20.006/20.006 YES 20.006/20.006 20.006/20.006 Problem 1: 20.006/20.006 20.006/20.006 20.006/20.006 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.006/20.006 Confluence Problem: 20.006/20.006 (VAR vNonEmpty x) 20.006/20.006 (REPLACEMENT-MAP 20.006/20.006 (a) 20.006/20.006 (f 1) 20.006/20.006 (b) 20.006/20.006 (fSNonEmpty) 20.006/20.006 ) 20.006/20.006 (RULES 20.006/20.006 a -> a | f(a) ->* a 20.006/20.006 f(x) -> a | x ->* b 20.006/20.006 ) 20.006/20.006 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.006/20.006 20.006/20.006 20.006/20.006 Problem 1: 20.006/20.006 20.006/20.006 Inlining of Conditions Processor [STERN17]: 20.006/20.006 20.006/20.006 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.006/20.006 Confluence Problem: 20.006/20.006 (VAR vNonEmpty x) 20.006/20.006 (REPLACEMENT-MAP 20.006/20.006 (a) 20.006/20.006 (f 1) 20.006/20.006 (b) 20.006/20.006 (fSNonEmpty) 20.006/20.006 ) 20.006/20.006 (RULES 20.006/20.006 a -> a | f(a) ->* a 20.006/20.006 f(x) -> a | x ->* b 20.006/20.006 ) 20.006/20.006 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.006/20.006 20.006/20.006 20.006/20.006 Problem 1: 20.006/20.006 20.006/20.006 Clean CTRS Processor: 20.006/20.006 20.006/20.006 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.006/20.006 Confluence Problem: 20.006/20.006 (VAR x) 20.006/20.006 (REPLACEMENT-MAP 20.006/20.006 (a) 20.006/20.006 (f 1) 20.006/20.006 (b) 20.006/20.006 (fSNonEmpty) 20.006/20.006 ) 20.006/20.006 (RULES 20.006/20.006 f(x) -> a | x ->* b 20.006/20.006 ) 20.006/20.006 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.006/20.006 20.006/20.006 CRule InfChecker Info: 20.006/20.006 a -> a | f(a) ->* a 20.006/20.006 Rule deleted 20.006/20.006 Proof: 20.006/20.006 SAME_LHS_AND_RHS 20.006/20.006 20.006/20.006 CRule InfChecker Info: 20.006/20.006 f(x) -> a | x ->* b 20.006/20.006 Rule remains 20.006/20.006 Proof: 20.006/20.006 NO 20.006/20.006 20.006/20.006 Problem 1: 20.006/20.006 20.006/20.006 Infeasibility Problem: 20.006/20.006 [(VAR vNonEmpty x vNonEmpty x) 20.006/20.006 (STRATEGY CONTEXTSENSITIVE 20.006/20.006 (f 1) 20.006/20.006 (a) 20.006/20.006 (b) 20.006/20.006 (fSNonEmpty) 20.006/20.006 ) 20.006/20.006 (RULES 20.006/20.006 f(x) -> a | x ->* b 20.006/20.006 )] 20.006/20.006 20.006/20.006 Infeasibility Conditions: 20.006/20.006 x ->* b 20.006/20.006 20.006/20.006 Problem 1: 20.006/20.006 20.006/20.006 Obtaining a proof using Prover9: 20.006/20.006 20.006/20.006 -> Prover9 Output: 20.006/20.006 ============================== Prover9 =============================== 20.006/20.006 Prover9 (64) version 2009-11A, November 2009. 20.006/20.006 Process 79176 was started by ubuntu on ubuntu, 20.006/20.006 Wed Mar 9 10:22:50 2022 20.006/20.006 The command was "./prover9 -f /tmp/prover979167-0.in". 20.006/20.006 ============================== end of head =========================== 20.006/20.006 20.006/20.006 ============================== INPUT ================================= 20.006/20.006 20.006/20.006 % Reading from file /tmp/prover979167-0.in 20.006/20.006 20.006/20.006 assign(max_seconds,20). 20.006/20.006 20.006/20.006 formulas(assumptions). 20.006/20.006 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 20.006/20.006 ->*_s0(x1,b) -> ->_s0(f(x1),a) # label(replacement). 20.006/20.006 ->*_s0(x,x) # label(reflexivity). 20.006/20.006 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.006/20.006 end_of_list. 20.006/20.006 20.006/20.006 formulas(goals). 20.006/20.006 (exists x3 ->*_s0(x3,b)) # label(goal). 20.006/20.006 end_of_list. 20.006/20.006 20.006/20.006 ============================== end of input ========================== 20.006/20.006 20.006/20.006 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.006/20.006 20.006/20.006 % Formulas that are not ordinary clauses: 20.006/20.006 1 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 20.006/20.006 2 ->*_s0(x1,b) -> ->_s0(f(x1),a) # label(replacement) # label(non_clause). [assumption]. 20.006/20.006 3 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.006/20.006 4 (exists x3 ->*_s0(x3,b)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.006/20.006 20.006/20.006 ============================== end of process non-clausal formulas === 20.006/20.006 20.006/20.006 ============================== PROCESS INITIAL CLAUSES =============== 20.006/20.006 20.006/20.006 % Clauses before input processing: 20.006/20.006 20.006/20.006 formulas(usable). 20.006/20.006 end_of_list. 20.006/20.006 20.006/20.006 formulas(sos). 20.006/20.006 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 20.006/20.006 -->*_s0(x,b) | ->_s0(f(x),a) # label(replacement). [clausify(2)]. 20.006/20.006 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.006/20.006 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(3)]. 20.006/20.006 -->*_s0(x,b) # label(goal). [deny(4)]. 20.006/20.006 end_of_list. 20.006/20.006 20.006/20.006 formulas(demodulators). 20.006/20.006 end_of_list. 20.006/20.006 20.006/20.006 ============================== PREDICATE ELIMINATION ================= 20.006/20.006 20.006/20.006 No predicates eliminated. 20.006/20.006 20.006/20.006 ============================== end predicate elimination ============= 20.006/20.006 20.006/20.006 Auto_denials: 20.006/20.006 % copying label goal to answer in negative clause 20.006/20.006 20.006/20.006 Term ordering decisions: 20.006/20.006 Predicate symbol precedence: predicate_order([ ->*_s0, ->_s0 ]). 20.006/20.006 Function symbol precedence: function_order([ a, b, f ]). 20.006/20.006 After inverse_order: (no changes). 20.006/20.006 Unfolding symbols: (none). 20.006/20.006 20.006/20.006 Auto_inference settings: 20.006/20.006 % set(neg_binary_resolution). % (HNE depth_diff=-2) 20.006/20.006 % clear(ordered_res). % (HNE depth_diff=-2) 20.006/20.006 % set(ur_resolution). % (HNE depth_diff=-2) 20.006/20.006 % set(ur_resolution) -> set(pos_ur_resolution). 20.006/20.006 % set(ur_resolution) -> set(neg_ur_resolution). 20.006/20.006 20.006/20.006 Auto_process settings: (no changes). 20.006/20.006 20.006/20.006 kept: 5 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 20.006/20.006 kept: 6 -->*_s0(x,b) | ->_s0(f(x),a) # label(replacement). [clausify(2)]. 20.006/20.006 kept: 7 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.006/20.006 kept: 8 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(3)]. 20.006/20.006 kept: 9 -->*_s0(x,b) # label(goal) # answer(goal). [deny(4)]. 20.006/20.006 20.006/20.006 ============================== PROOF ================================= 20.006/20.006 20.006/20.006 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.006/20.006 % Length of proof is 4. 20.006/20.006 % Level of proof is 2. 20.006/20.006 % Maximum clause weight is 3.000. 20.006/20.006 % Given clauses 0. 20.006/20.006 20.006/20.006 4 (exists x3 ->*_s0(x3,b)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.006/20.006 7 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.006/20.006 9 -->*_s0(x,b) # label(goal) # answer(goal). [deny(4)]. 20.006/20.006 10 $F # answer(goal). [resolve(9,a,7,a)]. 20.006/20.006 20.006/20.006 ============================== end of proof ========================== 20.006/20.006 20.006/20.006 ============================== STATISTICS ============================ 20.006/20.006 20.006/20.006 Given=0. Generated=5. Kept=5. proofs=1. 20.006/20.006 Usable=0. Sos=0. Demods=0. Limbo=4, Disabled=5. Hints=0. 20.006/20.006 Kept_by_rule=0, Deleted_by_rule=0. 20.006/20.006 Forward_subsumed=0. Back_subsumed=0. 20.006/20.006 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.006/20.006 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.006/20.006 Demod_attempts=0. Demod_rewrites=0. 20.006/20.006 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.006/20.006 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 20.006/20.006 Megabytes=0.03. 20.006/20.006 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.006/20.006 20.006/20.006 ============================== end of statistics ===================== 20.006/20.006 20.006/20.006 ============================== end of search ========================= 20.006/20.006 20.006/20.006 THEOREM PROVED 20.006/20.006 20.006/20.006 Exiting with 1 proof. 20.006/20.006 20.006/20.006 Process 79176 exit (max_proofs) Wed Mar 9 10:22:50 2022 20.006/20.006 20.006/20.006 20.006/20.006 The problem is feasible. 20.006/20.006 20.006/20.006 20.006/20.006 Problem 1: 20.006/20.006 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.006/20.006 Confluence Problem: 20.006/20.006 (VAR x) 20.006/20.006 (REPLACEMENT-MAP 20.006/20.006 (a) 20.006/20.006 (f 1) 20.006/20.006 (b) 20.006/20.006 (fSNonEmpty) 20.006/20.006 ) 20.006/20.006 (RULES 20.006/20.006 f(x) -> a | x ->* b 20.006/20.006 ) 20.006/20.006 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.006/20.006 20.006/20.006 Critical Pairs Processor: 20.006/20.006 -> Rules: 20.006/20.006 f(x) -> a | x ->* b 20.006/20.006 -> Vars: 20.006/20.006 "x" 20.006/20.006 20.006/20.006 -> Rlps: 20.006/20.006 crule: f(x) -> a | x ->* b, id: 1, possubterms: f(x)-> [] 20.006/20.006 20.006/20.006 -> Unifications: 20.006/20.006 20.006/20.006 20.006/20.006 -> Critical pairs info: 20.006/20.006 20.006/20.006 20.006/20.006 -> Problem conclusions: 20.006/20.006 Left linear, Right linear, Linear 20.006/20.006 Weakly orthogonal, Almost orthogonal, Orthogonal 20.006/20.006 CTRS Type: 1 20.006/20.006 Deterministic, Strongly deterministic 20.006/20.006 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.006/20.006 Maybe right-stable CTRS, Overlay CTRS 20.006/20.006 Maybe normal CTRS, Maybe almost normal CTRS 20.006/20.006 Maybe terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.006/20.006 Maybe level confluent 20.006/20.006 Maybe confluent 20.006/20.006 20.006/20.006 Problem 1: 20.006/20.006 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.006/20.006 Confluence Problem: 20.006/20.006 (VAR x) 20.006/20.006 (REPLACEMENT-MAP 20.006/20.006 (a) 20.006/20.006 (f 1) 20.006/20.006 (b) 20.006/20.006 (fSNonEmpty) 20.006/20.006 ) 20.006/20.006 (RULES 20.006/20.006 f(x) -> a | x ->* b 20.006/20.006 ) 20.006/20.006 Critical Pairs: 20.006/20.006 20.006/20.006 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.006/20.006 20.006/20.006 Right Stable Processor: 20.006/20.006 Right-stable CTRS 20.006/20.006 Justification: 20.006/20.006 20.006/20.006 -> Term right-stability conditions: 20.006/20.006 Term: b 20.006/20.006 Right-stable term 20.006/20.006 Linear constructor form 20.006/20.006 Don't know if term is a ground normal form (not needed) 20.006/20.006 Right-stability condition achieved 20.006/20.006 A call to InfChecker wasn't needed 20.006/20.006 20.006/20.006 -> Problem conclusions: 20.006/20.006 Left linear, Right linear, Linear 20.006/20.006 Weakly orthogonal, Almost orthogonal, Orthogonal 20.006/20.006 CTRS Type: 1 20.006/20.006 Deterministic, Strongly deterministic 20.006/20.006 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.006/20.006 Right-stable CTRS, Overlay CTRS 20.006/20.006 Maybe normal CTRS, Almost normal CTRS 20.006/20.006 Maybe terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 20.006/20.006 Level confluent 20.006/20.006 Confluent 20.006/20.006 20.006/20.006 The problem is joinable. 20.006/20.006 0.27user 0.06system 0:20.06elapsed 1%CPU (0avgtext+0avgdata 13148maxresident)k 20.006/20.006 8inputs+0outputs (0major+13945minor)pagefaults 0swaps