46.014/46.014 YES 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 46.014/46.014 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.014/46.014 Confluence Problem: 46.014/46.014 (VAR vNonEmpty:S x:S) 46.014/46.014 (STRATEGY CONTEXTSENSITIVE 46.014/46.014 (even 1) 46.014/46.014 (odd 1) 46.014/46.014 (0) 46.014/46.014 (fSNonEmpty) 46.014/46.014 (false) 46.014/46.014 (s 1) 46.014/46.014 (true) 46.014/46.014 ) 46.014/46.014 (RULES 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x:S)) -> false | odd(x:S) ->* false 46.014/46.014 even(s(x:S)) -> true | odd(x:S) ->* true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x:S)) -> false | even(x:S) ->* false 46.014/46.014 odd(s(x:S)) -> true | even(x:S) ->* true 46.014/46.014 ) 46.014/46.014 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.014/46.014 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 Inlining of Conditions Processor [STERN17]: 46.014/46.014 46.014/46.014 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.014/46.014 Confluence Problem: 46.014/46.014 (VAR vNonEmpty:S x:S) 46.014/46.014 (STRATEGY CONTEXTSENSITIVE 46.014/46.014 (even 1) 46.014/46.014 (odd 1) 46.014/46.014 (0) 46.014/46.014 (fSNonEmpty) 46.014/46.014 (false) 46.014/46.014 (s 1) 46.014/46.014 (true) 46.014/46.014 ) 46.014/46.014 (RULES 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x:S)) -> false | odd(x:S) ->* false 46.014/46.014 even(s(x:S)) -> true | odd(x:S) ->* true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x:S)) -> false | even(x:S) ->* false 46.014/46.014 odd(s(x:S)) -> true | even(x:S) ->* true 46.014/46.014 ) 46.014/46.014 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.014/46.014 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 Clean CTRS Processor: 46.014/46.014 46.014/46.014 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.014/46.014 Confluence Problem: 46.014/46.014 (VAR x:S) 46.014/46.014 (STRATEGY CONTEXTSENSITIVE 46.014/46.014 (even 1) 46.014/46.014 (odd 1) 46.014/46.014 (0) 46.014/46.014 (fSNonEmpty) 46.014/46.014 (false) 46.014/46.014 (s 1) 46.014/46.014 (true) 46.014/46.014 ) 46.014/46.014 (RULES 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x:S)) -> false | odd(x:S) ->* false 46.014/46.014 even(s(x:S)) -> true | odd(x:S) ->* true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x:S)) -> false | even(x:S) ->* false 46.014/46.014 odd(s(x:S)) -> true | even(x:S) ->* true 46.014/46.014 ) 46.014/46.014 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.014/46.014 46.014/46.014 CRule InfChecker Info: 46.014/46.014 even(0) -> true 46.014/46.014 Rule remains 46.014/46.014 Proof: 46.014/46.014 NO_CONDS 46.014/46.014 46.014/46.014 CRule InfChecker Info: 46.014/46.014 even(s(x:S)) -> false | odd(x:S) ->* false 46.014/46.014 Rule remains 46.014/46.014 Proof: 46.014/46.014 NO 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 Infeasibility Problem: 46.014/46.014 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 46.014/46.014 (STRATEGY CONTEXTSENSITIVE 46.014/46.014 (even 1) 46.014/46.014 (odd 1) 46.014/46.014 (0) 46.014/46.014 (fSNonEmpty) 46.014/46.014 (false) 46.014/46.014 (s 1) 46.014/46.014 (true) 46.014/46.014 ) 46.014/46.014 (RULES 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x:S:S)) -> false | odd(x:S:S) ->* false 46.014/46.014 even(s(x:S:S)) -> true | odd(x:S:S) ->* true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x:S:S)) -> false | even(x:S:S) ->* false 46.014/46.014 odd(s(x:S:S)) -> true | even(x:S:S) ->* true 46.014/46.014 )] 46.014/46.014 46.014/46.014 Infeasibility Conditions: 46.014/46.014 odd(x:S:S) ->* false 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 Obtaining a proof using Prover9: 46.014/46.014 46.014/46.014 -> Prover9 Output: 46.014/46.014 ============================== Prover9 =============================== 46.014/46.014 Prover9 (64) version 2009-11A, November 2009. 46.014/46.014 Process 42737 was started by ubuntu on ubuntu, 46.014/46.014 Wed Jul 14 10:26:53 2021 46.014/46.014 The command was "./prover9 -f /tmp/prover942728-0.in". 46.014/46.014 ============================== end of head =========================== 46.014/46.014 46.014/46.014 ============================== INPUT ================================= 46.014/46.014 46.014/46.014 % Reading from file /tmp/prover942728-0.in 46.014/46.014 46.014/46.014 assign(max_seconds,20). 46.014/46.014 46.014/46.014 formulas(assumptions). 46.014/46.014 ->*_s0(x,x) # label(reflexivity). 46.014/46.014 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 46.014/46.014 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence). 46.014/46.014 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence). 46.014/46.014 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 46.014/46.014 ->_s0(even(0),true) # label(replacement). 46.014/46.014 ->*_s0(odd(x2),false) -> ->_s0(even(s(x2)),false) # label(replacement). 46.014/46.014 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement). 46.014/46.014 ->_s0(odd(0),false) # label(replacement). 46.014/46.014 ->*_s0(even(x2),false) -> ->_s0(odd(s(x2)),false) # label(replacement). 46.014/46.014 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(goals). 46.014/46.014 (exists x2 ->*_s0(odd(x2),false)) # label(goal). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== end of input ========================== 46.014/46.014 46.014/46.014 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46.014/46.014 46.014/46.014 % Formulas that are not ordinary clauses: 46.014/46.014 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.014/46.014 2 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 3 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 5 ->*_s0(odd(x2),false) -> ->_s0(even(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 6 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 7 ->*_s0(even(x2),false) -> ->_s0(odd(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 9 (exists x2 ->*_s0(odd(x2),false)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.014/46.014 46.014/46.014 ============================== end of process non-clausal formulas === 46.014/46.014 46.014/46.014 ============================== PROCESS INITIAL CLAUSES =============== 46.014/46.014 46.014/46.014 % Clauses before input processing: 46.014/46.014 46.014/46.014 formulas(usable). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(sos). 46.014/46.014 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 -->*_s0(odd(x),false) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.014/46.014 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.014/46.014 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 -->*_s0(even(x),false) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.014/46.014 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.014/46.014 -->*_s0(odd(x),false) # label(goal). [deny(9)]. 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(demodulators). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== PREDICATE ELIMINATION ================= 46.014/46.014 46.014/46.014 No predicates eliminated. 46.014/46.014 46.014/46.014 ============================== end predicate elimination ============= 46.014/46.014 46.014/46.014 Auto_denials: 46.014/46.014 % copying label goal to answer in negative clause 46.014/46.014 46.014/46.014 Term ordering decisions: 46.014/46.014 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 46.014/46.014 Function symbol precedence: function_order([ false, true, 0, even, odd, s ]). 46.014/46.014 After inverse_order: (no changes). 46.014/46.014 Unfolding symbols: (none). 46.014/46.014 46.014/46.014 Auto_inference settings: 46.014/46.014 % set(neg_binary_resolution). % (HNE depth_diff=-7) 46.014/46.014 % clear(ordered_res). % (HNE depth_diff=-7) 46.014/46.014 % set(ur_resolution). % (HNE depth_diff=-7) 46.014/46.014 % set(ur_resolution) -> set(pos_ur_resolution). 46.014/46.014 % set(ur_resolution) -> set(neg_ur_resolution). 46.014/46.014 46.014/46.014 Auto_process settings: (no changes). 46.014/46.014 46.014/46.014 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 kept: 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 kept: 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 kept: 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 kept: 16 -->*_s0(odd(x),false) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.014/46.014 kept: 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.014/46.014 kept: 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 kept: 19 -->*_s0(even(x),false) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.014/46.014 kept: 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.014/46.014 kept: 21 -->*_s0(odd(x),false) # label(goal) # answer(goal). [deny(9)]. 46.014/46.014 46.014/46.014 ============================== end of process initial clauses ======== 46.014/46.014 46.014/46.014 ============================== CLAUSES FOR SEARCH ==================== 46.014/46.014 46.014/46.014 % Clauses after input processing: 46.014/46.014 46.014/46.014 formulas(usable). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(sos). 46.014/46.014 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.014/46.014 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 19 -->*_s0(even(x),false) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.014/46.014 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.014/46.014 21 -->*_s0(odd(x),false) # label(goal) # answer(goal). [deny(9)]. 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(demodulators). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== end of clauses for search ============= 46.014/46.014 46.014/46.014 ============================== SEARCH ================================ 46.014/46.014 46.014/46.014 % Starting search at 0.00 seconds. 46.014/46.014 46.014/46.014 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 46.014/46.014 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 46.014/46.014 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 46.014/46.014 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 46.014/46.014 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 46.014/46.014 given #6 (I,wt=4): 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 46.014/46.014 given #7 (I,wt=9): 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.014/46.014 46.014/46.014 given #8 (I,wt=4): 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 46.014/46.014 ============================== PROOF ================================= 46.014/46.014 46.014/46.014 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 46.014/46.014 % Length of proof is 8. 46.014/46.014 % Level of proof is 3. 46.014/46.014 % Maximum clause weight is 9.000. 46.014/46.014 % Given clauses 8. 46.014/46.014 46.014/46.014 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.014/46.014 9 (exists x2 ->*_s0(odd(x2),false)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.014/46.014 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 21 -->*_s0(odd(x),false) # label(goal) # answer(goal). [deny(9)]. 46.014/46.014 29 ->*_s0(odd(0),false). [ur(11,a,18,a,b,10,a)]. 46.014/46.014 30 $F # answer(goal). [resolve(29,a,21,a)]. 46.014/46.014 46.014/46.014 ============================== end of proof ========================== 46.014/46.014 46.014/46.014 ============================== STATISTICS ============================ 46.014/46.014 46.014/46.014 Given=8. Generated=20. Kept=20. proofs=1. 46.014/46.014 Usable=8. Sos=7. Demods=0. Limbo=3, Disabled=13. Hints=0. 46.014/46.014 Kept_by_rule=0, Deleted_by_rule=0. 46.014/46.014 Forward_subsumed=0. Back_subsumed=1. 46.014/46.014 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 46.014/46.014 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 46.014/46.014 Demod_attempts=0. Demod_rewrites=0. 46.014/46.014 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 46.014/46.014 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=10. 46.014/46.014 Megabytes=0.07. 46.014/46.014 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46.014/46.014 46.014/46.014 ============================== end of statistics ===================== 46.014/46.014 46.014/46.014 ============================== end of search ========================= 46.014/46.014 46.014/46.014 THEOREM PROVED 46.014/46.014 46.014/46.014 Exiting with 1 proof. 46.014/46.014 46.014/46.014 Process 42737 exit (max_proofs) Wed Jul 14 10:26:53 2021 46.014/46.014 46.014/46.014 46.014/46.014 The problem is feasible. 46.014/46.014 46.014/46.014 46.014/46.014 CRule InfChecker Info: 46.014/46.014 even(s(x:S)) -> true | odd(x:S) ->* true 46.014/46.014 Rule remains 46.014/46.014 Proof: 46.014/46.014 NO 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 Infeasibility Problem: 46.014/46.014 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 46.014/46.014 (STRATEGY CONTEXTSENSITIVE 46.014/46.014 (even 1) 46.014/46.014 (odd 1) 46.014/46.014 (0) 46.014/46.014 (fSNonEmpty) 46.014/46.014 (false) 46.014/46.014 (s 1) 46.014/46.014 (true) 46.014/46.014 ) 46.014/46.014 (RULES 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x:S:S)) -> false | odd(x:S:S) ->* false 46.014/46.014 even(s(x:S:S)) -> true | odd(x:S:S) ->* true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x:S:S)) -> false | even(x:S:S) ->* false 46.014/46.014 odd(s(x:S:S)) -> true | even(x:S:S) ->* true 46.014/46.014 )] 46.014/46.014 46.014/46.014 Infeasibility Conditions: 46.014/46.014 odd(x:S:S) ->* true 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 Obtaining a proof using Prover9: 46.014/46.014 46.014/46.014 -> Prover9 Output: 46.014/46.014 ============================== Prover9 =============================== 46.014/46.014 Prover9 (64) version 2009-11A, November 2009. 46.014/46.014 Process 42760 was started by ubuntu on ubuntu, 46.014/46.014 Wed Jul 14 10:26:53 2021 46.014/46.014 The command was "./prover9 -f /tmp/prover942750-0.in". 46.014/46.014 ============================== end of head =========================== 46.014/46.014 46.014/46.014 ============================== INPUT ================================= 46.014/46.014 46.014/46.014 % Reading from file /tmp/prover942750-0.in 46.014/46.014 46.014/46.014 assign(max_seconds,20). 46.014/46.014 46.014/46.014 formulas(assumptions). 46.014/46.014 ->*_s0(x,x) # label(reflexivity). 46.014/46.014 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 46.014/46.014 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence). 46.014/46.014 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence). 46.014/46.014 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 46.014/46.014 ->_s0(even(0),true) # label(replacement). 46.014/46.014 ->*_s0(odd(x2),false) -> ->_s0(even(s(x2)),false) # label(replacement). 46.014/46.014 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement). 46.014/46.014 ->_s0(odd(0),false) # label(replacement). 46.014/46.014 ->*_s0(even(x2),false) -> ->_s0(odd(s(x2)),false) # label(replacement). 46.014/46.014 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(goals). 46.014/46.014 (exists x2 ->*_s0(odd(x2),true)) # label(goal). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== end of input ========================== 46.014/46.014 46.014/46.014 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46.014/46.014 46.014/46.014 % Formulas that are not ordinary clauses: 46.014/46.014 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.014/46.014 2 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 3 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 5 ->*_s0(odd(x2),false) -> ->_s0(even(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 6 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 7 ->*_s0(even(x2),false) -> ->_s0(odd(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 9 (exists x2 ->*_s0(odd(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.014/46.014 46.014/46.014 ============================== end of process non-clausal formulas === 46.014/46.014 46.014/46.014 ============================== PROCESS INITIAL CLAUSES =============== 46.014/46.014 46.014/46.014 % Clauses before input processing: 46.014/46.014 46.014/46.014 formulas(usable). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(sos). 46.014/46.014 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 -->*_s0(odd(x),false) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.014/46.014 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.014/46.014 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 -->*_s0(even(x),false) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.014/46.014 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.014/46.014 -->*_s0(odd(x),true) # label(goal). [deny(9)]. 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(demodulators). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== PREDICATE ELIMINATION ================= 46.014/46.014 46.014/46.014 No predicates eliminated. 46.014/46.014 46.014/46.014 ============================== end predicate elimination ============= 46.014/46.014 46.014/46.014 Auto_denials: 46.014/46.014 % copying label goal to answer in negative clause 46.014/46.014 46.014/46.014 Term ordering decisions: 46.014/46.014 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 46.014/46.014 Function symbol precedence: function_order([ false, true, 0, even, odd, s ]). 46.014/46.014 After inverse_order: (no changes). 46.014/46.014 Unfolding symbols: (none). 46.014/46.014 46.014/46.014 Auto_inference settings: 46.014/46.014 % set(neg_binary_resolution). % (HNE depth_diff=-7) 46.014/46.014 % clear(ordered_res). % (HNE depth_diff=-7) 46.014/46.014 % set(ur_resolution). % (HNE depth_diff=-7) 46.014/46.014 % set(ur_resolution) -> set(pos_ur_resolution). 46.014/46.014 % set(ur_resolution) -> set(neg_ur_resolution). 46.014/46.014 46.014/46.014 Auto_process settings: (no changes). 46.014/46.014 46.014/46.014 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 kept: 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 kept: 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 kept: 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 kept: 16 -->*_s0(odd(x),false) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.014/46.014 kept: 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.014/46.014 kept: 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 kept: 19 -->*_s0(even(x),false) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.014/46.014 kept: 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.014/46.014 kept: 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46.014/46.014 46.014/46.014 ============================== end of process initial clauses ======== 46.014/46.014 46.014/46.014 ============================== CLAUSES FOR SEARCH ==================== 46.014/46.014 46.014/46.014 % Clauses after input processing: 46.014/46.014 46.014/46.014 formulas(usable). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(sos). 46.014/46.014 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 16 -->*_s0(odd(x),false) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.014/46.014 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 19 -->*_s0(even(x),false) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.014/46.014 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.014/46.014 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(demodulators). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== end of clauses for search ============= 46.014/46.014 46.014/46.014 ============================== SEARCH ================================ 46.014/46.014 46.014/46.014 % Starting search at 0.00 seconds. 46.014/46.014 46.014/46.014 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 46.014/46.014 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 46.014/46.014 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 46.014/46.014 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 46.014/46.014 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 46.014/46.014 given #6 (I,wt=4): 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 46.014/46.014 given #7 (I,wt=9): 16 -->*_s0(odd(x),false) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.014/46.014 46.014/46.014 given #8 (I,wt=4): 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 46.014/46.014 given #9 (I,wt=9): 19 -->*_s0(even(x),false) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.014/46.014 46.014/46.014 given #10 (I,wt=9): 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.014/46.014 46.014/46.014 given #11 (I,wt=4): 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46.014/46.014 46.014/46.014 given #12 (A,wt=6): 22 ->_s0(s(even(0)),s(true)). [ur(14,a,15,a)]. 46.014/46.014 46.014/46.014 given #13 (F,wt=3): 32 -->*_s0(false,true) # answer(goal). [ur(11,a,18,a,c,21,a)]. 46.014/46.014 46.014/46.014 given #14 (F,wt=3): 38 -->_s0(false,true) # answer(goal). [ur(11,b,10,a,c,32,a)]. 46.014/46.014 46.014/46.014 given #15 (F,wt=4): 31 -->_s0(odd(x),true) # answer(goal). [ur(11,b,10,a,c,21,a)]. 46.014/46.014 46.014/46.014 ============================== PROOF ================================= 46.014/46.014 46.014/46.014 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 46.014/46.014 % Length of proof is 12. 46.014/46.014 % Level of proof is 4. 46.014/46.014 % Maximum clause weight is 9.000. 46.014/46.014 % Given clauses 15. 46.014/46.014 46.014/46.014 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.014/46.014 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 9 (exists x2 ->*_s0(odd(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.014/46.014 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.014/46.014 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46.014/46.014 25 ->*_s0(even(0),true). [ur(11,a,15,a,b,10,a)]. 46.014/46.014 31 -->_s0(odd(x),true) # answer(goal). [ur(11,b,10,a,c,21,a)]. 46.014/46.014 39 -->*_s0(even(x),true) # answer(goal). [resolve(31,a,20,b)]. 46.014/46.014 40 $F # answer(goal). [resolve(39,a,25,a)]. 46.014/46.014 46.014/46.014 ============================== end of proof ========================== 46.014/46.014 46.014/46.014 ============================== STATISTICS ============================ 46.014/46.014 46.014/46.014 Given=15. Generated=30. Kept=30. proofs=1. 46.014/46.014 Usable=15. Sos=13. Demods=0. Limbo=0, Disabled=13. Hints=0. 46.014/46.014 Kept_by_rule=0, Deleted_by_rule=0. 46.014/46.014 Forward_subsumed=0. Back_subsumed=1. 46.014/46.014 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 46.014/46.014 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 46.014/46.014 Demod_attempts=0. Demod_rewrites=0. 46.014/46.014 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 46.014/46.014 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=14. 46.014/46.014 Megabytes=0.09. 46.014/46.014 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46.014/46.014 46.014/46.014 ============================== end of statistics ===================== 46.014/46.014 46.014/46.014 ============================== end of search ========================= 46.014/46.014 46.014/46.014 THEOREM PROVED 46.014/46.014 46.014/46.014 Exiting with 1 proof. 46.014/46.014 46.014/46.014 Process 42760 exit (max_proofs) Wed Jul 14 10:26:53 2021 46.014/46.014 46.014/46.014 46.014/46.014 The problem is feasible. 46.014/46.014 46.014/46.014 46.014/46.014 CRule InfChecker Info: 46.014/46.014 odd(0) -> false 46.014/46.014 Rule remains 46.014/46.014 Proof: 46.014/46.014 NO_CONDS 46.014/46.014 46.014/46.014 CRule InfChecker Info: 46.014/46.014 odd(s(x:S)) -> false | even(x:S) ->* false 46.014/46.014 Rule remains 46.014/46.014 Proof: 46.014/46.014 NO 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 Infeasibility Problem: 46.014/46.014 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 46.014/46.014 (STRATEGY CONTEXTSENSITIVE 46.014/46.014 (even 1) 46.014/46.014 (odd 1) 46.014/46.014 (0) 46.014/46.014 (fSNonEmpty) 46.014/46.014 (false) 46.014/46.014 (s 1) 46.014/46.014 (true) 46.014/46.014 ) 46.014/46.014 (RULES 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x:S:S)) -> false | odd(x:S:S) ->* false 46.014/46.014 even(s(x:S:S)) -> true | odd(x:S:S) ->* true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x:S:S)) -> false | even(x:S:S) ->* false 46.014/46.014 odd(s(x:S:S)) -> true | even(x:S:S) ->* true 46.014/46.014 )] 46.014/46.014 46.014/46.014 Infeasibility Conditions: 46.014/46.014 even(x:S:S) ->* false 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 Obtaining a proof using Prover9: 46.014/46.014 46.014/46.014 -> Prover9 Output: 46.014/46.014 ============================== Prover9 =============================== 46.014/46.014 Prover9 (64) version 2009-11A, November 2009. 46.014/46.014 Process 42791 was started by ubuntu on ubuntu, 46.014/46.014 Wed Jul 14 10:26:53 2021 46.014/46.014 The command was "./prover9 -f /tmp/prover942772-0.in". 46.014/46.014 ============================== end of head =========================== 46.014/46.014 46.014/46.014 ============================== INPUT ================================= 46.014/46.014 46.014/46.014 % Reading from file /tmp/prover942772-0.in 46.014/46.014 46.014/46.014 assign(max_seconds,20). 46.014/46.014 46.014/46.014 formulas(assumptions). 46.014/46.014 ->*_s0(x,x) # label(reflexivity). 46.014/46.014 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 46.014/46.014 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence). 46.014/46.014 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence). 46.014/46.014 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 46.014/46.014 ->_s0(even(0),true) # label(replacement). 46.014/46.014 ->*_s0(odd(x2),false) -> ->_s0(even(s(x2)),false) # label(replacement). 46.014/46.014 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement). 46.014/46.014 ->_s0(odd(0),false) # label(replacement). 46.014/46.014 ->*_s0(even(x2),false) -> ->_s0(odd(s(x2)),false) # label(replacement). 46.014/46.014 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(goals). 46.014/46.014 (exists x2 ->*_s0(even(x2),false)) # label(goal). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== end of input ========================== 46.014/46.014 46.014/46.014 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46.014/46.014 46.014/46.014 % Formulas that are not ordinary clauses: 46.014/46.014 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.014/46.014 2 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 3 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 5 ->*_s0(odd(x2),false) -> ->_s0(even(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 6 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 7 ->*_s0(even(x2),false) -> ->_s0(odd(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 9 (exists x2 ->*_s0(even(x2),false)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.014/46.014 46.014/46.014 ============================== end of process non-clausal formulas === 46.014/46.014 46.014/46.014 ============================== PROCESS INITIAL CLAUSES =============== 46.014/46.014 46.014/46.014 % Clauses before input processing: 46.014/46.014 46.014/46.014 formulas(usable). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(sos). 46.014/46.014 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 -->*_s0(odd(x),false) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.014/46.014 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.014/46.014 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 -->*_s0(even(x),false) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.014/46.014 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.014/46.014 -->*_s0(even(x),false) # label(goal). [deny(9)]. 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(demodulators). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== PREDICATE ELIMINATION ================= 46.014/46.014 46.014/46.014 No predicates eliminated. 46.014/46.014 46.014/46.014 ============================== end predicate elimination ============= 46.014/46.014 46.014/46.014 Auto_denials: 46.014/46.014 % copying label goal to answer in negative clause 46.014/46.014 46.014/46.014 Term ordering decisions: 46.014/46.014 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 46.014/46.014 Function symbol precedence: function_order([ false, true, 0, even, odd, s ]). 46.014/46.014 After inverse_order: (no changes). 46.014/46.014 Unfolding symbols: (none). 46.014/46.014 46.014/46.014 Auto_inference settings: 46.014/46.014 % set(neg_binary_resolution). % (HNE depth_diff=-7) 46.014/46.014 % clear(ordered_res). % (HNE depth_diff=-7) 46.014/46.014 % set(ur_resolution). % (HNE depth_diff=-7) 46.014/46.014 % set(ur_resolution) -> set(pos_ur_resolution). 46.014/46.014 % set(ur_resolution) -> set(neg_ur_resolution). 46.014/46.014 46.014/46.014 Auto_process settings: (no changes). 46.014/46.014 46.014/46.014 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 kept: 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 kept: 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 kept: 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 kept: 16 -->*_s0(odd(x),false) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.014/46.014 kept: 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.014/46.014 kept: 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 kept: 19 -->*_s0(even(x),false) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.014/46.014 kept: 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.014/46.014 kept: 21 -->*_s0(even(x),false) # label(goal) # answer(goal). [deny(9)]. 46.014/46.014 46.014/46.014 ============================== end of process initial clauses ======== 46.014/46.014 46.014/46.014 ============================== CLAUSES FOR SEARCH ==================== 46.014/46.014 46.014/46.014 % Clauses after input processing: 46.014/46.014 46.014/46.014 formulas(usable). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(sos). 46.014/46.014 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 16 -->*_s0(odd(x),false) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.014/46.014 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.014/46.014 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.014/46.014 21 -->*_s0(even(x),false) # label(goal) # answer(goal). [deny(9)]. 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(demodulators). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== end of clauses for search ============= 46.014/46.014 46.014/46.014 ============================== SEARCH ================================ 46.014/46.014 46.014/46.014 % Starting search at 0.00 seconds. 46.014/46.014 46.014/46.014 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 46.014/46.014 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 46.014/46.014 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 46.014/46.014 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 46.014/46.014 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 46.014/46.014 given #6 (I,wt=4): 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 46.014/46.014 given #7 (I,wt=9): 16 -->*_s0(odd(x),false) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.014/46.014 46.014/46.014 given #8 (I,wt=9): 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.014/46.014 46.014/46.014 given #9 (I,wt=4): 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 46.014/46.014 given #10 (I,wt=9): 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.014/46.014 46.014/46.014 given #11 (I,wt=4): 21 -->*_s0(even(x),false) # label(goal) # answer(goal). [deny(9)]. 46.014/46.014 46.014/46.014 given #12 (A,wt=6): 22 ->_s0(s(even(0)),s(true)). [ur(14,a,15,a)]. 46.014/46.014 46.014/46.014 given #13 (F,wt=3): 32 -->*_s0(true,false) # answer(goal). [ur(11,a,15,a,c,21,a)]. 46.014/46.014 46.014/46.014 given #14 (F,wt=3): 38 -->_s0(true,false) # answer(goal). [ur(11,b,10,a,c,32,a)]. 46.014/46.014 46.014/46.014 given #15 (F,wt=4): 31 -->_s0(even(x),false) # answer(goal). [ur(11,b,10,a,c,21,a)]. 46.014/46.014 46.014/46.014 ============================== PROOF ================================= 46.014/46.014 46.014/46.014 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 46.014/46.014 % Length of proof is 12. 46.014/46.014 % Level of proof is 4. 46.014/46.014 % Maximum clause weight is 9.000. 46.014/46.014 % Given clauses 15. 46.014/46.014 46.014/46.014 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.014/46.014 5 ->*_s0(odd(x2),false) -> ->_s0(even(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 9 (exists x2 ->*_s0(even(x2),false)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.014/46.014 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 16 -->*_s0(odd(x),false) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.014/46.014 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 21 -->*_s0(even(x),false) # label(goal) # answer(goal). [deny(9)]. 46.014/46.014 29 ->*_s0(odd(0),false). [ur(11,a,18,a,b,10,a)]. 46.014/46.014 31 -->_s0(even(x),false) # answer(goal). [ur(11,b,10,a,c,21,a)]. 46.014/46.014 39 -->*_s0(odd(x),false) # answer(goal). [resolve(31,a,16,b)]. 46.014/46.014 40 $F # answer(goal). [resolve(39,a,29,a)]. 46.014/46.014 46.014/46.014 ============================== end of proof ========================== 46.014/46.014 46.014/46.014 ============================== STATISTICS ============================ 46.014/46.014 46.014/46.014 Given=15. Generated=30. Kept=30. proofs=1. 46.014/46.014 Usable=15. Sos=13. Demods=0. Limbo=0, Disabled=13. Hints=0. 46.014/46.014 Kept_by_rule=0, Deleted_by_rule=0. 46.014/46.014 Forward_subsumed=0. Back_subsumed=1. 46.014/46.014 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 46.014/46.014 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 46.014/46.014 Demod_attempts=0. Demod_rewrites=0. 46.014/46.014 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 46.014/46.014 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=14. 46.014/46.014 Megabytes=0.09. 46.014/46.014 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46.014/46.014 46.014/46.014 ============================== end of statistics ===================== 46.014/46.014 46.014/46.014 ============================== end of search ========================= 46.014/46.014 46.014/46.014 THEOREM PROVED 46.014/46.014 46.014/46.014 Exiting with 1 proof. 46.014/46.014 46.014/46.014 Process 42791 exit (max_proofs) Wed Jul 14 10:26:53 2021 46.014/46.014 46.014/46.014 46.014/46.014 The problem is feasible. 46.014/46.014 46.014/46.014 46.014/46.014 CRule InfChecker Info: 46.014/46.014 odd(s(x:S)) -> true | even(x:S) ->* true 46.014/46.014 Rule remains 46.014/46.014 Proof: 46.014/46.014 NO 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 Infeasibility Problem: 46.014/46.014 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 46.014/46.014 (STRATEGY CONTEXTSENSITIVE 46.014/46.014 (even 1) 46.014/46.014 (odd 1) 46.014/46.014 (0) 46.014/46.014 (fSNonEmpty) 46.014/46.014 (false) 46.014/46.014 (s 1) 46.014/46.014 (true) 46.014/46.014 ) 46.014/46.014 (RULES 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x:S:S)) -> false | odd(x:S:S) ->* false 46.014/46.014 even(s(x:S:S)) -> true | odd(x:S:S) ->* true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x:S:S)) -> false | even(x:S:S) ->* false 46.014/46.014 odd(s(x:S:S)) -> true | even(x:S:S) ->* true 46.014/46.014 )] 46.014/46.014 46.014/46.014 Infeasibility Conditions: 46.014/46.014 even(x:S:S) ->* true 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 Obtaining a proof using Prover9: 46.014/46.014 46.014/46.014 -> Prover9 Output: 46.014/46.014 ============================== Prover9 =============================== 46.014/46.014 Prover9 (64) version 2009-11A, November 2009. 46.014/46.014 Process 42804 was started by ubuntu on ubuntu, 46.014/46.014 Wed Jul 14 10:26:53 2021 46.014/46.014 The command was "./prover9 -f /tmp/prover942794-0.in". 46.014/46.014 ============================== end of head =========================== 46.014/46.014 46.014/46.014 ============================== INPUT ================================= 46.014/46.014 46.014/46.014 % Reading from file /tmp/prover942794-0.in 46.014/46.014 46.014/46.014 assign(max_seconds,20). 46.014/46.014 46.014/46.014 formulas(assumptions). 46.014/46.014 ->*_s0(x,x) # label(reflexivity). 46.014/46.014 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 46.014/46.014 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence). 46.014/46.014 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence). 46.014/46.014 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 46.014/46.014 ->_s0(even(0),true) # label(replacement). 46.014/46.014 ->*_s0(odd(x2),false) -> ->_s0(even(s(x2)),false) # label(replacement). 46.014/46.014 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement). 46.014/46.014 ->_s0(odd(0),false) # label(replacement). 46.014/46.014 ->*_s0(even(x2),false) -> ->_s0(odd(s(x2)),false) # label(replacement). 46.014/46.014 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(goals). 46.014/46.014 (exists x2 ->*_s0(even(x2),true)) # label(goal). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== end of input ========================== 46.014/46.014 46.014/46.014 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46.014/46.014 46.014/46.014 % Formulas that are not ordinary clauses: 46.014/46.014 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.014/46.014 2 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 3 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 5 ->*_s0(odd(x2),false) -> ->_s0(even(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 6 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 7 ->*_s0(even(x2),false) -> ->_s0(odd(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 9 (exists x2 ->*_s0(even(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.014/46.014 46.014/46.014 ============================== end of process non-clausal formulas === 46.014/46.014 46.014/46.014 ============================== PROCESS INITIAL CLAUSES =============== 46.014/46.014 46.014/46.014 % Clauses before input processing: 46.014/46.014 46.014/46.014 formulas(usable). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(sos). 46.014/46.014 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 -->*_s0(odd(x),false) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.014/46.014 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.014/46.014 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 -->*_s0(even(x),false) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.014/46.014 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.014/46.014 -->*_s0(even(x),true) # label(goal). [deny(9)]. 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(demodulators). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== PREDICATE ELIMINATION ================= 46.014/46.014 46.014/46.014 No predicates eliminated. 46.014/46.014 46.014/46.014 ============================== end predicate elimination ============= 46.014/46.014 46.014/46.014 Auto_denials: 46.014/46.014 % copying label goal to answer in negative clause 46.014/46.014 46.014/46.014 Term ordering decisions: 46.014/46.014 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 46.014/46.014 Function symbol precedence: function_order([ false, true, 0, even, odd, s ]). 46.014/46.014 After inverse_order: (no changes). 46.014/46.014 Unfolding symbols: (none). 46.014/46.014 46.014/46.014 Auto_inference settings: 46.014/46.014 % set(neg_binary_resolution). % (HNE depth_diff=-7) 46.014/46.014 % clear(ordered_res). % (HNE depth_diff=-7) 46.014/46.014 % set(ur_resolution). % (HNE depth_diff=-7) 46.014/46.014 % set(ur_resolution) -> set(pos_ur_resolution). 46.014/46.014 % set(ur_resolution) -> set(neg_ur_resolution). 46.014/46.014 46.014/46.014 Auto_process settings: (no changes). 46.014/46.014 46.014/46.014 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 kept: 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 kept: 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 kept: 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 kept: 16 -->*_s0(odd(x),false) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.014/46.014 kept: 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.014/46.014 kept: 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 kept: 19 -->*_s0(even(x),false) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.014/46.014 kept: 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.014/46.014 kept: 21 -->*_s0(even(x),true) # label(goal) # answer(goal). [deny(9)]. 46.014/46.014 46.014/46.014 ============================== end of process initial clauses ======== 46.014/46.014 46.014/46.014 ============================== CLAUSES FOR SEARCH ==================== 46.014/46.014 46.014/46.014 % Clauses after input processing: 46.014/46.014 46.014/46.014 formulas(usable). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(sos). 46.014/46.014 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 16 -->*_s0(odd(x),false) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.014/46.014 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.014/46.014 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.014/46.014 19 -->*_s0(even(x),false) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.014/46.014 21 -->*_s0(even(x),true) # label(goal) # answer(goal). [deny(9)]. 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(demodulators). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== end of clauses for search ============= 46.014/46.014 46.014/46.014 ============================== SEARCH ================================ 46.014/46.014 46.014/46.014 % Starting search at 0.00 seconds. 46.014/46.014 46.014/46.014 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 46.014/46.014 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 46.014/46.014 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.014/46.014 46.014/46.014 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.014/46.014 46.014/46.014 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.014/46.014 46.014/46.014 given #6 (I,wt=4): 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 46.014/46.014 ============================== PROOF ================================= 46.014/46.014 46.014/46.014 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 46.014/46.014 % Length of proof is 8. 46.014/46.014 % Level of proof is 3. 46.014/46.014 % Maximum clause weight is 9.000. 46.014/46.014 % Given clauses 6. 46.014/46.014 46.014/46.014 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.014/46.014 9 (exists x2 ->*_s0(even(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.014/46.014 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.014/46.014 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.014/46.014 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.014/46.014 21 -->*_s0(even(x),true) # label(goal) # answer(goal). [deny(9)]. 46.014/46.014 25 ->*_s0(even(0),true). [ur(11,a,15,a,b,10,a)]. 46.014/46.014 26 $F # answer(goal). [resolve(25,a,21,a)]. 46.014/46.014 46.014/46.014 ============================== end of proof ========================== 46.014/46.014 46.014/46.014 ============================== STATISTICS ============================ 46.014/46.014 46.014/46.014 Given=6. Generated=16. Kept=16. proofs=1. 46.014/46.014 Usable=6. Sos=5. Demods=0. Limbo=3, Disabled=13. Hints=0. 46.014/46.014 Kept_by_rule=0, Deleted_by_rule=0. 46.014/46.014 Forward_subsumed=0. Back_subsumed=1. 46.014/46.014 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 46.014/46.014 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 46.014/46.014 Demod_attempts=0. Demod_rewrites=0. 46.014/46.014 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 46.014/46.014 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=10. 46.014/46.014 Megabytes=0.07. 46.014/46.014 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46.014/46.014 46.014/46.014 ============================== end of statistics ===================== 46.014/46.014 46.014/46.014 ============================== end of search ========================= 46.014/46.014 46.014/46.014 THEOREM PROVED 46.014/46.014 46.014/46.014 Exiting with 1 proof. 46.014/46.014 46.014/46.014 Process 42804 exit (max_proofs) Wed Jul 14 10:26:53 2021 46.014/46.014 46.014/46.014 46.014/46.014 The problem is feasible. 46.014/46.014 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.014/46.014 Confluence Problem: 46.014/46.014 (VAR x:S) 46.014/46.014 (STRATEGY CONTEXTSENSITIVE 46.014/46.014 (even 1) 46.014/46.014 (odd 1) 46.014/46.014 (0) 46.014/46.014 (fSNonEmpty) 46.014/46.014 (false) 46.014/46.014 (s 1) 46.014/46.014 (true) 46.014/46.014 ) 46.014/46.014 (RULES 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x:S)) -> false | odd(x:S) ->* false 46.014/46.014 even(s(x:S)) -> true | odd(x:S) ->* true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x:S)) -> false | even(x:S) ->* false 46.014/46.014 odd(s(x:S)) -> true | even(x:S) ->* true 46.014/46.014 ) 46.014/46.014 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.014/46.014 46.014/46.014 Critical Pairs Processor: 46.014/46.014 -> Rules: 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x:S)) -> false | odd(x:S) ->* false 46.014/46.014 even(s(x:S)) -> true | odd(x:S) ->* true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x:S)) -> false | even(x:S) ->* false 46.014/46.014 odd(s(x:S)) -> true | even(x:S) ->* true 46.014/46.014 -> Vars: 46.014/46.014 "x", "x", "x", "x" 46.014/46.014 -> FVars: 46.014/46.014 "x2", "x3", "x4", "x5" 46.014/46.014 -> PVars: 46.014/46.014 "x": ["x2", "x3", "x4", "x5"] 46.014/46.014 46.014/46.014 -> Rlps: 46.014/46.014 crule: even(0) -> true, id: 1, possubterms: even(0)-> [], 0-> [1] 46.014/46.014 crule: even(s(x2:S)) -> false | odd(x2:S) ->* false, id: 2, possubterms: even(s(x2:S))-> [], s(x2:S)-> [1] 46.014/46.014 crule: even(s(x3:S)) -> true | odd(x3:S) ->* true, id: 3, possubterms: even(s(x3:S))-> [], s(x3:S)-> [1] 46.014/46.014 crule: odd(0) -> false, id: 4, possubterms: odd(0)-> [], 0-> [1] 46.014/46.014 crule: odd(s(x4:S)) -> false | even(x4:S) ->* false, id: 5, possubterms: odd(s(x4:S))-> [], s(x4:S)-> [1] 46.014/46.014 crule: odd(s(x5:S)) -> true | even(x5:S) ->* true, id: 6, possubterms: odd(s(x5:S))-> [], s(x5:S)-> [1] 46.014/46.014 46.014/46.014 -> Unifications: 46.014/46.014 R3 unifies with R2 at p: [], l: even(s(x3:S)), lp: even(s(x3:S)), conds: {odd(x3:S) ->* true, odd(x:S) ->* false}, sig: {x3:S -> x:S}, l': even(s(x:S)), r: true, r': false 46.014/46.014 R6 unifies with R5 at p: [], l: odd(s(x5:S)), lp: odd(s(x5:S)), conds: {even(x5:S) ->* true, even(x:S) ->* false}, sig: {x5:S -> x:S}, l': odd(s(x:S)), r: true, r': false 46.014/46.014 46.014/46.014 -> Critical pairs info: 46.014/46.014 | even(x:S) ->* true, even(x:S) ->* false => Not trivial, Overlay, N1 46.014/46.014 | odd(x:S) ->* true, odd(x:S) ->* false => Not trivial, Overlay, N2 46.014/46.014 46.014/46.014 -> Problem conclusions: 46.014/46.014 Left linear, Right linear, Linear 46.014/46.014 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 46.014/46.014 CTRS Type: 1 46.014/46.014 Deterministic, Strongly deterministic 46.014/46.014 Oriented CTRS, Properly oriented CTRS, Not join CTRS 46.014/46.014 Maybe right-stable CTRS, Overlay CTRS 46.014/46.014 Maybe normal CTRS, Maybe almost normal CTRS 46.014/46.014 Maybe terminating CTRS, Maybe joinable CCPs 46.014/46.014 Maybe level confluent 46.014/46.014 Maybe confluent 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 Clean Infeasible CCPs Processor: 46.014/46.014 Num of CPIs: 2 46.014/46.014 Timeout: 60 46.014/46.014 Timeout for each infeasibility problem: 60 s 46.014/46.014 | even(x:S) ->* true, even(x:S) ->* false => Not trivial, Overlay, N1 46.014/46.014 (PROBLEM INFEASIBILITY) 46.014/46.014 (VAR x) 46.014/46.014 (RULES 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x)) -> false | odd(x) ->* false 46.014/46.014 even(s(x)) -> true | odd(x) ->* true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x)) -> false | even(x) ->* false 46.014/46.014 odd(s(x)) -> true | even(x) ->* true) 46.014/46.014 (VAR x2) 46.014/46.014 (CONDITION even(x2) ->* true, even(x2) ->* false) 46.014/46.014 46.014/46.014 Proof: 46.014/46.014 YES 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 Infeasibility Problem: 46.014/46.014 [(VAR vNonEmpty:S x:S x2:S) 46.014/46.014 (STRATEGY CONTEXTSENSITIVE 46.014/46.014 (even 1) 46.014/46.014 (odd 1) 46.014/46.014 (0) 46.014/46.014 (fSNonEmpty) 46.014/46.014 (false) 46.014/46.014 (s 1) 46.014/46.014 (true) 46.014/46.014 ) 46.014/46.014 (RULES 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x:S)) -> false | odd(x:S) ->* false 46.014/46.014 even(s(x:S)) -> true | odd(x:S) ->* true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x:S)) -> false | even(x:S) ->* false 46.014/46.014 odd(s(x:S)) -> true | even(x:S) ->* true 46.014/46.014 )] 46.014/46.014 46.014/46.014 Infeasibility Conditions: 46.014/46.014 even(x2:S) ->* true, even(x2:S) ->* false 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 Obtaining a model using Mace4: 46.014/46.014 46.014/46.014 -> Usable Rules: 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x:S)) -> false | odd(x:S) ->* false 46.014/46.014 even(s(x:S)) -> true | odd(x:S) ->* true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x:S)) -> false | even(x:S) ->* false 46.014/46.014 odd(s(x:S)) -> true | even(x:S) ->* true 46.014/46.014 46.014/46.014 -> Mace4 Output: 46.014/46.014 ============================== Mace4 ================================= 46.014/46.014 Mace4 (64) version 2009-11A, November 2009. 46.014/46.014 Process 42833 was started by ubuntu on ubuntu, 46.014/46.014 Wed Jul 14 10:26:53 2021 46.014/46.014 The command was "./mace4 -c -f /tmp/mace442816-2.in". 46.014/46.014 ============================== end of head =========================== 46.014/46.014 46.014/46.014 ============================== INPUT ================================= 46.014/46.014 46.014/46.014 % Reading from file /tmp/mace442816-2.in 46.014/46.014 46.014/46.014 assign(max_seconds,20). 46.014/46.014 46.014/46.014 formulas(assumptions). 46.014/46.014 ->(x1,y) -> ->(even(x1),even(y)) # label(congruence). 46.014/46.014 ->(x1,y) -> ->(odd(x1),odd(y)) # label(congruence). 46.014/46.014 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 46.014/46.014 ->(even(0),true) # label(replacement). 46.014/46.014 ->*(odd(x1),false) -> ->(even(s(x1)),false) # label(replacement). 46.014/46.014 ->*(odd(x1),true) -> ->(even(s(x1)),true) # label(replacement). 46.014/46.014 ->(odd(0),false) # label(replacement). 46.014/46.014 ->*(even(x1),false) -> ->(odd(s(x1)),false) # label(replacement). 46.014/46.014 ->*(even(x1),true) -> ->(odd(s(x1)),true) # label(replacement). 46.014/46.014 ->*(x,x) # label(reflexivity). 46.014/46.014 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(goals). 46.014/46.014 (exists x2 (->*(even(x2),true) & ->*(even(x2),false))) # label(goal). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== end of input ========================== 46.014/46.014 46.014/46.014 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46.014/46.014 46.014/46.014 % Formulas that are not ordinary clauses: 46.014/46.014 1 ->(x1,y) -> ->(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 2 ->(x1,y) -> ->(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 3 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 4 ->*(odd(x1),false) -> ->(even(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 5 ->*(odd(x1),true) -> ->(even(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 6 ->*(even(x1),false) -> ->(odd(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 7 ->*(even(x1),true) -> ->(odd(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 8 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.014/46.014 9 (exists x2 (->*(even(x2),true) & ->*(even(x2),false))) # label(goal) # label(non_clause) # label(goal). [goal]. 46.014/46.014 46.014/46.014 ============================== end of process non-clausal formulas === 46.014/46.014 46.014/46.014 ============================== CLAUSES FOR SEARCH ==================== 46.014/46.014 46.014/46.014 formulas(mace4_clauses). 46.014/46.014 -->(x,y) | ->(even(x),even(y)) # label(congruence). 46.014/46.014 -->(x,y) | ->(odd(x),odd(y)) # label(congruence). 46.014/46.014 -->(x,y) | ->(s(x),s(y)) # label(congruence). 46.014/46.014 ->(even(0),true) # label(replacement). 46.014/46.014 -->*(odd(x),false) | ->(even(s(x)),false) # label(replacement). 46.014/46.014 -->*(odd(x),true) | ->(even(s(x)),true) # label(replacement). 46.014/46.014 ->(odd(0),false) # label(replacement). 46.014/46.014 -->*(even(x),false) | ->(odd(s(x)),false) # label(replacement). 46.014/46.014 -->*(even(x),true) | ->(odd(s(x)),true) # label(replacement). 46.014/46.014 ->*(x,x) # label(reflexivity). 46.014/46.014 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 46.014/46.014 -->*(even(x),true) | -->*(even(x),false) # label(goal). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== end of clauses for search ============= 46.014/46.014 46.014/46.014 % There are no natural numbers in the input. 46.014/46.014 46.014/46.014 ============================== DOMAIN SIZE 2 ========================= 46.014/46.014 46.014/46.014 ============================== MODEL ================================= 46.014/46.014 46.014/46.014 interpretation( 2, [number=1, seconds=0], [ 46.014/46.014 46.014/46.014 function(0, [ 0 ]), 46.014/46.014 46.014/46.014 function(false, [ 0 ]), 46.014/46.014 46.014/46.014 function(true, [ 1 ]), 46.014/46.014 46.014/46.014 function(even(_), [ 1, 0 ]), 46.014/46.014 46.014/46.014 function(odd(_), [ 0, 1 ]), 46.014/46.014 46.014/46.014 function(s(_), [ 1, 0 ]), 46.014/46.014 46.014/46.014 relation(->*(_,_), [ 46.014/46.014 1, 0, 46.014/46.014 0, 1 ]), 46.014/46.014 46.014/46.014 relation(->(_,_), [ 46.014/46.014 1, 0, 46.014/46.014 0, 1 ]) 46.014/46.014 ]). 46.014/46.014 46.014/46.014 ============================== end of model ========================== 46.014/46.014 46.014/46.014 ============================== STATISTICS ============================ 46.014/46.014 46.014/46.014 For domain size 2. 46.014/46.014 46.014/46.014 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 46.014/46.014 Ground clauses: seen=34, kept=30. 46.014/46.014 Selections=7, assignments=11, propagations=17, current_models=1. 46.014/46.014 Rewrite_terms=118, rewrite_bools=63, indexes=24. 46.014/46.014 Rules_from_neg_clauses=3, cross_offs=3. 46.014/46.014 46.014/46.014 ============================== end of statistics ===================== 46.014/46.014 46.014/46.014 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46.014/46.014 46.014/46.014 Exiting with 1 model. 46.014/46.014 46.014/46.014 Process 42833 exit (max_models) Wed Jul 14 10:26:53 2021 46.014/46.014 The process finished Wed Jul 14 10:26:53 2021 46.014/46.014 46.014/46.014 46.014/46.014 Mace4 cooked interpretation: 46.014/46.014 46.014/46.014 46.014/46.014 46.014/46.014 The problem is infeasible. 46.014/46.014 46.014/46.014 46.014/46.014 46.014/46.014 46.014/46.014 | odd(x:S) ->* true, odd(x:S) ->* false => Not trivial, Overlay, N2 46.014/46.014 (PROBLEM INFEASIBILITY) 46.014/46.014 (VAR x) 46.014/46.014 (RULES 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x)) -> false | odd(x) ->* false 46.014/46.014 even(s(x)) -> true | odd(x) ->* true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x)) -> false | even(x) ->* false 46.014/46.014 odd(s(x)) -> true | even(x) ->* true) 46.014/46.014 (VAR x2) 46.014/46.014 (CONDITION odd(x2) ->* true, odd(x2) ->* false) 46.014/46.014 46.014/46.014 Proof: 46.014/46.014 YES 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 Infeasibility Problem: 46.014/46.014 [(VAR vNonEmpty:S x:S x2:S) 46.014/46.014 (STRATEGY CONTEXTSENSITIVE 46.014/46.014 (even 1) 46.014/46.014 (odd 1) 46.014/46.014 (0) 46.014/46.014 (fSNonEmpty) 46.014/46.014 (false) 46.014/46.014 (s 1) 46.014/46.014 (true) 46.014/46.014 ) 46.014/46.014 (RULES 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x:S)) -> false | odd(x:S) ->* false 46.014/46.014 even(s(x:S)) -> true | odd(x:S) ->* true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x:S)) -> false | even(x:S) ->* false 46.014/46.014 odd(s(x:S)) -> true | even(x:S) ->* true 46.014/46.014 )] 46.014/46.014 46.014/46.014 Infeasibility Conditions: 46.014/46.014 odd(x2:S) ->* true, odd(x2:S) ->* false 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 Obtaining a model using Mace4: 46.014/46.014 46.014/46.014 -> Usable Rules: 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x:S)) -> false | odd(x:S) ->* false 46.014/46.014 even(s(x:S)) -> true | odd(x:S) ->* true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x:S)) -> false | even(x:S) ->* false 46.014/46.014 odd(s(x:S)) -> true | even(x:S) ->* true 46.014/46.014 46.014/46.014 -> Mace4 Output: 46.014/46.014 ============================== Mace4 ================================= 46.014/46.014 Mace4 (64) version 2009-11A, November 2009. 46.014/46.014 Process 42913 was started by ubuntu on ubuntu, 46.014/46.014 Wed Jul 14 10:27:13 2021 46.014/46.014 The command was "./mace4 -c -f /tmp/mace442894-2.in". 46.014/46.014 ============================== end of head =========================== 46.014/46.014 46.014/46.014 ============================== INPUT ================================= 46.014/46.014 46.014/46.014 % Reading from file /tmp/mace442894-2.in 46.014/46.014 46.014/46.014 assign(max_seconds,20). 46.014/46.014 46.014/46.014 formulas(assumptions). 46.014/46.014 ->(x1,y) -> ->(even(x1),even(y)) # label(congruence). 46.014/46.014 ->(x1,y) -> ->(odd(x1),odd(y)) # label(congruence). 46.014/46.014 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 46.014/46.014 ->(even(0),true) # label(replacement). 46.014/46.014 ->*(odd(x1),false) -> ->(even(s(x1)),false) # label(replacement). 46.014/46.014 ->*(odd(x1),true) -> ->(even(s(x1)),true) # label(replacement). 46.014/46.014 ->(odd(0),false) # label(replacement). 46.014/46.014 ->*(even(x1),false) -> ->(odd(s(x1)),false) # label(replacement). 46.014/46.014 ->*(even(x1),true) -> ->(odd(s(x1)),true) # label(replacement). 46.014/46.014 ->*(x,x) # label(reflexivity). 46.014/46.014 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 formulas(goals). 46.014/46.014 (exists x2 (->*(odd(x2),true) & ->*(odd(x2),false))) # label(goal). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== end of input ========================== 46.014/46.014 46.014/46.014 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46.014/46.014 46.014/46.014 % Formulas that are not ordinary clauses: 46.014/46.014 1 ->(x1,y) -> ->(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 2 ->(x1,y) -> ->(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 3 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46.014/46.014 4 ->*(odd(x1),false) -> ->(even(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 5 ->*(odd(x1),true) -> ->(even(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 6 ->*(even(x1),false) -> ->(odd(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 7 ->*(even(x1),true) -> ->(odd(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 46.014/46.014 8 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.014/46.014 9 (exists x2 (->*(odd(x2),true) & ->*(odd(x2),false))) # label(goal) # label(non_clause) # label(goal). [goal]. 46.014/46.014 46.014/46.014 ============================== end of process non-clausal formulas === 46.014/46.014 46.014/46.014 ============================== CLAUSES FOR SEARCH ==================== 46.014/46.014 46.014/46.014 formulas(mace4_clauses). 46.014/46.014 -->(x,y) | ->(even(x),even(y)) # label(congruence). 46.014/46.014 -->(x,y) | ->(odd(x),odd(y)) # label(congruence). 46.014/46.014 -->(x,y) | ->(s(x),s(y)) # label(congruence). 46.014/46.014 ->(even(0),true) # label(replacement). 46.014/46.014 -->*(odd(x),false) | ->(even(s(x)),false) # label(replacement). 46.014/46.014 -->*(odd(x),true) | ->(even(s(x)),true) # label(replacement). 46.014/46.014 ->(odd(0),false) # label(replacement). 46.014/46.014 -->*(even(x),false) | ->(odd(s(x)),false) # label(replacement). 46.014/46.014 -->*(even(x),true) | ->(odd(s(x)),true) # label(replacement). 46.014/46.014 ->*(x,x) # label(reflexivity). 46.014/46.014 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 46.014/46.014 -->*(odd(x),true) | -->*(odd(x),false) # label(goal). 46.014/46.014 end_of_list. 46.014/46.014 46.014/46.014 ============================== end of clauses for search ============= 46.014/46.014 46.014/46.014 % There are no natural numbers in the input. 46.014/46.014 46.014/46.014 ============================== DOMAIN SIZE 2 ========================= 46.014/46.014 46.014/46.014 ============================== MODEL ================================= 46.014/46.014 46.014/46.014 interpretation( 2, [number=1, seconds=0], [ 46.014/46.014 46.014/46.014 function(0, [ 0 ]), 46.014/46.014 46.014/46.014 function(false, [ 0 ]), 46.014/46.014 46.014/46.014 function(true, [ 1 ]), 46.014/46.014 46.014/46.014 function(even(_), [ 1, 0 ]), 46.014/46.014 46.014/46.014 function(odd(_), [ 0, 1 ]), 46.014/46.014 46.014/46.014 function(s(_), [ 1, 0 ]), 46.014/46.014 46.014/46.014 relation(->*(_,_), [ 46.014/46.014 1, 0, 46.014/46.014 0, 1 ]), 46.014/46.014 46.014/46.014 relation(->(_,_), [ 46.014/46.014 1, 0, 46.014/46.014 0, 1 ]) 46.014/46.014 ]). 46.014/46.014 46.014/46.014 ============================== end of model ========================== 46.014/46.014 46.014/46.014 ============================== STATISTICS ============================ 46.014/46.014 46.014/46.014 For domain size 2. 46.014/46.014 46.014/46.014 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 46.014/46.014 Ground clauses: seen=34, kept=30. 46.014/46.014 Selections=11, assignments=18, propagations=22, current_models=1. 46.014/46.014 Rewrite_terms=142, rewrite_bools=102, indexes=32. 46.014/46.014 Rules_from_neg_clauses=2, cross_offs=2. 46.014/46.014 46.014/46.014 ============================== end of statistics ===================== 46.014/46.014 46.014/46.014 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46.014/46.014 46.014/46.014 Exiting with 1 model. 46.014/46.014 46.014/46.014 Process 42913 exit (max_models) Wed Jul 14 10:27:13 2021 46.014/46.014 The process finished Wed Jul 14 10:27:13 2021 46.014/46.014 46.014/46.014 46.014/46.014 Mace4 cooked interpretation: 46.014/46.014 46.014/46.014 46.014/46.014 46.014/46.014 The problem is infeasible. 46.014/46.014 46.014/46.014 46.014/46.014 46.014/46.014 46.014/46.014 -> Problem conclusions: 46.014/46.014 Left linear, Right linear, Linear 46.014/46.014 Weakly orthogonal, Almost orthogonal, Orthogonal 46.014/46.014 CTRS Type: 1 46.014/46.014 Deterministic, Strongly deterministic 46.014/46.014 Oriented CTRS, Properly oriented CTRS, Not join CTRS 46.014/46.014 Maybe right-stable CTRS, Overlay CTRS 46.014/46.014 Maybe normal CTRS, Maybe almost normal CTRS 46.014/46.014 Maybe terminating CTRS, Joinable CCPs 46.014/46.014 Maybe level confluent 46.014/46.014 Maybe confluent 46.014/46.014 46.014/46.014 Resulting CCPs: 46.014/46.014 No CCPs left 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 Underlying TRS Termination Processor: 46.014/46.014 46.014/46.014 Resulting Underlying TRS: 46.014/46.014 (VAR x:S) 46.014/46.014 (STRATEGY CONTEXTSENSITIVE 46.014/46.014 (even 1) 46.014/46.014 (odd 1) 46.014/46.014 (false) 46.014/46.014 (true) 46.014/46.014 ) 46.014/46.014 (RULES 46.014/46.014 even(0) -> true 46.014/46.014 even(s(x:S)) -> false 46.014/46.014 even(s(x:S)) -> true 46.014/46.014 odd(0) -> false 46.014/46.014 odd(s(x:S)) -> false 46.014/46.014 odd(s(x:S)) -> true 46.014/46.014 ) 46.014/46.014 Underlying TRS terminating? 46.014/46.014 YES 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 (VAR vu95NonEmpty:S xu58S:S) 46.014/46.014 (RULES 46.014/46.014 even(num0) -> ftrue 46.014/46.014 even(s(xu58S:S)) -> ffalse 46.014/46.014 even(s(xu58S:S)) -> ftrue 46.014/46.014 odd(num0) -> ffalse 46.014/46.014 odd(s(xu58S:S)) -> ffalse 46.014/46.014 odd(s(xu58S:S)) -> ftrue 46.014/46.014 ) 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 Dependency Pairs Processor: 46.014/46.014 -> Pairs: 46.014/46.014 Empty 46.014/46.014 -> Rules: 46.014/46.014 even(num0) -> ftrue 46.014/46.014 even(s(xu58S:S)) -> ffalse 46.014/46.014 even(s(xu58S:S)) -> ftrue 46.014/46.014 odd(num0) -> ffalse 46.014/46.014 odd(s(xu58S:S)) -> ffalse 46.014/46.014 odd(s(xu58S:S)) -> ftrue 46.014/46.014 46.014/46.014 Problem 1: 46.014/46.014 46.014/46.014 SCC Processor: 46.014/46.014 -> Pairs: 46.014/46.014 Empty 46.014/46.014 -> Rules: 46.014/46.014 even(num0) -> ftrue 46.014/46.014 even(s(xu58S:S)) -> ffalse 46.014/46.014 even(s(xu58S:S)) -> ftrue 46.014/46.014 odd(num0) -> ffalse 46.014/46.014 odd(s(xu58S:S)) -> ffalse 46.014/46.014 odd(s(xu58S:S)) -> ftrue 46.014/46.014 ->Strongly Connected Components: 46.014/46.014 There is no strongly connected component 46.014/46.014 46.014/46.014 The problem is finite. 46.014/46.014 46.014/46.014 46.014/46.014 -> Problem conclusions: 46.014/46.014 Left linear, Right linear, Linear 46.014/46.014 Weakly orthogonal, Almost orthogonal, Orthogonal 46.014/46.014 CTRS Type: 1 46.014/46.014 Deterministic, Strongly deterministic 46.014/46.014 Oriented CTRS, Properly oriented CTRS, Not join CTRS 46.014/46.014 Maybe right-stable CTRS, Overlay CTRS 46.014/46.014 Maybe normal CTRS, Maybe almost normal CTRS 46.014/46.014 Terminating CTRS, Joinable CCPs 46.014/46.014 Maybe level confluent 46.014/46.014 Confluent 46.014/46.014 46.014/46.014 The problem is joinable. 46.014/46.014 84.47user 2.08system 0:46.14elapsed 187%CPU (0avgtext+0avgdata 142928maxresident)k 46.014/46.014 0inputs+0outputs (0major+184779minor)pagefaults 0swaps