20.036/20.036 YES 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 20.036/20.036 20.036/20.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.036/20.036 Confluence Problem: 20.036/20.036 (VAR vNonEmpty x) 20.036/20.036 (REPLACEMENT-MAP 20.036/20.036 (f 1) 20.036/20.036 (a) 20.036/20.036 (b) 20.036/20.036 (fSNonEmpty) 20.036/20.036 ) 20.036/20.036 (RULES 20.036/20.036 f(x) -> a | x ->* a 20.036/20.036 f(x) -> b | x ->* b 20.036/20.036 ) 20.036/20.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.036/20.036 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 20.036/20.036 Inlining of Conditions Processor [STERN17]: 20.036/20.036 20.036/20.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.036/20.036 Confluence Problem: 20.036/20.036 (VAR vNonEmpty x) 20.036/20.036 (REPLACEMENT-MAP 20.036/20.036 (f 1) 20.036/20.036 (a) 20.036/20.036 (b) 20.036/20.036 (fSNonEmpty) 20.036/20.036 ) 20.036/20.036 (RULES 20.036/20.036 f(x) -> a | x ->* a 20.036/20.036 f(x) -> b | x ->* b 20.036/20.036 ) 20.036/20.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.036/20.036 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 20.036/20.036 Clean CTRS Processor: 20.036/20.036 20.036/20.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.036/20.036 Confluence Problem: 20.036/20.036 (VAR x) 20.036/20.036 (REPLACEMENT-MAP 20.036/20.036 (f 1) 20.036/20.036 (a) 20.036/20.036 (b) 20.036/20.036 (fSNonEmpty) 20.036/20.036 ) 20.036/20.036 (RULES 20.036/20.036 f(x) -> a | x ->* a 20.036/20.036 f(x) -> b | x ->* b 20.036/20.036 ) 20.036/20.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.036/20.036 20.036/20.036 CRule InfChecker Info: 20.036/20.036 f(x) -> a | x ->* a 20.036/20.036 Rule remains 20.036/20.036 Proof: 20.036/20.036 NO 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 20.036/20.036 Infeasibility Problem: 20.036/20.036 [(VAR vNonEmpty x vNonEmpty x) 20.036/20.036 (STRATEGY CONTEXTSENSITIVE 20.036/20.036 (f 1) 20.036/20.036 (a) 20.036/20.036 (b) 20.036/20.036 (fSNonEmpty) 20.036/20.036 ) 20.036/20.036 (RULES 20.036/20.036 f(x) -> a | x ->* a 20.036/20.036 f(x) -> b | x ->* b 20.036/20.036 )] 20.036/20.036 20.036/20.036 Infeasibility Conditions: 20.036/20.036 x ->* a 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 20.036/20.036 Obtaining a proof using Prover9: 20.036/20.036 20.036/20.036 -> Prover9 Output: 20.036/20.036 ============================== Prover9 =============================== 20.036/20.036 Prover9 (64) version 2009-11A, November 2009. 20.036/20.036 Process 76841 was started by ubuntu on ubuntu, 20.036/20.036 Wed Mar 9 10:17:18 2022 20.036/20.036 The command was "./prover9 -f /tmp/prover976831-0.in". 20.036/20.036 ============================== end of head =========================== 20.036/20.036 20.036/20.036 ============================== INPUT ================================= 20.036/20.036 20.036/20.036 % Reading from file /tmp/prover976831-0.in 20.036/20.036 20.036/20.036 assign(max_seconds,20). 20.036/20.036 20.036/20.036 formulas(assumptions). 20.036/20.036 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 20.036/20.036 ->*_s0(x1,a) -> ->_s0(f(x1),a) # label(replacement). 20.036/20.036 ->*_s0(x1,b) -> ->_s0(f(x1),b) # label(replacement). 20.036/20.036 ->*_s0(x,x) # label(reflexivity). 20.036/20.036 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 formulas(goals). 20.036/20.036 (exists x3 ->*_s0(x3,a)) # label(goal). 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 ============================== end of input ========================== 20.036/20.036 20.036/20.036 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.036/20.036 20.036/20.036 % Formulas that are not ordinary clauses: 20.036/20.036 1 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 20.036/20.036 2 ->*_s0(x1,a) -> ->_s0(f(x1),a) # label(replacement) # label(non_clause). [assumption]. 20.036/20.036 3 ->*_s0(x1,b) -> ->_s0(f(x1),b) # label(replacement) # label(non_clause). [assumption]. 20.036/20.036 4 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.036/20.036 5 (exists x3 ->*_s0(x3,a)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.036/20.036 20.036/20.036 ============================== end of process non-clausal formulas === 20.036/20.036 20.036/20.036 ============================== PROCESS INITIAL CLAUSES =============== 20.036/20.036 20.036/20.036 % Clauses before input processing: 20.036/20.036 20.036/20.036 formulas(usable). 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 formulas(sos). 20.036/20.036 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 20.036/20.036 -->*_s0(x,a) | ->_s0(f(x),a) # label(replacement). [clausify(2)]. 20.036/20.036 -->*_s0(x,b) | ->_s0(f(x),b) # label(replacement). [clausify(3)]. 20.036/20.036 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.036/20.036 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(4)]. 20.036/20.036 -->*_s0(x,a) # label(goal). [deny(5)]. 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 formulas(demodulators). 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 ============================== PREDICATE ELIMINATION ================= 20.036/20.036 20.036/20.036 No predicates eliminated. 20.036/20.036 20.036/20.036 ============================== end predicate elimination ============= 20.036/20.036 20.036/20.036 Auto_denials: 20.036/20.036 % copying label goal to answer in negative clause 20.036/20.036 20.036/20.036 Term ordering decisions: 20.036/20.036 Predicate symbol precedence: predicate_order([ ->*_s0, ->_s0 ]). 20.036/20.036 Function symbol precedence: function_order([ a, b, f ]). 20.036/20.036 After inverse_order: (no changes). 20.036/20.036 Unfolding symbols: (none). 20.036/20.036 20.036/20.036 Auto_inference settings: 20.036/20.036 % set(neg_binary_resolution). % (HNE depth_diff=-3) 20.036/20.036 % clear(ordered_res). % (HNE depth_diff=-3) 20.036/20.036 % set(ur_resolution). % (HNE depth_diff=-3) 20.036/20.036 % set(ur_resolution) -> set(pos_ur_resolution). 20.036/20.036 % set(ur_resolution) -> set(neg_ur_resolution). 20.036/20.036 20.036/20.036 Auto_process settings: (no changes). 20.036/20.036 20.036/20.036 kept: 6 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 20.036/20.036 kept: 7 -->*_s0(x,a) | ->_s0(f(x),a) # label(replacement). [clausify(2)]. 20.036/20.036 kept: 8 -->*_s0(x,b) | ->_s0(f(x),b) # label(replacement). [clausify(3)]. 20.036/20.036 kept: 9 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.036/20.036 kept: 10 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(4)]. 20.036/20.036 kept: 11 -->*_s0(x,a) # label(goal) # answer(goal). [deny(5)]. 20.036/20.036 20.036/20.036 ============================== PROOF ================================= 20.036/20.036 20.036/20.036 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.036/20.036 % Length of proof is 4. 20.036/20.036 % Level of proof is 2. 20.036/20.036 % Maximum clause weight is 3.000. 20.036/20.036 % Given clauses 0. 20.036/20.036 20.036/20.036 5 (exists x3 ->*_s0(x3,a)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.036/20.036 9 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.036/20.036 11 -->*_s0(x,a) # label(goal) # answer(goal). [deny(5)]. 20.036/20.036 12 $F # answer(goal). [resolve(11,a,9,a)]. 20.036/20.036 20.036/20.036 ============================== end of proof ========================== 20.036/20.036 20.036/20.036 ============================== STATISTICS ============================ 20.036/20.036 20.036/20.036 Given=0. Generated=6. Kept=6. proofs=1. 20.036/20.036 Usable=0. Sos=0. Demods=0. Limbo=5, Disabled=6. Hints=0. 20.036/20.036 Kept_by_rule=0, Deleted_by_rule=0. 20.036/20.036 Forward_subsumed=0. Back_subsumed=0. 20.036/20.036 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.036/20.036 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.036/20.036 Demod_attempts=0. Demod_rewrites=0. 20.036/20.036 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.036/20.036 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 20.036/20.036 Megabytes=0.04. 20.036/20.036 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.036/20.036 20.036/20.036 ============================== end of statistics ===================== 20.036/20.036 20.036/20.036 ============================== end of search ========================= 20.036/20.036 20.036/20.036 THEOREM PROVED 20.036/20.036 20.036/20.036 Exiting with 1 proof. 20.036/20.036 20.036/20.036 Process 76841 exit (max_proofs) Wed Mar 9 10:17:18 2022 20.036/20.036 20.036/20.036 20.036/20.036 The problem is feasible. 20.036/20.036 20.036/20.036 20.036/20.036 CRule InfChecker Info: 20.036/20.036 f(x) -> b | x ->* b 20.036/20.036 Rule remains 20.036/20.036 Proof: 20.036/20.036 NO 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 20.036/20.036 Infeasibility Problem: 20.036/20.036 [(VAR vNonEmpty x vNonEmpty x) 20.036/20.036 (STRATEGY CONTEXTSENSITIVE 20.036/20.036 (f 1) 20.036/20.036 (a) 20.036/20.036 (b) 20.036/20.036 (fSNonEmpty) 20.036/20.036 ) 20.036/20.036 (RULES 20.036/20.036 f(x) -> a | x ->* a 20.036/20.036 f(x) -> b | x ->* b 20.036/20.036 )] 20.036/20.036 20.036/20.036 Infeasibility Conditions: 20.036/20.036 x ->* b 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 20.036/20.036 Obtaining a proof using Prover9: 20.036/20.036 20.036/20.036 -> Prover9 Output: 20.036/20.036 ============================== Prover9 =============================== 20.036/20.036 Prover9 (64) version 2009-11A, November 2009. 20.036/20.036 Process 76863 was started by ubuntu on ubuntu, 20.036/20.036 Wed Mar 9 10:17:18 2022 20.036/20.036 The command was "./prover9 -f /tmp/prover976853-0.in". 20.036/20.036 ============================== end of head =========================== 20.036/20.036 20.036/20.036 ============================== INPUT ================================= 20.036/20.036 20.036/20.036 % Reading from file /tmp/prover976853-0.in 20.036/20.036 20.036/20.036 assign(max_seconds,20). 20.036/20.036 20.036/20.036 formulas(assumptions). 20.036/20.036 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 20.036/20.036 ->*_s0(x1,a) -> ->_s0(f(x1),a) # label(replacement). 20.036/20.036 ->*_s0(x1,b) -> ->_s0(f(x1),b) # label(replacement). 20.036/20.036 ->*_s0(x,x) # label(reflexivity). 20.036/20.036 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 formulas(goals). 20.036/20.036 (exists x3 ->*_s0(x3,b)) # label(goal). 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 ============================== end of input ========================== 20.036/20.036 20.036/20.036 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.036/20.036 20.036/20.036 % Formulas that are not ordinary clauses: 20.036/20.036 1 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 20.036/20.036 2 ->*_s0(x1,a) -> ->_s0(f(x1),a) # label(replacement) # label(non_clause). [assumption]. 20.036/20.036 3 ->*_s0(x1,b) -> ->_s0(f(x1),b) # label(replacement) # label(non_clause). [assumption]. 20.036/20.036 4 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.036/20.036 5 (exists x3 ->*_s0(x3,b)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.036/20.036 20.036/20.036 ============================== end of process non-clausal formulas === 20.036/20.036 20.036/20.036 ============================== PROCESS INITIAL CLAUSES =============== 20.036/20.036 20.036/20.036 % Clauses before input processing: 20.036/20.036 20.036/20.036 formulas(usable). 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 formulas(sos). 20.036/20.036 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 20.036/20.036 -->*_s0(x,a) | ->_s0(f(x),a) # label(replacement). [clausify(2)]. 20.036/20.036 -->*_s0(x,b) | ->_s0(f(x),b) # label(replacement). [clausify(3)]. 20.036/20.036 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.036/20.036 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(4)]. 20.036/20.036 -->*_s0(x,b) # label(goal). [deny(5)]. 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 formulas(demodulators). 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 ============================== PREDICATE ELIMINATION ================= 20.036/20.036 20.036/20.036 No predicates eliminated. 20.036/20.036 20.036/20.036 ============================== end predicate elimination ============= 20.036/20.036 20.036/20.036 Auto_denials: 20.036/20.036 % copying label goal to answer in negative clause 20.036/20.036 20.036/20.036 Term ordering decisions: 20.036/20.036 Predicate symbol precedence: predicate_order([ ->*_s0, ->_s0 ]). 20.036/20.036 Function symbol precedence: function_order([ a, b, f ]). 20.036/20.036 After inverse_order: (no changes). 20.036/20.036 Unfolding symbols: (none). 20.036/20.036 20.036/20.036 Auto_inference settings: 20.036/20.036 % set(neg_binary_resolution). % (HNE depth_diff=-3) 20.036/20.036 % clear(ordered_res). % (HNE depth_diff=-3) 20.036/20.036 % set(ur_resolution). % (HNE depth_diff=-3) 20.036/20.036 % set(ur_resolution) -> set(pos_ur_resolution). 20.036/20.036 % set(ur_resolution) -> set(neg_ur_resolution). 20.036/20.036 20.036/20.036 Auto_process settings: (no changes). 20.036/20.036 20.036/20.036 kept: 6 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 20.036/20.036 kept: 7 -->*_s0(x,a) | ->_s0(f(x),a) # label(replacement). [clausify(2)]. 20.036/20.036 kept: 8 -->*_s0(x,b) | ->_s0(f(x),b) # label(replacement). [clausify(3)]. 20.036/20.036 kept: 9 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.036/20.036 kept: 10 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(4)]. 20.036/20.036 kept: 11 -->*_s0(x,b) # label(goal) # answer(goal). [deny(5)]. 20.036/20.036 20.036/20.036 ============================== PROOF ================================= 20.036/20.036 20.036/20.036 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.036/20.036 % Length of proof is 4. 20.036/20.036 % Level of proof is 2. 20.036/20.036 % Maximum clause weight is 3.000. 20.036/20.036 % Given clauses 0. 20.036/20.036 20.036/20.036 5 (exists x3 ->*_s0(x3,b)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.036/20.036 9 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.036/20.036 11 -->*_s0(x,b) # label(goal) # answer(goal). [deny(5)]. 20.036/20.036 12 $F # answer(goal). [resolve(11,a,9,a)]. 20.036/20.036 20.036/20.036 ============================== end of proof ========================== 20.036/20.036 20.036/20.036 ============================== STATISTICS ============================ 20.036/20.036 20.036/20.036 Given=0. Generated=6. Kept=6. proofs=1. 20.036/20.036 Usable=0. Sos=0. Demods=0. Limbo=5, Disabled=6. Hints=0. 20.036/20.036 Kept_by_rule=0, Deleted_by_rule=0. 20.036/20.036 Forward_subsumed=0. Back_subsumed=0. 20.036/20.036 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.036/20.036 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.036/20.036 Demod_attempts=0. Demod_rewrites=0. 20.036/20.036 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.036/20.036 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 20.036/20.036 Megabytes=0.04. 20.036/20.036 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.036/20.036 20.036/20.036 ============================== end of statistics ===================== 20.036/20.036 20.036/20.036 ============================== end of search ========================= 20.036/20.036 20.036/20.036 THEOREM PROVED 20.036/20.036 20.036/20.036 Exiting with 1 proof. 20.036/20.036 20.036/20.036 Process 76863 exit (max_proofs) Wed Mar 9 10:17:18 2022 20.036/20.036 20.036/20.036 20.036/20.036 The problem is feasible. 20.036/20.036 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.036/20.036 Confluence Problem: 20.036/20.036 (VAR x) 20.036/20.036 (REPLACEMENT-MAP 20.036/20.036 (f 1) 20.036/20.036 (a) 20.036/20.036 (b) 20.036/20.036 (fSNonEmpty) 20.036/20.036 ) 20.036/20.036 (RULES 20.036/20.036 f(x) -> a | x ->* a 20.036/20.036 f(x) -> b | x ->* b 20.036/20.036 ) 20.036/20.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.036/20.036 20.036/20.036 Critical Pairs Processor: 20.036/20.036 -> Rules: 20.036/20.036 f(x) -> a | x ->* a 20.036/20.036 f(x) -> b | x ->* b 20.036/20.036 -> Vars: 20.036/20.036 "x" 20.036/20.036 20.036/20.036 -> Rlps: 20.036/20.036 crule: f(x) -> a | x ->* a, id: 1, possubterms: f(x)-> [] 20.036/20.036 crule: f(x) -> b | x ->* b, id: 2, possubterms: f(x)-> [] 20.036/20.036 20.036/20.036 -> Unifications: 20.036/20.036 R2 unifies with R1 at p: [], l: f(x), lp: f(x), conds: {x ->* b, x' ->* a}, sig: {x -> x'}, l': f(x'), r: b, r': a 20.036/20.036 20.036/20.036 -> Critical pairs info: 20.036/20.036 | x' ->* b, x' ->* a => Not trivial, Overlay, NW2, N1 20.036/20.036 20.036/20.036 -> Problem conclusions: 20.036/20.036 Left linear, Right linear, Linear 20.036/20.036 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.036/20.036 CTRS Type: 1 20.036/20.036 Deterministic, Strongly deterministic 20.036/20.036 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.036/20.036 Maybe right-stable CTRS, Overlay CTRS 20.036/20.036 Maybe normal CTRS, Maybe almost normal CTRS 20.036/20.036 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.036/20.036 Maybe level confluent 20.036/20.036 Maybe confluent 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 Clean Infeasible CCPs Processor: 20.036/20.036 Num of CPIs: 1 20.036/20.036 Timeout: 60 20.036/20.036 Timeout for each infeasibility problem: 60 s 20.036/20.036 | x' ->* b, x' ->* a => Not trivial, Overlay, NW2, N1 20.036/20.036 (CONDITIONTYPE ORIENTED) 20.036/20.036 (VAR x x1) 20.036/20.036 (RULES 20.036/20.036 f(x) -> a | x ->* a 20.036/20.036 f(x) -> b | x ->* b) 20.036/20.036 (VAR x4) 20.036/20.036 (CONDITION x4 ->* b, x4 ->* a) 20.036/20.036 20.036/20.036 Proof: 20.036/20.036 YES 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 20.036/20.036 Infeasibility Problem: 20.036/20.036 [(VAR vNonEmpty x x1 vNonEmpty x4) 20.036/20.036 (STRATEGY CONTEXTSENSITIVE 20.036/20.036 (f 1) 20.036/20.036 (a) 20.036/20.036 (b) 20.036/20.036 (fSNonEmpty) 20.036/20.036 ) 20.036/20.036 (RULES 20.036/20.036 f(x) -> a | x ->* a 20.036/20.036 f(x) -> b | x ->* b 20.036/20.036 )] 20.036/20.036 20.036/20.036 Infeasibility Conditions: 20.036/20.036 x4 ->* b, x4 ->* a 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 20.036/20.036 Obtaining a model using Mace4: 20.036/20.036 20.036/20.036 -> Usable Rules: 20.036/20.036 f(x) -> a | x ->* a 20.036/20.036 f(x) -> b | x ->* b 20.036/20.036 20.036/20.036 -> Mace4 Output: 20.036/20.036 ============================== Mace4 ================================= 20.036/20.036 Mace4 (64) version 2009-11A, November 2009. 20.036/20.036 Process 76902 was started by ubuntu on ubuntu, 20.036/20.036 Wed Mar 9 10:17:18 2022 20.036/20.036 The command was "./mace4 -c -f /tmp/mace476879-2.in". 20.036/20.036 ============================== end of head =========================== 20.036/20.036 20.036/20.036 ============================== INPUT ================================= 20.036/20.036 20.036/20.036 % Reading from file /tmp/mace476879-2.in 20.036/20.036 20.036/20.036 assign(max_seconds,20). 20.036/20.036 20.036/20.036 formulas(assumptions). 20.036/20.036 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 20.036/20.036 ->*(x1,a) -> ->(f(x1),a) # label(replacement). 20.036/20.036 ->*(x1,b) -> ->(f(x1),b) # label(replacement). 20.036/20.036 ->*(x,x) # label(reflexivity). 20.036/20.036 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 formulas(goals). 20.036/20.036 (exists x4 (->*(x4,b) & ->*(x4,a))) # label(goal). 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 ============================== end of input ========================== 20.036/20.036 20.036/20.036 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.036/20.036 20.036/20.036 % Formulas that are not ordinary clauses: 20.036/20.036 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 20.036/20.036 2 ->*(x1,a) -> ->(f(x1),a) # label(replacement) # label(non_clause). [assumption]. 20.036/20.036 3 ->*(x1,b) -> ->(f(x1),b) # label(replacement) # label(non_clause). [assumption]. 20.036/20.036 4 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.036/20.036 5 (exists x4 (->*(x4,b) & ->*(x4,a))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.036/20.036 20.036/20.036 ============================== end of process non-clausal formulas === 20.036/20.036 20.036/20.036 ============================== CLAUSES FOR SEARCH ==================== 20.036/20.036 20.036/20.036 formulas(mace4_clauses). 20.036/20.036 -->(x,y) | ->(f(x),f(y)) # label(congruence). 20.036/20.036 -->*(x,a) | ->(f(x),a) # label(replacement). 20.036/20.036 -->*(x,b) | ->(f(x),b) # label(replacement). 20.036/20.036 ->*(x,x) # label(reflexivity). 20.036/20.036 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 20.036/20.036 -->*(x,b) | -->*(x,a) # label(goal). 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 ============================== end of clauses for search ============= 20.036/20.036 20.036/20.036 % There are no natural numbers in the input. 20.036/20.036 20.036/20.036 ============================== DOMAIN SIZE 2 ========================= 20.036/20.036 20.036/20.036 ============================== MODEL ================================= 20.036/20.036 20.036/20.036 interpretation( 2, [number=1, seconds=0], [ 20.036/20.036 20.036/20.036 function(a, [ 0 ]), 20.036/20.036 20.036/20.036 function(b, [ 1 ]), 20.036/20.036 20.036/20.036 function(f(_), [ 0, 1 ]), 20.036/20.036 20.036/20.036 relation(->*(_,_), [ 20.036/20.036 1, 0, 20.036/20.036 0, 1 ]), 20.036/20.036 20.036/20.036 relation(->(_,_), [ 20.036/20.036 1, 0, 20.036/20.036 0, 1 ]) 20.036/20.036 ]). 20.036/20.036 20.036/20.036 ============================== end of model ========================== 20.036/20.036 20.036/20.036 ============================== STATISTICS ============================ 20.036/20.036 20.036/20.036 For domain size 2. 20.036/20.036 20.036/20.036 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.036/20.036 Ground clauses: seen=20, kept=16. 20.036/20.036 Selections=1, assignments=1, propagations=11, current_models=1. 20.036/20.036 Rewrite_terms=24, rewrite_bools=25, indexes=6. 20.036/20.036 Rules_from_neg_clauses=3, cross_offs=3. 20.036/20.036 20.036/20.036 ============================== end of statistics ===================== 20.036/20.036 20.036/20.036 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.036/20.036 20.036/20.036 Exiting with 1 model. 20.036/20.036 20.036/20.036 Process 76902 exit (max_models) Wed Mar 9 10:17:18 2022 20.036/20.036 The process finished Wed Mar 9 10:17:18 2022 20.036/20.036 20.036/20.036 20.036/20.036 Mace4 cooked interpretation: 20.036/20.036 20.036/20.036 20.036/20.036 20.036/20.036 The problem is infeasible. 20.036/20.036 20.036/20.036 20.036/20.036 20.036/20.036 20.036/20.036 -> Problem conclusions: 20.036/20.036 Left linear, Right linear, Linear 20.036/20.036 Weakly orthogonal, Almost orthogonal, Orthogonal 20.036/20.036 CTRS Type: 1 20.036/20.036 Deterministic, Strongly deterministic 20.036/20.036 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.036/20.036 Maybe right-stable CTRS, Overlay CTRS 20.036/20.036 Maybe normal CTRS, Maybe almost normal CTRS 20.036/20.036 Maybe terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.036/20.036 Maybe level confluent 20.036/20.036 Maybe confluent 20.036/20.036 20.036/20.036 Resulting CCPs: 20.036/20.036 No CCPs left 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 Underlying TRS Termination Processor: 20.036/20.036 20.036/20.036 Resulting Underlying TRS: 20.036/20.036 (VAR x x') 20.036/20.036 (STRATEGY CONTEXTSENSITIVE 20.036/20.036 (f 1) 20.036/20.036 (a) 20.036/20.036 (b) 20.036/20.036 ) 20.036/20.036 (RULES 20.036/20.036 f(x) -> a 20.036/20.036 f(x) -> b 20.036/20.036 ) 20.036/20.036 Underlying TRS terminating? 20.036/20.036 YES 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 20.036/20.036 (VAR vu95NonEmpty x xu39) 20.036/20.036 (RULES 20.036/20.036 f(x) -> a 20.036/20.036 f(x) -> b 20.036/20.036 ) 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 20.036/20.036 Dependency Pairs Processor: 20.036/20.036 -> Pairs: 20.036/20.036 Empty 20.036/20.036 -> Rules: 20.036/20.036 f(x) -> a 20.036/20.036 f(x) -> b 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 20.036/20.036 SCC Processor: 20.036/20.036 -> Pairs: 20.036/20.036 Empty 20.036/20.036 -> Rules: 20.036/20.036 f(x) -> a 20.036/20.036 f(x) -> b 20.036/20.036 ->Strongly Connected Components: 20.036/20.036 There is no strongly connected component 20.036/20.036 20.036/20.036 The problem is finite. 20.036/20.036 20.036/20.036 20.036/20.036 -> Problem conclusions: 20.036/20.036 Left linear, Right linear, Linear 20.036/20.036 Weakly orthogonal, Almost orthogonal, Orthogonal 20.036/20.036 CTRS Type: 1 20.036/20.036 Deterministic, Strongly deterministic 20.036/20.036 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.036/20.036 Maybe right-stable CTRS, Overlay CTRS 20.036/20.036 Maybe normal CTRS, Maybe almost normal CTRS 20.036/20.036 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.036/20.036 Maybe level confluent 20.036/20.036 Maybe confluent 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.036/20.036 Confluence Problem: 20.036/20.036 (VAR x x') 20.036/20.036 (REPLACEMENT-MAP 20.036/20.036 (f 1) 20.036/20.036 (a) 20.036/20.036 (b) 20.036/20.036 (fSNonEmpty) 20.036/20.036 ) 20.036/20.036 (RULES 20.036/20.036 f(x) -> a | x ->* a 20.036/20.036 f(x) -> b | x ->* b 20.036/20.036 ) 20.036/20.036 Critical Pairs: 20.036/20.036 20.036/20.036 Terminating:YES 20.036/20.036 20.036/20.036 -> Problem conclusions: 20.036/20.036 Left linear, Right linear, Linear 20.036/20.036 Weakly orthogonal, Almost orthogonal, Orthogonal 20.036/20.036 CTRS Type: 1 20.036/20.036 Deterministic, Strongly deterministic 20.036/20.036 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.036/20.036 Maybe right-stable CTRS, Overlay CTRS 20.036/20.036 Maybe normal CTRS, Maybe almost normal CTRS 20.036/20.036 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.036/20.036 Maybe level confluent 20.036/20.036 Maybe confluent 20.036/20.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.036/20.036 20.036/20.036 Right Stable Processor: 20.036/20.036 Right-stable CTRS 20.036/20.036 Justification: 20.036/20.036 20.036/20.036 -> Term right-stability conditions: 20.036/20.036 Term: a 20.036/20.036 Right-stable term 20.036/20.036 Linear constructor form 20.036/20.036 Don't know if term is a ground normal form (not needed) 20.036/20.036 Right-stability condition achieved 20.036/20.036 A call to InfChecker wasn't needed 20.036/20.036 20.036/20.036 20.036/20.036 -> Term right-stability conditions: 20.036/20.036 Term: b 20.036/20.036 Right-stable term 20.036/20.036 Linear constructor form 20.036/20.036 Don't know if term is a ground normal form (not needed) 20.036/20.036 Right-stability condition achieved 20.036/20.036 A call to InfChecker wasn't needed 20.036/20.036 20.036/20.036 -> Problem conclusions: 20.036/20.036 Left linear, Right linear, Linear 20.036/20.036 Weakly orthogonal, Almost orthogonal, Orthogonal 20.036/20.036 CTRS Type: 1 20.036/20.036 Deterministic, Strongly deterministic 20.036/20.036 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.036/20.036 Right-stable CTRS, Overlay CTRS 20.036/20.036 Maybe normal CTRS, Almost normal CTRS 20.036/20.036 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.036/20.036 Level confluent 20.036/20.036 Confluent 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.036/20.036 Confluence Problem: 20.036/20.036 (VAR x x') 20.036/20.036 (REPLACEMENT-MAP 20.036/20.036 (f 1) 20.036/20.036 (a) 20.036/20.036 (b) 20.036/20.036 (fSNonEmpty) 20.036/20.036 ) 20.036/20.036 (RULES 20.036/20.036 f(x) -> a | x ->* a 20.036/20.036 f(x) -> b | x ->* b 20.036/20.036 ) 20.036/20.036 Critical Pairs: 20.036/20.036 20.036/20.036 Terminating:YES 20.036/20.036 20.036/20.036 -> Problem conclusions: 20.036/20.036 Left linear, Right linear, Linear 20.036/20.036 Weakly orthogonal, Almost orthogonal, Orthogonal 20.036/20.036 CTRS Type: 1 20.036/20.036 Deterministic, Strongly deterministic 20.036/20.036 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.036/20.036 Right-stable CTRS, Overlay CTRS 20.036/20.036 Maybe normal CTRS, Almost normal CTRS 20.036/20.036 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.036/20.036 Level confluent 20.036/20.036 Confluent 20.036/20.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.036/20.036 20.036/20.036 Normal RConds Processor: 20.036/20.036 YES 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 20.036/20.036 Infeasibility Problem: 20.036/20.036 [(VAR vNonEmpty x x1 vNonEmpty z z1) 20.036/20.036 (STRATEGY CONTEXTSENSITIVE 20.036/20.036 (f 1) 20.036/20.036 (a) 20.036/20.036 (b) 20.036/20.036 (fSNonEmpty) 20.036/20.036 ) 20.036/20.036 (RULES 20.036/20.036 f(x) -> a | x ->* a 20.036/20.036 f(x) -> b | x ->* b 20.036/20.036 )] 20.036/20.036 20.036/20.036 Infeasibility Conditions: 20.036/20.036 a -> z, b -> z1 20.036/20.036 20.036/20.036 Problem 1: 20.036/20.036 20.036/20.036 Obtaining a model using Mace4: 20.036/20.036 20.036/20.036 -> Usable Rules: 20.036/20.036 Empty 20.036/20.036 20.036/20.036 -> Mace4 Output: 20.036/20.036 ============================== Mace4 ================================= 20.036/20.036 Mace4 (64) version 2009-11A, November 2009. 20.036/20.036 Process 76981 was started by ubuntu on ubuntu, 20.036/20.036 Wed Mar 9 10:17:39 2022 20.036/20.036 The command was "./mace4 -c -f /tmp/mace476968-2.in". 20.036/20.036 ============================== end of head =========================== 20.036/20.036 20.036/20.036 ============================== INPUT ================================= 20.036/20.036 20.036/20.036 % Reading from file /tmp/mace476968-2.in 20.036/20.036 20.036/20.036 assign(max_seconds,13). 20.036/20.036 20.036/20.036 formulas(assumptions). 20.036/20.036 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 formulas(goals). 20.036/20.036 (exists x4 exists x5 (->(a,x4) & ->(b,x5))) # label(goal). 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 ============================== end of input ========================== 20.036/20.036 20.036/20.036 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.036/20.036 20.036/20.036 % Formulas that are not ordinary clauses: 20.036/20.036 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 20.036/20.036 2 (exists x4 exists x5 (->(a,x4) & ->(b,x5))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.036/20.036 20.036/20.036 ============================== end of process non-clausal formulas === 20.036/20.036 20.036/20.036 ============================== CLAUSES FOR SEARCH ==================== 20.036/20.036 20.036/20.036 formulas(mace4_clauses). 20.036/20.036 -->(x,y) | ->(f(x),f(y)) # label(congruence). 20.036/20.036 -->(a,x) | -->(b,y) # label(goal). 20.036/20.036 end_of_list. 20.036/20.036 20.036/20.036 ============================== end of clauses for search ============= 20.036/20.036 20.036/20.036 % There are no natural numbers in the input. 20.036/20.036 20.036/20.036 ============================== DOMAIN SIZE 2 ========================= 20.036/20.036 20.036/20.036 ============================== MODEL ================================= 20.036/20.036 20.036/20.036 interpretation( 2, [number=1, seconds=0], [ 20.036/20.036 20.036/20.036 function(a, [ 0 ]), 20.036/20.036 20.036/20.036 function(b, [ 0 ]), 20.036/20.036 20.036/20.036 function(f(_), [ 0, 0 ]), 20.036/20.036 20.036/20.036 relation(->(_,_), [ 20.036/20.036 0, 0, 20.036/20.036 0, 0 ]) 20.036/20.036 ]). 20.036/20.036 20.036/20.036 ============================== end of model ========================== 20.036/20.036 20.036/20.036 ============================== STATISTICS ============================ 20.036/20.036 20.036/20.036 For domain size 2. 20.036/20.036 20.036/20.036 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.036/20.036 Ground clauses: seen=8, kept=8. 20.036/20.036 Selections=5, assignments=5, propagations=3, current_models=1. 20.036/20.036 Rewrite_terms=16, rewrite_bools=12, indexes=9. 20.036/20.036 Rules_from_neg_clauses=0, cross_offs=0. 20.036/20.036 20.036/20.036 ============================== end of statistics ===================== 20.036/20.036 20.036/20.036 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.036/20.036 20.036/20.036 Exiting with 1 model. 20.036/20.036 20.036/20.036 Process 76981 exit (max_models) Wed Mar 9 10:17:39 2022 20.036/20.036 The process finished Wed Mar 9 10:17:39 2022 20.036/20.036 20.036/20.036 20.036/20.036 Mace4 cooked interpretation: 20.036/20.036 20.036/20.036 20.036/20.036 20.036/20.036 The problem is infeasible. 20.036/20.036 20.036/20.036 20.036/20.036 -> Problem conclusions: 20.036/20.036 Left linear, Right linear, Linear 20.036/20.036 Weakly orthogonal, Almost orthogonal, Orthogonal 20.036/20.036 CTRS Type: 1 20.036/20.036 Deterministic, Strongly deterministic 20.036/20.036 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.036/20.036 Right-stable CTRS, Overlay CTRS 20.036/20.036 Normal CTRS, Almost normal CTRS 20.036/20.036 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.036/20.036 Level confluent 20.036/20.036 Confluent 20.036/20.036 20.036/20.036 The problem is joinable. 20.036/20.036 53.07user 1.08system 0:20.36elapsed 265%CPU (0avgtext+0avgdata 69480maxresident)k 20.036/20.036 8inputs+0outputs (0major+62606minor)pagefaults 0swaps