20.075/20.075 YES 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 20.075/20.075 20.075/20.075 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.075/20.075 Confluence Problem: 20.075/20.075 (VAR vNonEmpty x) 20.075/20.075 (REPLACEMENT-MAP 20.075/20.075 (f 1) 20.075/20.075 (g 1) 20.075/20.075 (a) 20.075/20.075 (b) 20.075/20.075 (c) 20.075/20.075 (fSNonEmpty) 20.075/20.075 ) 20.075/20.075 (RULES 20.075/20.075 f(g(x)) -> b | x ->* a 20.075/20.075 g(x) -> c | x ->* c 20.075/20.075 ) 20.075/20.075 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.075/20.075 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 20.075/20.075 Inlining of Conditions Processor [STERN17]: 20.075/20.075 20.075/20.075 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.075/20.075 Confluence Problem: 20.075/20.075 (VAR vNonEmpty x) 20.075/20.075 (REPLACEMENT-MAP 20.075/20.075 (f 1) 20.075/20.075 (g 1) 20.075/20.075 (a) 20.075/20.075 (b) 20.075/20.075 (c) 20.075/20.075 (fSNonEmpty) 20.075/20.075 ) 20.075/20.075 (RULES 20.075/20.075 f(g(x)) -> b | x ->* a 20.075/20.075 g(x) -> c | x ->* c 20.075/20.075 ) 20.075/20.075 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.075/20.075 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 20.075/20.075 Clean CTRS Processor: 20.075/20.075 20.075/20.075 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.075/20.075 Confluence Problem: 20.075/20.075 (VAR x) 20.075/20.075 (REPLACEMENT-MAP 20.075/20.075 (f 1) 20.075/20.075 (g 1) 20.075/20.075 (a) 20.075/20.075 (b) 20.075/20.075 (c) 20.075/20.075 (fSNonEmpty) 20.075/20.075 ) 20.075/20.075 (RULES 20.075/20.075 f(g(x)) -> b | x ->* a 20.075/20.075 g(x) -> c | x ->* c 20.075/20.075 ) 20.075/20.075 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.075/20.075 20.075/20.075 CRule InfChecker Info: 20.075/20.075 f(g(x)) -> b | x ->* a 20.075/20.075 Rule remains 20.075/20.075 Proof: 20.075/20.075 NO 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 20.075/20.075 Infeasibility Problem: 20.075/20.075 [(VAR vNonEmpty x vNonEmpty x) 20.075/20.075 (STRATEGY CONTEXTSENSITIVE 20.075/20.075 (f 1) 20.075/20.075 (g 1) 20.075/20.075 (a) 20.075/20.075 (b) 20.075/20.075 (c) 20.075/20.075 (fSNonEmpty) 20.075/20.075 ) 20.075/20.075 (RULES 20.075/20.075 f(g(x)) -> b | x ->* a 20.075/20.075 g(x) -> c | x ->* c 20.075/20.075 )] 20.075/20.075 20.075/20.075 Infeasibility Conditions: 20.075/20.075 x ->* a 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 20.075/20.075 Obtaining a proof using Prover9: 20.075/20.075 20.075/20.075 -> Prover9 Output: 20.075/20.075 ============================== Prover9 =============================== 20.075/20.075 Prover9 (64) version 2009-11A, November 2009. 20.075/20.075 Process 74205 was started by ubuntu on ubuntu, 20.075/20.075 Wed Mar 9 10:08:04 2022 20.075/20.075 The command was "./prover9 -f /tmp/prover974196-0.in". 20.075/20.075 ============================== end of head =========================== 20.075/20.075 20.075/20.075 ============================== INPUT ================================= 20.075/20.075 20.075/20.075 % Reading from file /tmp/prover974196-0.in 20.075/20.075 20.075/20.075 assign(max_seconds,20). 20.075/20.075 20.075/20.075 formulas(assumptions). 20.075/20.075 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 20.075/20.075 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 20.075/20.075 ->*_s0(x1,a) -> ->_s0(f(g(x1)),b) # label(replacement). 20.075/20.075 ->*_s0(x1,c) -> ->_s0(g(x1),c) # label(replacement). 20.075/20.075 ->*_s0(x,x) # label(reflexivity). 20.075/20.075 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 formulas(goals). 20.075/20.075 (exists x3 ->*_s0(x3,a)) # label(goal). 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 ============================== end of input ========================== 20.075/20.075 20.075/20.075 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.075/20.075 20.075/20.075 % Formulas that are not ordinary clauses: 20.075/20.075 1 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 20.075/20.075 2 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 20.075/20.075 3 ->*_s0(x1,a) -> ->_s0(f(g(x1)),b) # label(replacement) # label(non_clause). [assumption]. 20.075/20.075 4 ->*_s0(x1,c) -> ->_s0(g(x1),c) # label(replacement) # label(non_clause). [assumption]. 20.075/20.075 5 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.075/20.075 6 (exists x3 ->*_s0(x3,a)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.075/20.075 20.075/20.075 ============================== end of process non-clausal formulas === 20.075/20.075 20.075/20.075 ============================== PROCESS INITIAL CLAUSES =============== 20.075/20.075 20.075/20.075 % Clauses before input processing: 20.075/20.075 20.075/20.075 formulas(usable). 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 formulas(sos). 20.075/20.075 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 20.075/20.075 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(2)]. 20.075/20.075 -->*_s0(x,a) | ->_s0(f(g(x)),b) # label(replacement). [clausify(3)]. 20.075/20.075 -->*_s0(x,c) | ->_s0(g(x),c) # label(replacement). [clausify(4)]. 20.075/20.075 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.075/20.075 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(5)]. 20.075/20.075 -->*_s0(x,a) # label(goal). [deny(6)]. 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 formulas(demodulators). 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 ============================== PREDICATE ELIMINATION ================= 20.075/20.075 20.075/20.075 No predicates eliminated. 20.075/20.075 20.075/20.075 ============================== end predicate elimination ============= 20.075/20.075 20.075/20.075 Auto_denials: 20.075/20.075 % copying label goal to answer in negative clause 20.075/20.075 20.075/20.075 Term ordering decisions: 20.075/20.075 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.075/20.075 Function symbol precedence: function_order([ c, a, b, g, f ]). 20.075/20.075 After inverse_order: (no changes). 20.075/20.075 Unfolding symbols: (none). 20.075/20.075 20.075/20.075 Auto_inference settings: 20.075/20.075 % set(neg_binary_resolution). % (HNE depth_diff=-5) 20.075/20.075 % clear(ordered_res). % (HNE depth_diff=-5) 20.075/20.075 % set(ur_resolution). % (HNE depth_diff=-5) 20.075/20.075 % set(ur_resolution) -> set(pos_ur_resolution). 20.075/20.075 % set(ur_resolution) -> set(neg_ur_resolution). 20.075/20.075 20.075/20.075 Auto_process settings: (no changes). 20.075/20.075 20.075/20.075 kept: 7 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 20.075/20.075 kept: 8 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(2)]. 20.075/20.075 kept: 9 -->*_s0(x,a) | ->_s0(f(g(x)),b) # label(replacement). [clausify(3)]. 20.075/20.075 kept: 10 -->*_s0(x,c) | ->_s0(g(x),c) # label(replacement). [clausify(4)]. 20.075/20.075 kept: 11 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.075/20.075 kept: 12 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(5)]. 20.075/20.075 kept: 13 -->*_s0(x,a) # label(goal) # answer(goal). [deny(6)]. 20.075/20.075 20.075/20.075 ============================== PROOF ================================= 20.075/20.075 20.075/20.075 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.075/20.075 % Length of proof is 4. 20.075/20.075 % Level of proof is 2. 20.075/20.075 % Maximum clause weight is 3.000. 20.075/20.075 % Given clauses 0. 20.075/20.075 20.075/20.075 6 (exists x3 ->*_s0(x3,a)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.075/20.075 11 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.075/20.075 13 -->*_s0(x,a) # label(goal) # answer(goal). [deny(6)]. 20.075/20.075 14 $F # answer(goal). [resolve(13,a,11,a)]. 20.075/20.075 20.075/20.075 ============================== end of proof ========================== 20.075/20.075 20.075/20.075 ============================== STATISTICS ============================ 20.075/20.075 20.075/20.075 Given=0. Generated=7. Kept=7. proofs=1. 20.075/20.075 Usable=0. Sos=0. Demods=0. Limbo=6, Disabled=7. Hints=0. 20.075/20.075 Kept_by_rule=0, Deleted_by_rule=0. 20.075/20.075 Forward_subsumed=0. Back_subsumed=0. 20.075/20.075 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.075/20.075 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.075/20.075 Demod_attempts=0. Demod_rewrites=0. 20.075/20.075 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.075/20.075 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 20.075/20.075 Megabytes=0.04. 20.075/20.075 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.075/20.075 20.075/20.075 ============================== end of statistics ===================== 20.075/20.075 20.075/20.075 ============================== end of search ========================= 20.075/20.075 20.075/20.075 THEOREM PROVED 20.075/20.075 20.075/20.075 Exiting with 1 proof. 20.075/20.075 20.075/20.075 Process 74205 exit (max_proofs) Wed Mar 9 10:08:04 2022 20.075/20.075 20.075/20.075 20.075/20.075 The problem is feasible. 20.075/20.075 20.075/20.075 20.075/20.075 CRule InfChecker Info: 20.075/20.075 g(x) -> c | x ->* c 20.075/20.075 Rule remains 20.075/20.075 Proof: 20.075/20.075 NO 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 20.075/20.075 Infeasibility Problem: 20.075/20.075 [(VAR vNonEmpty x vNonEmpty x) 20.075/20.075 (STRATEGY CONTEXTSENSITIVE 20.075/20.075 (f 1) 20.075/20.075 (g 1) 20.075/20.075 (a) 20.075/20.075 (b) 20.075/20.075 (c) 20.075/20.075 (fSNonEmpty) 20.075/20.075 ) 20.075/20.075 (RULES 20.075/20.075 f(g(x)) -> b | x ->* a 20.075/20.075 g(x) -> c | x ->* c 20.075/20.075 )] 20.075/20.075 20.075/20.075 Infeasibility Conditions: 20.075/20.075 x ->* c 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 20.075/20.075 Obtaining a proof using Prover9: 20.075/20.075 20.075/20.075 -> Prover9 Output: 20.075/20.075 ============================== Prover9 =============================== 20.075/20.075 Prover9 (64) version 2009-11A, November 2009. 20.075/20.075 Process 74228 was started by ubuntu on ubuntu, 20.075/20.075 Wed Mar 9 10:08:04 2022 20.075/20.075 The command was "./prover9 -f /tmp/prover974218-0.in". 20.075/20.075 ============================== end of head =========================== 20.075/20.075 20.075/20.075 ============================== INPUT ================================= 20.075/20.075 20.075/20.075 % Reading from file /tmp/prover974218-0.in 20.075/20.075 20.075/20.075 assign(max_seconds,20). 20.075/20.075 20.075/20.075 formulas(assumptions). 20.075/20.075 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 20.075/20.075 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 20.075/20.075 ->*_s0(x1,a) -> ->_s0(f(g(x1)),b) # label(replacement). 20.075/20.075 ->*_s0(x1,c) -> ->_s0(g(x1),c) # label(replacement). 20.075/20.075 ->*_s0(x,x) # label(reflexivity). 20.075/20.075 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 formulas(goals). 20.075/20.075 (exists x3 ->*_s0(x3,c)) # label(goal). 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 ============================== end of input ========================== 20.075/20.075 20.075/20.075 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.075/20.075 20.075/20.075 % Formulas that are not ordinary clauses: 20.075/20.075 1 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 20.075/20.075 2 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 20.075/20.075 3 ->*_s0(x1,a) -> ->_s0(f(g(x1)),b) # label(replacement) # label(non_clause). [assumption]. 20.075/20.075 4 ->*_s0(x1,c) -> ->_s0(g(x1),c) # label(replacement) # label(non_clause). [assumption]. 20.075/20.075 5 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.075/20.075 6 (exists x3 ->*_s0(x3,c)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.075/20.075 20.075/20.075 ============================== end of process non-clausal formulas === 20.075/20.075 20.075/20.075 ============================== PROCESS INITIAL CLAUSES =============== 20.075/20.075 20.075/20.075 % Clauses before input processing: 20.075/20.075 20.075/20.075 formulas(usable). 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 formulas(sos). 20.075/20.075 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 20.075/20.075 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(2)]. 20.075/20.075 -->*_s0(x,a) | ->_s0(f(g(x)),b) # label(replacement). [clausify(3)]. 20.075/20.075 -->*_s0(x,c) | ->_s0(g(x),c) # label(replacement). [clausify(4)]. 20.075/20.075 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.075/20.075 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(5)]. 20.075/20.075 -->*_s0(x,c) # label(goal). [deny(6)]. 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 formulas(demodulators). 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 ============================== PREDICATE ELIMINATION ================= 20.075/20.075 20.075/20.075 No predicates eliminated. 20.075/20.075 20.075/20.075 ============================== end predicate elimination ============= 20.075/20.075 20.075/20.075 Auto_denials: 20.075/20.075 % copying label goal to answer in negative clause 20.075/20.075 20.075/20.075 Term ordering decisions: 20.075/20.075 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.075/20.075 Function symbol precedence: function_order([ c, a, b, g, f ]). 20.075/20.075 After inverse_order: (no changes). 20.075/20.075 Unfolding symbols: (none). 20.075/20.075 20.075/20.075 Auto_inference settings: 20.075/20.075 % set(neg_binary_resolution). % (HNE depth_diff=-5) 20.075/20.075 % clear(ordered_res). % (HNE depth_diff=-5) 20.075/20.075 % set(ur_resolution). % (HNE depth_diff=-5) 20.075/20.075 % set(ur_resolution) -> set(pos_ur_resolution). 20.075/20.075 % set(ur_resolution) -> set(neg_ur_resolution). 20.075/20.075 20.075/20.075 Auto_process settings: (no changes). 20.075/20.075 20.075/20.075 kept: 7 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 20.075/20.075 kept: 8 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(2)]. 20.075/20.075 kept: 9 -->*_s0(x,a) | ->_s0(f(g(x)),b) # label(replacement). [clausify(3)]. 20.075/20.075 kept: 10 -->*_s0(x,c) | ->_s0(g(x),c) # label(replacement). [clausify(4)]. 20.075/20.075 kept: 11 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.075/20.075 kept: 12 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(5)]. 20.075/20.075 kept: 13 -->*_s0(x,c) # label(goal) # answer(goal). [deny(6)]. 20.075/20.075 20.075/20.075 ============================== PROOF ================================= 20.075/20.075 20.075/20.075 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.075/20.075 % Length of proof is 4. 20.075/20.075 % Level of proof is 2. 20.075/20.075 % Maximum clause weight is 3.000. 20.075/20.075 % Given clauses 0. 20.075/20.075 20.075/20.075 6 (exists x3 ->*_s0(x3,c)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.075/20.075 11 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.075/20.075 13 -->*_s0(x,c) # label(goal) # answer(goal). [deny(6)]. 20.075/20.075 14 $F # answer(goal). [resolve(13,a,11,a)]. 20.075/20.075 20.075/20.075 ============================== end of proof ========================== 20.075/20.075 20.075/20.075 ============================== STATISTICS ============================ 20.075/20.075 20.075/20.075 Given=0. Generated=7. Kept=7. proofs=1. 20.075/20.075 Usable=0. Sos=0. Demods=0. Limbo=6, Disabled=7. Hints=0. 20.075/20.075 Kept_by_rule=0, Deleted_by_rule=0. 20.075/20.075 Forward_subsumed=0. Back_subsumed=0. 20.075/20.075 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.075/20.075 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.075/20.075 Demod_attempts=0. Demod_rewrites=0. 20.075/20.075 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.075/20.075 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 20.075/20.075 Megabytes=0.04. 20.075/20.075 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.075/20.075 20.075/20.075 ============================== end of statistics ===================== 20.075/20.075 20.075/20.075 ============================== end of search ========================= 20.075/20.075 20.075/20.075 THEOREM PROVED 20.075/20.075 20.075/20.075 Exiting with 1 proof. 20.075/20.075 20.075/20.075 Process 74228 exit (max_proofs) Wed Mar 9 10:08:04 2022 20.075/20.075 20.075/20.075 20.075/20.075 The problem is feasible. 20.075/20.075 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.075/20.075 Confluence Problem: 20.075/20.075 (VAR x) 20.075/20.075 (REPLACEMENT-MAP 20.075/20.075 (f 1) 20.075/20.075 (g 1) 20.075/20.075 (a) 20.075/20.075 (b) 20.075/20.075 (c) 20.075/20.075 (fSNonEmpty) 20.075/20.075 ) 20.075/20.075 (RULES 20.075/20.075 f(g(x)) -> b | x ->* a 20.075/20.075 g(x) -> c | x ->* c 20.075/20.075 ) 20.075/20.075 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.075/20.075 20.075/20.075 Critical Pairs Processor: 20.075/20.075 -> Rules: 20.075/20.075 f(g(x)) -> b | x ->* a 20.075/20.075 g(x) -> c | x ->* c 20.075/20.075 -> Vars: 20.075/20.075 "x" 20.075/20.075 20.075/20.075 -> Rlps: 20.075/20.075 crule: f(g(x)) -> b | x ->* a, id: 1, possubterms: f(g(x))-> [], g(x)-> [1] 20.075/20.075 crule: g(x) -> c | x ->* c, id: 2, possubterms: g(x)-> [] 20.075/20.075 20.075/20.075 -> Unifications: 20.075/20.075 R1 unifies with R2 at p: [1], l: f(g(x)), lp: g(x), conds: {x ->* a, x' ->* c}, sig: {x -> x'}, l': g(x'), r: b, r': c 20.075/20.075 20.075/20.075 -> Critical pairs info: 20.075/20.075 | x' ->* a, x' ->* c => Not trivial, Not overlay, NW2, N1 20.075/20.075 20.075/20.075 -> Problem conclusions: 20.075/20.075 Left linear, Right linear, Linear 20.075/20.075 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.075/20.075 CTRS Type: 1 20.075/20.075 Deterministic, Strongly deterministic 20.075/20.075 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.075/20.075 Maybe right-stable CTRS, Not overlay CTRS 20.075/20.075 Maybe normal CTRS, Maybe almost normal CTRS 20.075/20.075 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.075/20.075 Maybe level confluent 20.075/20.075 Maybe confluent 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 Clean Infeasible CCPs Processor: 20.075/20.075 Num of CPIs: 1 20.075/20.075 Timeout: 60 20.075/20.075 Timeout for each infeasibility problem: 60 s 20.075/20.075 | x' ->* a, x' ->* c => Not trivial, Not overlay, NW2, N1 20.075/20.075 (CONDITIONTYPE ORIENTED) 20.075/20.075 (VAR x x1) 20.075/20.075 (RULES 20.075/20.075 f(g(x)) -> b | x ->* a 20.075/20.075 g(x) -> c | x ->* c) 20.075/20.075 (VAR x4) 20.075/20.075 (CONDITION x4 ->* a, x4 ->* c) 20.075/20.075 20.075/20.075 Proof: 20.075/20.075 YES 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 20.075/20.075 Infeasibility Problem: 20.075/20.075 [(VAR vNonEmpty x x1 vNonEmpty x4) 20.075/20.075 (STRATEGY CONTEXTSENSITIVE 20.075/20.075 (f 1) 20.075/20.075 (g 1) 20.075/20.075 (a) 20.075/20.075 (b) 20.075/20.075 (c) 20.075/20.075 (fSNonEmpty) 20.075/20.075 ) 20.075/20.075 (RULES 20.075/20.075 f(g(x)) -> b | x ->* a 20.075/20.075 g(x) -> c | x ->* c 20.075/20.075 )] 20.075/20.075 20.075/20.075 Infeasibility Conditions: 20.075/20.075 x4 ->* a, x4 ->* c 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 20.075/20.075 Obtaining a model using Mace4: 20.075/20.075 20.075/20.075 -> Usable Rules: 20.075/20.075 f(g(x)) -> b | x ->* a 20.075/20.075 g(x) -> c | x ->* c 20.075/20.075 20.075/20.075 -> Mace4 Output: 20.075/20.075 ============================== Mace4 ================================= 20.075/20.075 Mace4 (64) version 2009-11A, November 2009. 20.075/20.075 Process 74266 was started by ubuntu on ubuntu, 20.075/20.075 Wed Mar 9 10:08:04 2022 20.075/20.075 The command was "./mace4 -c -f /tmp/mace474244-2.in". 20.075/20.075 ============================== end of head =========================== 20.075/20.075 20.075/20.075 ============================== INPUT ================================= 20.075/20.075 20.075/20.075 % Reading from file /tmp/mace474244-2.in 20.075/20.075 20.075/20.075 assign(max_seconds,20). 20.075/20.075 20.075/20.075 formulas(assumptions). 20.075/20.075 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 20.075/20.075 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence). 20.075/20.075 ->*(x1,a) -> ->(f(g(x1)),b) # label(replacement). 20.075/20.075 ->*(x1,c) -> ->(g(x1),c) # label(replacement). 20.075/20.075 ->*(x,x) # label(reflexivity). 20.075/20.075 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 formulas(goals). 20.075/20.075 (exists x4 (->*(x4,a) & ->*(x4,c))) # label(goal). 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 ============================== end of input ========================== 20.075/20.075 20.075/20.075 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.075/20.075 20.075/20.075 % Formulas that are not ordinary clauses: 20.075/20.075 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 20.075/20.075 2 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 20.075/20.075 3 ->*(x1,a) -> ->(f(g(x1)),b) # label(replacement) # label(non_clause). [assumption]. 20.075/20.075 4 ->*(x1,c) -> ->(g(x1),c) # label(replacement) # label(non_clause). [assumption]. 20.075/20.075 5 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.075/20.075 6 (exists x4 (->*(x4,a) & ->*(x4,c))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.075/20.075 20.075/20.075 ============================== end of process non-clausal formulas === 20.075/20.075 20.075/20.075 ============================== CLAUSES FOR SEARCH ==================== 20.075/20.075 20.075/20.075 formulas(mace4_clauses). 20.075/20.075 -->(x,y) | ->(f(x),f(y)) # label(congruence). 20.075/20.075 -->(x,y) | ->(g(x),g(y)) # label(congruence). 20.075/20.075 -->*(x,a) | ->(f(g(x)),b) # label(replacement). 20.075/20.075 -->*(x,c) | ->(g(x),c) # label(replacement). 20.075/20.075 ->*(x,x) # label(reflexivity). 20.075/20.075 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 20.075/20.075 -->*(x,a) | -->*(x,c) # label(goal). 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 ============================== end of clauses for search ============= 20.075/20.075 20.075/20.075 % There are no natural numbers in the input. 20.075/20.075 20.075/20.075 ============================== DOMAIN SIZE 2 ========================= 20.075/20.075 20.075/20.075 ============================== MODEL ================================= 20.075/20.075 20.075/20.075 interpretation( 2, [number=1, seconds=0], [ 20.075/20.075 20.075/20.075 function(a, [ 0 ]), 20.075/20.075 20.075/20.075 function(b, [ 0 ]), 20.075/20.075 20.075/20.075 function(c, [ 1 ]), 20.075/20.075 20.075/20.075 function(f(_), [ 0, 0 ]), 20.075/20.075 20.075/20.075 function(g(_), [ 0, 1 ]), 20.075/20.075 20.075/20.075 relation(->*(_,_), [ 20.075/20.075 1, 0, 20.075/20.075 0, 1 ]), 20.075/20.075 20.075/20.075 relation(->(_,_), [ 20.075/20.075 1, 0, 20.075/20.075 0, 1 ]) 20.075/20.075 ]). 20.075/20.075 20.075/20.075 ============================== end of model ========================== 20.075/20.075 20.075/20.075 ============================== STATISTICS ============================ 20.075/20.075 20.075/20.075 For domain size 2. 20.075/20.075 20.075/20.075 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.075/20.075 Ground clauses: seen=24, kept=20. 20.075/20.075 Selections=5, assignments=5, propagations=10, current_models=1. 20.075/20.075 Rewrite_terms=34, rewrite_bools=30, indexes=7. 20.075/20.075 Rules_from_neg_clauses=2, cross_offs=2. 20.075/20.075 20.075/20.075 ============================== end of statistics ===================== 20.075/20.075 20.075/20.075 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.075/20.075 20.075/20.075 Exiting with 1 model. 20.075/20.075 20.075/20.075 Process 74266 exit (max_models) Wed Mar 9 10:08:04 2022 20.075/20.075 The process finished Wed Mar 9 10:08:04 2022 20.075/20.075 20.075/20.075 20.075/20.075 Mace4 cooked interpretation: 20.075/20.075 20.075/20.075 20.075/20.075 20.075/20.075 The problem is infeasible. 20.075/20.075 20.075/20.075 20.075/20.075 20.075/20.075 20.075/20.075 -> Problem conclusions: 20.075/20.075 Left linear, Right linear, Linear 20.075/20.075 Weakly orthogonal, Almost orthogonal, Orthogonal 20.075/20.075 CTRS Type: 1 20.075/20.075 Deterministic, Strongly deterministic 20.075/20.075 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.075/20.075 Maybe right-stable CTRS, Overlay CTRS 20.075/20.075 Maybe normal CTRS, Maybe almost normal CTRS 20.075/20.075 Maybe terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.075/20.075 Maybe level confluent 20.075/20.075 Maybe confluent 20.075/20.075 20.075/20.075 Resulting CCPs: 20.075/20.075 No CCPs left 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 Underlying TRS Termination Processor: 20.075/20.075 20.075/20.075 Resulting Underlying TRS: 20.075/20.075 (VAR x x') 20.075/20.075 (STRATEGY CONTEXTSENSITIVE 20.075/20.075 (f 1) 20.075/20.075 (g 1) 20.075/20.075 (b) 20.075/20.075 (c) 20.075/20.075 ) 20.075/20.075 (RULES 20.075/20.075 f(g(x)) -> b 20.075/20.075 g(x) -> c 20.075/20.075 ) 20.075/20.075 Underlying TRS terminating? 20.075/20.075 YES 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 20.075/20.075 (VAR vu95NonEmpty x xu39) 20.075/20.075 (RULES 20.075/20.075 f(g(x)) -> b 20.075/20.075 g(x) -> c 20.075/20.075 ) 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 20.075/20.075 Dependency Pairs Processor: 20.075/20.075 -> Pairs: 20.075/20.075 Empty 20.075/20.075 -> Rules: 20.075/20.075 f(g(x)) -> b 20.075/20.075 g(x) -> c 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 20.075/20.075 SCC Processor: 20.075/20.075 -> Pairs: 20.075/20.075 Empty 20.075/20.075 -> Rules: 20.075/20.075 f(g(x)) -> b 20.075/20.075 g(x) -> c 20.075/20.075 ->Strongly Connected Components: 20.075/20.075 There is no strongly connected component 20.075/20.075 20.075/20.075 The problem is finite. 20.075/20.075 20.075/20.075 20.075/20.075 -> Problem conclusions: 20.075/20.075 Left linear, Right linear, Linear 20.075/20.075 Weakly orthogonal, Almost orthogonal, Orthogonal 20.075/20.075 CTRS Type: 1 20.075/20.075 Deterministic, Strongly deterministic 20.075/20.075 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.075/20.075 Maybe right-stable CTRS, Overlay CTRS 20.075/20.075 Maybe normal CTRS, Maybe almost normal CTRS 20.075/20.075 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.075/20.075 Maybe level confluent 20.075/20.075 Maybe confluent 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.075/20.075 Confluence Problem: 20.075/20.075 (VAR x x') 20.075/20.075 (REPLACEMENT-MAP 20.075/20.075 (f 1) 20.075/20.075 (g 1) 20.075/20.075 (a) 20.075/20.075 (b) 20.075/20.075 (c) 20.075/20.075 (fSNonEmpty) 20.075/20.075 ) 20.075/20.075 (RULES 20.075/20.075 f(g(x)) -> b | x ->* a 20.075/20.075 g(x) -> c | x ->* c 20.075/20.075 ) 20.075/20.075 Critical Pairs: 20.075/20.075 20.075/20.075 Terminating:YES 20.075/20.075 20.075/20.075 -> Problem conclusions: 20.075/20.075 Left linear, Right linear, Linear 20.075/20.075 Weakly orthogonal, Almost orthogonal, Orthogonal 20.075/20.075 CTRS Type: 1 20.075/20.075 Deterministic, Strongly deterministic 20.075/20.075 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.075/20.075 Maybe right-stable CTRS, Overlay CTRS 20.075/20.075 Maybe normal CTRS, Maybe almost normal CTRS 20.075/20.075 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.075/20.075 Maybe level confluent 20.075/20.075 Maybe confluent 20.075/20.075 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.075/20.075 20.075/20.075 Right Stable Processor: 20.075/20.075 Right-stable CTRS 20.075/20.075 Justification: 20.075/20.075 20.075/20.075 -> Term right-stability conditions: 20.075/20.075 Term: a 20.075/20.075 Right-stable term 20.075/20.075 Linear constructor form 20.075/20.075 Don't know if term is a ground normal form (not needed) 20.075/20.075 Right-stability condition achieved 20.075/20.075 A call to InfChecker wasn't needed 20.075/20.075 20.075/20.075 20.075/20.075 -> Term right-stability conditions: 20.075/20.075 Term: c 20.075/20.075 Right-stable term 20.075/20.075 Linear constructor form 20.075/20.075 Don't know if term is a ground normal form (not needed) 20.075/20.075 Right-stability condition achieved 20.075/20.075 A call to InfChecker wasn't needed 20.075/20.075 20.075/20.075 -> Problem conclusions: 20.075/20.075 Left linear, Right linear, Linear 20.075/20.075 Weakly orthogonal, Almost orthogonal, Orthogonal 20.075/20.075 CTRS Type: 1 20.075/20.075 Deterministic, Strongly deterministic 20.075/20.075 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.075/20.075 Right-stable CTRS, Overlay CTRS 20.075/20.075 Maybe normal CTRS, Almost normal CTRS 20.075/20.075 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.075/20.075 Level confluent 20.075/20.075 Confluent 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.075/20.075 Confluence Problem: 20.075/20.075 (VAR x x') 20.075/20.075 (REPLACEMENT-MAP 20.075/20.075 (f 1) 20.075/20.075 (g 1) 20.075/20.075 (a) 20.075/20.075 (b) 20.075/20.075 (c) 20.075/20.075 (fSNonEmpty) 20.075/20.075 ) 20.075/20.075 (RULES 20.075/20.075 f(g(x)) -> b | x ->* a 20.075/20.075 g(x) -> c | x ->* c 20.075/20.075 ) 20.075/20.075 Critical Pairs: 20.075/20.075 20.075/20.075 Terminating:YES 20.075/20.075 20.075/20.075 -> Problem conclusions: 20.075/20.075 Left linear, Right linear, Linear 20.075/20.075 Weakly orthogonal, Almost orthogonal, Orthogonal 20.075/20.075 CTRS Type: 1 20.075/20.075 Deterministic, Strongly deterministic 20.075/20.075 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.075/20.075 Right-stable CTRS, Overlay CTRS 20.075/20.075 Maybe normal CTRS, Almost normal CTRS 20.075/20.075 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.075/20.075 Level confluent 20.075/20.075 Confluent 20.075/20.075 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.075/20.075 20.075/20.075 Normal RConds Processor: 20.075/20.075 YES 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 20.075/20.075 Infeasibility Problem: 20.075/20.075 [(VAR vNonEmpty x x1 vNonEmpty z z1) 20.075/20.075 (STRATEGY CONTEXTSENSITIVE 20.075/20.075 (f 1) 20.075/20.075 (g 1) 20.075/20.075 (a) 20.075/20.075 (b) 20.075/20.075 (c) 20.075/20.075 (fSNonEmpty) 20.075/20.075 ) 20.075/20.075 (RULES 20.075/20.075 f(g(x)) -> b | x ->* a 20.075/20.075 g(x) -> c | x ->* c 20.075/20.075 )] 20.075/20.075 20.075/20.075 Infeasibility Conditions: 20.075/20.075 a -> z, c -> z1 20.075/20.075 20.075/20.075 Problem 1: 20.075/20.075 20.075/20.075 Obtaining a model using Mace4: 20.075/20.075 20.075/20.075 -> Usable Rules: 20.075/20.075 Empty 20.075/20.075 20.075/20.075 -> Mace4 Output: 20.075/20.075 ============================== Mace4 ================================= 20.075/20.075 Mace4 (64) version 2009-11A, November 2009. 20.075/20.075 Process 74333 was started by ubuntu on ubuntu, 20.075/20.075 Wed Mar 9 10:08:25 2022 20.075/20.075 The command was "./mace4 -c -f /tmp/mace474320-2.in". 20.075/20.075 ============================== end of head =========================== 20.075/20.075 20.075/20.075 ============================== INPUT ================================= 20.075/20.075 20.075/20.075 % Reading from file /tmp/mace474320-2.in 20.075/20.075 20.075/20.075 assign(max_seconds,13). 20.075/20.075 20.075/20.075 formulas(assumptions). 20.075/20.075 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 20.075/20.075 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence). 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 formulas(goals). 20.075/20.075 (exists x4 exists x5 (->(a,x4) & ->(c,x5))) # label(goal). 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 ============================== end of input ========================== 20.075/20.075 20.075/20.075 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.075/20.075 20.075/20.075 % Formulas that are not ordinary clauses: 20.075/20.075 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 20.075/20.075 2 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 20.075/20.075 3 (exists x4 exists x5 (->(a,x4) & ->(c,x5))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.075/20.075 20.075/20.075 ============================== end of process non-clausal formulas === 20.075/20.075 20.075/20.075 ============================== CLAUSES FOR SEARCH ==================== 20.075/20.075 20.075/20.075 formulas(mace4_clauses). 20.075/20.075 -->(x,y) | ->(f(x),f(y)) # label(congruence). 20.075/20.075 -->(x,y) | ->(g(x),g(y)) # label(congruence). 20.075/20.075 -->(a,x) | -->(c,y) # label(goal). 20.075/20.075 end_of_list. 20.075/20.075 20.075/20.075 ============================== end of clauses for search ============= 20.075/20.075 20.075/20.075 % There are no natural numbers in the input. 20.075/20.075 20.075/20.075 ============================== DOMAIN SIZE 2 ========================= 20.075/20.075 20.075/20.075 ============================== MODEL ================================= 20.075/20.075 20.075/20.075 interpretation( 2, [number=1, seconds=0], [ 20.075/20.075 20.075/20.075 function(a, [ 0 ]), 20.075/20.075 20.075/20.075 function(c, [ 0 ]), 20.075/20.075 20.075/20.075 function(f(_), [ 0, 0 ]), 20.075/20.075 20.075/20.075 function(g(_), [ 0, 0 ]), 20.075/20.075 20.075/20.075 relation(->(_,_), [ 20.075/20.075 0, 0, 20.075/20.075 0, 0 ]) 20.075/20.075 ]). 20.075/20.075 20.075/20.075 ============================== end of model ========================== 20.075/20.075 20.075/20.075 ============================== STATISTICS ============================ 20.075/20.075 20.075/20.075 For domain size 2. 20.075/20.075 20.075/20.075 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.075/20.075 Ground clauses: seen=12, kept=12. 20.075/20.075 Selections=7, assignments=7, propagations=3, current_models=1. 20.075/20.075 Rewrite_terms=24, rewrite_bools=17, indexes=10. 20.075/20.075 Rules_from_neg_clauses=0, cross_offs=0. 20.075/20.075 20.075/20.075 ============================== end of statistics ===================== 20.075/20.075 20.075/20.075 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.075/20.075 20.075/20.075 Exiting with 1 model. 20.075/20.075 20.075/20.075 Process 74333 exit (max_models) Wed Mar 9 10:08:25 2022 20.075/20.075 The process finished Wed Mar 9 10:08:25 2022 20.075/20.075 20.075/20.075 20.075/20.075 Mace4 cooked interpretation: 20.075/20.075 20.075/20.075 20.075/20.075 20.075/20.075 The problem is infeasible. 20.075/20.075 20.075/20.075 20.075/20.075 -> Problem conclusions: 20.075/20.075 Left linear, Right linear, Linear 20.075/20.075 Weakly orthogonal, Almost orthogonal, Orthogonal 20.075/20.075 CTRS Type: 1 20.075/20.075 Deterministic, Strongly deterministic 20.075/20.075 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.075/20.075 Right-stable CTRS, Overlay CTRS 20.075/20.075 Normal CTRS, Almost normal CTRS 20.075/20.075 Terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.075/20.075 Level confluent 20.075/20.075 Confluent 20.075/20.075 20.075/20.075 The problem is joinable. 20.075/20.075 48.19user 0.97system 0:20.75elapsed 236%CPU (0avgtext+0avgdata 261956maxresident)k 20.075/20.075 8inputs+0outputs (0major+125168minor)pagefaults 0swaps