25.036/25.036 YES 25.036/25.036 25.036/25.036 Problem 1: 25.036/25.036 25.036/25.036 25.036/25.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.036/25.036 Confluence Problem: 25.036/25.036 (VAR vNonEmpty:S x:S) 25.036/25.036 (STRATEGY CONTEXTSENSITIVE 25.036/25.036 (f 1) 25.036/25.036 (g 1) 25.036/25.036 (a) 25.036/25.036 (b) 25.036/25.036 (c) 25.036/25.036 (fSNonEmpty) 25.036/25.036 ) 25.036/25.036 (RULES 25.036/25.036 f(g(x:S)) -> b | x:S ->* a 25.036/25.036 g(x:S) -> c | x:S ->* c 25.036/25.036 ) 25.036/25.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.036/25.036 25.036/25.036 25.036/25.036 Problem 1: 25.036/25.036 25.036/25.036 Inlining of Conditions Processor [STERN17]: 25.036/25.036 25.036/25.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.036/25.036 Confluence Problem: 25.036/25.036 (VAR vNonEmpty:S x:S) 25.036/25.036 (STRATEGY CONTEXTSENSITIVE 25.036/25.036 (f 1) 25.036/25.036 (g 1) 25.036/25.036 (a) 25.036/25.036 (b) 25.036/25.036 (c) 25.036/25.036 (fSNonEmpty) 25.036/25.036 ) 25.036/25.036 (RULES 25.036/25.036 f(g(x:S)) -> b | x:S ->* a 25.036/25.036 g(x:S) -> c | x:S ->* c 25.036/25.036 ) 25.036/25.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.036/25.036 25.036/25.036 25.036/25.036 Problem 1: 25.036/25.036 25.036/25.036 Clean CTRS Processor: 25.036/25.036 25.036/25.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.036/25.036 Confluence Problem: 25.036/25.036 (VAR x:S) 25.036/25.036 (STRATEGY CONTEXTSENSITIVE 25.036/25.036 (f 1) 25.036/25.036 (g 1) 25.036/25.036 (a) 25.036/25.036 (b) 25.036/25.036 (c) 25.036/25.036 (fSNonEmpty) 25.036/25.036 ) 25.036/25.036 (RULES 25.036/25.036 f(g(x:S)) -> b | x:S ->* a 25.036/25.036 g(x:S) -> c | x:S ->* c 25.036/25.036 ) 25.036/25.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.036/25.036 25.036/25.036 CRule InfChecker Info: 25.036/25.036 f(g(x:S)) -> b | x:S ->* a 25.036/25.036 Rule remains 25.036/25.036 Proof: 25.036/25.036 NO 25.036/25.036 25.036/25.036 Problem 1: 25.036/25.036 25.036/25.036 Infeasibility Problem: 25.036/25.036 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 25.036/25.036 (STRATEGY CONTEXTSENSITIVE 25.036/25.036 (f 1) 25.036/25.036 (g 1) 25.036/25.036 (a) 25.036/25.036 (b) 25.036/25.036 (c) 25.036/25.036 (fSNonEmpty) 25.036/25.036 ) 25.036/25.036 (RULES 25.036/25.036 f(g(x:S:S)) -> b | x:S:S ->* a 25.036/25.036 g(x:S:S) -> c | x:S:S ->* c 25.036/25.036 )] 25.036/25.036 25.036/25.036 Infeasibility Conditions: 25.036/25.036 x:S:S ->* a 25.036/25.036 25.036/25.036 Problem 1: 25.036/25.036 25.036/25.036 Obtaining a proof using Prover9: 25.036/25.036 25.036/25.036 -> Prover9 Output: 25.036/25.036 ============================== Prover9 =============================== 25.036/25.036 Prover9 (64) version 2009-11A, November 2009. 25.036/25.036 Process 117092 was started by ubuntu on ubuntu, 25.036/25.036 Fri Jul 16 15:28:52 2021 25.036/25.036 The command was "./prover9 -f /tmp/prover9117083-0.in". 25.036/25.036 ============================== end of head =========================== 25.036/25.036 25.036/25.036 ============================== INPUT ================================= 25.036/25.036 25.036/25.036 % Reading from file /tmp/prover9117083-0.in 25.036/25.036 25.036/25.036 assign(max_seconds,20). 25.036/25.036 25.036/25.036 formulas(assumptions). 25.036/25.036 ->*_s0(x,x) # label(reflexivity). 25.036/25.036 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.036/25.036 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 25.036/25.036 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 25.036/25.036 ->*_s0(x2,a) -> ->_s0(f(g(x2)),b) # label(replacement). 25.036/25.036 ->*_s0(x2,c) -> ->_s0(g(x2),c) # label(replacement). 25.036/25.036 end_of_list. 25.036/25.036 25.036/25.036 formulas(goals). 25.036/25.036 (exists x2 ->*_s0(x2,a)) # label(goal). 25.036/25.036 end_of_list. 25.036/25.036 25.036/25.036 ============================== end of input ========================== 25.036/25.036 25.036/25.036 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.036/25.036 25.036/25.036 % Formulas that are not ordinary clauses: 25.036/25.036 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.036/25.036 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.036/25.036 3 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.036/25.036 4 ->*_s0(x2,a) -> ->_s0(f(g(x2)),b) # label(replacement) # label(non_clause). [assumption]. 25.036/25.036 5 ->*_s0(x2,c) -> ->_s0(g(x2),c) # label(replacement) # label(non_clause). [assumption]. 25.036/25.036 6 (exists x2 ->*_s0(x2,a)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.036/25.036 25.036/25.036 ============================== end of process non-clausal formulas === 25.036/25.036 25.036/25.036 ============================== PROCESS INITIAL CLAUSES =============== 25.036/25.036 25.036/25.036 % Clauses before input processing: 25.036/25.036 25.036/25.036 formulas(usable). 25.036/25.036 end_of_list. 25.036/25.036 25.036/25.036 formulas(sos). 25.036/25.036 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.036/25.036 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.036/25.036 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.036/25.036 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.036/25.036 -->*_s0(x,a) | ->_s0(f(g(x)),b) # label(replacement). [clausify(4)]. 25.036/25.036 -->*_s0(x,c) | ->_s0(g(x),c) # label(replacement). [clausify(5)]. 25.036/25.036 -->*_s0(x,a) # label(goal). [deny(6)]. 25.036/25.036 end_of_list. 25.036/25.036 25.036/25.036 formulas(demodulators). 25.036/25.036 end_of_list. 25.036/25.036 25.036/25.036 ============================== PREDICATE ELIMINATION ================= 25.036/25.036 25.036/25.036 No predicates eliminated. 25.036/25.036 25.036/25.036 ============================== end predicate elimination ============= 25.036/25.036 25.036/25.036 Auto_denials: 25.036/25.036 % copying label goal to answer in negative clause 25.036/25.036 25.036/25.036 Term ordering decisions: 25.036/25.036 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.036/25.036 Function symbol precedence: function_order([ c, a, b, g, f ]). 25.036/25.036 After inverse_order: (no changes). 25.036/25.036 Unfolding symbols: (none). 25.036/25.036 25.036/25.036 Auto_inference settings: 25.036/25.036 % set(neg_binary_resolution). % (HNE depth_diff=-5) 25.036/25.036 % clear(ordered_res). % (HNE depth_diff=-5) 25.036/25.036 % set(ur_resolution). % (HNE depth_diff=-5) 25.036/25.036 % set(ur_resolution) -> set(pos_ur_resolution). 25.036/25.036 % set(ur_resolution) -> set(neg_ur_resolution). 25.036/25.036 25.036/25.036 Auto_process settings: (no changes). 25.036/25.036 25.036/25.036 kept: 7 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.036/25.036 kept: 8 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.036/25.036 kept: 9 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.036/25.036 kept: 10 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.036/25.036 kept: 11 -->*_s0(x,a) | ->_s0(f(g(x)),b) # label(replacement). [clausify(4)]. 25.036/25.036 kept: 12 -->*_s0(x,c) | ->_s0(g(x),c) # label(replacement). [clausify(5)]. 25.036/25.036 kept: 13 -->*_s0(x,a) # label(goal) # answer(goal). [deny(6)]. 25.036/25.036 25.036/25.036 ============================== PROOF ================================= 25.036/25.036 25.036/25.036 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.036/25.036 % Length of proof is 4. 25.036/25.036 % Level of proof is 2. 25.036/25.036 % Maximum clause weight is 3.000. 25.036/25.036 % Given clauses 0. 25.036/25.036 25.036/25.036 6 (exists x2 ->*_s0(x2,a)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.036/25.036 7 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.036/25.036 13 -->*_s0(x,a) # label(goal) # answer(goal). [deny(6)]. 25.036/25.036 14 $F # answer(goal). [resolve(13,a,7,a)]. 25.036/25.036 25.036/25.036 ============================== end of proof ========================== 25.036/25.036 25.036/25.036 ============================== STATISTICS ============================ 25.036/25.036 25.036/25.036 Given=0. Generated=7. Kept=7. proofs=1. 25.036/25.036 Usable=0. Sos=0. Demods=0. Limbo=6, Disabled=7. Hints=0. 25.036/25.036 Kept_by_rule=0, Deleted_by_rule=0. 25.036/25.036 Forward_subsumed=0. Back_subsumed=0. 25.036/25.036 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.036/25.036 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.036/25.036 Demod_attempts=0. Demod_rewrites=0. 25.036/25.036 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.036/25.036 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 25.036/25.036 Megabytes=0.04. 25.036/25.036 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.036/25.036 25.036/25.036 ============================== end of statistics ===================== 25.036/25.036 25.036/25.036 ============================== end of search ========================= 25.036/25.036 25.036/25.036 THEOREM PROVED 25.036/25.036 25.036/25.036 Exiting with 1 proof. 25.036/25.036 25.036/25.036 Process 117092 exit (max_proofs) Fri Jul 16 15:28:52 2021 25.036/25.036 25.036/25.036 25.036/25.036 The problem is feasible. 25.036/25.036 25.036/25.036 25.036/25.036 CRule InfChecker Info: 25.036/25.036 g(x:S) -> c | x:S ->* c 25.036/25.036 Rule remains 25.036/25.036 Proof: 25.036/25.036 NO 25.036/25.036 25.036/25.036 Problem 1: 25.036/25.036 25.036/25.036 Infeasibility Problem: 25.036/25.036 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 25.036/25.036 (STRATEGY CONTEXTSENSITIVE 25.036/25.036 (f 1) 25.036/25.036 (g 1) 25.036/25.036 (a) 25.036/25.036 (b) 25.036/25.036 (c) 25.036/25.036 (fSNonEmpty) 25.036/25.036 ) 25.036/25.036 (RULES 25.036/25.036 f(g(x:S:S)) -> b | x:S:S ->* a 25.036/25.036 g(x:S:S) -> c | x:S:S ->* c 25.036/25.036 )] 25.036/25.036 25.036/25.036 Infeasibility Conditions: 25.036/25.036 x:S:S ->* c 25.036/25.036 25.036/25.036 Problem 1: 25.036/25.036 25.036/25.036 Obtaining a proof using Prover9: 25.036/25.036 25.036/25.036 -> Prover9 Output: 25.036/25.036 ============================== Prover9 =============================== 25.036/25.036 Prover9 (64) version 2009-11A, November 2009. 25.036/25.036 Process 117114 was started by ubuntu on ubuntu, 25.036/25.036 Fri Jul 16 15:28:52 2021 25.036/25.036 The command was "./prover9 -f /tmp/prover9117105-0.in". 25.036/25.036 ============================== end of head =========================== 25.036/25.036 25.036/25.036 ============================== INPUT ================================= 25.036/25.036 25.036/25.036 % Reading from file /tmp/prover9117105-0.in 25.036/25.036 25.036/25.036 assign(max_seconds,20). 25.036/25.036 25.036/25.036 formulas(assumptions). 25.036/25.036 ->*_s0(x,x) # label(reflexivity). 25.036/25.036 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.036/25.036 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 25.036/25.036 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 25.036/25.036 ->*_s0(x2,a) -> ->_s0(f(g(x2)),b) # label(replacement). 25.036/25.036 ->*_s0(x2,c) -> ->_s0(g(x2),c) # label(replacement). 25.036/25.036 end_of_list. 25.036/25.036 25.036/25.036 formulas(goals). 25.036/25.036 (exists x2 ->*_s0(x2,c)) # label(goal). 25.036/25.036 end_of_list. 25.036/25.036 25.036/25.036 ============================== end of input ========================== 25.036/25.036 25.036/25.036 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.036/25.036 25.036/25.036 % Formulas that are not ordinary clauses: 25.036/25.036 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.036/25.036 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.036/25.036 3 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.036/25.036 4 ->*_s0(x2,a) -> ->_s0(f(g(x2)),b) # label(replacement) # label(non_clause). [assumption]. 25.036/25.036 5 ->*_s0(x2,c) -> ->_s0(g(x2),c) # label(replacement) # label(non_clause). [assumption]. 25.036/25.036 6 (exists x2 ->*_s0(x2,c)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.036/25.036 25.036/25.036 ============================== end of process non-clausal formulas === 25.036/25.036 25.036/25.036 ============================== PROCESS INITIAL CLAUSES =============== 25.036/25.036 25.036/25.036 % Clauses before input processing: 25.036/25.036 25.036/25.036 formulas(usable). 25.036/25.036 end_of_list. 25.036/25.036 25.036/25.036 formulas(sos). 25.036/25.036 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.036/25.036 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.036/25.036 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.036/25.036 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.036/25.036 -->*_s0(x,a) | ->_s0(f(g(x)),b) # label(replacement). [clausify(4)]. 25.036/25.036 -->*_s0(x,c) | ->_s0(g(x),c) # label(replacement). [clausify(5)]. 25.036/25.036 -->*_s0(x,c) # label(goal). [deny(6)]. 25.036/25.036 end_of_list. 25.036/25.036 25.036/25.036 formulas(demodulators). 25.036/25.036 end_of_list. 25.036/25.036 25.036/25.036 ============================== PREDICATE ELIMINATION ================= 25.036/25.036 25.036/25.036 No predicates eliminated. 25.036/25.036 25.036/25.036 ============================== end predicate elimination ============= 25.036/25.036 25.036/25.036 Auto_denials: 25.036/25.036 % copying label goal to answer in negative clause 25.036/25.036 25.036/25.036 Term ordering decisions: 25.036/25.036 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.036/25.036 Function symbol precedence: function_order([ c, a, b, g, f ]). 25.036/25.036 After inverse_order: (no changes). 25.036/25.036 Unfolding symbols: (none). 25.036/25.036 25.036/25.036 Auto_inference settings: 25.036/25.036 % set(neg_binary_resolution). % (HNE depth_diff=-5) 25.036/25.036 % clear(ordered_res). % (HNE depth_diff=-5) 25.036/25.036 % set(ur_resolution). % (HNE depth_diff=-5) 25.036/25.036 % set(ur_resolution) -> set(pos_ur_resolution). 25.036/25.036 % set(ur_resolution) -> set(neg_ur_resolution). 25.036/25.036 25.036/25.036 Auto_process settings: (no changes). 25.036/25.036 25.036/25.036 kept: 7 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.036/25.036 kept: 8 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.036/25.036 kept: 9 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.036/25.036 kept: 10 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.036/25.036 kept: 11 -->*_s0(x,a) | ->_s0(f(g(x)),b) # label(replacement). [clausify(4)]. 25.036/25.036 kept: 12 -->*_s0(x,c) | ->_s0(g(x),c) # label(replacement). [clausify(5)]. 25.036/25.036 kept: 13 -->*_s0(x,c) # label(goal) # answer(goal). [deny(6)]. 25.036/25.036 25.036/25.036 ============================== PROOF ================================= 25.036/25.036 25.036/25.036 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.036/25.036 % Length of proof is 4. 25.036/25.036 % Level of proof is 2. 25.036/25.036 % Maximum clause weight is 3.000. 25.036/25.036 % Given clauses 0. 25.036/25.036 25.036/25.036 6 (exists x2 ->*_s0(x2,c)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.036/25.036 7 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.036/25.036 13 -->*_s0(x,c) # label(goal) # answer(goal). [deny(6)]. 25.036/25.036 14 $F # answer(goal). [resolve(13,a,7,a)]. 25.036/25.036 25.036/25.036 ============================== end of proof ========================== 25.036/25.036 25.036/25.036 ============================== STATISTICS ============================ 25.036/25.036 25.036/25.036 Given=0. Generated=7. Kept=7. proofs=1. 25.036/25.036 Usable=0. Sos=0. Demods=0. Limbo=6, Disabled=7. Hints=0. 25.036/25.036 Kept_by_rule=0, Deleted_by_rule=0. 25.036/25.036 Forward_subsumed=0. Back_subsumed=0. 25.036/25.036 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.036/25.036 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.036/25.036 Demod_attempts=0. Demod_rewrites=0. 25.036/25.036 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.036/25.036 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 25.036/25.036 Megabytes=0.04. 25.036/25.036 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.036/25.036 25.036/25.036 ============================== end of statistics ===================== 25.036/25.036 25.036/25.036 ============================== end of search ========================= 25.036/25.036 25.036/25.036 THEOREM PROVED 25.036/25.036 25.036/25.036 Exiting with 1 proof. 25.036/25.036 25.036/25.036 Process 117114 exit (max_proofs) Fri Jul 16 15:28:52 2021 25.036/25.036 25.036/25.036 25.036/25.036 The problem is feasible. 25.036/25.036 25.036/25.036 25.036/25.036 Problem 1: 25.036/25.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.036/25.036 Confluence Problem: 25.036/25.036 (VAR x:S) 25.036/25.036 (STRATEGY CONTEXTSENSITIVE 25.036/25.036 (f 1) 25.036/25.036 (g 1) 25.036/25.036 (a) 25.036/25.036 (b) 25.036/25.036 (c) 25.036/25.036 (fSNonEmpty) 25.036/25.036 ) 25.036/25.036 (RULES 25.036/25.036 f(g(x:S)) -> b | x:S ->* a 25.036/25.036 g(x:S) -> c | x:S ->* c 25.036/25.036 ) 25.036/25.036 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.036/25.036 25.036/25.036 Critical Pairs Processor: 25.036/25.036 -> Rules: 25.036/25.036 f(g(x:S)) -> b | x:S ->* a 25.036/25.036 g(x:S) -> c | x:S ->* c 25.036/25.036 -> Vars: 25.036/25.036 "x", "x" 25.036/25.036 -> FVars: 25.036/25.036 "x2", "x3" 25.036/25.036 -> PVars: 25.036/25.036 "x": ["x2", "x3"] 25.036/25.036 25.036/25.036 -> Rlps: 25.036/25.036 crule: f(g(x2:S)) -> b | x2:S ->* a, id: 1, possubterms: f(g(x2:S))-> [], g(x2:S)-> [1] 25.036/25.036 crule: g(x3:S) -> c | x3:S ->* c, id: 2, possubterms: g(x3:S)-> [] 25.036/25.036 25.036/25.036 -> Unifications: 25.036/25.036 R1 unifies with R2 at p: [1], l: f(g(x2:S)), lp: g(x2:S), conds: {x2:S ->* a, x:S ->* c}, sig: {x2:S -> x:S}, l': g(x:S), r: b, r': c 25.036/25.036 25.036/25.036 -> Critical pairs info: 25.036/25.036 | x:S ->* a, x:S ->* c => Not trivial, Not overlay, N1 25.036/25.036 25.036/25.036 -> Problem conclusions: 25.036/25.036 Left linear, Right linear, Linear 25.036/25.036 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 25.036/25.036 CTRS Type: 1 25.036/25.036 Deterministic, Strongly deterministic 25.036/25.036 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.036/25.036 Maybe right-stable CTRS, Not overlay CTRS 25.036/25.036 Maybe normal CTRS, Maybe almost normal CTRS 25.036/25.036 Maybe terminating CTRS, Maybe joinable CCPs 25.036/25.036 Maybe level confluent 25.036/25.036 Maybe confluent 25.036/25.036 25.036/25.036 Problem 1: 25.036/25.036 Clean Infeasible CCPs Processor: 25.036/25.036 Num of CPIs: 1 25.036/25.036 Timeout: 30 25.036/25.036 Timeout for each infeasibility problem: 30 s 25.036/25.036 | x:S ->* a, x:S ->* c => Not trivial, Not overlay, N1 25.036/25.036 (PROBLEM INFEASIBILITY) 25.036/25.036 (VAR x) 25.036/25.036 (RULES 25.036/25.036 f(g(x)) -> b | x ->* a 25.036/25.036 g(x) -> c | x ->* c) 25.036/25.036 (VAR x2) 25.036/25.036 (CONDITION x2 ->* a, x2 ->* c) 25.036/25.036 25.036/25.036 Proof: 25.036/25.036 YES 25.036/25.036 25.036/25.036 Problem 1: 25.036/25.036 25.036/25.036 Infeasibility Problem: 25.036/25.036 [(VAR vNonEmpty:S x:S x2:S) 25.036/25.036 (STRATEGY CONTEXTSENSITIVE 25.036/25.036 (f 1) 25.036/25.036 (g 1) 25.036/25.036 (a) 25.036/25.036 (b) 25.036/25.036 (c) 25.036/25.036 (fSNonEmpty) 25.036/25.036 ) 25.036/25.036 (RULES 25.036/25.036 f(g(x:S)) -> b | x:S ->* a 25.036/25.036 g(x:S) -> c | x:S ->* c 25.036/25.036 )] 25.036/25.036 25.036/25.036 Infeasibility Conditions: 25.036/25.036 x2:S ->* a, x2:S ->* c 25.036/25.036 25.036/25.036 Problem 1: 25.036/25.036 25.036/25.036 Obtaining a model using Mace4: 25.036/25.036 25.036/25.036 -> Usable Rules: 25.036/25.036 f(g(x:S)) -> b | x:S ->* a 25.036/25.036 g(x:S) -> c | x:S ->* c 25.036/25.036 25.036/25.036 -> Mace4 Output: 25.036/25.036 ============================== Mace4 ================================= 25.036/25.036 Mace4 (64) version 2009-11A, November 2009. 25.036/25.036 Process 117148 was started by ubuntu on ubuntu, 25.036/25.036 Fri Jul 16 15:28:52 2021 25.036/25.036 The command was "./mace4 -c -f /tmp/mace4117131-2.in". 25.036/25.036 ============================== end of head =========================== 25.036/25.036 25.036/25.036 ============================== INPUT ================================= 25.036/25.036 25.036/25.036 % Reading from file /tmp/mace4117131-2.in 25.036/25.036 25.036/25.036 assign(max_seconds,10). 25.036/25.036 25.036/25.036 formulas(assumptions). 25.036/25.036 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 25.036/25.036 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence). 25.036/25.036 ->*(x1,a) -> ->(f(g(x1)),b) # label(replacement). 25.036/25.036 ->*(x1,c) -> ->(g(x1),c) # label(replacement). 25.036/25.036 ->*(x,x) # label(reflexivity). 25.036/25.036 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 25.036/25.036 end_of_list. 25.036/25.036 25.036/25.036 formulas(goals). 25.036/25.036 (exists x2 (->*(x2,a) & ->*(x2,c))) # label(goal). 25.036/25.036 end_of_list. 25.036/25.036 25.036/25.036 ============================== end of input ========================== 25.036/25.036 25.036/25.036 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.036/25.036 25.036/25.036 % Formulas that are not ordinary clauses: 25.036/25.036 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.036/25.036 2 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.036/25.036 3 ->*(x1,a) -> ->(f(g(x1)),b) # label(replacement) # label(non_clause). [assumption]. 25.036/25.036 4 ->*(x1,c) -> ->(g(x1),c) # label(replacement) # label(non_clause). [assumption]. 25.036/25.036 5 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.036/25.036 6 (exists x2 (->*(x2,a) & ->*(x2,c))) # label(goal) # label(non_clause) # label(goal). [goal]. 25.036/25.036 25.036/25.036 ============================== end of process non-clausal formulas === 25.036/25.036 25.036/25.036 ============================== CLAUSES FOR SEARCH ==================== 25.036/25.036 25.036/25.036 formulas(mace4_clauses). 25.036/25.036 -->(x,y) | ->(f(x),f(y)) # label(congruence). 25.036/25.036 -->(x,y) | ->(g(x),g(y)) # label(congruence). 25.036/25.036 -->*(x,a) | ->(f(g(x)),b) # label(replacement). 25.036/25.036 -->*(x,c) | ->(g(x),c) # label(replacement). 25.036/25.036 ->*(x,x) # label(reflexivity). 25.036/25.036 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 25.036/25.036 -->*(x,a) | -->*(x,c) # label(goal). 25.036/25.036 end_of_list. 25.036/25.036 25.036/25.036 ============================== end of clauses for search ============= 25.036/25.036 25.036/25.036 % There are no natural numbers in the input. 25.036/25.036 25.036/25.036 ============================== DOMAIN SIZE 2 ========================= 25.036/25.036 25.036/25.036 ============================== MODEL ================================= 25.036/25.036 25.036/25.036 interpretation( 2, [number=1, seconds=0], [ 25.036/25.036 25.036/25.036 function(a, [ 0 ]), 25.036/25.036 25.036/25.036 function(b, [ 0 ]), 25.036/25.036 25.036/25.036 function(c, [ 1 ]), 25.036/25.036 25.036/25.036 function(f(_), [ 0, 0 ]), 25.036/25.036 25.036/25.036 function(g(_), [ 0, 1 ]), 25.036/25.036 25.036/25.036 relation(->*(_,_), [ 25.036/25.036 1, 0, 25.036/25.036 0, 1 ]), 25.036/25.036 25.036/25.036 relation(->(_,_), [ 25.036/25.036 1, 0, 25.036/25.036 0, 1 ]) 25.036/25.036 ]). 25.036/25.036 25.036/25.036 ============================== end of model ========================== 25.036/25.036 25.036/25.036 ============================== STATISTICS ============================ 25.036/25.036 25.036/25.036 For domain size 2. 25.036/25.036 25.036/25.036 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 25.036/25.036 Ground clauses: seen=24, kept=20. 25.036/25.036 Selections=5, assignments=5, propagations=10, current_models=1. 25.036/25.036 Rewrite_terms=34, rewrite_bools=30, indexes=7. 25.036/25.036 Rules_from_neg_clauses=2, cross_offs=2. 25.036/25.036 25.036/25.036 ============================== end of statistics ===================== 25.036/25.036 25.036/25.036 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.036/25.036 25.036/25.036 Exiting with 1 model. 25.036/25.036 25.036/25.036 Process 117148 exit (max_models) Fri Jul 16 15:28:52 2021 25.036/25.036 The process finished Fri Jul 16 15:28:52 2021 25.036/25.036 25.036/25.036 25.036/25.036 Mace4 cooked interpretation: 25.036/25.036 25.036/25.036 25.036/25.036 25.036/25.036 The problem is infeasible. 25.036/25.036 25.036/25.036 25.036/25.036 25.036/25.036 25.036/25.036 -> Problem conclusions: 25.036/25.036 Left linear, Right linear, Linear 25.036/25.036 Weakly orthogonal, Almost orthogonal, Orthogonal 25.036/25.036 CTRS Type: 1 25.036/25.036 Deterministic, Strongly deterministic 25.036/25.036 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.036/25.036 Maybe right-stable CTRS, Overlay CTRS 25.036/25.036 Maybe normal CTRS, Maybe almost normal CTRS 25.036/25.036 Maybe terminating CTRS, Joinable CCPs 25.036/25.036 Maybe level confluent 25.036/25.036 Maybe confluent 25.036/25.036 25.036/25.036 Resulting CCPs: 25.036/25.036 No CCPs left 25.036/25.036 25.036/25.036 Problem 1: 25.036/25.036 Underlying TRS Termination Processor: 25.036/25.036 25.036/25.036 Resulting Underlying TRS: 25.036/25.036 (VAR x:S) 25.036/25.036 (STRATEGY CONTEXTSENSITIVE 25.036/25.036 (f 1) 25.036/25.036 (g 1) 25.036/25.036 (b) 25.036/25.036 (c) 25.036/25.036 ) 25.036/25.036 (RULES 25.036/25.036 f(g(x:S)) -> b 25.036/25.036 g(x:S) -> c 25.036/25.036 ) 25.036/25.036 Underlying TRS terminating? 25.036/25.036 YES 25.036/25.036 25.036/25.036 Problem 1: 25.036/25.036 25.036/25.036 (VAR vu95NonEmpty:S xu58S:S) 25.036/25.036 (RULES 25.036/25.036 f(g(xu58S:S)) -> b 25.036/25.036 g(xu58S:S) -> c 25.036/25.036 ) 25.036/25.036 25.036/25.036 Problem 1: 25.036/25.036 25.036/25.036 Dependency Pairs Processor: 25.036/25.036 -> Pairs: 25.036/25.036 Empty 25.036/25.036 -> Rules: 25.036/25.036 f(g(xu58S:S)) -> b 25.036/25.036 g(xu58S:S) -> c 25.036/25.036 25.036/25.036 Problem 1: 25.036/25.036 25.036/25.036 SCC Processor: 25.036/25.036 -> Pairs: 25.036/25.036 Empty 25.036/25.036 -> Rules: 25.036/25.036 f(g(xu58S:S)) -> b 25.036/25.036 g(xu58S:S) -> c 25.036/25.036 ->Strongly Connected Components: 25.036/25.036 There is no strongly connected component 25.036/25.036 25.036/25.036 The problem is finite. 25.036/25.036 25.036/25.036 25.036/25.036 -> Problem conclusions: 25.036/25.036 Left linear, Right linear, Linear 25.036/25.036 Weakly orthogonal, Almost orthogonal, Orthogonal 25.036/25.036 CTRS Type: 1 25.036/25.036 Deterministic, Strongly deterministic 25.036/25.036 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.036/25.036 Maybe right-stable CTRS, Overlay CTRS 25.036/25.036 Maybe normal CTRS, Maybe almost normal CTRS 25.036/25.036 Terminating CTRS, Joinable CCPs 25.036/25.036 Maybe level confluent 25.036/25.036 Confluent 25.036/25.036 25.036/25.036 The problem is joinable. 25.036/25.036 46.47user 1.58system 0:25.36elapsed 189%CPU (0avgtext+0avgdata 130916maxresident)k 25.036/25.036 8inputs+0outputs (0major+95906minor)pagefaults 0swaps