25.035/25.035 NO 25.035/25.035 25.035/25.035 Problem 1: 25.035/25.035 25.035/25.035 25.035/25.035 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.035/25.035 Confluence Problem: 25.035/25.035 (VAR vNonEmpty:S x:S y:S) 25.035/25.035 (STRATEGY CONTEXTSENSITIVE 25.035/25.035 (a) 25.035/25.035 (b) 25.035/25.035 (e) 25.035/25.035 (f 1 2) 25.035/25.035 (g 1) 25.035/25.035 (h 1) 25.035/25.035 (d) 25.035/25.035 (fSNonEmpty) 25.035/25.035 (s 1) 25.035/25.035 ) 25.035/25.035 (RULES 25.035/25.035 a -> d 25.035/25.035 b -> d 25.035/25.035 e -> e 25.035/25.035 f(x:S,y:S) -> g(x:S) | a ->* d 25.035/25.035 f(x:S,y:S) -> h(x:S) | b ->* d 25.035/25.035 g(s(x:S)) -> x:S 25.035/25.035 h(s(x:S)) -> x:S 25.035/25.035 ) 25.035/25.035 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.035/25.035 25.035/25.035 25.035/25.035 Problem 1: 25.035/25.035 25.035/25.035 Inlining of Conditions Processor [STERN17]: 25.035/25.035 25.035/25.035 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.035/25.035 Confluence Problem: 25.035/25.035 (VAR vNonEmpty:S x:S y:S) 25.035/25.035 (STRATEGY CONTEXTSENSITIVE 25.035/25.035 (a) 25.035/25.035 (b) 25.035/25.035 (e) 25.035/25.035 (f 1 2) 25.035/25.035 (g 1) 25.035/25.035 (h 1) 25.035/25.035 (d) 25.035/25.035 (fSNonEmpty) 25.035/25.035 (s 1) 25.035/25.035 ) 25.035/25.035 (RULES 25.035/25.035 a -> d 25.035/25.035 b -> d 25.035/25.035 e -> e 25.035/25.035 f(x:S,y:S) -> g(x:S) | a ->* d 25.035/25.035 f(x:S,y:S) -> h(x:S) | b ->* d 25.035/25.035 g(s(x:S)) -> x:S 25.035/25.035 h(s(x:S)) -> x:S 25.035/25.035 ) 25.035/25.035 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.035/25.035 25.035/25.035 25.035/25.035 Problem 1: 25.035/25.035 25.035/25.035 Clean CTRS Processor: 25.035/25.035 25.035/25.035 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.035/25.035 Confluence Problem: 25.035/25.035 (VAR x:S y:S) 25.035/25.035 (STRATEGY CONTEXTSENSITIVE 25.035/25.035 (a) 25.035/25.035 (b) 25.035/25.035 (e) 25.035/25.035 (f 1 2) 25.035/25.035 (g 1) 25.035/25.035 (h 1) 25.035/25.035 (d) 25.035/25.035 (fSNonEmpty) 25.035/25.035 (s 1) 25.035/25.035 ) 25.035/25.035 (RULES 25.035/25.035 a -> d 25.035/25.035 b -> d 25.035/25.035 f(x:S,y:S) -> g(x:S) | a ->* d 25.035/25.035 f(x:S,y:S) -> h(x:S) | b ->* d 25.035/25.035 g(s(x:S)) -> x:S 25.035/25.035 h(s(x:S)) -> x:S 25.035/25.035 ) 25.035/25.035 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.035/25.035 25.035/25.035 CRule InfChecker Info: 25.035/25.035 a -> d 25.035/25.035 Rule remains 25.035/25.035 Proof: 25.035/25.035 NO_CONDS 25.035/25.035 25.035/25.035 CRule InfChecker Info: 25.035/25.035 b -> d 25.035/25.035 Rule remains 25.035/25.035 Proof: 25.035/25.035 NO_CONDS 25.035/25.035 25.035/25.035 CRule InfChecker Info: 25.035/25.035 e -> e 25.035/25.035 Rule deleted 25.035/25.035 Proof: 25.035/25.035 SAME_LHS_AND_RHS 25.035/25.035 25.035/25.035 CRule InfChecker Info: 25.035/25.035 f(x:S,y:S) -> g(x:S) | a ->* d 25.035/25.035 Rule remains 25.035/25.035 Proof: 25.035/25.035 NO 25.035/25.035 25.035/25.035 Problem 1: 25.035/25.035 25.035/25.035 Infeasibility Problem: 25.035/25.035 [(VAR vNonEmpty:S x:S:S y:S:S) 25.035/25.035 (STRATEGY CONTEXTSENSITIVE 25.035/25.035 (a) 25.035/25.035 (b) 25.035/25.035 (f 1 2) 25.035/25.035 (g 1) 25.035/25.035 (h 1) 25.035/25.035 (d) 25.035/25.035 (fSNonEmpty) 25.035/25.035 (s 1) 25.035/25.035 ) 25.035/25.035 (RULES 25.035/25.035 a -> d 25.035/25.035 b -> d 25.035/25.035 f(x:S:S,y:S:S) -> g(x:S:S) | a ->* d 25.035/25.035 f(x:S:S,y:S:S) -> h(x:S:S) | b ->* d 25.035/25.035 g(s(x:S:S)) -> x:S:S 25.035/25.035 h(s(x:S:S)) -> x:S:S 25.035/25.035 )] 25.035/25.035 25.035/25.035 Infeasibility Conditions: 25.035/25.035 a ->* d 25.035/25.035 25.035/25.035 Problem 1: 25.035/25.035 25.035/25.035 Obtaining a proof using Prover9: 25.035/25.035 25.035/25.035 -> Prover9 Output: 25.035/25.035 ============================== Prover9 =============================== 25.035/25.035 Prover9 (64) version 2009-11A, November 2009. 25.035/25.035 Process 55282 was started by ubuntu on ubuntu, 25.035/25.035 Wed Jul 14 11:15:20 2021 25.035/25.035 The command was "./prover9 -f /tmp/prover955273-0.in". 25.035/25.035 ============================== end of head =========================== 25.035/25.035 25.035/25.035 ============================== INPUT ================================= 25.035/25.035 25.035/25.035 % Reading from file /tmp/prover955273-0.in 25.035/25.035 25.035/25.035 assign(max_seconds,20). 25.035/25.035 25.035/25.035 formulas(assumptions). 25.035/25.035 ->*_s0(x,x) # label(reflexivity). 25.035/25.035 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.035/25.035 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 25.035/25.035 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 25.035/25.035 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 25.035/25.035 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 25.035/25.035 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 25.035/25.035 ->_s0(a,d) # label(replacement). 25.035/25.035 ->_s0(b,d) # label(replacement). 25.035/25.035 ->*_s0(a,d) -> ->_s0(f(x1,x2),g(x1)) # label(replacement). 25.035/25.035 ->*_s0(b,d) -> ->_s0(f(x1,x2),h(x1)) # label(replacement). 25.035/25.035 ->_s0(g(s(x1)),x1) # label(replacement). 25.035/25.035 ->_s0(h(s(x1)),x1) # label(replacement). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(goals). 25.035/25.035 ->*_s0(a,d) # label(goal). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 ============================== end of input ========================== 25.035/25.035 25.035/25.035 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.035/25.035 25.035/25.035 % Formulas that are not ordinary clauses: 25.035/25.035 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.035/25.035 2 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 3 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 4 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 5 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 6 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 7 ->*_s0(a,d) -> ->_s0(f(x1,x2),g(x1)) # label(replacement) # label(non_clause). [assumption]. 25.035/25.035 8 ->*_s0(b,d) -> ->_s0(f(x1,x2),h(x1)) # label(replacement) # label(non_clause). [assumption]. 25.035/25.035 9 ->*_s0(a,d) # label(goal) # label(non_clause) # label(goal). [goal]. 25.035/25.035 25.035/25.035 ============================== end of process non-clausal formulas === 25.035/25.035 25.035/25.035 ============================== PROCESS INITIAL CLAUSES =============== 25.035/25.035 25.035/25.035 % Clauses before input processing: 25.035/25.035 25.035/25.035 formulas(usable). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(sos). 25.035/25.035 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.035/25.035 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.035/25.035 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.035/25.035 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.035/25.035 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.035/25.035 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.035/25.035 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.035/25.035 ->_s0(a,d) # label(replacement). [assumption]. 25.035/25.035 ->_s0(b,d) # label(replacement). [assumption]. 25.035/25.035 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.035/25.035 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.035/25.035 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 -->*_s0(a,d) # label(goal). [deny(9)]. 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(demodulators). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 ============================== PREDICATE ELIMINATION ================= 25.035/25.035 25.035/25.035 No predicates eliminated. 25.035/25.035 25.035/25.035 ============================== end predicate elimination ============= 25.035/25.035 25.035/25.035 Auto_denials: 25.035/25.035 % copying label goal to answer in negative clause 25.035/25.035 25.035/25.035 Term ordering decisions: 25.035/25.035 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.035/25.035 Function symbol precedence: function_order([ d, a, b, f, g, h, s ]). 25.035/25.035 After inverse_order: (no changes). 25.035/25.035 Unfolding symbols: (none). 25.035/25.035 25.035/25.035 Auto_inference settings: 25.035/25.035 % set(neg_binary_resolution). % (HNE depth_diff=-7) 25.035/25.035 % clear(ordered_res). % (HNE depth_diff=-7) 25.035/25.035 % set(ur_resolution). % (HNE depth_diff=-7) 25.035/25.035 % set(ur_resolution) -> set(pos_ur_resolution). 25.035/25.035 % set(ur_resolution) -> set(neg_ur_resolution). 25.035/25.035 25.035/25.035 Auto_process settings: (no changes). 25.035/25.035 25.035/25.035 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.035/25.035 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.035/25.035 kept: 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.035/25.035 kept: 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.035/25.035 kept: 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.035/25.035 kept: 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.035/25.035 kept: 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.035/25.035 kept: 17 ->_s0(a,d) # label(replacement). [assumption]. 25.035/25.035 kept: 18 ->_s0(b,d) # label(replacement). [assumption]. 25.035/25.035 kept: 19 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.035/25.035 kept: 20 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.035/25.035 kept: 21 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 kept: 22 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 kept: 23 -->*_s0(a,d) # label(goal) # answer(goal). [deny(9)]. 25.035/25.035 25.035/25.035 ============================== end of process initial clauses ======== 25.035/25.035 25.035/25.035 ============================== CLAUSES FOR SEARCH ==================== 25.035/25.035 25.035/25.035 % Clauses after input processing: 25.035/25.035 25.035/25.035 formulas(usable). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(sos). 25.035/25.035 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.035/25.035 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.035/25.035 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.035/25.035 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.035/25.035 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.035/25.035 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.035/25.035 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.035/25.035 17 ->_s0(a,d) # label(replacement). [assumption]. 25.035/25.035 18 ->_s0(b,d) # label(replacement). [assumption]. 25.035/25.035 20 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.035/25.035 21 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 22 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 23 -->*_s0(a,d) # label(goal) # answer(goal). [deny(9)]. 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(demodulators). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 ============================== end of clauses for search ============= 25.035/25.035 25.035/25.035 ============================== SEARCH ================================ 25.035/25.035 25.035/25.035 % Starting search at 0.00 seconds. 25.035/25.035 25.035/25.035 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.035/25.035 25.035/25.035 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.035/25.035 25.035/25.035 given #3 (I,wt=10): 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.035/25.035 25.035/25.035 given #4 (I,wt=10): 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.035/25.035 25.035/25.035 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.035/25.035 25.035/25.035 given #6 (I,wt=8): 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.035/25.035 25.035/25.035 given #7 (I,wt=8): 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.035/25.035 25.035/25.035 given #8 (I,wt=3): 17 ->_s0(a,d) # label(replacement). [assumption]. 25.035/25.035 25.035/25.035 ============================== PROOF ================================= 25.035/25.035 25.035/25.035 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.035/25.035 % Length of proof is 8. 25.035/25.035 % Level of proof is 3. 25.035/25.035 % Maximum clause weight is 9.000. 25.035/25.035 % Given clauses 8. 25.035/25.035 25.035/25.035 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.035/25.035 9 ->*_s0(a,d) # label(goal) # label(non_clause) # label(goal). [goal]. 25.035/25.035 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.035/25.035 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.035/25.035 17 ->_s0(a,d) # label(replacement). [assumption]. 25.035/25.035 23 -->*_s0(a,d) # label(goal) # answer(goal). [deny(9)]. 25.035/25.035 29 ->*_s0(a,d). [ur(11,a,17,a,b,10,a)]. 25.035/25.035 30 $F # answer(goal). [resolve(29,a,23,a)]. 25.035/25.035 25.035/25.035 ============================== end of proof ========================== 25.035/25.035 25.035/25.035 ============================== STATISTICS ============================ 25.035/25.035 25.035/25.035 Given=8. Generated=20. Kept=20. proofs=1. 25.035/25.035 Usable=8. Sos=5. Demods=0. Limbo=5, Disabled=15. Hints=0. 25.035/25.035 Kept_by_rule=0, Deleted_by_rule=0. 25.035/25.035 Forward_subsumed=0. Back_subsumed=1. 25.035/25.035 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.035/25.035 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.035/25.035 Demod_attempts=0. Demod_rewrites=0. 25.035/25.035 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.035/25.035 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=12. 25.035/25.035 Megabytes=0.07. 25.035/25.035 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.035/25.035 25.035/25.035 ============================== end of statistics ===================== 25.035/25.035 25.035/25.035 ============================== end of search ========================= 25.035/25.035 25.035/25.035 THEOREM PROVED 25.035/25.035 25.035/25.035 Exiting with 1 proof. 25.035/25.035 25.035/25.035 Process 55282 exit (max_proofs) Wed Jul 14 11:15:20 2021 25.035/25.035 25.035/25.035 25.035/25.035 The problem is feasible. 25.035/25.035 25.035/25.035 25.035/25.035 CRule InfChecker Info: 25.035/25.035 f(x:S,y:S) -> h(x:S) | b ->* d 25.035/25.035 Rule remains 25.035/25.035 Proof: 25.035/25.035 NO 25.035/25.035 25.035/25.035 Problem 1: 25.035/25.035 25.035/25.035 Infeasibility Problem: 25.035/25.035 [(VAR vNonEmpty:S x:S:S y:S:S) 25.035/25.035 (STRATEGY CONTEXTSENSITIVE 25.035/25.035 (a) 25.035/25.035 (b) 25.035/25.035 (f 1 2) 25.035/25.035 (g 1) 25.035/25.035 (h 1) 25.035/25.035 (d) 25.035/25.035 (fSNonEmpty) 25.035/25.035 (s 1) 25.035/25.035 ) 25.035/25.035 (RULES 25.035/25.035 a -> d 25.035/25.035 b -> d 25.035/25.035 f(x:S:S,y:S:S) -> g(x:S:S) | a ->* d 25.035/25.035 f(x:S:S,y:S:S) -> h(x:S:S) | b ->* d 25.035/25.035 g(s(x:S:S)) -> x:S:S 25.035/25.035 h(s(x:S:S)) -> x:S:S 25.035/25.035 )] 25.035/25.035 25.035/25.035 Infeasibility Conditions: 25.035/25.035 b ->* d 25.035/25.035 25.035/25.035 Problem 1: 25.035/25.035 25.035/25.035 Obtaining a proof using Prover9: 25.035/25.035 25.035/25.035 -> Prover9 Output: 25.035/25.035 ============================== Prover9 =============================== 25.035/25.035 Prover9 (64) version 2009-11A, November 2009. 25.035/25.035 Process 55304 was started by ubuntu on ubuntu, 25.035/25.035 Wed Jul 14 11:15:20 2021 25.035/25.035 The command was "./prover9 -f /tmp/prover955295-0.in". 25.035/25.035 ============================== end of head =========================== 25.035/25.035 25.035/25.035 ============================== INPUT ================================= 25.035/25.035 25.035/25.035 % Reading from file /tmp/prover955295-0.in 25.035/25.035 25.035/25.035 assign(max_seconds,20). 25.035/25.035 25.035/25.035 formulas(assumptions). 25.035/25.035 ->*_s0(x,x) # label(reflexivity). 25.035/25.035 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.035/25.035 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 25.035/25.035 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 25.035/25.035 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 25.035/25.035 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 25.035/25.035 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 25.035/25.035 ->_s0(a,d) # label(replacement). 25.035/25.035 ->_s0(b,d) # label(replacement). 25.035/25.035 ->*_s0(a,d) -> ->_s0(f(x1,x2),g(x1)) # label(replacement). 25.035/25.035 ->*_s0(b,d) -> ->_s0(f(x1,x2),h(x1)) # label(replacement). 25.035/25.035 ->_s0(g(s(x1)),x1) # label(replacement). 25.035/25.035 ->_s0(h(s(x1)),x1) # label(replacement). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(goals). 25.035/25.035 ->*_s0(b,d) # label(goal). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 ============================== end of input ========================== 25.035/25.035 25.035/25.035 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.035/25.035 25.035/25.035 % Formulas that are not ordinary clauses: 25.035/25.035 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.035/25.035 2 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 3 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 4 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 5 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 6 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 7 ->*_s0(a,d) -> ->_s0(f(x1,x2),g(x1)) # label(replacement) # label(non_clause). [assumption]. 25.035/25.035 8 ->*_s0(b,d) -> ->_s0(f(x1,x2),h(x1)) # label(replacement) # label(non_clause). [assumption]. 25.035/25.035 9 ->*_s0(b,d) # label(goal) # label(non_clause) # label(goal). [goal]. 25.035/25.035 25.035/25.035 ============================== end of process non-clausal formulas === 25.035/25.035 25.035/25.035 ============================== PROCESS INITIAL CLAUSES =============== 25.035/25.035 25.035/25.035 % Clauses before input processing: 25.035/25.035 25.035/25.035 formulas(usable). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(sos). 25.035/25.035 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.035/25.035 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.035/25.035 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.035/25.035 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.035/25.035 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.035/25.035 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.035/25.035 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.035/25.035 ->_s0(a,d) # label(replacement). [assumption]. 25.035/25.035 ->_s0(b,d) # label(replacement). [assumption]. 25.035/25.035 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.035/25.035 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.035/25.035 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 -->*_s0(b,d) # label(goal). [deny(9)]. 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(demodulators). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 ============================== PREDICATE ELIMINATION ================= 25.035/25.035 25.035/25.035 No predicates eliminated. 25.035/25.035 25.035/25.035 ============================== end predicate elimination ============= 25.035/25.035 25.035/25.035 Auto_denials: 25.035/25.035 % copying label goal to answer in negative clause 25.035/25.035 25.035/25.035 Term ordering decisions: 25.035/25.035 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.035/25.035 Function symbol precedence: function_order([ d, a, b, f, g, h, s ]). 25.035/25.035 After inverse_order: (no changes). 25.035/25.035 Unfolding symbols: (none). 25.035/25.035 25.035/25.035 Auto_inference settings: 25.035/25.035 % set(neg_binary_resolution). % (HNE depth_diff=-7) 25.035/25.035 % clear(ordered_res). % (HNE depth_diff=-7) 25.035/25.035 % set(ur_resolution). % (HNE depth_diff=-7) 25.035/25.035 % set(ur_resolution) -> set(pos_ur_resolution). 25.035/25.035 % set(ur_resolution) -> set(neg_ur_resolution). 25.035/25.035 25.035/25.035 Auto_process settings: (no changes). 25.035/25.035 25.035/25.035 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.035/25.035 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.035/25.035 kept: 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.035/25.035 kept: 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.035/25.035 kept: 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.035/25.035 kept: 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.035/25.035 kept: 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.035/25.035 kept: 17 ->_s0(a,d) # label(replacement). [assumption]. 25.035/25.035 kept: 18 ->_s0(b,d) # label(replacement). [assumption]. 25.035/25.035 kept: 19 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.035/25.035 kept: 20 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.035/25.035 kept: 21 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 kept: 22 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 kept: 23 -->*_s0(b,d) # label(goal) # answer(goal). [deny(9)]. 25.035/25.035 25.035/25.035 ============================== end of process initial clauses ======== 25.035/25.035 25.035/25.035 ============================== CLAUSES FOR SEARCH ==================== 25.035/25.035 25.035/25.035 % Clauses after input processing: 25.035/25.035 25.035/25.035 formulas(usable). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(sos). 25.035/25.035 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.035/25.035 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.035/25.035 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.035/25.035 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.035/25.035 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.035/25.035 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.035/25.035 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.035/25.035 17 ->_s0(a,d) # label(replacement). [assumption]. 25.035/25.035 18 ->_s0(b,d) # label(replacement). [assumption]. 25.035/25.035 19 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.035/25.035 21 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 22 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 23 -->*_s0(b,d) # label(goal) # answer(goal). [deny(9)]. 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(demodulators). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 ============================== end of clauses for search ============= 25.035/25.035 25.035/25.035 ============================== SEARCH ================================ 25.035/25.035 25.035/25.035 % Starting search at 0.00 seconds. 25.035/25.035 25.035/25.035 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.035/25.035 25.035/25.035 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.035/25.035 25.035/25.035 given #3 (I,wt=10): 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.035/25.035 25.035/25.035 given #4 (I,wt=10): 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.035/25.035 25.035/25.035 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.035/25.035 25.035/25.035 given #6 (I,wt=8): 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.035/25.035 25.035/25.035 given #7 (I,wt=8): 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.035/25.035 25.035/25.035 given #8 (I,wt=3): 17 ->_s0(a,d) # label(replacement). [assumption]. 25.035/25.035 25.035/25.035 given #9 (I,wt=3): 18 ->_s0(b,d) # label(replacement). [assumption]. 25.035/25.035 25.035/25.035 ============================== PROOF ================================= 25.035/25.035 25.035/25.035 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.035/25.035 % Length of proof is 8. 25.035/25.035 % Level of proof is 3. 25.035/25.035 % Maximum clause weight is 9.000. 25.035/25.035 % Given clauses 9. 25.035/25.035 25.035/25.035 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.035/25.035 9 ->*_s0(b,d) # label(goal) # label(non_clause) # label(goal). [goal]. 25.035/25.035 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.035/25.035 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.035/25.035 18 ->_s0(b,d) # label(replacement). [assumption]. 25.035/25.035 23 -->*_s0(b,d) # label(goal) # answer(goal). [deny(9)]. 25.035/25.035 35 ->*_s0(b,d). [ur(11,a,18,a,b,10,a)]. 25.035/25.035 36 $F # answer(goal). [resolve(35,a,23,a)]. 25.035/25.035 25.035/25.035 ============================== end of proof ========================== 25.035/25.035 25.035/25.035 ============================== STATISTICS ============================ 25.035/25.035 25.035/25.035 Given=9. Generated=26. Kept=26. proofs=1. 25.035/25.035 Usable=9. Sos=10. Demods=0. Limbo=5, Disabled=15. Hints=0. 25.035/25.035 Kept_by_rule=0, Deleted_by_rule=0. 25.035/25.035 Forward_subsumed=0. Back_subsumed=1. 25.035/25.035 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.035/25.035 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.035/25.035 Demod_attempts=0. Demod_rewrites=0. 25.035/25.035 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.035/25.035 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=12. 25.035/25.035 Megabytes=0.08. 25.035/25.035 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.035/25.035 25.035/25.035 ============================== end of statistics ===================== 25.035/25.035 25.035/25.035 ============================== end of search ========================= 25.035/25.035 25.035/25.035 THEOREM PROVED 25.035/25.035 25.035/25.035 Exiting with 1 proof. 25.035/25.035 25.035/25.035 Process 55304 exit (max_proofs) Wed Jul 14 11:15:20 2021 25.035/25.035 25.035/25.035 25.035/25.035 The problem is feasible. 25.035/25.035 25.035/25.035 25.035/25.035 CRule InfChecker Info: 25.035/25.035 g(s(x:S)) -> x:S 25.035/25.035 Rule remains 25.035/25.035 Proof: 25.035/25.035 NO_CONDS 25.035/25.035 25.035/25.035 CRule InfChecker Info: 25.035/25.035 h(s(x:S)) -> x:S 25.035/25.035 Rule remains 25.035/25.035 Proof: 25.035/25.035 NO_CONDS 25.035/25.035 25.035/25.035 Problem 1: 25.035/25.035 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.035/25.035 Confluence Problem: 25.035/25.035 (VAR x:S y:S) 25.035/25.035 (STRATEGY CONTEXTSENSITIVE 25.035/25.035 (a) 25.035/25.035 (b) 25.035/25.035 (e) 25.035/25.035 (f 1 2) 25.035/25.035 (g 1) 25.035/25.035 (h 1) 25.035/25.035 (d) 25.035/25.035 (fSNonEmpty) 25.035/25.035 (s 1) 25.035/25.035 ) 25.035/25.035 (RULES 25.035/25.035 a -> d 25.035/25.035 b -> d 25.035/25.035 f(x:S,y:S) -> g(x:S) | a ->* d 25.035/25.035 f(x:S,y:S) -> h(x:S) | b ->* d 25.035/25.035 g(s(x:S)) -> x:S 25.035/25.035 h(s(x:S)) -> x:S 25.035/25.035 ) 25.035/25.035 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.035/25.035 25.035/25.035 Critical Pairs Processor: 25.035/25.035 -> Rules: 25.035/25.035 a -> d 25.035/25.035 b -> d 25.035/25.035 f(x:S,y:S) -> g(x:S) | a ->* d 25.035/25.035 f(x:S,y:S) -> h(x:S) | b ->* d 25.035/25.035 g(s(x:S)) -> x:S 25.035/25.035 h(s(x:S)) -> x:S 25.035/25.035 -> Vars: 25.035/25.035 "x", "y", "x", "y", "x", "x" 25.035/25.035 -> FVars: 25.035/25.035 "x3", "x4", "x5", "x6", "x7", "x8" 25.035/25.035 -> PVars: 25.035/25.035 "x": ["x3", "x5", "x7", "x8"], "y": ["x4", "x6"] 25.035/25.035 25.035/25.035 -> Rlps: 25.035/25.035 crule: a -> d, id: 1, possubterms: a-> [] 25.035/25.035 crule: b -> d, id: 2, possubterms: b-> [] 25.035/25.035 crule: f(x3:S,x4:S) -> g(x3:S) | a ->* d, id: 3, possubterms: f(x3:S,x4:S)-> [] 25.035/25.035 crule: f(x5:S,x6:S) -> h(x5:S) | b ->* d, id: 4, possubterms: f(x5:S,x6:S)-> [] 25.035/25.035 crule: g(s(x7:S)) -> x7:S, id: 5, possubterms: g(s(x7:S))-> [], s(x7:S)-> [1] 25.035/25.035 crule: h(s(x8:S)) -> x8:S, id: 6, possubterms: h(s(x8:S))-> [], s(x8:S)-> [1] 25.035/25.035 25.035/25.035 -> Unifications: 25.035/25.035 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) 25.035/25.035 25.035/25.035 -> Critical pairs info: 25.035/25.035 | b ->* d, a ->* d => Not trivial, Overlay, N1 25.035/25.035 25.035/25.035 -> Problem conclusions: 25.035/25.035 Left linear, Right linear, Linear 25.035/25.035 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 25.035/25.035 CTRS Type: 1 25.035/25.035 Deterministic, Strongly deterministic 25.035/25.035 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.035/25.035 Maybe right-stable CTRS, Overlay CTRS 25.035/25.035 Maybe normal CTRS, Maybe almost normal CTRS 25.035/25.035 Maybe terminating CTRS, Maybe joinable CCPs 25.035/25.035 Maybe level confluent 25.035/25.035 Maybe confluent 25.035/25.035 25.035/25.035 Problem 1: 25.035/25.035 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.035/25.035 Confluence Problem: 25.035/25.035 (VAR x:S y:S) 25.035/25.035 (STRATEGY CONTEXTSENSITIVE 25.035/25.035 (a) 25.035/25.035 (b) 25.035/25.035 (e) 25.035/25.035 (f 1 2) 25.035/25.035 (g 1) 25.035/25.035 (h 1) 25.035/25.035 (d) 25.035/25.035 (fSNonEmpty) 25.035/25.035 (s 1) 25.035/25.035 ) 25.035/25.035 (RULES 25.035/25.035 a -> d 25.035/25.035 b -> d 25.035/25.035 f(x:S,y:S) -> g(x:S) | a ->* d 25.035/25.035 f(x:S,y:S) -> h(x:S) | b ->* d 25.035/25.035 g(s(x:S)) -> x:S 25.035/25.035 h(s(x:S)) -> x:S 25.035/25.035 ) 25.035/25.035 Critical Pairs: 25.035/25.035 | b ->* d, a ->* d => Not trivial, Overlay, N1 25.035/25.035 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.035/25.035 25.035/25.035 Conditional Critical Pairs Distributor Processor 25.035/25.035 25.035/25.035 Problem 1: 25.035/25.035 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.035/25.035 Confluence Problem: 25.035/25.035 (VAR x:S y:S) 25.035/25.035 (STRATEGY CONTEXTSENSITIVE 25.035/25.035 (a) 25.035/25.035 (b) 25.035/25.035 (e) 25.035/25.035 (f 1 2) 25.035/25.035 (g 1) 25.035/25.035 (h 1) 25.035/25.035 (d) 25.035/25.035 (fSNonEmpty) 25.035/25.035 (s 1) 25.035/25.035 ) 25.035/25.035 (RULES 25.035/25.035 a -> d 25.035/25.035 b -> d 25.035/25.035 f(x:S,y:S) -> g(x:S) | a ->* d 25.035/25.035 f(x:S,y:S) -> h(x:S) | b ->* d 25.035/25.035 g(s(x:S)) -> x:S 25.035/25.035 h(s(x:S)) -> x:S 25.035/25.035 ) 25.035/25.035 Critical Pairs: 25.035/25.035 | b ->* d, a ->* d => Not trivial, Overlay, N1 25.035/25.035 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.035/25.035 25.035/25.035 Infeasible Convergence Constantized CCP Processor: 25.035/25.035 25.035/25.035 Infeasible constantized conditions? 25.035/25.035 NO 25.035/25.035 25.035/25.035 Problem 1: 25.035/25.035 25.035/25.035 Infeasibility Problem: 25.035/25.035 [(VAR vNonEmpty:S x:S y:S) 25.035/25.035 (STRATEGY CONTEXTSENSITIVE 25.035/25.035 (a) 25.035/25.035 (b) 25.035/25.035 (f 1 2) 25.035/25.035 (g 1) 25.035/25.035 (h 1) 25.035/25.035 (d) 25.035/25.035 (fSNonEmpty) 25.035/25.035 (s 1) 25.035/25.035 ) 25.035/25.035 (RULES 25.035/25.035 a -> d 25.035/25.035 b -> d 25.035/25.035 f(x:S,y:S) -> g(x:S) | a ->* d 25.035/25.035 f(x:S,y:S) -> h(x:S) | b ->* d 25.035/25.035 g(s(x:S)) -> x:S 25.035/25.035 h(s(x:S)) -> x:S 25.035/25.035 )] 25.035/25.035 25.035/25.035 Infeasibility Conditions: 25.035/25.035 b ->* d, a ->* d 25.035/25.035 25.035/25.035 Problem 1: 25.035/25.035 25.035/25.035 Obtaining a proof using Prover9: 25.035/25.035 25.035/25.035 -> Prover9 Output: 25.035/25.035 ============================== Prover9 =============================== 25.035/25.035 Prover9 (64) version 2009-11A, November 2009. 25.035/25.035 Process 55346 was started by ubuntu on ubuntu, 25.035/25.035 Wed Jul 14 11:15:20 2021 25.035/25.035 The command was "./prover9 -f /tmp/prover955320-0.in". 25.035/25.035 ============================== end of head =========================== 25.035/25.035 25.035/25.035 ============================== INPUT ================================= 25.035/25.035 25.035/25.035 % Reading from file /tmp/prover955320-0.in 25.035/25.035 25.035/25.035 assign(max_seconds,20). 25.035/25.035 25.035/25.035 formulas(assumptions). 25.035/25.035 ->*_s0(x,x) # label(reflexivity). 25.035/25.035 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.035/25.035 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 25.035/25.035 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 25.035/25.035 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 25.035/25.035 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 25.035/25.035 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 25.035/25.035 ->_s0(a,d) # label(replacement). 25.035/25.035 ->_s0(b,d) # label(replacement). 25.035/25.035 ->*_s0(a,d) -> ->_s0(f(x1,x2),g(x1)) # label(replacement). 25.035/25.035 ->*_s0(b,d) -> ->_s0(f(x1,x2),h(x1)) # label(replacement). 25.035/25.035 ->_s0(g(s(x1)),x1) # label(replacement). 25.035/25.035 ->_s0(h(s(x1)),x1) # label(replacement). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(goals). 25.035/25.035 ->*_s0(b,d) & ->*_s0(a,d) # label(goal). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 ============================== end of input ========================== 25.035/25.035 25.035/25.035 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.035/25.035 25.035/25.035 % Formulas that are not ordinary clauses: 25.035/25.035 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.035/25.035 2 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 3 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 4 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 5 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 6 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 7 ->*_s0(a,d) -> ->_s0(f(x1,x2),g(x1)) # label(replacement) # label(non_clause). [assumption]. 25.035/25.035 8 ->*_s0(b,d) -> ->_s0(f(x1,x2),h(x1)) # label(replacement) # label(non_clause). [assumption]. 25.035/25.035 9 ->*_s0(b,d) & ->*_s0(a,d) # label(goal) # label(non_clause) # label(goal). [goal]. 25.035/25.035 25.035/25.035 ============================== end of process non-clausal formulas === 25.035/25.035 25.035/25.035 ============================== PROCESS INITIAL CLAUSES =============== 25.035/25.035 25.035/25.035 % Clauses before input processing: 25.035/25.035 25.035/25.035 formulas(usable). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(sos). 25.035/25.035 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.035/25.035 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.035/25.035 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.035/25.035 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.035/25.035 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.035/25.035 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.035/25.035 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.035/25.035 ->_s0(a,d) # label(replacement). [assumption]. 25.035/25.035 ->_s0(b,d) # label(replacement). [assumption]. 25.035/25.035 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.035/25.035 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.035/25.035 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 -->*_s0(b,d) | -->*_s0(a,d) # label(goal). [deny(9)]. 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(demodulators). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 ============================== PREDICATE ELIMINATION ================= 25.035/25.035 25.035/25.035 No predicates eliminated. 25.035/25.035 25.035/25.035 ============================== end predicate elimination ============= 25.035/25.035 25.035/25.035 Auto_denials: 25.035/25.035 % copying label goal to answer in negative clause 25.035/25.035 25.035/25.035 Term ordering decisions: 25.035/25.035 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.035/25.035 Function symbol precedence: function_order([ d, a, b, f, g, h, s ]). 25.035/25.035 After inverse_order: (no changes). 25.035/25.035 Unfolding symbols: (none). 25.035/25.035 25.035/25.035 Auto_inference settings: 25.035/25.035 % set(neg_binary_resolution). % (HNE depth_diff=-7) 25.035/25.035 % clear(ordered_res). % (HNE depth_diff=-7) 25.035/25.035 % set(ur_resolution). % (HNE depth_diff=-7) 25.035/25.035 % set(ur_resolution) -> set(pos_ur_resolution). 25.035/25.035 % set(ur_resolution) -> set(neg_ur_resolution). 25.035/25.035 25.035/25.035 Auto_process settings: 25.035/25.035 % set(unit_deletion). % (Horn set with negative nonunits) 25.035/25.035 25.035/25.035 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.035/25.035 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.035/25.035 kept: 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.035/25.035 kept: 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.035/25.035 kept: 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.035/25.035 kept: 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.035/25.035 kept: 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.035/25.035 kept: 17 ->_s0(a,d) # label(replacement). [assumption]. 25.035/25.035 kept: 18 ->_s0(b,d) # label(replacement). [assumption]. 25.035/25.035 kept: 19 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.035/25.035 kept: 20 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.035/25.035 kept: 21 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 kept: 22 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 kept: 23 -->*_s0(b,d) | -->*_s0(a,d) # label(goal) # answer(goal). [deny(9)]. 25.035/25.035 25.035/25.035 ============================== end of process initial clauses ======== 25.035/25.035 25.035/25.035 ============================== CLAUSES FOR SEARCH ==================== 25.035/25.035 25.035/25.035 % Clauses after input processing: 25.035/25.035 25.035/25.035 formulas(usable). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(sos). 25.035/25.035 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.035/25.035 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.035/25.035 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.035/25.035 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.035/25.035 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.035/25.035 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.035/25.035 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.035/25.035 17 ->_s0(a,d) # label(replacement). [assumption]. 25.035/25.035 18 ->_s0(b,d) # label(replacement). [assumption]. 25.035/25.035 19 -->*_s0(a,d) | ->_s0(f(x,y),g(x)) # label(replacement). [clausify(7)]. 25.035/25.035 20 -->*_s0(b,d) | ->_s0(f(x,y),h(x)) # label(replacement). [clausify(8)]. 25.035/25.035 21 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 22 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 25.035/25.035 23 -->*_s0(b,d) | -->*_s0(a,d) # label(goal) # answer(goal). [deny(9)]. 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(demodulators). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 ============================== end of clauses for search ============= 25.035/25.035 25.035/25.035 ============================== SEARCH ================================ 25.035/25.035 25.035/25.035 % Starting search at 0.01 seconds. 25.035/25.035 25.035/25.035 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.035/25.035 25.035/25.035 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.035/25.035 25.035/25.035 given #3 (I,wt=10): 12 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.035/25.035 25.035/25.035 given #4 (I,wt=10): 13 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.035/25.035 25.035/25.035 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(4)]. 25.035/25.035 25.035/25.035 given #6 (I,wt=8): 15 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(5)]. 25.035/25.035 25.035/25.035 given #7 (I,wt=8): 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.035/25.035 25.035/25.035 given #8 (I,wt=3): 17 ->_s0(a,d) # label(replacement). [assumption]. 25.035/25.035 25.035/25.035 given #9 (I,wt=3): 18 ->_s0(b,d) # label(replacement). [assumption]. 25.035/25.035 25.035/25.035 ============================== PROOF ================================= 25.035/25.035 25.035/25.035 % Proof 1 at 0.01 (+ 0.00) seconds: goal. 25.035/25.035 % Length of proof is 10. 25.035/25.035 % Level of proof is 4. 25.035/25.035 % Maximum clause weight is 9.000. 25.035/25.035 % Given clauses 9. 25.035/25.035 25.035/25.035 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.035/25.035 9 ->*_s0(b,d) & ->*_s0(a,d) # label(goal) # label(non_clause) # label(goal). [goal]. 25.035/25.035 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.035/25.035 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.035/25.035 17 ->_s0(a,d) # label(replacement). [assumption]. 25.035/25.035 18 ->_s0(b,d) # label(replacement). [assumption]. 25.035/25.035 23 -->*_s0(b,d) | -->*_s0(a,d) # label(goal) # answer(goal). [deny(9)]. 25.035/25.035 29 ->*_s0(a,d). [ur(11,a,17,a,b,10,a)]. 25.035/25.035 30 -->*_s0(b,d) # answer(goal). [back_unit_del(23),unit_del(b,29)]. 25.035/25.035 37 $F # answer(goal). [ur(11,a,18,a,b,10,a),unit_del(a,30)]. 25.035/25.035 25.035/25.035 ============================== end of proof ========================== 25.035/25.035 25.035/25.035 ============================== STATISTICS ============================ 25.035/25.035 25.035/25.035 Given=9. Generated=28. Kept=27. proofs=1. 25.035/25.035 Usable=9. Sos=10. Demods=0. Limbo=5, Disabled=17. Hints=0. 25.035/25.035 Kept_by_rule=0, Deleted_by_rule=0. 25.035/25.035 Forward_subsumed=0. Back_subsumed=1. 25.035/25.035 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.035/25.035 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=2. 25.035/25.035 Demod_attempts=0. Demod_rewrites=0. 25.035/25.035 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.035/25.035 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=13. 25.035/25.035 Megabytes=0.08. 25.035/25.035 User_CPU=0.01, System_CPU=0.00, Wall_clock=0. 25.035/25.035 25.035/25.035 ============================== end of statistics ===================== 25.035/25.035 25.035/25.035 ============================== end of search ========================= 25.035/25.035 25.035/25.035 THEOREM PROVED 25.035/25.035 25.035/25.035 Exiting with 1 proof. 25.035/25.035 25.035/25.035 Process 55346 exit (max_proofs) Wed Jul 14 11:15:20 2021 25.035/25.035 25.035/25.035 25.035/25.035 The problem is feasible. 25.035/25.035 25.035/25.035 Infeasible constantized CCP convergence? 25.035/25.035 YES 25.035/25.035 25.035/25.035 Problem 1: 25.035/25.035 25.035/25.035 Infeasibility Problem: 25.035/25.035 [(VAR vNonEmpty:S x:S y:S z:S) 25.035/25.035 (STRATEGY CONTEXTSENSITIVE 25.035/25.035 (a) 25.035/25.035 (b) 25.035/25.035 (f 1 2) 25.035/25.035 (g 1) 25.035/25.035 (h 1) 25.035/25.035 (cx) 25.035/25.035 (d) 25.035/25.035 (fSNonEmpty) 25.035/25.035 (s 1) 25.035/25.035 ) 25.035/25.035 (RULES 25.035/25.035 a -> d 25.035/25.035 b -> d 25.035/25.035 f(x:S,y:S) -> g(x:S) | a ->* d 25.035/25.035 f(x:S,y:S) -> h(x:S) | b ->* d 25.035/25.035 g(s(x:S)) -> x:S 25.035/25.035 h(s(x:S)) -> x:S 25.035/25.035 )] 25.035/25.035 25.035/25.035 Infeasibility Conditions: 25.035/25.035 g(cx) ->* z:S, h(cx) ->* z:S 25.035/25.035 25.035/25.035 Problem 1: 25.035/25.035 25.035/25.035 Obtaining a model using Mace4: 25.035/25.035 25.035/25.035 -> Usable Rules: 25.035/25.035 g(s(x:S)) -> x:S 25.035/25.035 h(s(x:S)) -> x:S 25.035/25.035 25.035/25.035 -> Mace4 Output: 25.035/25.035 ============================== Mace4 ================================= 25.035/25.035 Mace4 (64) version 2009-11A, November 2009. 25.035/25.035 Process 55369 was started by ubuntu on ubuntu, 25.035/25.035 Wed Jul 14 11:15:20 2021 25.035/25.035 The command was "./mace4 -c -f /tmp/mace455354-2.in". 25.035/25.035 ============================== end of head =========================== 25.035/25.035 25.035/25.035 ============================== INPUT ================================= 25.035/25.035 25.035/25.035 % Reading from file /tmp/mace455354-2.in 25.035/25.035 25.035/25.035 assign(max_seconds,20). 25.035/25.035 25.035/25.035 formulas(assumptions). 25.035/25.035 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). 25.035/25.035 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). 25.035/25.035 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence). 25.035/25.035 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence). 25.035/25.035 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 25.035/25.035 ->(g(s(x1)),x1) # label(replacement). 25.035/25.035 ->(h(s(x1)),x1) # label(replacement). 25.035/25.035 ->*(x,x) # label(reflexivity). 25.035/25.035 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 formulas(goals). 25.035/25.035 (exists x3 (->*(g(cx),x3) & ->*(h(cx),x3))) # label(goal). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 ============================== end of input ========================== 25.035/25.035 25.035/25.035 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.035/25.035 25.035/25.035 % Formulas that are not ordinary clauses: 25.035/25.035 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 3 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 4 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 5 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 25.035/25.035 6 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.035/25.035 7 (exists x3 (->*(g(cx),x3) & ->*(h(cx),x3))) # label(goal) # label(non_clause) # label(goal). [goal]. 25.035/25.035 25.035/25.035 ============================== end of process non-clausal formulas === 25.035/25.035 25.035/25.035 ============================== CLAUSES FOR SEARCH ==================== 25.035/25.035 25.035/25.035 formulas(mace4_clauses). 25.035/25.035 -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). 25.035/25.035 -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). 25.035/25.035 -->(x,y) | ->(g(x),g(y)) # label(congruence). 25.035/25.035 -->(x,y) | ->(h(x),h(y)) # label(congruence). 25.035/25.035 -->(x,y) | ->(s(x),s(y)) # label(congruence). 25.035/25.035 ->(g(s(x)),x) # label(replacement). 25.035/25.035 ->(h(s(x)),x) # label(replacement). 25.035/25.035 ->*(x,x) # label(reflexivity). 25.035/25.035 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 25.035/25.035 -->*(g(cx),x) | -->*(h(cx),x) # label(goal). 25.035/25.035 end_of_list. 25.035/25.035 25.035/25.035 ============================== end of clauses for search ============= 25.035/25.035 25.035/25.035 % There are no natural numbers in the input. 25.035/25.035 25.035/25.035 ============================== DOMAIN SIZE 2 ========================= 25.035/25.035 25.035/25.035 ============================== STATISTICS ============================ 25.035/25.035 25.035/25.035 For domain size 2. 25.035/25.035 25.035/25.035 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 25.035/25.035 Ground clauses: seen=44, kept=40. 25.035/25.035 Selections=10, assignments=19, propagations=32, current_models=0. 25.035/25.035 Rewrite_terms=140, rewrite_bools=148, indexes=30. 25.035/25.035 Rules_from_neg_clauses=8, cross_offs=8. 25.035/25.035 25.035/25.035 ============================== end of statistics ===================== 25.035/25.035 25.035/25.035 ============================== DOMAIN SIZE 3 ========================= 25.035/25.035 25.035/25.035 ============================== MODEL ================================= 25.035/25.035 25.035/25.035 interpretation( 3, [number=1, seconds=0], [ 25.035/25.035 25.035/25.035 function(cx, [ 0 ]), 25.035/25.035 25.035/25.035 function(s(_), [ 1, 2, 2 ]), 25.035/25.035 25.035/25.035 function(g(_), [ 0, 0, 2 ]), 25.035/25.035 25.035/25.035 function(h(_), [ 1, 0, 2 ]), 25.035/25.035 25.035/25.035 function(f(_,_), [ 25.035/25.035 0, 0, 0, 25.035/25.035 0, 0, 0, 25.035/25.035 0, 0, 0 ]), 25.035/25.035 25.035/25.035 relation(->*(_,_), [ 25.035/25.035 1, 0, 0, 25.035/25.035 0, 1, 0, 25.035/25.035 1, 1, 1 ]), 25.035/25.035 25.035/25.035 relation(->(_,_), [ 25.035/25.035 1, 0, 0, 25.035/25.035 0, 1, 0, 25.035/25.035 1, 1, 1 ]) 25.035/25.035 ]). 25.035/25.035 25.035/25.035 ============================== end of model ========================== 25.035/25.035 25.035/25.035 ============================== STATISTICS ============================ 25.035/25.035 25.035/25.035 For domain size 3. 25.035/25.035 25.035/25.035 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 25.035/25.035 Ground clauses: seen=120, kept=111. 25.035/25.035 Selections=58, assignments=146, propagations=45, current_models=1. 25.035/25.035 Rewrite_terms=894, rewrite_bools=522, indexes=47. 25.035/25.035 Rules_from_neg_clauses=8, cross_offs=26. 25.035/25.035 25.035/25.035 ============================== end of statistics ===================== 25.035/25.035 25.035/25.035 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.035/25.035 25.035/25.035 Exiting with 1 model. 25.035/25.035 25.035/25.035 Process 55369 exit (max_models) Wed Jul 14 11:15:20 2021 25.035/25.035 The process finished Wed Jul 14 11:15:20 2021 25.035/25.035 25.035/25.035 25.035/25.035 Mace4 cooked interpretation: 25.035/25.035 25.035/25.035 25.035/25.035 25.035/25.035 The problem is infeasible. 25.035/25.035 25.035/25.035 25.035/25.035 The problem is not joinable. 25.035/25.035 17.87user 0.62system 0:25.35elapsed 72%CPU (0avgtext+0avgdata 62300maxresident)k 25.035/25.035 8inputs+0outputs (0major+44254minor)pagefaults 0swaps