20.043/20.043 YES 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 20.043/20.043 20.043/20.043 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.043/20.043 Confluence Problem: 20.043/20.043 (VAR vNonEmpty x) 20.043/20.043 (REPLACEMENT-MAP 20.043/20.043 (e 1) 20.043/20.043 (0) 20.043/20.043 (fSNonEmpty) 20.043/20.043 (false) 20.043/20.043 (s 1) 20.043/20.043 (true) 20.043/20.043 ) 20.043/20.043 (RULES 20.043/20.043 e(0) -> true 20.043/20.043 e(s(x)) -> false | e(x) ->* true 20.043/20.043 e(s(x)) -> true | e(x) ->* false 20.043/20.043 ) 20.043/20.043 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.043/20.043 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 20.043/20.043 Inlining of Conditions Processor [STERN17]: 20.043/20.043 20.043/20.043 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.043/20.043 Confluence Problem: 20.043/20.043 (VAR vNonEmpty x) 20.043/20.043 (REPLACEMENT-MAP 20.043/20.043 (e 1) 20.043/20.043 (0) 20.043/20.043 (fSNonEmpty) 20.043/20.043 (false) 20.043/20.043 (s 1) 20.043/20.043 (true) 20.043/20.043 ) 20.043/20.043 (RULES 20.043/20.043 e(0) -> true 20.043/20.043 e(s(x)) -> false | e(x) ->* true 20.043/20.043 e(s(x)) -> true | e(x) ->* false 20.043/20.043 ) 20.043/20.043 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.043/20.043 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 20.043/20.043 Clean CTRS Processor: 20.043/20.043 20.043/20.043 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.043/20.043 Confluence Problem: 20.043/20.043 (VAR x) 20.043/20.043 (REPLACEMENT-MAP 20.043/20.043 (e 1) 20.043/20.043 (0) 20.043/20.043 (fSNonEmpty) 20.043/20.043 (false) 20.043/20.043 (s 1) 20.043/20.043 (true) 20.043/20.043 ) 20.043/20.043 (RULES 20.043/20.043 e(0) -> true 20.043/20.043 e(s(x)) -> false | e(x) ->* true 20.043/20.043 e(s(x)) -> true | e(x) ->* false 20.043/20.043 ) 20.043/20.043 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.043/20.043 20.043/20.043 CRule InfChecker Info: 20.043/20.043 e(0) -> true 20.043/20.043 Rule remains 20.043/20.043 Proof: 20.043/20.043 NO_CONDS 20.043/20.043 20.043/20.043 CRule InfChecker Info: 20.043/20.043 e(s(x)) -> false | e(x) ->* true 20.043/20.043 Rule remains 20.043/20.043 Proof: 20.043/20.043 NO 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 20.043/20.043 Infeasibility Problem: 20.043/20.043 [(VAR vNonEmpty x vNonEmpty x) 20.043/20.043 (STRATEGY CONTEXTSENSITIVE 20.043/20.043 (e 1) 20.043/20.043 (0) 20.043/20.043 (fSNonEmpty) 20.043/20.043 (false) 20.043/20.043 (s 1) 20.043/20.043 (true) 20.043/20.043 ) 20.043/20.043 (RULES 20.043/20.043 e(0) -> true 20.043/20.043 e(s(x)) -> false | e(x) ->* true 20.043/20.043 e(s(x)) -> true | e(x) ->* false 20.043/20.043 )] 20.043/20.043 20.043/20.043 Infeasibility Conditions: 20.043/20.043 e(x) ->* true 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 20.043/20.043 Obtaining a proof using Prover9: 20.043/20.043 20.043/20.043 -> Prover9 Output: 20.043/20.043 ============================== Prover9 =============================== 20.043/20.043 Prover9 (64) version 2009-11A, November 2009. 20.043/20.043 Process 63055 was started by ubuntu on ubuntu, 20.043/20.043 Wed Mar 9 09:55:25 2022 20.043/20.043 The command was "./prover9 -f /tmp/prover963046-0.in". 20.043/20.043 ============================== end of head =========================== 20.043/20.043 20.043/20.043 ============================== INPUT ================================= 20.043/20.043 20.043/20.043 % Reading from file /tmp/prover963046-0.in 20.043/20.043 20.043/20.043 assign(max_seconds,20). 20.043/20.043 20.043/20.043 formulas(assumptions). 20.043/20.043 ->_s0(x1,y) -> ->_s0(e(x1),e(y)) # label(congruence). 20.043/20.043 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 20.043/20.043 ->_s0(e(0),true) # label(replacement). 20.043/20.043 ->*_s0(e(x1),true) -> ->_s0(e(s(x1)),false) # label(replacement). 20.043/20.043 ->*_s0(e(x1),false) -> ->_s0(e(s(x1)),true) # label(replacement). 20.043/20.043 ->*_s0(x,x) # label(reflexivity). 20.043/20.043 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 formulas(goals). 20.043/20.043 (exists x3 ->*_s0(e(x3),true)) # label(goal). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 ============================== end of input ========================== 20.043/20.043 20.043/20.043 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.043/20.043 20.043/20.043 % Formulas that are not ordinary clauses: 20.043/20.043 1 ->_s0(x1,y) -> ->_s0(e(x1),e(y)) # label(congruence) # label(non_clause). [assumption]. 20.043/20.043 2 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 20.043/20.043 3 ->*_s0(e(x1),true) -> ->_s0(e(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 20.043/20.043 4 ->*_s0(e(x1),false) -> ->_s0(e(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 20.043/20.043 5 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.043/20.043 6 (exists x3 ->*_s0(e(x3),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.043/20.043 20.043/20.043 ============================== end of process non-clausal formulas === 20.043/20.043 20.043/20.043 ============================== PROCESS INITIAL CLAUSES =============== 20.043/20.043 20.043/20.043 % Clauses before input processing: 20.043/20.043 20.043/20.043 formulas(usable). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 formulas(sos). 20.043/20.043 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(1)]. 20.043/20.043 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(2)]. 20.043/20.043 ->_s0(e(0),true) # label(replacement). [assumption]. 20.043/20.043 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(3)]. 20.043/20.043 -->*_s0(e(x),false) | ->_s0(e(s(x)),true) # label(replacement). [clausify(4)]. 20.043/20.043 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.043/20.043 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(5)]. 20.043/20.043 -->*_s0(e(x),true) # label(goal). [deny(6)]. 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 formulas(demodulators). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 ============================== PREDICATE ELIMINATION ================= 20.043/20.043 20.043/20.043 No predicates eliminated. 20.043/20.043 20.043/20.043 ============================== end predicate elimination ============= 20.043/20.043 20.043/20.043 Auto_denials: 20.043/20.043 % copying label goal to answer in negative clause 20.043/20.043 20.043/20.043 Term ordering decisions: 20.043/20.043 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.043/20.043 Function symbol precedence: function_order([ true, false, 0, e, s ]). 20.043/20.043 After inverse_order: (no changes). 20.043/20.043 Unfolding symbols: (none). 20.043/20.043 20.043/20.043 Auto_inference settings: 20.043/20.043 % set(neg_binary_resolution). % (HNE depth_diff=-4) 20.043/20.043 % clear(ordered_res). % (HNE depth_diff=-4) 20.043/20.043 % set(ur_resolution). % (HNE depth_diff=-4) 20.043/20.043 % set(ur_resolution) -> set(pos_ur_resolution). 20.043/20.043 % set(ur_resolution) -> set(neg_ur_resolution). 20.043/20.043 20.043/20.043 Auto_process settings: (no changes). 20.043/20.043 20.043/20.043 kept: 7 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(1)]. 20.043/20.043 kept: 8 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(2)]. 20.043/20.043 kept: 9 ->_s0(e(0),true) # label(replacement). [assumption]. 20.043/20.043 kept: 10 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(3)]. 20.043/20.043 kept: 11 -->*_s0(e(x),false) | ->_s0(e(s(x)),true) # label(replacement). [clausify(4)]. 20.043/20.043 kept: 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.043/20.043 kept: 13 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(5)]. 20.043/20.043 kept: 14 -->*_s0(e(x),true) # label(goal) # answer(goal). [deny(6)]. 20.043/20.043 20.043/20.043 ============================== end of process initial clauses ======== 20.043/20.043 20.043/20.043 ============================== CLAUSES FOR SEARCH ==================== 20.043/20.043 20.043/20.043 % Clauses after input processing: 20.043/20.043 20.043/20.043 formulas(usable). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 formulas(sos). 20.043/20.043 7 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(1)]. 20.043/20.043 8 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(2)]. 20.043/20.043 9 ->_s0(e(0),true) # label(replacement). [assumption]. 20.043/20.043 11 -->*_s0(e(x),false) | ->_s0(e(s(x)),true) # label(replacement). [clausify(4)]. 20.043/20.043 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.043/20.043 13 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(5)]. 20.043/20.043 14 -->*_s0(e(x),true) # label(goal) # answer(goal). [deny(6)]. 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 formulas(demodulators). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 ============================== end of clauses for search ============= 20.043/20.043 20.043/20.043 ============================== SEARCH ================================ 20.043/20.043 20.043/20.043 % Starting search at 0.00 seconds. 20.043/20.043 20.043/20.043 given #1 (I,wt=8): 7 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(1)]. 20.043/20.043 20.043/20.043 given #2 (I,wt=8): 8 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(2)]. 20.043/20.043 20.043/20.043 given #3 (I,wt=4): 9 ->_s0(e(0),true) # label(replacement). [assumption]. 20.043/20.043 20.043/20.043 given #4 (I,wt=9): 11 -->*_s0(e(x),false) | ->_s0(e(s(x)),true) # label(replacement). [clausify(4)]. 20.043/20.043 20.043/20.043 given #5 (I,wt=3): 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.043/20.043 20.043/20.043 given #6 (I,wt=9): 13 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(5)]. 20.043/20.043 20.043/20.043 ============================== PROOF ================================= 20.043/20.043 20.043/20.043 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.043/20.043 % Length of proof is 8. 20.043/20.043 % Level of proof is 3. 20.043/20.043 % Maximum clause weight is 9.000. 20.043/20.043 % Given clauses 6. 20.043/20.043 20.043/20.043 5 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.043/20.043 6 (exists x3 ->*_s0(e(x3),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.043/20.043 9 ->_s0(e(0),true) # label(replacement). [assumption]. 20.043/20.043 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.043/20.043 13 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(5)]. 20.043/20.043 14 -->*_s0(e(x),true) # label(goal) # answer(goal). [deny(6)]. 20.043/20.043 17 ->*_s0(e(0),true). [ur(13,a,9,a,b,12,a)]. 20.043/20.043 18 $F # answer(goal). [resolve(17,a,14,a)]. 20.043/20.043 20.043/20.043 ============================== end of proof ========================== 20.043/20.043 20.043/20.043 ============================== STATISTICS ============================ 20.043/20.043 20.043/20.043 Given=6. Generated=11. Kept=11. proofs=1. 20.043/20.043 Usable=6. Sos=3. Demods=0. Limbo=0, Disabled=9. Hints=0. 20.043/20.043 Kept_by_rule=0, Deleted_by_rule=0. 20.043/20.043 Forward_subsumed=0. Back_subsumed=1. 20.043/20.043 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.043/20.043 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.043/20.043 Demod_attempts=0. Demod_rewrites=0. 20.043/20.043 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.043/20.043 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=7. 20.043/20.043 Megabytes=0.06. 20.043/20.043 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.043/20.043 20.043/20.043 ============================== end of statistics ===================== 20.043/20.043 20.043/20.043 ============================== end of search ========================= 20.043/20.043 20.043/20.043 THEOREM PROVED 20.043/20.043 20.043/20.043 Exiting with 1 proof. 20.043/20.043 20.043/20.043 Process 63055 exit (max_proofs) Wed Mar 9 09:55:25 2022 20.043/20.043 20.043/20.043 20.043/20.043 The problem is feasible. 20.043/20.043 20.043/20.043 20.043/20.043 CRule InfChecker Info: 20.043/20.043 e(s(x)) -> true | e(x) ->* false 20.043/20.043 Rule remains 20.043/20.043 Proof: 20.043/20.043 NO 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 20.043/20.043 Infeasibility Problem: 20.043/20.043 [(VAR vNonEmpty x vNonEmpty x) 20.043/20.043 (STRATEGY CONTEXTSENSITIVE 20.043/20.043 (e 1) 20.043/20.043 (0) 20.043/20.043 (fSNonEmpty) 20.043/20.043 (false) 20.043/20.043 (s 1) 20.043/20.043 (true) 20.043/20.043 ) 20.043/20.043 (RULES 20.043/20.043 e(0) -> true 20.043/20.043 e(s(x)) -> false | e(x) ->* true 20.043/20.043 e(s(x)) -> true | e(x) ->* false 20.043/20.043 )] 20.043/20.043 20.043/20.043 Infeasibility Conditions: 20.043/20.043 e(x) ->* false 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 20.043/20.043 Obtaining a proof using Prover9: 20.043/20.043 20.043/20.043 -> Prover9 Output: 20.043/20.043 ============================== Prover9 =============================== 20.043/20.043 Prover9 (64) version 2009-11A, November 2009. 20.043/20.043 Process 63077 was started by ubuntu on ubuntu, 20.043/20.043 Wed Mar 9 09:55:25 2022 20.043/20.043 The command was "./prover9 -f /tmp/prover963068-0.in". 20.043/20.043 ============================== end of head =========================== 20.043/20.043 20.043/20.043 ============================== INPUT ================================= 20.043/20.043 20.043/20.043 % Reading from file /tmp/prover963068-0.in 20.043/20.043 20.043/20.043 assign(max_seconds,20). 20.043/20.043 20.043/20.043 formulas(assumptions). 20.043/20.043 ->_s0(x1,y) -> ->_s0(e(x1),e(y)) # label(congruence). 20.043/20.043 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 20.043/20.043 ->_s0(e(0),true) # label(replacement). 20.043/20.043 ->*_s0(e(x1),true) -> ->_s0(e(s(x1)),false) # label(replacement). 20.043/20.043 ->*_s0(e(x1),false) -> ->_s0(e(s(x1)),true) # label(replacement). 20.043/20.043 ->*_s0(x,x) # label(reflexivity). 20.043/20.043 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 formulas(goals). 20.043/20.043 (exists x3 ->*_s0(e(x3),false)) # label(goal). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 ============================== end of input ========================== 20.043/20.043 20.043/20.043 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.043/20.043 20.043/20.043 % Formulas that are not ordinary clauses: 20.043/20.043 1 ->_s0(x1,y) -> ->_s0(e(x1),e(y)) # label(congruence) # label(non_clause). [assumption]. 20.043/20.043 2 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 20.043/20.043 3 ->*_s0(e(x1),true) -> ->_s0(e(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 20.043/20.043 4 ->*_s0(e(x1),false) -> ->_s0(e(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 20.043/20.043 5 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.043/20.043 6 (exists x3 ->*_s0(e(x3),false)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.043/20.043 20.043/20.043 ============================== end of process non-clausal formulas === 20.043/20.043 20.043/20.043 ============================== PROCESS INITIAL CLAUSES =============== 20.043/20.043 20.043/20.043 % Clauses before input processing: 20.043/20.043 20.043/20.043 formulas(usable). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 formulas(sos). 20.043/20.043 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(1)]. 20.043/20.043 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(2)]. 20.043/20.043 ->_s0(e(0),true) # label(replacement). [assumption]. 20.043/20.043 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(3)]. 20.043/20.043 -->*_s0(e(x),false) | ->_s0(e(s(x)),true) # label(replacement). [clausify(4)]. 20.043/20.043 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.043/20.043 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(5)]. 20.043/20.043 -->*_s0(e(x),false) # label(goal). [deny(6)]. 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 formulas(demodulators). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 ============================== PREDICATE ELIMINATION ================= 20.043/20.043 20.043/20.043 No predicates eliminated. 20.043/20.043 20.043/20.043 ============================== end predicate elimination ============= 20.043/20.043 20.043/20.043 Auto_denials: 20.043/20.043 % copying label goal to answer in negative clause 20.043/20.043 20.043/20.043 Term ordering decisions: 20.043/20.043 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.043/20.043 Function symbol precedence: function_order([ true, false, 0, e, s ]). 20.043/20.043 After inverse_order: (no changes). 20.043/20.043 Unfolding symbols: (none). 20.043/20.043 20.043/20.043 Auto_inference settings: 20.043/20.043 % set(neg_binary_resolution). % (HNE depth_diff=-4) 20.043/20.043 % clear(ordered_res). % (HNE depth_diff=-4) 20.043/20.043 % set(ur_resolution). % (HNE depth_diff=-4) 20.043/20.043 % set(ur_resolution) -> set(pos_ur_resolution). 20.043/20.043 % set(ur_resolution) -> set(neg_ur_resolution). 20.043/20.043 20.043/20.043 Auto_process settings: (no changes). 20.043/20.043 20.043/20.043 kept: 7 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(1)]. 20.043/20.043 kept: 8 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(2)]. 20.043/20.043 kept: 9 ->_s0(e(0),true) # label(replacement). [assumption]. 20.043/20.043 kept: 10 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(3)]. 20.043/20.043 kept: 11 -->*_s0(e(x),false) | ->_s0(e(s(x)),true) # label(replacement). [clausify(4)]. 20.043/20.043 kept: 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.043/20.043 kept: 13 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(5)]. 20.043/20.043 kept: 14 -->*_s0(e(x),false) # label(goal) # answer(goal). [deny(6)]. 20.043/20.043 20.043/20.043 ============================== end of process initial clauses ======== 20.043/20.043 20.043/20.043 ============================== CLAUSES FOR SEARCH ==================== 20.043/20.043 20.043/20.043 % Clauses after input processing: 20.043/20.043 20.043/20.043 formulas(usable). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 formulas(sos). 20.043/20.043 7 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(1)]. 20.043/20.043 8 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(2)]. 20.043/20.043 9 ->_s0(e(0),true) # label(replacement). [assumption]. 20.043/20.043 10 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(3)]. 20.043/20.043 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.043/20.043 13 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(5)]. 20.043/20.043 14 -->*_s0(e(x),false) # label(goal) # answer(goal). [deny(6)]. 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 formulas(demodulators). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 ============================== end of clauses for search ============= 20.043/20.043 20.043/20.043 ============================== SEARCH ================================ 20.043/20.043 20.043/20.043 % Starting search at 0.00 seconds. 20.043/20.043 20.043/20.043 given #1 (I,wt=8): 7 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(1)]. 20.043/20.043 20.043/20.043 given #2 (I,wt=8): 8 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(2)]. 20.043/20.043 20.043/20.043 given #3 (I,wt=4): 9 ->_s0(e(0),true) # label(replacement). [assumption]. 20.043/20.043 20.043/20.043 given #4 (I,wt=9): 10 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(3)]. 20.043/20.043 20.043/20.043 given #5 (I,wt=3): 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.043/20.043 20.043/20.043 given #6 (I,wt=9): 13 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(5)]. 20.043/20.043 20.043/20.043 given #7 (I,wt=4): 14 -->*_s0(e(x),false) # label(goal) # answer(goal). [deny(6)]. 20.043/20.043 20.043/20.043 given #8 (A,wt=6): 15 ->_s0(s(e(0)),s(true)). [ur(8,a,9,a)]. 20.043/20.043 20.043/20.043 given #9 (F,wt=3): 20 -->*_s0(true,false) # answer(goal). [ur(13,a,9,a,c,14,a)]. 20.043/20.043 20.043/20.043 given #10 (F,wt=3): 25 -->_s0(true,false) # answer(goal). [ur(13,b,12,a,c,20,a)]. 20.043/20.043 20.043/20.043 given #11 (F,wt=4): 19 -->_s0(e(x),false) # answer(goal). [ur(13,b,12,a,c,14,a)]. 20.043/20.043 20.043/20.043 ============================== PROOF ================================= 20.043/20.043 20.043/20.043 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.043/20.043 % Length of proof is 12. 20.043/20.043 % Level of proof is 4. 20.043/20.043 % Maximum clause weight is 9.000. 20.043/20.043 % Given clauses 11. 20.043/20.043 20.043/20.043 3 ->*_s0(e(x1),true) -> ->_s0(e(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 20.043/20.043 5 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.043/20.043 6 (exists x3 ->*_s0(e(x3),false)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.043/20.043 9 ->_s0(e(0),true) # label(replacement). [assumption]. 20.043/20.043 10 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(3)]. 20.043/20.043 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.043/20.043 13 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(5)]. 20.043/20.043 14 -->*_s0(e(x),false) # label(goal) # answer(goal). [deny(6)]. 20.043/20.043 17 ->*_s0(e(0),true). [ur(13,a,9,a,b,12,a)]. 20.043/20.043 19 -->_s0(e(x),false) # answer(goal). [ur(13,b,12,a,c,14,a)]. 20.043/20.043 26 -->*_s0(e(x),true) # answer(goal). [resolve(19,a,10,b)]. 20.043/20.043 27 $F # answer(goal). [resolve(26,a,17,a)]. 20.043/20.043 20.043/20.043 ============================== end of proof ========================== 20.043/20.043 20.043/20.043 ============================== STATISTICS ============================ 20.043/20.043 20.043/20.043 Given=11. Generated=20. Kept=20. proofs=1. 20.043/20.043 Usable=11. Sos=7. Demods=0. Limbo=0, Disabled=9. Hints=0. 20.043/20.043 Kept_by_rule=0, Deleted_by_rule=0. 20.043/20.043 Forward_subsumed=0. Back_subsumed=1. 20.043/20.043 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.043/20.043 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.043/20.043 Demod_attempts=0. Demod_rewrites=0. 20.043/20.043 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.043/20.043 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=11. 20.043/20.043 Megabytes=0.07. 20.043/20.043 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.043/20.043 20.043/20.043 ============================== end of statistics ===================== 20.043/20.043 20.043/20.043 ============================== end of search ========================= 20.043/20.043 20.043/20.043 THEOREM PROVED 20.043/20.043 20.043/20.043 Exiting with 1 proof. 20.043/20.043 20.043/20.043 Process 63077 exit (max_proofs) Wed Mar 9 09:55:25 2022 20.043/20.043 20.043/20.043 20.043/20.043 The problem is feasible. 20.043/20.043 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.043/20.043 Confluence Problem: 20.043/20.043 (VAR x) 20.043/20.043 (REPLACEMENT-MAP 20.043/20.043 (e 1) 20.043/20.043 (0) 20.043/20.043 (fSNonEmpty) 20.043/20.043 (false) 20.043/20.043 (s 1) 20.043/20.043 (true) 20.043/20.043 ) 20.043/20.043 (RULES 20.043/20.043 e(0) -> true 20.043/20.043 e(s(x)) -> false | e(x) ->* true 20.043/20.043 e(s(x)) -> true | e(x) ->* false 20.043/20.043 ) 20.043/20.043 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.043/20.043 20.043/20.043 Critical Pairs Processor: 20.043/20.043 -> Rules: 20.043/20.043 e(0) -> true 20.043/20.043 e(s(x)) -> false | e(x) ->* true 20.043/20.043 e(s(x)) -> true | e(x) ->* false 20.043/20.043 -> Vars: 20.043/20.043 "x" 20.043/20.043 20.043/20.043 -> Rlps: 20.043/20.043 crule: e(0) -> true, id: 1, possubterms: e(0)-> [], 0-> [1] 20.043/20.043 crule: e(s(x)) -> false | e(x) ->* true, id: 2, possubterms: e(s(x))-> [], s(x)-> [1] 20.043/20.043 crule: e(s(x)) -> true | e(x) ->* false, id: 3, possubterms: e(s(x))-> [], s(x)-> [1] 20.043/20.043 20.043/20.043 -> Unifications: 20.043/20.043 R3 unifies with R2 at p: [], l: e(s(x)), lp: e(s(x)), conds: {e(x) ->* false, e(x') ->* true}, sig: {x -> x'}, l': e(s(x')), r: true, r': false 20.043/20.043 20.043/20.043 -> Critical pairs info: 20.043/20.043 | e(x') ->* false, e(x') ->* true => Not trivial, Overlay, NW2, N1 20.043/20.043 20.043/20.043 -> Problem conclusions: 20.043/20.043 Left linear, Right linear, Linear 20.043/20.043 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.043/20.043 CTRS Type: 1 20.043/20.043 Deterministic, Strongly deterministic 20.043/20.043 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.043/20.043 Maybe right-stable CTRS, Overlay CTRS 20.043/20.043 Maybe normal CTRS, Maybe almost normal CTRS 20.043/20.043 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.043/20.043 Maybe level confluent 20.043/20.043 Maybe confluent 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 Clean Infeasible CCPs Processor: 20.043/20.043 Num of CPIs: 1 20.043/20.043 Timeout: 60 20.043/20.043 Timeout for each infeasibility problem: 60 s 20.043/20.043 | e(x') ->* false, e(x') ->* true => Not trivial, Overlay, NW2, N1 20.043/20.043 (CONDITIONTYPE ORIENTED) 20.043/20.043 (VAR x x1) 20.043/20.043 (RULES 20.043/20.043 e(0) -> true 20.043/20.043 e(s(x)) -> false | e(x) ->* true 20.043/20.043 e(s(x)) -> true | e(x) ->* false) 20.043/20.043 (VAR x4) 20.043/20.043 (CONDITION e(x4) ->* false, e(x4) ->* true) 20.043/20.043 20.043/20.043 Proof: 20.043/20.043 YES 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 20.043/20.043 Infeasibility Problem: 20.043/20.043 [(VAR vNonEmpty x x1 vNonEmpty x4) 20.043/20.043 (STRATEGY CONTEXTSENSITIVE 20.043/20.043 (e 1) 20.043/20.043 (0) 20.043/20.043 (fSNonEmpty) 20.043/20.043 (false) 20.043/20.043 (s 1) 20.043/20.043 (true) 20.043/20.043 ) 20.043/20.043 (RULES 20.043/20.043 e(0) -> true 20.043/20.043 e(s(x)) -> false | e(x) ->* true 20.043/20.043 e(s(x)) -> true | e(x) ->* false 20.043/20.043 )] 20.043/20.043 20.043/20.043 Infeasibility Conditions: 20.043/20.043 e(x4) ->* false, e(x4) ->* true 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 20.043/20.043 Obtaining a model using Mace4: 20.043/20.043 20.043/20.043 -> Usable Rules: 20.043/20.043 e(0) -> true 20.043/20.043 e(s(x)) -> false | e(x) ->* true 20.043/20.043 e(s(x)) -> true | e(x) ->* false 20.043/20.043 20.043/20.043 -> Mace4 Output: 20.043/20.043 ============================== Mace4 ================================= 20.043/20.043 Mace4 (64) version 2009-11A, November 2009. 20.043/20.043 Process 63113 was started by ubuntu on ubuntu, 20.043/20.043 Wed Mar 9 09:55:25 2022 20.043/20.043 The command was "./mace4 -c -f /tmp/mace463094-2.in". 20.043/20.043 ============================== end of head =========================== 20.043/20.043 20.043/20.043 ============================== INPUT ================================= 20.043/20.043 20.043/20.043 % Reading from file /tmp/mace463094-2.in 20.043/20.043 20.043/20.043 assign(max_seconds,20). 20.043/20.043 20.043/20.043 formulas(assumptions). 20.043/20.043 ->(x1,y) -> ->(e(x1),e(y)) # label(congruence). 20.043/20.043 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 20.043/20.043 ->(e(0),true) # label(replacement). 20.043/20.043 ->*(e(x1),true) -> ->(e(s(x1)),false) # label(replacement). 20.043/20.043 ->*(e(x1),false) -> ->(e(s(x1)),true) # label(replacement). 20.043/20.043 ->*(x,x) # label(reflexivity). 20.043/20.043 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 formulas(goals). 20.043/20.043 (exists x4 (->*(e(x4),false) & ->*(e(x4),true))) # label(goal). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 ============================== end of input ========================== 20.043/20.043 20.043/20.043 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.043/20.043 20.043/20.043 % Formulas that are not ordinary clauses: 20.043/20.043 1 ->(x1,y) -> ->(e(x1),e(y)) # label(congruence) # label(non_clause). [assumption]. 20.043/20.043 2 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 20.043/20.043 3 ->*(e(x1),true) -> ->(e(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 20.043/20.043 4 ->*(e(x1),false) -> ->(e(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 20.043/20.043 5 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.043/20.043 6 (exists x4 (->*(e(x4),false) & ->*(e(x4),true))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.043/20.043 20.043/20.043 ============================== end of process non-clausal formulas === 20.043/20.043 20.043/20.043 ============================== CLAUSES FOR SEARCH ==================== 20.043/20.043 20.043/20.043 formulas(mace4_clauses). 20.043/20.043 -->(x,y) | ->(e(x),e(y)) # label(congruence). 20.043/20.043 -->(x,y) | ->(s(x),s(y)) # label(congruence). 20.043/20.043 ->(e(0),true) # label(replacement). 20.043/20.043 -->*(e(x),true) | ->(e(s(x)),false) # label(replacement). 20.043/20.043 -->*(e(x),false) | ->(e(s(x)),true) # label(replacement). 20.043/20.043 ->*(x,x) # label(reflexivity). 20.043/20.043 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 20.043/20.043 -->*(e(x),false) | -->*(e(x),true) # label(goal). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 ============================== end of clauses for search ============= 20.043/20.043 20.043/20.043 % There are no natural numbers in the input. 20.043/20.043 20.043/20.043 ============================== DOMAIN SIZE 2 ========================= 20.043/20.043 20.043/20.043 ============================== MODEL ================================= 20.043/20.043 20.043/20.043 interpretation( 2, [number=1, seconds=0], [ 20.043/20.043 20.043/20.043 function(0, [ 0 ]), 20.043/20.043 20.043/20.043 function(false, [ 0 ]), 20.043/20.043 20.043/20.043 function(true, [ 1 ]), 20.043/20.043 20.043/20.043 function(e(_), [ 1, 0 ]), 20.043/20.043 20.043/20.043 function(s(_), [ 1, 0 ]), 20.043/20.043 20.043/20.043 relation(->*(_,_), [ 20.043/20.043 1, 0, 20.043/20.043 0, 1 ]), 20.043/20.043 20.043/20.043 relation(->(_,_), [ 20.043/20.043 1, 0, 20.043/20.043 0, 1 ]) 20.043/20.043 ]). 20.043/20.043 20.043/20.043 ============================== end of model ========================== 20.043/20.043 20.043/20.043 ============================== STATISTICS ============================ 20.043/20.043 20.043/20.043 For domain size 2. 20.043/20.043 20.043/20.043 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.043/20.043 Ground clauses: seen=25, kept=21. 20.043/20.043 Selections=7, assignments=11, propagations=14, current_models=1. 20.043/20.043 Rewrite_terms=79, rewrite_bools=46, indexes=19. 20.043/20.043 Rules_from_neg_clauses=1, cross_offs=1. 20.043/20.043 20.043/20.043 ============================== end of statistics ===================== 20.043/20.043 20.043/20.043 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.043/20.043 20.043/20.043 Exiting with 1 model. 20.043/20.043 20.043/20.043 Process 63113 exit (max_models) Wed Mar 9 09:55:25 2022 20.043/20.043 The process finished Wed Mar 9 09:55:25 2022 20.043/20.043 20.043/20.043 20.043/20.043 Mace4 cooked interpretation: 20.043/20.043 20.043/20.043 20.043/20.043 20.043/20.043 The problem is infeasible. 20.043/20.043 20.043/20.043 20.043/20.043 20.043/20.043 20.043/20.043 -> Problem conclusions: 20.043/20.043 Left linear, Right linear, Linear 20.043/20.043 Weakly orthogonal, Almost orthogonal, Orthogonal 20.043/20.043 CTRS Type: 1 20.043/20.043 Deterministic, Strongly deterministic 20.043/20.043 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.043/20.043 Maybe right-stable CTRS, Overlay CTRS 20.043/20.043 Maybe normal CTRS, Maybe almost normal CTRS 20.043/20.043 Maybe terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.043/20.043 Maybe level confluent 20.043/20.043 Maybe confluent 20.043/20.043 20.043/20.043 Resulting CCPs: 20.043/20.043 No CCPs left 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 Underlying TRS Termination Processor: 20.043/20.043 20.043/20.043 Resulting Underlying TRS: 20.043/20.043 (VAR x x') 20.043/20.043 (STRATEGY CONTEXTSENSITIVE 20.043/20.043 (e 1) 20.043/20.043 (false) 20.043/20.043 (true) 20.043/20.043 ) 20.043/20.043 (RULES 20.043/20.043 e(0) -> true 20.043/20.043 e(s(x)) -> false 20.043/20.043 e(s(x)) -> true 20.043/20.043 ) 20.043/20.043 Underlying TRS terminating? 20.043/20.043 YES 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 20.043/20.043 (VAR vu95NonEmpty x xu39) 20.043/20.043 (RULES 20.043/20.043 e(num0) -> ftrue 20.043/20.043 e(s(x)) -> ffalse 20.043/20.043 e(s(x)) -> ftrue 20.043/20.043 ) 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 20.043/20.043 Dependency Pairs Processor: 20.043/20.043 -> Pairs: 20.043/20.043 Empty 20.043/20.043 -> Rules: 20.043/20.043 e(num0) -> ftrue 20.043/20.043 e(s(x)) -> ffalse 20.043/20.043 e(s(x)) -> ftrue 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 20.043/20.043 SCC Processor: 20.043/20.043 -> Pairs: 20.043/20.043 Empty 20.043/20.043 -> Rules: 20.043/20.043 e(num0) -> ftrue 20.043/20.043 e(s(x)) -> ffalse 20.043/20.043 e(s(x)) -> ftrue 20.043/20.043 ->Strongly Connected Components: 20.043/20.043 There is no strongly connected component 20.043/20.043 20.043/20.043 The problem is finite. 20.043/20.043 20.043/20.043 20.043/20.043 -> Problem conclusions: 20.043/20.043 Left linear, Right linear, Linear 20.043/20.043 Weakly orthogonal, Almost orthogonal, Orthogonal 20.043/20.043 CTRS Type: 1 20.043/20.043 Deterministic, Strongly deterministic 20.043/20.043 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.043/20.043 Maybe right-stable CTRS, Overlay CTRS 20.043/20.043 Maybe normal CTRS, Maybe almost normal CTRS 20.043/20.043 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.043/20.043 Maybe level confluent 20.043/20.043 Maybe confluent 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.043/20.043 Confluence Problem: 20.043/20.043 (VAR x x') 20.043/20.043 (REPLACEMENT-MAP 20.043/20.043 (e 1) 20.043/20.043 (0) 20.043/20.043 (fSNonEmpty) 20.043/20.043 (false) 20.043/20.043 (s 1) 20.043/20.043 (true) 20.043/20.043 ) 20.043/20.043 (RULES 20.043/20.043 e(0) -> true 20.043/20.043 e(s(x)) -> false | e(x) ->* true 20.043/20.043 e(s(x)) -> true | e(x) ->* false 20.043/20.043 ) 20.043/20.043 Critical Pairs: 20.043/20.043 20.043/20.043 Terminating:YES 20.043/20.043 20.043/20.043 -> Problem conclusions: 20.043/20.043 Left linear, Right linear, Linear 20.043/20.043 Weakly orthogonal, Almost orthogonal, Orthogonal 20.043/20.043 CTRS Type: 1 20.043/20.043 Deterministic, Strongly deterministic 20.043/20.043 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.043/20.043 Maybe right-stable CTRS, Overlay CTRS 20.043/20.043 Maybe normal CTRS, Maybe almost normal CTRS 20.043/20.043 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.043/20.043 Maybe level confluent 20.043/20.043 Maybe confluent 20.043/20.043 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.043/20.043 20.043/20.043 Right Stable Processor: 20.043/20.043 Right-stable CTRS 20.043/20.043 Justification: 20.043/20.043 20.043/20.043 -> Term right-stability conditions: 20.043/20.043 Term: true 20.043/20.043 Right-stable term 20.043/20.043 Linear constructor form 20.043/20.043 Don't know if term is a ground normal form (not needed) 20.043/20.043 Right-stability condition achieved 20.043/20.043 A call to InfChecker wasn't needed 20.043/20.043 20.043/20.043 20.043/20.043 -> Term right-stability conditions: 20.043/20.043 Term: false 20.043/20.043 Right-stable term 20.043/20.043 Linear constructor form 20.043/20.043 Don't know if term is a ground normal form (not needed) 20.043/20.043 Right-stability condition achieved 20.043/20.043 A call to InfChecker wasn't needed 20.043/20.043 20.043/20.043 -> Problem conclusions: 20.043/20.043 Left linear, Right linear, Linear 20.043/20.043 Weakly orthogonal, Almost orthogonal, Orthogonal 20.043/20.043 CTRS Type: 1 20.043/20.043 Deterministic, Strongly deterministic 20.043/20.043 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.043/20.043 Right-stable CTRS, Overlay CTRS 20.043/20.043 Maybe normal CTRS, Almost normal CTRS 20.043/20.043 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.043/20.043 Level confluent 20.043/20.043 Confluent 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.043/20.043 Confluence Problem: 20.043/20.043 (VAR x x') 20.043/20.043 (REPLACEMENT-MAP 20.043/20.043 (e 1) 20.043/20.043 (0) 20.043/20.043 (fSNonEmpty) 20.043/20.043 (false) 20.043/20.043 (s 1) 20.043/20.043 (true) 20.043/20.043 ) 20.043/20.043 (RULES 20.043/20.043 e(0) -> true 20.043/20.043 e(s(x)) -> false | e(x) ->* true 20.043/20.043 e(s(x)) -> true | e(x) ->* false 20.043/20.043 ) 20.043/20.043 Critical Pairs: 20.043/20.043 20.043/20.043 Terminating:YES 20.043/20.043 20.043/20.043 -> Problem conclusions: 20.043/20.043 Left linear, Right linear, Linear 20.043/20.043 Weakly orthogonal, Almost orthogonal, Orthogonal 20.043/20.043 CTRS Type: 1 20.043/20.043 Deterministic, Strongly deterministic 20.043/20.043 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.043/20.043 Right-stable CTRS, Overlay CTRS 20.043/20.043 Maybe normal CTRS, Almost normal CTRS 20.043/20.043 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.043/20.043 Level confluent 20.043/20.043 Confluent 20.043/20.043 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.043/20.043 20.043/20.043 Normal RConds Processor: 20.043/20.043 YES 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 20.043/20.043 Infeasibility Problem: 20.043/20.043 [(VAR vNonEmpty x x1 vNonEmpty z z1) 20.043/20.043 (STRATEGY CONTEXTSENSITIVE 20.043/20.043 (e 1) 20.043/20.043 (0) 20.043/20.043 (fSNonEmpty) 20.043/20.043 (false) 20.043/20.043 (s 1) 20.043/20.043 (true) 20.043/20.043 ) 20.043/20.043 (RULES 20.043/20.043 e(0) -> true 20.043/20.043 e(s(x)) -> false | e(x) ->* true 20.043/20.043 e(s(x)) -> true | e(x) ->* false 20.043/20.043 )] 20.043/20.043 20.043/20.043 Infeasibility Conditions: 20.043/20.043 false -> z, true -> z1 20.043/20.043 20.043/20.043 Problem 1: 20.043/20.043 20.043/20.043 Obtaining a model using Mace4: 20.043/20.043 20.043/20.043 -> Usable Rules: 20.043/20.043 Empty 20.043/20.043 20.043/20.043 -> Mace4 Output: 20.043/20.043 ============================== Mace4 ================================= 20.043/20.043 Mace4 (64) version 2009-11A, November 2009. 20.043/20.043 Process 63197 was started by ubuntu on ubuntu, 20.043/20.043 Wed Mar 9 09:55:45 2022 20.043/20.043 The command was "./mace4 -c -f /tmp/mace463184-2.in". 20.043/20.043 ============================== end of head =========================== 20.043/20.043 20.043/20.043 ============================== INPUT ================================= 20.043/20.043 20.043/20.043 % Reading from file /tmp/mace463184-2.in 20.043/20.043 20.043/20.043 assign(max_seconds,13). 20.043/20.043 20.043/20.043 formulas(assumptions). 20.043/20.043 ->(x1,y) -> ->(e(x1),e(y)) # label(congruence). 20.043/20.043 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 formulas(goals). 20.043/20.043 (exists x4 exists x5 (->(false,x4) & ->(true,x5))) # label(goal). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 ============================== end of input ========================== 20.043/20.043 20.043/20.043 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.043/20.043 20.043/20.043 % Formulas that are not ordinary clauses: 20.043/20.043 1 ->(x1,y) -> ->(e(x1),e(y)) # label(congruence) # label(non_clause). [assumption]. 20.043/20.043 2 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 20.043/20.043 3 (exists x4 exists x5 (->(false,x4) & ->(true,x5))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.043/20.043 20.043/20.043 ============================== end of process non-clausal formulas === 20.043/20.043 20.043/20.043 ============================== CLAUSES FOR SEARCH ==================== 20.043/20.043 20.043/20.043 formulas(mace4_clauses). 20.043/20.043 -->(x,y) | ->(e(x),e(y)) # label(congruence). 20.043/20.043 -->(x,y) | ->(s(x),s(y)) # label(congruence). 20.043/20.043 -->(false,x) | -->(true,y) # label(goal). 20.043/20.043 end_of_list. 20.043/20.043 20.043/20.043 ============================== end of clauses for search ============= 20.043/20.043 20.043/20.043 % There are no natural numbers in the input. 20.043/20.043 20.043/20.043 ============================== DOMAIN SIZE 2 ========================= 20.043/20.043 20.043/20.043 ============================== MODEL ================================= 20.043/20.043 20.043/20.043 interpretation( 2, [number=1, seconds=0], [ 20.043/20.043 20.043/20.043 function(false, [ 0 ]), 20.043/20.043 20.043/20.043 function(true, [ 0 ]), 20.043/20.043 20.043/20.043 function(e(_), [ 0, 0 ]), 20.043/20.043 20.043/20.043 function(s(_), [ 0, 0 ]), 20.043/20.043 20.043/20.043 relation(->(_,_), [ 20.043/20.043 0, 0, 20.043/20.043 0, 0 ]) 20.043/20.043 ]). 20.043/20.043 20.043/20.043 ============================== end of model ========================== 20.043/20.043 20.043/20.043 ============================== STATISTICS ============================ 20.043/20.043 20.043/20.043 For domain size 2. 20.043/20.043 20.043/20.043 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.043/20.043 Ground clauses: seen=12, kept=12. 20.043/20.043 Selections=7, assignments=7, propagations=3, current_models=1. 20.043/20.043 Rewrite_terms=24, rewrite_bools=17, indexes=10. 20.043/20.043 Rules_from_neg_clauses=0, cross_offs=0. 20.043/20.043 20.043/20.043 ============================== end of statistics ===================== 20.043/20.043 20.043/20.043 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.043/20.043 20.043/20.043 Exiting with 1 model. 20.043/20.043 20.043/20.043 Process 63197 exit (max_models) Wed Mar 9 09:55:45 2022 20.043/20.043 The process finished Wed Mar 9 09:55:45 2022 20.043/20.043 20.043/20.043 20.043/20.043 Mace4 cooked interpretation: 20.043/20.043 20.043/20.043 20.043/20.043 20.043/20.043 The problem is infeasible. 20.043/20.043 20.043/20.043 20.043/20.043 -> Problem conclusions: 20.043/20.043 Left linear, Right linear, Linear 20.043/20.043 Weakly orthogonal, Almost orthogonal, Orthogonal 20.043/20.043 CTRS Type: 1 20.043/20.043 Deterministic, Strongly deterministic 20.043/20.043 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.043/20.043 Right-stable CTRS, Overlay CTRS 20.043/20.043 Normal CTRS, Almost normal CTRS 20.043/20.043 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.043/20.043 Level confluent 20.043/20.043 Confluent 20.043/20.043 20.043/20.043 The problem is joinable. 20.043/20.043 55.11user 1.34system 0:20.43elapsed 276%CPU (0avgtext+0avgdata 75532maxresident)k 20.043/20.043 8inputs+0outputs (0major+76481minor)pagefaults 0swaps