20.004/20.004 YES 20.004/20.004 20.004/20.004 Problem 1: 20.004/20.004 20.004/20.004 20.004/20.004 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.004/20.004 Confluence Problem: 20.004/20.004 (VAR vNonEmpty:S x:S) 20.004/20.004 (STRATEGY CONTEXTSENSITIVE 20.004/20.004 (a) 20.004/20.004 (f 1) 20.004/20.004 (b) 20.004/20.004 (fSNonEmpty) 20.004/20.004 ) 20.004/20.004 (RULES 20.004/20.004 a -> a | f(a) ->* a 20.004/20.004 f(x:S) -> a | x:S ->* b 20.004/20.004 ) 20.004/20.004 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.004/20.004 20.004/20.004 20.004/20.004 Problem 1: 20.004/20.004 20.004/20.004 Inlining of Conditions Processor [STERN17]: 20.004/20.004 20.004/20.004 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.004/20.004 Confluence Problem: 20.004/20.004 (VAR vNonEmpty:S x:S) 20.004/20.004 (STRATEGY CONTEXTSENSITIVE 20.004/20.004 (a) 20.004/20.004 (f 1) 20.004/20.004 (b) 20.004/20.004 (fSNonEmpty) 20.004/20.004 ) 20.004/20.004 (RULES 20.004/20.004 a -> a | f(a) ->* a 20.004/20.004 f(x:S) -> a | x:S ->* b 20.004/20.004 ) 20.004/20.004 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.004/20.004 20.004/20.004 20.004/20.004 Problem 1: 20.004/20.004 20.004/20.004 Clean CTRS Processor: 20.004/20.004 20.004/20.004 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.004/20.004 Confluence Problem: 20.004/20.004 (VAR x:S) 20.004/20.004 (STRATEGY CONTEXTSENSITIVE 20.004/20.004 (a) 20.004/20.004 (f 1) 20.004/20.004 (b) 20.004/20.004 (fSNonEmpty) 20.004/20.004 ) 20.004/20.004 (RULES 20.004/20.004 f(x:S) -> a | x:S ->* b 20.004/20.004 ) 20.004/20.004 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.004/20.004 20.004/20.004 CRule InfChecker Info: 20.004/20.004 a -> a | f(a) ->* a 20.004/20.004 Rule deleted 20.004/20.004 Proof: 20.004/20.004 SAME_LHS_AND_RHS 20.004/20.004 20.004/20.004 CRule InfChecker Info: 20.004/20.004 f(x:S) -> a | x:S ->* b 20.004/20.004 Rule remains 20.004/20.004 Proof: 20.004/20.004 NO 20.004/20.004 20.004/20.004 Problem 1: 20.004/20.004 20.004/20.004 Infeasibility Problem: 20.004/20.004 [(VAR vNonEmpty:S x:S:S) 20.004/20.004 (STRATEGY CONTEXTSENSITIVE 20.004/20.004 (f 1) 20.004/20.004 (a) 20.004/20.004 (b) 20.004/20.004 (fSNonEmpty) 20.004/20.004 ) 20.004/20.004 (RULES 20.004/20.004 f(x:S:S) -> a | x:S:S ->* b 20.004/20.004 )] 20.004/20.004 20.004/20.004 Infeasibility Conditions: 20.004/20.004 x:S:S ->* b 20.004/20.004 20.004/20.004 Problem 1: 20.004/20.004 20.004/20.004 Obtaining a proof using Prover9: 20.004/20.004 20.004/20.004 -> Prover9 Output: 20.004/20.004 ============================== Prover9 =============================== 20.004/20.004 Prover9 (64) version 2009-11A, November 2009. 20.004/20.004 Process 121832 was started by ubuntu on ubuntu, 20.004/20.004 Fri Jul 16 15:42:22 2021 20.004/20.004 The command was "./prover9 -f /tmp/prover9121822-0.in". 20.004/20.004 ============================== end of head =========================== 20.004/20.004 20.004/20.004 ============================== INPUT ================================= 20.004/20.004 20.004/20.004 % Reading from file /tmp/prover9121822-0.in 20.004/20.004 20.004/20.004 assign(max_seconds,20). 20.004/20.004 20.004/20.004 formulas(assumptions). 20.004/20.004 ->*_s0(x,x) # label(reflexivity). 20.004/20.004 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.004/20.004 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 20.004/20.004 ->*_s0(x1,b) -> ->_s0(f(x1),a) # label(replacement). 20.004/20.004 end_of_list. 20.004/20.004 20.004/20.004 formulas(goals). 20.004/20.004 (exists x1 ->*_s0(x1,b)) # label(goal). 20.004/20.004 end_of_list. 20.004/20.004 20.004/20.004 ============================== end of input ========================== 20.004/20.004 20.004/20.004 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.004/20.004 20.004/20.004 % Formulas that are not ordinary clauses: 20.004/20.004 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.004/20.004 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 20.004/20.004 3 ->*_s0(x1,b) -> ->_s0(f(x1),a) # label(replacement) # label(non_clause). [assumption]. 20.004/20.004 4 (exists x1 ->*_s0(x1,b)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.004/20.004 20.004/20.004 ============================== end of process non-clausal formulas === 20.004/20.004 20.004/20.004 ============================== PROCESS INITIAL CLAUSES =============== 20.004/20.004 20.004/20.004 % Clauses before input processing: 20.004/20.004 20.004/20.004 formulas(usable). 20.004/20.004 end_of_list. 20.004/20.004 20.004/20.004 formulas(sos). 20.004/20.004 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.004/20.004 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 20.004/20.004 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 20.004/20.004 -->*_s0(x,b) | ->_s0(f(x),a) # label(replacement). [clausify(3)]. 20.004/20.004 -->*_s0(x,b) # label(goal). [deny(4)]. 20.004/20.004 end_of_list. 20.004/20.004 20.004/20.004 formulas(demodulators). 20.004/20.004 end_of_list. 20.004/20.004 20.004/20.004 ============================== PREDICATE ELIMINATION ================= 20.004/20.004 20.004/20.004 No predicates eliminated. 20.004/20.004 20.004/20.004 ============================== end predicate elimination ============= 20.004/20.004 20.004/20.004 Auto_denials: 20.004/20.004 % copying label goal to answer in negative clause 20.004/20.004 20.004/20.004 Term ordering decisions: 20.004/20.004 Predicate symbol precedence: predicate_order([ ->*_s0, ->_s0 ]). 20.004/20.004 Function symbol precedence: function_order([ a, b, f ]). 20.004/20.004 After inverse_order: (no changes). 20.004/20.004 Unfolding symbols: (none). 20.004/20.004 20.004/20.004 Auto_inference settings: 20.004/20.004 % set(neg_binary_resolution). % (HNE depth_diff=-2) 20.004/20.004 % clear(ordered_res). % (HNE depth_diff=-2) 20.004/20.004 % set(ur_resolution). % (HNE depth_diff=-2) 20.004/20.004 % set(ur_resolution) -> set(pos_ur_resolution). 20.004/20.004 % set(ur_resolution) -> set(neg_ur_resolution). 20.004/20.004 20.004/20.004 Auto_process settings: (no changes). 20.004/20.004 20.004/20.004 kept: 5 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.004/20.004 kept: 6 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 20.004/20.004 kept: 7 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 20.004/20.004 kept: 8 -->*_s0(x,b) | ->_s0(f(x),a) # label(replacement). [clausify(3)]. 20.004/20.004 kept: 9 -->*_s0(x,b) # label(goal) # answer(goal). [deny(4)]. 20.004/20.004 20.004/20.004 ============================== PROOF ================================= 20.004/20.004 20.004/20.004 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.004/20.004 % Length of proof is 4. 20.004/20.004 % Level of proof is 2. 20.004/20.004 % Maximum clause weight is 3.000. 20.004/20.004 % Given clauses 0. 20.004/20.004 20.004/20.004 4 (exists x1 ->*_s0(x1,b)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.004/20.004 5 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.004/20.004 9 -->*_s0(x,b) # label(goal) # answer(goal). [deny(4)]. 20.004/20.004 10 $F # answer(goal). [resolve(9,a,5,a)]. 20.004/20.004 20.004/20.004 ============================== end of proof ========================== 20.004/20.004 20.004/20.004 ============================== STATISTICS ============================ 20.004/20.004 20.004/20.004 Given=0. Generated=5. Kept=5. proofs=1. 20.004/20.004 Usable=0. Sos=0. Demods=0. Limbo=4, Disabled=5. Hints=0. 20.004/20.004 Kept_by_rule=0, Deleted_by_rule=0. 20.004/20.004 Forward_subsumed=0. Back_subsumed=0. 20.004/20.004 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.004/20.004 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 20.004/20.004 Demod_attempts=0. Demod_rewrites=0. 20.004/20.004 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.004/20.004 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 20.004/20.004 Megabytes=0.03. 20.004/20.004 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.004/20.004 20.004/20.004 ============================== end of statistics ===================== 20.004/20.004 20.004/20.004 ============================== end of search ========================= 20.004/20.004 20.004/20.004 THEOREM PROVED 20.004/20.004 20.004/20.004 Exiting with 1 proof. 20.004/20.004 20.004/20.004 Process 121832 exit (max_proofs) Fri Jul 16 15:42:22 2021 20.004/20.004 20.004/20.004 20.004/20.004 The problem is feasible. 20.004/20.004 20.004/20.004 20.004/20.004 Problem 1: 20.004/20.004 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.004/20.004 Confluence Problem: 20.004/20.004 (VAR x:S) 20.004/20.004 (STRATEGY CONTEXTSENSITIVE 20.004/20.004 (a) 20.004/20.004 (f 1) 20.004/20.004 (b) 20.004/20.004 (fSNonEmpty) 20.004/20.004 ) 20.004/20.004 (RULES 20.004/20.004 f(x:S) -> a | x:S ->* b 20.004/20.004 ) 20.004/20.004 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.004/20.004 20.004/20.004 Critical Pairs Processor: 20.004/20.004 -> Rules: 20.004/20.004 f(x:S) -> a | x:S ->* b 20.004/20.004 -> Vars: 20.004/20.004 "x" 20.004/20.004 -> FVars: 20.004/20.004 "x2" 20.004/20.004 -> PVars: 20.004/20.004 "x": ["x2"] 20.004/20.004 20.004/20.004 -> Rlps: 20.004/20.004 crule: f(x2:S) -> a | x2:S ->* b, id: 1, possubterms: f(x2:S)-> [] 20.004/20.004 20.004/20.004 -> Unifications: 20.004/20.004 20.004/20.004 20.004/20.004 -> Critical pairs info: 20.004/20.004 20.004/20.004 20.004/20.004 -> Problem conclusions: 20.004/20.004 Left linear, Right linear, Linear 20.004/20.004 Weakly orthogonal, Almost orthogonal, Orthogonal 20.004/20.004 CTRS Type: 1 20.004/20.004 Deterministic, Strongly deterministic 20.004/20.004 Oriented CTRS, Properly oriented CTRS, Not join CTRS 20.004/20.004 Maybe right-stable CTRS, Overlay CTRS 20.004/20.004 Maybe normal CTRS, Maybe almost normal CTRS 20.004/20.004 Maybe terminating CTRS, Joinable CCPs 20.004/20.004 Maybe level confluent 20.004/20.004 Maybe confluent 20.004/20.004 20.004/20.004 Problem 1: 20.004/20.004 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.004/20.004 Confluence Problem: 20.004/20.004 (VAR x:S) 20.004/20.004 (STRATEGY CONTEXTSENSITIVE 20.004/20.004 (a) 20.004/20.004 (f 1) 20.004/20.004 (b) 20.004/20.004 (fSNonEmpty) 20.004/20.004 ) 20.004/20.004 (RULES 20.004/20.004 f(x:S) -> a | x:S ->* b 20.004/20.004 ) 20.004/20.004 Critical Pairs: 20.004/20.004 20.004/20.004 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.004/20.004 20.004/20.004 Right Stable Processor: 20.004/20.004 Right-stable CTRS 20.004/20.004 Justification: 20.004/20.004 20.004/20.004 -> Term right-stability conditions: 20.004/20.004 Term: b 20.004/20.004 Right-stable term 20.004/20.004 Linear constructor form 20.004/20.004 Don't know if term is a ground normal form (not needed) 20.004/20.004 Right-stability condition achieved 20.004/20.004 A call to InfChecker wasn't needed 20.004/20.004 20.004/20.004 -> Problem conclusions: 20.004/20.004 Left linear, Right linear, Linear 20.004/20.004 Weakly orthogonal, Almost orthogonal, Orthogonal 20.004/20.004 CTRS Type: 1 20.004/20.004 Deterministic, Strongly deterministic 20.004/20.004 Oriented CTRS, Properly oriented CTRS, Not join CTRS 20.004/20.004 Right-stable CTRS, Overlay CTRS 20.004/20.004 Maybe normal CTRS, Almost normal CTRS 20.004/20.004 Maybe terminating CTRS, Joinable CCPs 20.004/20.004 Level confluent 20.004/20.004 Confluent 20.004/20.004 20.004/20.004 The problem is joinable. 20.004/20.004 0.26user 0.02system 0:20.04elapsed 1%CPU (0avgtext+0avgdata 12892maxresident)k 20.004/20.004 0inputs+0outputs (0major+13782minor)pagefaults 0swaps