25.022/25.022 YES 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 Confluence Problem: 25.022/25.022 (VAR vNonEmpty:S x:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (f 1) 25.022/25.022 (a) 25.022/25.022 (b) 25.022/25.022 (fSNonEmpty) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 f(x:S) -> a | x:S ->* a 25.022/25.022 f(x:S) -> b | x:S ->* b 25.022/25.022 ) 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 Inlining of Conditions Processor [STERN17]: 25.022/25.022 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 Confluence Problem: 25.022/25.022 (VAR vNonEmpty:S x:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (f 1) 25.022/25.022 (a) 25.022/25.022 (b) 25.022/25.022 (fSNonEmpty) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 f(x:S) -> a | x:S ->* a 25.022/25.022 f(x:S) -> b | x:S ->* b 25.022/25.022 ) 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 Clean CTRS Processor: 25.022/25.022 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 Confluence Problem: 25.022/25.022 (VAR x:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (f 1) 25.022/25.022 (a) 25.022/25.022 (b) 25.022/25.022 (fSNonEmpty) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 f(x:S) -> a | x:S ->* a 25.022/25.022 f(x:S) -> b | x:S ->* b 25.022/25.022 ) 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 25.022/25.022 CRule InfChecker Info: 25.022/25.022 f(x:S) -> a | x:S ->* a 25.022/25.022 Rule remains 25.022/25.022 Proof: 25.022/25.022 NO 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 Infeasibility Problem: 25.022/25.022 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (f 1) 25.022/25.022 (a) 25.022/25.022 (b) 25.022/25.022 (fSNonEmpty) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 f(x:S:S) -> a | x:S:S ->* a 25.022/25.022 f(x:S:S) -> b | x:S:S ->* b 25.022/25.022 )] 25.022/25.022 25.022/25.022 Infeasibility Conditions: 25.022/25.022 x:S:S ->* a 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 Obtaining a proof using Prover9: 25.022/25.022 25.022/25.022 -> Prover9 Output: 25.022/25.022 ============================== Prover9 =============================== 25.022/25.022 Prover9 (64) version 2009-11A, November 2009. 25.022/25.022 Process 119282 was started by ubuntu on ubuntu, 25.022/25.022 Fri Jul 16 15:35:23 2021 25.022/25.022 The command was "./prover9 -f /tmp/prover9119273-0.in". 25.022/25.022 ============================== end of head =========================== 25.022/25.022 25.022/25.022 ============================== INPUT ================================= 25.022/25.022 25.022/25.022 % Reading from file /tmp/prover9119273-0.in 25.022/25.022 25.022/25.022 assign(max_seconds,20). 25.022/25.022 25.022/25.022 formulas(assumptions). 25.022/25.022 ->*_s0(x,x) # label(reflexivity). 25.022/25.022 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.022/25.022 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 25.022/25.022 ->*_s0(x2,a) -> ->_s0(f(x2),a) # label(replacement). 25.022/25.022 ->*_s0(x2,b) -> ->_s0(f(x2),b) # label(replacement). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 formulas(goals). 25.022/25.022 (exists x2 ->*_s0(x2,a)) # label(goal). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 ============================== end of input ========================== 25.022/25.022 25.022/25.022 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.022/25.022 25.022/25.022 % Formulas that are not ordinary clauses: 25.022/25.022 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.022/25.022 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.022/25.022 3 ->*_s0(x2,a) -> ->_s0(f(x2),a) # label(replacement) # label(non_clause). [assumption]. 25.022/25.022 4 ->*_s0(x2,b) -> ->_s0(f(x2),b) # label(replacement) # label(non_clause). [assumption]. 25.022/25.022 5 (exists x2 ->*_s0(x2,a)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.022/25.022 25.022/25.022 ============================== end of process non-clausal formulas === 25.022/25.022 25.022/25.022 ============================== PROCESS INITIAL CLAUSES =============== 25.022/25.022 25.022/25.022 % Clauses before input processing: 25.022/25.022 25.022/25.022 formulas(usable). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 formulas(sos). 25.022/25.022 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.022/25.022 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.022/25.022 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.022/25.022 -->*_s0(x,a) | ->_s0(f(x),a) # label(replacement). [clausify(3)]. 25.022/25.022 -->*_s0(x,b) | ->_s0(f(x),b) # label(replacement). [clausify(4)]. 25.022/25.022 -->*_s0(x,a) # label(goal). [deny(5)]. 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 formulas(demodulators). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 ============================== PREDICATE ELIMINATION ================= 25.022/25.022 25.022/25.022 No predicates eliminated. 25.022/25.022 25.022/25.022 ============================== end predicate elimination ============= 25.022/25.022 25.022/25.022 Auto_denials: 25.022/25.022 % copying label goal to answer in negative clause 25.022/25.022 25.022/25.022 Term ordering decisions: 25.022/25.022 Predicate symbol precedence: predicate_order([ ->*_s0, ->_s0 ]). 25.022/25.022 Function symbol precedence: function_order([ a, b, f ]). 25.022/25.022 After inverse_order: (no changes). 25.022/25.022 Unfolding symbols: (none). 25.022/25.022 25.022/25.022 Auto_inference settings: 25.022/25.022 % set(neg_binary_resolution). % (HNE depth_diff=-3) 25.022/25.022 % clear(ordered_res). % (HNE depth_diff=-3) 25.022/25.022 % set(ur_resolution). % (HNE depth_diff=-3) 25.022/25.022 % set(ur_resolution) -> set(pos_ur_resolution). 25.022/25.022 % set(ur_resolution) -> set(neg_ur_resolution). 25.022/25.022 25.022/25.022 Auto_process settings: (no changes). 25.022/25.022 25.022/25.022 kept: 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.022/25.022 kept: 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.022/25.022 kept: 8 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.022/25.022 kept: 9 -->*_s0(x,a) | ->_s0(f(x),a) # label(replacement). [clausify(3)]. 25.022/25.022 kept: 10 -->*_s0(x,b) | ->_s0(f(x),b) # label(replacement). [clausify(4)]. 25.022/25.022 kept: 11 -->*_s0(x,a) # label(goal) # answer(goal). [deny(5)]. 25.022/25.022 25.022/25.022 ============================== PROOF ================================= 25.022/25.022 25.022/25.022 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.022/25.022 % Length of proof is 4. 25.022/25.022 % Level of proof is 2. 25.022/25.022 % Maximum clause weight is 3.000. 25.022/25.022 % Given clauses 0. 25.022/25.022 25.022/25.022 5 (exists x2 ->*_s0(x2,a)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.022/25.022 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.022/25.022 11 -->*_s0(x,a) # label(goal) # answer(goal). [deny(5)]. 25.022/25.022 12 $F # answer(goal). [resolve(11,a,6,a)]. 25.022/25.022 25.022/25.022 ============================== end of proof ========================== 25.022/25.022 25.022/25.022 ============================== STATISTICS ============================ 25.022/25.022 25.022/25.022 Given=0. Generated=6. Kept=6. proofs=1. 25.022/25.022 Usable=0. Sos=0. Demods=0. Limbo=5, Disabled=6. Hints=0. 25.022/25.022 Kept_by_rule=0, Deleted_by_rule=0. 25.022/25.022 Forward_subsumed=0. Back_subsumed=0. 25.022/25.022 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.022/25.022 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.022/25.022 Demod_attempts=0. Demod_rewrites=0. 25.022/25.022 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.022/25.022 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 25.022/25.022 Megabytes=0.04. 25.022/25.022 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.022/25.022 25.022/25.022 ============================== end of statistics ===================== 25.022/25.022 25.022/25.022 ============================== end of search ========================= 25.022/25.022 25.022/25.022 THEOREM PROVED 25.022/25.022 25.022/25.022 Exiting with 1 proof. 25.022/25.022 25.022/25.022 Process 119282 exit (max_proofs) Fri Jul 16 15:35:23 2021 25.022/25.022 25.022/25.022 25.022/25.022 The problem is feasible. 25.022/25.022 25.022/25.022 25.022/25.022 CRule InfChecker Info: 25.022/25.022 f(x:S) -> b | x:S ->* b 25.022/25.022 Rule remains 25.022/25.022 Proof: 25.022/25.022 NO 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 Infeasibility Problem: 25.022/25.022 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (f 1) 25.022/25.022 (a) 25.022/25.022 (b) 25.022/25.022 (fSNonEmpty) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 f(x:S:S) -> a | x:S:S ->* a 25.022/25.022 f(x:S:S) -> b | x:S:S ->* b 25.022/25.022 )] 25.022/25.022 25.022/25.022 Infeasibility Conditions: 25.022/25.022 x:S:S ->* b 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 Obtaining a proof using Prover9: 25.022/25.022 25.022/25.022 -> Prover9 Output: 25.022/25.022 ============================== Prover9 =============================== 25.022/25.022 Prover9 (64) version 2009-11A, November 2009. 25.022/25.022 Process 119304 was started by ubuntu on ubuntu, 25.022/25.022 Fri Jul 16 15:35:23 2021 25.022/25.022 The command was "./prover9 -f /tmp/prover9119294-0.in". 25.022/25.022 ============================== end of head =========================== 25.022/25.022 25.022/25.022 ============================== INPUT ================================= 25.022/25.022 25.022/25.022 % Reading from file /tmp/prover9119294-0.in 25.022/25.022 25.022/25.022 assign(max_seconds,20). 25.022/25.022 25.022/25.022 formulas(assumptions). 25.022/25.022 ->*_s0(x,x) # label(reflexivity). 25.022/25.022 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.022/25.022 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 25.022/25.022 ->*_s0(x2,a) -> ->_s0(f(x2),a) # label(replacement). 25.022/25.022 ->*_s0(x2,b) -> ->_s0(f(x2),b) # label(replacement). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 formulas(goals). 25.022/25.022 (exists x2 ->*_s0(x2,b)) # label(goal). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 ============================== end of input ========================== 25.022/25.022 25.022/25.022 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.022/25.022 25.022/25.022 % Formulas that are not ordinary clauses: 25.022/25.022 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.022/25.022 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.022/25.022 3 ->*_s0(x2,a) -> ->_s0(f(x2),a) # label(replacement) # label(non_clause). [assumption]. 25.022/25.022 4 ->*_s0(x2,b) -> ->_s0(f(x2),b) # label(replacement) # label(non_clause). [assumption]. 25.022/25.022 5 (exists x2 ->*_s0(x2,b)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.022/25.022 25.022/25.022 ============================== end of process non-clausal formulas === 25.022/25.022 25.022/25.022 ============================== PROCESS INITIAL CLAUSES =============== 25.022/25.022 25.022/25.022 % Clauses before input processing: 25.022/25.022 25.022/25.022 formulas(usable). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 formulas(sos). 25.022/25.022 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.022/25.022 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.022/25.022 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.022/25.022 -->*_s0(x,a) | ->_s0(f(x),a) # label(replacement). [clausify(3)]. 25.022/25.022 -->*_s0(x,b) | ->_s0(f(x),b) # label(replacement). [clausify(4)]. 25.022/25.022 -->*_s0(x,b) # label(goal). [deny(5)]. 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 formulas(demodulators). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 ============================== PREDICATE ELIMINATION ================= 25.022/25.022 25.022/25.022 No predicates eliminated. 25.022/25.022 25.022/25.022 ============================== end predicate elimination ============= 25.022/25.022 25.022/25.022 Auto_denials: 25.022/25.022 % copying label goal to answer in negative clause 25.022/25.022 25.022/25.022 Term ordering decisions: 25.022/25.022 Predicate symbol precedence: predicate_order([ ->*_s0, ->_s0 ]). 25.022/25.022 Function symbol precedence: function_order([ a, b, f ]). 25.022/25.022 After inverse_order: (no changes). 25.022/25.022 Unfolding symbols: (none). 25.022/25.022 25.022/25.022 Auto_inference settings: 25.022/25.022 % set(neg_binary_resolution). % (HNE depth_diff=-3) 25.022/25.022 % clear(ordered_res). % (HNE depth_diff=-3) 25.022/25.022 % set(ur_resolution). % (HNE depth_diff=-3) 25.022/25.022 % set(ur_resolution) -> set(pos_ur_resolution). 25.022/25.022 % set(ur_resolution) -> set(neg_ur_resolution). 25.022/25.022 25.022/25.022 Auto_process settings: (no changes). 25.022/25.022 25.022/25.022 kept: 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.022/25.022 kept: 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.022/25.022 kept: 8 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.022/25.022 kept: 9 -->*_s0(x,a) | ->_s0(f(x),a) # label(replacement). [clausify(3)]. 25.022/25.022 kept: 10 -->*_s0(x,b) | ->_s0(f(x),b) # label(replacement). [clausify(4)]. 25.022/25.022 kept: 11 -->*_s0(x,b) # label(goal) # answer(goal). [deny(5)]. 25.022/25.022 25.022/25.022 ============================== PROOF ================================= 25.022/25.022 25.022/25.022 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.022/25.022 % Length of proof is 4. 25.022/25.022 % Level of proof is 2. 25.022/25.022 % Maximum clause weight is 3.000. 25.022/25.022 % Given clauses 0. 25.022/25.022 25.022/25.022 5 (exists x2 ->*_s0(x2,b)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.022/25.022 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.022/25.022 11 -->*_s0(x,b) # label(goal) # answer(goal). [deny(5)]. 25.022/25.022 12 $F # answer(goal). [resolve(11,a,6,a)]. 25.022/25.022 25.022/25.022 ============================== end of proof ========================== 25.022/25.022 25.022/25.022 ============================== STATISTICS ============================ 25.022/25.022 25.022/25.022 Given=0. Generated=6. Kept=6. proofs=1. 25.022/25.022 Usable=0. Sos=0. Demods=0. Limbo=5, Disabled=6. Hints=0. 25.022/25.022 Kept_by_rule=0, Deleted_by_rule=0. 25.022/25.022 Forward_subsumed=0. Back_subsumed=0. 25.022/25.022 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.022/25.022 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.022/25.022 Demod_attempts=0. Demod_rewrites=0. 25.022/25.022 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.022/25.022 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 25.022/25.022 Megabytes=0.04. 25.022/25.022 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.022/25.022 25.022/25.022 ============================== end of statistics ===================== 25.022/25.022 25.022/25.022 ============================== end of search ========================= 25.022/25.022 25.022/25.022 THEOREM PROVED 25.022/25.022 25.022/25.022 Exiting with 1 proof. 25.022/25.022 25.022/25.022 Process 119304 exit (max_proofs) Fri Jul 16 15:35:23 2021 25.022/25.022 25.022/25.022 25.022/25.022 The problem is feasible. 25.022/25.022 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 Confluence Problem: 25.022/25.022 (VAR x:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (f 1) 25.022/25.022 (a) 25.022/25.022 (b) 25.022/25.022 (fSNonEmpty) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 f(x:S) -> a | x:S ->* a 25.022/25.022 f(x:S) -> b | x:S ->* b 25.022/25.022 ) 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 25.022/25.022 Critical Pairs Processor: 25.022/25.022 -> Rules: 25.022/25.022 f(x:S) -> a | x:S ->* a 25.022/25.022 f(x:S) -> b | x:S ->* b 25.022/25.022 -> Vars: 25.022/25.022 "x", "x" 25.022/25.022 -> FVars: 25.022/25.022 "x2", "x3" 25.022/25.022 -> PVars: 25.022/25.022 "x": ["x2", "x3"] 25.022/25.022 25.022/25.022 -> Rlps: 25.022/25.022 crule: f(x2:S) -> a | x2:S ->* a, id: 1, possubterms: f(x2:S)-> [] 25.022/25.022 crule: f(x3:S) -> b | x3:S ->* b, id: 2, possubterms: f(x3:S)-> [] 25.022/25.022 25.022/25.022 -> Unifications: 25.022/25.022 R2 unifies with R1 at p: [], l: f(x3:S), lp: f(x3:S), conds: {x3:S ->* b, x:S ->* a}, sig: {x3:S -> x:S}, l': f(x:S), r: b, r': a 25.022/25.022 25.022/25.022 -> Critical pairs info: 25.022/25.022 | x:S ->* b, x:S ->* a => Not trivial, Overlay, N1 25.022/25.022 25.022/25.022 -> Problem conclusions: 25.022/25.022 Left linear, Right linear, Linear 25.022/25.022 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 25.022/25.022 CTRS Type: 1 25.022/25.022 Deterministic, Strongly deterministic 25.022/25.022 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.022/25.022 Maybe right-stable CTRS, Overlay CTRS 25.022/25.022 Maybe normal CTRS, Maybe almost normal CTRS 25.022/25.022 Maybe terminating CTRS, Maybe joinable CCPs 25.022/25.022 Maybe level confluent 25.022/25.022 Maybe confluent 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 Clean Infeasible CCPs Processor: 25.022/25.022 Num of CPIs: 1 25.022/25.022 Timeout: 30 25.022/25.022 Timeout for each infeasibility problem: 30 s 25.022/25.022 | x:S ->* b, x:S ->* a => Not trivial, Overlay, N1 25.022/25.022 (PROBLEM INFEASIBILITY) 25.022/25.022 (VAR x) 25.022/25.022 (RULES 25.022/25.022 f(x) -> a | x ->* a 25.022/25.022 f(x) -> b | x ->* b) 25.022/25.022 (VAR x2) 25.022/25.022 (CONDITION x2 ->* b, x2 ->* a) 25.022/25.022 25.022/25.022 Proof: 25.022/25.022 YES 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 Infeasibility Problem: 25.022/25.022 [(VAR vNonEmpty:S x:S x2:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (f 1) 25.022/25.022 (a) 25.022/25.022 (b) 25.022/25.022 (fSNonEmpty) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 f(x:S) -> a | x:S ->* a 25.022/25.022 f(x:S) -> b | x:S ->* b 25.022/25.022 )] 25.022/25.022 25.022/25.022 Infeasibility Conditions: 25.022/25.022 x2:S ->* b, x2:S ->* a 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 Obtaining a model using Mace4: 25.022/25.022 25.022/25.022 -> Usable Rules: 25.022/25.022 f(x:S) -> a | x:S ->* a 25.022/25.022 f(x:S) -> b | x:S ->* b 25.022/25.022 25.022/25.022 -> Mace4 Output: 25.022/25.022 ============================== Mace4 ================================= 25.022/25.022 Mace4 (64) version 2009-11A, November 2009. 25.022/25.022 Process 119341 was started by ubuntu on ubuntu, 25.022/25.022 Fri Jul 16 15:35:24 2021 25.022/25.022 The command was "./mace4 -c -f /tmp/mace4119321-2.in". 25.022/25.022 ============================== end of head =========================== 25.022/25.022 25.022/25.022 ============================== INPUT ================================= 25.022/25.022 25.022/25.022 % Reading from file /tmp/mace4119321-2.in 25.022/25.022 25.022/25.022 assign(max_seconds,10). 25.022/25.022 25.022/25.022 formulas(assumptions). 25.022/25.022 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 25.022/25.022 ->*(x1,a) -> ->(f(x1),a) # label(replacement). 25.022/25.022 ->*(x1,b) -> ->(f(x1),b) # label(replacement). 25.022/25.022 ->*(x,x) # label(reflexivity). 25.022/25.022 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 formulas(goals). 25.022/25.022 (exists x2 (->*(x2,b) & ->*(x2,a))) # label(goal). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 ============================== end of input ========================== 25.022/25.022 25.022/25.022 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.022/25.022 25.022/25.022 % Formulas that are not ordinary clauses: 25.022/25.022 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.022/25.022 2 ->*(x1,a) -> ->(f(x1),a) # label(replacement) # label(non_clause). [assumption]. 25.022/25.022 3 ->*(x1,b) -> ->(f(x1),b) # label(replacement) # label(non_clause). [assumption]. 25.022/25.022 4 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.022/25.022 5 (exists x2 (->*(x2,b) & ->*(x2,a))) # label(goal) # label(non_clause) # label(goal). [goal]. 25.022/25.022 25.022/25.022 ============================== end of process non-clausal formulas === 25.022/25.022 25.022/25.022 ============================== CLAUSES FOR SEARCH ==================== 25.022/25.022 25.022/25.022 formulas(mace4_clauses). 25.022/25.022 -->(x,y) | ->(f(x),f(y)) # label(congruence). 25.022/25.022 -->*(x,a) | ->(f(x),a) # label(replacement). 25.022/25.022 -->*(x,b) | ->(f(x),b) # label(replacement). 25.022/25.022 ->*(x,x) # label(reflexivity). 25.022/25.022 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 25.022/25.022 -->*(x,b) | -->*(x,a) # label(goal). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 ============================== end of clauses for search ============= 25.022/25.022 25.022/25.022 % There are no natural numbers in the input. 25.022/25.022 25.022/25.022 ============================== DOMAIN SIZE 2 ========================= 25.022/25.022 25.022/25.022 ============================== MODEL ================================= 25.022/25.022 25.022/25.022 interpretation( 2, [number=1, seconds=0], [ 25.022/25.022 25.022/25.022 function(a, [ 0 ]), 25.022/25.022 25.022/25.022 function(b, [ 1 ]), 25.022/25.022 25.022/25.022 function(f(_), [ 0, 1 ]), 25.022/25.022 25.022/25.022 relation(->*(_,_), [ 25.022/25.022 1, 0, 25.022/25.022 0, 1 ]), 25.022/25.022 25.022/25.022 relation(->(_,_), [ 25.022/25.022 1, 0, 25.022/25.022 0, 1 ]) 25.022/25.022 ]). 25.022/25.022 25.022/25.022 ============================== end of model ========================== 25.022/25.022 25.022/25.022 ============================== STATISTICS ============================ 25.022/25.022 25.022/25.022 For domain size 2. 25.022/25.022 25.022/25.022 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 25.022/25.022 Ground clauses: seen=20, kept=16. 25.022/25.022 Selections=1, assignments=1, propagations=11, current_models=1. 25.022/25.022 Rewrite_terms=24, rewrite_bools=25, indexes=6. 25.022/25.022 Rules_from_neg_clauses=3, cross_offs=3. 25.022/25.022 25.022/25.022 ============================== end of statistics ===================== 25.022/25.022 25.022/25.022 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.022/25.022 25.022/25.022 Exiting with 1 model. 25.022/25.022 25.022/25.022 Process 119341 exit (max_models) Fri Jul 16 15:35:24 2021 25.022/25.022 The process finished Fri Jul 16 15:35:24 2021 25.022/25.022 25.022/25.022 25.022/25.022 Mace4 cooked interpretation: 25.022/25.022 25.022/25.022 25.022/25.022 25.022/25.022 The problem is infeasible. 25.022/25.022 25.022/25.022 25.022/25.022 25.022/25.022 25.022/25.022 -> Problem conclusions: 25.022/25.022 Left linear, Right linear, Linear 25.022/25.022 Weakly orthogonal, Almost orthogonal, Orthogonal 25.022/25.022 CTRS Type: 1 25.022/25.022 Deterministic, Strongly deterministic 25.022/25.022 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.022/25.022 Maybe right-stable CTRS, Overlay CTRS 25.022/25.022 Maybe normal CTRS, Maybe almost normal CTRS 25.022/25.022 Maybe terminating CTRS, Joinable CCPs 25.022/25.022 Maybe level confluent 25.022/25.022 Maybe confluent 25.022/25.022 25.022/25.022 Resulting CCPs: 25.022/25.022 No CCPs left 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 Underlying TRS Termination Processor: 25.022/25.022 25.022/25.022 Resulting Underlying TRS: 25.022/25.022 (VAR x:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (f 1) 25.022/25.022 (a) 25.022/25.022 (b) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 f(x:S) -> a 25.022/25.022 f(x:S) -> b 25.022/25.022 ) 25.022/25.022 Underlying TRS terminating? 25.022/25.022 YES 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 (VAR vu95NonEmpty:S xu58S:S) 25.022/25.022 (RULES 25.022/25.022 f(xu58S:S) -> a 25.022/25.022 f(xu58S:S) -> b 25.022/25.022 ) 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 Dependency Pairs Processor: 25.022/25.022 -> Pairs: 25.022/25.022 Empty 25.022/25.022 -> Rules: 25.022/25.022 f(xu58S:S) -> a 25.022/25.022 f(xu58S:S) -> b 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 SCC Processor: 25.022/25.022 -> Pairs: 25.022/25.022 Empty 25.022/25.022 -> Rules: 25.022/25.022 f(xu58S:S) -> a 25.022/25.022 f(xu58S:S) -> b 25.022/25.022 ->Strongly Connected Components: 25.022/25.022 There is no strongly connected component 25.022/25.022 25.022/25.022 The problem is finite. 25.022/25.022 25.022/25.022 25.022/25.022 -> Problem conclusions: 25.022/25.022 Left linear, Right linear, Linear 25.022/25.022 Weakly orthogonal, Almost orthogonal, Orthogonal 25.022/25.022 CTRS Type: 1 25.022/25.022 Deterministic, Strongly deterministic 25.022/25.022 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.022/25.022 Maybe right-stable CTRS, Overlay CTRS 25.022/25.022 Maybe normal CTRS, Maybe almost normal CTRS 25.022/25.022 Terminating CTRS, Joinable CCPs 25.022/25.022 Maybe level confluent 25.022/25.022 Confluent 25.022/25.022 25.022/25.022 The problem is joinable. 25.022/25.022 39.87user 1.07system 0:25.22elapsed 162%CPU (0avgtext+0avgdata 106440maxresident)k 25.022/25.022 0inputs+0outputs (0major+94683minor)pagefaults 0swaps