25.033/25.033 YES 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 Confluence Problem: 25.033/25.033 (VAR vNonEmpty:S x:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (f 1) 25.033/25.033 (g 1) 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (c) 25.033/25.033 (fSNonEmpty) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 f(g(x:S)) -> b | x:S ->* a 25.033/25.033 g(x:S) -> c | x:S ->* c 25.033/25.033 ) 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Inlining of Conditions Processor [STERN17]: 25.033/25.033 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 Confluence Problem: 25.033/25.033 (VAR vNonEmpty:S x:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (f 1) 25.033/25.033 (g 1) 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (c) 25.033/25.033 (fSNonEmpty) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 f(g(x:S)) -> b | x:S ->* a 25.033/25.033 g(x:S) -> c | x:S ->* c 25.033/25.033 ) 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Clean CTRS Processor: 25.033/25.033 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 Confluence Problem: 25.033/25.033 (VAR x:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (f 1) 25.033/25.033 (g 1) 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (c) 25.033/25.033 (fSNonEmpty) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 f(g(x:S)) -> b | x:S ->* a 25.033/25.033 g(x:S) -> c | x:S ->* c 25.033/25.033 ) 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 25.033/25.033 CRule InfChecker Info: 25.033/25.033 f(g(x:S)) -> b | x:S ->* a 25.033/25.033 Rule remains 25.033/25.033 Proof: 25.033/25.033 NO 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Infeasibility Problem: 25.033/25.033 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (f 1) 25.033/25.033 (g 1) 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (c) 25.033/25.033 (fSNonEmpty) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 f(g(x:S:S)) -> b | x:S:S ->* a 25.033/25.033 g(x:S:S) -> c | x:S:S ->* c 25.033/25.033 )] 25.033/25.033 25.033/25.033 Infeasibility Conditions: 25.033/25.033 x:S:S ->* a 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Obtaining a proof using Prover9: 25.033/25.033 25.033/25.033 -> Prover9 Output: 25.033/25.033 ============================== Prover9 =============================== 25.033/25.033 Prover9 (64) version 2009-11A, November 2009. 25.033/25.033 Process 56789 was started by ubuntu on ubuntu, 25.033/25.033 Wed Jul 14 11:21:06 2021 25.033/25.033 The command was "./prover9 -f /tmp/prover956780-0.in". 25.033/25.033 ============================== end of head =========================== 25.033/25.033 25.033/25.033 ============================== INPUT ================================= 25.033/25.033 25.033/25.033 % Reading from file /tmp/prover956780-0.in 25.033/25.033 25.033/25.033 assign(max_seconds,20). 25.033/25.033 25.033/25.033 formulas(assumptions). 25.033/25.033 ->*_s0(x,x) # label(reflexivity). 25.033/25.033 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.033/25.033 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 25.033/25.033 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 25.033/25.033 ->*_s0(x2,a) -> ->_s0(f(g(x2)),b) # label(replacement). 25.033/25.033 ->*_s0(x2,c) -> ->_s0(g(x2),c) # label(replacement). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(goals). 25.033/25.033 (exists x2 ->*_s0(x2,a)) # label(goal). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== end of input ========================== 25.033/25.033 25.033/25.033 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.033/25.033 25.033/25.033 % Formulas that are not ordinary clauses: 25.033/25.033 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.033/25.033 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 3 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 4 ->*_s0(x2,a) -> ->_s0(f(g(x2)),b) # label(replacement) # label(non_clause). [assumption]. 25.033/25.033 5 ->*_s0(x2,c) -> ->_s0(g(x2),c) # label(replacement) # label(non_clause). [assumption]. 25.033/25.033 6 (exists x2 ->*_s0(x2,a)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.033/25.033 25.033/25.033 ============================== end of process non-clausal formulas === 25.033/25.033 25.033/25.033 ============================== PROCESS INITIAL CLAUSES =============== 25.033/25.033 25.033/25.033 % Clauses before input processing: 25.033/25.033 25.033/25.033 formulas(usable). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(sos). 25.033/25.033 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.033/25.033 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.033/25.033 -->*_s0(x,a) | ->_s0(f(g(x)),b) # label(replacement). [clausify(4)]. 25.033/25.033 -->*_s0(x,c) | ->_s0(g(x),c) # label(replacement). [clausify(5)]. 25.033/25.033 -->*_s0(x,a) # label(goal). [deny(6)]. 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(demodulators). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== PREDICATE ELIMINATION ================= 25.033/25.033 25.033/25.033 No predicates eliminated. 25.033/25.033 25.033/25.033 ============================== end predicate elimination ============= 25.033/25.033 25.033/25.033 Auto_denials: 25.033/25.033 % copying label goal to answer in negative clause 25.033/25.033 25.033/25.033 Term ordering decisions: 25.033/25.033 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.033/25.033 Function symbol precedence: function_order([ c, a, b, g, f ]). 25.033/25.033 After inverse_order: (no changes). 25.033/25.033 Unfolding symbols: (none). 25.033/25.033 25.033/25.033 Auto_inference settings: 25.033/25.033 % set(neg_binary_resolution). % (HNE depth_diff=-5) 25.033/25.033 % clear(ordered_res). % (HNE depth_diff=-5) 25.033/25.033 % set(ur_resolution). % (HNE depth_diff=-5) 25.033/25.033 % set(ur_resolution) -> set(pos_ur_resolution). 25.033/25.033 % set(ur_resolution) -> set(neg_ur_resolution). 25.033/25.033 25.033/25.033 Auto_process settings: (no changes). 25.033/25.033 25.033/25.033 kept: 7 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 kept: 8 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 kept: 9 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.033/25.033 kept: 10 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.033/25.033 kept: 11 -->*_s0(x,a) | ->_s0(f(g(x)),b) # label(replacement). [clausify(4)]. 25.033/25.033 kept: 12 -->*_s0(x,c) | ->_s0(g(x),c) # label(replacement). [clausify(5)]. 25.033/25.033 kept: 13 -->*_s0(x,a) # label(goal) # answer(goal). [deny(6)]. 25.033/25.033 25.033/25.033 ============================== PROOF ================================= 25.033/25.033 25.033/25.033 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.033/25.033 % Length of proof is 4. 25.033/25.033 % Level of proof is 2. 25.033/25.033 % Maximum clause weight is 3.000. 25.033/25.033 % Given clauses 0. 25.033/25.033 25.033/25.033 6 (exists x2 ->*_s0(x2,a)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.033/25.033 7 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 13 -->*_s0(x,a) # label(goal) # answer(goal). [deny(6)]. 25.033/25.033 14 $F # answer(goal). [resolve(13,a,7,a)]. 25.033/25.033 25.033/25.033 ============================== end of proof ========================== 25.033/25.033 25.033/25.033 ============================== STATISTICS ============================ 25.033/25.033 25.033/25.033 Given=0. Generated=7. Kept=7. proofs=1. 25.033/25.033 Usable=0. Sos=0. Demods=0. Limbo=6, Disabled=7. Hints=0. 25.033/25.033 Kept_by_rule=0, Deleted_by_rule=0. 25.033/25.033 Forward_subsumed=0. Back_subsumed=0. 25.033/25.033 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.033/25.033 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.033/25.033 Demod_attempts=0. Demod_rewrites=0. 25.033/25.033 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.033/25.033 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 25.033/25.033 Megabytes=0.04. 25.033/25.033 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.033/25.033 25.033/25.033 ============================== end of statistics ===================== 25.033/25.033 25.033/25.033 ============================== end of search ========================= 25.033/25.033 25.033/25.033 THEOREM PROVED 25.033/25.033 25.033/25.033 Exiting with 1 proof. 25.033/25.033 25.033/25.033 Process 56789 exit (max_proofs) Wed Jul 14 11:21:06 2021 25.033/25.033 25.033/25.033 25.033/25.033 The problem is feasible. 25.033/25.033 25.033/25.033 25.033/25.033 CRule InfChecker Info: 25.033/25.033 g(x:S) -> c | x:S ->* c 25.033/25.033 Rule remains 25.033/25.033 Proof: 25.033/25.033 NO 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Infeasibility Problem: 25.033/25.033 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (f 1) 25.033/25.033 (g 1) 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (c) 25.033/25.033 (fSNonEmpty) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 f(g(x:S:S)) -> b | x:S:S ->* a 25.033/25.033 g(x:S:S) -> c | x:S:S ->* c 25.033/25.033 )] 25.033/25.033 25.033/25.033 Infeasibility Conditions: 25.033/25.033 x:S:S ->* c 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Obtaining a proof using Prover9: 25.033/25.033 25.033/25.033 -> Prover9 Output: 25.033/25.033 ============================== Prover9 =============================== 25.033/25.033 Prover9 (64) version 2009-11A, November 2009. 25.033/25.033 Process 56811 was started by ubuntu on ubuntu, 25.033/25.033 Wed Jul 14 11:21:06 2021 25.033/25.033 The command was "./prover9 -f /tmp/prover956802-0.in". 25.033/25.033 ============================== end of head =========================== 25.033/25.033 25.033/25.033 ============================== INPUT ================================= 25.033/25.033 25.033/25.033 % Reading from file /tmp/prover956802-0.in 25.033/25.033 25.033/25.033 assign(max_seconds,20). 25.033/25.033 25.033/25.033 formulas(assumptions). 25.033/25.033 ->*_s0(x,x) # label(reflexivity). 25.033/25.033 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.033/25.033 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 25.033/25.033 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 25.033/25.033 ->*_s0(x2,a) -> ->_s0(f(g(x2)),b) # label(replacement). 25.033/25.033 ->*_s0(x2,c) -> ->_s0(g(x2),c) # label(replacement). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(goals). 25.033/25.033 (exists x2 ->*_s0(x2,c)) # label(goal). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== end of input ========================== 25.033/25.033 25.033/25.033 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.033/25.033 25.033/25.033 % Formulas that are not ordinary clauses: 25.033/25.033 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.033/25.033 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 3 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 4 ->*_s0(x2,a) -> ->_s0(f(g(x2)),b) # label(replacement) # label(non_clause). [assumption]. 25.033/25.033 5 ->*_s0(x2,c) -> ->_s0(g(x2),c) # label(replacement) # label(non_clause). [assumption]. 25.033/25.033 6 (exists x2 ->*_s0(x2,c)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.033/25.033 25.033/25.033 ============================== end of process non-clausal formulas === 25.033/25.033 25.033/25.033 ============================== PROCESS INITIAL CLAUSES =============== 25.033/25.033 25.033/25.033 % Clauses before input processing: 25.033/25.033 25.033/25.033 formulas(usable). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(sos). 25.033/25.033 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.033/25.033 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.033/25.033 -->*_s0(x,a) | ->_s0(f(g(x)),b) # label(replacement). [clausify(4)]. 25.033/25.033 -->*_s0(x,c) | ->_s0(g(x),c) # label(replacement). [clausify(5)]. 25.033/25.033 -->*_s0(x,c) # label(goal). [deny(6)]. 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(demodulators). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== PREDICATE ELIMINATION ================= 25.033/25.033 25.033/25.033 No predicates eliminated. 25.033/25.033 25.033/25.033 ============================== end predicate elimination ============= 25.033/25.033 25.033/25.033 Auto_denials: 25.033/25.033 % copying label goal to answer in negative clause 25.033/25.033 25.033/25.033 Term ordering decisions: 25.033/25.033 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.033/25.033 Function symbol precedence: function_order([ c, a, b, g, f ]). 25.033/25.033 After inverse_order: (no changes). 25.033/25.033 Unfolding symbols: (none). 25.033/25.033 25.033/25.033 Auto_inference settings: 25.033/25.033 % set(neg_binary_resolution). % (HNE depth_diff=-5) 25.033/25.033 % clear(ordered_res). % (HNE depth_diff=-5) 25.033/25.033 % set(ur_resolution). % (HNE depth_diff=-5) 25.033/25.033 % set(ur_resolution) -> set(pos_ur_resolution). 25.033/25.033 % set(ur_resolution) -> set(neg_ur_resolution). 25.033/25.033 25.033/25.033 Auto_process settings: (no changes). 25.033/25.033 25.033/25.033 kept: 7 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 kept: 8 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.033/25.033 kept: 9 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.033/25.033 kept: 10 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.033/25.033 kept: 11 -->*_s0(x,a) | ->_s0(f(g(x)),b) # label(replacement). [clausify(4)]. 25.033/25.033 kept: 12 -->*_s0(x,c) | ->_s0(g(x),c) # label(replacement). [clausify(5)]. 25.033/25.033 kept: 13 -->*_s0(x,c) # label(goal) # answer(goal). [deny(6)]. 25.033/25.033 25.033/25.033 ============================== PROOF ================================= 25.033/25.033 25.033/25.033 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.033/25.033 % Length of proof is 4. 25.033/25.033 % Level of proof is 2. 25.033/25.033 % Maximum clause weight is 3.000. 25.033/25.033 % Given clauses 0. 25.033/25.033 25.033/25.033 6 (exists x2 ->*_s0(x2,c)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.033/25.033 7 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.033/25.033 13 -->*_s0(x,c) # label(goal) # answer(goal). [deny(6)]. 25.033/25.033 14 $F # answer(goal). [resolve(13,a,7,a)]. 25.033/25.033 25.033/25.033 ============================== end of proof ========================== 25.033/25.033 25.033/25.033 ============================== STATISTICS ============================ 25.033/25.033 25.033/25.033 Given=0. Generated=7. Kept=7. proofs=1. 25.033/25.033 Usable=0. Sos=0. Demods=0. Limbo=6, Disabled=7. Hints=0. 25.033/25.033 Kept_by_rule=0, Deleted_by_rule=0. 25.033/25.033 Forward_subsumed=0. Back_subsumed=0. 25.033/25.033 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.033/25.033 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.033/25.033 Demod_attempts=0. Demod_rewrites=0. 25.033/25.033 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.033/25.033 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 25.033/25.033 Megabytes=0.04. 25.033/25.033 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.033/25.033 25.033/25.033 ============================== end of statistics ===================== 25.033/25.033 25.033/25.033 ============================== end of search ========================= 25.033/25.033 25.033/25.033 THEOREM PROVED 25.033/25.033 25.033/25.033 Exiting with 1 proof. 25.033/25.033 25.033/25.033 Process 56811 exit (max_proofs) Wed Jul 14 11:21:06 2021 25.033/25.033 25.033/25.033 25.033/25.033 The problem is feasible. 25.033/25.033 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 Confluence Problem: 25.033/25.033 (VAR x:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (f 1) 25.033/25.033 (g 1) 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (c) 25.033/25.033 (fSNonEmpty) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 f(g(x:S)) -> b | x:S ->* a 25.033/25.033 g(x:S) -> c | x:S ->* c 25.033/25.033 ) 25.033/25.033 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.033/25.033 25.033/25.033 Critical Pairs Processor: 25.033/25.033 -> Rules: 25.033/25.033 f(g(x:S)) -> b | x:S ->* a 25.033/25.033 g(x:S) -> c | x:S ->* c 25.033/25.033 -> Vars: 25.033/25.033 "x", "x" 25.033/25.033 -> FVars: 25.033/25.033 "x2", "x3" 25.033/25.033 -> PVars: 25.033/25.033 "x": ["x2", "x3"] 25.033/25.033 25.033/25.033 -> Rlps: 25.033/25.033 crule: f(g(x2:S)) -> b | x2:S ->* a, id: 1, possubterms: f(g(x2:S))-> [], g(x2:S)-> [1] 25.033/25.033 crule: g(x3:S) -> c | x3:S ->* c, id: 2, possubterms: g(x3:S)-> [] 25.033/25.033 25.033/25.033 -> Unifications: 25.033/25.033 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 25.033/25.033 25.033/25.033 -> Critical pairs info: 25.033/25.033 | x:S ->* a, x:S ->* c => Not trivial, Not overlay, N1 25.033/25.033 25.033/25.033 -> Problem conclusions: 25.033/25.033 Left linear, Right linear, Linear 25.033/25.033 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 25.033/25.033 CTRS Type: 1 25.033/25.033 Deterministic, Strongly deterministic 25.033/25.033 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.033/25.033 Maybe right-stable CTRS, Not overlay CTRS 25.033/25.033 Maybe normal CTRS, Maybe almost normal CTRS 25.033/25.033 Maybe terminating CTRS, Maybe joinable CCPs 25.033/25.033 Maybe level confluent 25.033/25.033 Maybe confluent 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 Clean Infeasible CCPs Processor: 25.033/25.033 Num of CPIs: 1 25.033/25.033 Timeout: 60 25.033/25.033 Timeout for each infeasibility problem: 60 s 25.033/25.033 | x:S ->* a, x:S ->* c => Not trivial, Not overlay, N1 25.033/25.033 (PROBLEM INFEASIBILITY) 25.033/25.033 (VAR x) 25.033/25.033 (RULES 25.033/25.033 f(g(x)) -> b | x ->* a 25.033/25.033 g(x) -> c | x ->* c) 25.033/25.033 (VAR x2) 25.033/25.033 (CONDITION x2 ->* a, x2 ->* c) 25.033/25.033 25.033/25.033 Proof: 25.033/25.033 YES 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Infeasibility Problem: 25.033/25.033 [(VAR vNonEmpty:S x:S x2:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (f 1) 25.033/25.033 (g 1) 25.033/25.033 (a) 25.033/25.033 (b) 25.033/25.033 (c) 25.033/25.033 (fSNonEmpty) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 f(g(x:S)) -> b | x:S ->* a 25.033/25.033 g(x:S) -> c | x:S ->* c 25.033/25.033 )] 25.033/25.033 25.033/25.033 Infeasibility Conditions: 25.033/25.033 x2:S ->* a, x2:S ->* c 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Obtaining a model using Mace4: 25.033/25.033 25.033/25.033 -> Usable Rules: 25.033/25.033 f(g(x:S)) -> b | x:S ->* a 25.033/25.033 g(x:S) -> c | x:S ->* c 25.033/25.033 25.033/25.033 -> Mace4 Output: 25.033/25.033 ============================== Mace4 ================================= 25.033/25.033 Mace4 (64) version 2009-11A, November 2009. 25.033/25.033 Process 56851 was started by ubuntu on ubuntu, 25.033/25.033 Wed Jul 14 11:21:06 2021 25.033/25.033 The command was "./mace4 -c -f /tmp/mace456828-2.in". 25.033/25.033 ============================== end of head =========================== 25.033/25.033 25.033/25.033 ============================== INPUT ================================= 25.033/25.033 25.033/25.033 % Reading from file /tmp/mace456828-2.in 25.033/25.033 25.033/25.033 assign(max_seconds,20). 25.033/25.033 25.033/25.033 formulas(assumptions). 25.033/25.033 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 25.033/25.033 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence). 25.033/25.033 ->*(x1,a) -> ->(f(g(x1)),b) # label(replacement). 25.033/25.033 ->*(x1,c) -> ->(g(x1),c) # label(replacement). 25.033/25.033 ->*(x,x) # label(reflexivity). 25.033/25.033 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 formulas(goals). 25.033/25.033 (exists x2 (->*(x2,a) & ->*(x2,c))) # label(goal). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== end of input ========================== 25.033/25.033 25.033/25.033 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.033/25.033 25.033/25.033 % Formulas that are not ordinary clauses: 25.033/25.033 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 2 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.033/25.033 3 ->*(x1,a) -> ->(f(g(x1)),b) # label(replacement) # label(non_clause). [assumption]. 25.033/25.033 4 ->*(x1,c) -> ->(g(x1),c) # label(replacement) # label(non_clause). [assumption]. 25.033/25.033 5 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.033/25.033 6 (exists x2 (->*(x2,a) & ->*(x2,c))) # label(goal) # label(non_clause) # label(goal). [goal]. 25.033/25.033 25.033/25.033 ============================== end of process non-clausal formulas === 25.033/25.033 25.033/25.033 ============================== CLAUSES FOR SEARCH ==================== 25.033/25.033 25.033/25.033 formulas(mace4_clauses). 25.033/25.033 -->(x,y) | ->(f(x),f(y)) # label(congruence). 25.033/25.033 -->(x,y) | ->(g(x),g(y)) # label(congruence). 25.033/25.033 -->*(x,a) | ->(f(g(x)),b) # label(replacement). 25.033/25.033 -->*(x,c) | ->(g(x),c) # label(replacement). 25.033/25.033 ->*(x,x) # label(reflexivity). 25.033/25.033 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 25.033/25.033 -->*(x,a) | -->*(x,c) # label(goal). 25.033/25.033 end_of_list. 25.033/25.033 25.033/25.033 ============================== end of clauses for search ============= 25.033/25.033 25.033/25.033 % There are no natural numbers in the input. 25.033/25.033 25.033/25.033 ============================== DOMAIN SIZE 2 ========================= 25.033/25.033 25.033/25.033 ============================== MODEL ================================= 25.033/25.033 25.033/25.033 interpretation( 2, [number=1, seconds=0], [ 25.033/25.033 25.033/25.033 function(a, [ 0 ]), 25.033/25.033 25.033/25.033 function(b, [ 0 ]), 25.033/25.033 25.033/25.033 function(c, [ 1 ]), 25.033/25.033 25.033/25.033 function(f(_), [ 0, 0 ]), 25.033/25.033 25.033/25.033 function(g(_), [ 0, 1 ]), 25.033/25.033 25.033/25.033 relation(->*(_,_), [ 25.033/25.033 1, 0, 25.033/25.033 0, 1 ]), 25.033/25.033 25.033/25.033 relation(->(_,_), [ 25.033/25.033 1, 0, 25.033/25.033 0, 1 ]) 25.033/25.033 ]). 25.033/25.033 25.033/25.033 ============================== end of model ========================== 25.033/25.033 25.033/25.033 ============================== STATISTICS ============================ 25.033/25.033 25.033/25.033 For domain size 2. 25.033/25.033 25.033/25.033 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 25.033/25.033 Ground clauses: seen=24, kept=20. 25.033/25.033 Selections=5, assignments=5, propagations=10, current_models=1. 25.033/25.033 Rewrite_terms=34, rewrite_bools=30, indexes=7. 25.033/25.033 Rules_from_neg_clauses=2, cross_offs=2. 25.033/25.033 25.033/25.033 ============================== end of statistics ===================== 25.033/25.033 25.033/25.033 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.033/25.033 25.033/25.033 Exiting with 1 model. 25.033/25.033 25.033/25.033 Process 56851 exit (max_models) Wed Jul 14 11:21:06 2021 25.033/25.033 The process finished Wed Jul 14 11:21:06 2021 25.033/25.033 25.033/25.033 25.033/25.033 Mace4 cooked interpretation: 25.033/25.033 25.033/25.033 25.033/25.033 25.033/25.033 The problem is infeasible. 25.033/25.033 25.033/25.033 25.033/25.033 25.033/25.033 25.033/25.033 -> Problem conclusions: 25.033/25.033 Left linear, Right linear, Linear 25.033/25.033 Weakly orthogonal, Almost orthogonal, Orthogonal 25.033/25.033 CTRS Type: 1 25.033/25.033 Deterministic, Strongly deterministic 25.033/25.033 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.033/25.033 Maybe right-stable CTRS, Overlay CTRS 25.033/25.033 Maybe normal CTRS, Maybe almost normal CTRS 25.033/25.033 Maybe terminating CTRS, Joinable CCPs 25.033/25.033 Maybe level confluent 25.033/25.033 Maybe confluent 25.033/25.033 25.033/25.033 Resulting CCPs: 25.033/25.033 No CCPs left 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 Underlying TRS Termination Processor: 25.033/25.033 25.033/25.033 Resulting Underlying TRS: 25.033/25.033 (VAR x:S) 25.033/25.033 (STRATEGY CONTEXTSENSITIVE 25.033/25.033 (f 1) 25.033/25.033 (g 1) 25.033/25.033 (b) 25.033/25.033 (c) 25.033/25.033 ) 25.033/25.033 (RULES 25.033/25.033 f(g(x:S)) -> b 25.033/25.033 g(x:S) -> c 25.033/25.033 ) 25.033/25.033 Underlying TRS terminating? 25.033/25.033 YES 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 (VAR vu95NonEmpty:S xu58S:S) 25.033/25.033 (RULES 25.033/25.033 f(g(xu58S:S)) -> b 25.033/25.033 g(xu58S:S) -> c 25.033/25.033 ) 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 Dependency Pairs Processor: 25.033/25.033 -> Pairs: 25.033/25.033 Empty 25.033/25.033 -> Rules: 25.033/25.033 f(g(xu58S:S)) -> b 25.033/25.033 g(xu58S:S) -> c 25.033/25.033 25.033/25.033 Problem 1: 25.033/25.033 25.033/25.033 SCC Processor: 25.033/25.033 -> Pairs: 25.033/25.033 Empty 25.033/25.033 -> Rules: 25.033/25.033 f(g(xu58S:S)) -> b 25.033/25.033 g(xu58S:S) -> c 25.033/25.033 ->Strongly Connected Components: 25.033/25.033 There is no strongly connected component 25.033/25.033 25.033/25.033 The problem is finite. 25.033/25.033 25.033/25.033 25.033/25.033 -> Problem conclusions: 25.033/25.033 Left linear, Right linear, Linear 25.033/25.033 Weakly orthogonal, Almost orthogonal, Orthogonal 25.033/25.033 CTRS Type: 1 25.033/25.033 Deterministic, Strongly deterministic 25.033/25.033 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.033/25.033 Maybe right-stable CTRS, Overlay CTRS 25.033/25.033 Maybe normal CTRS, Maybe almost normal CTRS 25.033/25.033 Terminating CTRS, Joinable CCPs 25.033/25.033 Maybe level confluent 25.033/25.033 Confluent 25.033/25.033 25.033/25.033 The problem is joinable. 25.033/25.033 47.76user 1.42system 0:25.33elapsed 194%CPU (0avgtext+0avgdata 72556maxresident)k 25.033/25.033 8inputs+0outputs (0major+80049minor)pagefaults 0swaps