0.015/0.015 YES 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 Confluence Problem: 0.015/0.015 (VAR vNonEmpty x) 0.015/0.015 (REPLACEMENT-MAP 0.015/0.015 (a) 0.015/0.015 (b) 0.015/0.015 (f 1) 0.015/0.015 (c) 0.015/0.015 (fSNonEmpty) 0.015/0.015 ) 0.015/0.015 (RULES 0.015/0.015 a -> c 0.015/0.015 b -> c 0.015/0.015 f(x) -> x | a ->* x 0.015/0.015 ) 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 Inlining of Conditions Processor [STERN17]: 0.015/0.015 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 Confluence Problem: 0.015/0.015 (VAR vNonEmpty x) 0.015/0.015 (REPLACEMENT-MAP 0.015/0.015 (a) 0.015/0.015 (b) 0.015/0.015 (f 1) 0.015/0.015 (c) 0.015/0.015 (fSNonEmpty) 0.015/0.015 ) 0.015/0.015 (RULES 0.015/0.015 a -> c 0.015/0.015 b -> c 0.015/0.015 f(x) -> x | a ->* x 0.015/0.015 ) 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 Clean CTRS Processor: 0.015/0.015 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 Confluence Problem: 0.015/0.015 (VAR x) 0.015/0.015 (REPLACEMENT-MAP 0.015/0.015 (a) 0.015/0.015 (b) 0.015/0.015 (f 1) 0.015/0.015 (c) 0.015/0.015 (fSNonEmpty) 0.015/0.015 ) 0.015/0.015 (RULES 0.015/0.015 a -> c 0.015/0.015 b -> c 0.015/0.015 f(x) -> x | a ->* x 0.015/0.015 ) 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 0.015/0.015 CRule InfChecker Info: 0.015/0.015 a -> c 0.015/0.015 Rule remains 0.015/0.015 Proof: 0.015/0.015 NO_CONDS 0.015/0.015 0.015/0.015 CRule InfChecker Info: 0.015/0.015 b -> c 0.015/0.015 Rule remains 0.015/0.015 Proof: 0.015/0.015 NO_CONDS 0.015/0.015 0.015/0.015 CRule InfChecker Info: 0.015/0.015 f(x) -> x | a ->* x 0.015/0.015 Rule remains 0.015/0.015 Proof: 0.015/0.015 NO 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 Infeasibility Problem: 0.015/0.015 [(VAR vNonEmpty x vNonEmpty x) 0.015/0.015 (STRATEGY CONTEXTSENSITIVE 0.015/0.015 (a) 0.015/0.015 (b) 0.015/0.015 (f 1) 0.015/0.015 (c) 0.015/0.015 (fSNonEmpty) 0.015/0.015 ) 0.015/0.015 (RULES 0.015/0.015 a -> c 0.015/0.015 b -> c 0.015/0.015 f(x) -> x | a ->* x 0.015/0.015 )] 0.015/0.015 0.015/0.015 Infeasibility Conditions: 0.015/0.015 a ->* x 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 Obtaining a proof using Prover9: 0.015/0.015 0.015/0.015 -> Prover9 Output: 0.015/0.015 ============================== Prover9 =============================== 0.015/0.015 Prover9 (64) version 2009-11A, November 2009. 0.015/0.015 Process 60448 was started by ubuntu on ubuntu, 0.015/0.015 Wed Mar 9 09:44:39 2022 0.015/0.015 The command was "./prover9 -f /tmp/prover960439-0.in". 0.015/0.015 ============================== end of head =========================== 0.015/0.015 0.015/0.015 ============================== INPUT ================================= 0.015/0.015 0.015/0.015 % Reading from file /tmp/prover960439-0.in 0.015/0.015 0.015/0.015 assign(max_seconds,20). 0.015/0.015 0.015/0.015 formulas(assumptions). 0.015/0.015 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 0.015/0.015 ->_s0(a,c) # label(replacement). 0.015/0.015 ->*_s0(x,x) # label(reflexivity). 0.015/0.015 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 formulas(goals). 0.015/0.015 (exists x3 ->*_s0(a,x3)) # label(goal). 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 ============================== end of input ========================== 0.015/0.015 0.015/0.015 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.015/0.015 0.015/0.015 % Formulas that are not ordinary clauses: 0.015/0.015 1 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 2 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 0.015/0.015 3 (exists x3 ->*_s0(a,x3)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.015/0.015 0.015/0.015 ============================== end of process non-clausal formulas === 0.015/0.015 0.015/0.015 ============================== PROCESS INITIAL CLAUSES =============== 0.015/0.015 0.015/0.015 % Clauses before input processing: 0.015/0.015 0.015/0.015 formulas(usable). 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 formulas(sos). 0.015/0.015 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 0.015/0.015 ->_s0(a,c) # label(replacement). [assumption]. 0.015/0.015 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.015/0.015 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(2)]. 0.015/0.015 -->*_s0(a,x) # label(goal). [deny(3)]. 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 formulas(demodulators). 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 ============================== PREDICATE ELIMINATION ================= 0.015/0.015 0.015/0.015 No predicates eliminated. 0.015/0.015 0.015/0.015 ============================== end predicate elimination ============= 0.015/0.015 0.015/0.015 Auto_denials: 0.015/0.015 % copying label goal to answer in negative clause 0.015/0.015 0.015/0.015 Term ordering decisions: 0.015/0.015 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 0.015/0.015 Function symbol precedence: function_order([ a, c, f ]). 0.015/0.015 After inverse_order: (no changes). 0.015/0.015 Unfolding symbols: (none). 0.015/0.015 0.015/0.015 Auto_inference settings: 0.015/0.015 % set(neg_binary_resolution). % (HNE depth_diff=-1) 0.015/0.015 % clear(ordered_res). % (HNE depth_diff=-1) 0.015/0.015 % set(ur_resolution). % (HNE depth_diff=-1) 0.015/0.015 % set(ur_resolution) -> set(pos_ur_resolution). 0.015/0.015 % set(ur_resolution) -> set(neg_ur_resolution). 0.015/0.015 0.015/0.015 Auto_process settings: (no changes). 0.015/0.015 0.015/0.015 kept: 4 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 0.015/0.015 kept: 5 ->_s0(a,c) # label(replacement). [assumption]. 0.015/0.015 kept: 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.015/0.015 kept: 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(2)]. 0.015/0.015 kept: 8 -->*_s0(a,x) # label(goal) # answer(goal). [deny(3)]. 0.015/0.015 0.015/0.015 ============================== PROOF ================================= 0.015/0.015 0.015/0.015 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 0.015/0.015 % Length of proof is 4. 0.015/0.015 % Level of proof is 2. 0.015/0.015 % Maximum clause weight is 3.000. 0.015/0.015 % Given clauses 0. 0.015/0.015 0.015/0.015 3 (exists x3 ->*_s0(a,x3)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.015/0.015 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.015/0.015 8 -->*_s0(a,x) # label(goal) # answer(goal). [deny(3)]. 0.015/0.015 9 $F # answer(goal). [resolve(8,a,6,a)]. 0.015/0.015 0.015/0.015 ============================== end of proof ========================== 0.015/0.015 0.015/0.015 ============================== STATISTICS ============================ 0.015/0.015 0.015/0.015 Given=0. Generated=5. Kept=5. proofs=1. 0.015/0.015 Usable=0. Sos=0. Demods=0. Limbo=4, Disabled=5. Hints=0. 0.015/0.015 Kept_by_rule=0, Deleted_by_rule=0. 0.015/0.015 Forward_subsumed=0. Back_subsumed=0. 0.015/0.015 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 0.015/0.015 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 0.015/0.015 Demod_attempts=0. Demod_rewrites=0. 0.015/0.015 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 0.015/0.015 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 0.015/0.015 Megabytes=0.03. 0.015/0.015 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.015/0.015 0.015/0.015 ============================== end of statistics ===================== 0.015/0.015 0.015/0.015 ============================== end of search ========================= 0.015/0.015 0.015/0.015 THEOREM PROVED 0.015/0.015 0.015/0.015 Exiting with 1 proof. 0.015/0.015 0.015/0.015 Process 60448 exit (max_proofs) Wed Mar 9 09:44:39 2022 0.015/0.015 0.015/0.015 0.015/0.015 The problem is feasible. 0.015/0.015 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 Confluence Problem: 0.015/0.015 (VAR x) 0.015/0.015 (REPLACEMENT-MAP 0.015/0.015 (a) 0.015/0.015 (b) 0.015/0.015 (f 1) 0.015/0.015 (c) 0.015/0.015 (fSNonEmpty) 0.015/0.015 ) 0.015/0.015 (RULES 0.015/0.015 a -> c 0.015/0.015 b -> c 0.015/0.015 f(x) -> x | a ->* x 0.015/0.015 ) 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 0.015/0.015 Critical Pairs Processor: 0.015/0.015 -> Rules: 0.015/0.015 a -> c 0.015/0.015 b -> c 0.015/0.015 f(x) -> x | a ->* x 0.015/0.015 -> Vars: 0.015/0.015 "x" 0.015/0.015 0.015/0.015 -> Rlps: 0.015/0.015 crule: a -> c, id: 1, possubterms: a-> [] 0.015/0.015 crule: b -> c, id: 2, possubterms: b-> [] 0.015/0.015 crule: f(x) -> x | a ->* x, id: 3, possubterms: f(x)-> [] 0.015/0.015 0.015/0.015 -> Unifications: 0.015/0.015 0.015/0.015 0.015/0.015 -> Critical pairs info: 0.015/0.015 0.015/0.015 0.015/0.015 -> Problem conclusions: 0.015/0.015 Left linear, Right linear, Linear 0.015/0.015 Weakly orthogonal, Almost orthogonal, Orthogonal 0.015/0.015 CTRS Type: 1 0.015/0.015 Deterministic, Strongly deterministic 0.015/0.015 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.015/0.015 Maybe right-stable CTRS, Overlay CTRS 0.015/0.015 Maybe normal CTRS, Maybe almost normal CTRS 0.015/0.015 Maybe terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 0.015/0.015 Maybe level confluent 0.015/0.015 Maybe confluent 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 Clean Infeasible CCPs Processor: 0.015/0.015 0.015/0.015 0.015/0.015 -> Problem conclusions: 0.015/0.015 Left linear, Right linear, Linear 0.015/0.015 Weakly orthogonal, Almost orthogonal, Orthogonal 0.015/0.015 CTRS Type: 1 0.015/0.015 Deterministic, Strongly deterministic 0.015/0.015 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.015/0.015 Maybe right-stable CTRS, Overlay CTRS 0.015/0.015 Maybe normal CTRS, Maybe almost normal CTRS 0.015/0.015 Maybe terminating CTRS, Maybe operational terminating CTRS, Joinable CCPs 0.015/0.015 Maybe level confluent 0.015/0.015 Maybe confluent 0.015/0.015 0.015/0.015 Resulting CCPs: 0.015/0.015 No CCPs left 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 Operation Termination Processor: 0.015/0.015 Operational terminating? 0.015/0.015 YES 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 (VAR vu95NonEmpty x) 0.015/0.015 (RULES 0.015/0.015 a -> c 0.015/0.015 b -> c 0.015/0.015 f(x) -> x | a ->* x 0.015/0.015 ) 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 Valid CTRS Processor: 0.015/0.015 -> Rules: 0.015/0.015 a -> c 0.015/0.015 b -> c 0.015/0.015 f(x) -> x | a ->* x 0.015/0.015 -> The system is a deterministic 3-CTRS. 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 2D Dependency Pair Processor: 0.015/0.015 0.015/0.015 Conditional Termination Problem 1: 0.015/0.015 -> Pairs: 0.015/0.015 Empty 0.015/0.015 -> QPairs: 0.015/0.015 Empty 0.015/0.015 -> Rules: 0.015/0.015 a -> c 0.015/0.015 b -> c 0.015/0.015 f(x) -> x | a ->* x 0.015/0.015 0.015/0.015 Conditional Termination Problem 2: 0.015/0.015 -> Pairs: 0.015/0.015 F(x) -> A 0.015/0.015 -> QPairs: 0.015/0.015 Empty 0.015/0.015 -> Rules: 0.015/0.015 a -> c 0.015/0.015 b -> c 0.015/0.015 f(x) -> x | a ->* x 0.015/0.015 0.015/0.015 0.015/0.015 The problem is decomposed in 2 subproblems. 0.015/0.015 0.015/0.015 Problem 1.1: 0.015/0.015 0.015/0.015 SCC Processor: 0.015/0.015 -> Pairs: 0.015/0.015 Empty 0.015/0.015 -> QPairs: 0.015/0.015 Empty 0.015/0.015 -> Rules: 0.015/0.015 a -> c 0.015/0.015 b -> c 0.015/0.015 f(x) -> x | a ->* x 0.015/0.015 ->Strongly Connected Components: 0.015/0.015 There is no strongly connected component 0.015/0.015 0.015/0.015 The problem is finite. 0.015/0.015 0.015/0.015 Problem 1.2: 0.015/0.015 0.015/0.015 SCC Processor: 0.015/0.015 -> Pairs: 0.015/0.015 F(x) -> A 0.015/0.015 -> QPairs: 0.015/0.015 Empty 0.015/0.015 -> Rules: 0.015/0.015 a -> c 0.015/0.015 b -> c 0.015/0.015 f(x) -> x | a ->* x 0.015/0.015 ->Strongly Connected Components: 0.015/0.015 There is no strongly connected component 0.015/0.015 0.015/0.015 The problem is finite. 0.015/0.015 0.015/0.015 0.015/0.015 -> Problem conclusions: 0.015/0.015 Left linear, Right linear, Linear 0.015/0.015 Weakly orthogonal, Almost orthogonal, Orthogonal 0.015/0.015 CTRS Type: 1 0.015/0.015 Deterministic, Strongly deterministic 0.015/0.015 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.015/0.015 Maybe right-stable CTRS, Overlay CTRS 0.015/0.015 Maybe normal CTRS, Maybe almost normal CTRS 0.015/0.015 Terminating CTRS, Operational terminating CTRS, Joinable CCPs 0.015/0.015 Maybe level confluent 0.015/0.015 Confluent 0.015/0.015 0.015/0.015 The problem is joinable. 0.015/0.015 0.08user 0.03system 0:00.15elapsed 80%CPU (0avgtext+0avgdata 11252maxresident)k 0.015/0.015 8inputs+0outputs (0major+12214minor)pagefaults 0swaps