45.089/45.089 YES 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 45.089/45.089 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 45.089/45.089 Confluence Problem: 45.089/45.089 (VAR vNonEmpty:S x:S) 45.089/45.089 (STRATEGY CONTEXTSENSITIVE 45.089/45.089 (e 1) 45.089/45.089 (o 1) 45.089/45.089 (0) 45.089/45.089 (fSNonEmpty) 45.089/45.089 (false) 45.089/45.089 (s 1) 45.089/45.089 (true) 45.089/45.089 ) 45.089/45.089 (RULES 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x:S)) -> false | e(x:S) ->* true 45.089/45.089 e(s(x:S)) -> true | o(x:S) ->* true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x:S)) -> false | o(x:S) ->* true 45.089/45.089 o(s(x:S)) -> true | e(x:S) ->* true 45.089/45.089 ) 45.089/45.089 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 45.089/45.089 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 Inlining of Conditions Processor [STERN17]: 45.089/45.089 45.089/45.089 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 45.089/45.089 Confluence Problem: 45.089/45.089 (VAR vNonEmpty:S x:S) 45.089/45.089 (STRATEGY CONTEXTSENSITIVE 45.089/45.089 (e 1) 45.089/45.089 (o 1) 45.089/45.089 (0) 45.089/45.089 (fSNonEmpty) 45.089/45.089 (false) 45.089/45.089 (s 1) 45.089/45.089 (true) 45.089/45.089 ) 45.089/45.089 (RULES 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x:S)) -> false | e(x:S) ->* true 45.089/45.089 e(s(x:S)) -> true | o(x:S) ->* true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x:S)) -> false | o(x:S) ->* true 45.089/45.089 o(s(x:S)) -> true | e(x:S) ->* true 45.089/45.089 ) 45.089/45.089 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 45.089/45.089 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 Clean CTRS Processor: 45.089/45.089 45.089/45.089 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 45.089/45.089 Confluence Problem: 45.089/45.089 (VAR x:S) 45.089/45.089 (STRATEGY CONTEXTSENSITIVE 45.089/45.089 (e 1) 45.089/45.089 (o 1) 45.089/45.089 (0) 45.089/45.089 (fSNonEmpty) 45.089/45.089 (false) 45.089/45.089 (s 1) 45.089/45.089 (true) 45.089/45.089 ) 45.089/45.089 (RULES 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x:S)) -> false | e(x:S) ->* true 45.089/45.089 e(s(x:S)) -> true | o(x:S) ->* true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x:S)) -> false | o(x:S) ->* true 45.089/45.089 o(s(x:S)) -> true | e(x:S) ->* true 45.089/45.089 ) 45.089/45.089 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 45.089/45.089 45.089/45.089 CRule InfChecker Info: 45.089/45.089 e(0) -> true 45.089/45.089 Rule remains 45.089/45.089 Proof: 45.089/45.089 NO_CONDS 45.089/45.089 45.089/45.089 CRule InfChecker Info: 45.089/45.089 e(s(x:S)) -> false | e(x:S) ->* true 45.089/45.089 Rule remains 45.089/45.089 Proof: 45.089/45.089 NO 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 Infeasibility Problem: 45.089/45.089 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 45.089/45.089 (STRATEGY CONTEXTSENSITIVE 45.089/45.089 (e 1) 45.089/45.089 (o 1) 45.089/45.089 (0) 45.089/45.089 (fSNonEmpty) 45.089/45.089 (false) 45.089/45.089 (s 1) 45.089/45.089 (true) 45.089/45.089 ) 45.089/45.089 (RULES 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x:S:S)) -> false | e(x:S:S) ->* true 45.089/45.089 e(s(x:S:S)) -> true | o(x:S:S) ->* true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x:S:S)) -> false | o(x:S:S) ->* true 45.089/45.089 o(s(x:S:S)) -> true | e(x:S:S) ->* true 45.089/45.089 )] 45.089/45.089 45.089/45.089 Infeasibility Conditions: 45.089/45.089 e(x:S:S) ->* true 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 Obtaining a proof using Prover9: 45.089/45.089 45.089/45.089 -> Prover9 Output: 45.089/45.089 ============================== Prover9 =============================== 45.089/45.089 Prover9 (64) version 2009-11A, November 2009. 45.089/45.089 Process 118485 was started by ubuntu on ubuntu, 45.089/45.089 Fri Jul 16 15:32:42 2021 45.089/45.089 The command was "./prover9 -f /tmp/prover9118476-0.in". 45.089/45.089 ============================== end of head =========================== 45.089/45.089 45.089/45.089 ============================== INPUT ================================= 45.089/45.089 45.089/45.089 % Reading from file /tmp/prover9118476-0.in 45.089/45.089 45.089/45.089 assign(max_seconds,20). 45.089/45.089 45.089/45.089 formulas(assumptions). 45.089/45.089 ->*_s0(x,x) # label(reflexivity). 45.089/45.089 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 45.089/45.089 ->_s0(x1,y) -> ->_s0(e(x1),e(y)) # label(congruence). 45.089/45.089 ->_s0(x1,y) -> ->_s0(o(x1),o(y)) # label(congruence). 45.089/45.089 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 45.089/45.089 ->_s0(e(0),true) # label(replacement). 45.089/45.089 ->*_s0(e(x2),true) -> ->_s0(e(s(x2)),false) # label(replacement). 45.089/45.089 ->*_s0(o(x2),true) -> ->_s0(e(s(x2)),true) # label(replacement). 45.089/45.089 ->_s0(o(0),false) # label(replacement). 45.089/45.089 ->*_s0(o(x2),true) -> ->_s0(o(s(x2)),false) # label(replacement). 45.089/45.089 ->*_s0(e(x2),true) -> ->_s0(o(s(x2)),true) # label(replacement). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(goals). 45.089/45.089 (exists x2 ->*_s0(e(x2),true)) # label(goal). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== end of input ========================== 45.089/45.089 45.089/45.089 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 45.089/45.089 45.089/45.089 % Formulas that are not ordinary clauses: 45.089/45.089 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 45.089/45.089 2 ->_s0(x1,y) -> ->_s0(e(x1),e(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 3 ->_s0(x1,y) -> ->_s0(o(x1),o(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 5 ->*_s0(e(x2),true) -> ->_s0(e(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 6 ->*_s0(o(x2),true) -> ->_s0(e(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 7 ->*_s0(o(x2),true) -> ->_s0(o(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 8 ->*_s0(e(x2),true) -> ->_s0(o(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 9 (exists x2 ->*_s0(e(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 45.089/45.089 45.089/45.089 ============================== end of process non-clausal formulas === 45.089/45.089 45.089/45.089 ============================== PROCESS INITIAL CLAUSES =============== 45.089/45.089 45.089/45.089 % Clauses before input processing: 45.089/45.089 45.089/45.089 formulas(usable). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(sos). 45.089/45.089 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(5)]. 45.089/45.089 -->*_s0(o(x),true) | ->_s0(e(s(x)),true) # label(replacement). [clausify(6)]. 45.089/45.089 ->_s0(o(0),false) # label(replacement). [assumption]. 45.089/45.089 -->*_s0(o(x),true) | ->_s0(o(s(x)),false) # label(replacement). [clausify(7)]. 45.089/45.089 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement). [clausify(8)]. 45.089/45.089 -->*_s0(e(x),true) # label(goal). [deny(9)]. 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(demodulators). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== PREDICATE ELIMINATION ================= 45.089/45.089 45.089/45.089 No predicates eliminated. 45.089/45.089 45.089/45.089 ============================== end predicate elimination ============= 45.089/45.089 45.089/45.089 Auto_denials: 45.089/45.089 % copying label goal to answer in negative clause 45.089/45.089 45.089/45.089 Term ordering decisions: 45.089/45.089 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 45.089/45.089 Function symbol precedence: function_order([ true, false, 0, e, o, s ]). 45.089/45.089 After inverse_order: (no changes). 45.089/45.089 Unfolding symbols: (none). 45.089/45.089 45.089/45.089 Auto_inference settings: 45.089/45.089 % set(neg_binary_resolution). % (HNE depth_diff=-7) 45.089/45.089 % clear(ordered_res). % (HNE depth_diff=-7) 45.089/45.089 % set(ur_resolution). % (HNE depth_diff=-7) 45.089/45.089 % set(ur_resolution) -> set(pos_ur_resolution). 45.089/45.089 % set(ur_resolution) -> set(neg_ur_resolution). 45.089/45.089 45.089/45.089 Auto_process settings: (no changes). 45.089/45.089 45.089/45.089 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 kept: 12 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 kept: 13 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 kept: 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 kept: 16 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(5)]. 45.089/45.089 kept: 17 -->*_s0(o(x),true) | ->_s0(e(s(x)),true) # label(replacement). [clausify(6)]. 45.089/45.089 kept: 18 ->_s0(o(0),false) # label(replacement). [assumption]. 45.089/45.089 kept: 19 -->*_s0(o(x),true) | ->_s0(o(s(x)),false) # label(replacement). [clausify(7)]. 45.089/45.089 kept: 20 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement). [clausify(8)]. 45.089/45.089 kept: 21 -->*_s0(e(x),true) # label(goal) # answer(goal). [deny(9)]. 45.089/45.089 45.089/45.089 ============================== end of process initial clauses ======== 45.089/45.089 45.089/45.089 ============================== CLAUSES FOR SEARCH ==================== 45.089/45.089 45.089/45.089 % Clauses after input processing: 45.089/45.089 45.089/45.089 formulas(usable). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(sos). 45.089/45.089 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 12 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 13 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 17 -->*_s0(o(x),true) | ->_s0(e(s(x)),true) # label(replacement). [clausify(6)]. 45.089/45.089 18 ->_s0(o(0),false) # label(replacement). [assumption]. 45.089/45.089 19 -->*_s0(o(x),true) | ->_s0(o(s(x)),false) # label(replacement). [clausify(7)]. 45.089/45.089 21 -->*_s0(e(x),true) # label(goal) # answer(goal). [deny(9)]. 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(demodulators). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== end of clauses for search ============= 45.089/45.089 45.089/45.089 ============================== SEARCH ================================ 45.089/45.089 45.089/45.089 % Starting search at 0.00 seconds. 45.089/45.089 45.089/45.089 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 45.089/45.089 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 45.089/45.089 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 45.089/45.089 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 45.089/45.089 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 45.089/45.089 given #6 (I,wt=4): 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 45.089/45.089 ============================== PROOF ================================= 45.089/45.089 45.089/45.089 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 45.089/45.089 % Length of proof is 8. 45.089/45.089 % Level of proof is 3. 45.089/45.089 % Maximum clause weight is 9.000. 45.089/45.089 % Given clauses 6. 45.089/45.089 45.089/45.089 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 45.089/45.089 9 (exists x2 ->*_s0(e(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 45.089/45.089 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 21 -->*_s0(e(x),true) # label(goal) # answer(goal). [deny(9)]. 45.089/45.089 25 ->*_s0(e(0),true). [ur(11,a,15,a,b,10,a)]. 45.089/45.089 26 $F # answer(goal). [resolve(25,a,21,a)]. 45.089/45.089 45.089/45.089 ============================== end of proof ========================== 45.089/45.089 45.089/45.089 ============================== STATISTICS ============================ 45.089/45.089 45.089/45.089 Given=6. Generated=16. Kept=16. proofs=1. 45.089/45.089 Usable=6. Sos=4. Demods=0. Limbo=3, Disabled=14. Hints=0. 45.089/45.089 Kept_by_rule=0, Deleted_by_rule=0. 45.089/45.089 Forward_subsumed=0. Back_subsumed=2. 45.089/45.089 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 45.089/45.089 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 45.089/45.089 Demod_attempts=0. Demod_rewrites=0. 45.089/45.089 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 45.089/45.089 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=11. 45.089/45.089 Megabytes=0.07. 45.089/45.089 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 45.089/45.089 45.089/45.089 ============================== end of statistics ===================== 45.089/45.089 45.089/45.089 ============================== end of search ========================= 45.089/45.089 45.089/45.089 THEOREM PROVED 45.089/45.089 45.089/45.089 Exiting with 1 proof. 45.089/45.089 45.089/45.089 Process 118485 exit (max_proofs) Fri Jul 16 15:32:42 2021 45.089/45.089 45.089/45.089 45.089/45.089 The problem is feasible. 45.089/45.089 45.089/45.089 45.089/45.089 CRule InfChecker Info: 45.089/45.089 e(s(x:S)) -> true | o(x:S) ->* true 45.089/45.089 Rule remains 45.089/45.089 Proof: 45.089/45.089 NO 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 Infeasibility Problem: 45.089/45.089 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 45.089/45.089 (STRATEGY CONTEXTSENSITIVE 45.089/45.089 (e 1) 45.089/45.089 (o 1) 45.089/45.089 (0) 45.089/45.089 (fSNonEmpty) 45.089/45.089 (false) 45.089/45.089 (s 1) 45.089/45.089 (true) 45.089/45.089 ) 45.089/45.089 (RULES 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x:S:S)) -> false | e(x:S:S) ->* true 45.089/45.089 e(s(x:S:S)) -> true | o(x:S:S) ->* true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x:S:S)) -> false | o(x:S:S) ->* true 45.089/45.089 o(s(x:S:S)) -> true | e(x:S:S) ->* true 45.089/45.089 )] 45.089/45.089 45.089/45.089 Infeasibility Conditions: 45.089/45.089 o(x:S:S) ->* true 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 Obtaining a proof using Prover9: 45.089/45.089 45.089/45.089 -> Prover9 Output: 45.089/45.089 ============================== Prover9 =============================== 45.089/45.089 Prover9 (64) version 2009-11A, November 2009. 45.089/45.089 Process 118508 was started by ubuntu on ubuntu, 45.089/45.089 Fri Jul 16 15:32:42 2021 45.089/45.089 The command was "./prover9 -f /tmp/prover9118497-0.in". 45.089/45.089 ============================== end of head =========================== 45.089/45.089 45.089/45.089 ============================== INPUT ================================= 45.089/45.089 45.089/45.089 % Reading from file /tmp/prover9118497-0.in 45.089/45.089 45.089/45.089 assign(max_seconds,20). 45.089/45.089 45.089/45.089 formulas(assumptions). 45.089/45.089 ->*_s0(x,x) # label(reflexivity). 45.089/45.089 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 45.089/45.089 ->_s0(x1,y) -> ->_s0(e(x1),e(y)) # label(congruence). 45.089/45.089 ->_s0(x1,y) -> ->_s0(o(x1),o(y)) # label(congruence). 45.089/45.089 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 45.089/45.089 ->_s0(e(0),true) # label(replacement). 45.089/45.089 ->*_s0(e(x2),true) -> ->_s0(e(s(x2)),false) # label(replacement). 45.089/45.089 ->*_s0(o(x2),true) -> ->_s0(e(s(x2)),true) # label(replacement). 45.089/45.089 ->_s0(o(0),false) # label(replacement). 45.089/45.089 ->*_s0(o(x2),true) -> ->_s0(o(s(x2)),false) # label(replacement). 45.089/45.089 ->*_s0(e(x2),true) -> ->_s0(o(s(x2)),true) # label(replacement). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(goals). 45.089/45.089 (exists x2 ->*_s0(o(x2),true)) # label(goal). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== end of input ========================== 45.089/45.089 45.089/45.089 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 45.089/45.089 45.089/45.089 % Formulas that are not ordinary clauses: 45.089/45.089 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 45.089/45.089 2 ->_s0(x1,y) -> ->_s0(e(x1),e(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 3 ->_s0(x1,y) -> ->_s0(o(x1),o(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 5 ->*_s0(e(x2),true) -> ->_s0(e(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 6 ->*_s0(o(x2),true) -> ->_s0(e(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 7 ->*_s0(o(x2),true) -> ->_s0(o(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 8 ->*_s0(e(x2),true) -> ->_s0(o(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 9 (exists x2 ->*_s0(o(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 45.089/45.089 45.089/45.089 ============================== end of process non-clausal formulas === 45.089/45.089 45.089/45.089 ============================== PROCESS INITIAL CLAUSES =============== 45.089/45.089 45.089/45.089 % Clauses before input processing: 45.089/45.089 45.089/45.089 formulas(usable). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(sos). 45.089/45.089 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(5)]. 45.089/45.089 -->*_s0(o(x),true) | ->_s0(e(s(x)),true) # label(replacement). [clausify(6)]. 45.089/45.089 ->_s0(o(0),false) # label(replacement). [assumption]. 45.089/45.089 -->*_s0(o(x),true) | ->_s0(o(s(x)),false) # label(replacement). [clausify(7)]. 45.089/45.089 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement). [clausify(8)]. 45.089/45.089 -->*_s0(o(x),true) # label(goal). [deny(9)]. 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(demodulators). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== PREDICATE ELIMINATION ================= 45.089/45.089 45.089/45.089 No predicates eliminated. 45.089/45.089 45.089/45.089 ============================== end predicate elimination ============= 45.089/45.089 45.089/45.089 Auto_denials: 45.089/45.089 % copying label goal to answer in negative clause 45.089/45.089 45.089/45.089 Term ordering decisions: 45.089/45.089 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 45.089/45.089 Function symbol precedence: function_order([ true, false, 0, e, o, s ]). 45.089/45.089 After inverse_order: (no changes). 45.089/45.089 Unfolding symbols: (none). 45.089/45.089 45.089/45.089 Auto_inference settings: 45.089/45.089 % set(neg_binary_resolution). % (HNE depth_diff=-7) 45.089/45.089 % clear(ordered_res). % (HNE depth_diff=-7) 45.089/45.089 % set(ur_resolution). % (HNE depth_diff=-7) 45.089/45.089 % set(ur_resolution) -> set(pos_ur_resolution). 45.089/45.089 % set(ur_resolution) -> set(neg_ur_resolution). 45.089/45.089 45.089/45.089 Auto_process settings: (no changes). 45.089/45.089 45.089/45.089 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 kept: 12 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 kept: 13 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 kept: 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 kept: 16 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(5)]. 45.089/45.089 kept: 17 -->*_s0(o(x),true) | ->_s0(e(s(x)),true) # label(replacement). [clausify(6)]. 45.089/45.089 kept: 18 ->_s0(o(0),false) # label(replacement). [assumption]. 45.089/45.089 kept: 19 -->*_s0(o(x),true) | ->_s0(o(s(x)),false) # label(replacement). [clausify(7)]. 45.089/45.089 kept: 20 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement). [clausify(8)]. 45.089/45.089 kept: 21 -->*_s0(o(x),true) # label(goal) # answer(goal). [deny(9)]. 45.089/45.089 45.089/45.089 ============================== end of process initial clauses ======== 45.089/45.089 45.089/45.089 ============================== CLAUSES FOR SEARCH ==================== 45.089/45.089 45.089/45.089 % Clauses after input processing: 45.089/45.089 45.089/45.089 formulas(usable). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(sos). 45.089/45.089 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 12 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 13 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 16 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(5)]. 45.089/45.089 18 ->_s0(o(0),false) # label(replacement). [assumption]. 45.089/45.089 20 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement). [clausify(8)]. 45.089/45.089 21 -->*_s0(o(x),true) # label(goal) # answer(goal). [deny(9)]. 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(demodulators). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== end of clauses for search ============= 45.089/45.089 45.089/45.089 ============================== SEARCH ================================ 45.089/45.089 45.089/45.089 % Starting search at 0.00 seconds. 45.089/45.089 45.089/45.089 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 45.089/45.089 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 45.089/45.089 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 45.089/45.089 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 45.089/45.089 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 45.089/45.089 given #6 (I,wt=4): 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 45.089/45.089 given #7 (I,wt=9): 16 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(5)]. 45.089/45.089 45.089/45.089 given #8 (I,wt=4): 18 ->_s0(o(0),false) # label(replacement). [assumption]. 45.089/45.089 45.089/45.089 given #9 (I,wt=9): 20 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement). [clausify(8)]. 45.089/45.089 45.089/45.089 given #10 (I,wt=4): 21 -->*_s0(o(x),true) # label(goal) # answer(goal). [deny(9)]. 45.089/45.089 45.089/45.089 given #11 (A,wt=6): 22 ->_s0(s(e(0)),s(true)). [ur(14,a,15,a)]. 45.089/45.089 45.089/45.089 given #12 (F,wt=3): 32 -->*_s0(false,true) # answer(goal). [ur(11,a,18,a,c,21,a)]. 45.089/45.089 45.089/45.089 given #13 (F,wt=3): 38 -->_s0(false,true) # answer(goal). [ur(11,b,10,a,c,32,a)]. 45.089/45.089 45.089/45.089 given #14 (F,wt=4): 31 -->_s0(o(x),true) # answer(goal). [ur(11,b,10,a,c,21,a)]. 45.089/45.089 45.089/45.089 ============================== PROOF ================================= 45.089/45.089 45.089/45.089 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 45.089/45.089 % Length of proof is 12. 45.089/45.089 % Level of proof is 4. 45.089/45.089 % Maximum clause weight is 9.000. 45.089/45.089 % Given clauses 14. 45.089/45.089 45.089/45.089 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 45.089/45.089 8 ->*_s0(e(x2),true) -> ->_s0(o(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 9 (exists x2 ->*_s0(o(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 45.089/45.089 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 20 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement). [clausify(8)]. 45.089/45.089 21 -->*_s0(o(x),true) # label(goal) # answer(goal). [deny(9)]. 45.089/45.089 25 ->*_s0(e(0),true). [ur(11,a,15,a,b,10,a)]. 45.089/45.089 31 -->_s0(o(x),true) # answer(goal). [ur(11,b,10,a,c,21,a)]. 45.089/45.089 39 -->*_s0(e(x),true) # answer(goal). [resolve(31,a,20,b)]. 45.089/45.089 40 $F # answer(goal). [resolve(39,a,25,a)]. 45.089/45.089 45.089/45.089 ============================== end of proof ========================== 45.089/45.089 45.089/45.089 ============================== STATISTICS ============================ 45.089/45.089 45.089/45.089 Given=14. Generated=30. Kept=30. proofs=1. 45.089/45.089 Usable=14. Sos=13. Demods=0. Limbo=0, Disabled=14. Hints=0. 45.089/45.089 Kept_by_rule=0, Deleted_by_rule=0. 45.089/45.089 Forward_subsumed=0. Back_subsumed=2. 45.089/45.089 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 45.089/45.089 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 45.089/45.089 Demod_attempts=0. Demod_rewrites=0. 45.089/45.089 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 45.089/45.089 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=15. 45.089/45.089 Megabytes=0.09. 45.089/45.089 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 45.089/45.089 45.089/45.089 ============================== end of statistics ===================== 45.089/45.089 45.089/45.089 ============================== end of search ========================= 45.089/45.089 45.089/45.089 THEOREM PROVED 45.089/45.089 45.089/45.089 Exiting with 1 proof. 45.089/45.089 45.089/45.089 Process 118508 exit (max_proofs) Fri Jul 16 15:32:42 2021 45.089/45.089 45.089/45.089 45.089/45.089 The problem is feasible. 45.089/45.089 45.089/45.089 45.089/45.089 CRule InfChecker Info: 45.089/45.089 o(0) -> false 45.089/45.089 Rule remains 45.089/45.089 Proof: 45.089/45.089 NO_CONDS 45.089/45.089 45.089/45.089 CRule InfChecker Info: 45.089/45.089 o(s(x:S)) -> false | o(x:S) ->* true 45.089/45.089 Rule remains 45.089/45.089 Proof: 45.089/45.089 NO 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 Infeasibility Problem: 45.089/45.089 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 45.089/45.089 (STRATEGY CONTEXTSENSITIVE 45.089/45.089 (e 1) 45.089/45.089 (o 1) 45.089/45.089 (0) 45.089/45.089 (fSNonEmpty) 45.089/45.089 (false) 45.089/45.089 (s 1) 45.089/45.089 (true) 45.089/45.089 ) 45.089/45.089 (RULES 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x:S:S)) -> false | e(x:S:S) ->* true 45.089/45.089 e(s(x:S:S)) -> true | o(x:S:S) ->* true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x:S:S)) -> false | o(x:S:S) ->* true 45.089/45.089 o(s(x:S:S)) -> true | e(x:S:S) ->* true 45.089/45.089 )] 45.089/45.089 45.089/45.089 Infeasibility Conditions: 45.089/45.089 o(x:S:S) ->* true 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 Obtaining a proof using Prover9: 45.089/45.089 45.089/45.089 -> Prover9 Output: 45.089/45.089 ============================== Prover9 =============================== 45.089/45.089 Prover9 (64) version 2009-11A, November 2009. 45.089/45.089 Process 118532 was started by ubuntu on ubuntu, 45.089/45.089 Fri Jul 16 15:32:42 2021 45.089/45.089 The command was "./prover9 -f /tmp/prover9118520-0.in". 45.089/45.089 ============================== end of head =========================== 45.089/45.089 45.089/45.089 ============================== INPUT ================================= 45.089/45.089 45.089/45.089 % Reading from file /tmp/prover9118520-0.in 45.089/45.089 45.089/45.089 assign(max_seconds,20). 45.089/45.089 45.089/45.089 formulas(assumptions). 45.089/45.089 ->*_s0(x,x) # label(reflexivity). 45.089/45.089 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 45.089/45.089 ->_s0(x1,y) -> ->_s0(e(x1),e(y)) # label(congruence). 45.089/45.089 ->_s0(x1,y) -> ->_s0(o(x1),o(y)) # label(congruence). 45.089/45.089 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 45.089/45.089 ->_s0(e(0),true) # label(replacement). 45.089/45.089 ->*_s0(e(x2),true) -> ->_s0(e(s(x2)),false) # label(replacement). 45.089/45.089 ->*_s0(o(x2),true) -> ->_s0(e(s(x2)),true) # label(replacement). 45.089/45.089 ->_s0(o(0),false) # label(replacement). 45.089/45.089 ->*_s0(o(x2),true) -> ->_s0(o(s(x2)),false) # label(replacement). 45.089/45.089 ->*_s0(e(x2),true) -> ->_s0(o(s(x2)),true) # label(replacement). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(goals). 45.089/45.089 (exists x2 ->*_s0(o(x2),true)) # label(goal). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== end of input ========================== 45.089/45.089 45.089/45.089 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 45.089/45.089 45.089/45.089 % Formulas that are not ordinary clauses: 45.089/45.089 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 45.089/45.089 2 ->_s0(x1,y) -> ->_s0(e(x1),e(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 3 ->_s0(x1,y) -> ->_s0(o(x1),o(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 5 ->*_s0(e(x2),true) -> ->_s0(e(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 6 ->*_s0(o(x2),true) -> ->_s0(e(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 7 ->*_s0(o(x2),true) -> ->_s0(o(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 8 ->*_s0(e(x2),true) -> ->_s0(o(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 9 (exists x2 ->*_s0(o(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 45.089/45.089 45.089/45.089 ============================== end of process non-clausal formulas === 45.089/45.089 45.089/45.089 ============================== PROCESS INITIAL CLAUSES =============== 45.089/45.089 45.089/45.089 % Clauses before input processing: 45.089/45.089 45.089/45.089 formulas(usable). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(sos). 45.089/45.089 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(5)]. 45.089/45.089 -->*_s0(o(x),true) | ->_s0(e(s(x)),true) # label(replacement). [clausify(6)]. 45.089/45.089 ->_s0(o(0),false) # label(replacement). [assumption]. 45.089/45.089 -->*_s0(o(x),true) | ->_s0(o(s(x)),false) # label(replacement). [clausify(7)]. 45.089/45.089 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement). [clausify(8)]. 45.089/45.089 -->*_s0(o(x),true) # label(goal). [deny(9)]. 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(demodulators). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== PREDICATE ELIMINATION ================= 45.089/45.089 45.089/45.089 No predicates eliminated. 45.089/45.089 45.089/45.089 ============================== end predicate elimination ============= 45.089/45.089 45.089/45.089 Auto_denials: 45.089/45.089 % copying label goal to answer in negative clause 45.089/45.089 45.089/45.089 Term ordering decisions: 45.089/45.089 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 45.089/45.089 Function symbol precedence: function_order([ true, false, 0, e, o, s ]). 45.089/45.089 After inverse_order: (no changes). 45.089/45.089 Unfolding symbols: (none). 45.089/45.089 45.089/45.089 Auto_inference settings: 45.089/45.089 % set(neg_binary_resolution). % (HNE depth_diff=-7) 45.089/45.089 % clear(ordered_res). % (HNE depth_diff=-7) 45.089/45.089 % set(ur_resolution). % (HNE depth_diff=-7) 45.089/45.089 % set(ur_resolution) -> set(pos_ur_resolution). 45.089/45.089 % set(ur_resolution) -> set(neg_ur_resolution). 45.089/45.089 45.089/45.089 Auto_process settings: (no changes). 45.089/45.089 45.089/45.089 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 kept: 12 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 kept: 13 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 kept: 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 kept: 16 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(5)]. 45.089/45.089 kept: 17 -->*_s0(o(x),true) | ->_s0(e(s(x)),true) # label(replacement). [clausify(6)]. 45.089/45.089 kept: 18 ->_s0(o(0),false) # label(replacement). [assumption]. 45.089/45.089 kept: 19 -->*_s0(o(x),true) | ->_s0(o(s(x)),false) # label(replacement). [clausify(7)]. 45.089/45.089 kept: 20 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement). [clausify(8)]. 45.089/45.089 kept: 21 -->*_s0(o(x),true) # label(goal) # answer(goal). [deny(9)]. 45.089/45.089 45.089/45.089 ============================== end of process initial clauses ======== 45.089/45.089 45.089/45.089 ============================== CLAUSES FOR SEARCH ==================== 45.089/45.089 45.089/45.089 % Clauses after input processing: 45.089/45.089 45.089/45.089 formulas(usable). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(sos). 45.089/45.089 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 12 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 13 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 16 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(5)]. 45.089/45.089 18 ->_s0(o(0),false) # label(replacement). [assumption]. 45.089/45.089 20 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement). [clausify(8)]. 45.089/45.089 21 -->*_s0(o(x),true) # label(goal) # answer(goal). [deny(9)]. 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(demodulators). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== end of clauses for search ============= 45.089/45.089 45.089/45.089 ============================== SEARCH ================================ 45.089/45.089 45.089/45.089 % Starting search at 0.00 seconds. 45.089/45.089 45.089/45.089 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 45.089/45.089 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 45.089/45.089 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 45.089/45.089 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 45.089/45.089 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 45.089/45.089 given #6 (I,wt=4): 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 45.089/45.089 given #7 (I,wt=9): 16 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(5)]. 45.089/45.089 45.089/45.089 given #8 (I,wt=4): 18 ->_s0(o(0),false) # label(replacement). [assumption]. 45.089/45.089 45.089/45.089 given #9 (I,wt=9): 20 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement). [clausify(8)]. 45.089/45.089 45.089/45.089 given #10 (I,wt=4): 21 -->*_s0(o(x),true) # label(goal) # answer(goal). [deny(9)]. 45.089/45.089 45.089/45.089 given #11 (A,wt=6): 22 ->_s0(s(e(0)),s(true)). [ur(14,a,15,a)]. 45.089/45.089 45.089/45.089 given #12 (F,wt=3): 32 -->*_s0(false,true) # answer(goal). [ur(11,a,18,a,c,21,a)]. 45.089/45.089 45.089/45.089 given #13 (F,wt=3): 38 -->_s0(false,true) # answer(goal). [ur(11,b,10,a,c,32,a)]. 45.089/45.089 45.089/45.089 given #14 (F,wt=4): 31 -->_s0(o(x),true) # answer(goal). [ur(11,b,10,a,c,21,a)]. 45.089/45.089 45.089/45.089 ============================== PROOF ================================= 45.089/45.089 45.089/45.089 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 45.089/45.089 % Length of proof is 12. 45.089/45.089 % Level of proof is 4. 45.089/45.089 % Maximum clause weight is 9.000. 45.089/45.089 % Given clauses 14. 45.089/45.089 45.089/45.089 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 45.089/45.089 8 ->*_s0(e(x2),true) -> ->_s0(o(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 9 (exists x2 ->*_s0(o(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 45.089/45.089 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 20 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement). [clausify(8)]. 45.089/45.089 21 -->*_s0(o(x),true) # label(goal) # answer(goal). [deny(9)]. 45.089/45.089 25 ->*_s0(e(0),true). [ur(11,a,15,a,b,10,a)]. 45.089/45.089 31 -->_s0(o(x),true) # answer(goal). [ur(11,b,10,a,c,21,a)]. 45.089/45.089 39 -->*_s0(e(x),true) # answer(goal). [resolve(31,a,20,b)]. 45.089/45.089 40 $F # answer(goal). [resolve(39,a,25,a)]. 45.089/45.089 45.089/45.089 ============================== end of proof ========================== 45.089/45.089 45.089/45.089 ============================== STATISTICS ============================ 45.089/45.089 45.089/45.089 Given=14. Generated=30. Kept=30. proofs=1. 45.089/45.089 Usable=14. Sos=13. Demods=0. Limbo=0, Disabled=14. Hints=0. 45.089/45.089 Kept_by_rule=0, Deleted_by_rule=0. 45.089/45.089 Forward_subsumed=0. Back_subsumed=2. 45.089/45.089 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 45.089/45.089 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 45.089/45.089 Demod_attempts=0. Demod_rewrites=0. 45.089/45.089 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 45.089/45.089 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=15. 45.089/45.089 Megabytes=0.09. 45.089/45.089 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 45.089/45.089 45.089/45.089 ============================== end of statistics ===================== 45.089/45.089 45.089/45.089 ============================== end of search ========================= 45.089/45.089 45.089/45.089 THEOREM PROVED 45.089/45.089 45.089/45.089 Exiting with 1 proof. 45.089/45.089 45.089/45.089 Process 118532 exit (max_proofs) Fri Jul 16 15:32:42 2021 45.089/45.089 45.089/45.089 45.089/45.089 The problem is feasible. 45.089/45.089 45.089/45.089 45.089/45.089 CRule InfChecker Info: 45.089/45.089 o(s(x:S)) -> true | e(x:S) ->* true 45.089/45.089 Rule remains 45.089/45.089 Proof: 45.089/45.089 NO 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 Infeasibility Problem: 45.089/45.089 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 45.089/45.089 (STRATEGY CONTEXTSENSITIVE 45.089/45.089 (e 1) 45.089/45.089 (o 1) 45.089/45.089 (0) 45.089/45.089 (fSNonEmpty) 45.089/45.089 (false) 45.089/45.089 (s 1) 45.089/45.089 (true) 45.089/45.089 ) 45.089/45.089 (RULES 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x:S:S)) -> false | e(x:S:S) ->* true 45.089/45.089 e(s(x:S:S)) -> true | o(x:S:S) ->* true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x:S:S)) -> false | o(x:S:S) ->* true 45.089/45.089 o(s(x:S:S)) -> true | e(x:S:S) ->* true 45.089/45.089 )] 45.089/45.089 45.089/45.089 Infeasibility Conditions: 45.089/45.089 e(x:S:S) ->* true 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 Obtaining a proof using Prover9: 45.089/45.089 45.089/45.089 -> Prover9 Output: 45.089/45.089 ============================== Prover9 =============================== 45.089/45.089 Prover9 (64) version 2009-11A, November 2009. 45.089/45.089 Process 118545 was started by ubuntu on ubuntu, 45.089/45.089 Fri Jul 16 15:32:42 2021 45.089/45.089 The command was "./prover9 -f /tmp/prover9118538-0.in". 45.089/45.089 ============================== end of head =========================== 45.089/45.089 45.089/45.089 ============================== INPUT ================================= 45.089/45.089 45.089/45.089 % Reading from file /tmp/prover9118538-0.in 45.089/45.089 45.089/45.089 assign(max_seconds,20). 45.089/45.089 45.089/45.089 formulas(assumptions). 45.089/45.089 ->*_s0(x,x) # label(reflexivity). 45.089/45.089 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 45.089/45.089 ->_s0(x1,y) -> ->_s0(e(x1),e(y)) # label(congruence). 45.089/45.089 ->_s0(x1,y) -> ->_s0(o(x1),o(y)) # label(congruence). 45.089/45.089 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 45.089/45.089 ->_s0(e(0),true) # label(replacement). 45.089/45.089 ->*_s0(e(x2),true) -> ->_s0(e(s(x2)),false) # label(replacement). 45.089/45.089 ->*_s0(o(x2),true) -> ->_s0(e(s(x2)),true) # label(replacement). 45.089/45.089 ->_s0(o(0),false) # label(replacement). 45.089/45.089 ->*_s0(o(x2),true) -> ->_s0(o(s(x2)),false) # label(replacement). 45.089/45.089 ->*_s0(e(x2),true) -> ->_s0(o(s(x2)),true) # label(replacement). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(goals). 45.089/45.089 (exists x2 ->*_s0(e(x2),true)) # label(goal). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== end of input ========================== 45.089/45.089 45.089/45.089 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 45.089/45.089 45.089/45.089 % Formulas that are not ordinary clauses: 45.089/45.089 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 45.089/45.089 2 ->_s0(x1,y) -> ->_s0(e(x1),e(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 3 ->_s0(x1,y) -> ->_s0(o(x1),o(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 5 ->*_s0(e(x2),true) -> ->_s0(e(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 6 ->*_s0(o(x2),true) -> ->_s0(e(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 7 ->*_s0(o(x2),true) -> ->_s0(o(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 8 ->*_s0(e(x2),true) -> ->_s0(o(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 9 (exists x2 ->*_s0(e(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 45.089/45.089 45.089/45.089 ============================== end of process non-clausal formulas === 45.089/45.089 45.089/45.089 ============================== PROCESS INITIAL CLAUSES =============== 45.089/45.089 45.089/45.089 % Clauses before input processing: 45.089/45.089 45.089/45.089 formulas(usable). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(sos). 45.089/45.089 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(5)]. 45.089/45.089 -->*_s0(o(x),true) | ->_s0(e(s(x)),true) # label(replacement). [clausify(6)]. 45.089/45.089 ->_s0(o(0),false) # label(replacement). [assumption]. 45.089/45.089 -->*_s0(o(x),true) | ->_s0(o(s(x)),false) # label(replacement). [clausify(7)]. 45.089/45.089 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement). [clausify(8)]. 45.089/45.089 -->*_s0(e(x),true) # label(goal). [deny(9)]. 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(demodulators). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== PREDICATE ELIMINATION ================= 45.089/45.089 45.089/45.089 No predicates eliminated. 45.089/45.089 45.089/45.089 ============================== end predicate elimination ============= 45.089/45.089 45.089/45.089 Auto_denials: 45.089/45.089 % copying label goal to answer in negative clause 45.089/45.089 45.089/45.089 Term ordering decisions: 45.089/45.089 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 45.089/45.089 Function symbol precedence: function_order([ true, false, 0, e, o, s ]). 45.089/45.089 After inverse_order: (no changes). 45.089/45.089 Unfolding symbols: (none). 45.089/45.089 45.089/45.089 Auto_inference settings: 45.089/45.089 % set(neg_binary_resolution). % (HNE depth_diff=-7) 45.089/45.089 % clear(ordered_res). % (HNE depth_diff=-7) 45.089/45.089 % set(ur_resolution). % (HNE depth_diff=-7) 45.089/45.089 % set(ur_resolution) -> set(pos_ur_resolution). 45.089/45.089 % set(ur_resolution) -> set(neg_ur_resolution). 45.089/45.089 45.089/45.089 Auto_process settings: (no changes). 45.089/45.089 45.089/45.089 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 kept: 12 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 kept: 13 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 kept: 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 kept: 16 -->*_s0(e(x),true) | ->_s0(e(s(x)),false) # label(replacement). [clausify(5)]. 45.089/45.089 kept: 17 -->*_s0(o(x),true) | ->_s0(e(s(x)),true) # label(replacement). [clausify(6)]. 45.089/45.089 kept: 18 ->_s0(o(0),false) # label(replacement). [assumption]. 45.089/45.089 kept: 19 -->*_s0(o(x),true) | ->_s0(o(s(x)),false) # label(replacement). [clausify(7)]. 45.089/45.089 kept: 20 -->*_s0(e(x),true) | ->_s0(o(s(x)),true) # label(replacement). [clausify(8)]. 45.089/45.089 kept: 21 -->*_s0(e(x),true) # label(goal) # answer(goal). [deny(9)]. 45.089/45.089 45.089/45.089 ============================== end of process initial clauses ======== 45.089/45.089 45.089/45.089 ============================== CLAUSES FOR SEARCH ==================== 45.089/45.089 45.089/45.089 % Clauses after input processing: 45.089/45.089 45.089/45.089 formulas(usable). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(sos). 45.089/45.089 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 12 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 13 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 17 -->*_s0(o(x),true) | ->_s0(e(s(x)),true) # label(replacement). [clausify(6)]. 45.089/45.089 18 ->_s0(o(0),false) # label(replacement). [assumption]. 45.089/45.089 19 -->*_s0(o(x),true) | ->_s0(o(s(x)),false) # label(replacement). [clausify(7)]. 45.089/45.089 21 -->*_s0(e(x),true) # label(goal) # answer(goal). [deny(9)]. 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(demodulators). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== end of clauses for search ============= 45.089/45.089 45.089/45.089 ============================== SEARCH ================================ 45.089/45.089 45.089/45.089 % Starting search at 0.00 seconds. 45.089/45.089 45.089/45.089 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 45.089/45.089 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 45.089/45.089 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(e(x),e(y)) # label(congruence). [clausify(2)]. 45.089/45.089 45.089/45.089 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(o(x),o(y)) # label(congruence). [clausify(3)]. 45.089/45.089 45.089/45.089 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 45.089/45.089 45.089/45.089 given #6 (I,wt=4): 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 45.089/45.089 ============================== PROOF ================================= 45.089/45.089 45.089/45.089 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 45.089/45.089 % Length of proof is 8. 45.089/45.089 % Level of proof is 3. 45.089/45.089 % Maximum clause weight is 9.000. 45.089/45.089 % Given clauses 6. 45.089/45.089 45.089/45.089 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 45.089/45.089 9 (exists x2 ->*_s0(e(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 45.089/45.089 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 45.089/45.089 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 45.089/45.089 15 ->_s0(e(0),true) # label(replacement). [assumption]. 45.089/45.089 21 -->*_s0(e(x),true) # label(goal) # answer(goal). [deny(9)]. 45.089/45.089 25 ->*_s0(e(0),true). [ur(11,a,15,a,b,10,a)]. 45.089/45.089 26 $F # answer(goal). [resolve(25,a,21,a)]. 45.089/45.089 45.089/45.089 ============================== end of proof ========================== 45.089/45.089 45.089/45.089 ============================== STATISTICS ============================ 45.089/45.089 45.089/45.089 Given=6. Generated=16. Kept=16. proofs=1. 45.089/45.089 Usable=6. Sos=4. Demods=0. Limbo=3, Disabled=14. Hints=0. 45.089/45.089 Kept_by_rule=0, Deleted_by_rule=0. 45.089/45.089 Forward_subsumed=0. Back_subsumed=2. 45.089/45.089 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 45.089/45.089 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 45.089/45.089 Demod_attempts=0. Demod_rewrites=0. 45.089/45.089 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 45.089/45.089 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=11. 45.089/45.089 Megabytes=0.07. 45.089/45.089 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 45.089/45.089 45.089/45.089 ============================== end of statistics ===================== 45.089/45.089 45.089/45.089 ============================== end of search ========================= 45.089/45.089 45.089/45.089 THEOREM PROVED 45.089/45.089 45.089/45.089 Exiting with 1 proof. 45.089/45.089 45.089/45.089 Process 118545 exit (max_proofs) Fri Jul 16 15:32:42 2021 45.089/45.089 45.089/45.089 45.089/45.089 The problem is feasible. 45.089/45.089 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 45.089/45.089 Confluence Problem: 45.089/45.089 (VAR x:S) 45.089/45.089 (STRATEGY CONTEXTSENSITIVE 45.089/45.089 (e 1) 45.089/45.089 (o 1) 45.089/45.089 (0) 45.089/45.089 (fSNonEmpty) 45.089/45.089 (false) 45.089/45.089 (s 1) 45.089/45.089 (true) 45.089/45.089 ) 45.089/45.089 (RULES 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x:S)) -> false | e(x:S) ->* true 45.089/45.089 e(s(x:S)) -> true | o(x:S) ->* true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x:S)) -> false | o(x:S) ->* true 45.089/45.089 o(s(x:S)) -> true | e(x:S) ->* true 45.089/45.089 ) 45.089/45.089 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 45.089/45.089 45.089/45.089 Critical Pairs Processor: 45.089/45.089 -> Rules: 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x:S)) -> false | e(x:S) ->* true 45.089/45.089 e(s(x:S)) -> true | o(x:S) ->* true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x:S)) -> false | o(x:S) ->* true 45.089/45.089 o(s(x:S)) -> true | e(x:S) ->* true 45.089/45.089 -> Vars: 45.089/45.089 "x", "x", "x", "x" 45.089/45.089 -> FVars: 45.089/45.089 "x2", "x3", "x4", "x5" 45.089/45.089 -> PVars: 45.089/45.089 "x": ["x2", "x3", "x4", "x5"] 45.089/45.089 45.089/45.089 -> Rlps: 45.089/45.089 crule: e(0) -> true, id: 1, possubterms: e(0)-> [], 0-> [1] 45.089/45.089 crule: e(s(x2:S)) -> false | e(x2:S) ->* true, id: 2, possubterms: e(s(x2:S))-> [], s(x2:S)-> [1] 45.089/45.089 crule: e(s(x3:S)) -> true | o(x3:S) ->* true, id: 3, possubterms: e(s(x3:S))-> [], s(x3:S)-> [1] 45.089/45.089 crule: o(0) -> false, id: 4, possubterms: o(0)-> [], 0-> [1] 45.089/45.089 crule: o(s(x4:S)) -> false | o(x4:S) ->* true, id: 5, possubterms: o(s(x4:S))-> [], s(x4:S)-> [1] 45.089/45.089 crule: o(s(x5:S)) -> true | e(x5:S) ->* true, id: 6, possubterms: o(s(x5:S))-> [], s(x5:S)-> [1] 45.089/45.089 45.089/45.089 -> Unifications: 45.089/45.089 R3 unifies with R2 at p: [], l: e(s(x3:S)), lp: e(s(x3:S)), conds: {o(x3:S) ->* true, e(x:S) ->* true}, sig: {x3:S -> x:S}, l': e(s(x:S)), r: true, r': false 45.089/45.089 R6 unifies with R5 at p: [], l: o(s(x5:S)), lp: o(s(x5:S)), conds: {e(x5:S) ->* true, o(x:S) ->* true}, sig: {x5:S -> x:S}, l': o(s(x:S)), r: true, r': false 45.089/45.089 45.089/45.089 -> Critical pairs info: 45.089/45.089 | e(x:S) ->* true, o(x:S) ->* true => Not trivial, Overlay, N1 45.089/45.089 | o(x:S) ->* true, e(x:S) ->* true => Not trivial, Overlay, N2 45.089/45.089 45.089/45.089 -> Problem conclusions: 45.089/45.089 Left linear, Right linear, Linear 45.089/45.089 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 45.089/45.089 CTRS Type: 1 45.089/45.089 Deterministic, Strongly deterministic 45.089/45.089 Oriented CTRS, Properly oriented CTRS, Not join CTRS 45.089/45.089 Maybe right-stable CTRS, Overlay CTRS 45.089/45.089 Maybe normal CTRS, Maybe almost normal CTRS 45.089/45.089 Maybe terminating CTRS, Maybe joinable CCPs 45.089/45.089 Maybe level confluent 45.089/45.089 Maybe confluent 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 Clean Infeasible CCPs Processor: 45.089/45.089 Num of CPIs: 2 45.089/45.089 Timeout: 30 45.089/45.089 Timeout for each infeasibility problem: 30 s 45.089/45.089 | e(x:S) ->* true, o(x:S) ->* true => Not trivial, Overlay, N1 45.089/45.089 (PROBLEM INFEASIBILITY) 45.089/45.089 (VAR x) 45.089/45.089 (RULES 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x)) -> false | e(x) ->* true 45.089/45.089 e(s(x)) -> true | o(x) ->* true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x)) -> false | o(x) ->* true 45.089/45.089 o(s(x)) -> true | e(x) ->* true) 45.089/45.089 (VAR x2) 45.089/45.089 (CONDITION e(x2) ->* true, o(x2) ->* true) 45.089/45.089 45.089/45.089 Proof: 45.089/45.089 YES 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 Infeasibility Problem: 45.089/45.089 [(VAR vNonEmpty:S x:S x2:S) 45.089/45.089 (STRATEGY CONTEXTSENSITIVE 45.089/45.089 (e 1) 45.089/45.089 (o 1) 45.089/45.089 (0) 45.089/45.089 (fSNonEmpty) 45.089/45.089 (false) 45.089/45.089 (s 1) 45.089/45.089 (true) 45.089/45.089 ) 45.089/45.089 (RULES 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x:S)) -> false | e(x:S) ->* true 45.089/45.089 e(s(x:S)) -> true | o(x:S) ->* true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x:S)) -> false | o(x:S) ->* true 45.089/45.089 o(s(x:S)) -> true | e(x:S) ->* true 45.089/45.089 )] 45.089/45.089 45.089/45.089 Infeasibility Conditions: 45.089/45.089 e(x2:S) ->* true, o(x2:S) ->* true 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 Obtaining a model using Mace4: 45.089/45.089 45.089/45.089 -> Usable Rules: 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x:S)) -> false | e(x:S) ->* true 45.089/45.089 e(s(x:S)) -> true | o(x:S) ->* true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x:S)) -> false | o(x:S) ->* true 45.089/45.089 o(s(x:S)) -> true | e(x:S) ->* true 45.089/45.089 45.089/45.089 -> Mace4 Output: 45.089/45.089 ============================== Mace4 ================================= 45.089/45.089 Mace4 (64) version 2009-11A, November 2009. 45.089/45.089 Process 118589 was started by ubuntu on ubuntu, 45.089/45.089 Fri Jul 16 15:32:42 2021 45.089/45.089 The command was "./mace4 -c -f /tmp/mace4118563-2.in". 45.089/45.089 ============================== end of head =========================== 45.089/45.089 45.089/45.089 ============================== INPUT ================================= 45.089/45.089 45.089/45.089 % Reading from file /tmp/mace4118563-2.in 45.089/45.089 45.089/45.089 assign(max_seconds,10). 45.089/45.089 45.089/45.089 formulas(assumptions). 45.089/45.089 ->(x1,y) -> ->(e(x1),e(y)) # label(congruence). 45.089/45.089 ->(x1,y) -> ->(o(x1),o(y)) # label(congruence). 45.089/45.089 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 45.089/45.089 ->(e(0),true) # label(replacement). 45.089/45.089 ->*(e(x1),true) -> ->(e(s(x1)),false) # label(replacement). 45.089/45.089 ->*(o(x1),true) -> ->(e(s(x1)),true) # label(replacement). 45.089/45.089 ->(o(0),false) # label(replacement). 45.089/45.089 ->*(o(x1),true) -> ->(o(s(x1)),false) # label(replacement). 45.089/45.089 ->*(e(x1),true) -> ->(o(s(x1)),true) # label(replacement). 45.089/45.089 ->*(x,x) # label(reflexivity). 45.089/45.089 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(goals). 45.089/45.089 (exists x2 (->*(e(x2),true) & ->*(o(x2),true))) # label(goal). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== end of input ========================== 45.089/45.089 45.089/45.089 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 45.089/45.089 45.089/45.089 % Formulas that are not ordinary clauses: 45.089/45.089 1 ->(x1,y) -> ->(e(x1),e(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 2 ->(x1,y) -> ->(o(x1),o(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 3 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 4 ->*(e(x1),true) -> ->(e(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 5 ->*(o(x1),true) -> ->(e(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 6 ->*(o(x1),true) -> ->(o(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 7 ->*(e(x1),true) -> ->(o(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 8 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 45.089/45.089 9 (exists x2 (->*(e(x2),true) & ->*(o(x2),true))) # label(goal) # label(non_clause) # label(goal). [goal]. 45.089/45.089 45.089/45.089 ============================== end of process non-clausal formulas === 45.089/45.089 45.089/45.089 ============================== CLAUSES FOR SEARCH ==================== 45.089/45.089 45.089/45.089 formulas(mace4_clauses). 45.089/45.089 -->(x,y) | ->(e(x),e(y)) # label(congruence). 45.089/45.089 -->(x,y) | ->(o(x),o(y)) # label(congruence). 45.089/45.089 -->(x,y) | ->(s(x),s(y)) # label(congruence). 45.089/45.089 ->(e(0),true) # label(replacement). 45.089/45.089 -->*(e(x),true) | ->(e(s(x)),false) # label(replacement). 45.089/45.089 -->*(o(x),true) | ->(e(s(x)),true) # label(replacement). 45.089/45.089 ->(o(0),false) # label(replacement). 45.089/45.089 -->*(o(x),true) | ->(o(s(x)),false) # label(replacement). 45.089/45.089 -->*(e(x),true) | ->(o(s(x)),true) # label(replacement). 45.089/45.089 ->*(x,x) # label(reflexivity). 45.089/45.089 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 45.089/45.089 -->*(e(x),true) | -->*(o(x),true) # label(goal). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== end of clauses for search ============= 45.089/45.089 45.089/45.089 % There are no natural numbers in the input. 45.089/45.089 45.089/45.089 ============================== DOMAIN SIZE 2 ========================= 45.089/45.089 45.089/45.089 ============================== MODEL ================================= 45.089/45.089 45.089/45.089 interpretation( 2, [number=1, seconds=0], [ 45.089/45.089 45.089/45.089 function(0, [ 0 ]), 45.089/45.089 45.089/45.089 function(false, [ 0 ]), 45.089/45.089 45.089/45.089 function(true, [ 1 ]), 45.089/45.089 45.089/45.089 function(e(_), [ 1, 0 ]), 45.089/45.089 45.089/45.089 function(o(_), [ 0, 1 ]), 45.089/45.089 45.089/45.089 function(s(_), [ 1, 0 ]), 45.089/45.089 45.089/45.089 relation(->*(_,_), [ 45.089/45.089 1, 0, 45.089/45.089 0, 1 ]), 45.089/45.089 45.089/45.089 relation(->(_,_), [ 45.089/45.089 1, 0, 45.089/45.089 0, 1 ]) 45.089/45.089 ]). 45.089/45.089 45.089/45.089 ============================== end of model ========================== 45.089/45.089 45.089/45.089 ============================== STATISTICS ============================ 45.089/45.089 45.089/45.089 For domain size 2. 45.089/45.089 45.089/45.089 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 45.089/45.089 Ground clauses: seen=34, kept=30. 45.089/45.089 Selections=8, assignments=12, propagations=22, current_models=1. 45.089/45.089 Rewrite_terms=133, rewrite_bools=79, indexes=27. 45.089/45.089 Rules_from_neg_clauses=6, cross_offs=6. 45.089/45.089 45.089/45.089 ============================== end of statistics ===================== 45.089/45.089 45.089/45.089 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 45.089/45.089 45.089/45.089 Exiting with 1 model. 45.089/45.089 45.089/45.089 Process 118589 exit (max_models) Fri Jul 16 15:32:42 2021 45.089/45.089 The process finished Fri Jul 16 15:32:42 2021 45.089/45.089 45.089/45.089 45.089/45.089 Mace4 cooked interpretation: 45.089/45.089 45.089/45.089 45.089/45.089 45.089/45.089 The problem is infeasible. 45.089/45.089 45.089/45.089 45.089/45.089 45.089/45.089 45.089/45.089 | o(x:S) ->* true, e(x:S) ->* true => Not trivial, Overlay, N2 45.089/45.089 (PROBLEM INFEASIBILITY) 45.089/45.089 (VAR x) 45.089/45.089 (RULES 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x)) -> false | e(x) ->* true 45.089/45.089 e(s(x)) -> true | o(x) ->* true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x)) -> false | o(x) ->* true 45.089/45.089 o(s(x)) -> true | e(x) ->* true) 45.089/45.089 (VAR x2) 45.089/45.089 (CONDITION o(x2) ->* true, e(x2) ->* true) 45.089/45.089 45.089/45.089 Proof: 45.089/45.089 YES 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 Infeasibility Problem: 45.089/45.089 [(VAR vNonEmpty:S x:S x2:S) 45.089/45.089 (STRATEGY CONTEXTSENSITIVE 45.089/45.089 (e 1) 45.089/45.089 (o 1) 45.089/45.089 (0) 45.089/45.089 (fSNonEmpty) 45.089/45.089 (false) 45.089/45.089 (s 1) 45.089/45.089 (true) 45.089/45.089 ) 45.089/45.089 (RULES 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x:S)) -> false | e(x:S) ->* true 45.089/45.089 e(s(x:S)) -> true | o(x:S) ->* true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x:S)) -> false | o(x:S) ->* true 45.089/45.089 o(s(x:S)) -> true | e(x:S) ->* true 45.089/45.089 )] 45.089/45.089 45.089/45.089 Infeasibility Conditions: 45.089/45.089 o(x2:S) ->* true, e(x2:S) ->* true 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 Obtaining a model using Mace4: 45.089/45.089 45.089/45.089 -> Usable Rules: 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x:S)) -> false | e(x:S) ->* true 45.089/45.089 e(s(x:S)) -> true | o(x:S) ->* true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x:S)) -> false | o(x:S) ->* true 45.089/45.089 o(s(x:S)) -> true | e(x:S) ->* true 45.089/45.089 45.089/45.089 -> Mace4 Output: 45.089/45.089 ============================== Mace4 ================================= 45.089/45.089 Mace4 (64) version 2009-11A, November 2009. 45.089/45.089 Process 118627 was started by ubuntu on ubuntu, 45.089/45.089 Fri Jul 16 15:33:02 2021 45.089/45.089 The command was "./mace4 -c -f /tmp/mace4118614-2.in". 45.089/45.089 ============================== end of head =========================== 45.089/45.089 45.089/45.089 ============================== INPUT ================================= 45.089/45.089 45.089/45.089 % Reading from file /tmp/mace4118614-2.in 45.089/45.089 45.089/45.089 assign(max_seconds,10). 45.089/45.089 45.089/45.089 formulas(assumptions). 45.089/45.089 ->(x1,y) -> ->(e(x1),e(y)) # label(congruence). 45.089/45.089 ->(x1,y) -> ->(o(x1),o(y)) # label(congruence). 45.089/45.089 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 45.089/45.089 ->(e(0),true) # label(replacement). 45.089/45.089 ->*(e(x1),true) -> ->(e(s(x1)),false) # label(replacement). 45.089/45.089 ->*(o(x1),true) -> ->(e(s(x1)),true) # label(replacement). 45.089/45.089 ->(o(0),false) # label(replacement). 45.089/45.089 ->*(o(x1),true) -> ->(o(s(x1)),false) # label(replacement). 45.089/45.089 ->*(e(x1),true) -> ->(o(s(x1)),true) # label(replacement). 45.089/45.089 ->*(x,x) # label(reflexivity). 45.089/45.089 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 formulas(goals). 45.089/45.089 (exists x2 (->*(o(x2),true) & ->*(e(x2),true))) # label(goal). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== end of input ========================== 45.089/45.089 45.089/45.089 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 45.089/45.089 45.089/45.089 % Formulas that are not ordinary clauses: 45.089/45.089 1 ->(x1,y) -> ->(e(x1),e(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 2 ->(x1,y) -> ->(o(x1),o(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 3 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 45.089/45.089 4 ->*(e(x1),true) -> ->(e(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 5 ->*(o(x1),true) -> ->(e(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 6 ->*(o(x1),true) -> ->(o(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 7 ->*(e(x1),true) -> ->(o(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 45.089/45.089 8 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 45.089/45.089 9 (exists x2 (->*(o(x2),true) & ->*(e(x2),true))) # label(goal) # label(non_clause) # label(goal). [goal]. 45.089/45.089 45.089/45.089 ============================== end of process non-clausal formulas === 45.089/45.089 45.089/45.089 ============================== CLAUSES FOR SEARCH ==================== 45.089/45.089 45.089/45.089 formulas(mace4_clauses). 45.089/45.089 -->(x,y) | ->(e(x),e(y)) # label(congruence). 45.089/45.089 -->(x,y) | ->(o(x),o(y)) # label(congruence). 45.089/45.089 -->(x,y) | ->(s(x),s(y)) # label(congruence). 45.089/45.089 ->(e(0),true) # label(replacement). 45.089/45.089 -->*(e(x),true) | ->(e(s(x)),false) # label(replacement). 45.089/45.089 -->*(o(x),true) | ->(e(s(x)),true) # label(replacement). 45.089/45.089 ->(o(0),false) # label(replacement). 45.089/45.089 -->*(o(x),true) | ->(o(s(x)),false) # label(replacement). 45.089/45.089 -->*(e(x),true) | ->(o(s(x)),true) # label(replacement). 45.089/45.089 ->*(x,x) # label(reflexivity). 45.089/45.089 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 45.089/45.089 -->*(o(x),true) | -->*(e(x),true) # label(goal). 45.089/45.089 end_of_list. 45.089/45.089 45.089/45.089 ============================== end of clauses for search ============= 45.089/45.089 45.089/45.089 % There are no natural numbers in the input. 45.089/45.089 45.089/45.089 ============================== DOMAIN SIZE 2 ========================= 45.089/45.089 45.089/45.089 ============================== MODEL ================================= 45.089/45.089 45.089/45.089 interpretation( 2, [number=1, seconds=0], [ 45.089/45.089 45.089/45.089 function(0, [ 0 ]), 45.089/45.089 45.089/45.089 function(false, [ 0 ]), 45.089/45.089 45.089/45.089 function(true, [ 1 ]), 45.089/45.089 45.089/45.089 function(e(_), [ 1, 0 ]), 45.089/45.089 45.089/45.089 function(o(_), [ 0, 1 ]), 45.089/45.089 45.089/45.089 function(s(_), [ 1, 0 ]), 45.089/45.089 45.089/45.089 relation(->*(_,_), [ 45.089/45.089 1, 0, 45.089/45.089 0, 1 ]), 45.089/45.089 45.089/45.089 relation(->(_,_), [ 45.089/45.089 1, 0, 45.089/45.089 0, 1 ]) 45.089/45.089 ]). 45.089/45.089 45.089/45.089 ============================== end of model ========================== 45.089/45.089 45.089/45.089 ============================== STATISTICS ============================ 45.089/45.089 45.089/45.089 For domain size 2. 45.089/45.089 45.089/45.089 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 45.089/45.089 Ground clauses: seen=34, kept=30. 45.089/45.089 Selections=8, assignments=12, propagations=22, current_models=1. 45.089/45.089 Rewrite_terms=133, rewrite_bools=79, indexes=27. 45.089/45.089 Rules_from_neg_clauses=6, cross_offs=6. 45.089/45.089 45.089/45.089 ============================== end of statistics ===================== 45.089/45.089 45.089/45.089 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 45.089/45.089 45.089/45.089 Exiting with 1 model. 45.089/45.089 45.089/45.089 Process 118627 exit (max_models) Fri Jul 16 15:33:02 2021 45.089/45.089 The process finished Fri Jul 16 15:33:02 2021 45.089/45.089 45.089/45.089 45.089/45.089 Mace4 cooked interpretation: 45.089/45.089 45.089/45.089 45.089/45.089 45.089/45.089 The problem is infeasible. 45.089/45.089 45.089/45.089 45.089/45.089 45.089/45.089 45.089/45.089 -> Problem conclusions: 45.089/45.089 Left linear, Right linear, Linear 45.089/45.089 Weakly orthogonal, Almost orthogonal, Orthogonal 45.089/45.089 CTRS Type: 1 45.089/45.089 Deterministic, Strongly deterministic 45.089/45.089 Oriented CTRS, Properly oriented CTRS, Not join CTRS 45.089/45.089 Maybe right-stable CTRS, Overlay CTRS 45.089/45.089 Maybe normal CTRS, Maybe almost normal CTRS 45.089/45.089 Maybe terminating CTRS, Joinable CCPs 45.089/45.089 Maybe level confluent 45.089/45.089 Maybe confluent 45.089/45.089 45.089/45.089 Resulting CCPs: 45.089/45.089 No CCPs left 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 Underlying TRS Termination Processor: 45.089/45.089 45.089/45.089 Resulting Underlying TRS: 45.089/45.089 (VAR x:S) 45.089/45.089 (STRATEGY CONTEXTSENSITIVE 45.089/45.089 (e 1) 45.089/45.089 (o 1) 45.089/45.089 (false) 45.089/45.089 (true) 45.089/45.089 ) 45.089/45.089 (RULES 45.089/45.089 e(0) -> true 45.089/45.089 e(s(x:S)) -> false 45.089/45.089 e(s(x:S)) -> true 45.089/45.089 o(0) -> false 45.089/45.089 o(s(x:S)) -> false 45.089/45.089 o(s(x:S)) -> true 45.089/45.089 ) 45.089/45.089 Underlying TRS terminating? 45.089/45.089 YES 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 (VAR vu95NonEmpty:S xu58S:S) 45.089/45.089 (RULES 45.089/45.089 e(num0) -> ftrue 45.089/45.089 e(s(xu58S:S)) -> ffalse 45.089/45.089 e(s(xu58S:S)) -> ftrue 45.089/45.089 o(num0) -> ffalse 45.089/45.089 o(s(xu58S:S)) -> ffalse 45.089/45.089 o(s(xu58S:S)) -> ftrue 45.089/45.089 ) 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 Dependency Pairs Processor: 45.089/45.089 -> Pairs: 45.089/45.089 Empty 45.089/45.089 -> Rules: 45.089/45.089 e(num0) -> ftrue 45.089/45.089 e(s(xu58S:S)) -> ffalse 45.089/45.089 e(s(xu58S:S)) -> ftrue 45.089/45.089 o(num0) -> ffalse 45.089/45.089 o(s(xu58S:S)) -> ffalse 45.089/45.089 o(s(xu58S:S)) -> ftrue 45.089/45.089 45.089/45.089 Problem 1: 45.089/45.089 45.089/45.089 SCC Processor: 45.089/45.089 -> Pairs: 45.089/45.089 Empty 45.089/45.089 -> Rules: 45.089/45.089 e(num0) -> ftrue 45.089/45.089 e(s(xu58S:S)) -> ffalse 45.089/45.089 e(s(xu58S:S)) -> ftrue 45.089/45.089 o(num0) -> ffalse 45.089/45.089 o(s(xu58S:S)) -> ffalse 45.089/45.089 o(s(xu58S:S)) -> ftrue 45.089/45.089 ->Strongly Connected Components: 45.089/45.089 There is no strongly connected component 45.089/45.089 45.089/45.089 The problem is finite. 45.089/45.089 45.089/45.089 45.089/45.089 -> Problem conclusions: 45.089/45.089 Left linear, Right linear, Linear 45.089/45.089 Weakly orthogonal, Almost orthogonal, Orthogonal 45.089/45.089 CTRS Type: 1 45.089/45.089 Deterministic, Strongly deterministic 45.089/45.089 Oriented CTRS, Properly oriented CTRS, Not join CTRS 45.089/45.089 Maybe right-stable CTRS, Overlay CTRS 45.089/45.089 Maybe normal CTRS, Maybe almost normal CTRS 45.089/45.089 Terminating CTRS, Joinable CCPs 45.089/45.089 Maybe level confluent 45.089/45.089 Confluent 45.089/45.089 45.089/45.089 The problem is joinable. 45.089/45.089 73.22user 1.90system 0:45.89elapsed 163%CPU (0avgtext+0avgdata 138048maxresident)k 45.089/45.089 8inputs+0outputs (0major+173251minor)pagefaults 0swaps