46/46 YES 46/46 46/46 Problem 1: 46/46 46/46 46/46 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46/46 Confluence Problem: 46/46 (VAR vNonEmpty:S x:S) 46/46 (STRATEGY CONTEXTSENSITIVE 46/46 (even 1) 46/46 (odd 1) 46/46 (0) 46/46 (fSNonEmpty) 46/46 (false) 46/46 (s 1) 46/46 (true) 46/46 ) 46/46 (RULES 46/46 even(0) -> true 46/46 even(s(x:S)) -> false | even(x:S) ->* true 46/46 even(s(x:S)) -> true | odd(x:S) ->* true 46/46 odd(0) -> false 46/46 odd(s(x:S)) -> false | odd(x:S) ->* true 46/46 odd(s(x:S)) -> true | even(x:S) ->* true 46/46 ) 46/46 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46/46 46/46 46/46 Problem 1: 46/46 46/46 Inlining of Conditions Processor [STERN17]: 46/46 46/46 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46/46 Confluence Problem: 46/46 (VAR vNonEmpty:S x:S) 46/46 (STRATEGY CONTEXTSENSITIVE 46/46 (even 1) 46/46 (odd 1) 46/46 (0) 46/46 (fSNonEmpty) 46/46 (false) 46/46 (s 1) 46/46 (true) 46/46 ) 46/46 (RULES 46/46 even(0) -> true 46/46 even(s(x:S)) -> false | even(x:S) ->* true 46/46 even(s(x:S)) -> true | odd(x:S) ->* true 46/46 odd(0) -> false 46/46 odd(s(x:S)) -> false | odd(x:S) ->* true 46/46 odd(s(x:S)) -> true | even(x:S) ->* true 46/46 ) 46/46 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46/46 46/46 46/46 Problem 1: 46/46 46/46 Clean CTRS Processor: 46/46 46/46 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46/46 Confluence Problem: 46/46 (VAR x:S) 46/46 (STRATEGY CONTEXTSENSITIVE 46/46 (even 1) 46/46 (odd 1) 46/46 (0) 46/46 (fSNonEmpty) 46/46 (false) 46/46 (s 1) 46/46 (true) 46/46 ) 46/46 (RULES 46/46 even(0) -> true 46/46 even(s(x:S)) -> false | even(x:S) ->* true 46/46 even(s(x:S)) -> true | odd(x:S) ->* true 46/46 odd(0) -> false 46/46 odd(s(x:S)) -> false | odd(x:S) ->* true 46/46 odd(s(x:S)) -> true | even(x:S) ->* true 46/46 ) 46/46 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46/46 46/46 CRule InfChecker Info: 46/46 even(0) -> true 46/46 Rule remains 46/46 Proof: 46/46 NO_CONDS 46/46 46/46 CRule InfChecker Info: 46/46 even(s(x:S)) -> false | even(x:S) ->* true 46/46 Rule remains 46/46 Proof: 46/46 NO 46/46 46/46 Problem 1: 46/46 46/46 Infeasibility Problem: 46/46 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 46/46 (STRATEGY CONTEXTSENSITIVE 46/46 (even 1) 46/46 (odd 1) 46/46 (0) 46/46 (fSNonEmpty) 46/46 (false) 46/46 (s 1) 46/46 (true) 46/46 ) 46/46 (RULES 46/46 even(0) -> true 46/46 even(s(x:S:S)) -> false | even(x:S:S) ->* true 46/46 even(s(x:S:S)) -> true | odd(x:S:S) ->* true 46/46 odd(0) -> false 46/46 odd(s(x:S:S)) -> false | odd(x:S:S) ->* true 46/46 odd(s(x:S:S)) -> true | even(x:S:S) ->* true 46/46 )] 46/46 46/46 Infeasibility Conditions: 46/46 even(x:S:S) ->* true 46/46 46/46 Problem 1: 46/46 46/46 Obtaining a proof using Prover9: 46/46 46/46 -> Prover9 Output: 46/46 ============================== Prover9 =============================== 46/46 Prover9 (64) version 2009-11A, November 2009. 46/46 Process 116789 was started by ubuntu on ubuntu, 46/46 Fri Jul 16 15:28:06 2021 46/46 The command was "./prover9 -f /tmp/prover9116780-0.in". 46/46 ============================== end of head =========================== 46/46 46/46 ============================== INPUT ================================= 46/46 46/46 % Reading from file /tmp/prover9116780-0.in 46/46 46/46 assign(max_seconds,20). 46/46 46/46 formulas(assumptions). 46/46 ->*_s0(x,x) # label(reflexivity). 46/46 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 46/46 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence). 46/46 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence). 46/46 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 46/46 ->_s0(even(0),true) # label(replacement). 46/46 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement). 46/46 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement). 46/46 ->_s0(odd(0),false) # label(replacement). 46/46 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement). 46/46 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement). 46/46 end_of_list. 46/46 46/46 formulas(goals). 46/46 (exists x2 ->*_s0(even(x2),true)) # label(goal). 46/46 end_of_list. 46/46 46/46 ============================== end of input ========================== 46/46 46/46 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46/46 46/46 % Formulas that are not ordinary clauses: 46/46 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46/46 2 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 3 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 5 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46/46 6 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46/46 7 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46/46 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46/46 9 (exists x2 ->*_s0(even(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46/46 46/46 ============================== end of process non-clausal formulas === 46/46 46/46 ============================== PROCESS INITIAL CLAUSES =============== 46/46 46/46 % Clauses before input processing: 46/46 46/46 formulas(usable). 46/46 end_of_list. 46/46 46/46 formulas(sos). 46/46 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46/46 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46/46 ->_s0(odd(0),false) # label(replacement). [assumption]. 46/46 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46/46 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46/46 -->*_s0(even(x),true) # label(goal). [deny(9)]. 46/46 end_of_list. 46/46 46/46 formulas(demodulators). 46/46 end_of_list. 46/46 46/46 ============================== PREDICATE ELIMINATION ================= 46/46 46/46 No predicates eliminated. 46/46 46/46 ============================== end predicate elimination ============= 46/46 46/46 Auto_denials: 46/46 % copying label goal to answer in negative clause 46/46 46/46 Term ordering decisions: 46/46 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 46/46 Function symbol precedence: function_order([ true, false, 0, even, odd, s ]). 46/46 After inverse_order: (no changes). 46/46 Unfolding symbols: (none). 46/46 46/46 Auto_inference settings: 46/46 % set(neg_binary_resolution). % (HNE depth_diff=-7) 46/46 % clear(ordered_res). % (HNE depth_diff=-7) 46/46 % set(ur_resolution). % (HNE depth_diff=-7) 46/46 % set(ur_resolution) -> set(pos_ur_resolution). 46/46 % set(ur_resolution) -> set(neg_ur_resolution). 46/46 46/46 Auto_process settings: (no changes). 46/46 46/46 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 kept: 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 kept: 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 kept: 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 kept: 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46/46 kept: 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46/46 kept: 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46/46 kept: 19 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46/46 kept: 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46/46 kept: 21 -->*_s0(even(x),true) # label(goal) # answer(goal). [deny(9)]. 46/46 46/46 ============================== end of process initial clauses ======== 46/46 46/46 ============================== CLAUSES FOR SEARCH ==================== 46/46 46/46 % Clauses after input processing: 46/46 46/46 formulas(usable). 46/46 end_of_list. 46/46 46/46 formulas(sos). 46/46 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46/46 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46/46 19 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46/46 21 -->*_s0(even(x),true) # label(goal) # answer(goal). [deny(9)]. 46/46 end_of_list. 46/46 46/46 formulas(demodulators). 46/46 end_of_list. 46/46 46/46 ============================== end of clauses for search ============= 46/46 46/46 ============================== SEARCH ================================ 46/46 46/46 % Starting search at 0.00 seconds. 46/46 46/46 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 46/46 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 46/46 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 46/46 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 46/46 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 46/46 given #6 (I,wt=4): 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 46/46 ============================== PROOF ================================= 46/46 46/46 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 46/46 % Length of proof is 8. 46/46 % Level of proof is 3. 46/46 % Maximum clause weight is 9.000. 46/46 % Given clauses 6. 46/46 46/46 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46/46 9 (exists x2 ->*_s0(even(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46/46 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 21 -->*_s0(even(x),true) # label(goal) # answer(goal). [deny(9)]. 46/46 25 ->*_s0(even(0),true). [ur(11,a,15,a,b,10,a)]. 46/46 26 $F # answer(goal). [resolve(25,a,21,a)]. 46/46 46/46 ============================== end of proof ========================== 46/46 46/46 ============================== STATISTICS ============================ 46/46 46/46 Given=6. Generated=16. Kept=16. proofs=1. 46/46 Usable=6. Sos=4. Demods=0. Limbo=3, Disabled=14. Hints=0. 46/46 Kept_by_rule=0, Deleted_by_rule=0. 46/46 Forward_subsumed=0. Back_subsumed=2. 46/46 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 46/46 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 46/46 Demod_attempts=0. Demod_rewrites=0. 46/46 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 46/46 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=11. 46/46 Megabytes=0.07. 46/46 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46/46 46/46 ============================== end of statistics ===================== 46/46 46/46 ============================== end of search ========================= 46/46 46/46 THEOREM PROVED 46/46 46/46 Exiting with 1 proof. 46/46 46/46 Process 116789 exit (max_proofs) Fri Jul 16 15:28:06 2021 46/46 46/46 46/46 The problem is feasible. 46/46 46/46 46/46 CRule InfChecker Info: 46/46 even(s(x:S)) -> true | odd(x:S) ->* true 46/46 Rule remains 46/46 Proof: 46/46 NO 46/46 46/46 Problem 1: 46/46 46/46 Infeasibility Problem: 46/46 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 46/46 (STRATEGY CONTEXTSENSITIVE 46/46 (even 1) 46/46 (odd 1) 46/46 (0) 46/46 (fSNonEmpty) 46/46 (false) 46/46 (s 1) 46/46 (true) 46/46 ) 46/46 (RULES 46/46 even(0) -> true 46/46 even(s(x:S:S)) -> false | even(x:S:S) ->* true 46/46 even(s(x:S:S)) -> true | odd(x:S:S) ->* true 46/46 odd(0) -> false 46/46 odd(s(x:S:S)) -> false | odd(x:S:S) ->* true 46/46 odd(s(x:S:S)) -> true | even(x:S:S) ->* true 46/46 )] 46/46 46/46 Infeasibility Conditions: 46/46 odd(x:S:S) ->* true 46/46 46/46 Problem 1: 46/46 46/46 Obtaining a proof using Prover9: 46/46 46/46 -> Prover9 Output: 46/46 ============================== Prover9 =============================== 46/46 Prover9 (64) version 2009-11A, November 2009. 46/46 Process 116811 was started by ubuntu on ubuntu, 46/46 Fri Jul 16 15:28:06 2021 46/46 The command was "./prover9 -f /tmp/prover9116801-0.in". 46/46 ============================== end of head =========================== 46/46 46/46 ============================== INPUT ================================= 46/46 46/46 % Reading from file /tmp/prover9116801-0.in 46/46 46/46 assign(max_seconds,20). 46/46 46/46 formulas(assumptions). 46/46 ->*_s0(x,x) # label(reflexivity). 46/46 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 46/46 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence). 46/46 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence). 46/46 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 46/46 ->_s0(even(0),true) # label(replacement). 46/46 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement). 46/46 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement). 46/46 ->_s0(odd(0),false) # label(replacement). 46/46 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement). 46/46 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement). 46/46 end_of_list. 46/46 46/46 formulas(goals). 46/46 (exists x2 ->*_s0(odd(x2),true)) # label(goal). 46/46 end_of_list. 46/46 46/46 ============================== end of input ========================== 46/46 46/46 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46/46 46/46 % Formulas that are not ordinary clauses: 46/46 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46/46 2 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 3 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 5 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46/46 6 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46/46 7 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46/46 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46/46 9 (exists x2 ->*_s0(odd(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46/46 46/46 ============================== end of process non-clausal formulas === 46/46 46/46 ============================== PROCESS INITIAL CLAUSES =============== 46/46 46/46 % Clauses before input processing: 46/46 46/46 formulas(usable). 46/46 end_of_list. 46/46 46/46 formulas(sos). 46/46 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46/46 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46/46 ->_s0(odd(0),false) # label(replacement). [assumption]. 46/46 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46/46 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46/46 -->*_s0(odd(x),true) # label(goal). [deny(9)]. 46/46 end_of_list. 46/46 46/46 formulas(demodulators). 46/46 end_of_list. 46/46 46/46 ============================== PREDICATE ELIMINATION ================= 46/46 46/46 No predicates eliminated. 46/46 46/46 ============================== end predicate elimination ============= 46/46 46/46 Auto_denials: 46/46 % copying label goal to answer in negative clause 46/46 46/46 Term ordering decisions: 46/46 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 46/46 Function symbol precedence: function_order([ true, false, 0, even, odd, s ]). 46/46 After inverse_order: (no changes). 46/46 Unfolding symbols: (none). 46/46 46/46 Auto_inference settings: 46/46 % set(neg_binary_resolution). % (HNE depth_diff=-7) 46/46 % clear(ordered_res). % (HNE depth_diff=-7) 46/46 % set(ur_resolution). % (HNE depth_diff=-7) 46/46 % set(ur_resolution) -> set(pos_ur_resolution). 46/46 % set(ur_resolution) -> set(neg_ur_resolution). 46/46 46/46 Auto_process settings: (no changes). 46/46 46/46 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 kept: 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 kept: 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 kept: 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 kept: 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46/46 kept: 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46/46 kept: 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46/46 kept: 19 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46/46 kept: 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46/46 kept: 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46/46 46/46 ============================== end of process initial clauses ======== 46/46 46/46 ============================== CLAUSES FOR SEARCH ==================== 46/46 46/46 % Clauses after input processing: 46/46 46/46 formulas(usable). 46/46 end_of_list. 46/46 46/46 formulas(sos). 46/46 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46/46 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46/46 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46/46 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46/46 end_of_list. 46/46 46/46 formulas(demodulators). 46/46 end_of_list. 46/46 46/46 ============================== end of clauses for search ============= 46/46 46/46 ============================== SEARCH ================================ 46/46 46/46 % Starting search at 0.00 seconds. 46/46 46/46 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 46/46 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 46/46 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 46/46 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 46/46 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 46/46 given #6 (I,wt=4): 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 46/46 given #7 (I,wt=9): 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46/46 46/46 given #8 (I,wt=4): 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46/46 46/46 given #9 (I,wt=9): 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46/46 46/46 given #10 (I,wt=4): 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46/46 46/46 given #11 (A,wt=6): 22 ->_s0(s(even(0)),s(true)). [ur(14,a,15,a)]. 46/46 46/46 given #12 (F,wt=3): 32 -->*_s0(false,true) # answer(goal). [ur(11,a,18,a,c,21,a)]. 46/46 46/46 given #13 (F,wt=3): 38 -->_s0(false,true) # answer(goal). [ur(11,b,10,a,c,32,a)]. 46/46 46/46 given #14 (F,wt=4): 31 -->_s0(odd(x),true) # answer(goal). [ur(11,b,10,a,c,21,a)]. 46/46 46/46 ============================== PROOF ================================= 46/46 46/46 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 46/46 % Length of proof is 12. 46/46 % Level of proof is 4. 46/46 % Maximum clause weight is 9.000. 46/46 % Given clauses 14. 46/46 46/46 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46/46 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46/46 9 (exists x2 ->*_s0(odd(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46/46 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46/46 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46/46 25 ->*_s0(even(0),true). [ur(11,a,15,a,b,10,a)]. 46/46 31 -->_s0(odd(x),true) # answer(goal). [ur(11,b,10,a,c,21,a)]. 46/46 39 -->*_s0(even(x),true) # answer(goal). [resolve(31,a,20,b)]. 46/46 40 $F # answer(goal). [resolve(39,a,25,a)]. 46/46 46/46 ============================== end of proof ========================== 46/46 46/46 ============================== STATISTICS ============================ 46/46 46/46 Given=14. Generated=30. Kept=30. proofs=1. 46/46 Usable=14. Sos=13. Demods=0. Limbo=0, Disabled=14. Hints=0. 46/46 Kept_by_rule=0, Deleted_by_rule=0. 46/46 Forward_subsumed=0. Back_subsumed=2. 46/46 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 46/46 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 46/46 Demod_attempts=0. Demod_rewrites=0. 46/46 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 46/46 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=15. 46/46 Megabytes=0.09. 46/46 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46/46 46/46 ============================== end of statistics ===================== 46/46 46/46 ============================== end of search ========================= 46/46 46/46 THEOREM PROVED 46/46 46/46 Exiting with 1 proof. 46/46 46/46 Process 116811 exit (max_proofs) Fri Jul 16 15:28:06 2021 46/46 46/46 46/46 The problem is feasible. 46/46 46/46 46/46 CRule InfChecker Info: 46/46 odd(0) -> false 46/46 Rule remains 46/46 Proof: 46/46 NO_CONDS 46/46 46/46 CRule InfChecker Info: 46/46 odd(s(x:S)) -> false | odd(x:S) ->* true 46/46 Rule remains 46/46 Proof: 46/46 NO 46/46 46/46 Problem 1: 46/46 46/46 Infeasibility Problem: 46/46 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 46/46 (STRATEGY CONTEXTSENSITIVE 46/46 (even 1) 46/46 (odd 1) 46/46 (0) 46/46 (fSNonEmpty) 46/46 (false) 46/46 (s 1) 46/46 (true) 46/46 ) 46/46 (RULES 46/46 even(0) -> true 46/46 even(s(x:S:S)) -> false | even(x:S:S) ->* true 46/46 even(s(x:S:S)) -> true | odd(x:S:S) ->* true 46/46 odd(0) -> false 46/46 odd(s(x:S:S)) -> false | odd(x:S:S) ->* true 46/46 odd(s(x:S:S)) -> true | even(x:S:S) ->* true 46/46 )] 46/46 46/46 Infeasibility Conditions: 46/46 odd(x:S:S) ->* true 46/46 46/46 Problem 1: 46/46 46/46 Obtaining a proof using Prover9: 46/46 46/46 -> Prover9 Output: 46/46 ============================== Prover9 =============================== 46/46 Prover9 (64) version 2009-11A, November 2009. 46/46 Process 116843 was started by ubuntu on ubuntu, 46/46 Fri Jul 16 15:28:06 2021 46/46 The command was "./prover9 -f /tmp/prover9116824-0.in". 46/46 ============================== end of head =========================== 46/46 46/46 ============================== INPUT ================================= 46/46 46/46 % Reading from file /tmp/prover9116824-0.in 46/46 46/46 assign(max_seconds,20). 46/46 46/46 formulas(assumptions). 46/46 ->*_s0(x,x) # label(reflexivity). 46/46 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 46/46 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence). 46/46 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence). 46/46 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 46/46 ->_s0(even(0),true) # label(replacement). 46/46 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement). 46/46 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement). 46/46 ->_s0(odd(0),false) # label(replacement). 46/46 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement). 46/46 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement). 46/46 end_of_list. 46/46 46/46 formulas(goals). 46/46 (exists x2 ->*_s0(odd(x2),true)) # label(goal). 46/46 end_of_list. 46/46 46/46 ============================== end of input ========================== 46/46 46/46 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46/46 46/46 % Formulas that are not ordinary clauses: 46/46 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46/46 2 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 3 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 5 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46/46 6 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46/46 7 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46/46 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46/46 9 (exists x2 ->*_s0(odd(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46/46 46/46 ============================== end of process non-clausal formulas === 46/46 46/46 ============================== PROCESS INITIAL CLAUSES =============== 46/46 46/46 % Clauses before input processing: 46/46 46/46 formulas(usable). 46/46 end_of_list. 46/46 46/46 formulas(sos). 46/46 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46/46 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46/46 ->_s0(odd(0),false) # label(replacement). [assumption]. 46/46 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46/46 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46/46 -->*_s0(odd(x),true) # label(goal). [deny(9)]. 46/46 end_of_list. 46/46 46/46 formulas(demodulators). 46/46 end_of_list. 46/46 46/46 ============================== PREDICATE ELIMINATION ================= 46/46 46/46 No predicates eliminated. 46/46 46/46 ============================== end predicate elimination ============= 46/46 46/46 Auto_denials: 46/46 % copying label goal to answer in negative clause 46/46 46/46 Term ordering decisions: 46/46 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 46/46 Function symbol precedence: function_order([ true, false, 0, even, odd, s ]). 46/46 After inverse_order: (no changes). 46/46 Unfolding symbols: (none). 46/46 46/46 Auto_inference settings: 46/46 % set(neg_binary_resolution). % (HNE depth_diff=-7) 46/46 % clear(ordered_res). % (HNE depth_diff=-7) 46/46 % set(ur_resolution). % (HNE depth_diff=-7) 46/46 % set(ur_resolution) -> set(pos_ur_resolution). 46/46 % set(ur_resolution) -> set(neg_ur_resolution). 46/46 46/46 Auto_process settings: (no changes). 46/46 46/46 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 kept: 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 kept: 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 kept: 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 kept: 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46/46 kept: 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46/46 kept: 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46/46 kept: 19 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46/46 kept: 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46/46 kept: 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46/46 46/46 ============================== end of process initial clauses ======== 46/46 46/46 ============================== CLAUSES FOR SEARCH ==================== 46/46 46/46 % Clauses after input processing: 46/46 46/46 formulas(usable). 46/46 end_of_list. 46/46 46/46 formulas(sos). 46/46 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46/46 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46/46 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46/46 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46/46 end_of_list. 46/46 46/46 formulas(demodulators). 46/46 end_of_list. 46/46 46/46 ============================== end of clauses for search ============= 46/46 46/46 ============================== SEARCH ================================ 46/46 46/46 % Starting search at 0.00 seconds. 46/46 46/46 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 46/46 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 46/46 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 46/46 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 46/46 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 46/46 given #6 (I,wt=4): 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 46/46 given #7 (I,wt=9): 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46/46 46/46 given #8 (I,wt=4): 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46/46 46/46 given #9 (I,wt=9): 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46/46 46/46 given #10 (I,wt=4): 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46/46 46/46 given #11 (A,wt=6): 22 ->_s0(s(even(0)),s(true)). [ur(14,a,15,a)]. 46/46 46/46 given #12 (F,wt=3): 32 -->*_s0(false,true) # answer(goal). [ur(11,a,18,a,c,21,a)]. 46/46 46/46 given #13 (F,wt=3): 38 -->_s0(false,true) # answer(goal). [ur(11,b,10,a,c,32,a)]. 46/46 46/46 given #14 (F,wt=4): 31 -->_s0(odd(x),true) # answer(goal). [ur(11,b,10,a,c,21,a)]. 46/46 46/46 ============================== PROOF ================================= 46/46 46/46 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 46/46 % Length of proof is 12. 46/46 % Level of proof is 4. 46/46 % Maximum clause weight is 9.000. 46/46 % Given clauses 14. 46/46 46/46 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46/46 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46/46 9 (exists x2 ->*_s0(odd(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46/46 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46/46 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46/46 25 ->*_s0(even(0),true). [ur(11,a,15,a,b,10,a)]. 46/46 31 -->_s0(odd(x),true) # answer(goal). [ur(11,b,10,a,c,21,a)]. 46/46 39 -->*_s0(even(x),true) # answer(goal). [resolve(31,a,20,b)]. 46/46 40 $F # answer(goal). [resolve(39,a,25,a)]. 46/46 46/46 ============================== end of proof ========================== 46/46 46/46 ============================== STATISTICS ============================ 46/46 46/46 Given=14. Generated=30. Kept=30. proofs=1. 46/46 Usable=14. Sos=13. Demods=0. Limbo=0, Disabled=14. Hints=0. 46/46 Kept_by_rule=0, Deleted_by_rule=0. 46/46 Forward_subsumed=0. Back_subsumed=2. 46/46 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 46/46 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 46/46 Demod_attempts=0. Demod_rewrites=0. 46/46 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 46/46 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=15. 46/46 Megabytes=0.09. 46/46 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46/46 46/46 ============================== end of statistics ===================== 46/46 46/46 ============================== end of search ========================= 46/46 46/46 THEOREM PROVED 46/46 46/46 Exiting with 1 proof. 46/46 46/46 Process 116843 exit (max_proofs) Fri Jul 16 15:28:06 2021 46/46 46/46 46/46 The problem is feasible. 46/46 46/46 46/46 CRule InfChecker Info: 46/46 odd(s(x:S)) -> true | even(x:S) ->* true 46/46 Rule remains 46/46 Proof: 46/46 NO 46/46 46/46 Problem 1: 46/46 46/46 Infeasibility Problem: 46/46 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 46/46 (STRATEGY CONTEXTSENSITIVE 46/46 (even 1) 46/46 (odd 1) 46/46 (0) 46/46 (fSNonEmpty) 46/46 (false) 46/46 (s 1) 46/46 (true) 46/46 ) 46/46 (RULES 46/46 even(0) -> true 46/46 even(s(x:S:S)) -> false | even(x:S:S) ->* true 46/46 even(s(x:S:S)) -> true | odd(x:S:S) ->* true 46/46 odd(0) -> false 46/46 odd(s(x:S:S)) -> false | odd(x:S:S) ->* true 46/46 odd(s(x:S:S)) -> true | even(x:S:S) ->* true 46/46 )] 46/46 46/46 Infeasibility Conditions: 46/46 even(x:S:S) ->* true 46/46 46/46 Problem 1: 46/46 46/46 Obtaining a proof using Prover9: 46/46 46/46 -> Prover9 Output: 46/46 ============================== Prover9 =============================== 46/46 Prover9 (64) version 2009-11A, November 2009. 46/46 Process 116859 was started by ubuntu on ubuntu, 46/46 Fri Jul 16 15:28:06 2021 46/46 The command was "./prover9 -f /tmp/prover9116846-0.in". 46/46 ============================== end of head =========================== 46/46 46/46 ============================== INPUT ================================= 46/46 46/46 % Reading from file /tmp/prover9116846-0.in 46/46 46/46 assign(max_seconds,20). 46/46 46/46 formulas(assumptions). 46/46 ->*_s0(x,x) # label(reflexivity). 46/46 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 46/46 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence). 46/46 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence). 46/46 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 46/46 ->_s0(even(0),true) # label(replacement). 46/46 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement). 46/46 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement). 46/46 ->_s0(odd(0),false) # label(replacement). 46/46 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement). 46/46 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement). 46/46 end_of_list. 46/46 46/46 formulas(goals). 46/46 (exists x2 ->*_s0(even(x2),true)) # label(goal). 46/46 end_of_list. 46/46 46/46 ============================== end of input ========================== 46/46 46/46 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46/46 46/46 % Formulas that are not ordinary clauses: 46/46 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46/46 2 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 3 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 5 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46/46 6 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46/46 7 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46/46 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46/46 9 (exists x2 ->*_s0(even(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46/46 46/46 ============================== end of process non-clausal formulas === 46/46 46/46 ============================== PROCESS INITIAL CLAUSES =============== 46/46 46/46 % Clauses before input processing: 46/46 46/46 formulas(usable). 46/46 end_of_list. 46/46 46/46 formulas(sos). 46/46 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46/46 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46/46 ->_s0(odd(0),false) # label(replacement). [assumption]. 46/46 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46/46 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46/46 -->*_s0(even(x),true) # label(goal). [deny(9)]. 46/46 end_of_list. 46/46 46/46 formulas(demodulators). 46/46 end_of_list. 46/46 46/46 ============================== PREDICATE ELIMINATION ================= 46/46 46/46 No predicates eliminated. 46/46 46/46 ============================== end predicate elimination ============= 46/46 46/46 Auto_denials: 46/46 % copying label goal to answer in negative clause 46/46 46/46 Term ordering decisions: 46/46 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 46/46 Function symbol precedence: function_order([ true, false, 0, even, odd, s ]). 46/46 After inverse_order: (no changes). 46/46 Unfolding symbols: (none). 46/46 46/46 Auto_inference settings: 46/46 % set(neg_binary_resolution). % (HNE depth_diff=-7) 46/46 % clear(ordered_res). % (HNE depth_diff=-7) 46/46 % set(ur_resolution). % (HNE depth_diff=-7) 46/46 % set(ur_resolution) -> set(pos_ur_resolution). 46/46 % set(ur_resolution) -> set(neg_ur_resolution). 46/46 46/46 Auto_process settings: (no changes). 46/46 46/46 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 kept: 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 kept: 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 kept: 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 kept: 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46/46 kept: 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46/46 kept: 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46/46 kept: 19 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46/46 kept: 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46/46 kept: 21 -->*_s0(even(x),true) # label(goal) # answer(goal). [deny(9)]. 46/46 46/46 ============================== end of process initial clauses ======== 46/46 46/46 ============================== CLAUSES FOR SEARCH ==================== 46/46 46/46 % Clauses after input processing: 46/46 46/46 formulas(usable). 46/46 end_of_list. 46/46 46/46 formulas(sos). 46/46 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46/46 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46/46 19 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46/46 21 -->*_s0(even(x),true) # label(goal) # answer(goal). [deny(9)]. 46/46 end_of_list. 46/46 46/46 formulas(demodulators). 46/46 end_of_list. 46/46 46/46 ============================== end of clauses for search ============= 46/46 46/46 ============================== SEARCH ================================ 46/46 46/46 % Starting search at 0.00 seconds. 46/46 46/46 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 46/46 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 46/46 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46/46 46/46 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46/46 46/46 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46/46 46/46 given #6 (I,wt=4): 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 46/46 ============================== PROOF ================================= 46/46 46/46 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 46/46 % Length of proof is 8. 46/46 % Level of proof is 3. 46/46 % Maximum clause weight is 9.000. 46/46 % Given clauses 6. 46/46 46/46 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46/46 9 (exists x2 ->*_s0(even(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46/46 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46/46 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46/46 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46/46 21 -->*_s0(even(x),true) # label(goal) # answer(goal). [deny(9)]. 46/46 25 ->*_s0(even(0),true). [ur(11,a,15,a,b,10,a)]. 46/46 26 $F # answer(goal). [resolve(25,a,21,a)]. 46/46 46/46 ============================== end of proof ========================== 46/46 46/46 ============================== STATISTICS ============================ 46/46 46/46 Given=6. Generated=16. Kept=16. proofs=1. 46/46 Usable=6. Sos=4. Demods=0. Limbo=3, Disabled=14. Hints=0. 46/46 Kept_by_rule=0, Deleted_by_rule=0. 46/46 Forward_subsumed=0. Back_subsumed=2. 46/46 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 46/46 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 46/46 Demod_attempts=0. Demod_rewrites=0. 46/46 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 46/46 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=11. 46/46 Megabytes=0.07. 46/46 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46/46 46/46 ============================== end of statistics ===================== 46/46 46/46 ============================== end of search ========================= 46/46 46/46 THEOREM PROVED 46/46 46/46 Exiting with 1 proof. 46/46 46/46 Process 116859 exit (max_proofs) Fri Jul 16 15:28:06 2021 46/46 46/46 46/46 The problem is feasible. 46/46 46/46 46/46 Problem 1: 46/46 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46/46 Confluence Problem: 46/46 (VAR x:S) 46/46 (STRATEGY CONTEXTSENSITIVE 46/46 (even 1) 46/46 (odd 1) 46/46 (0) 46/46 (fSNonEmpty) 46/46 (false) 46/46 (s 1) 46/46 (true) 46/46 ) 46/46 (RULES 46/46 even(0) -> true 46/46 even(s(x:S)) -> false | even(x:S) ->* true 46/46 even(s(x:S)) -> true | odd(x:S) ->* true 46/46 odd(0) -> false 46/46 odd(s(x:S)) -> false | odd(x:S) ->* true 46/46 odd(s(x:S)) -> true | even(x:S) ->* true 46/46 ) 46/46 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46/46 46/46 Critical Pairs Processor: 46/46 -> Rules: 46/46 even(0) -> true 46/46 even(s(x:S)) -> false | even(x:S) ->* true 46/46 even(s(x:S)) -> true | odd(x:S) ->* true 46/46 odd(0) -> false 46/46 odd(s(x:S)) -> false | odd(x:S) ->* true 46/46 odd(s(x:S)) -> true | even(x:S) ->* true 46/46 -> Vars: 46/46 "x", "x", "x", "x" 46/46 -> FVars: 46/46 "x2", "x3", "x4", "x5" 46/46 -> PVars: 46/46 "x": ["x2", "x3", "x4", "x5"] 46/46 46/46 -> Rlps: 46/46 crule: even(0) -> true, id: 1, possubterms: even(0)-> [], 0-> [1] 46/46 crule: even(s(x2:S)) -> false | even(x2:S) ->* true, id: 2, possubterms: even(s(x2:S))-> [], s(x2:S)-> [1] 46/46 crule: even(s(x3:S)) -> true | odd(x3:S) ->* true, id: 3, possubterms: even(s(x3:S))-> [], s(x3:S)-> [1] 46/46 crule: odd(0) -> false, id: 4, possubterms: odd(0)-> [], 0-> [1] 46/46 crule: odd(s(x4:S)) -> false | odd(x4:S) ->* true, id: 5, possubterms: odd(s(x4:S))-> [], s(x4:S)-> [1] 46/46 crule: odd(s(x5:S)) -> true | even(x5:S) ->* true, id: 6, possubterms: odd(s(x5:S))-> [], s(x5:S)-> [1] 46/46 46/46 -> Unifications: 46/46 R3 unifies with R2 at p: [], l: even(s(x3:S)), lp: even(s(x3:S)), conds: {odd(x3:S) ->* true, even(x:S) ->* true}, sig: {x3:S -> x:S}, l': even(s(x:S)), r: true, r': false 46/46 R6 unifies with R5 at p: [], l: odd(s(x5:S)), lp: odd(s(x5:S)), conds: {even(x5:S) ->* true, odd(x:S) ->* true}, sig: {x5:S -> x:S}, l': odd(s(x:S)), r: true, r': false 46/46 46/46 -> Critical pairs info: 46/46 | even(x:S) ->* true, odd(x:S) ->* true => Not trivial, Overlay, N1 46/46 | odd(x:S) ->* true, even(x:S) ->* true => Not trivial, Overlay, N2 46/46 46/46 -> Problem conclusions: 46/46 Left linear, Right linear, Linear 46/46 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 46/46 CTRS Type: 1 46/46 Deterministic, Strongly deterministic 46/46 Oriented CTRS, Properly oriented CTRS, Not join CTRS 46/46 Maybe right-stable CTRS, Overlay CTRS 46/46 Maybe normal CTRS, Maybe almost normal CTRS 46/46 Maybe terminating CTRS, Maybe joinable CCPs 46/46 Maybe level confluent 46/46 Maybe confluent 46/46 46/46 Problem 1: 46/46 Clean Infeasible CCPs Processor: 46/46 Num of CPIs: 2 46/46 Timeout: 30 46/46 Timeout for each infeasibility problem: 30 s 46/46 | even(x:S) ->* true, odd(x:S) ->* true => Not trivial, Overlay, N1 46/46 (PROBLEM INFEASIBILITY) 46/46 (VAR x) 46/46 (RULES 46/46 even(0) -> true 46/46 even(s(x)) -> false | even(x) ->* true 46/46 even(s(x)) -> true | odd(x) ->* true 46/46 odd(0) -> false 46/46 odd(s(x)) -> false | odd(x) ->* true 46/46 odd(s(x)) -> true | even(x) ->* true) 46/46 (VAR x2) 46/46 (CONDITION even(x2) ->* true, odd(x2) ->* true) 46/46 46/46 Proof: 46/46 YES 46/46 46/46 Problem 1: 46/46 46/46 Infeasibility Problem: 46/46 [(VAR vNonEmpty:S x:S x2:S) 46/46 (STRATEGY CONTEXTSENSITIVE 46/46 (even 1) 46/46 (odd 1) 46/46 (0) 46/46 (fSNonEmpty) 46/46 (false) 46/46 (s 1) 46/46 (true) 46/46 ) 46/46 (RULES 46/46 even(0) -> true 46/46 even(s(x:S)) -> false | even(x:S) ->* true 46/46 even(s(x:S)) -> true | odd(x:S) ->* true 46/46 odd(0) -> false 46/46 odd(s(x:S)) -> false | odd(x:S) ->* true 46/46 odd(s(x:S)) -> true | even(x:S) ->* true 46/46 )] 46/46 46/46 Infeasibility Conditions: 46/46 even(x2:S) ->* true, odd(x2:S) ->* true 46/46 46/46 Problem 1: 46/46 46/46 Obtaining a model using Mace4: 46/46 46/46 -> Usable Rules: 46/46 even(0) -> true 46/46 even(s(x:S)) -> false | even(x:S) ->* true 46/46 even(s(x:S)) -> true | odd(x:S) ->* true 46/46 odd(0) -> false 46/46 odd(s(x:S)) -> false | odd(x:S) ->* true 46/46 odd(s(x:S)) -> true | even(x:S) ->* true 46/46 46/46 -> Mace4 Output: 46/46 ============================== Mace4 ================================= 46/46 Mace4 (64) version 2009-11A, November 2009. 46/46 Process 116902 was started by ubuntu on ubuntu, 46/46 Fri Jul 16 15:28:06 2021 46/46 The command was "./mace4 -c -f /tmp/mace4116878-2.in". 46/46 ============================== end of head =========================== 46/46 46/46 ============================== INPUT ================================= 46/46 46/46 % Reading from file /tmp/mace4116878-2.in 46/46 46/46 assign(max_seconds,10). 46/46 46/46 formulas(assumptions). 46/46 ->(x1,y) -> ->(even(x1),even(y)) # label(congruence). 46/46 ->(x1,y) -> ->(odd(x1),odd(y)) # label(congruence). 46/46 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 46/46 ->(even(0),true) # label(replacement). 46/46 ->*(even(x1),true) -> ->(even(s(x1)),false) # label(replacement). 46/46 ->*(odd(x1),true) -> ->(even(s(x1)),true) # label(replacement). 46/46 ->(odd(0),false) # label(replacement). 46/46 ->*(odd(x1),true) -> ->(odd(s(x1)),false) # label(replacement). 46/46 ->*(even(x1),true) -> ->(odd(s(x1)),true) # label(replacement). 46/46 ->*(x,x) # label(reflexivity). 46/46 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 46/46 end_of_list. 46/46 46/46 formulas(goals). 46/46 (exists x2 (->*(even(x2),true) & ->*(odd(x2),true))) # label(goal). 46/46 end_of_list. 46/46 46/46 ============================== end of input ========================== 46/46 46/46 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46/46 46/46 % Formulas that are not ordinary clauses: 46/46 1 ->(x1,y) -> ->(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 2 ->(x1,y) -> ->(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 3 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 4 ->*(even(x1),true) -> ->(even(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 46/46 5 ->*(odd(x1),true) -> ->(even(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 46/46 6 ->*(odd(x1),true) -> ->(odd(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 46/46 7 ->*(even(x1),true) -> ->(odd(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 46/46 8 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 46/46 9 (exists x2 (->*(even(x2),true) & ->*(odd(x2),true))) # label(goal) # label(non_clause) # label(goal). [goal]. 46/46 46/46 ============================== end of process non-clausal formulas === 46/46 46/46 ============================== CLAUSES FOR SEARCH ==================== 46/46 46/46 formulas(mace4_clauses). 46/46 -->(x,y) | ->(even(x),even(y)) # label(congruence). 46/46 -->(x,y) | ->(odd(x),odd(y)) # label(congruence). 46/46 -->(x,y) | ->(s(x),s(y)) # label(congruence). 46/46 ->(even(0),true) # label(replacement). 46/46 -->*(even(x),true) | ->(even(s(x)),false) # label(replacement). 46/46 -->*(odd(x),true) | ->(even(s(x)),true) # label(replacement). 46/46 ->(odd(0),false) # label(replacement). 46/46 -->*(odd(x),true) | ->(odd(s(x)),false) # label(replacement). 46/46 -->*(even(x),true) | ->(odd(s(x)),true) # label(replacement). 46/46 ->*(x,x) # label(reflexivity). 46/46 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 46/46 -->*(even(x),true) | -->*(odd(x),true) # label(goal). 46/46 end_of_list. 46/46 46/46 ============================== end of clauses for search ============= 46/46 46/46 % There are no natural numbers in the input. 46/46 46/46 ============================== DOMAIN SIZE 2 ========================= 46/46 46/46 ============================== MODEL ================================= 46/46 46/46 interpretation( 2, [number=1, seconds=0], [ 46/46 46/46 function(0, [ 0 ]), 46/46 46/46 function(false, [ 0 ]), 46/46 46/46 function(true, [ 1 ]), 46/46 46/46 function(even(_), [ 1, 0 ]), 46/46 46/46 function(odd(_), [ 0, 1 ]), 46/46 46/46 function(s(_), [ 1, 0 ]), 46/46 46/46 relation(->*(_,_), [ 46/46 1, 0, 46/46 0, 1 ]), 46/46 46/46 relation(->(_,_), [ 46/46 1, 0, 46/46 0, 1 ]) 46/46 ]). 46/46 46/46 ============================== end of model ========================== 46/46 46/46 ============================== STATISTICS ============================ 46/46 46/46 For domain size 2. 46/46 46/46 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 46/46 Ground clauses: seen=34, kept=30. 46/46 Selections=8, assignments=12, propagations=22, current_models=1. 46/46 Rewrite_terms=133, rewrite_bools=79, indexes=27. 46/46 Rules_from_neg_clauses=6, cross_offs=6. 46/46 46/46 ============================== end of statistics ===================== 46/46 46/46 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46/46 46/46 Exiting with 1 model. 46/46 46/46 Process 116902 exit (max_models) Fri Jul 16 15:28:06 2021 46/46 The process finished Fri Jul 16 15:28:06 2021 46/46 46/46 46/46 Mace4 cooked interpretation: 46/46 46/46 46/46 46/46 The problem is infeasible. 46/46 46/46 46/46 46/46 46/46 | odd(x:S) ->* true, even(x:S) ->* true => Not trivial, Overlay, N2 46/46 (PROBLEM INFEASIBILITY) 46/46 (VAR x) 46/46 (RULES 46/46 even(0) -> true 46/46 even(s(x)) -> false | even(x) ->* true 46/46 even(s(x)) -> true | odd(x) ->* true 46/46 odd(0) -> false 46/46 odd(s(x)) -> false | odd(x) ->* true 46/46 odd(s(x)) -> true | even(x) ->* true) 46/46 (VAR x2) 46/46 (CONDITION odd(x2) ->* true, even(x2) ->* true) 46/46 46/46 Proof: 46/46 YES 46/46 46/46 Problem 1: 46/46 46/46 Infeasibility Problem: 46/46 [(VAR vNonEmpty:S x:S x2:S) 46/46 (STRATEGY CONTEXTSENSITIVE 46/46 (even 1) 46/46 (odd 1) 46/46 (0) 46/46 (fSNonEmpty) 46/46 (false) 46/46 (s 1) 46/46 (true) 46/46 ) 46/46 (RULES 46/46 even(0) -> true 46/46 even(s(x:S)) -> false | even(x:S) ->* true 46/46 even(s(x:S)) -> true | odd(x:S) ->* true 46/46 odd(0) -> false 46/46 odd(s(x:S)) -> false | odd(x:S) ->* true 46/46 odd(s(x:S)) -> true | even(x:S) ->* true 46/46 )] 46/46 46/46 Infeasibility Conditions: 46/46 odd(x2:S) ->* true, even(x2:S) ->* true 46/46 46/46 Problem 1: 46/46 46/46 Obtaining a model using Mace4: 46/46 46/46 -> Usable Rules: 46/46 even(0) -> true 46/46 even(s(x:S)) -> false | even(x:S) ->* true 46/46 even(s(x:S)) -> true | odd(x:S) ->* true 46/46 odd(0) -> false 46/46 odd(s(x:S)) -> false | odd(x:S) ->* true 46/46 odd(s(x:S)) -> true | even(x:S) ->* true 46/46 46/46 -> Mace4 Output: 46/46 ============================== Mace4 ================================= 46/46 Mace4 (64) version 2009-11A, November 2009. 46/46 Process 116942 was started by ubuntu on ubuntu, 46/46 Fri Jul 16 15:28:26 2021 46/46 The command was "./mace4 -c -f /tmp/mace4116929-2.in". 46/46 ============================== end of head =========================== 46/46 46/46 ============================== INPUT ================================= 46/46 46/46 % Reading from file /tmp/mace4116929-2.in 46/46 46/46 assign(max_seconds,10). 46/46 46/46 formulas(assumptions). 46/46 ->(x1,y) -> ->(even(x1),even(y)) # label(congruence). 46/46 ->(x1,y) -> ->(odd(x1),odd(y)) # label(congruence). 46/46 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 46/46 ->(even(0),true) # label(replacement). 46/46 ->*(even(x1),true) -> ->(even(s(x1)),false) # label(replacement). 46/46 ->*(odd(x1),true) -> ->(even(s(x1)),true) # label(replacement). 46/46 ->(odd(0),false) # label(replacement). 46/46 ->*(odd(x1),true) -> ->(odd(s(x1)),false) # label(replacement). 46/46 ->*(even(x1),true) -> ->(odd(s(x1)),true) # label(replacement). 46/46 ->*(x,x) # label(reflexivity). 46/46 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 46/46 end_of_list. 46/46 46/46 formulas(goals). 46/46 (exists x2 (->*(odd(x2),true) & ->*(even(x2),true))) # label(goal). 46/46 end_of_list. 46/46 46/46 ============================== end of input ========================== 46/46 46/46 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46/46 46/46 % Formulas that are not ordinary clauses: 46/46 1 ->(x1,y) -> ->(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 2 ->(x1,y) -> ->(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 3 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46/46 4 ->*(even(x1),true) -> ->(even(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 46/46 5 ->*(odd(x1),true) -> ->(even(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 46/46 6 ->*(odd(x1),true) -> ->(odd(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 46/46 7 ->*(even(x1),true) -> ->(odd(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 46/46 8 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 46/46 9 (exists x2 (->*(odd(x2),true) & ->*(even(x2),true))) # label(goal) # label(non_clause) # label(goal). [goal]. 46/46 46/46 ============================== end of process non-clausal formulas === 46/46 46/46 ============================== CLAUSES FOR SEARCH ==================== 46/46 46/46 formulas(mace4_clauses). 46/46 -->(x,y) | ->(even(x),even(y)) # label(congruence). 46/46 -->(x,y) | ->(odd(x),odd(y)) # label(congruence). 46/46 -->(x,y) | ->(s(x),s(y)) # label(congruence). 46/46 ->(even(0),true) # label(replacement). 46/46 -->*(even(x),true) | ->(even(s(x)),false) # label(replacement). 46/46 -->*(odd(x),true) | ->(even(s(x)),true) # label(replacement). 46/46 ->(odd(0),false) # label(replacement). 46/46 -->*(odd(x),true) | ->(odd(s(x)),false) # label(replacement). 46/46 -->*(even(x),true) | ->(odd(s(x)),true) # label(replacement). 46/46 ->*(x,x) # label(reflexivity). 46/46 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 46/46 -->*(odd(x),true) | -->*(even(x),true) # label(goal). 46/46 end_of_list. 46/46 46/46 ============================== end of clauses for search ============= 46/46 46/46 % There are no natural numbers in the input. 46/46 46/46 ============================== DOMAIN SIZE 2 ========================= 46/46 46/46 ============================== MODEL ================================= 46/46 46/46 interpretation( 2, [number=1, seconds=0], [ 46/46 46/46 function(0, [ 0 ]), 46/46 46/46 function(false, [ 0 ]), 46/46 46/46 function(true, [ 1 ]), 46/46 46/46 function(even(_), [ 1, 0 ]), 46/46 46/46 function(odd(_), [ 0, 1 ]), 46/46 46/46 function(s(_), [ 1, 0 ]), 46/46 46/46 relation(->*(_,_), [ 46/46 1, 0, 46/46 0, 1 ]), 46/46 46/46 relation(->(_,_), [ 46/46 1, 0, 46/46 0, 1 ]) 46/46 ]). 46/46 46/46 ============================== end of model ========================== 46/46 46/46 ============================== STATISTICS ============================ 46/46 46/46 For domain size 2. 46/46 46/46 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 46/46 Ground clauses: seen=34, kept=30. 46/46 Selections=8, assignments=12, propagations=22, current_models=1. 46/46 Rewrite_terms=133, rewrite_bools=79, indexes=27. 46/46 Rules_from_neg_clauses=6, cross_offs=6. 46/46 46/46 ============================== end of statistics ===================== 46/46 46/46 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46/46 46/46 Exiting with 1 model. 46/46 46/46 Process 116942 exit (max_models) Fri Jul 16 15:28:26 2021 46/46 The process finished Fri Jul 16 15:28:26 2021 46/46 46/46 46/46 Mace4 cooked interpretation: 46/46 46/46 46/46 46/46 The problem is infeasible. 46/46 46/46 46/46 46/46 46/46 -> Problem conclusions: 46/46 Left linear, Right linear, Linear 46/46 Weakly orthogonal, Almost orthogonal, Orthogonal 46/46 CTRS Type: 1 46/46 Deterministic, Strongly deterministic 46/46 Oriented CTRS, Properly oriented CTRS, Not join CTRS 46/46 Maybe right-stable CTRS, Overlay CTRS 46/46 Maybe normal CTRS, Maybe almost normal CTRS 46/46 Maybe terminating CTRS, Joinable CCPs 46/46 Maybe level confluent 46/46 Maybe confluent 46/46 46/46 Resulting CCPs: 46/46 No CCPs left 46/46 46/46 Problem 1: 46/46 Underlying TRS Termination Processor: 46/46 46/46 Resulting Underlying TRS: 46/46 (VAR x:S) 46/46 (STRATEGY CONTEXTSENSITIVE 46/46 (even 1) 46/46 (odd 1) 46/46 (false) 46/46 (true) 46/46 ) 46/46 (RULES 46/46 even(0) -> true 46/46 even(s(x:S)) -> false 46/46 even(s(x:S)) -> true 46/46 odd(0) -> false 46/46 odd(s(x:S)) -> false 46/46 odd(s(x:S)) -> true 46/46 ) 46/46 Underlying TRS terminating? 46/46 YES 46/46 46/46 Problem 1: 46/46 46/46 (VAR vu95NonEmpty:S xu58S:S) 46/46 (RULES 46/46 even(num0) -> ftrue 46/46 even(s(xu58S:S)) -> ffalse 46/46 even(s(xu58S:S)) -> ftrue 46/46 odd(num0) -> ffalse 46/46 odd(s(xu58S:S)) -> ffalse 46/46 odd(s(xu58S:S)) -> ftrue 46/46 ) 46/46 46/46 Problem 1: 46/46 46/46 Dependency Pairs Processor: 46/46 -> Pairs: 46/46 Empty 46/46 -> Rules: 46/46 even(num0) -> ftrue 46/46 even(s(xu58S:S)) -> ffalse 46/46 even(s(xu58S:S)) -> ftrue 46/46 odd(num0) -> ffalse 46/46 odd(s(xu58S:S)) -> ffalse 46/46 odd(s(xu58S:S)) -> ftrue 46/46 46/46 Problem 1: 46/46 46/46 SCC Processor: 46/46 -> Pairs: 46/46 Empty 46/46 -> Rules: 46/46 even(num0) -> ftrue 46/46 even(s(xu58S:S)) -> ffalse 46/46 even(s(xu58S:S)) -> ftrue 46/46 odd(num0) -> ffalse 46/46 odd(s(xu58S:S)) -> ffalse 46/46 odd(s(xu58S:S)) -> ftrue 46/46 ->Strongly Connected Components: 46/46 There is no strongly connected component 46/46 46/46 The problem is finite. 46/46 46/46 46/46 -> Problem conclusions: 46/46 Left linear, Right linear, Linear 46/46 Weakly orthogonal, Almost orthogonal, Orthogonal 46/46 CTRS Type: 1 46/46 Deterministic, Strongly deterministic 46/46 Oriented CTRS, Properly oriented CTRS, Not join CTRS 46/46 Maybe right-stable CTRS, Overlay CTRS 46/46 Maybe normal CTRS, Maybe almost normal CTRS 46/46 Terminating CTRS, Joinable CCPs 46/46 Maybe level confluent 46/46 Confluent 46/46 46/46 The problem is joinable. 46/46 71.35user 1.97system 0:46.00elapsed 159%CPU (0avgtext+0avgdata 135976maxresident)k 46/46 8inputs+0outputs (0major+173015minor)pagefaults 0swaps