20.051/20.051 YES 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 20.051/20.051 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 Confluence Problem: 20.051/20.051 (VAR vNonEmpty x y) 20.051/20.051 (REPLACEMENT-MAP 20.051/20.051 (f 1, 2) 20.051/20.051 (g 1) 20.051/20.051 (h 1) 20.051/20.051 (a) 20.051/20.051 (c 1) 20.051/20.051 (fSNonEmpty) 20.051/20.051 (s 1) 20.051/20.051 ) 20.051/20.051 (RULES 20.051/20.051 f(x,y) -> g(s(x)) | c(g(x)) ->* c(a) 20.051/20.051 f(x,y) -> h(s(x)) | c(h(x)) ->* c(a) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 ) 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 20.051/20.051 Inlining of Conditions Processor [STERN17]: 20.051/20.051 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 Confluence Problem: 20.051/20.051 (VAR vNonEmpty x y) 20.051/20.051 (REPLACEMENT-MAP 20.051/20.051 (f 1, 2) 20.051/20.051 (g 1) 20.051/20.051 (h 1) 20.051/20.051 (a) 20.051/20.051 (c 1) 20.051/20.051 (fSNonEmpty) 20.051/20.051 (s 1) 20.051/20.051 ) 20.051/20.051 (RULES 20.051/20.051 f(x,y) -> g(s(x)) | c(g(x)) ->* c(a) 20.051/20.051 f(x,y) -> h(s(x)) | c(h(x)) ->* c(a) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 ) 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 20.051/20.051 Clean CTRS Processor: 20.051/20.051 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 Confluence Problem: 20.051/20.051 (VAR x y) 20.051/20.051 (REPLACEMENT-MAP 20.051/20.051 (f 1, 2) 20.051/20.051 (g 1) 20.051/20.051 (h 1) 20.051/20.051 (a) 20.051/20.051 (c 1) 20.051/20.051 (fSNonEmpty) 20.051/20.051 (s 1) 20.051/20.051 ) 20.051/20.051 (RULES 20.051/20.051 f(x,y) -> g(s(x)) | c(g(x)) ->* c(a) 20.051/20.051 f(x,y) -> h(s(x)) | c(h(x)) ->* c(a) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 ) 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 20.051/20.051 CRule InfChecker Info: 20.051/20.051 f(x,y) -> g(s(x)) | c(g(x)) ->* c(a) 20.051/20.051 Rule remains 20.051/20.051 Proof: 20.051/20.051 NO 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 20.051/20.051 Infeasibility Problem: 20.051/20.051 [(VAR vNonEmpty x y vNonEmpty x) 20.051/20.051 (STRATEGY CONTEXTSENSITIVE 20.051/20.051 (f 1 2) 20.051/20.051 (g 1) 20.051/20.051 (h 1) 20.051/20.051 (a) 20.051/20.051 (c 1) 20.051/20.051 (fSNonEmpty) 20.051/20.051 (s 1) 20.051/20.051 ) 20.051/20.051 (RULES 20.051/20.051 f(x,y) -> g(s(x)) | c(g(x)) ->* c(a) 20.051/20.051 f(x,y) -> h(s(x)) | c(h(x)) ->* c(a) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 )] 20.051/20.051 20.051/20.051 Infeasibility Conditions: 20.051/20.051 c(g(x)) ->* c(a) 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 20.051/20.051 Obtaining a proof using Prover9: 20.051/20.051 20.051/20.051 -> Prover9 Output: 20.051/20.051 ============================== Prover9 =============================== 20.051/20.051 Prover9 (64) version 2009-11A, November 2009. 20.051/20.051 Process 72365 was started by ubuntu on ubuntu, 20.051/20.051 Wed Mar 9 10:03:03 2022 20.051/20.051 The command was "./prover9 -f /tmp/prover972355-0.in". 20.051/20.051 ============================== end of head =========================== 20.051/20.051 20.051/20.051 ============================== INPUT ================================= 20.051/20.051 20.051/20.051 % Reading from file /tmp/prover972355-0.in 20.051/20.051 20.051/20.051 assign(max_seconds,20). 20.051/20.051 20.051/20.051 formulas(assumptions). 20.051/20.051 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 20.051/20.051 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 20.051/20.051 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 20.051/20.051 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 20.051/20.051 ->_s0(x1,y) -> ->_s0(c(x1),c(y)) # label(congruence). 20.051/20.051 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 20.051/20.051 ->*_s0(c(g(x1)),c(a)) -> ->_s0(f(x1,x2),g(s(x1))) # label(replacement). 20.051/20.051 ->*_s0(c(h(x1)),c(a)) -> ->_s0(f(x1,x2),h(s(x1))) # label(replacement). 20.051/20.051 ->_s0(g(s(x1)),x1) # label(replacement). 20.051/20.051 ->_s0(h(s(x1)),x1) # label(replacement). 20.051/20.051 ->*_s0(x,x) # label(reflexivity). 20.051/20.051 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(goals). 20.051/20.051 (exists x4 ->*_s0(c(g(x4)),c(a))) # label(goal). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 ============================== end of input ========================== 20.051/20.051 20.051/20.051 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.051/20.051 20.051/20.051 % Formulas that are not ordinary clauses: 20.051/20.051 1 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 2 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 3 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 4 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 5 ->_s0(x1,y) -> ->_s0(c(x1),c(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 6 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 7 ->*_s0(c(g(x1)),c(a)) -> ->_s0(f(x1,x2),g(s(x1))) # label(replacement) # label(non_clause). [assumption]. 20.051/20.051 8 ->*_s0(c(h(x1)),c(a)) -> ->_s0(f(x1,x2),h(s(x1))) # label(replacement) # label(non_clause). [assumption]. 20.051/20.051 9 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.051/20.051 10 (exists x4 ->*_s0(c(g(x4)),c(a))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.051/20.051 20.051/20.051 ============================== end of process non-clausal formulas === 20.051/20.051 20.051/20.051 ============================== PROCESS INITIAL CLAUSES =============== 20.051/20.051 20.051/20.051 % Clauses before input processing: 20.051/20.051 20.051/20.051 formulas(usable). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(sos). 20.051/20.051 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.051/20.051 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.051/20.051 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.051/20.051 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.051/20.051 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence). [clausify(5)]. 20.051/20.051 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 20.051/20.051 -->*_s0(c(g(x)),c(a)) | ->_s0(f(x,y),g(s(x))) # label(replacement). [clausify(7)]. 20.051/20.051 -->*_s0(c(h(x)),c(a)) | ->_s0(f(x,y),h(s(x))) # label(replacement). [clausify(8)]. 20.051/20.051 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.051/20.051 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 20.051/20.051 -->*_s0(c(g(x)),c(a)) # label(goal). [deny(10)]. 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(demodulators). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 ============================== PREDICATE ELIMINATION ================= 20.051/20.051 20.051/20.051 No predicates eliminated. 20.051/20.051 20.051/20.051 ============================== end predicate elimination ============= 20.051/20.051 20.051/20.051 Auto_denials: 20.051/20.051 % copying label goal to answer in negative clause 20.051/20.051 20.051/20.051 Term ordering decisions: 20.051/20.051 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.051/20.051 Function symbol precedence: function_order([ a, f, c, s, g, h ]). 20.051/20.051 After inverse_order: (no changes). 20.051/20.051 Unfolding symbols: (none). 20.051/20.051 20.051/20.051 Auto_inference settings: 20.051/20.051 % set(neg_binary_resolution). % (HNE depth_diff=-6) 20.051/20.051 % clear(ordered_res). % (HNE depth_diff=-6) 20.051/20.051 % set(ur_resolution). % (HNE depth_diff=-6) 20.051/20.051 % set(ur_resolution) -> set(pos_ur_resolution). 20.051/20.051 % set(ur_resolution) -> set(neg_ur_resolution). 20.051/20.051 20.051/20.051 Auto_process settings: (no changes). 20.051/20.051 20.051/20.051 kept: 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.051/20.051 kept: 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.051/20.051 kept: 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.051/20.051 kept: 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.051/20.051 kept: 15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence). [clausify(5)]. 20.051/20.051 kept: 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 20.051/20.051 kept: 17 -->*_s0(c(g(x)),c(a)) | ->_s0(f(x,y),g(s(x))) # label(replacement). [clausify(7)]. 20.051/20.051 kept: 18 -->*_s0(c(h(x)),c(a)) | ->_s0(f(x,y),h(s(x))) # label(replacement). [clausify(8)]. 20.051/20.051 kept: 19 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 kept: 20 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 kept: 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.051/20.051 kept: 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 20.051/20.051 kept: 23 -->*_s0(c(g(x)),c(a)) # label(goal) # answer(goal). [deny(10)]. 20.051/20.051 20.051/20.051 ============================== end of process initial clauses ======== 20.051/20.051 20.051/20.051 ============================== CLAUSES FOR SEARCH ==================== 20.051/20.051 20.051/20.051 % Clauses after input processing: 20.051/20.051 20.051/20.051 formulas(usable). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(sos). 20.051/20.051 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.051/20.051 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.051/20.051 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.051/20.051 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.051/20.051 15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence). [clausify(5)]. 20.051/20.051 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 20.051/20.051 18 -->*_s0(c(h(x)),c(a)) | ->_s0(f(x,y),h(s(x))) # label(replacement). [clausify(8)]. 20.051/20.051 19 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 20 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.051/20.051 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 20.051/20.051 23 -->*_s0(c(g(x)),c(a)) # label(goal) # answer(goal). [deny(10)]. 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(demodulators). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 ============================== end of clauses for search ============= 20.051/20.051 20.051/20.051 ============================== SEARCH ================================ 20.051/20.051 20.051/20.051 % Starting search at 0.00 seconds. 20.051/20.051 20.051/20.051 given #1 (I,wt=10): 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.051/20.051 20.051/20.051 given #2 (I,wt=10): 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.051/20.051 20.051/20.051 given #3 (I,wt=8): 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.051/20.051 20.051/20.051 given #4 (I,wt=8): 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.051/20.051 20.051/20.051 given #5 (I,wt=8): 15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence). [clausify(5)]. 20.051/20.051 20.051/20.051 given #6 (I,wt=8): 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 20.051/20.051 20.051/20.051 given #7 (I,wt=13): 18 -->*_s0(c(h(x)),c(a)) | ->_s0(f(x,y),h(s(x))) # label(replacement). [clausify(8)]. 20.051/20.051 20.051/20.051 given #8 (I,wt=5): 19 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 20.051/20.051 given #9 (I,wt=5): 20 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 20.051/20.051 given #10 (I,wt=3): 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.051/20.051 20.051/20.051 given #11 (I,wt=9): 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 20.051/20.051 20.051/20.051 given #12 (I,wt=6): 23 -->*_s0(c(g(x)),c(a)) # label(goal) # answer(goal). [deny(10)]. 20.051/20.051 20.051/20.051 ============================== PROOF ================================= 20.051/20.051 20.051/20.051 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.051/20.051 % Length of proof is 11. 20.051/20.051 % Level of proof is 3. 20.051/20.051 % Maximum clause weight is 9.000. 20.051/20.051 % Given clauses 12. 20.051/20.051 20.051/20.051 5 ->_s0(x1,y) -> ->_s0(c(x1),c(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 9 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.051/20.051 10 (exists x4 ->*_s0(c(g(x4)),c(a))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.051/20.051 15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence). [clausify(5)]. 20.051/20.051 19 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.051/20.051 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 20.051/20.051 23 -->*_s0(c(g(x)),c(a)) # label(goal) # answer(goal). [deny(10)]. 20.051/20.051 25 ->_s0(c(g(s(x))),c(x)). [ur(15,a,19,a)]. 20.051/20.051 39 -->_s0(c(g(x)),c(a)) # answer(goal). [ur(22,b,21,a,c,23,a)]. 20.051/20.051 40 $F # answer(goal). [resolve(39,a,25,a)]. 20.051/20.051 20.051/20.051 ============================== end of proof ========================== 20.051/20.051 20.051/20.051 ============================== STATISTICS ============================ 20.051/20.051 20.051/20.051 Given=12. Generated=29. Kept=29. proofs=1. 20.051/20.051 Usable=12. Sos=14. Demods=0. Limbo=1, Disabled=14. Hints=0. 20.051/20.051 Kept_by_rule=0, Deleted_by_rule=0. 20.051/20.051 Forward_subsumed=0. Back_subsumed=1. 20.051/20.051 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.051/20.051 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.051/20.051 Demod_attempts=0. Demod_rewrites=0. 20.051/20.051 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.051/20.051 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=15. 20.051/20.051 Megabytes=0.10. 20.051/20.051 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.051/20.051 20.051/20.051 ============================== end of statistics ===================== 20.051/20.051 20.051/20.051 ============================== end of search ========================= 20.051/20.051 20.051/20.051 THEOREM PROVED 20.051/20.051 20.051/20.051 Exiting with 1 proof. 20.051/20.051 20.051/20.051 Process 72365 exit (max_proofs) Wed Mar 9 10:03:03 2022 20.051/20.051 20.051/20.051 20.051/20.051 The problem is feasible. 20.051/20.051 20.051/20.051 20.051/20.051 CRule InfChecker Info: 20.051/20.051 f(x,y) -> h(s(x)) | c(h(x)) ->* c(a) 20.051/20.051 Rule remains 20.051/20.051 Proof: 20.051/20.051 NO 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 20.051/20.051 Infeasibility Problem: 20.051/20.051 [(VAR vNonEmpty x y vNonEmpty x) 20.051/20.051 (STRATEGY CONTEXTSENSITIVE 20.051/20.051 (f 1 2) 20.051/20.051 (g 1) 20.051/20.051 (h 1) 20.051/20.051 (a) 20.051/20.051 (c 1) 20.051/20.051 (fSNonEmpty) 20.051/20.051 (s 1) 20.051/20.051 ) 20.051/20.051 (RULES 20.051/20.051 f(x,y) -> g(s(x)) | c(g(x)) ->* c(a) 20.051/20.051 f(x,y) -> h(s(x)) | c(h(x)) ->* c(a) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 )] 20.051/20.051 20.051/20.051 Infeasibility Conditions: 20.051/20.051 c(h(x)) ->* c(a) 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 20.051/20.051 Obtaining a proof using Prover9: 20.051/20.051 20.051/20.051 -> Prover9 Output: 20.051/20.051 ============================== Prover9 =============================== 20.051/20.051 Prover9 (64) version 2009-11A, November 2009. 20.051/20.051 Process 72389 was started by ubuntu on ubuntu, 20.051/20.051 Wed Mar 9 10:03:03 2022 20.051/20.051 The command was "./prover9 -f /tmp/prover972379-0.in". 20.051/20.051 ============================== end of head =========================== 20.051/20.051 20.051/20.051 ============================== INPUT ================================= 20.051/20.051 20.051/20.051 % Reading from file /tmp/prover972379-0.in 20.051/20.051 20.051/20.051 assign(max_seconds,20). 20.051/20.051 20.051/20.051 formulas(assumptions). 20.051/20.051 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 20.051/20.051 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 20.051/20.051 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 20.051/20.051 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 20.051/20.051 ->_s0(x1,y) -> ->_s0(c(x1),c(y)) # label(congruence). 20.051/20.051 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 20.051/20.051 ->*_s0(c(g(x1)),c(a)) -> ->_s0(f(x1,x2),g(s(x1))) # label(replacement). 20.051/20.051 ->*_s0(c(h(x1)),c(a)) -> ->_s0(f(x1,x2),h(s(x1))) # label(replacement). 20.051/20.051 ->_s0(g(s(x1)),x1) # label(replacement). 20.051/20.051 ->_s0(h(s(x1)),x1) # label(replacement). 20.051/20.051 ->*_s0(x,x) # label(reflexivity). 20.051/20.051 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(goals). 20.051/20.051 (exists x4 ->*_s0(c(h(x4)),c(a))) # label(goal). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 ============================== end of input ========================== 20.051/20.051 20.051/20.051 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.051/20.051 20.051/20.051 % Formulas that are not ordinary clauses: 20.051/20.051 1 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 2 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 3 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 4 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 5 ->_s0(x1,y) -> ->_s0(c(x1),c(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 6 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 7 ->*_s0(c(g(x1)),c(a)) -> ->_s0(f(x1,x2),g(s(x1))) # label(replacement) # label(non_clause). [assumption]. 20.051/20.051 8 ->*_s0(c(h(x1)),c(a)) -> ->_s0(f(x1,x2),h(s(x1))) # label(replacement) # label(non_clause). [assumption]. 20.051/20.051 9 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.051/20.051 10 (exists x4 ->*_s0(c(h(x4)),c(a))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.051/20.051 20.051/20.051 ============================== end of process non-clausal formulas === 20.051/20.051 20.051/20.051 ============================== PROCESS INITIAL CLAUSES =============== 20.051/20.051 20.051/20.051 % Clauses before input processing: 20.051/20.051 20.051/20.051 formulas(usable). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(sos). 20.051/20.051 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.051/20.051 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.051/20.051 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.051/20.051 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.051/20.051 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence). [clausify(5)]. 20.051/20.051 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 20.051/20.051 -->*_s0(c(g(x)),c(a)) | ->_s0(f(x,y),g(s(x))) # label(replacement). [clausify(7)]. 20.051/20.051 -->*_s0(c(h(x)),c(a)) | ->_s0(f(x,y),h(s(x))) # label(replacement). [clausify(8)]. 20.051/20.051 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.051/20.051 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 20.051/20.051 -->*_s0(c(h(x)),c(a)) # label(goal). [deny(10)]. 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(demodulators). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 ============================== PREDICATE ELIMINATION ================= 20.051/20.051 20.051/20.051 No predicates eliminated. 20.051/20.051 20.051/20.051 ============================== end predicate elimination ============= 20.051/20.051 20.051/20.051 Auto_denials: 20.051/20.051 % copying label goal to answer in negative clause 20.051/20.051 20.051/20.051 Term ordering decisions: 20.051/20.051 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.051/20.051 Function symbol precedence: function_order([ a, f, c, s, g, h ]). 20.051/20.051 After inverse_order: (no changes). 20.051/20.051 Unfolding symbols: (none). 20.051/20.051 20.051/20.051 Auto_inference settings: 20.051/20.051 % set(neg_binary_resolution). % (HNE depth_diff=-6) 20.051/20.051 % clear(ordered_res). % (HNE depth_diff=-6) 20.051/20.051 % set(ur_resolution). % (HNE depth_diff=-6) 20.051/20.051 % set(ur_resolution) -> set(pos_ur_resolution). 20.051/20.051 % set(ur_resolution) -> set(neg_ur_resolution). 20.051/20.051 20.051/20.051 Auto_process settings: (no changes). 20.051/20.051 20.051/20.051 kept: 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.051/20.051 kept: 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.051/20.051 kept: 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.051/20.051 kept: 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.051/20.051 kept: 15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence). [clausify(5)]. 20.051/20.051 kept: 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 20.051/20.051 kept: 17 -->*_s0(c(g(x)),c(a)) | ->_s0(f(x,y),g(s(x))) # label(replacement). [clausify(7)]. 20.051/20.051 kept: 18 -->*_s0(c(h(x)),c(a)) | ->_s0(f(x,y),h(s(x))) # label(replacement). [clausify(8)]. 20.051/20.051 kept: 19 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 kept: 20 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 kept: 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.051/20.051 kept: 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 20.051/20.051 kept: 23 -->*_s0(c(h(x)),c(a)) # label(goal) # answer(goal). [deny(10)]. 20.051/20.051 20.051/20.051 ============================== end of process initial clauses ======== 20.051/20.051 20.051/20.051 ============================== CLAUSES FOR SEARCH ==================== 20.051/20.051 20.051/20.051 % Clauses after input processing: 20.051/20.051 20.051/20.051 formulas(usable). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(sos). 20.051/20.051 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.051/20.051 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.051/20.051 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.051/20.051 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.051/20.051 15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence). [clausify(5)]. 20.051/20.051 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 20.051/20.051 17 -->*_s0(c(g(x)),c(a)) | ->_s0(f(x,y),g(s(x))) # label(replacement). [clausify(7)]. 20.051/20.051 19 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 20 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.051/20.051 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 20.051/20.051 23 -->*_s0(c(h(x)),c(a)) # label(goal) # answer(goal). [deny(10)]. 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(demodulators). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 ============================== end of clauses for search ============= 20.051/20.051 20.051/20.051 ============================== SEARCH ================================ 20.051/20.051 20.051/20.051 % Starting search at 0.00 seconds. 20.051/20.051 20.051/20.051 given #1 (I,wt=10): 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.051/20.051 20.051/20.051 given #2 (I,wt=10): 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.051/20.051 20.051/20.051 given #3 (I,wt=8): 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.051/20.051 20.051/20.051 given #4 (I,wt=8): 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.051/20.051 20.051/20.051 given #5 (I,wt=8): 15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence). [clausify(5)]. 20.051/20.051 20.051/20.051 given #6 (I,wt=8): 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 20.051/20.051 20.051/20.051 given #7 (I,wt=13): 17 -->*_s0(c(g(x)),c(a)) | ->_s0(f(x,y),g(s(x))) # label(replacement). [clausify(7)]. 20.051/20.051 20.051/20.051 given #8 (I,wt=5): 19 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 20.051/20.051 given #9 (I,wt=5): 20 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 20.051/20.051 given #10 (I,wt=3): 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.051/20.051 20.051/20.051 given #11 (I,wt=9): 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 20.051/20.051 20.051/20.051 given #12 (I,wt=6): 23 -->*_s0(c(h(x)),c(a)) # label(goal) # answer(goal). [deny(10)]. 20.051/20.051 20.051/20.051 ============================== PROOF ================================= 20.051/20.051 20.051/20.051 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.051/20.051 % Length of proof is 11. 20.051/20.051 % Level of proof is 3. 20.051/20.051 % Maximum clause weight is 9.000. 20.051/20.051 % Given clauses 12. 20.051/20.051 20.051/20.051 5 ->_s0(x1,y) -> ->_s0(c(x1),c(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 9 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.051/20.051 10 (exists x4 ->*_s0(c(h(x4)),c(a))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.051/20.051 15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence). [clausify(5)]. 20.051/20.051 20 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.051/20.051 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 20.051/20.051 23 -->*_s0(c(h(x)),c(a)) # label(goal) # answer(goal). [deny(10)]. 20.051/20.051 31 ->_s0(c(h(s(x))),c(x)). [ur(15,a,20,a)]. 20.051/20.051 39 -->_s0(c(h(x)),c(a)) # answer(goal). [ur(22,b,21,a,c,23,a)]. 20.051/20.051 40 $F # answer(goal). [resolve(39,a,31,a)]. 20.051/20.051 20.051/20.051 ============================== end of proof ========================== 20.051/20.051 20.051/20.051 ============================== STATISTICS ============================ 20.051/20.051 20.051/20.051 Given=12. Generated=29. Kept=29. proofs=1. 20.051/20.051 Usable=12. Sos=14. Demods=0. Limbo=1, Disabled=14. Hints=0. 20.051/20.051 Kept_by_rule=0, Deleted_by_rule=0. 20.051/20.051 Forward_subsumed=0. Back_subsumed=1. 20.051/20.051 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.051/20.051 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.051/20.051 Demod_attempts=0. Demod_rewrites=0. 20.051/20.051 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.051/20.051 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=15. 20.051/20.051 Megabytes=0.10. 20.051/20.051 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.051/20.051 20.051/20.051 ============================== end of statistics ===================== 20.051/20.051 20.051/20.051 ============================== end of search ========================= 20.051/20.051 20.051/20.051 THEOREM PROVED 20.051/20.051 20.051/20.051 Exiting with 1 proof. 20.051/20.051 20.051/20.051 Process 72389 exit (max_proofs) Wed Mar 9 10:03:03 2022 20.051/20.051 20.051/20.051 20.051/20.051 The problem is feasible. 20.051/20.051 20.051/20.051 20.051/20.051 CRule InfChecker Info: 20.051/20.051 g(s(x)) -> x 20.051/20.051 Rule remains 20.051/20.051 Proof: 20.051/20.051 NO_CONDS 20.051/20.051 20.051/20.051 CRule InfChecker Info: 20.051/20.051 h(s(x)) -> x 20.051/20.051 Rule remains 20.051/20.051 Proof: 20.051/20.051 NO_CONDS 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 Confluence Problem: 20.051/20.051 (VAR x y) 20.051/20.051 (REPLACEMENT-MAP 20.051/20.051 (f 1, 2) 20.051/20.051 (g 1) 20.051/20.051 (h 1) 20.051/20.051 (a) 20.051/20.051 (c 1) 20.051/20.051 (fSNonEmpty) 20.051/20.051 (s 1) 20.051/20.051 ) 20.051/20.051 (RULES 20.051/20.051 f(x,y) -> g(s(x)) | c(g(x)) ->* c(a) 20.051/20.051 f(x,y) -> h(s(x)) | c(h(x)) ->* c(a) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 ) 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 20.051/20.051 Critical Pairs Processor: 20.051/20.051 -> Rules: 20.051/20.051 f(x,y) -> g(s(x)) | c(g(x)) ->* c(a) 20.051/20.051 f(x,y) -> h(s(x)) | c(h(x)) ->* c(a) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 -> Vars: 20.051/20.051 "x", "y" 20.051/20.051 20.051/20.051 -> Rlps: 20.051/20.051 crule: f(x,y) -> g(s(x)) | c(g(x)) ->* c(a), id: 1, possubterms: f(x,y)-> [] 20.051/20.051 crule: f(x,y) -> h(s(x)) | c(h(x)) ->* c(a), id: 2, possubterms: f(x,y)-> [] 20.051/20.051 crule: g(s(x)) -> x, id: 3, possubterms: g(s(x))-> [], s(x)-> [1] 20.051/20.051 crule: h(s(x)) -> x, id: 4, possubterms: h(s(x))-> [], s(x)-> [1] 20.051/20.051 20.051/20.051 -> Unifications: 20.051/20.051 R2 unifies with R1 at p: [], l: f(x,y), lp: f(x,y), conds: {c(h(x)) ->* c(a), c(g(x')) ->* c(a)}, sig: {x -> x',y -> y'}, l': f(x',y'), r: h(s(x)), r': g(s(x')) 20.051/20.051 20.051/20.051 -> Critical pairs info: 20.051/20.051 | c(h(x')) ->* c(a), c(g(x')) ->* c(a) => Not trivial, Overlay, NW0, N1 20.051/20.051 20.051/20.051 -> Problem conclusions: 20.051/20.051 Left linear, Right linear, Linear 20.051/20.051 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.051/20.051 CTRS Type: 1 20.051/20.051 Deterministic, Strongly deterministic 20.051/20.051 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.051/20.051 Maybe right-stable CTRS, Overlay CTRS 20.051/20.051 Maybe normal CTRS, Maybe almost normal CTRS 20.051/20.051 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.051/20.051 Maybe level confluent 20.051/20.051 Maybe confluent 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 Clean Infeasible CCPs Processor: 20.051/20.051 Num of CPIs: 1 20.051/20.051 Timeout: 60 20.051/20.051 Timeout for each infeasibility problem: 60 s 20.051/20.051 | c(h(x')) ->* c(a), c(g(x')) ->* c(a) => Not trivial, Overlay, Feasible conditions, NW0, N1 20.051/20.051 (CONDITIONTYPE ORIENTED) 20.051/20.051 (VAR x y x1) 20.051/20.051 (RULES 20.051/20.051 f(x,y) -> g(s(x)) | c(g(x)) ->* c(a) 20.051/20.051 f(x,y) -> h(s(x)) | c(h(x)) ->* c(a) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x) 20.051/20.051 (VAR x7) 20.051/20.051 (CONDITION c(h(x7)) ->* c(a), c(g(x7)) ->* c(a)) 20.051/20.051 20.051/20.051 Proof: 20.051/20.051 NO 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 20.051/20.051 Infeasibility Problem: 20.051/20.051 [(VAR vNonEmpty x y x1 vNonEmpty x7) 20.051/20.051 (STRATEGY CONTEXTSENSITIVE 20.051/20.051 (f 1 2) 20.051/20.051 (g 1) 20.051/20.051 (h 1) 20.051/20.051 (a) 20.051/20.051 (c 1) 20.051/20.051 (fSNonEmpty) 20.051/20.051 (s 1) 20.051/20.051 ) 20.051/20.051 (RULES 20.051/20.051 f(x,y) -> g(s(x)) | c(g(x)) ->* c(a) 20.051/20.051 f(x,y) -> h(s(x)) | c(h(x)) ->* c(a) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 )] 20.051/20.051 20.051/20.051 Infeasibility Conditions: 20.051/20.051 c(h(x7)) ->* c(a), c(g(x7)) ->* c(a) 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 20.051/20.051 Obtaining a proof using Prover9: 20.051/20.051 20.051/20.051 -> Prover9 Output: 20.051/20.051 ============================== Prover9 =============================== 20.051/20.051 Prover9 (64) version 2009-11A, November 2009. 20.051/20.051 Process 72418 was started by ubuntu on ubuntu, 20.051/20.051 Wed Mar 9 10:03:03 2022 20.051/20.051 The command was "./prover9 -f /tmp/prover972405-0.in". 20.051/20.051 ============================== end of head =========================== 20.051/20.051 20.051/20.051 ============================== INPUT ================================= 20.051/20.051 20.051/20.051 % Reading from file /tmp/prover972405-0.in 20.051/20.051 20.051/20.051 assign(max_seconds,20). 20.051/20.051 20.051/20.051 formulas(assumptions). 20.051/20.051 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 20.051/20.051 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 20.051/20.051 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 20.051/20.051 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence). 20.051/20.051 ->_s0(x1,y) -> ->_s0(c(x1),c(y)) # label(congruence). 20.051/20.051 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 20.051/20.051 ->*_s0(c(g(x1)),c(a)) -> ->_s0(f(x1,x2),g(s(x1))) # label(replacement). 20.051/20.051 ->*_s0(c(h(x1)),c(a)) -> ->_s0(f(x1,x2),h(s(x1))) # label(replacement). 20.051/20.051 ->_s0(g(s(x1)),x1) # label(replacement). 20.051/20.051 ->_s0(h(s(x1)),x1) # label(replacement). 20.051/20.051 ->*_s0(x,x) # label(reflexivity). 20.051/20.051 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(goals). 20.051/20.051 (exists x5 (->*_s0(c(h(x5)),c(a)) & ->*_s0(c(g(x5)),c(a)))) # label(goal). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 ============================== end of input ========================== 20.051/20.051 20.051/20.051 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.051/20.051 20.051/20.051 % Formulas that are not ordinary clauses: 20.051/20.051 1 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 2 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 3 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 4 ->_s0(x1,y) -> ->_s0(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 5 ->_s0(x1,y) -> ->_s0(c(x1),c(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 6 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 7 ->*_s0(c(g(x1)),c(a)) -> ->_s0(f(x1,x2),g(s(x1))) # label(replacement) # label(non_clause). [assumption]. 20.051/20.051 8 ->*_s0(c(h(x1)),c(a)) -> ->_s0(f(x1,x2),h(s(x1))) # label(replacement) # label(non_clause). [assumption]. 20.051/20.051 9 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.051/20.051 10 (exists x5 (->*_s0(c(h(x5)),c(a)) & ->*_s0(c(g(x5)),c(a)))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.051/20.051 20.051/20.051 ============================== end of process non-clausal formulas === 20.051/20.051 20.051/20.051 ============================== PROCESS INITIAL CLAUSES =============== 20.051/20.051 20.051/20.051 % Clauses before input processing: 20.051/20.051 20.051/20.051 formulas(usable). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(sos). 20.051/20.051 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.051/20.051 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.051/20.051 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.051/20.051 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.051/20.051 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence). [clausify(5)]. 20.051/20.051 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 20.051/20.051 -->*_s0(c(g(x)),c(a)) | ->_s0(f(x,y),g(s(x))) # label(replacement). [clausify(7)]. 20.051/20.051 -->*_s0(c(h(x)),c(a)) | ->_s0(f(x,y),h(s(x))) # label(replacement). [clausify(8)]. 20.051/20.051 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.051/20.051 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 20.051/20.051 -->*_s0(c(h(x)),c(a)) | -->*_s0(c(g(x)),c(a)) # label(goal). [deny(10)]. 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(demodulators). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 ============================== PREDICATE ELIMINATION ================= 20.051/20.051 20.051/20.051 No predicates eliminated. 20.051/20.051 20.051/20.051 ============================== end predicate elimination ============= 20.051/20.051 20.051/20.051 Auto_denials: 20.051/20.051 % copying label goal to answer in negative clause 20.051/20.051 20.051/20.051 Term ordering decisions: 20.051/20.051 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.051/20.051 Function symbol precedence: function_order([ a, f, c, s, g, h ]). 20.051/20.051 After inverse_order: (no changes). 20.051/20.051 Unfolding symbols: (none). 20.051/20.051 20.051/20.051 Auto_inference settings: 20.051/20.051 % set(neg_binary_resolution). % (HNE depth_diff=-6) 20.051/20.051 % clear(ordered_res). % (HNE depth_diff=-6) 20.051/20.051 % set(ur_resolution). % (HNE depth_diff=-6) 20.051/20.051 % set(ur_resolution) -> set(pos_ur_resolution). 20.051/20.051 % set(ur_resolution) -> set(neg_ur_resolution). 20.051/20.051 20.051/20.051 Auto_process settings: 20.051/20.051 % set(unit_deletion). % (Horn set with negative nonunits) 20.051/20.051 20.051/20.051 kept: 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.051/20.051 kept: 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.051/20.051 kept: 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.051/20.051 kept: 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.051/20.051 kept: 15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence). [clausify(5)]. 20.051/20.051 kept: 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 20.051/20.051 kept: 17 -->*_s0(c(g(x)),c(a)) | ->_s0(f(x,y),g(s(x))) # label(replacement). [clausify(7)]. 20.051/20.051 kept: 18 -->*_s0(c(h(x)),c(a)) | ->_s0(f(x,y),h(s(x))) # label(replacement). [clausify(8)]. 20.051/20.051 kept: 19 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 kept: 20 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 kept: 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.051/20.051 kept: 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 20.051/20.051 kept: 23 -->*_s0(c(h(x)),c(a)) | -->*_s0(c(g(x)),c(a)) # label(goal) # answer(goal). [deny(10)]. 20.051/20.051 20.051/20.051 ============================== end of process initial clauses ======== 20.051/20.051 20.051/20.051 ============================== CLAUSES FOR SEARCH ==================== 20.051/20.051 20.051/20.051 % Clauses after input processing: 20.051/20.051 20.051/20.051 formulas(usable). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(sos). 20.051/20.051 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.051/20.051 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.051/20.051 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.051/20.051 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.051/20.051 15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence). [clausify(5)]. 20.051/20.051 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 20.051/20.051 17 -->*_s0(c(g(x)),c(a)) | ->_s0(f(x,y),g(s(x))) # label(replacement). [clausify(7)]. 20.051/20.051 18 -->*_s0(c(h(x)),c(a)) | ->_s0(f(x,y),h(s(x))) # label(replacement). [clausify(8)]. 20.051/20.051 19 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 20 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.051/20.051 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 20.051/20.051 23 -->*_s0(c(h(x)),c(a)) | -->*_s0(c(g(x)),c(a)) # label(goal) # answer(goal). [deny(10)]. 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(demodulators). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 ============================== end of clauses for search ============= 20.051/20.051 20.051/20.051 ============================== SEARCH ================================ 20.051/20.051 20.051/20.051 % Starting search at 0.00 seconds. 20.051/20.051 20.051/20.051 given #1 (I,wt=10): 11 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.051/20.051 20.051/20.051 given #2 (I,wt=10): 12 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.051/20.051 20.051/20.051 given #3 (I,wt=8): 13 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 20.051/20.051 20.051/20.051 given #4 (I,wt=8): 14 -->_s0(x,y) | ->_s0(h(x),h(y)) # label(congruence). [clausify(4)]. 20.051/20.051 20.051/20.051 given #5 (I,wt=8): 15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence). [clausify(5)]. 20.051/20.051 20.051/20.051 given #6 (I,wt=8): 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 20.051/20.051 20.051/20.051 given #7 (I,wt=13): 17 -->*_s0(c(g(x)),c(a)) | ->_s0(f(x,y),g(s(x))) # label(replacement). [clausify(7)]. 20.051/20.051 20.051/20.051 given #8 (I,wt=13): 18 -->*_s0(c(h(x)),c(a)) | ->_s0(f(x,y),h(s(x))) # label(replacement). [clausify(8)]. 20.051/20.051 20.051/20.051 given #9 (I,wt=5): 19 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 20.051/20.051 given #10 (I,wt=5): 20 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 20.051/20.051 given #11 (I,wt=3): 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.051/20.051 20.051/20.051 given #12 (I,wt=9): 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 20.051/20.051 20.051/20.051 given #13 (I,wt=12): 23 -->*_s0(c(h(x)),c(a)) | -->*_s0(c(g(x)),c(a)) # label(goal) # answer(goal). [deny(10)]. 20.051/20.051 20.051/20.051 given #14 (A,wt=7): 24 ->_s0(s(g(s(x))),s(x)). [ur(16,a,19,a)]. 20.051/20.051 20.051/20.051 given #15 (F,wt=15): 38 -->*_s0(c(g(x)),c(a)) | -->_s0(c(h(x)),y) | -->*_s0(y,c(a)) # answer(goal). [resolve(23,a,22,c)]. 20.051/20.051 20.051/20.051 given #16 (F,wt=12): 49 -->*_s0(c(g(x)),c(a)) | -->_s0(c(h(x)),c(a)) # answer(goal). [resolve(38,c,21,a)]. 20.051/20.051 20.051/20.051 given #17 (F,wt=10): 51 -->*_s0(c(g(x)),c(a)) | -->_s0(h(x),a) # answer(goal). [resolve(49,b,15,b)]. 20.051/20.051 20.051/20.051 given #18 (F,wt=7): 53 -->*_s0(c(g(s(a))),c(a)) # answer(goal). [resolve(51,b,20,a)]. 20.051/20.051 20.051/20.051 ============================== PROOF ================================= 20.051/20.051 20.051/20.051 % Proof 1 at 0.01 (+ 0.00) seconds: goal. 20.051/20.051 % Length of proof is 15. 20.051/20.051 % Level of proof is 6. 20.051/20.051 % Maximum clause weight is 15.000. 20.051/20.051 % Given clauses 18. 20.051/20.051 20.051/20.051 5 ->_s0(x1,y) -> ->_s0(c(x1),c(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 9 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.051/20.051 10 (exists x5 (->*_s0(c(h(x5)),c(a)) & ->*_s0(c(g(x5)),c(a)))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.051/20.051 15 -->_s0(x,y) | ->_s0(c(x),c(y)) # label(congruence). [clausify(5)]. 20.051/20.051 19 ->_s0(g(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 20 ->_s0(h(s(x)),x) # label(replacement). [assumption]. 20.051/20.051 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.051/20.051 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(9)]. 20.051/20.051 23 -->*_s0(c(h(x)),c(a)) | -->*_s0(c(g(x)),c(a)) # label(goal) # answer(goal). [deny(10)]. 20.051/20.051 25 ->_s0(c(g(s(x))),c(x)). [ur(15,a,19,a)]. 20.051/20.051 38 -->*_s0(c(g(x)),c(a)) | -->_s0(c(h(x)),y) | -->*_s0(y,c(a)) # answer(goal). [resolve(23,a,22,c)]. 20.051/20.051 49 -->*_s0(c(g(x)),c(a)) | -->_s0(c(h(x)),c(a)) # answer(goal). [resolve(38,c,21,a)]. 20.051/20.051 51 -->*_s0(c(g(x)),c(a)) | -->_s0(h(x),a) # answer(goal). [resolve(49,b,15,b)]. 20.051/20.051 53 -->*_s0(c(g(s(a))),c(a)) # answer(goal). [resolve(51,b,20,a)]. 20.051/20.051 55 $F # answer(goal). [ur(22,b,21,a,c,53,a),unit_del(a,25)]. 20.051/20.051 20.051/20.051 ============================== end of proof ========================== 20.051/20.051 20.051/20.051 ============================== STATISTICS ============================ 20.051/20.051 20.051/20.051 Given=18. Generated=47. Kept=44. proofs=1. 20.051/20.051 Usable=18. Sos=25. Demods=0. Limbo=1, Disabled=13. Hints=0. 20.051/20.051 Kept_by_rule=0, Deleted_by_rule=0. 20.051/20.051 Forward_subsumed=2. Back_subsumed=0. 20.051/20.051 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.051/20.051 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.051/20.051 Demod_attempts=0. Demod_rewrites=0. 20.051/20.051 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.051/20.051 Nonunit_fsub_feature_tests=14. Nonunit_bsub_feature_tests=48. 20.051/20.051 Megabytes=0.12. 20.051/20.051 User_CPU=0.01, System_CPU=0.00, Wall_clock=0. 20.051/20.051 20.051/20.051 ============================== end of statistics ===================== 20.051/20.051 20.051/20.051 ============================== end of search ========================= 20.051/20.051 20.051/20.051 THEOREM PROVED 20.051/20.051 20.051/20.051 Exiting with 1 proof. 20.051/20.051 20.051/20.051 Process 72418 exit (max_proofs) Wed Mar 9 10:03:03 2022 20.051/20.051 20.051/20.051 20.051/20.051 The problem is feasible. 20.051/20.051 20.051/20.051 20.051/20.051 20.051/20.051 20.051/20.051 -> Problem conclusions: 20.051/20.051 Left linear, Right linear, Linear 20.051/20.051 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.051/20.051 CTRS Type: 1 20.051/20.051 Deterministic, Strongly deterministic 20.051/20.051 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.051/20.051 Maybe right-stable CTRS, Overlay CTRS 20.051/20.051 Maybe normal CTRS, Maybe almost normal CTRS 20.051/20.051 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.051/20.051 Maybe level confluent 20.051/20.051 Maybe confluent 20.051/20.051 20.051/20.051 Resulting CCPs: 20.051/20.051 | c(h(x')) ->* c(a), c(g(x')) ->* c(a) => Not trivial, Overlay, Feasible conditions, NW0, N1 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 Underlying TRS Termination Processor: 20.051/20.051 20.051/20.051 Resulting Underlying TRS: 20.051/20.051 (VAR x y x') 20.051/20.051 (STRATEGY CONTEXTSENSITIVE 20.051/20.051 (f 1 2) 20.051/20.051 (g 1) 20.051/20.051 (h 1) 20.051/20.051 ) 20.051/20.051 (RULES 20.051/20.051 f(x,y) -> g(s(x)) 20.051/20.051 f(x,y) -> h(s(x)) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 ) 20.051/20.051 Underlying TRS terminating? 20.051/20.051 YES 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 20.051/20.051 (VAR vu95NonEmpty x y xu39) 20.051/20.051 (RULES 20.051/20.051 f(x,y) -> g(s(x)) 20.051/20.051 f(x,y) -> h(s(x)) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 ) 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 20.051/20.051 Dependency Pairs Processor: 20.051/20.051 -> Pairs: 20.051/20.051 F(x,y) -> G(s(x)) 20.051/20.051 F(x,y) -> H(s(x)) 20.051/20.051 -> Rules: 20.051/20.051 f(x,y) -> g(s(x)) 20.051/20.051 f(x,y) -> h(s(x)) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 20.051/20.051 SCC Processor: 20.051/20.051 -> Pairs: 20.051/20.051 F(x,y) -> G(s(x)) 20.051/20.051 F(x,y) -> H(s(x)) 20.051/20.051 -> Rules: 20.051/20.051 f(x,y) -> g(s(x)) 20.051/20.051 f(x,y) -> h(s(x)) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 ->Strongly Connected Components: 20.051/20.051 There is no strongly connected component 20.051/20.051 20.051/20.051 The problem is finite. 20.051/20.051 20.051/20.051 20.051/20.051 -> Problem conclusions: 20.051/20.051 Left linear, Right linear, Linear 20.051/20.051 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.051/20.051 CTRS Type: 1 20.051/20.051 Deterministic, Strongly deterministic 20.051/20.051 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.051/20.051 Maybe right-stable CTRS, Overlay CTRS 20.051/20.051 Maybe normal CTRS, Maybe almost normal CTRS 20.051/20.051 Terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.051/20.051 Maybe level confluent 20.051/20.051 Maybe confluent 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 Confluence Problem: 20.051/20.051 (VAR x y x') 20.051/20.051 (REPLACEMENT-MAP 20.051/20.051 (f 1, 2) 20.051/20.051 (g 1) 20.051/20.051 (h 1) 20.051/20.051 (a) 20.051/20.051 (c 1) 20.051/20.051 (fSNonEmpty) 20.051/20.051 (s 1) 20.051/20.051 ) 20.051/20.051 (RULES 20.051/20.051 f(x,y) -> g(s(x)) | c(g(x)) ->* c(a) 20.051/20.051 f(x,y) -> h(s(x)) | c(h(x)) ->* c(a) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 ) 20.051/20.051 Critical Pairs: 20.051/20.051 | c(h(x')) ->* c(a), c(g(x')) ->* c(a) => Not trivial, Overlay, Feasible conditions, NW0, N1 20.051/20.051 Terminating:YES 20.051/20.051 20.051/20.051 -> Problem conclusions: 20.051/20.051 Left linear, Right linear, Linear 20.051/20.051 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.051/20.051 CTRS Type: 1 20.051/20.051 Deterministic, Strongly deterministic 20.051/20.051 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.051/20.051 Maybe right-stable CTRS, Overlay CTRS 20.051/20.051 Maybe normal CTRS, Maybe almost normal CTRS 20.051/20.051 Terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.051/20.051 Maybe level confluent 20.051/20.051 Maybe confluent 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 20.051/20.051 Right Stable Processor: 20.051/20.051 Right-stable CTRS 20.051/20.051 Justification: 20.051/20.051 20.051/20.051 -> Term right-stability conditions: 20.051/20.051 Term: c(a) 20.051/20.051 Right-stable term 20.051/20.051 Linear constructor form 20.051/20.051 Don't know if term is a ground normal form (not needed) 20.051/20.051 Right-stability condition achieved 20.051/20.051 A call to InfChecker wasn't needed 20.051/20.051 20.051/20.051 20.051/20.051 -> Term right-stability conditions: 20.051/20.051 Term: c(a) 20.051/20.051 Right-stable term 20.051/20.051 Linear constructor form 20.051/20.051 Don't know if term is a ground normal form (not needed) 20.051/20.051 Right-stability condition achieved 20.051/20.051 A call to InfChecker wasn't needed 20.051/20.051 20.051/20.051 -> Problem conclusions: 20.051/20.051 Left linear, Right linear, Linear 20.051/20.051 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.051/20.051 CTRS Type: 1 20.051/20.051 Deterministic, Strongly deterministic 20.051/20.051 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.051/20.051 Right-stable CTRS, Overlay CTRS 20.051/20.051 Maybe normal CTRS, Almost normal CTRS 20.051/20.051 Terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.051/20.051 Maybe level confluent 20.051/20.051 Maybe confluent 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 Confluence Problem: 20.051/20.051 (VAR x y x') 20.051/20.051 (REPLACEMENT-MAP 20.051/20.051 (f 1, 2) 20.051/20.051 (g 1) 20.051/20.051 (h 1) 20.051/20.051 (a) 20.051/20.051 (c 1) 20.051/20.051 (fSNonEmpty) 20.051/20.051 (s 1) 20.051/20.051 ) 20.051/20.051 (RULES 20.051/20.051 f(x,y) -> g(s(x)) | c(g(x)) ->* c(a) 20.051/20.051 f(x,y) -> h(s(x)) | c(h(x)) ->* c(a) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 ) 20.051/20.051 Critical Pairs: 20.051/20.051 | c(h(x')) ->* c(a), c(g(x')) ->* c(a) => Not trivial, Overlay, Feasible conditions, NW0, N1 20.051/20.051 Terminating:YES 20.051/20.051 20.051/20.051 -> Problem conclusions: 20.051/20.051 Left linear, Right linear, Linear 20.051/20.051 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.051/20.051 CTRS Type: 1 20.051/20.051 Deterministic, Strongly deterministic 20.051/20.051 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.051/20.051 Right-stable CTRS, Overlay CTRS 20.051/20.051 Maybe normal CTRS, Almost normal CTRS 20.051/20.051 Terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.051/20.051 Maybe level confluent 20.051/20.051 Maybe confluent 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 20.051/20.051 Normal RConds Processor: 20.051/20.051 YES 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 20.051/20.051 Infeasibility Problem: 20.051/20.051 [(VAR vNonEmpty x y x1 vNonEmpty z) 20.051/20.051 (STRATEGY CONTEXTSENSITIVE 20.051/20.051 (f 1 2) 20.051/20.051 (g 1) 20.051/20.051 (h 1) 20.051/20.051 (a) 20.051/20.051 (c 1) 20.051/20.051 (fSNonEmpty) 20.051/20.051 (s 1) 20.051/20.051 ) 20.051/20.051 (RULES 20.051/20.051 f(x,y) -> g(s(x)) | c(g(x)) ->* c(a) 20.051/20.051 f(x,y) -> h(s(x)) | c(h(x)) ->* c(a) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 )] 20.051/20.051 20.051/20.051 Infeasibility Conditions: 20.051/20.051 c(a) -> z 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 20.051/20.051 Obtaining a model using Mace4: 20.051/20.051 20.051/20.051 -> Usable Rules: 20.051/20.051 Empty 20.051/20.051 20.051/20.051 -> Mace4 Output: 20.051/20.051 ============================== Mace4 ================================= 20.051/20.051 Mace4 (64) version 2009-11A, November 2009. 20.051/20.051 Process 72491 was started by ubuntu on ubuntu, 20.051/20.051 Wed Mar 9 10:03:23 2022 20.051/20.051 The command was "./mace4 -c -f /tmp/mace472478-2.in". 20.051/20.051 ============================== end of head =========================== 20.051/20.051 20.051/20.051 ============================== INPUT ================================= 20.051/20.051 20.051/20.051 % Reading from file /tmp/mace472478-2.in 20.051/20.051 20.051/20.051 assign(max_seconds,13). 20.051/20.051 20.051/20.051 formulas(assumptions). 20.051/20.051 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). 20.051/20.051 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). 20.051/20.051 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence). 20.051/20.051 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence). 20.051/20.051 ->(x1,y) -> ->(c(x1),c(y)) # label(congruence). 20.051/20.051 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(goals). 20.051/20.051 (exists x5 ->(c(a),x5)) # label(goal). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 ============================== end of input ========================== 20.051/20.051 20.051/20.051 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.051/20.051 20.051/20.051 % Formulas that are not ordinary clauses: 20.051/20.051 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 3 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 4 ->(x1,y) -> ->(h(x1),h(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 5 ->(x1,y) -> ->(c(x1),c(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 6 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 20.051/20.051 7 (exists x5 ->(c(a),x5)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.051/20.051 20.051/20.051 ============================== end of process non-clausal formulas === 20.051/20.051 20.051/20.051 ============================== CLAUSES FOR SEARCH ==================== 20.051/20.051 20.051/20.051 formulas(mace4_clauses). 20.051/20.051 -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). 20.051/20.051 -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). 20.051/20.051 -->(x,y) | ->(g(x),g(y)) # label(congruence). 20.051/20.051 -->(x,y) | ->(h(x),h(y)) # label(congruence). 20.051/20.051 -->(x,y) | ->(c(x),c(y)) # label(congruence). 20.051/20.051 -->(x,y) | ->(s(x),s(y)) # label(congruence). 20.051/20.051 -->(c(a),x) # label(goal). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 ============================== end of clauses for search ============= 20.051/20.051 20.051/20.051 % There are no natural numbers in the input. 20.051/20.051 20.051/20.051 ============================== DOMAIN SIZE 2 ========================= 20.051/20.051 20.051/20.051 ============================== MODEL ================================= 20.051/20.051 20.051/20.051 interpretation( 2, [number=1, seconds=0], [ 20.051/20.051 20.051/20.051 function(a, [ 0 ]), 20.051/20.051 20.051/20.051 function(g(_), [ 0, 0 ]), 20.051/20.051 20.051/20.051 function(h(_), [ 0, 0 ]), 20.051/20.051 20.051/20.051 function(c(_), [ 0, 0 ]), 20.051/20.051 20.051/20.051 function(s(_), [ 0, 0 ]), 20.051/20.051 20.051/20.051 function(f(_,_), [ 20.051/20.051 0, 0, 20.051/20.051 0, 0 ]), 20.051/20.051 20.051/20.051 relation(->(_,_), [ 20.051/20.051 0, 0, 20.051/20.051 0, 0 ]) 20.051/20.051 ]). 20.051/20.051 20.051/20.051 ============================== end of model ========================== 20.051/20.051 20.051/20.051 ============================== STATISTICS ============================ 20.051/20.051 20.051/20.051 For domain size 2. 20.051/20.051 20.051/20.051 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.051/20.051 Ground clauses: seen=34, kept=34. 20.051/20.051 Selections=13, assignments=13, propagations=4, current_models=1. 20.051/20.051 Rewrite_terms=68, rewrite_bools=39, indexes=6. 20.051/20.051 Rules_from_neg_clauses=0, cross_offs=0. 20.051/20.051 20.051/20.051 ============================== end of statistics ===================== 20.051/20.051 20.051/20.051 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.051/20.051 20.051/20.051 Exiting with 1 model. 20.051/20.051 20.051/20.051 Process 72491 exit (max_models) Wed Mar 9 10:03:23 2022 20.051/20.051 The process finished Wed Mar 9 10:03:23 2022 20.051/20.051 20.051/20.051 20.051/20.051 Mace4 cooked interpretation: 20.051/20.051 20.051/20.051 20.051/20.051 20.051/20.051 The problem is infeasible. 20.051/20.051 20.051/20.051 20.051/20.051 -> Problem conclusions: 20.051/20.051 Left linear, Right linear, Linear 20.051/20.051 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.051/20.051 CTRS Type: 1 20.051/20.051 Deterministic, Strongly deterministic 20.051/20.051 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.051/20.051 Right-stable CTRS, Overlay CTRS 20.051/20.051 Normal CTRS, Almost normal CTRS 20.051/20.051 Terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.051/20.051 Maybe level confluent 20.051/20.051 Maybe confluent 20.051/20.051 20.051/20.051 Problem 1: 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 Confluence Problem: 20.051/20.051 (VAR x y x') 20.051/20.051 (REPLACEMENT-MAP 20.051/20.051 (f 1, 2) 20.051/20.051 (g 1) 20.051/20.051 (h 1) 20.051/20.051 (a) 20.051/20.051 (c 1) 20.051/20.051 (fSNonEmpty) 20.051/20.051 (s 1) 20.051/20.051 ) 20.051/20.051 (RULES 20.051/20.051 f(x,y) -> g(s(x)) | c(g(x)) ->* c(a) 20.051/20.051 f(x,y) -> h(s(x)) | c(h(x)) ->* c(a) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 ) 20.051/20.051 Critical Pairs: 20.051/20.051 | c(h(x')) ->* c(a), c(g(x')) ->* c(a) => Not trivial, Overlay, Feasible conditions, NW0, N1 20.051/20.051 Terminating:YES 20.051/20.051 20.051/20.051 -> Problem conclusions: 20.051/20.051 Left linear, Right linear, Linear 20.051/20.051 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.051/20.051 CTRS Type: 1 20.051/20.051 Deterministic, Strongly deterministic 20.051/20.051 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.051/20.051 Right-stable CTRS, Overlay CTRS 20.051/20.051 Normal CTRS, Almost normal CTRS 20.051/20.051 Terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.051/20.051 Maybe level confluent 20.051/20.051 Maybe confluent 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 20.051/20.051 Conditional Critical Pairs Distributor Processor 20.051/20.051 Problem 1.1: 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 Confluence Problem: 20.051/20.051 (VAR x y x') 20.051/20.051 (REPLACEMENT-MAP 20.051/20.051 (f 1, 2) 20.051/20.051 (g 1) 20.051/20.051 (h 1) 20.051/20.051 (a) 20.051/20.051 (c 1) 20.051/20.051 (fSNonEmpty) 20.051/20.051 (s 1) 20.051/20.051 ) 20.051/20.051 (RULES 20.051/20.051 f(x,y) -> g(s(x)) | c(g(x)) ->* c(a) 20.051/20.051 f(x,y) -> h(s(x)) | c(h(x)) ->* c(a) 20.051/20.051 g(s(x)) -> x 20.051/20.051 h(s(x)) -> x 20.051/20.051 ) 20.051/20.051 Critical Pairs: 20.051/20.051 | c(h(x')) ->* c(a), c(g(x')) ->* c(a) => Not trivial, Overlay, Feasible conditions, NW0, N1 20.051/20.051 Terminating:YES 20.051/20.051 20.051/20.051 -> Problem conclusions: 20.051/20.051 Left linear, Right linear, Linear 20.051/20.051 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.051/20.051 CTRS Type: 1 20.051/20.051 Deterministic, Strongly deterministic 20.051/20.051 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.051/20.051 Right-stable CTRS, Overlay CTRS 20.051/20.051 Normal CTRS, Almost normal CTRS 20.051/20.051 Terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.051/20.051 Maybe level confluent 20.051/20.051 Maybe confluent 20.051/20.051 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.051/20.051 20.051/20.051 Prover9 CCP Convergence Checker Processor: 20.051/20.051 Proof: 20.051/20.051 ============================== Prover9 =============================== 20.051/20.051 Prover9 (64) version 2009-11A, November 2009. 20.051/20.051 Process 72504 was started by ubuntu on ubuntu, 20.051/20.051 Wed Mar 9 10:03:23 2022 20.051/20.051 The command was "./prover9 -t 60 -f /tmp/prover972317-20.in". 20.051/20.051 ============================== end of head =========================== 20.051/20.051 20.051/20.051 ============================== INPUT ================================= 20.051/20.051 20.051/20.051 % Reading from file /tmp/prover972317-20.in 20.051/20.051 20.051/20.051 20.051/20.051 formulas(sos). 20.051/20.051 reds(x,x). 20.051/20.051 red(x,y) & reds(y,z) -> reds(x,z). 20.051/20.051 red(x,y) -> red(f(x,z1),f(y,z1)). 20.051/20.051 red(x,y) -> red(f(z,x),f(z,y)). 20.051/20.051 red(x,y) -> red(g(x),g(y)). 20.051/20.051 red(x,y) -> red(h(x),h(y)). 20.051/20.051 red(x,y) -> red(c(x),c(y)). 20.051/20.051 red(x,y) -> red(s(x),s(y)). 20.051/20.051 (all x all y (reds(c(g(x)),c(a)) -> red(f(x,y),g(s(x))))). 20.051/20.051 (all x all y (reds(c(h(x)),c(a)) -> red(f(x,y),h(s(x))))). 20.051/20.051 (all x red(g(s(x)),x)). 20.051/20.051 (all x red(h(s(x)),x)). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(goals). 20.051/20.051 (all x1 exists z exists z1 exists z2 (reds(c(h(x1)),c(a)) & reds(c(g(x1)),c(a)) -> reds(g(s(x1)),z) & reds(h(s(x1)),z))). 20.051/20.051 end_of_list. 20.051/20.051 assign(max_megs,-1). 20.051/20.051 set(quiet). 20.051/20.051 20.051/20.051 ============================== end of input ========================== 20.051/20.051 20.051/20.051 % From the command line: assign(max_seconds, 60). 20.051/20.051 20.051/20.051 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.051/20.051 20.051/20.051 % Formulas that are not ordinary clauses: 20.051/20.051 1 red(x,y) & reds(y,z) -> reds(x,z) # label(non_clause). [assumption]. 20.051/20.051 2 red(x,y) -> red(f(x,z1),f(y,z1)) # label(non_clause). [assumption]. 20.051/20.051 3 red(x,y) -> red(f(z,x),f(z,y)) # label(non_clause). [assumption]. 20.051/20.051 4 red(x,y) -> red(g(x),g(y)) # label(non_clause). [assumption]. 20.051/20.051 5 red(x,y) -> red(h(x),h(y)) # label(non_clause). [assumption]. 20.051/20.051 6 red(x,y) -> red(c(x),c(y)) # label(non_clause). [assumption]. 20.051/20.051 7 red(x,y) -> red(s(x),s(y)) # label(non_clause). [assumption]. 20.051/20.051 8 (all x all y (reds(c(g(x)),c(a)) -> red(f(x,y),g(s(x))))) # label(non_clause). [assumption]. 20.051/20.051 9 (all x all y (reds(c(h(x)),c(a)) -> red(f(x,y),h(s(x))))) # label(non_clause). [assumption]. 20.051/20.051 10 (all x red(g(s(x)),x)) # label(non_clause). [assumption]. 20.051/20.051 11 (all x red(h(s(x)),x)) # label(non_clause). [assumption]. 20.051/20.051 12 (all x1 exists z exists z1 exists z2 (reds(c(h(x1)),c(a)) & reds(c(g(x1)),c(a)) -> reds(g(s(x1)),z) & reds(h(s(x1)),z))) # label(non_clause) # label(goal). [goal]. 20.051/20.051 20.051/20.051 ============================== end of process non-clausal formulas === 20.051/20.051 20.051/20.051 ============================== PROCESS INITIAL CLAUSES =============== 20.051/20.051 20.051/20.051 % Clauses before input processing: 20.051/20.051 20.051/20.051 formulas(usable). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(sos). 20.051/20.051 reds(x,x). [assumption]. 20.051/20.051 -red(x,y) | -reds(y,z) | reds(x,z). [clausify(1)]. 20.051/20.051 -red(x,y) | red(f(x,z),f(y,z)). [clausify(2)]. 20.051/20.051 -red(x,y) | red(f(z,x),f(z,y)). [clausify(3)]. 20.051/20.051 -red(x,y) | red(g(x),g(y)). [clausify(4)]. 20.051/20.051 -red(x,y) | red(h(x),h(y)). [clausify(5)]. 20.051/20.051 -red(x,y) | red(c(x),c(y)). [clausify(6)]. 20.051/20.051 -red(x,y) | red(s(x),s(y)). [clausify(7)]. 20.051/20.051 -reds(c(g(x)),c(a)) | red(f(x,y),g(s(x))). [clausify(8)]. 20.051/20.051 -reds(c(h(x)),c(a)) | red(f(x,y),h(s(x))). [clausify(9)]. 20.051/20.051 red(g(s(x)),x). [clausify(10)]. 20.051/20.051 red(h(s(x)),x). [clausify(11)]. 20.051/20.051 reds(c(h(c1)),c(a)). [deny(12)]. 20.051/20.051 reds(c(g(c1)),c(a)). [deny(12)]. 20.051/20.051 -reds(g(s(c1)),x) | -reds(h(s(c1)),x). [deny(12)]. 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(demodulators). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 ============================== PREDICATE ELIMINATION ================= 20.051/20.051 20.051/20.051 ============================== end predicate elimination ============= 20.051/20.051 20.051/20.051 Auto_denials: (no changes). 20.051/20.051 20.051/20.051 Term ordering decisions: 20.051/20.051 20.051/20.051 kept: 13 reds(x,x). [assumption]. 20.051/20.051 kept: 14 -red(x,y) | -reds(y,z) | reds(x,z). [clausify(1)]. 20.051/20.051 kept: 15 -red(x,y) | red(f(x,z),f(y,z)). [clausify(2)]. 20.051/20.051 kept: 16 -red(x,y) | red(f(z,x),f(z,y)). [clausify(3)]. 20.051/20.051 kept: 17 -red(x,y) | red(g(x),g(y)). [clausify(4)]. 20.051/20.051 kept: 18 -red(x,y) | red(h(x),h(y)). [clausify(5)]. 20.051/20.051 kept: 19 -red(x,y) | red(c(x),c(y)). [clausify(6)]. 20.051/20.051 kept: 20 -red(x,y) | red(s(x),s(y)). [clausify(7)]. 20.051/20.051 kept: 21 -reds(c(g(x)),c(a)) | red(f(x,y),g(s(x))). [clausify(8)]. 20.051/20.051 kept: 22 -reds(c(h(x)),c(a)) | red(f(x,y),h(s(x))). [clausify(9)]. 20.051/20.051 kept: 23 red(g(s(x)),x). [clausify(10)]. 20.051/20.051 kept: 24 red(h(s(x)),x). [clausify(11)]. 20.051/20.051 kept: 25 reds(c(h(c1)),c(a)). [deny(12)]. 20.051/20.051 kept: 26 reds(c(g(c1)),c(a)). [deny(12)]. 20.051/20.051 kept: 27 -reds(g(s(c1)),x) | -reds(h(s(c1)),x). [deny(12)]. 20.051/20.051 20.051/20.051 ============================== end of process initial clauses ======== 20.051/20.051 20.051/20.051 ============================== CLAUSES FOR SEARCH ==================== 20.051/20.051 20.051/20.051 % Clauses after input processing: 20.051/20.051 20.051/20.051 formulas(usable). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(sos). 20.051/20.051 13 reds(x,x). [assumption]. 20.051/20.051 14 -red(x,y) | -reds(y,z) | reds(x,z). [clausify(1)]. 20.051/20.051 15 -red(x,y) | red(f(x,z),f(y,z)). [clausify(2)]. 20.051/20.051 16 -red(x,y) | red(f(z,x),f(z,y)). [clausify(3)]. 20.051/20.051 17 -red(x,y) | red(g(x),g(y)). [clausify(4)]. 20.051/20.051 18 -red(x,y) | red(h(x),h(y)). [clausify(5)]. 20.051/20.051 19 -red(x,y) | red(c(x),c(y)). [clausify(6)]. 20.051/20.051 20 -red(x,y) | red(s(x),s(y)). [clausify(7)]. 20.051/20.051 21 -reds(c(g(x)),c(a)) | red(f(x,y),g(s(x))). [clausify(8)]. 20.051/20.051 22 -reds(c(h(x)),c(a)) | red(f(x,y),h(s(x))). [clausify(9)]. 20.051/20.051 23 red(g(s(x)),x). [clausify(10)]. 20.051/20.051 24 red(h(s(x)),x). [clausify(11)]. 20.051/20.051 25 reds(c(h(c1)),c(a)). [deny(12)]. 20.051/20.051 26 reds(c(g(c1)),c(a)). [deny(12)]. 20.051/20.051 27 -reds(g(s(c1)),x) | -reds(h(s(c1)),x). [deny(12)]. 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 formulas(demodulators). 20.051/20.051 end_of_list. 20.051/20.051 20.051/20.051 ============================== end of clauses for search ============= 20.051/20.051 20.051/20.051 ============================== SEARCH ================================ 20.051/20.051 20.051/20.051 % Starting search at 0.00 seconds. 20.051/20.051 20.051/20.051 given #1 (I,wt=3): 13 reds(x,x). [assumption]. 20.051/20.051 20.051/20.051 given #2 (I,wt=9): 14 -red(x,y) | -reds(y,z) | reds(x,z). [clausify(1)]. 20.051/20.051 20.051/20.051 given #3 (I,wt=10): 15 -red(x,y) | red(f(x,z),f(y,z)). [clausify(2)]. 20.051/20.051 20.051/20.051 given #4 (I,wt=10): 16 -red(x,y) | red(f(z,x),f(z,y)). [clausify(3)]. 20.051/20.051 20.051/20.051 given #5 (I,wt=8): 17 -red(x,y) | red(g(x),g(y)). [clausify(4)]. 20.051/20.051 20.051/20.051 given #6 (I,wt=8): 18 -red(x,y) | red(h(x),h(y)). [clausify(5)]. 20.051/20.051 20.051/20.051 given #7 (I,wt=8): 19 -red(x,y) | red(c(x),c(y)). [clausify(6)]. 20.051/20.051 20.051/20.051 given #8 (I,wt=8): 20 -red(x,y) | red(s(x),s(y)). [clausify(7)]. 20.051/20.051 20.051/20.051 given #9 (I,wt=13): 21 -reds(c(g(x)),c(a)) | red(f(x,y),g(s(x))). [clausify(8)]. 20.051/20.051 20.051/20.051 given #10 (I,wt=13): 22 -reds(c(h(x)),c(a)) | red(f(x,y),h(s(x))). [clausify(9)]. 20.051/20.051 20.051/20.051 given #11 (I,wt=5): 23 red(g(s(x)),x). [clausify(10)]. 20.051/20.051 20.051/20.051 given #12 (I,wt=5): 24 red(h(s(x)),x). [clausify(11)]. 20.051/20.051 20.051/20.051 given #13 (I,wt=6): 25 reds(c(h(c1)),c(a)). [deny(12)]. 20.051/20.051 20.051/20.051 given #14 (I,wt=6): 26 reds(c(g(c1)),c(a)). [deny(12)]. 20.051/20.051 20.051/20.051 given #15 (I,wt=10): 27 -reds(g(s(c1)),x) | -reds(h(s(c1)),x). [deny(12)]. 20.051/20.051 20.051/20.051 given #16 (A,wt=7): 28 red(s(g(s(x))),s(x)). [ur(20,a,23,a)]. 20.051/20.051 20.051/20.051 given #17 (F,wt=7): 49 -reds(h(s(c1)),g(s(c1))). [resolve(27,a,13,a)]. 20.051/20.051 20.051/20.051 given #18 (F,wt=5): 60 -reds(c1,g(s(c1))). [ur(14,a,24,a,c,49,a)]. 20.051/20.051 20.051/20.051 given #19 (F,wt=5): 62 -red(c1,g(s(c1))). [ur(14,b,13,a,c,60,a)]. 20.051/20.051 20.051/20.051 given #20 (F,wt=7): 51 -reds(g(s(c1)),h(s(c1))). [resolve(27,b,13,a)]. 20.051/20.051 20.051/20.051 given #21 (T,wt=5): 34 reds(g(s(x)),x). [ur(14,a,23,a,b,13,a)]. 20.051/20.051 20.051/20.051 ============================== PROOF ================================= 20.051/20.051 20.051/20.051 % Proof 1 at 0.00 (+ 0.00) seconds. 20.051/20.051 % Length of proof is 12. 20.051/20.051 % Level of proof is 3. 20.051/20.051 % Maximum clause weight is 10.000. 20.051/20.051 % Given clauses 21. 20.051/20.051 20.051/20.051 1 red(x,y) & reds(y,z) -> reds(x,z) # label(non_clause). [assumption]. 20.051/20.051 10 (all x red(g(s(x)),x)) # label(non_clause). [assumption]. 20.051/20.051 11 (all x red(h(s(x)),x)) # label(non_clause). [assumption]. 20.051/20.051 12 (all x1 exists z exists z1 exists z2 (reds(c(h(x1)),c(a)) & reds(c(g(x1)),c(a)) -> reds(g(s(x1)),z) & reds(h(s(x1)),z))) # label(non_clause) # label(goal). [goal]. 20.051/20.051 13 reds(x,x). [assumption]. 20.051/20.051 14 -red(x,y) | -reds(y,z) | reds(x,z). [clausify(1)]. 20.051/20.051 23 red(g(s(x)),x). [clausify(10)]. 20.051/20.051 24 red(h(s(x)),x). [clausify(11)]. 20.051/20.051 27 -reds(g(s(c1)),x) | -reds(h(s(c1)),x). [deny(12)]. 20.051/20.051 34 reds(g(s(x)),x). [ur(14,a,23,a,b,13,a)]. 20.051/20.051 41 reds(h(s(x)),x). [ur(14,a,24,a,b,13,a)]. 20.051/20.051 66 $F. [resolve(34,a,27,a),unit_del(a,41)]. 20.051/20.051 20.051/20.051 ============================== end of proof ========================== 20.051/20.051 20.051/20.051 ============================== STATISTICS ============================ 20.051/20.051 20.051/20.051 Given=21. Generated=57. Kept=53. proofs=1. 20.051/20.051 Usable=21. Sos=32. Demods=0. Limbo=0, Disabled=15. Hints=0. 20.051/20.051 Kept_by_rule=0, Deleted_by_rule=0. 20.051/20.051 Forward_subsumed=3. Back_subsumed=0. 20.051/20.051 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.051/20.051 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.051/20.051 Demod_attempts=0. Demod_rewrites=0. 20.051/20.051 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.051/20.051 Nonunit_fsub_feature_tests=6. Nonunit_bsub_feature_tests=56. 20.051/20.051 Megabytes=0.14. 20.051/20.051 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.051/20.051 20.051/20.051 ============================== end of statistics ===================== 20.051/20.051 20.051/20.051 ============================== end of search ========================= 20.051/20.051 20.051/20.051 THEOREM PROVED 20.051/20.051 20.051/20.051 Exiting with 1 proof. 20.051/20.051 20.051/20.051 Process 72504 exit (max_proofs) Wed Mar 9 10:03:23 2022 20.051/20.051 20.051/20.051 20.051/20.051 The problem is joinable. 20.051/20.051 20.04user 0.39system 0:20.51elapsed 99%CPU (0avgtext+0avgdata 64176maxresident)k 20.051/20.051 8inputs+0outputs (0major+43739minor)pagefaults 0swaps