20.091/20.091 YES 20.091/20.091 20.091/20.091 Problem 1: 20.091/20.091 20.091/20.091 20.091/20.091 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.091/20.091 Confluence Problem: 20.091/20.091 (VAR vNonEmpty:S x:S) 20.091/20.091 (STRATEGY CONTEXTSENSITIVE 20.091/20.091 (f 1) 20.091/20.091 (a) 20.091/20.091 (b) 20.091/20.091 (fSNonEmpty) 20.091/20.091 ) 20.091/20.091 (RULES 20.091/20.091 f(x:S) -> a | a ->* x:S 20.091/20.091 f(x:S) -> b | b ->* x:S 20.091/20.091 ) 20.091/20.091 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.091/20.091 20.091/20.091 20.091/20.091 Problem 1: 20.091/20.091 20.091/20.091 Inlining of Conditions Processor [STERN17]: 20.091/20.091 20.091/20.091 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.091/20.091 Confluence Problem: 20.091/20.091 (VAR vNonEmpty:S x:S) 20.091/20.091 (STRATEGY CONTEXTSENSITIVE 20.091/20.091 (f 1) 20.091/20.091 (a) 20.091/20.091 (b) 20.091/20.091 (fSNonEmpty) 20.091/20.091 ) 20.091/20.091 (RULES 20.091/20.091 f(x:S) -> a | a ->* x:S 20.091/20.091 f(x:S) -> b | b ->* x:S 20.091/20.091 ) 20.091/20.091 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.091/20.091 20.091/20.091 20.091/20.091 Problem 1: 20.091/20.091 20.091/20.091 Clean CTRS Processor: 20.091/20.091 20.091/20.091 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.091/20.091 Confluence Problem: 20.091/20.091 (VAR x:S) 20.091/20.091 (STRATEGY CONTEXTSENSITIVE 20.091/20.091 (f 1) 20.091/20.091 (a) 20.091/20.091 (b) 20.091/20.091 (fSNonEmpty) 20.091/20.091 ) 20.091/20.091 (RULES 20.091/20.091 f(x:S) -> a | a ->* x:S 20.091/20.091 f(x:S) -> b | b ->* x:S 20.091/20.091 ) 20.091/20.091 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.091/20.091 20.091/20.091 CRule InfChecker Info: 20.091/20.091 f(x:S) -> a | a ->* x:S 20.091/20.091 Rule remains 20.091/20.091 Proof: 20.091/20.091 NO 20.091/20.091 20.091/20.091 Problem 1: 20.091/20.091 20.091/20.091 Infeasibility Problem: 20.091/20.091 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 20.091/20.091 (STRATEGY CONTEXTSENSITIVE 20.091/20.091 (f 1) 20.091/20.091 (a) 20.091/20.091 (b) 20.091/20.091 (fSNonEmpty) 20.091/20.091 ) 20.091/20.091 (RULES 20.091/20.091 f(x:S:S) -> a | a ->* x:S:S 20.091/20.091 f(x:S:S) -> b | b ->* x:S:S 20.091/20.091 )] 20.091/20.091 20.091/20.091 Infeasibility Conditions: 20.091/20.091 a ->* x:S:S 20.091/20.091 20.091/20.091 Problem 1: 20.091/20.091 20.091/20.091 Obtaining a proof using Prover9: 20.091/20.091 20.091/20.091 -> Prover9 Output: 20.091/20.091 ============================== Prover9 =============================== 20.091/20.091 Prover9 (64) version 2009-11A, November 2009. 20.091/20.091 Process 56977 was started by ubuntu on ubuntu, 20.091/20.091 Wed Jul 14 11:21:27 2021 20.091/20.091 The command was "./prover9 -f /tmp/prover956968-0.in". 20.091/20.091 ============================== end of head =========================== 20.091/20.091 20.091/20.091 ============================== INPUT ================================= 20.091/20.091 20.091/20.091 % Reading from file /tmp/prover956968-0.in 20.091/20.091 20.091/20.091 assign(max_seconds,20). 20.091/20.091 20.091/20.091 formulas(assumptions). 20.091/20.091 ->*_s0(x,x) # label(reflexivity). 20.091/20.091 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.091/20.091 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 20.091/20.091 ->*_s0(a,x2) -> ->_s0(f(x2),a) # label(replacement). 20.091/20.091 ->*_s0(b,x2) -> ->_s0(f(x2),b) # label(replacement). 20.091/20.091 end_of_list. 20.091/20.091 20.091/20.091 formulas(goals). 20.091/20.091 (exists x2 ->*_s0(a,x2)) # label(goal). 20.091/20.091 end_of_list. 20.091/20.091 20.091/20.091 ============================== end of input ========================== 20.091/20.091 20.091/20.091 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.091/20.091 20.091/20.091 % Formulas that are not ordinary clauses: 20.091/20.091 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.091/20.091 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 20.091/20.091 3 ->*_s0(a,x2) -> ->_s0(f(x2),a) # label(replacement) # label(non_clause). [assumption]. 20.091/20.091 4 ->*_s0(b,x2) -> ->_s0(f(x2),b) # label(replacement) # label(non_clause). [assumption]. 20.091/20.091 5 (exists x2 ->*_s0(a,x2)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.091/20.091 20.091/20.091 ============================== end of process non-clausal formulas === 20.091/20.091 20.091/20.091 ============================== PROCESS INITIAL CLAUSES =============== 20.091/20.091 20.091/20.091 % Clauses before input processing: 20.091/20.091 20.091/20.091 formulas(usable). 20.091/20.091 end_of_list. 20.091/20.091 20.091/20.091 formulas(sos). 20.091/20.091 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.091/20.091 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 20.091/20.091 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 20.091/20.091 -->*_s0(a,x) | ->_s0(f(x),a) # label(replacement). [clausify(3)]. 20.091/20.091 -->*_s0(b,x) | ->_s0(f(x),b) # label(replacement). [clausify(4)]. 20.091/20.091 -->*_s0(a,x) # label(goal). [deny(5)]. 20.091/20.091 end_of_list. 20.091/20.091 20.091/20.091 formulas(demodulators). 20.091/20.091 end_of_list. 20.091/20.091 20.091/20.091 ============================== PREDICATE ELIMINATION ================= 20.091/20.091 20.091/20.091 No predicates eliminated. 20.091/20.091 20.091/20.091 ============================== end predicate elimination ============= 20.091/20.091 20.091/20.091 Auto_denials: 20.091/20.091 % copying label goal to answer in negative clause 20.091/20.091 20.091/20.091 Term ordering decisions: 20.091/20.091 Predicate symbol precedence: predicate_order([ ->*_s0, ->_s0 ]). 20.091/20.091 Function symbol precedence: function_order([ a, b, f ]). 20.091/20.091 After inverse_order: (no changes). 20.091/20.091 Unfolding symbols: (none). 20.091/20.091 20.091/20.091 Auto_inference settings: 20.091/20.091 % set(neg_binary_resolution). % (HNE depth_diff=-3) 20.091/20.091 % clear(ordered_res). % (HNE depth_diff=-3) 20.091/20.091 % set(ur_resolution). % (HNE depth_diff=-3) 20.091/20.091 % set(ur_resolution) -> set(pos_ur_resolution). 20.091/20.091 % set(ur_resolution) -> set(neg_ur_resolution). 20.091/20.091 20.091/20.091 Auto_process settings: (no changes). 20.091/20.091 20.091/20.091 kept: 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.091/20.091 kept: 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 20.091/20.091 kept: 8 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 20.091/20.091 kept: 9 -->*_s0(a,x) | ->_s0(f(x),a) # label(replacement). [clausify(3)]. 20.091/20.091 kept: 10 -->*_s0(b,x) | ->_s0(f(x),b) # label(replacement). [clausify(4)]. 20.091/20.091 kept: 11 -->*_s0(a,x) # label(goal) # answer(goal). [deny(5)]. 20.091/20.091 20.091/20.091 ============================== PROOF ================================= 20.091/20.091 20.091/20.091 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.091/20.091 % Length of proof is 4. 20.091/20.091 % Level of proof is 2. 20.091/20.091 % Maximum clause weight is 3.000. 20.091/20.091 % Given clauses 0. 20.091/20.091 20.091/20.091 5 (exists x2 ->*_s0(a,x2)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.091/20.091 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.091/20.091 11 -->*_s0(a,x) # label(goal) # answer(goal). [deny(5)]. 20.091/20.091 12 $F # answer(goal). [resolve(11,a,6,a)]. 20.091/20.091 20.091/20.091 ============================== end of proof ========================== 20.091/20.091 20.091/20.091 ============================== STATISTICS ============================ 20.091/20.091 20.091/20.091 Given=0. Generated=6. Kept=6. proofs=1. 20.091/20.091 Usable=0. Sos=0. Demods=0. Limbo=5, Disabled=6. Hints=0. 20.091/20.091 Kept_by_rule=0, Deleted_by_rule=0. 20.091/20.091 Forward_subsumed=0. Back_subsumed=0. 20.091/20.091 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.091/20.091 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.091/20.091 Demod_attempts=0. Demod_rewrites=0. 20.091/20.091 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.091/20.091 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 20.091/20.091 Megabytes=0.04. 20.091/20.091 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.091/20.091 20.091/20.091 ============================== end of statistics ===================== 20.091/20.091 20.091/20.091 ============================== end of search ========================= 20.091/20.091 20.091/20.091 THEOREM PROVED 20.091/20.091 20.091/20.091 Exiting with 1 proof. 20.091/20.091 20.091/20.091 Process 56977 exit (max_proofs) Wed Jul 14 11:21:27 2021 20.091/20.091 20.091/20.091 20.091/20.091 The problem is feasible. 20.091/20.091 20.091/20.091 20.091/20.091 CRule InfChecker Info: 20.091/20.091 f(x:S) -> b | b ->* x:S 20.091/20.091 Rule remains 20.091/20.091 Proof: 20.091/20.091 NO 20.091/20.091 20.091/20.091 Problem 1: 20.091/20.091 20.091/20.091 Infeasibility Problem: 20.091/20.091 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 20.091/20.091 (STRATEGY CONTEXTSENSITIVE 20.091/20.091 (f 1) 20.091/20.091 (a) 20.091/20.091 (b) 20.091/20.091 (fSNonEmpty) 20.091/20.091 ) 20.091/20.091 (RULES 20.091/20.091 f(x:S:S) -> a | a ->* x:S:S 20.091/20.091 f(x:S:S) -> b | b ->* x:S:S 20.091/20.091 )] 20.091/20.091 20.091/20.091 Infeasibility Conditions: 20.091/20.091 b ->* x:S:S 20.091/20.091 20.091/20.091 Problem 1: 20.091/20.091 20.091/20.091 Obtaining a proof using Prover9: 20.091/20.091 20.091/20.091 -> Prover9 Output: 20.091/20.091 ============================== Prover9 =============================== 20.091/20.091 Prover9 (64) version 2009-11A, November 2009. 20.091/20.091 Process 57000 was started by ubuntu on ubuntu, 20.091/20.091 Wed Jul 14 11:21:27 2021 20.091/20.091 The command was "./prover9 -f /tmp/prover956989-0.in". 20.091/20.091 ============================== end of head =========================== 20.091/20.091 20.091/20.091 ============================== INPUT ================================= 20.091/20.091 20.091/20.091 % Reading from file /tmp/prover956989-0.in 20.091/20.091 20.091/20.091 assign(max_seconds,20). 20.091/20.091 20.091/20.091 formulas(assumptions). 20.091/20.091 ->*_s0(x,x) # label(reflexivity). 20.091/20.091 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.091/20.091 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 20.091/20.091 ->*_s0(a,x2) -> ->_s0(f(x2),a) # label(replacement). 20.091/20.091 ->*_s0(b,x2) -> ->_s0(f(x2),b) # label(replacement). 20.091/20.091 end_of_list. 20.091/20.091 20.091/20.091 formulas(goals). 20.091/20.091 (exists x2 ->*_s0(b,x2)) # label(goal). 20.091/20.091 end_of_list. 20.091/20.091 20.091/20.091 ============================== end of input ========================== 20.091/20.091 20.091/20.091 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.091/20.091 20.091/20.091 % Formulas that are not ordinary clauses: 20.091/20.091 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.091/20.091 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 20.091/20.091 3 ->*_s0(a,x2) -> ->_s0(f(x2),a) # label(replacement) # label(non_clause). [assumption]. 20.091/20.091 4 ->*_s0(b,x2) -> ->_s0(f(x2),b) # label(replacement) # label(non_clause). [assumption]. 20.091/20.091 5 (exists x2 ->*_s0(b,x2)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.091/20.091 20.091/20.091 ============================== end of process non-clausal formulas === 20.091/20.091 20.091/20.091 ============================== PROCESS INITIAL CLAUSES =============== 20.091/20.091 20.091/20.091 % Clauses before input processing: 20.091/20.091 20.091/20.091 formulas(usable). 20.091/20.091 end_of_list. 20.091/20.091 20.091/20.091 formulas(sos). 20.091/20.091 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.091/20.091 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 20.091/20.091 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 20.091/20.091 -->*_s0(a,x) | ->_s0(f(x),a) # label(replacement). [clausify(3)]. 20.091/20.091 -->*_s0(b,x) | ->_s0(f(x),b) # label(replacement). [clausify(4)]. 20.091/20.091 -->*_s0(b,x) # label(goal). [deny(5)]. 20.091/20.091 end_of_list. 20.091/20.091 20.091/20.091 formulas(demodulators). 20.091/20.091 end_of_list. 20.091/20.091 20.091/20.091 ============================== PREDICATE ELIMINATION ================= 20.091/20.091 20.091/20.091 No predicates eliminated. 20.091/20.091 20.091/20.091 ============================== end predicate elimination ============= 20.091/20.091 20.091/20.091 Auto_denials: 20.091/20.091 % copying label goal to answer in negative clause 20.091/20.091 20.091/20.091 Term ordering decisions: 20.091/20.091 Predicate symbol precedence: predicate_order([ ->*_s0, ->_s0 ]). 20.091/20.091 Function symbol precedence: function_order([ a, b, f ]). 20.091/20.091 After inverse_order: (no changes). 20.091/20.091 Unfolding symbols: (none). 20.091/20.091 20.091/20.091 Auto_inference settings: 20.091/20.091 % set(neg_binary_resolution). % (HNE depth_diff=-3) 20.091/20.091 % clear(ordered_res). % (HNE depth_diff=-3) 20.091/20.091 % set(ur_resolution). % (HNE depth_diff=-3) 20.091/20.091 % set(ur_resolution) -> set(pos_ur_resolution). 20.091/20.091 % set(ur_resolution) -> set(neg_ur_resolution). 20.091/20.091 20.091/20.091 Auto_process settings: (no changes). 20.091/20.091 20.091/20.091 kept: 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.091/20.091 kept: 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 20.091/20.091 kept: 8 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 20.091/20.091 kept: 9 -->*_s0(a,x) | ->_s0(f(x),a) # label(replacement). [clausify(3)]. 20.091/20.091 kept: 10 -->*_s0(b,x) | ->_s0(f(x),b) # label(replacement). [clausify(4)]. 20.091/20.091 kept: 11 -->*_s0(b,x) # label(goal) # answer(goal). [deny(5)]. 20.091/20.091 20.091/20.091 ============================== PROOF ================================= 20.091/20.091 20.091/20.091 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.091/20.091 % Length of proof is 4. 20.091/20.091 % Level of proof is 2. 20.091/20.091 % Maximum clause weight is 3.000. 20.091/20.091 % Given clauses 0. 20.091/20.091 20.091/20.091 5 (exists x2 ->*_s0(b,x2)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.091/20.091 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.091/20.091 11 -->*_s0(b,x) # label(goal) # answer(goal). [deny(5)]. 20.091/20.091 12 $F # answer(goal). [resolve(11,a,6,a)]. 20.091/20.091 20.091/20.091 ============================== end of proof ========================== 20.091/20.091 20.091/20.091 ============================== STATISTICS ============================ 20.091/20.091 20.091/20.091 Given=0. Generated=6. Kept=6. proofs=1. 20.091/20.091 Usable=0. Sos=0. Demods=0. Limbo=5, Disabled=6. Hints=0. 20.091/20.091 Kept_by_rule=0, Deleted_by_rule=0. 20.091/20.091 Forward_subsumed=0. Back_subsumed=0. 20.091/20.091 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.091/20.091 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.091/20.091 Demod_attempts=0. Demod_rewrites=0. 20.091/20.091 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.091/20.091 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 20.091/20.091 Megabytes=0.04. 20.091/20.091 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.091/20.091 20.091/20.091 ============================== end of statistics ===================== 20.091/20.091 20.091/20.091 ============================== end of search ========================= 20.091/20.091 20.091/20.091 THEOREM PROVED 20.091/20.091 20.091/20.091 Exiting with 1 proof. 20.091/20.091 20.091/20.091 Process 57000 exit (max_proofs) Wed Jul 14 11:21:27 2021 20.091/20.091 20.091/20.091 20.091/20.091 The problem is feasible. 20.091/20.091 20.091/20.091 20.091/20.091 Problem 1: 20.091/20.091 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.091/20.091 Confluence Problem: 20.091/20.091 (VAR x:S) 20.091/20.091 (STRATEGY CONTEXTSENSITIVE 20.091/20.091 (f 1) 20.091/20.091 (a) 20.091/20.091 (b) 20.091/20.091 (fSNonEmpty) 20.091/20.091 ) 20.091/20.091 (RULES 20.091/20.091 f(x:S) -> a | a ->* x:S 20.091/20.091 f(x:S) -> b | b ->* x:S 20.091/20.091 ) 20.091/20.091 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.091/20.091 20.091/20.091 Critical Pairs Processor: 20.091/20.091 -> Rules: 20.091/20.091 f(x:S) -> a | a ->* x:S 20.091/20.091 f(x:S) -> b | b ->* x:S 20.091/20.091 -> Vars: 20.091/20.091 "x", "x" 20.091/20.091 -> FVars: 20.091/20.091 "x2", "x3" 20.091/20.091 -> PVars: 20.091/20.091 "x": ["x2", "x3"] 20.091/20.091 20.091/20.091 -> Rlps: 20.091/20.091 crule: f(x2:S) -> a | a ->* x2:S, id: 1, possubterms: f(x2:S)-> [] 20.091/20.091 crule: f(x3:S) -> b | b ->* x3:S, id: 2, possubterms: f(x3:S)-> [] 20.091/20.091 20.091/20.091 -> Unifications: 20.091/20.091 R2 unifies with R1 at p: [], l: f(x3:S), lp: f(x3:S), conds: {b ->* x3:S, a ->* x:S}, sig: {x3:S -> x:S}, l': f(x:S), r: b, r': a 20.091/20.091 20.091/20.091 -> Critical pairs info: 20.091/20.091 | b ->* x:S, a ->* x:S => Not trivial, Overlay, N1 20.091/20.091 20.091/20.091 -> Problem conclusions: 20.091/20.091 Left linear, Right linear, Linear 20.091/20.091 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.091/20.091 CTRS Type: 1 20.091/20.091 Deterministic, Strongly deterministic 20.091/20.091 Oriented CTRS, Properly oriented CTRS, Not join CTRS 20.091/20.091 Maybe right-stable CTRS, Overlay CTRS 20.091/20.091 Maybe normal CTRS, Maybe almost normal CTRS 20.091/20.091 Maybe terminating CTRS, Maybe joinable CCPs 20.091/20.091 Maybe level confluent 20.091/20.091 Maybe confluent 20.091/20.091 20.091/20.091 Problem 1: 20.091/20.091 Clean Infeasible CCPs Processor: 20.091/20.091 Num of CPIs: 1 20.091/20.091 Timeout: 60 20.091/20.091 Timeout for each infeasibility problem: 60 s 20.091/20.091 | b ->* x:S, a ->* x:S => Not trivial, Overlay, N1 20.091/20.091 (PROBLEM INFEASIBILITY) 20.091/20.091 (VAR x) 20.091/20.091 (RULES 20.091/20.091 f(x) -> a | a ->* x 20.091/20.091 f(x) -> b | b ->* x) 20.091/20.091 (VAR x2) 20.091/20.091 (CONDITION b ->* x2, a ->* x2) 20.091/20.091 20.091/20.091 Proof: 20.091/20.091 YES 20.091/20.091 20.091/20.091 Problem 1: 20.091/20.091 20.091/20.091 Infeasibility Problem: 20.091/20.091 [(VAR vNonEmpty:S x:S x2:S) 20.091/20.091 (STRATEGY CONTEXTSENSITIVE 20.091/20.091 (f 1) 20.091/20.091 (a) 20.091/20.091 (b) 20.091/20.091 (fSNonEmpty) 20.091/20.091 ) 20.091/20.091 (RULES 20.091/20.091 f(x:S) -> a | a ->* x:S 20.091/20.091 f(x:S) -> b | b ->* x:S 20.091/20.091 )] 20.091/20.091 20.091/20.091 Infeasibility Conditions: 20.091/20.091 b ->* x2:S, a ->* x2:S 20.091/20.091 20.091/20.091 Problem 1: 20.091/20.091 20.091/20.091 Obtaining a model using Mace4: 20.091/20.091 20.091/20.091 -> Usable Rules: 20.091/20.091 Empty 20.091/20.091 20.091/20.091 -> Mace4 Output: 20.091/20.091 ============================== Mace4 ================================= 20.091/20.091 Mace4 (64) version 2009-11A, November 2009. 20.091/20.091 Process 57048 was started by ubuntu on ubuntu, 20.091/20.091 Wed Jul 14 11:21:27 2021 20.091/20.091 The command was "./mace4 -c -f /tmp/mace457017-2.in". 20.091/20.091 ============================== end of head =========================== 20.091/20.091 20.091/20.091 ============================== INPUT ================================= 20.091/20.091 20.091/20.091 % Reading from file /tmp/mace457017-2.in 20.091/20.091 20.091/20.091 assign(max_seconds,20). 20.091/20.091 20.091/20.091 formulas(assumptions). 20.091/20.091 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 20.091/20.091 ->*(x,x) # label(reflexivity). 20.091/20.091 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 20.091/20.091 end_of_list. 20.091/20.091 20.091/20.091 formulas(goals). 20.091/20.091 (exists x2 (->*(b,x2) & ->*(a,x2))) # label(goal). 20.091/20.091 end_of_list. 20.091/20.091 20.091/20.091 ============================== end of input ========================== 20.091/20.091 20.091/20.091 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.091/20.091 20.091/20.091 % Formulas that are not ordinary clauses: 20.091/20.091 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 20.091/20.091 2 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.091/20.091 3 (exists x2 (->*(b,x2) & ->*(a,x2))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.091/20.091 20.091/20.091 ============================== end of process non-clausal formulas === 20.091/20.091 20.091/20.091 ============================== CLAUSES FOR SEARCH ==================== 20.091/20.091 20.091/20.091 formulas(mace4_clauses). 20.091/20.091 -->(x,y) | ->(f(x),f(y)) # label(congruence). 20.091/20.091 ->*(x,x) # label(reflexivity). 20.091/20.091 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 20.091/20.091 -->*(b,x) | -->*(a,x) # label(goal). 20.091/20.091 end_of_list. 20.091/20.091 20.091/20.091 ============================== end of clauses for search ============= 20.091/20.091 20.091/20.091 % There are no natural numbers in the input. 20.091/20.091 20.091/20.091 ============================== DOMAIN SIZE 2 ========================= 20.091/20.091 20.091/20.091 ============================== MODEL ================================= 20.091/20.091 20.091/20.091 interpretation( 2, [number=1, seconds=0], [ 20.091/20.091 20.091/20.091 function(a, [ 0 ]), 20.091/20.091 20.091/20.091 function(b, [ 1 ]), 20.091/20.091 20.091/20.091 function(f(_), [ 0, 0 ]), 20.091/20.091 20.091/20.091 relation(->*(_,_), [ 20.091/20.091 1, 0, 20.091/20.091 0, 1 ]), 20.091/20.091 20.091/20.091 relation(->(_,_), [ 20.091/20.091 0, 0, 20.091/20.091 0, 0 ]) 20.091/20.091 ]). 20.091/20.091 20.091/20.091 ============================== end of model ========================== 20.091/20.091 20.091/20.091 ============================== STATISTICS ============================ 20.091/20.091 20.091/20.091 For domain size 2. 20.091/20.091 20.091/20.091 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.091/20.091 Ground clauses: seen=16, kept=12. 20.091/20.091 Selections=4, assignments=4, propagations=8, current_models=1. 20.091/20.091 Rewrite_terms=12, rewrite_bools=20, indexes=3. 20.091/20.091 Rules_from_neg_clauses=1, cross_offs=1. 20.091/20.091 20.091/20.091 ============================== end of statistics ===================== 20.091/20.091 20.091/20.091 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.091/20.091 20.091/20.091 Exiting with 1 model. 20.091/20.091 20.091/20.091 Process 57048 exit (max_models) Wed Jul 14 11:21:27 2021 20.091/20.091 The process finished Wed Jul 14 11:21:27 2021 20.091/20.091 20.091/20.091 20.091/20.091 Mace4 cooked interpretation: 20.091/20.091 20.091/20.091 20.091/20.091 20.091/20.091 The problem is infeasible. 20.091/20.091 20.091/20.091 20.091/20.091 20.091/20.091 20.091/20.091 -> Problem conclusions: 20.091/20.091 Left linear, Right linear, Linear 20.091/20.091 Weakly orthogonal, Almost orthogonal, Orthogonal 20.091/20.091 CTRS Type: 1 20.091/20.091 Deterministic, Strongly deterministic 20.091/20.091 Oriented CTRS, Properly oriented CTRS, Not join CTRS 20.091/20.091 Maybe right-stable CTRS, Overlay CTRS 20.091/20.091 Maybe normal CTRS, Maybe almost normal CTRS 20.091/20.091 Maybe terminating CTRS, Joinable CCPs 20.091/20.091 Maybe level confluent 20.091/20.091 Maybe confluent 20.091/20.091 20.091/20.091 Resulting CCPs: 20.091/20.091 No CCPs left 20.091/20.091 20.091/20.091 Problem 1: 20.091/20.091 Underlying TRS Termination Processor: 20.091/20.091 20.091/20.091 Resulting Underlying TRS: 20.091/20.091 (VAR x:S) 20.091/20.091 (STRATEGY CONTEXTSENSITIVE 20.091/20.091 (f 1) 20.091/20.091 (a) 20.091/20.091 (b) 20.091/20.091 ) 20.091/20.091 (RULES 20.091/20.091 f(x:S) -> a 20.091/20.091 f(x:S) -> b 20.091/20.091 ) 20.091/20.091 Underlying TRS terminating? 20.091/20.091 YES 20.091/20.091 20.091/20.091 Problem 1: 20.091/20.091 20.091/20.091 (VAR vu95NonEmpty:S xu58S:S) 20.091/20.091 (RULES 20.091/20.091 f(xu58S:S) -> a 20.091/20.091 f(xu58S:S) -> b 20.091/20.091 ) 20.091/20.091 20.091/20.091 Problem 1: 20.091/20.091 20.091/20.091 Dependency Pairs Processor: 20.091/20.091 -> Pairs: 20.091/20.091 Empty 20.091/20.091 -> Rules: 20.091/20.091 f(xu58S:S) -> a 20.091/20.091 f(xu58S:S) -> b 20.091/20.091 20.091/20.091 Problem 1: 20.091/20.091 20.091/20.091 SCC Processor: 20.091/20.091 -> Pairs: 20.091/20.091 Empty 20.091/20.091 -> Rules: 20.091/20.091 f(xu58S:S) -> a 20.091/20.091 f(xu58S:S) -> b 20.091/20.091 ->Strongly Connected Components: 20.091/20.091 There is no strongly connected component 20.091/20.091 20.091/20.091 The problem is finite. 20.091/20.091 20.091/20.091 20.091/20.091 -> Problem conclusions: 20.091/20.091 Left linear, Right linear, Linear 20.091/20.091 Weakly orthogonal, Almost orthogonal, Orthogonal 20.091/20.091 CTRS Type: 1 20.091/20.091 Deterministic, Strongly deterministic 20.091/20.091 Oriented CTRS, Properly oriented CTRS, Not join CTRS 20.091/20.091 Maybe right-stable CTRS, Overlay CTRS 20.091/20.091 Maybe normal CTRS, Maybe almost normal CTRS 20.091/20.091 Terminating CTRS, Joinable CCPs 20.091/20.091 Maybe level confluent 20.091/20.091 Confluent 20.091/20.091 20.091/20.091 The problem is joinable. 20.091/20.091 41.91user 1.15system 0:20.91elapsed 205%CPU (0avgtext+0avgdata 136020maxresident)k 20.091/20.091 8inputs+0outputs (0major+134089minor)pagefaults 0swaps