25.033/25.033 g(s(x:S)) -> x:S 25.033/25.033 h(s(x:S)) -> x:S 25.033/25.033 ) 25.033/25.033 Critical Pairs: 25.033/25.033 | b ->* d, a ->* d => Not trivial, Overlay, N1 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 25.033/25.033 Infeasible Convergence Constantized CCP Processor: 25.033/25.033 25.033/25.033 Infeasible constantized conditions? 25.033/25.033 NO 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Infeasibility Problem: 25.033/25.033 [(VAR vNonEmpty:S x:S y:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (f 1 2) 25.033/25.033 (g 1) 25.033/25.033 (h 1) 25.033/25.033 (d) 25.033/25.033 (fSNonEmpty) 25.033/25.033 (s 1) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 a -> d 25.033/25.033 b -> d 25.033/25.033 f(x:S,y:S) -> g(x:S) | a ->* d 25.033/25.033 f(x:S,y:S) -> h(x:S) | b ->* d 25.033/25.033 g(s(x:S)) -> x:S 25.033/25.033 h(s(x:S)) -> x:S 25.033/25.033 )] 25.033/25.033 25.033/25.033 Infeasibility Conditions: 25.033/25.033 b ->* d, a ->* d 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Obtaining a proof using Prover9: 25.033/25.033 25.033/25.033 -> Prover9 Output: 25.033/25.033 ============================== Prover9 =============================== 25.033/25.033 Prover9 (64) version 2009-11A, November 2009. 25.033/25.033 Process 115735 was started by ubuntu on ubuntu, 25.033/25.033 Fri Jul 16 15:24:22 2021 25.033/25.033 The command was "./prover9 -f /tmp/prover9115718-0.in". 25.033/25.033 ============================== end of head =========================== 25.033/25.033 25.033/25.033 ============================== INPUT ================================= 25.033/25.033 25.033/25.033 % Reading from file /tmp/prover9115718-0.in 25.033/25.033 25.033/25.033 assign(max_seconds,20). 25.033/25.033 25.033/25.033 formulas(assumptions). 25.033/25.033 ->*_s0(x,x) # label(reflexivity). 25.033/25.033 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.033/25.033 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 25.033/25.033 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 25.033/25.033 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 25.033/25.033 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 25.033/25.033 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 25.033/25.033 ->_s0(a,d) # label(replacement). 25.033/25.033 ->_s0(b,d) # label(replacement). 25.033/25.033 ->*_s0(a,d) -> ->_s0(f(x1,x2),g(x1)) # label(replacement). 25.033/25.033 ->*_s0(b,d) -> ->_s0(f(x1,x2),h(x1)) # label(replacement). 25.033/25.033 ->_s0(g(s(x1)),x1) # label(replacement). 25.033/25.033 ->_s0(h(s(x1)),x1) # label(replacement). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(goals). 25.033/25.033 ->*_s0(b,d) & ->*_s0(a,d) # label(goal). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== end of input ========================== 25.033/25.033 25.033/25.033 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.033/25.033 25.033/25.033 % Formulas that are not ordinary clauses: 25.033/25.033 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.033/25.033 2 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 3 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 4 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 5 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 6 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 7 ->*_s0(a,d) -> ->_s0(f(x1,x2),g(x1)) # label(replacement) # label(non_clause). [assumption]. 25.033/25.033 8 ->*_s0(b,d) -> ->_s0(f(x1,x2),h(x1)) # label(replacement) # label(non_clause). [assumption]. 25.033/25.033 9 ->*_s0(b,d) & ->*_s0(a,d) # label(goal) # label(non_clause) # label(goal). [goal]. 25.033/25.033 25.033/25.033 ============================== end of process non-clausal formulas === 25.033/25.033 25.033/25.033 ============================== PROCESS INITIAL CLAUSES =============== 25.033/25.033 25.033/25.033 % Clauses before input processing: 25.033/25.033 25.033/25.033 formulas(usable). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(sos). 25.033/25.033 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.033/25.033 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.033/25.033 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.033/25.033 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.033/25.033 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.033/25.033 ->_s0(a,d) # label(replacement). [assumption]. 25.033/25.033 ->_s0(b,d) # label(replacement). [assumption]. 25.033/25.033 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.033/25.033 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.033/25.033 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 -->*_s0(b,d) | -->*_s0(a,d) # label(goal). [deny(9)]. 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(demodulators). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== PREDICATE ELIMINATION ================= 25.033/25.033 25.033/25.033 No predicates eliminated. 25.033/25.033 25.033/25.033 ============================== end predicate elimination ============= 25.033/25.033 25.033/25.033 Auto_denials: 25.033/25.033 % copying label goal to answer in negative clause 25.033/25.033 25.033/25.033 Term ordering decisions: 25.033/25.033 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.033/25.033 Function symbol precedence: function_order([ d, a, b, f, g, h, s ]). 25.033/25.033 After inverse_order: (no changes). 25.033/25.033 Unfolding symbols: (none). 25.033/25.033 25.033/25.033 Auto_inference settings: 25.033/25.033 % set(neg_binary_resolution). % (HNE depth_diff=-7) 25.033/25.033 % clear(ordered_res). % (HNE depth_diff=-7) 25.033/25.033 % set(ur_resolution). % (HNE depth_diff=-7) 25.033/25.033 % set(ur_resolution) -> set(pos_ur_resolution). 25.033/25.033 % set(ur_resolution) -> set(neg_ur_resolution). 25.033/25.033 25.033/25.033 Auto_process settings: 25.033/25.033 % set(unit_deletion). % (Horn set with negative nonunits) 25.033/25.033 25.033/25.033 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 kept: 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.033/25.033 kept: 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.033/25.033 kept: 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.033/25.033 kept: 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.033/25.033 kept: 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.033/25.033 kept: 17 ->_s0(a,d) # label(replacement). [assumption]. 25.033/25.033 kept: 18 ->_s0(b,d) # label(replacement). [assumption]. 25.033/25.033 kept: 19 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.033/25.033 kept: 20 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.033/25.033 kept: 21 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 kept: 22 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 kept: 23 -->*_s0(b,d) | -->*_s0(a,d) # label(goal) # answer(goal). [deny(9)]. 25.033/25.033 25.033/25.033 ============================== end of process initial clauses ======== 25.033/25.033 25.033/25.033 ============================== CLAUSES FOR SEARCH ==================== 25.033/25.033 25.033/25.033 % Clauses after input processing: 25.033/25.033 25.033/25.033 formulas(usable). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(sos). 25.033/25.033 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.033/25.033 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.033/25.033 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.033/25.033 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.033/25.033 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.033/25.033 17 ->_s0(a,d) # label(replacement). [assumption]. 25.033/25.033 18 ->_s0(b,d) # label(replacement). [assumption]. 25.033/25.033 19 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.033/25.033 20 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.033/25.033 21 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 22 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.033/25.033 23 -->*_s0(b,d) | -->*_s0(a,d) # label(goal) # answer(goal). [deny(9)]. 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(demodulators). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== end of clauses for search ============= 25.033/25.033 25.033/25.033 ============================== SEARCH ================================ 25.033/25.033 25.033/25.033 % Starting search at 0.00 seconds. 25.033/25.033 25.033/25.033 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 25.033/25.033 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 25.033/25.033 given #3 (I,wt=10): 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.033/25.033 25.033/25.033 given #4 (I,wt=10): 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.033/25.033 25.033/25.033 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.033/25.033 25.033/25.033 given #6 (I,wt=8): 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.033/25.033 25.033/25.033 given #7 (I,wt=8): 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.033/25.033 25.033/25.033 given #8 (I,wt=3): 17 ->_s0(a,d) # label(replacement). [assumption]. 25.033/25.033 25.033/25.033 given #9 (I,wt=3): 18 ->_s0(b,d) # label(replacement). [assumption]. 25.033/25.033 25.033/25.033 ============================== PROOF ================================= 25.033/25.033 25.033/25.033 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.033/25.033 % Length of proof is 10. 25.033/25.033 % Level of proof is 4. 25.033/25.033 % Maximum clause weight is 9.000. 25.033/25.033 % Given clauses 9. 25.033/25.033 25.033/25.033 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.033/25.033 9 ->*_s0(b,d) & ->*_s0(a,d) # label(goal) # label(non_clause) # label(goal). [goal]. 25.033/25.033 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 17 ->_s0(a,d) # label(replacement). [assumption]. 25.033/25.033 18 ->_s0(b,d) # label(replacement). [assumption]. 25.033/25.033 23 -->*_s0(b,d) | -->*_s0(a,d) # label(goal) # answer(goal). [deny(9)]. 25.033/25.033 29 ->*_s0(a,d). [ur(11,a,17,a,b,10,a)]. 25.033/25.033 30 -->*_s0(b,d) # answer(goal). [back_unit_del(23),unit_del(b,29)]. 25.033/25.033 37 $F # answer(goal). [ur(11,a,18,a,b,10,a),unit_del(a,30)]. 25.033/25.033 25.033/25.033 ============================== end of proof ========================== 25.033/25.033 25.033/25.033 ============================== STATISTICS ============================ 25.033/25.033 25.033/25.033 Given=9. Generated=28. Kept=27. proofs=1. 25.033/25.033 Usable=9. Sos=10. Demods=0. Limbo=5, Disabled=17. Hints=0. 25.033/25.033 Kept_by_rule=0, Deleted_by_rule=0. 25.033/25.033 Forward_subsumed=0. Back_subsumed=1. 25.033/25.033 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.033/25.033 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=2. 25.033/25.033 Demod_attempts=0. Demod_rewrites=0. 25.033/25.033 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.033/25.033 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=13. 25.033/25.033 Megabytes=0.08. 25.033/25.033 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.033/25.033 25.033/25.033 ============================== end of statistics ===================== 25.033/25.033 25.033/25.033 ============================== end of search ========================= 25.033/25.033 25.033/25.033 THEOREM PROVED 25.033/25.033 25.033/25.033 Exiting with 1 proof. 25.033/25.033 25.033/25.033 Process 115735 exit (max_proofs) Fri Jul 16 15:24:22 2021 25.033/25.033 25.033/25.033 25.033/25.033 The problem is feasible. 25.033/25.033 25.033/25.033 Infeasible constantized CCP convergence? 25.033/25.033 YES 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Infeasibility Problem: 25.033/25.033 [(VAR vNonEmpty:S x:S y:S z:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (f 1 2) 25.033/25.033 (g 1) 25.033/25.033 (h 1) 25.033/25.033 (cx) 25.033/25.033 (d) 25.033/25.033 (fSNonEmpty) 25.033/25.033 (s 1) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 a -> d 25.033/25.033 b -> d 25.033/25.033 f(x:S,y:S) -> g(x:S) | a ->* d 25.033/25.033 f(x:S,y:S) -> h(x:S) | b ->* d 25.033/25.033 g(s(x:S)) -> x:S 25.033/25.033 h(s(x:S)) -> x:S 25.033/25.033 )] 25.033/25.033 25.033/25.033 Infeasibility Conditions: 25.033/25.033 g(cx) ->* z:S, h(cx) ->* z:S 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Obtaining a model using Mace4: 25.033/25.033 25.033/25.033 -> Usable Rules: 25.033/25.033 g(s(x:S)) -> x:S 25.033/25.033 h(s(x:S)) -> x:S 25.033/25.033 25.033/25.033 -> Mace4 Output: 25.033/25.033 ============================== Mace4 ================================= 25.033/25.033 Mace4 (64) version 2009-11A, November 2009. 25.033/25.033 Process 115766 was started by ubuntu on ubuntu, 25.033/25.033 Fri Jul 16 15:24:22 2021 25.033/25.033 The command was "./mace4 -c -f /tmp/mace4115751-2.in". 25.033/25.033 ============================== end of head =========================== 25.033/25.033 25.033/25.033 ============================== INPUT ================================= 25.033/25.033 25.033/25.033 % Reading from file /tmp/mace4115751-2.in 25.033/25.033 25.033/25.033 assign(max_seconds,10). 25.033/25.033 25.033/25.033 formulas(assumptions). 25.033/25.033 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). 25.033/25.033 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). 25.033/25.033 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence). 25.033/25.033 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence). 25.033/25.033 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 25.033/25.033 ->(g(s(x1)),x1) # label(replacement). 25.033/25.033 ->(h(s(x1)),x1) # label(replacement). 25.033/25.033 ->*(x,x) # label(reflexivity). 25.033/25.033 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(goals). 25.033/25.033 (exists x3 (->*(g(cx),x3) & ->*(h(cx),x3))) # label(goal). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== end of input ========================== 25.033/25.033 25.033/25.033 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.033/25.033 25.033/25.033 % Formulas that are not ordinary clauses: 25.033/25.033 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 3 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 4 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 5 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 6 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.033/25.033 7 (exists x3 (->*(g(cx),x3) & ->*(h(cx),x3))) # label(goal) # label(non_clause) # label(goal). [goal]. 25.033/25.033 25.033/25.033 ============================== end of process non-clausal formulas === 25.033/25.033 25.033/25.033 ============================== CLAUSES FOR SEARCH ==================== 25.033/25.033 25.033/25.033 formulas(mace4_clauses). 25.033/25.033 -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). 25.033/25.033 -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). 25.033/25.033 -->(x,y) | ->(g(x),g(y)) # label(congruence). 25.033/25.033 -->(x,y) | ->(h(x),h(y)) # label(congruence). 25.033/25.033 -->(x,y) | ->(s(x),s(y)) # label(congruence). 25.033/25.033 ->(g(s(x)),x) # label(replacement). 25.033/25.033 ->(h(s(x)),x) # label(replacement). 25.033/25.033 ->*(x,x) # label(reflexivity). 25.033/25.033 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 25.033/25.033 -->*(g(cx),x) | -->*(h(cx),x) # label(goal). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== end of clauses for search ============= 25.033/25.033 25.033/25.033 % There are no natural numbers in the input. 25.033/25.033 25.033/25.033 ============================== DOMAIN SIZE 2 ========================= 25.033/25.033 25.033/25.033 ============================== STATISTICS ============================ 25.033/25.033 25.033/25.033 For domain size 2. 25.033/25.033 25.033/25.033 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 25.033/25.033 Ground clauses: seen=44, kept=40. 25.033/25.033 Selections=10, assignments=19, propagations=32, current_models=0. 25.033/25.033 Rewrite_terms=140, rewrite_bools=148, indexes=30. 25.033/25.033 Rules_from_neg_clauses=8, cross_offs=8. 25.033/25.033 25.033/25.033 ============================== end of statistics ===================== 25.033/25.033 25.033/25.033 ============================== DOMAIN SIZE 3 ========================= 25.033/25.033 25.033/25.033 ============================== MODEL ================================= 25.033/25.033 25.033/25.033 interpretation( 3, [number=1, seconds=0], [ 25.033/25.033 25.033/25.033 function(cx, [ 0 ]), 25.033/25.033 25.033/25.033 function(s(_), [ 1, 2, 2 ]), 25.033/25.033 25.033/25.033 function(g(_), [ 0, 0, 2 ]), 25.033/25.033 25.033/25.033 function(h(_), [ 1, 0, 2 ]), 25.033/25.033 25.033/25.033 function(f(_,_), [ 25.033/25.033 0, 0, 0, 25.033/25.033 0, 0, 0, 25.033/25.033 0, 0, 0 ]), 25.033/25.033 25.033/25.033 relation(->*(_,_), [ 25.033/25.033 1, 0, 0, 25.033/25.033 0, 1, 0, 25.033/25.033 1, 1, 1 ]), 25.033/25.033 25.033/25.033 relation(->(_,_), [ 25.033/25.033 1, 0, 0, 25.033/25.033 0, 1, 0, 25.033/25.033 1, 1, 1 ]) 25.033/25.033 ]). 25.033/25.033 25.033/25.033 ============================== end of model ========================== 25.033/25.033 25.033/25.033 ============================== STATISTICS ============================ 25.033/25.033 25.033/25.033 For domain size 3. 25.033/25.033 25.033/25.033 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 25.033/25.033 Ground clauses: seen=120, kept=111. 25.033/25.033 Selections=58, assignments=146, propagations=45, current_models=1. 25.033/25.033 Rewrite_terms=894, rewrite_bools=522, indexes=47. 25.033/25.033 Rules_from_neg_clauses=8, cross_offs=26. 25.033/25.033 25.033/25.033 ============================== end of statistics ===================== 25.033/25.033 25.033/25.033 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.033/25.033 25.033/25.033 Exiting with 1 model. 25.033/25.033 25.033/25.033 Process 115766 exit (max_models) Fri Jul 16 15:24:22 2021 25.033/25.033 The process finished Fri Jul 16 15:24:22 2021 25.033/25.033 25.033/25.033 25.033/25.033 Mace4 cooked interpretation: 25.033/25.033 25.033/25.033 25.033/25.033 25.033/25.033 The problem is infeasible. 25.033/25.033 25.033/25.033 25.033/25.033 The problem is not joinable. 25.033/25.033 19.27user 0.73system 0:25.33elapsed 78%CPU (0avgtext+0avgdata 72492maxresident)k 25.033/25.033 8inputs+0outputs (0major+48118minor)pagefaults 0swaps