0.00/0.03 NO 0.00/0.03 0.00/0.03 Problem 1: 0.00/0.03 0.00/0.03 Infeasibility Problem: 0.00/0.03 (VAR vNonEmpty:S x:S) 0.00/0.03 (RULES 0.00/0.03 p(s(x:S)) -> x:S 0.00/0.03 pos(p(x:S)) -> ffalse | pos(x:S) ->* ffalse 0.00/0.03 pos(s(0)) -> ttrue 0.00/0.03 pos(s(x:S)) -> ttrue | pos(x:S) ->* ttrue 0.00/0.03 pos(0) -> ffalse 0.00/0.03 s(p(x:S)) -> x:S 0.00/0.03 ) 0.00/0.03 0.00/0.03 Infeasibility Conditions: 0.00/0.03 pos(p(x:S)) ->* ttrue 0.00/0.03 0.00/0.03 Problem 1: 0.00/0.03 0.00/0.03 Obtaining a proof using Prover9. Prover9 Output: 0.00/0.03 ============================== Prover9 =============================== 0.00/0.03 Prover9 (64) version 2009-11A, November 2009. 0.00/0.03 Process 11100 was started by sandbox on n005.star.cs.uiowa.edu, 0.00/0.03 Thu Jun 11 09:56:51 2020 0.00/0.03 The command was "./prover9 -f /tmp/prover91804289383846930886.in". 0.00/0.03 ============================== end of head =========================== 0.00/0.03 0.00/0.03 ============================== INPUT ================================= 0.00/0.03 0.00/0.03 % Reading from file /tmp/prover91804289383846930886.in 0.00/0.03 0.00/0.03 assign(max_seconds,300). 0.00/0.03 0.00/0.03 formulas(assumptions). 0.00/0.03 arrowStar_s0(x,x) # label(reflexivity). 0.00/0.03 arrow_s0(x,y) & arrowStar_s0(y,z) -> arrowStar_s0(x,z) # label(transitivity). 0.00/0.03 arrow_s0(x1,y) -> arrow_s0(f2(x1),f2(y)) # label(congruence). 0.00/0.03 arrow_s0(x1,y) -> arrow_s0(f3(x1),f3(y)) # label(congruence). 0.00/0.03 arrow_s0(x1,y) -> arrow_s0(f4(x1),f4(y)) # label(congruence). 0.00/0.03 arrow_s0(f2(f4(x1)),x1) # label(repacement). 0.00/0.03 arrowStar_s0(f3(x1),f7) -> arrow_s0(f3(f2(x1)),f7) # label(repacement). 0.00/0.03 arrow_s0(f3(f4(f5)),f8) # label(repacement). 0.00/0.03 arrowStar_s0(f3(x1),f8) -> arrow_s0(f3(f4(x1)),f8) # label(repacement). 0.00/0.03 arrow_s0(f3(f5),f7) # label(repacement). 0.00/0.03 arrow_s0(f4(f2(x1)),x1) # label(repacement). 0.00/0.03 end_of_list. 0.00/0.03 0.00/0.03 formulas(goals). 0.00/0.03 (exists x3 arrowStar_s0(f3(f2(x3)),f8)) # label(goal). 0.00/0.03 end_of_list. 0.00/0.03 0.00/0.03 ============================== end of input ========================== 0.00/0.03 0.00/0.03 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.00/0.03 0.00/0.03 % Formulas that are not ordinary clauses: 0.00/0.03 1 arrow_s0(x,y) & arrowStar_s0(y,z) -> arrowStar_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 0.00/0.03 2 arrow_s0(x1,y) -> arrow_s0(f2(x1),f2(y)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.03 3 arrow_s0(x1,y) -> arrow_s0(f3(x1),f3(y)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.03 4 arrow_s0(x1,y) -> arrow_s0(f4(x1),f4(y)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.03 5 arrowStar_s0(f3(x1),f7) -> arrow_s0(f3(f2(x1)),f7) # label(repacement) # label(non_clause). [assumption]. 0.00/0.03 6 arrowStar_s0(f3(x1),f8) -> arrow_s0(f3(f4(x1)),f8) # label(repacement) # label(non_clause). [assumption]. 0.00/0.03 7 (exists x3 arrowStar_s0(f3(f2(x3)),f8)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.00/0.03 0.00/0.03 ============================== end of process non-clausal formulas === 0.00/0.03 0.00/0.03 ============================== PROCESS INITIAL CLAUSES =============== 0.00/0.03 0.00/0.03 % Clauses before input processing: 0.00/0.03 0.00/0.03 formulas(usable). 0.00/0.03 end_of_list. 0.00/0.03 0.00/0.03 formulas(sos). 0.00/0.03 arrowStar_s0(x,x) # label(reflexivity). [assumption]. 0.00/0.03 -arrow_s0(x,y) | -arrowStar_s0(y,z) | arrowStar_s0(x,z) # label(transitivity). [clausify(1)]. 0.00/0.03 -arrow_s0(x,y) | arrow_s0(f2(x),f2(y)) # label(congruence). [clausify(2)]. 0.00/0.03 -arrow_s0(x,y) | arrow_s0(f3(x),f3(y)) # label(congruence). [clausify(3)]. 0.00/0.03 -arrow_s0(x,y) | arrow_s0(f4(x),f4(y)) # label(congruence). [clausify(4)]. 0.00/0.03 arrow_s0(f2(f4(x)),x) # label(repacement). [assumption]. 0.00/0.03 -arrowStar_s0(f3(x),f7) | arrow_s0(f3(f2(x)),f7) # label(repacement). [clausify(5)]. 0.00/0.03 arrow_s0(f3(f4(f5)),f8) # label(repacement). [assumption]. 0.00/0.03 -arrowStar_s0(f3(x),f8) | arrow_s0(f3(f4(x)),f8) # label(repacement). [clausify(6)]. 0.00/0.03 arrow_s0(f3(f5),f7) # label(repacement). [assumption]. 0.00/0.03 arrow_s0(f4(f2(x)),x) # label(repacement). [assumption]. 0.00/0.03 -arrowStar_s0(f3(f2(x)),f8) # label(goal). [deny(7)]. 0.00/0.03 end_of_list. 0.00/0.03 0.00/0.03 formulas(demodulators). 0.00/0.03 end_of_list. 0.00/0.03 0.00/0.03 ============================== PREDICATE ELIMINATION ================= 0.00/0.03 0.00/0.03 No predicates eliminated. 0.00/0.03 0.00/0.03 ============================== end predicate elimination ============= 0.00/0.03 0.00/0.03 Auto_denials: 0.00/0.03 % copying label goal to answer in negative clause 0.00/0.03 0.00/0.03 Term ordering decisions: 0.00/0.03 Predicate symbol precedence: predicate_order([ arrow_s0, arrowStar_s0 ]). 0.00/0.03 Function symbol precedence: function_order([ f7, f8, f5, f3, f4, f2 ]). 0.00/0.03 After inverse_order: (no changes). 0.00/0.03 Unfolding symbols: (none). 0.00/0.03 0.00/0.03 Auto_inference settings: 0.00/0.03 % set(neg_binary_resolution). % (HNE depth_diff=-5) 0.00/0.03 % clear(ordered_res). % (HNE depth_diff=-5) 0.00/0.03 % set(ur_resolution). % (HNE depth_diff=-5) 0.00/0.03 % set(ur_resolution) -> set(pos_ur_resolution). 0.00/0.03 % set(ur_resolution) -> set(neg_ur_resolution). 0.00/0.03 0.00/0.03 Auto_process settings: (no changes). 0.00/0.03 0.00/0.03 kept: 8 arrowStar_s0(x,x) # label(reflexivity). [assumption]. 0.00/0.03 kept: 9 -arrow_s0(x,y) | -arrowStar_s0(y,z) | arrowStar_s0(x,z) # label(transitivity). [clausify(1)]. 0.00/0.03 kept: 10 -arrow_s0(x,y) | arrow_s0(f2(x),f2(y)) # label(congruence). [clausify(2)]. 0.00/0.03 kept: 11 -arrow_s0(x,y) | arrow_s0(f3(x),f3(y)) # label(congruence). [clausify(3)]. 0.00/0.03 kept: 12 -arrow_s0(x,y) | arrow_s0(f4(x),f4(y)) # label(congruence). [clausify(4)]. 0.00/0.03 kept: 13 arrow_s0(f2(f4(x)),x) # label(repacement). [assumption]. 0.00/0.03 kept: 14 -arrowStar_s0(f3(x),f7) | arrow_s0(f3(f2(x)),f7) # label(repacement). [clausify(5)]. 0.00/0.03 kept: 15 arrow_s0(f3(f4(f5)),f8) # label(repacement). [assumption]. 0.00/0.03 kept: 16 -arrowStar_s0(f3(x),f8) | arrow_s0(f3(f4(x)),f8) # label(repacement). [clausify(6)]. 0.00/0.03 kept: 17 arrow_s0(f3(f5),f7) # label(repacement). [assumption]. 0.00/0.03 kept: 18 arrow_s0(f4(f2(x)),x) # label(repacement). [assumption]. 0.00/0.03 kept: 19 -arrowStar_s0(f3(f2(x)),f8) # label(goal) # answer(goal). [deny(7)]. 0.00/0.03 0.00/0.03 ============================== end of process initial clauses ======== 0.00/0.03 0.00/0.03 ============================== CLAUSES FOR SEARCH ==================== 0.00/0.03 0.00/0.03 % Clauses after input processing: 0.00/0.03 0.00/0.03 formulas(usable). 0.00/0.03 end_of_list. 0.00/0.03 0.00/0.03 formulas(sos). 0.00/0.03 8 arrowStar_s0(x,x) # label(reflexivity). [assumption]. 0.00/0.03 9 -arrow_s0(x,y) | -arrowStar_s0(y,z) | arrowStar_s0(x,z) # label(transitivity). [clausify(1)]. 0.00/0.03 10 -arrow_s0(x,y) | arrow_s0(f2(x),f2(y)) # label(congruence). [clausify(2)]. 0.00/0.03 11 -arrow_s0(x,y) | arrow_s0(f3(x),f3(y)) # label(congruence). [clausify(3)]. 0.00/0.03 12 -arrow_s0(x,y) | arrow_s0(f4(x),f4(y)) # label(congruence). [clausify(4)]. 0.00/0.03 13 arrow_s0(f2(f4(x)),x) # label(repacement). [assumption]. 0.00/0.03 14 -arrowStar_s0(f3(x),f7) | arrow_s0(f3(f2(x)),f7) # label(repacement). [clausify(5)]. 0.00/0.03 15 arrow_s0(f3(f4(f5)),f8) # label(repacement). [assumption]. 0.00/0.03 16 -arrowStar_s0(f3(x),f8) | arrow_s0(f3(f4(x)),f8) # label(repacement). [clausify(6)]. 0.00/0.03 17 arrow_s0(f3(f5),f7) # label(repacement). [assumption]. 0.00/0.03 18 arrow_s0(f4(f2(x)),x) # label(repacement). [assumption]. 0.00/0.03 19 -arrowStar_s0(f3(f2(x)),f8) # label(goal) # answer(goal). [deny(7)]. 0.00/0.03 end_of_list. 0.00/0.03 0.00/0.03 formulas(demodulators). 0.00/0.03 end_of_list. 0.00/0.03 0.00/0.03 ============================== end of clauses for search ============= 0.00/0.03 0.00/0.03 ============================== SEARCH ================================ 0.00/0.03 0.00/0.03 % Starting search at 0.00 seconds. 0.00/0.03 0.00/0.03 given #1 (I,wt=3): 8 arrowStar_s0(x,x) # label(reflexivity). [assumption]. 0.00/0.03 0.00/0.03 given #2 (I,wt=9): 9 -arrow_s0(x,y) | -arrowStar_s0(y,z) | arrowStar_s0(x,z) # label(transitivity). [clausify(1)]. 0.00/0.03 0.00/0.03 given #3 (I,wt=8): 10 -arrow_s0(x,y) | arrow_s0(f2(x),f2(y)) # label(congruence). [clausify(2)]. 0.00/0.03 0.00/0.03 given #4 (I,wt=8): 11 -arrow_s0(x,y) | arrow_s0(f3(x),f3(y)) # label(congruence). [clausify(3)]. 0.00/0.03 0.00/0.03 given #5 (I,wt=8): 12 -arrow_s0(x,y) | arrow_s0(f4(x),f4(y)) # label(congruence). [clausify(4)]. 0.00/0.03 0.00/0.03 given #6 (I,wt=5): 13 arrow_s0(f2(f4(x)),x) # label(repacement). [assumption]. 0.00/0.03 0.00/0.03 given #7 (I,wt=9): 14 -arrowStar_s0(f3(x),f7) | arrow_s0(f3(f2(x)),f7) # label(repacement). [clausify(5)]. 0.00/0.03 0.00/0.03 given #8 (I,wt=5): 15 arrow_s0(f3(f4(f5)),f8) # label(repacement). [assumption]. 0.00/0.03 0.00/0.03 given #9 (I,wt=9): 16 -arrowStar_s0(f3(x),f8) | arrow_s0(f3(f4(x)),f8) # label(repacement). [clausify(6)]. 0.00/0.03 0.00/0.03 given #10 (I,wt=4): 17 arrow_s0(f3(f5),f7) # label(repacement). [assumption]. 0.00/0.03 0.00/0.03 given #11 (I,wt=5): 18 arrow_s0(f4(f2(x)),x) # label(repacement). [assumption]. 0.00/0.03 0.00/0.03 given #12 (I,wt=5): 19 -arrowStar_s0(f3(f2(x)),f8) # label(goal) # answer(goal). [deny(7)]. 0.00/0.03 0.00/0.03 given #13 (A,wt=7): 20 arrow_s0(f3(f2(f4(x))),f3(x)). [ur(11,a,13,a)]. 0.00/0.03 0.00/0.03 ============================== PROOF ================================= 0.00/0.03 0.00/0.03 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 0.00/0.03 % Length of proof is 13. 0.00/0.03 % Level of proof is 4. 0.00/0.03 % Maximum clause weight is 9.000. 0.00/0.03 % Given clauses 13. 0.00/0.03 0.00/0.03 1 arrow_s0(x,y) & arrowStar_s0(y,z) -> arrowStar_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 0.00/0.03 3 arrow_s0(x1,y) -> arrow_s0(f3(x1),f3(y)) # label(congruence) # label(non_clause). [assumption]. 0.00/0.03 7 (exists x3 arrowStar_s0(f3(f2(x3)),f8)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.00/0.03 8 arrowStar_s0(x,x) # label(reflexivity). [assumption]. 0.00/0.03 9 -arrow_s0(x,y) | -arrowStar_s0(y,z) | arrowStar_s0(x,z) # label(transitivity). [clausify(1)]. 0.00/0.03 11 -arrow_s0(x,y) | arrow_s0(f3(x),f3(y)) # label(congruence). [clausify(3)]. 0.00/0.03 13 arrow_s0(f2(f4(x)),x) # label(repacement). [assumption]. 0.00/0.03 15 arrow_s0(f3(f4(f5)),f8) # label(repacement). [assumption]. 0.00/0.03 19 -arrowStar_s0(f3(f2(x)),f8) # label(goal) # answer(goal). [deny(7)]. 0.00/0.03 20 arrow_s0(f3(f2(f4(x))),f3(x)). [ur(11,a,13,a)]. 0.00/0.03 26 arrowStar_s0(f3(f4(f5)),f8). [ur(9,a,15,a,b,8,a)]. 0.00/0.03 40 -arrowStar_s0(f3(x),f8) # answer(goal). [ur(9,a,20,a,c,19,a)]. 0.00/0.03 41 \$F # answer(goal). [resolve(40,a,26,a)]. 0.00/0.03 0.00/0.03 ============================== end of proof ========================== 0.00/0.03 0.00/0.03 ============================== STATISTICS ============================ 0.00/0.03 0.00/0.03 Given=13. Generated=35. Kept=33. proofs=1. 0.00/0.03 Usable=13. Sos=15. Demods=0. Limbo=4, Disabled=12. Hints=0. 0.00/0.03 Kept_by_rule=0, Deleted_by_rule=0. 0.00/0.03 Forward_subsumed=2. Back_subsumed=0. 0.00/0.03 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 0.00/0.03 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 0.00/0.03 Demod_attempts=0. Demod_rewrites=0. 0.00/0.03 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 0.00/0.03 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=9. 0.00/0.03 Megabytes=0.09. 0.00/0.03 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.00/0.03 0.00/0.03 ============================== end of statistics ===================== 0.00/0.03 0.00/0.03 ============================== end of search ========================= 0.00/0.03 0.00/0.03 THEOREM PROVED 0.00/0.03 0.00/0.03 Exiting with 1 proof. 0.00/0.03 0.00/0.03 Process 11100 exit (max_proofs) Thu Jun 11 09:56:51 2020 0.00/0.03 0.00/0.03 0.00/0.03 The problem is feasible. 0.00/20.08 EOF