0.03/0.03 YES 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 0.03/0.03 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 Confluence Problem: 0.03/0.03 (VAR vNonEmpty) 0.03/0.03 (REPLACEMENT-MAP 0.03/0.03 (a) 0.03/0.03 (b) 0.03/0.03 (f 1) 0.03/0.03 (c) 0.03/0.03 (fSNonEmpty) 0.03/0.03 ) 0.03/0.03 (RULES 0.03/0.03 a -> c | b ->* c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 ) 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 0.03/0.03 Inlining of Conditions Processor [STERN17]: 0.03/0.03 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 Confluence Problem: 0.03/0.03 (VAR vNonEmpty) 0.03/0.03 (REPLACEMENT-MAP 0.03/0.03 (a) 0.03/0.03 (b) 0.03/0.03 (f 1) 0.03/0.03 (c) 0.03/0.03 (fSNonEmpty) 0.03/0.03 ) 0.03/0.03 (RULES 0.03/0.03 a -> c | b ->* c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 ) 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 0.03/0.03 Clean CTRS Processor: 0.03/0.03 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 Confluence Problem: 0.03/0.03 (REPLACEMENT-MAP 0.03/0.03 (a) 0.03/0.03 (b) 0.03/0.03 (f 1) 0.03/0.03 (c) 0.03/0.03 (fSNonEmpty) 0.03/0.03 ) 0.03/0.03 (RULES 0.03/0.03 a -> c | b ->* c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 ) 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 0.03/0.03 CRule InfChecker Info: 0.03/0.03 a -> c | b ->* c 0.03/0.03 Rule remains 0.03/0.03 Proof: 0.03/0.03 NO 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 0.03/0.03 Infeasibility Problem: 0.03/0.03 [(VAR vNonEmpty vNonEmpty) 0.03/0.03 (STRATEGY CONTEXTSENSITIVE 0.03/0.03 (a) 0.03/0.03 (b) 0.03/0.03 (f 1) 0.03/0.03 (c) 0.03/0.03 (fSNonEmpty) 0.03/0.03 ) 0.03/0.03 (RULES 0.03/0.03 a -> c | b ->* c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 )] 0.03/0.03 0.03/0.03 Infeasibility Conditions: 0.03/0.03 b ->* c 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 0.03/0.03 Obtaining a proof using Prover9: 0.03/0.03 0.03/0.03 -> Prover9 Output: 0.03/0.03 ============================== Prover9 =============================== 0.03/0.03 Prover9 (64) version 2009-11A, November 2009. 0.03/0.03 Process 77173 was started by ubuntu on ubuntu, 0.03/0.03 Wed Mar 9 10:17:59 2022 0.03/0.03 The command was "./prover9 -f /tmp/prover977163-0.in". 0.03/0.03 ============================== end of head =========================== 0.03/0.03 0.03/0.03 ============================== INPUT ================================= 0.03/0.03 0.03/0.03 % Reading from file /tmp/prover977163-0.in 0.03/0.03 0.03/0.03 assign(max_seconds,20). 0.03/0.03 0.03/0.03 formulas(assumptions). 0.03/0.03 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 0.03/0.03 ->_s0(b,c) # label(replacement). 0.03/0.03 ->*_s0(x,x) # label(reflexivity). 0.03/0.03 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 formulas(goals). 0.03/0.03 ->*_s0(b,c) # label(goal). 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 ============================== end of input ========================== 0.03/0.03 0.03/0.03 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.03/0.03 0.03/0.03 % Formulas that are not ordinary clauses: 0.03/0.03 1 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 0.03/0.03 2 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 0.03/0.03 3 ->*_s0(b,c) # label(goal) # label(non_clause) # label(goal). [goal]. 0.03/0.03 0.03/0.03 ============================== end of process non-clausal formulas === 0.03/0.03 0.03/0.03 ============================== PROCESS INITIAL CLAUSES =============== 0.03/0.03 0.03/0.03 % Clauses before input processing: 0.03/0.03 0.03/0.03 formulas(usable). 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 formulas(sos). 0.03/0.03 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 0.03/0.03 ->_s0(b,c) # label(replacement). [assumption]. 0.03/0.03 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.03/0.03 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(2)]. 0.03/0.03 -->*_s0(b,c) # label(goal). [deny(3)]. 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 formulas(demodulators). 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 ============================== PREDICATE ELIMINATION ================= 0.03/0.03 0.03/0.03 No predicates eliminated. 0.03/0.03 0.03/0.03 ============================== end predicate elimination ============= 0.03/0.03 0.03/0.03 Auto_denials: 0.03/0.03 % copying label goal to answer in negative clause 0.03/0.03 0.03/0.03 Term ordering decisions: 0.03/0.03 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 0.03/0.03 Function symbol precedence: function_order([ b, c, f ]). 0.03/0.03 After inverse_order: (no changes). 0.03/0.03 Unfolding symbols: (none). 0.03/0.03 0.03/0.03 Auto_inference settings: 0.03/0.03 % set(neg_binary_resolution). % (HNE depth_diff=-1) 0.03/0.03 % clear(ordered_res). % (HNE depth_diff=-1) 0.03/0.03 % set(ur_resolution). % (HNE depth_diff=-1) 0.03/0.03 % set(ur_resolution) -> set(pos_ur_resolution). 0.03/0.03 % set(ur_resolution) -> set(neg_ur_resolution). 0.03/0.03 0.03/0.03 Auto_process settings: (no changes). 0.03/0.03 0.03/0.03 kept: 4 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 0.03/0.03 kept: 5 ->_s0(b,c) # label(replacement). [assumption]. 0.03/0.03 kept: 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.03/0.03 kept: 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(2)]. 0.03/0.03 kept: 8 -->*_s0(b,c) # label(goal) # answer(goal). [deny(3)]. 0.03/0.03 0.03/0.03 ============================== end of process initial clauses ======== 0.03/0.03 0.03/0.03 ============================== CLAUSES FOR SEARCH ==================== 0.03/0.03 0.03/0.03 % Clauses after input processing: 0.03/0.03 0.03/0.03 formulas(usable). 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 formulas(sos). 0.03/0.03 4 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 0.03/0.03 5 ->_s0(b,c) # label(replacement). [assumption]. 0.03/0.03 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.03/0.03 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(2)]. 0.03/0.03 8 -->*_s0(b,c) # label(goal) # answer(goal). [deny(3)]. 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 formulas(demodulators). 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 ============================== end of clauses for search ============= 0.03/0.03 0.03/0.03 ============================== SEARCH ================================ 0.03/0.03 0.03/0.03 % Starting search at 0.00 seconds. 0.03/0.03 0.03/0.03 given #1 (I,wt=8): 4 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 0.03/0.03 0.03/0.03 given #2 (I,wt=3): 5 ->_s0(b,c) # label(replacement). [assumption]. 0.03/0.03 0.03/0.03 given #3 (I,wt=3): 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.03/0.03 0.03/0.03 given #4 (I,wt=9): 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(2)]. 0.03/0.03 0.03/0.03 ============================== PROOF ================================= 0.03/0.03 0.03/0.03 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 0.03/0.03 % Length of proof is 8. 0.03/0.03 % Level of proof is 3. 0.03/0.03 % Maximum clause weight is 9.000. 0.03/0.03 % Given clauses 4. 0.03/0.03 0.03/0.03 2 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 0.03/0.03 3 ->*_s0(b,c) # label(goal) # label(non_clause) # label(goal). [goal]. 0.03/0.03 5 ->_s0(b,c) # label(replacement). [assumption]. 0.03/0.03 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.03/0.03 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(2)]. 0.03/0.03 8 -->*_s0(b,c) # label(goal) # answer(goal). [deny(3)]. 0.03/0.03 10 ->*_s0(b,c). [ur(7,a,5,a,b,6,a)]. 0.03/0.03 11 $F # answer(goal). [resolve(10,a,8,a)]. 0.03/0.03 0.03/0.03 ============================== end of proof ========================== 0.03/0.03 0.03/0.03 ============================== STATISTICS ============================ 0.03/0.03 0.03/0.03 Given=4. Generated=7. Kept=7. proofs=1. 0.03/0.03 Usable=4. Sos=2. Demods=0. Limbo=0, Disabled=5. Hints=0. 0.03/0.03 Kept_by_rule=0, Deleted_by_rule=0. 0.03/0.03 Forward_subsumed=0. Back_subsumed=0. 0.03/0.03 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 0.03/0.03 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 0.03/0.03 Demod_attempts=0. Demod_rewrites=0. 0.03/0.03 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 0.03/0.03 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=3. 0.03/0.03 Megabytes=0.04. 0.03/0.03 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.03/0.03 0.03/0.03 ============================== end of statistics ===================== 0.03/0.03 0.03/0.03 ============================== end of search ========================= 0.03/0.03 0.03/0.03 THEOREM PROVED 0.03/0.03 0.03/0.03 Exiting with 1 proof. 0.03/0.03 0.03/0.03 Process 77173 exit (max_proofs) Wed Mar 9 10:17:59 2022 0.03/0.03 0.03/0.03 0.03/0.03 The problem is feasible. 0.03/0.03 0.03/0.03 0.03/0.03 CRule InfChecker Info: 0.03/0.03 b -> c 0.03/0.03 Rule remains 0.03/0.03 Proof: 0.03/0.03 NO_CONDS 0.03/0.03 0.03/0.03 CRule InfChecker Info: 0.03/0.03 f(b) -> f(a) 0.03/0.03 Rule remains 0.03/0.03 Proof: 0.03/0.03 NO_CONDS 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 Confluence Problem: 0.03/0.03 (REPLACEMENT-MAP 0.03/0.03 (a) 0.03/0.03 (b) 0.03/0.03 (f 1) 0.03/0.03 (c) 0.03/0.03 (fSNonEmpty) 0.03/0.03 ) 0.03/0.03 (RULES 0.03/0.03 a -> c | b ->* c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 ) 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 0.03/0.03 Critical Pairs Processor: 0.03/0.03 -> Rules: 0.03/0.03 a -> c | b ->* c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 -> Vars: 0.03/0.03 0.03/0.03 0.03/0.03 -> Rlps: 0.03/0.03 crule: a -> c | b ->* c, id: 1, possubterms: a-> [] 0.03/0.03 crule: b -> c, id: 2, possubterms: b-> [] 0.03/0.03 crule: f(b) -> f(a), id: 3, possubterms: f(b)-> [], b-> [1] 0.03/0.03 0.03/0.03 -> Unifications: 0.03/0.03 R3 unifies with R2 at p: [1], l: f(b), lp: b, conds: {}, sig: {}, l': b, r: f(a), r': c 0.03/0.03 0.03/0.03 -> Critical pairs info: 0.03/0.03 => Not trivial, Not overlay, NW1, N1 0.03/0.03 0.03/0.03 -> Problem conclusions: 0.03/0.03 Left linear, Right linear, Linear 0.03/0.03 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.03/0.03 CTRS Type: 1 0.03/0.03 Deterministic, Strongly deterministic 0.03/0.03 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.03/0.03 Maybe right-stable CTRS, Not overlay CTRS 0.03/0.03 Maybe normal CTRS, Maybe almost normal CTRS 0.03/0.03 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 0.03/0.03 Maybe level confluent 0.03/0.03 Maybe confluent 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 Clean Infeasible CCPs Processor: 0.03/0.03 Num of CPIs: 1 0.03/0.03 Timeout: 60 0.03/0.03 Timeout for each infeasibility problem: 60 s 0.03/0.03 => Not trivial, Not overlay, NW1, N1 0.03/0.03 0.03/0.03 Proof: 0.03/0.03 NO_CONDS 0.03/0.03 0.03/0.03 0.03/0.03 0.03/0.03 -> Problem conclusions: 0.03/0.03 Left linear, Right linear, Linear 0.03/0.03 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.03/0.03 CTRS Type: 1 0.03/0.03 Deterministic, Strongly deterministic 0.03/0.03 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.03/0.03 Maybe right-stable CTRS, Not overlay CTRS 0.03/0.03 Maybe normal CTRS, Maybe almost normal CTRS 0.03/0.03 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 0.03/0.03 Maybe level confluent 0.03/0.03 Maybe confluent 0.03/0.03 0.03/0.03 Resulting CCPs: 0.03/0.03 => Not trivial, Not overlay, NW1, N1 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 Underlying TRS Termination Processor: 0.03/0.03 0.03/0.03 Resulting Underlying TRS: 0.03/0.03 (STRATEGY CONTEXTSENSITIVE 0.03/0.03 (a) 0.03/0.03 (b) 0.03/0.03 (f 1) 0.03/0.03 (c) 0.03/0.03 ) 0.03/0.03 (RULES 0.03/0.03 a -> c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 ) 0.03/0.03 Underlying TRS terminating? 0.03/0.03 YES 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 0.03/0.03 (VAR vu95NonEmpty) 0.03/0.03 (RULES 0.03/0.03 a -> c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 ) 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 0.03/0.03 Dependency Pairs Processor: 0.03/0.03 -> Pairs: 0.03/0.03 F(b) -> A 0.03/0.03 F(b) -> F(a) 0.03/0.03 -> Rules: 0.03/0.03 a -> c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 0.03/0.03 SCC Processor: 0.03/0.03 -> Pairs: 0.03/0.03 F(b) -> A 0.03/0.03 F(b) -> F(a) 0.03/0.03 -> Rules: 0.03/0.03 a -> c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 ->Strongly Connected Components: 0.03/0.03 ->->Cycle: 0.03/0.03 ->->-> Pairs: 0.03/0.03 F(b) -> F(a) 0.03/0.03 ->->-> Rules: 0.03/0.03 a -> c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 0.03/0.03 Reduction Pair Processor: 0.03/0.03 -> Pairs: 0.03/0.03 F(b) -> F(a) 0.03/0.03 -> Rules: 0.03/0.03 a -> c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 -> Usable rules: 0.03/0.03 a -> c 0.03/0.03 ->Interpretation type: 0.03/0.03 Linear 0.03/0.03 ->Coefficients: 0.03/0.03 Natural Numbers 0.03/0.03 ->Dimension: 0.03/0.03 1 0.03/0.03 ->Bound: 0.03/0.03 2 0.03/0.03 ->Interpretation: 0.03/0.03 0.03/0.03 [a] = 1 0.03/0.03 [b] = 2 0.03/0.03 [f](X) = 0 0.03/0.03 [c] = 1 0.03/0.03 [fSNonEmpty] = 0 0.03/0.03 [A] = 0 0.03/0.03 [F](X) = 2.X 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 0.03/0.03 SCC Processor: 0.03/0.03 -> Pairs: 0.03/0.03 Empty 0.03/0.03 -> Rules: 0.03/0.03 a -> c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 ->Strongly Connected Components: 0.03/0.03 There is no strongly connected component 0.03/0.03 0.03/0.03 The problem is finite. 0.03/0.03 0.03/0.03 0.03/0.03 -> Problem conclusions: 0.03/0.03 Left linear, Right linear, Linear 0.03/0.03 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.03/0.03 CTRS Type: 1 0.03/0.03 Deterministic, Strongly deterministic 0.03/0.03 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.03/0.03 Maybe right-stable CTRS, Not overlay CTRS 0.03/0.03 Maybe normal CTRS, Maybe almost normal CTRS 0.03/0.03 Terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 0.03/0.03 Maybe level confluent 0.03/0.03 Maybe confluent 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 Confluence Problem: 0.03/0.03 (REPLACEMENT-MAP 0.03/0.03 (a) 0.03/0.03 (b) 0.03/0.03 (f 1) 0.03/0.03 (c) 0.03/0.03 (fSNonEmpty) 0.03/0.03 ) 0.03/0.03 (RULES 0.03/0.03 a -> c | b ->* c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 ) 0.03/0.03 Critical Pairs: 0.03/0.03 => Not trivial, Not overlay, NW1, N1 0.03/0.03 Terminating:YES 0.03/0.03 0.03/0.03 -> Problem conclusions: 0.03/0.03 Left linear, Right linear, Linear 0.03/0.03 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.03/0.03 CTRS Type: 1 0.03/0.03 Deterministic, Strongly deterministic 0.03/0.03 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.03/0.03 Maybe right-stable CTRS, Not overlay CTRS 0.03/0.03 Maybe normal CTRS, Maybe almost normal CTRS 0.03/0.03 Terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 0.03/0.03 Maybe level confluent 0.03/0.03 Maybe confluent 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 0.03/0.03 Right Stable Processor: 0.03/0.03 Right-stable CTRS 0.03/0.03 Justification: 0.03/0.03 0.03/0.03 -> Term right-stability conditions: 0.03/0.03 Term: c 0.03/0.03 Right-stable term 0.03/0.03 Linear constructor form 0.03/0.03 Don't know if term is a ground normal form (not needed) 0.03/0.03 Right-stability condition achieved 0.03/0.03 A call to InfChecker wasn't needed 0.03/0.03 0.03/0.03 -> Problem conclusions: 0.03/0.03 Left linear, Right linear, Linear 0.03/0.03 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.03/0.03 CTRS Type: 1 0.03/0.03 Deterministic, Strongly deterministic 0.03/0.03 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.03/0.03 Right-stable CTRS, Not overlay CTRS 0.03/0.03 Maybe normal CTRS, Almost normal CTRS 0.03/0.03 Terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 0.03/0.03 Maybe level confluent 0.03/0.03 Maybe confluent 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 Confluence Problem: 0.03/0.03 (REPLACEMENT-MAP 0.03/0.03 (a) 0.03/0.03 (b) 0.03/0.03 (f 1) 0.03/0.03 (c) 0.03/0.03 (fSNonEmpty) 0.03/0.03 ) 0.03/0.03 (RULES 0.03/0.03 a -> c | b ->* c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 ) 0.03/0.03 Critical Pairs: 0.03/0.03 => Not trivial, Not overlay, NW1, N1 0.03/0.03 Terminating:YES 0.03/0.03 0.03/0.03 -> Problem conclusions: 0.03/0.03 Left linear, Right linear, Linear 0.03/0.03 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.03/0.03 CTRS Type: 1 0.03/0.03 Deterministic, Strongly deterministic 0.03/0.03 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.03/0.03 Right-stable CTRS, Not overlay CTRS 0.03/0.03 Maybe normal CTRS, Almost normal CTRS 0.03/0.03 Terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 0.03/0.03 Maybe level confluent 0.03/0.03 Maybe confluent 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 0.03/0.03 Normal RConds Processor: 0.03/0.03 YES 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 0.03/0.03 Infeasibility Problem: 0.03/0.03 [(VAR vNonEmpty vNonEmpty z) 0.03/0.03 (STRATEGY CONTEXTSENSITIVE 0.03/0.03 (a) 0.03/0.03 (b) 0.03/0.03 (f 1) 0.03/0.03 (c) 0.03/0.03 (fSNonEmpty) 0.03/0.03 ) 0.03/0.03 (RULES 0.03/0.03 a -> c | b ->* c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 )] 0.03/0.03 0.03/0.03 Infeasibility Conditions: 0.03/0.03 c -> z 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 0.03/0.03 Obtaining a model using Mace4: 0.03/0.03 0.03/0.03 -> Usable Rules: 0.03/0.03 Empty 0.03/0.03 0.03/0.03 -> Mace4 Output: 0.03/0.03 ============================== Mace4 ================================= 0.03/0.03 Mace4 (64) version 2009-11A, November 2009. 0.03/0.03 Process 77315 was started by ubuntu on ubuntu, 0.03/0.03 Wed Mar 9 10:17:59 2022 0.03/0.03 The command was "./mace4 -c -f /tmp/mace477300-2.in". 0.03/0.03 ============================== end of head =========================== 0.03/0.03 0.03/0.03 ============================== INPUT ================================= 0.03/0.03 0.03/0.03 % Reading from file /tmp/mace477300-2.in 0.03/0.03 0.03/0.03 assign(max_seconds,13). 0.03/0.03 0.03/0.03 formulas(assumptions). 0.03/0.03 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 formulas(goals). 0.03/0.03 (exists x2 ->(c,x2)) # label(goal). 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 ============================== end of input ========================== 0.03/0.03 0.03/0.03 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.03/0.03 0.03/0.03 % Formulas that are not ordinary clauses: 0.03/0.03 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 0.03/0.03 2 (exists x2 ->(c,x2)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.03/0.03 0.03/0.03 ============================== end of process non-clausal formulas === 0.03/0.03 0.03/0.03 ============================== CLAUSES FOR SEARCH ==================== 0.03/0.03 0.03/0.03 formulas(mace4_clauses). 0.03/0.03 -->(x,y) | ->(f(x),f(y)) # label(congruence). 0.03/0.03 -->(c,x) # label(goal). 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 ============================== end of clauses for search ============= 0.03/0.03 0.03/0.03 % There are no natural numbers in the input. 0.03/0.03 0.03/0.03 ============================== DOMAIN SIZE 2 ========================= 0.03/0.03 0.03/0.03 ============================== MODEL ================================= 0.03/0.03 0.03/0.03 interpretation( 2, [number=1, seconds=0], [ 0.03/0.03 0.03/0.03 function(c, [ 0 ]), 0.03/0.03 0.03/0.03 function(f(_), [ 0, 0 ]), 0.03/0.03 0.03/0.03 relation(->(_,_), [ 0.03/0.03 0, 0, 0.03/0.03 0, 0 ]) 0.03/0.03 ]). 0.03/0.03 0.03/0.03 ============================== end of model ========================== 0.03/0.03 0.03/0.03 ============================== STATISTICS ============================ 0.03/0.03 0.03/0.03 For domain size 2. 0.03/0.03 0.03/0.03 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 0.03/0.03 Ground clauses: seen=6, kept=6. 0.03/0.03 Selections=3, assignments=3, propagations=4, current_models=1. 0.03/0.03 Rewrite_terms=10, rewrite_bools=8, indexes=2. 0.03/0.03 Rules_from_neg_clauses=0, cross_offs=0. 0.03/0.03 0.03/0.03 ============================== end of statistics ===================== 0.03/0.03 0.03/0.03 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.03/0.03 0.03/0.03 Exiting with 1 model. 0.03/0.03 0.03/0.03 Process 77315 exit (max_models) Wed Mar 9 10:17:59 2022 0.03/0.03 The process finished Wed Mar 9 10:17:59 2022 0.03/0.03 0.03/0.03 0.03/0.03 Mace4 cooked interpretation: 0.03/0.03 0.03/0.03 0.03/0.03 0.03/0.03 The problem is infeasible. 0.03/0.03 0.03/0.03 0.03/0.03 -> Problem conclusions: 0.03/0.03 Left linear, Right linear, Linear 0.03/0.03 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.03/0.03 CTRS Type: 1 0.03/0.03 Deterministic, Strongly deterministic 0.03/0.03 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.03/0.03 Right-stable CTRS, Not overlay CTRS 0.03/0.03 Normal CTRS, Almost normal CTRS 0.03/0.03 Terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 0.03/0.03 Maybe level confluent 0.03/0.03 Maybe confluent 0.03/0.03 0.03/0.03 Problem 1: 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 Confluence Problem: 0.03/0.03 (REPLACEMENT-MAP 0.03/0.03 (a) 0.03/0.03 (b) 0.03/0.03 (f 1) 0.03/0.03 (c) 0.03/0.03 (fSNonEmpty) 0.03/0.03 ) 0.03/0.03 (RULES 0.03/0.03 a -> c | b ->* c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 ) 0.03/0.03 Critical Pairs: 0.03/0.03 => Not trivial, Not overlay, NW1, N1 0.03/0.03 Terminating:YES 0.03/0.03 0.03/0.03 -> Problem conclusions: 0.03/0.03 Left linear, Right linear, Linear 0.03/0.03 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.03/0.03 CTRS Type: 1 0.03/0.03 Deterministic, Strongly deterministic 0.03/0.03 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.03/0.03 Right-stable CTRS, Not overlay CTRS 0.03/0.03 Normal CTRS, Almost normal CTRS 0.03/0.03 Terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 0.03/0.03 Maybe level confluent 0.03/0.03 Maybe confluent 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 0.03/0.03 Conditional Critical Pairs Distributor Processor 0.03/0.03 Problem 1.1: 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 Confluence Problem: 0.03/0.03 (REPLACEMENT-MAP 0.03/0.03 (a) 0.03/0.03 (b) 0.03/0.03 (f 1) 0.03/0.03 (c) 0.03/0.03 (fSNonEmpty) 0.03/0.03 ) 0.03/0.03 (RULES 0.03/0.03 a -> c | b ->* c 0.03/0.03 b -> c 0.03/0.03 f(b) -> f(a) 0.03/0.03 ) 0.03/0.03 Critical Pairs: 0.03/0.03 => Not trivial, Not overlay, NW1, N1 0.03/0.03 Terminating:YES 0.03/0.03 0.03/0.03 -> Problem conclusions: 0.03/0.03 Left linear, Right linear, Linear 0.03/0.03 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.03/0.03 CTRS Type: 1 0.03/0.03 Deterministic, Strongly deterministic 0.03/0.03 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.03/0.03 Right-stable CTRS, Not overlay CTRS 0.03/0.03 Normal CTRS, Almost normal CTRS 0.03/0.03 Terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 0.03/0.03 Maybe level confluent 0.03/0.03 Maybe confluent 0.03/0.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.03/0.03 0.03/0.03 Prover9 CCP Convergence Checker Processor: 0.03/0.03 Proof: 0.03/0.03 ============================== Prover9 =============================== 0.03/0.03 Prover9 (64) version 2009-11A, November 2009. 0.03/0.03 Process 77328 was started by ubuntu on ubuntu, 0.03/0.03 Wed Mar 9 10:18:00 2022 0.03/0.03 The command was "./prover9 -t 60 -f /tmp/prover977111-20.in". 0.03/0.03 ============================== end of head =========================== 0.03/0.03 0.03/0.03 ============================== INPUT ================================= 0.03/0.03 0.03/0.03 % Reading from file /tmp/prover977111-20.in 0.03/0.03 0.03/0.03 0.03/0.03 formulas(sos). 0.03/0.03 reds(x,x). 0.03/0.03 red(x,y) & reds(y,z) -> reds(x,z). 0.03/0.03 red(x,y) -> red(f(x),f(y)). 0.03/0.03 reds(b,c) -> red(a,c). 0.03/0.03 red(b,c). 0.03/0.03 red(f(b),f(a)). 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 formulas(goals). 0.03/0.03 (exists z (reds(f(c),z) & reds(f(a),z))). 0.03/0.03 end_of_list. 0.03/0.03 assign(max_megs,-1). 0.03/0.03 set(quiet). 0.03/0.03 0.03/0.03 ============================== end of input ========================== 0.03/0.03 0.03/0.03 % From the command line: assign(max_seconds, 60). 0.03/0.03 0.03/0.03 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.03/0.03 0.03/0.03 % Formulas that are not ordinary clauses: 0.03/0.03 1 red(x,y) & reds(y,z) -> reds(x,z) # label(non_clause). [assumption]. 0.03/0.03 2 red(x,y) -> red(f(x),f(y)) # label(non_clause). [assumption]. 0.03/0.03 3 reds(b,c) -> red(a,c) # label(non_clause). [assumption]. 0.03/0.03 4 (exists z (reds(f(c),z) & reds(f(a),z))) # label(non_clause) # label(goal). [goal]. 0.03/0.03 0.03/0.03 ============================== end of process non-clausal formulas === 0.03/0.03 0.03/0.03 ============================== PROCESS INITIAL CLAUSES =============== 0.03/0.03 0.03/0.03 % Clauses before input processing: 0.03/0.03 0.03/0.03 formulas(usable). 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 formulas(sos). 0.03/0.03 reds(x,x). [assumption]. 0.03/0.03 -red(x,y) | -reds(y,z) | reds(x,z). [clausify(1)]. 0.03/0.03 -red(x,y) | red(f(x),f(y)). [clausify(2)]. 0.03/0.03 -reds(b,c) | red(a,c). [clausify(3)]. 0.03/0.03 red(b,c). [assumption]. 0.03/0.03 red(f(b),f(a)). [assumption]. 0.03/0.03 -reds(f(c),x) | -reds(f(a),x). [deny(4)]. 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 formulas(demodulators). 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 ============================== PREDICATE ELIMINATION ================= 0.03/0.03 0.03/0.03 ============================== end predicate elimination ============= 0.03/0.03 0.03/0.03 Auto_denials: (no changes). 0.03/0.03 0.03/0.03 Term ordering decisions: 0.03/0.03 0.03/0.03 kept: 5 reds(x,x). [assumption]. 0.03/0.03 kept: 6 -red(x,y) | -reds(y,z) | reds(x,z). [clausify(1)]. 0.03/0.03 kept: 7 -red(x,y) | red(f(x),f(y)). [clausify(2)]. 0.03/0.03 kept: 8 -reds(b,c) | red(a,c). [clausify(3)]. 0.03/0.03 kept: 9 red(b,c). [assumption]. 0.03/0.03 kept: 10 red(f(b),f(a)). [assumption]. 0.03/0.03 kept: 11 -reds(f(c),x) | -reds(f(a),x). [deny(4)]. 0.03/0.03 0.03/0.03 ============================== end of process initial clauses ======== 0.03/0.03 0.03/0.03 ============================== CLAUSES FOR SEARCH ==================== 0.03/0.03 0.03/0.03 % Clauses after input processing: 0.03/0.03 0.03/0.03 formulas(usable). 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 formulas(sos). 0.03/0.03 5 reds(x,x). [assumption]. 0.03/0.03 6 -red(x,y) | -reds(y,z) | reds(x,z). [clausify(1)]. 0.03/0.03 7 -red(x,y) | red(f(x),f(y)). [clausify(2)]. 0.03/0.03 8 -reds(b,c) | red(a,c). [clausify(3)]. 0.03/0.03 9 red(b,c). [assumption]. 0.03/0.03 10 red(f(b),f(a)). [assumption]. 0.03/0.03 11 -reds(f(c),x) | -reds(f(a),x). [deny(4)]. 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 formulas(demodulators). 0.03/0.03 end_of_list. 0.03/0.03 0.03/0.03 ============================== end of clauses for search ============= 0.03/0.03 0.03/0.03 ============================== SEARCH ================================ 0.03/0.03 0.03/0.03 % Starting search at 0.00 seconds. 0.03/0.03 0.03/0.03 given #1 (I,wt=3): 5 reds(x,x). [assumption]. 0.03/0.03 0.03/0.03 given #2 (I,wt=9): 6 -red(x,y) | -reds(y,z) | reds(x,z). [clausify(1)]. 0.03/0.03 0.03/0.03 given #3 (I,wt=8): 7 -red(x,y) | red(f(x),f(y)). [clausify(2)]. 0.03/0.03 0.03/0.03 given #4 (I,wt=6): 8 -reds(b,c) | red(a,c). [clausify(3)]. 0.03/0.03 0.03/0.03 given #5 (I,wt=3): 9 red(b,c). [assumption]. 0.03/0.03 0.03/0.03 given #6 (I,wt=5): 10 red(f(b),f(a)). [assumption]. 0.03/0.03 0.03/0.03 given #7 (I,wt=8): 11 -reds(f(c),x) | -reds(f(a),x). [deny(4)]. 0.03/0.03 0.03/0.03 given #8 (A,wt=5): 12 red(f(b),f(c)). [ur(7,a,9,a)]. 0.03/0.03 0.03/0.03 given #9 (F,wt=5): 18 -reds(f(a),f(c)). [resolve(11,a,5,a)]. 0.03/0.03 0.03/0.03 given #10 (F,wt=5): 20 -reds(f(c),f(a)). [resolve(11,b,5,a)]. 0.03/0.03 0.03/0.03 given #11 (F,wt=5): 24 -red(f(a),f(c)). [ur(6,b,5,a,c,18,a)]. 0.03/0.03 0.03/0.03 ============================== PROOF ================================= 0.03/0.03 0.03/0.03 % Proof 1 at 0.00 (+ 0.00) seconds. 0.03/0.03 % Length of proof is 15. 0.03/0.03 % Level of proof is 4. 0.03/0.03 % Maximum clause weight is 9.000. 0.03/0.03 % Given clauses 11. 0.03/0.03 0.03/0.03 1 red(x,y) & reds(y,z) -> reds(x,z) # label(non_clause). [assumption]. 0.03/0.03 2 red(x,y) -> red(f(x),f(y)) # label(non_clause). [assumption]. 0.03/0.03 3 reds(b,c) -> red(a,c) # label(non_clause). [assumption]. 0.03/0.03 4 (exists z (reds(f(c),z) & reds(f(a),z))) # label(non_clause) # label(goal). [goal]. 0.03/0.03 5 reds(x,x). [assumption]. 0.03/0.03 6 -red(x,y) | -reds(y,z) | reds(x,z). [clausify(1)]. 0.03/0.03 7 -red(x,y) | red(f(x),f(y)). [clausify(2)]. 0.03/0.03 8 -reds(b,c) | red(a,c). [clausify(3)]. 0.03/0.03 9 red(b,c). [assumption]. 0.03/0.03 11 -reds(f(c),x) | -reds(f(a),x). [deny(4)]. 0.03/0.03 13 reds(b,c). [ur(6,a,9,a,b,5,a)]. 0.03/0.03 14 red(a,c). [back_unit_del(8),unit_del(a,13)]. 0.03/0.03 18 -reds(f(a),f(c)). [resolve(11,a,5,a)]. 0.03/0.03 24 -red(f(a),f(c)). [ur(6,b,5,a,c,18,a)]. 0.03/0.03 27 $F. [resolve(24,a,7,b),unit_del(a,14)]. 0.03/0.03 0.03/0.03 ============================== end of proof ========================== 0.03/0.03 0.03/0.03 ============================== STATISTICS ============================ 0.03/0.03 0.03/0.03 Given=11. Generated=25. Kept=22. proofs=1. 0.03/0.03 Usable=10. Sos=11. Demods=0. Limbo=0, Disabled=8. Hints=0. 0.03/0.03 Kept_by_rule=0, Deleted_by_rule=0. 0.03/0.03 Forward_subsumed=2. Back_subsumed=0. 0.03/0.03 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 0.03/0.03 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=1. 0.03/0.03 Demod_attempts=0. Demod_rewrites=0. 0.03/0.03 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 0.03/0.03 Nonunit_fsub_feature_tests=4. Nonunit_bsub_feature_tests=29. 0.03/0.03 Megabytes=0.06. 0.03/0.03 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.03/0.03 0.03/0.03 ============================== end of statistics ===================== 0.03/0.03 0.03/0.03 ============================== end of search ========================= 0.03/0.03 0.03/0.03 THEOREM PROVED 0.03/0.03 0.03/0.03 Exiting with 1 proof. 0.03/0.03 0.03/0.03 Process 77328 exit (max_proofs) Wed Mar 9 10:18:00 2022 0.03/0.03 0.03/0.03 0.03/0.03 The problem is joinable. 0.03/0.03 0.19user 0.06system 0:00.30elapsed 85%CPU (0avgtext+0avgdata 14928maxresident)k 0.03/0.03 8inputs+0outputs (0major+18751minor)pagefaults 0swaps