46.027/46.027 YES 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 46.027/46.027 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.027/46.027 Confluence Problem: 46.027/46.027 (VAR vNonEmpty:S x:S) 46.027/46.027 (STRATEGY CONTEXTSENSITIVE 46.027/46.027 (even 1) 46.027/46.027 (odd 1) 46.027/46.027 (0) 46.027/46.027 (fSNonEmpty) 46.027/46.027 (false) 46.027/46.027 (s 1) 46.027/46.027 (true) 46.027/46.027 ) 46.027/46.027 (RULES 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x:S)) -> false | even(x:S) ->* true 46.027/46.027 even(s(x:S)) -> true | odd(x:S) ->* true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x:S)) -> false | odd(x:S) ->* true 46.027/46.027 odd(s(x:S)) -> true | even(x:S) ->* true 46.027/46.027 ) 46.027/46.027 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.027/46.027 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 Inlining of Conditions Processor [STERN17]: 46.027/46.027 46.027/46.027 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.027/46.027 Confluence Problem: 46.027/46.027 (VAR vNonEmpty:S x:S) 46.027/46.027 (STRATEGY CONTEXTSENSITIVE 46.027/46.027 (even 1) 46.027/46.027 (odd 1) 46.027/46.027 (0) 46.027/46.027 (fSNonEmpty) 46.027/46.027 (false) 46.027/46.027 (s 1) 46.027/46.027 (true) 46.027/46.027 ) 46.027/46.027 (RULES 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x:S)) -> false | even(x:S) ->* true 46.027/46.027 even(s(x:S)) -> true | odd(x:S) ->* true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x:S)) -> false | odd(x:S) ->* true 46.027/46.027 odd(s(x:S)) -> true | even(x:S) ->* true 46.027/46.027 ) 46.027/46.027 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.027/46.027 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 Clean CTRS Processor: 46.027/46.027 46.027/46.027 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.027/46.027 Confluence Problem: 46.027/46.027 (VAR x:S) 46.027/46.027 (STRATEGY CONTEXTSENSITIVE 46.027/46.027 (even 1) 46.027/46.027 (odd 1) 46.027/46.027 (0) 46.027/46.027 (fSNonEmpty) 46.027/46.027 (false) 46.027/46.027 (s 1) 46.027/46.027 (true) 46.027/46.027 ) 46.027/46.027 (RULES 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x:S)) -> false | even(x:S) ->* true 46.027/46.027 even(s(x:S)) -> true | odd(x:S) ->* true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x:S)) -> false | odd(x:S) ->* true 46.027/46.027 odd(s(x:S)) -> true | even(x:S) ->* true 46.027/46.027 ) 46.027/46.027 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.027/46.027 46.027/46.027 CRule InfChecker Info: 46.027/46.027 even(0) -> true 46.027/46.027 Rule remains 46.027/46.027 Proof: 46.027/46.027 NO_CONDS 46.027/46.027 46.027/46.027 CRule InfChecker Info: 46.027/46.027 even(s(x:S)) -> false | even(x:S) ->* true 46.027/46.027 Rule remains 46.027/46.027 Proof: 46.027/46.027 NO 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 Infeasibility Problem: 46.027/46.027 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 46.027/46.027 (STRATEGY CONTEXTSENSITIVE 46.027/46.027 (even 1) 46.027/46.027 (odd 1) 46.027/46.027 (0) 46.027/46.027 (fSNonEmpty) 46.027/46.027 (false) 46.027/46.027 (s 1) 46.027/46.027 (true) 46.027/46.027 ) 46.027/46.027 (RULES 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x:S:S)) -> false | even(x:S:S) ->* true 46.027/46.027 even(s(x:S:S)) -> true | odd(x:S:S) ->* true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x:S:S)) -> false | odd(x:S:S) ->* true 46.027/46.027 odd(s(x:S:S)) -> true | even(x:S:S) ->* true 46.027/46.027 )] 46.027/46.027 46.027/46.027 Infeasibility Conditions: 46.027/46.027 even(x:S:S) ->* true 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 Obtaining a proof using Prover9: 46.027/46.027 46.027/46.027 -> Prover9 Output: 46.027/46.027 ============================== Prover9 =============================== 46.027/46.027 Prover9 (64) version 2009-11A, November 2009. 46.027/46.027 Process 56479 was started by ubuntu on ubuntu, 46.027/46.027 Wed Jul 14 11:20:18 2021 46.027/46.027 The command was "./prover9 -f /tmp/prover956470-0.in". 46.027/46.027 ============================== end of head =========================== 46.027/46.027 46.027/46.027 ============================== INPUT ================================= 46.027/46.027 46.027/46.027 % Reading from file /tmp/prover956470-0.in 46.027/46.027 46.027/46.027 assign(max_seconds,20). 46.027/46.027 46.027/46.027 formulas(assumptions). 46.027/46.027 ->*_s0(x,x) # label(reflexivity). 46.027/46.027 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 46.027/46.027 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence). 46.027/46.027 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence). 46.027/46.027 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 46.027/46.027 ->_s0(even(0),true) # label(replacement). 46.027/46.027 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement). 46.027/46.027 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement). 46.027/46.027 ->_s0(odd(0),false) # label(replacement). 46.027/46.027 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement). 46.027/46.027 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(goals). 46.027/46.027 (exists x2 ->*_s0(even(x2),true)) # label(goal). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== end of input ========================== 46.027/46.027 46.027/46.027 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46.027/46.027 46.027/46.027 % Formulas that are not ordinary clauses: 46.027/46.027 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.027/46.027 2 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 3 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 5 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 6 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 7 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 9 (exists x2 ->*_s0(even(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.027/46.027 46.027/46.027 ============================== end of process non-clausal formulas === 46.027/46.027 46.027/46.027 ============================== PROCESS INITIAL CLAUSES =============== 46.027/46.027 46.027/46.027 % Clauses before input processing: 46.027/46.027 46.027/46.027 formulas(usable). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(sos). 46.027/46.027 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.027/46.027 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.027/46.027 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.027/46.027 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.027/46.027 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.027/46.027 -->*_s0(even(x),true) # label(goal). [deny(9)]. 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(demodulators). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== PREDICATE ELIMINATION ================= 46.027/46.027 46.027/46.027 No predicates eliminated. 46.027/46.027 46.027/46.027 ============================== end predicate elimination ============= 46.027/46.027 46.027/46.027 Auto_denials: 46.027/46.027 % copying label goal to answer in negative clause 46.027/46.027 46.027/46.027 Term ordering decisions: 46.027/46.027 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 46.027/46.027 Function symbol precedence: function_order([ true, false, 0, even, odd, s ]). 46.027/46.027 After inverse_order: (no changes). 46.027/46.027 Unfolding symbols: (none). 46.027/46.027 46.027/46.027 Auto_inference settings: 46.027/46.027 % set(neg_binary_resolution). % (HNE depth_diff=-7) 46.027/46.027 % clear(ordered_res). % (HNE depth_diff=-7) 46.027/46.027 % set(ur_resolution). % (HNE depth_diff=-7) 46.027/46.027 % set(ur_resolution) -> set(pos_ur_resolution). 46.027/46.027 % set(ur_resolution) -> set(neg_ur_resolution). 46.027/46.027 46.027/46.027 Auto_process settings: (no changes). 46.027/46.027 46.027/46.027 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 kept: 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 kept: 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 kept: 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 kept: 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.027/46.027 kept: 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.027/46.027 kept: 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.027/46.027 kept: 19 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.027/46.027 kept: 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.027/46.027 kept: 21 -->*_s0(even(x),true) # label(goal) # answer(goal). [deny(9)]. 46.027/46.027 46.027/46.027 ============================== end of process initial clauses ======== 46.027/46.027 46.027/46.027 ============================== CLAUSES FOR SEARCH ==================== 46.027/46.027 46.027/46.027 % Clauses after input processing: 46.027/46.027 46.027/46.027 formulas(usable). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(sos). 46.027/46.027 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.027/46.027 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.027/46.027 19 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.027/46.027 21 -->*_s0(even(x),true) # label(goal) # answer(goal). [deny(9)]. 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(demodulators). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== end of clauses for search ============= 46.027/46.027 46.027/46.027 ============================== SEARCH ================================ 46.027/46.027 46.027/46.027 % Starting search at 0.00 seconds. 46.027/46.027 46.027/46.027 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 46.027/46.027 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 46.027/46.027 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 46.027/46.027 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 46.027/46.027 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 46.027/46.027 given #6 (I,wt=4): 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 46.027/46.027 ============================== PROOF ================================= 46.027/46.027 46.027/46.027 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 46.027/46.027 % Length of proof is 8. 46.027/46.027 % Level of proof is 3. 46.027/46.027 % Maximum clause weight is 9.000. 46.027/46.027 % Given clauses 6. 46.027/46.027 46.027/46.027 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.027/46.027 9 (exists x2 ->*_s0(even(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.027/46.027 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 21 -->*_s0(even(x),true) # label(goal) # answer(goal). [deny(9)]. 46.027/46.027 25 ->*_s0(even(0),true). [ur(11,a,15,a,b,10,a)]. 46.027/46.027 26 $F # answer(goal). [resolve(25,a,21,a)]. 46.027/46.027 46.027/46.027 ============================== end of proof ========================== 46.027/46.027 46.027/46.027 ============================== STATISTICS ============================ 46.027/46.027 46.027/46.027 Given=6. Generated=16. Kept=16. proofs=1. 46.027/46.027 Usable=6. Sos=4. Demods=0. Limbo=3, Disabled=14. Hints=0. 46.027/46.027 Kept_by_rule=0, Deleted_by_rule=0. 46.027/46.027 Forward_subsumed=0. Back_subsumed=2. 46.027/46.027 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 46.027/46.027 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 46.027/46.027 Demod_attempts=0. Demod_rewrites=0. 46.027/46.027 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 46.027/46.027 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=11. 46.027/46.027 Megabytes=0.07. 46.027/46.027 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46.027/46.027 46.027/46.027 ============================== end of statistics ===================== 46.027/46.027 46.027/46.027 ============================== end of search ========================= 46.027/46.027 46.027/46.027 THEOREM PROVED 46.027/46.027 46.027/46.027 Exiting with 1 proof. 46.027/46.027 46.027/46.027 Process 56479 exit (max_proofs) Wed Jul 14 11:20:18 2021 46.027/46.027 46.027/46.027 46.027/46.027 The problem is feasible. 46.027/46.027 46.027/46.027 46.027/46.027 CRule InfChecker Info: 46.027/46.027 even(s(x:S)) -> true | odd(x:S) ->* true 46.027/46.027 Rule remains 46.027/46.027 Proof: 46.027/46.027 NO 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 Infeasibility Problem: 46.027/46.027 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 46.027/46.027 (STRATEGY CONTEXTSENSITIVE 46.027/46.027 (even 1) 46.027/46.027 (odd 1) 46.027/46.027 (0) 46.027/46.027 (fSNonEmpty) 46.027/46.027 (false) 46.027/46.027 (s 1) 46.027/46.027 (true) 46.027/46.027 ) 46.027/46.027 (RULES 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x:S:S)) -> false | even(x:S:S) ->* true 46.027/46.027 even(s(x:S:S)) -> true | odd(x:S:S) ->* true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x:S:S)) -> false | odd(x:S:S) ->* true 46.027/46.027 odd(s(x:S:S)) -> true | even(x:S:S) ->* true 46.027/46.027 )] 46.027/46.027 46.027/46.027 Infeasibility Conditions: 46.027/46.027 odd(x:S:S) ->* true 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 Obtaining a proof using Prover9: 46.027/46.027 46.027/46.027 -> Prover9 Output: 46.027/46.027 ============================== Prover9 =============================== 46.027/46.027 Prover9 (64) version 2009-11A, November 2009. 46.027/46.027 Process 56501 was started by ubuntu on ubuntu, 46.027/46.027 Wed Jul 14 11:20:18 2021 46.027/46.027 The command was "./prover9 -f /tmp/prover956492-0.in". 46.027/46.027 ============================== end of head =========================== 46.027/46.027 46.027/46.027 ============================== INPUT ================================= 46.027/46.027 46.027/46.027 % Reading from file /tmp/prover956492-0.in 46.027/46.027 46.027/46.027 assign(max_seconds,20). 46.027/46.027 46.027/46.027 formulas(assumptions). 46.027/46.027 ->*_s0(x,x) # label(reflexivity). 46.027/46.027 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 46.027/46.027 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence). 46.027/46.027 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence). 46.027/46.027 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 46.027/46.027 ->_s0(even(0),true) # label(replacement). 46.027/46.027 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement). 46.027/46.027 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement). 46.027/46.027 ->_s0(odd(0),false) # label(replacement). 46.027/46.027 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement). 46.027/46.027 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(goals). 46.027/46.027 (exists x2 ->*_s0(odd(x2),true)) # label(goal). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== end of input ========================== 46.027/46.027 46.027/46.027 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46.027/46.027 46.027/46.027 % Formulas that are not ordinary clauses: 46.027/46.027 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.027/46.027 2 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 3 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 5 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 6 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 7 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 9 (exists x2 ->*_s0(odd(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.027/46.027 46.027/46.027 ============================== end of process non-clausal formulas === 46.027/46.027 46.027/46.027 ============================== PROCESS INITIAL CLAUSES =============== 46.027/46.027 46.027/46.027 % Clauses before input processing: 46.027/46.027 46.027/46.027 formulas(usable). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(sos). 46.027/46.027 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.027/46.027 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.027/46.027 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.027/46.027 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.027/46.027 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.027/46.027 -->*_s0(odd(x),true) # label(goal). [deny(9)]. 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(demodulators). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== PREDICATE ELIMINATION ================= 46.027/46.027 46.027/46.027 No predicates eliminated. 46.027/46.027 46.027/46.027 ============================== end predicate elimination ============= 46.027/46.027 46.027/46.027 Auto_denials: 46.027/46.027 % copying label goal to answer in negative clause 46.027/46.027 46.027/46.027 Term ordering decisions: 46.027/46.027 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 46.027/46.027 Function symbol precedence: function_order([ true, false, 0, even, odd, s ]). 46.027/46.027 After inverse_order: (no changes). 46.027/46.027 Unfolding symbols: (none). 46.027/46.027 46.027/46.027 Auto_inference settings: 46.027/46.027 % set(neg_binary_resolution). % (HNE depth_diff=-7) 46.027/46.027 % clear(ordered_res). % (HNE depth_diff=-7) 46.027/46.027 % set(ur_resolution). % (HNE depth_diff=-7) 46.027/46.027 % set(ur_resolution) -> set(pos_ur_resolution). 46.027/46.027 % set(ur_resolution) -> set(neg_ur_resolution). 46.027/46.027 46.027/46.027 Auto_process settings: (no changes). 46.027/46.027 46.027/46.027 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 kept: 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 kept: 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 kept: 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 kept: 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.027/46.027 kept: 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.027/46.027 kept: 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.027/46.027 kept: 19 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.027/46.027 kept: 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.027/46.027 kept: 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46.027/46.027 46.027/46.027 ============================== end of process initial clauses ======== 46.027/46.027 46.027/46.027 ============================== CLAUSES FOR SEARCH ==================== 46.027/46.027 46.027/46.027 % Clauses after input processing: 46.027/46.027 46.027/46.027 formulas(usable). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(sos). 46.027/46.027 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.027/46.027 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.027/46.027 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.027/46.027 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(demodulators). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== end of clauses for search ============= 46.027/46.027 46.027/46.027 ============================== SEARCH ================================ 46.027/46.027 46.027/46.027 % Starting search at 0.00 seconds. 46.027/46.027 46.027/46.027 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 46.027/46.027 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 46.027/46.027 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 46.027/46.027 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 46.027/46.027 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 46.027/46.027 given #6 (I,wt=4): 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 46.027/46.027 given #7 (I,wt=9): 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.027/46.027 46.027/46.027 given #8 (I,wt=4): 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.027/46.027 46.027/46.027 given #9 (I,wt=9): 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.027/46.027 46.027/46.027 given #10 (I,wt=4): 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46.027/46.027 46.027/46.027 given #11 (A,wt=6): 22 ->_s0(s(even(0)),s(true)). [ur(14,a,15,a)]. 46.027/46.027 46.027/46.027 given #12 (F,wt=3): 32 -->*_s0(false,true) # answer(goal). [ur(11,a,18,a,c,21,a)]. 46.027/46.027 46.027/46.027 given #13 (F,wt=3): 38 -->_s0(false,true) # answer(goal). [ur(11,b,10,a,c,32,a)]. 46.027/46.027 46.027/46.027 given #14 (F,wt=4): 31 -->_s0(odd(x),true) # answer(goal). [ur(11,b,10,a,c,21,a)]. 46.027/46.027 46.027/46.027 ============================== PROOF ================================= 46.027/46.027 46.027/46.027 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 46.027/46.027 % Length of proof is 12. 46.027/46.027 % Level of proof is 4. 46.027/46.027 % Maximum clause weight is 9.000. 46.027/46.027 % Given clauses 14. 46.027/46.027 46.027/46.027 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.027/46.027 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 9 (exists x2 ->*_s0(odd(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.027/46.027 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.027/46.027 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46.027/46.027 25 ->*_s0(even(0),true). [ur(11,a,15,a,b,10,a)]. 46.027/46.027 31 -->_s0(odd(x),true) # answer(goal). [ur(11,b,10,a,c,21,a)]. 46.027/46.027 39 -->*_s0(even(x),true) # answer(goal). [resolve(31,a,20,b)]. 46.027/46.027 40 $F # answer(goal). [resolve(39,a,25,a)]. 46.027/46.027 46.027/46.027 ============================== end of proof ========================== 46.027/46.027 46.027/46.027 ============================== STATISTICS ============================ 46.027/46.027 46.027/46.027 Given=14. Generated=30. Kept=30. proofs=1. 46.027/46.027 Usable=14. Sos=13. Demods=0. Limbo=0, Disabled=14. Hints=0. 46.027/46.027 Kept_by_rule=0, Deleted_by_rule=0. 46.027/46.027 Forward_subsumed=0. Back_subsumed=2. 46.027/46.027 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 46.027/46.027 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 46.027/46.027 Demod_attempts=0. Demod_rewrites=0. 46.027/46.027 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 46.027/46.027 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=15. 46.027/46.027 Megabytes=0.09. 46.027/46.027 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46.027/46.027 46.027/46.027 ============================== end of statistics ===================== 46.027/46.027 46.027/46.027 ============================== end of search ========================= 46.027/46.027 46.027/46.027 THEOREM PROVED 46.027/46.027 46.027/46.027 Exiting with 1 proof. 46.027/46.027 46.027/46.027 Process 56501 exit (max_proofs) Wed Jul 14 11:20:18 2021 46.027/46.027 46.027/46.027 46.027/46.027 The problem is feasible. 46.027/46.027 46.027/46.027 46.027/46.027 CRule InfChecker Info: 46.027/46.027 odd(0) -> false 46.027/46.027 Rule remains 46.027/46.027 Proof: 46.027/46.027 NO_CONDS 46.027/46.027 46.027/46.027 CRule InfChecker Info: 46.027/46.027 odd(s(x:S)) -> false | odd(x:S) ->* true 46.027/46.027 Rule remains 46.027/46.027 Proof: 46.027/46.027 NO 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 Infeasibility Problem: 46.027/46.027 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 46.027/46.027 (STRATEGY CONTEXTSENSITIVE 46.027/46.027 (even 1) 46.027/46.027 (odd 1) 46.027/46.027 (0) 46.027/46.027 (fSNonEmpty) 46.027/46.027 (false) 46.027/46.027 (s 1) 46.027/46.027 (true) 46.027/46.027 ) 46.027/46.027 (RULES 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x:S:S)) -> false | even(x:S:S) ->* true 46.027/46.027 even(s(x:S:S)) -> true | odd(x:S:S) ->* true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x:S:S)) -> false | odd(x:S:S) ->* true 46.027/46.027 odd(s(x:S:S)) -> true | even(x:S:S) ->* true 46.027/46.027 )] 46.027/46.027 46.027/46.027 Infeasibility Conditions: 46.027/46.027 odd(x:S:S) ->* true 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 Obtaining a proof using Prover9: 46.027/46.027 46.027/46.027 -> Prover9 Output: 46.027/46.027 ============================== Prover9 =============================== 46.027/46.027 Prover9 (64) version 2009-11A, November 2009. 46.027/46.027 Process 56523 was started by ubuntu on ubuntu, 46.027/46.027 Wed Jul 14 11:20:18 2021 46.027/46.027 The command was "./prover9 -f /tmp/prover956514-0.in". 46.027/46.027 ============================== end of head =========================== 46.027/46.027 46.027/46.027 ============================== INPUT ================================= 46.027/46.027 46.027/46.027 % Reading from file /tmp/prover956514-0.in 46.027/46.027 46.027/46.027 assign(max_seconds,20). 46.027/46.027 46.027/46.027 formulas(assumptions). 46.027/46.027 ->*_s0(x,x) # label(reflexivity). 46.027/46.027 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 46.027/46.027 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence). 46.027/46.027 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence). 46.027/46.027 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 46.027/46.027 ->_s0(even(0),true) # label(replacement). 46.027/46.027 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement). 46.027/46.027 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement). 46.027/46.027 ->_s0(odd(0),false) # label(replacement). 46.027/46.027 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement). 46.027/46.027 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(goals). 46.027/46.027 (exists x2 ->*_s0(odd(x2),true)) # label(goal). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== end of input ========================== 46.027/46.027 46.027/46.027 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46.027/46.027 46.027/46.027 % Formulas that are not ordinary clauses: 46.027/46.027 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.027/46.027 2 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 3 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 5 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 6 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 7 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 9 (exists x2 ->*_s0(odd(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.027/46.027 46.027/46.027 ============================== end of process non-clausal formulas === 46.027/46.027 46.027/46.027 ============================== PROCESS INITIAL CLAUSES =============== 46.027/46.027 46.027/46.027 % Clauses before input processing: 46.027/46.027 46.027/46.027 formulas(usable). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(sos). 46.027/46.027 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.027/46.027 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.027/46.027 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.027/46.027 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.027/46.027 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.027/46.027 -->*_s0(odd(x),true) # label(goal). [deny(9)]. 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(demodulators). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== PREDICATE ELIMINATION ================= 46.027/46.027 46.027/46.027 No predicates eliminated. 46.027/46.027 46.027/46.027 ============================== end predicate elimination ============= 46.027/46.027 46.027/46.027 Auto_denials: 46.027/46.027 % copying label goal to answer in negative clause 46.027/46.027 46.027/46.027 Term ordering decisions: 46.027/46.027 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 46.027/46.027 Function symbol precedence: function_order([ true, false, 0, even, odd, s ]). 46.027/46.027 After inverse_order: (no changes). 46.027/46.027 Unfolding symbols: (none). 46.027/46.027 46.027/46.027 Auto_inference settings: 46.027/46.027 % set(neg_binary_resolution). % (HNE depth_diff=-7) 46.027/46.027 % clear(ordered_res). % (HNE depth_diff=-7) 46.027/46.027 % set(ur_resolution). % (HNE depth_diff=-7) 46.027/46.027 % set(ur_resolution) -> set(pos_ur_resolution). 46.027/46.027 % set(ur_resolution) -> set(neg_ur_resolution). 46.027/46.027 46.027/46.027 Auto_process settings: (no changes). 46.027/46.027 46.027/46.027 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 kept: 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 kept: 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 kept: 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 kept: 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.027/46.027 kept: 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.027/46.027 kept: 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.027/46.027 kept: 19 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.027/46.027 kept: 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.027/46.027 kept: 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46.027/46.027 46.027/46.027 ============================== end of process initial clauses ======== 46.027/46.027 46.027/46.027 ============================== CLAUSES FOR SEARCH ==================== 46.027/46.027 46.027/46.027 % Clauses after input processing: 46.027/46.027 46.027/46.027 formulas(usable). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(sos). 46.027/46.027 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.027/46.027 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.027/46.027 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.027/46.027 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(demodulators). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== end of clauses for search ============= 46.027/46.027 46.027/46.027 ============================== SEARCH ================================ 46.027/46.027 46.027/46.027 % Starting search at 0.00 seconds. 46.027/46.027 46.027/46.027 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 46.027/46.027 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 46.027/46.027 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 46.027/46.027 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 46.027/46.027 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 46.027/46.027 given #6 (I,wt=4): 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 46.027/46.027 given #7 (I,wt=9): 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.027/46.027 46.027/46.027 given #8 (I,wt=4): 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.027/46.027 46.027/46.027 given #9 (I,wt=9): 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.027/46.027 46.027/46.027 given #10 (I,wt=4): 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46.027/46.027 46.027/46.027 given #11 (A,wt=6): 22 ->_s0(s(even(0)),s(true)). [ur(14,a,15,a)]. 46.027/46.027 46.027/46.027 given #12 (F,wt=3): 32 -->*_s0(false,true) # answer(goal). [ur(11,a,18,a,c,21,a)]. 46.027/46.027 46.027/46.027 given #13 (F,wt=3): 38 -->_s0(false,true) # answer(goal). [ur(11,b,10,a,c,32,a)]. 46.027/46.027 46.027/46.027 given #14 (F,wt=4): 31 -->_s0(odd(x),true) # answer(goal). [ur(11,b,10,a,c,21,a)]. 46.027/46.027 46.027/46.027 ============================== PROOF ================================= 46.027/46.027 46.027/46.027 % Proof 1 at 0.01 (+ 0.00) seconds: goal. 46.027/46.027 % Length of proof is 12. 46.027/46.027 % Level of proof is 4. 46.027/46.027 % Maximum clause weight is 9.000. 46.027/46.027 % Given clauses 14. 46.027/46.027 46.027/46.027 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.027/46.027 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 9 (exists x2 ->*_s0(odd(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.027/46.027 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.027/46.027 21 -->*_s0(odd(x),true) # label(goal) # answer(goal). [deny(9)]. 46.027/46.027 25 ->*_s0(even(0),true). [ur(11,a,15,a,b,10,a)]. 46.027/46.027 31 -->_s0(odd(x),true) # answer(goal). [ur(11,b,10,a,c,21,a)]. 46.027/46.027 39 -->*_s0(even(x),true) # answer(goal). [resolve(31,a,20,b)]. 46.027/46.027 40 $F # answer(goal). [resolve(39,a,25,a)]. 46.027/46.027 46.027/46.027 ============================== end of proof ========================== 46.027/46.027 46.027/46.027 ============================== STATISTICS ============================ 46.027/46.027 46.027/46.027 Given=14. Generated=30. Kept=30. proofs=1. 46.027/46.027 Usable=14. Sos=13. Demods=0. Limbo=0, Disabled=14. Hints=0. 46.027/46.027 Kept_by_rule=0, Deleted_by_rule=0. 46.027/46.027 Forward_subsumed=0. Back_subsumed=2. 46.027/46.027 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 46.027/46.027 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 46.027/46.027 Demod_attempts=0. Demod_rewrites=0. 46.027/46.027 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 46.027/46.027 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=15. 46.027/46.027 Megabytes=0.09. 46.027/46.027 User_CPU=0.01, System_CPU=0.00, Wall_clock=0. 46.027/46.027 46.027/46.027 ============================== end of statistics ===================== 46.027/46.027 46.027/46.027 ============================== end of search ========================= 46.027/46.027 46.027/46.027 THEOREM PROVED 46.027/46.027 46.027/46.027 Exiting with 1 proof. 46.027/46.027 46.027/46.027 Process 56523 exit (max_proofs) Wed Jul 14 11:20:18 2021 46.027/46.027 46.027/46.027 46.027/46.027 The problem is feasible. 46.027/46.027 46.027/46.027 46.027/46.027 CRule InfChecker Info: 46.027/46.027 odd(s(x:S)) -> true | even(x:S) ->* true 46.027/46.027 Rule remains 46.027/46.027 Proof: 46.027/46.027 NO 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 Infeasibility Problem: 46.027/46.027 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 46.027/46.027 (STRATEGY CONTEXTSENSITIVE 46.027/46.027 (even 1) 46.027/46.027 (odd 1) 46.027/46.027 (0) 46.027/46.027 (fSNonEmpty) 46.027/46.027 (false) 46.027/46.027 (s 1) 46.027/46.027 (true) 46.027/46.027 ) 46.027/46.027 (RULES 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x:S:S)) -> false | even(x:S:S) ->* true 46.027/46.027 even(s(x:S:S)) -> true | odd(x:S:S) ->* true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x:S:S)) -> false | odd(x:S:S) ->* true 46.027/46.027 odd(s(x:S:S)) -> true | even(x:S:S) ->* true 46.027/46.027 )] 46.027/46.027 46.027/46.027 Infeasibility Conditions: 46.027/46.027 even(x:S:S) ->* true 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 Obtaining a proof using Prover9: 46.027/46.027 46.027/46.027 -> Prover9 Output: 46.027/46.027 ============================== Prover9 =============================== 46.027/46.027 Prover9 (64) version 2009-11A, November 2009. 46.027/46.027 Process 56544 was started by ubuntu on ubuntu, 46.027/46.027 Wed Jul 14 11:20:18 2021 46.027/46.027 The command was "./prover9 -f /tmp/prover956536-0.in". 46.027/46.027 ============================== end of head =========================== 46.027/46.027 46.027/46.027 ============================== INPUT ================================= 46.027/46.027 46.027/46.027 % Reading from file /tmp/prover956536-0.in 46.027/46.027 46.027/46.027 assign(max_seconds,20). 46.027/46.027 46.027/46.027 formulas(assumptions). 46.027/46.027 ->*_s0(x,x) # label(reflexivity). 46.027/46.027 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 46.027/46.027 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence). 46.027/46.027 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence). 46.027/46.027 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 46.027/46.027 ->_s0(even(0),true) # label(replacement). 46.027/46.027 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement). 46.027/46.027 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement). 46.027/46.027 ->_s0(odd(0),false) # label(replacement). 46.027/46.027 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement). 46.027/46.027 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(goals). 46.027/46.027 (exists x2 ->*_s0(even(x2),true)) # label(goal). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== end of input ========================== 46.027/46.027 46.027/46.027 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46.027/46.027 46.027/46.027 % Formulas that are not ordinary clauses: 46.027/46.027 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.027/46.027 2 ->_s0(x1,y) -> ->_s0(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 3 ->_s0(x1,y) -> ->_s0(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 4 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 5 ->*_s0(even(x2),true) -> ->_s0(even(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 6 ->*_s0(odd(x2),true) -> ->_s0(even(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 7 ->*_s0(odd(x2),true) -> ->_s0(odd(s(x2)),false) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 8 ->*_s0(even(x2),true) -> ->_s0(odd(s(x2)),true) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 9 (exists x2 ->*_s0(even(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.027/46.027 46.027/46.027 ============================== end of process non-clausal formulas === 46.027/46.027 46.027/46.027 ============================== PROCESS INITIAL CLAUSES =============== 46.027/46.027 46.027/46.027 % Clauses before input processing: 46.027/46.027 46.027/46.027 formulas(usable). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(sos). 46.027/46.027 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.027/46.027 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.027/46.027 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.027/46.027 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.027/46.027 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.027/46.027 -->*_s0(even(x),true) # label(goal). [deny(9)]. 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(demodulators). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== PREDICATE ELIMINATION ================= 46.027/46.027 46.027/46.027 No predicates eliminated. 46.027/46.027 46.027/46.027 ============================== end predicate elimination ============= 46.027/46.027 46.027/46.027 Auto_denials: 46.027/46.027 % copying label goal to answer in negative clause 46.027/46.027 46.027/46.027 Term ordering decisions: 46.027/46.027 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 46.027/46.027 Function symbol precedence: function_order([ true, false, 0, even, odd, s ]). 46.027/46.027 After inverse_order: (no changes). 46.027/46.027 Unfolding symbols: (none). 46.027/46.027 46.027/46.027 Auto_inference settings: 46.027/46.027 % set(neg_binary_resolution). % (HNE depth_diff=-7) 46.027/46.027 % clear(ordered_res). % (HNE depth_diff=-7) 46.027/46.027 % set(ur_resolution). % (HNE depth_diff=-7) 46.027/46.027 % set(ur_resolution) -> set(pos_ur_resolution). 46.027/46.027 % set(ur_resolution) -> set(neg_ur_resolution). 46.027/46.027 46.027/46.027 Auto_process settings: (no changes). 46.027/46.027 46.027/46.027 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 kept: 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 kept: 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 kept: 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 kept: 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 kept: 16 -->*_s0(even(x),true) | ->_s0(even(s(x)),false) # label(replacement). [clausify(5)]. 46.027/46.027 kept: 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.027/46.027 kept: 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.027/46.027 kept: 19 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.027/46.027 kept: 20 -->*_s0(even(x),true) | ->_s0(odd(s(x)),true) # label(replacement). [clausify(8)]. 46.027/46.027 kept: 21 -->*_s0(even(x),true) # label(goal) # answer(goal). [deny(9)]. 46.027/46.027 46.027/46.027 ============================== end of process initial clauses ======== 46.027/46.027 46.027/46.027 ============================== CLAUSES FOR SEARCH ==================== 46.027/46.027 46.027/46.027 % Clauses after input processing: 46.027/46.027 46.027/46.027 formulas(usable). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(sos). 46.027/46.027 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 17 -->*_s0(odd(x),true) | ->_s0(even(s(x)),true) # label(replacement). [clausify(6)]. 46.027/46.027 18 ->_s0(odd(0),false) # label(replacement). [assumption]. 46.027/46.027 19 -->*_s0(odd(x),true) | ->_s0(odd(s(x)),false) # label(replacement). [clausify(7)]. 46.027/46.027 21 -->*_s0(even(x),true) # label(goal) # answer(goal). [deny(9)]. 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(demodulators). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== end of clauses for search ============= 46.027/46.027 46.027/46.027 ============================== SEARCH ================================ 46.027/46.027 46.027/46.027 % Starting search at 0.00 seconds. 46.027/46.027 46.027/46.027 given #1 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 46.027/46.027 given #2 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 46.027/46.027 given #3 (I,wt=8): 12 -->_s0(x,y) | ->_s0(even(x),even(y)) # label(congruence). [clausify(2)]. 46.027/46.027 46.027/46.027 given #4 (I,wt=8): 13 -->_s0(x,y) | ->_s0(odd(x),odd(y)) # label(congruence). [clausify(3)]. 46.027/46.027 46.027/46.027 given #5 (I,wt=8): 14 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(4)]. 46.027/46.027 46.027/46.027 given #6 (I,wt=4): 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 46.027/46.027 ============================== PROOF ================================= 46.027/46.027 46.027/46.027 % Proof 1 at 0.00 (+ 0.01) seconds: goal. 46.027/46.027 % Length of proof is 8. 46.027/46.027 % Level of proof is 3. 46.027/46.027 % Maximum clause weight is 9.000. 46.027/46.027 % Given clauses 6. 46.027/46.027 46.027/46.027 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.027/46.027 9 (exists x2 ->*_s0(even(x2),true)) # label(goal) # label(non_clause) # label(goal). [goal]. 46.027/46.027 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 46.027/46.027 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 46.027/46.027 15 ->_s0(even(0),true) # label(replacement). [assumption]. 46.027/46.027 21 -->*_s0(even(x),true) # label(goal) # answer(goal). [deny(9)]. 46.027/46.027 25 ->*_s0(even(0),true). [ur(11,a,15,a,b,10,a)]. 46.027/46.027 26 $F # answer(goal). [resolve(25,a,21,a)]. 46.027/46.027 46.027/46.027 ============================== end of proof ========================== 46.027/46.027 46.027/46.027 ============================== STATISTICS ============================ 46.027/46.027 46.027/46.027 Given=6. Generated=16. Kept=16. proofs=1. 46.027/46.027 Usable=6. Sos=4. Demods=0. Limbo=3, Disabled=14. Hints=0. 46.027/46.027 Kept_by_rule=0, Deleted_by_rule=0. 46.027/46.027 Forward_subsumed=0. Back_subsumed=2. 46.027/46.027 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 46.027/46.027 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 46.027/46.027 Demod_attempts=0. Demod_rewrites=0. 46.027/46.027 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 46.027/46.027 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=11. 46.027/46.027 Megabytes=0.07. 46.027/46.027 User_CPU=0.00, System_CPU=0.01, Wall_clock=0. 46.027/46.027 46.027/46.027 ============================== end of statistics ===================== 46.027/46.027 46.027/46.027 ============================== end of search ========================= 46.027/46.027 46.027/46.027 THEOREM PROVED 46.027/46.027 46.027/46.027 Exiting with 1 proof. 46.027/46.027 46.027/46.027 Process 56544 exit (max_proofs) Wed Jul 14 11:20:18 2021 46.027/46.027 46.027/46.027 46.027/46.027 The problem is feasible. 46.027/46.027 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.027/46.027 Confluence Problem: 46.027/46.027 (VAR x:S) 46.027/46.027 (STRATEGY CONTEXTSENSITIVE 46.027/46.027 (even 1) 46.027/46.027 (odd 1) 46.027/46.027 (0) 46.027/46.027 (fSNonEmpty) 46.027/46.027 (false) 46.027/46.027 (s 1) 46.027/46.027 (true) 46.027/46.027 ) 46.027/46.027 (RULES 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x:S)) -> false | even(x:S) ->* true 46.027/46.027 even(s(x:S)) -> true | odd(x:S) ->* true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x:S)) -> false | odd(x:S) ->* true 46.027/46.027 odd(s(x:S)) -> true | even(x:S) ->* true 46.027/46.027 ) 46.027/46.027 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 46.027/46.027 46.027/46.027 Critical Pairs Processor: 46.027/46.027 -> Rules: 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x:S)) -> false | even(x:S) ->* true 46.027/46.027 even(s(x:S)) -> true | odd(x:S) ->* true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x:S)) -> false | odd(x:S) ->* true 46.027/46.027 odd(s(x:S)) -> true | even(x:S) ->* true 46.027/46.027 -> Vars: 46.027/46.027 "x", "x", "x", "x" 46.027/46.027 -> FVars: 46.027/46.027 "x2", "x3", "x4", "x5" 46.027/46.027 -> PVars: 46.027/46.027 "x": ["x2", "x3", "x4", "x5"] 46.027/46.027 46.027/46.027 -> Rlps: 46.027/46.027 crule: even(0) -> true, id: 1, possubterms: even(0)-> [], 0-> [1] 46.027/46.027 crule: even(s(x2:S)) -> false | even(x2:S) ->* true, id: 2, possubterms: even(s(x2:S))-> [], s(x2:S)-> [1] 46.027/46.027 crule: even(s(x3:S)) -> true | odd(x3:S) ->* true, id: 3, possubterms: even(s(x3:S))-> [], s(x3:S)-> [1] 46.027/46.027 crule: odd(0) -> false, id: 4, possubterms: odd(0)-> [], 0-> [1] 46.027/46.027 crule: odd(s(x4:S)) -> false | odd(x4:S) ->* true, id: 5, possubterms: odd(s(x4:S))-> [], s(x4:S)-> [1] 46.027/46.027 crule: odd(s(x5:S)) -> true | even(x5:S) ->* true, id: 6, possubterms: odd(s(x5:S))-> [], s(x5:S)-> [1] 46.027/46.027 46.027/46.027 -> Unifications: 46.027/46.027 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 46.027/46.027 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 46.027/46.027 46.027/46.027 -> Critical pairs info: 46.027/46.027 | even(x:S) ->* true, odd(x:S) ->* true => Not trivial, Overlay, N1 46.027/46.027 | odd(x:S) ->* true, even(x:S) ->* true => Not trivial, Overlay, N2 46.027/46.027 46.027/46.027 -> Problem conclusions: 46.027/46.027 Left linear, Right linear, Linear 46.027/46.027 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 46.027/46.027 CTRS Type: 1 46.027/46.027 Deterministic, Strongly deterministic 46.027/46.027 Oriented CTRS, Properly oriented CTRS, Not join CTRS 46.027/46.027 Maybe right-stable CTRS, Overlay CTRS 46.027/46.027 Maybe normal CTRS, Maybe almost normal CTRS 46.027/46.027 Maybe terminating CTRS, Maybe joinable CCPs 46.027/46.027 Maybe level confluent 46.027/46.027 Maybe confluent 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 Clean Infeasible CCPs Processor: 46.027/46.027 Num of CPIs: 2 46.027/46.027 Timeout: 60 46.027/46.027 Timeout for each infeasibility problem: 60 s 46.027/46.027 | even(x:S) ->* true, odd(x:S) ->* true => Not trivial, Overlay, N1 46.027/46.027 (PROBLEM INFEASIBILITY) 46.027/46.027 (VAR x) 46.027/46.027 (RULES 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x)) -> false | even(x) ->* true 46.027/46.027 even(s(x)) -> true | odd(x) ->* true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x)) -> false | odd(x) ->* true 46.027/46.027 odd(s(x)) -> true | even(x) ->* true) 46.027/46.027 (VAR x2) 46.027/46.027 (CONDITION even(x2) ->* true, odd(x2) ->* true) 46.027/46.027 46.027/46.027 Proof: 46.027/46.027 YES 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 Infeasibility Problem: 46.027/46.027 [(VAR vNonEmpty:S x:S x2:S) 46.027/46.027 (STRATEGY CONTEXTSENSITIVE 46.027/46.027 (even 1) 46.027/46.027 (odd 1) 46.027/46.027 (0) 46.027/46.027 (fSNonEmpty) 46.027/46.027 (false) 46.027/46.027 (s 1) 46.027/46.027 (true) 46.027/46.027 ) 46.027/46.027 (RULES 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x:S)) -> false | even(x:S) ->* true 46.027/46.027 even(s(x:S)) -> true | odd(x:S) ->* true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x:S)) -> false | odd(x:S) ->* true 46.027/46.027 odd(s(x:S)) -> true | even(x:S) ->* true 46.027/46.027 )] 46.027/46.027 46.027/46.027 Infeasibility Conditions: 46.027/46.027 even(x2:S) ->* true, odd(x2:S) ->* true 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 Obtaining a model using Mace4: 46.027/46.027 46.027/46.027 -> Usable Rules: 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x:S)) -> false | even(x:S) ->* true 46.027/46.027 even(s(x:S)) -> true | odd(x:S) ->* true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x:S)) -> false | odd(x:S) ->* true 46.027/46.027 odd(s(x:S)) -> true | even(x:S) ->* true 46.027/46.027 46.027/46.027 -> Mace4 Output: 46.027/46.027 ============================== Mace4 ================================= 46.027/46.027 Mace4 (64) version 2009-11A, November 2009. 46.027/46.027 Process 56594 was started by ubuntu on ubuntu, 46.027/46.027 Wed Jul 14 11:20:18 2021 46.027/46.027 The command was "./mace4 -c -f /tmp/mace456565-2.in". 46.027/46.027 ============================== end of head =========================== 46.027/46.027 46.027/46.027 ============================== INPUT ================================= 46.027/46.027 46.027/46.027 % Reading from file /tmp/mace456565-2.in 46.027/46.027 46.027/46.027 assign(max_seconds,20). 46.027/46.027 46.027/46.027 formulas(assumptions). 46.027/46.027 ->(x1,y) -> ->(even(x1),even(y)) # label(congruence). 46.027/46.027 ->(x1,y) -> ->(odd(x1),odd(y)) # label(congruence). 46.027/46.027 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 46.027/46.027 ->(even(0),true) # label(replacement). 46.027/46.027 ->*(even(x1),true) -> ->(even(s(x1)),false) # label(replacement). 46.027/46.027 ->*(odd(x1),true) -> ->(even(s(x1)),true) # label(replacement). 46.027/46.027 ->(odd(0),false) # label(replacement). 46.027/46.027 ->*(odd(x1),true) -> ->(odd(s(x1)),false) # label(replacement). 46.027/46.027 ->*(even(x1),true) -> ->(odd(s(x1)),true) # label(replacement). 46.027/46.027 ->*(x,x) # label(reflexivity). 46.027/46.027 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(goals). 46.027/46.027 (exists x2 (->*(even(x2),true) & ->*(odd(x2),true))) # label(goal). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== end of input ========================== 46.027/46.027 46.027/46.027 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46.027/46.027 46.027/46.027 % Formulas that are not ordinary clauses: 46.027/46.027 1 ->(x1,y) -> ->(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 2 ->(x1,y) -> ->(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 3 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 4 ->*(even(x1),true) -> ->(even(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 5 ->*(odd(x1),true) -> ->(even(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 6 ->*(odd(x1),true) -> ->(odd(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 7 ->*(even(x1),true) -> ->(odd(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 8 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.027/46.027 9 (exists x2 (->*(even(x2),true) & ->*(odd(x2),true))) # label(goal) # label(non_clause) # label(goal). [goal]. 46.027/46.027 46.027/46.027 ============================== end of process non-clausal formulas === 46.027/46.027 46.027/46.027 ============================== CLAUSES FOR SEARCH ==================== 46.027/46.027 46.027/46.027 formulas(mace4_clauses). 46.027/46.027 -->(x,y) | ->(even(x),even(y)) # label(congruence). 46.027/46.027 -->(x,y) | ->(odd(x),odd(y)) # label(congruence). 46.027/46.027 -->(x,y) | ->(s(x),s(y)) # label(congruence). 46.027/46.027 ->(even(0),true) # label(replacement). 46.027/46.027 -->*(even(x),true) | ->(even(s(x)),false) # label(replacement). 46.027/46.027 -->*(odd(x),true) | ->(even(s(x)),true) # label(replacement). 46.027/46.027 ->(odd(0),false) # label(replacement). 46.027/46.027 -->*(odd(x),true) | ->(odd(s(x)),false) # label(replacement). 46.027/46.027 -->*(even(x),true) | ->(odd(s(x)),true) # label(replacement). 46.027/46.027 ->*(x,x) # label(reflexivity). 46.027/46.027 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 46.027/46.027 -->*(even(x),true) | -->*(odd(x),true) # label(goal). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== end of clauses for search ============= 46.027/46.027 46.027/46.027 % There are no natural numbers in the input. 46.027/46.027 46.027/46.027 ============================== DOMAIN SIZE 2 ========================= 46.027/46.027 46.027/46.027 ============================== MODEL ================================= 46.027/46.027 46.027/46.027 interpretation( 2, [number=1, seconds=0], [ 46.027/46.027 46.027/46.027 function(0, [ 0 ]), 46.027/46.027 46.027/46.027 function(false, [ 0 ]), 46.027/46.027 46.027/46.027 function(true, [ 1 ]), 46.027/46.027 46.027/46.027 function(even(_), [ 1, 0 ]), 46.027/46.027 46.027/46.027 function(odd(_), [ 0, 1 ]), 46.027/46.027 46.027/46.027 function(s(_), [ 1, 0 ]), 46.027/46.027 46.027/46.027 relation(->*(_,_), [ 46.027/46.027 1, 0, 46.027/46.027 0, 1 ]), 46.027/46.027 46.027/46.027 relation(->(_,_), [ 46.027/46.027 1, 0, 46.027/46.027 0, 1 ]) 46.027/46.027 ]). 46.027/46.027 46.027/46.027 ============================== end of model ========================== 46.027/46.027 46.027/46.027 ============================== STATISTICS ============================ 46.027/46.027 46.027/46.027 For domain size 2. 46.027/46.027 46.027/46.027 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 46.027/46.027 Ground clauses: seen=34, kept=30. 46.027/46.027 Selections=8, assignments=12, propagations=22, current_models=1. 46.027/46.027 Rewrite_terms=133, rewrite_bools=79, indexes=27. 46.027/46.027 Rules_from_neg_clauses=6, cross_offs=6. 46.027/46.027 46.027/46.027 ============================== end of statistics ===================== 46.027/46.027 46.027/46.027 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46.027/46.027 46.027/46.027 Exiting with 1 model. 46.027/46.027 46.027/46.027 Process 56594 exit (max_models) Wed Jul 14 11:20:18 2021 46.027/46.027 The process finished Wed Jul 14 11:20:18 2021 46.027/46.027 46.027/46.027 46.027/46.027 Mace4 cooked interpretation: 46.027/46.027 46.027/46.027 46.027/46.027 46.027/46.027 The problem is infeasible. 46.027/46.027 46.027/46.027 46.027/46.027 46.027/46.027 46.027/46.027 | odd(x:S) ->* true, even(x:S) ->* true => Not trivial, Overlay, N2 46.027/46.027 (PROBLEM INFEASIBILITY) 46.027/46.027 (VAR x) 46.027/46.027 (RULES 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x)) -> false | even(x) ->* true 46.027/46.027 even(s(x)) -> true | odd(x) ->* true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x)) -> false | odd(x) ->* true 46.027/46.027 odd(s(x)) -> true | even(x) ->* true) 46.027/46.027 (VAR x2) 46.027/46.027 (CONDITION odd(x2) ->* true, even(x2) ->* true) 46.027/46.027 46.027/46.027 Proof: 46.027/46.027 YES 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 Infeasibility Problem: 46.027/46.027 [(VAR vNonEmpty:S x:S x2:S) 46.027/46.027 (STRATEGY CONTEXTSENSITIVE 46.027/46.027 (even 1) 46.027/46.027 (odd 1) 46.027/46.027 (0) 46.027/46.027 (fSNonEmpty) 46.027/46.027 (false) 46.027/46.027 (s 1) 46.027/46.027 (true) 46.027/46.027 ) 46.027/46.027 (RULES 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x:S)) -> false | even(x:S) ->* true 46.027/46.027 even(s(x:S)) -> true | odd(x:S) ->* true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x:S)) -> false | odd(x:S) ->* true 46.027/46.027 odd(s(x:S)) -> true | even(x:S) ->* true 46.027/46.027 )] 46.027/46.027 46.027/46.027 Infeasibility Conditions: 46.027/46.027 odd(x2:S) ->* true, even(x2:S) ->* true 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 Obtaining a model using Mace4: 46.027/46.027 46.027/46.027 -> Usable Rules: 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x:S)) -> false | even(x:S) ->* true 46.027/46.027 even(s(x:S)) -> true | odd(x:S) ->* true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x:S)) -> false | odd(x:S) ->* true 46.027/46.027 odd(s(x:S)) -> true | even(x:S) ->* true 46.027/46.027 46.027/46.027 -> Mace4 Output: 46.027/46.027 ============================== Mace4 ================================= 46.027/46.027 Mace4 (64) version 2009-11A, November 2009. 46.027/46.027 Process 56637 was started by ubuntu on ubuntu, 46.027/46.027 Wed Jul 14 11:20:38 2021 46.027/46.027 The command was "./mace4 -c -f /tmp/mace456624-2.in". 46.027/46.027 ============================== end of head =========================== 46.027/46.027 46.027/46.027 ============================== INPUT ================================= 46.027/46.027 46.027/46.027 % Reading from file /tmp/mace456624-2.in 46.027/46.027 46.027/46.027 assign(max_seconds,20). 46.027/46.027 46.027/46.027 formulas(assumptions). 46.027/46.027 ->(x1,y) -> ->(even(x1),even(y)) # label(congruence). 46.027/46.027 ->(x1,y) -> ->(odd(x1),odd(y)) # label(congruence). 46.027/46.027 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 46.027/46.027 ->(even(0),true) # label(replacement). 46.027/46.027 ->*(even(x1),true) -> ->(even(s(x1)),false) # label(replacement). 46.027/46.027 ->*(odd(x1),true) -> ->(even(s(x1)),true) # label(replacement). 46.027/46.027 ->(odd(0),false) # label(replacement). 46.027/46.027 ->*(odd(x1),true) -> ->(odd(s(x1)),false) # label(replacement). 46.027/46.027 ->*(even(x1),true) -> ->(odd(s(x1)),true) # label(replacement). 46.027/46.027 ->*(x,x) # label(reflexivity). 46.027/46.027 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 formulas(goals). 46.027/46.027 (exists x2 (->*(odd(x2),true) & ->*(even(x2),true))) # label(goal). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== end of input ========================== 46.027/46.027 46.027/46.027 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 46.027/46.027 46.027/46.027 % Formulas that are not ordinary clauses: 46.027/46.027 1 ->(x1,y) -> ->(even(x1),even(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 2 ->(x1,y) -> ->(odd(x1),odd(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 3 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 46.027/46.027 4 ->*(even(x1),true) -> ->(even(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 5 ->*(odd(x1),true) -> ->(even(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 6 ->*(odd(x1),true) -> ->(odd(s(x1)),false) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 7 ->*(even(x1),true) -> ->(odd(s(x1)),true) # label(replacement) # label(non_clause). [assumption]. 46.027/46.027 8 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 46.027/46.027 9 (exists x2 (->*(odd(x2),true) & ->*(even(x2),true))) # label(goal) # label(non_clause) # label(goal). [goal]. 46.027/46.027 46.027/46.027 ============================== end of process non-clausal formulas === 46.027/46.027 46.027/46.027 ============================== CLAUSES FOR SEARCH ==================== 46.027/46.027 46.027/46.027 formulas(mace4_clauses). 46.027/46.027 -->(x,y) | ->(even(x),even(y)) # label(congruence). 46.027/46.027 -->(x,y) | ->(odd(x),odd(y)) # label(congruence). 46.027/46.027 -->(x,y) | ->(s(x),s(y)) # label(congruence). 46.027/46.027 ->(even(0),true) # label(replacement). 46.027/46.027 -->*(even(x),true) | ->(even(s(x)),false) # label(replacement). 46.027/46.027 -->*(odd(x),true) | ->(even(s(x)),true) # label(replacement). 46.027/46.027 ->(odd(0),false) # label(replacement). 46.027/46.027 -->*(odd(x),true) | ->(odd(s(x)),false) # label(replacement). 46.027/46.027 -->*(even(x),true) | ->(odd(s(x)),true) # label(replacement). 46.027/46.027 ->*(x,x) # label(reflexivity). 46.027/46.027 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 46.027/46.027 -->*(odd(x),true) | -->*(even(x),true) # label(goal). 46.027/46.027 end_of_list. 46.027/46.027 46.027/46.027 ============================== end of clauses for search ============= 46.027/46.027 46.027/46.027 % There are no natural numbers in the input. 46.027/46.027 46.027/46.027 ============================== DOMAIN SIZE 2 ========================= 46.027/46.027 46.027/46.027 ============================== MODEL ================================= 46.027/46.027 46.027/46.027 interpretation( 2, [number=1, seconds=0], [ 46.027/46.027 46.027/46.027 function(0, [ 0 ]), 46.027/46.027 46.027/46.027 function(false, [ 0 ]), 46.027/46.027 46.027/46.027 function(true, [ 1 ]), 46.027/46.027 46.027/46.027 function(even(_), [ 1, 0 ]), 46.027/46.027 46.027/46.027 function(odd(_), [ 0, 1 ]), 46.027/46.027 46.027/46.027 function(s(_), [ 1, 0 ]), 46.027/46.027 46.027/46.027 relation(->*(_,_), [ 46.027/46.027 1, 0, 46.027/46.027 0, 1 ]), 46.027/46.027 46.027/46.027 relation(->(_,_), [ 46.027/46.027 1, 0, 46.027/46.027 0, 1 ]) 46.027/46.027 ]). 46.027/46.027 46.027/46.027 ============================== end of model ========================== 46.027/46.027 46.027/46.027 ============================== STATISTICS ============================ 46.027/46.027 46.027/46.027 For domain size 2. 46.027/46.027 46.027/46.027 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 46.027/46.027 Ground clauses: seen=34, kept=30. 46.027/46.027 Selections=8, assignments=12, propagations=22, current_models=1. 46.027/46.027 Rewrite_terms=133, rewrite_bools=79, indexes=27. 46.027/46.027 Rules_from_neg_clauses=6, cross_offs=6. 46.027/46.027 46.027/46.027 ============================== end of statistics ===================== 46.027/46.027 46.027/46.027 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 46.027/46.027 46.027/46.027 Exiting with 1 model. 46.027/46.027 46.027/46.027 Process 56637 exit (max_models) Wed Jul 14 11:20:38 2021 46.027/46.027 The process finished Wed Jul 14 11:20:38 2021 46.027/46.027 46.027/46.027 46.027/46.027 Mace4 cooked interpretation: 46.027/46.027 46.027/46.027 46.027/46.027 46.027/46.027 The problem is infeasible. 46.027/46.027 46.027/46.027 46.027/46.027 46.027/46.027 46.027/46.027 -> Problem conclusions: 46.027/46.027 Left linear, Right linear, Linear 46.027/46.027 Weakly orthogonal, Almost orthogonal, Orthogonal 46.027/46.027 CTRS Type: 1 46.027/46.027 Deterministic, Strongly deterministic 46.027/46.027 Oriented CTRS, Properly oriented CTRS, Not join CTRS 46.027/46.027 Maybe right-stable CTRS, Overlay CTRS 46.027/46.027 Maybe normal CTRS, Maybe almost normal CTRS 46.027/46.027 Maybe terminating CTRS, Joinable CCPs 46.027/46.027 Maybe level confluent 46.027/46.027 Maybe confluent 46.027/46.027 46.027/46.027 Resulting CCPs: 46.027/46.027 No CCPs left 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 Underlying TRS Termination Processor: 46.027/46.027 46.027/46.027 Resulting Underlying TRS: 46.027/46.027 (VAR x:S) 46.027/46.027 (STRATEGY CONTEXTSENSITIVE 46.027/46.027 (even 1) 46.027/46.027 (odd 1) 46.027/46.027 (false) 46.027/46.027 (true) 46.027/46.027 ) 46.027/46.027 (RULES 46.027/46.027 even(0) -> true 46.027/46.027 even(s(x:S)) -> false 46.027/46.027 even(s(x:S)) -> true 46.027/46.027 odd(0) -> false 46.027/46.027 odd(s(x:S)) -> false 46.027/46.027 odd(s(x:S)) -> true 46.027/46.027 ) 46.027/46.027 Underlying TRS terminating? 46.027/46.027 YES 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 (VAR vu95NonEmpty:S xu58S:S) 46.027/46.027 (RULES 46.027/46.027 even(num0) -> ftrue 46.027/46.027 even(s(xu58S:S)) -> ffalse 46.027/46.027 even(s(xu58S:S)) -> ftrue 46.027/46.027 odd(num0) -> ffalse 46.027/46.027 odd(s(xu58S:S)) -> ffalse 46.027/46.027 odd(s(xu58S:S)) -> ftrue 46.027/46.027 ) 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 Dependency Pairs Processor: 46.027/46.027 -> Pairs: 46.027/46.027 Empty 46.027/46.027 -> Rules: 46.027/46.027 even(num0) -> ftrue 46.027/46.027 even(s(xu58S:S)) -> ffalse 46.027/46.027 even(s(xu58S:S)) -> ftrue 46.027/46.027 odd(num0) -> ffalse 46.027/46.027 odd(s(xu58S:S)) -> ffalse 46.027/46.027 odd(s(xu58S:S)) -> ftrue 46.027/46.027 46.027/46.027 Problem 1: 46.027/46.027 46.027/46.027 SCC Processor: 46.027/46.027 -> Pairs: 46.027/46.027 Empty 46.027/46.027 -> Rules: 46.027/46.027 even(num0) -> ftrue 46.027/46.027 even(s(xu58S:S)) -> ffalse 46.027/46.027 even(s(xu58S:S)) -> ftrue 46.027/46.027 odd(num0) -> ffalse 46.027/46.027 odd(s(xu58S:S)) -> ffalse 46.027/46.027 odd(s(xu58S:S)) -> ftrue 46.027/46.027 ->Strongly Connected Components: 46.027/46.027 There is no strongly connected component 46.027/46.027 46.027/46.027 The problem is finite. 46.027/46.027 46.027/46.027 46.027/46.027 -> Problem conclusions: 46.027/46.027 Left linear, Right linear, Linear 46.027/46.027 Weakly orthogonal, Almost orthogonal, Orthogonal 46.027/46.027 CTRS Type: 1 46.027/46.027 Deterministic, Strongly deterministic 46.027/46.027 Oriented CTRS, Properly oriented CTRS, Not join CTRS 46.027/46.027 Maybe right-stable CTRS, Overlay CTRS 46.027/46.027 Maybe normal CTRS, Maybe almost normal CTRS 46.027/46.027 Terminating CTRS, Joinable CCPs 46.027/46.027 Maybe level confluent 46.027/46.027 Confluent 46.027/46.027 46.027/46.027 The problem is joinable. 46.027/46.027 63.85user 1.97system 0:46.27elapsed 142%CPU (0avgtext+0avgdata 118472maxresident)k 46.027/46.027 8inputs+0outputs (0major+147651minor)pagefaults 0swaps