0.007/0.007 YES 0.007/0.007 0.007/0.007 Problem 1: 0.007/0.007 0.007/0.007 0.007/0.007 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.007/0.007 Confluence Problem: 0.007/0.007 (VAR vNonEmpty:S x:S y:S) 0.007/0.007 (STRATEGY CONTEXTSENSITIVE 0.007/0.007 (a) 0.007/0.007 (f 1 2) 0.007/0.007 (b) 0.007/0.007 (c) 0.007/0.007 (fSNonEmpty) 0.007/0.007 ) 0.007/0.007 (RULES 0.007/0.007 a -> b 0.007/0.007 f(c,x:S) -> a | f(x:S,a) ->*<- f(b,b) 0.007/0.007 f(y:S,y:S) -> b 0.007/0.007 ) 0.007/0.007 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.007/0.007 0.007/0.007 0.007/0.007 Problem 1: 0.007/0.007 0.007/0.007 Inlining of Conditions Processor [STERN17]: 0.007/0.007 0.007/0.007 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.007/0.007 Confluence Problem: 0.007/0.007 (VAR vNonEmpty:S x:S y:S) 0.007/0.007 (STRATEGY CONTEXTSENSITIVE 0.007/0.007 (a) 0.007/0.007 (f 1 2) 0.007/0.007 (b) 0.007/0.007 (c) 0.007/0.007 (fSNonEmpty) 0.007/0.007 ) 0.007/0.007 (RULES 0.007/0.007 a -> b 0.007/0.007 f(c,x:S) -> a | f(x:S,a) ->*<- f(b,b) 0.007/0.007 f(y:S,y:S) -> b 0.007/0.007 ) 0.007/0.007 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.007/0.007 0.007/0.007 0.007/0.007 Problem 1: 0.007/0.007 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.007/0.007 Confluence Problem: 0.007/0.007 (VAR vNonEmpty:S x:S y:S) 0.007/0.007 (STRATEGY CONTEXTSENSITIVE 0.007/0.007 (a) 0.007/0.007 (f 1 2) 0.007/0.007 (b) 0.007/0.007 (c) 0.007/0.007 (fSNonEmpty) 0.007/0.007 ) 0.007/0.007 (RULES 0.007/0.007 a -> b 0.007/0.007 f(c,x:S) -> a | f(x:S,a) ->*<- f(b,b) 0.007/0.007 f(y:S,y:S) -> b 0.007/0.007 ) 0.007/0.007 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.007/0.007 0.007/0.007 Critical Pairs Processor: 0.007/0.007 -> Rules: 0.007/0.007 a -> b 0.007/0.007 f(c,x:S) -> a | f(x:S,a) ->*<- f(b,b) 0.007/0.007 f(y:S,y:S) -> b 0.007/0.007 -> Vars: 0.007/0.007 "x", "y" 0.007/0.007 -> FVars: 0.007/0.007 "x3", "x4" 0.007/0.007 -> PVars: 0.007/0.007 "x": ["x3"], "y": ["x4"] 0.007/0.007 0.007/0.007 -> Rlps: 0.007/0.007 crule: a -> b, id: 1, possubterms: a-> [] 0.007/0.007 crule: f(c,x3:S) -> a | f(x3:S,a) ->*<- f(b,b), id: 2, possubterms: f(c,x3:S)-> [], c-> [1] 0.007/0.007 crule: f(x4:S,x4:S) -> b, id: 3, possubterms: f(x4:S,x4:S)-> [] 0.007/0.007 0.007/0.007 -> Unifications: 0.007/0.007 R3 unifies with R2 at p: [], l: f(x4:S,x4:S), lp: f(x4:S,x4:S), conds: {f(x:S,a) ->*<- f(b,b)}, sig: {x:S -> c,x4:S -> c}, l': f(c,x:S), r: b, r': a 0.007/0.007 0.007/0.007 -> Critical pairs info: 0.007/0.007 | f(c,a) ->*<- f(b,b) => Not trivial, Overlay, N1 0.007/0.007 0.007/0.007 -> Problem conclusions: 0.007/0.007 Not left linear, Right linear, Not linear 0.007/0.007 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.007/0.007 CTRS Type: 1 0.007/0.007 Deterministic, Strongly deterministic 0.007/0.007 Not oriented CTRS, Properly oriented CTRS, Join CTRS 0.007/0.007 Maybe right-stable CTRS, Overlay CTRS 0.007/0.007 Normal CTRS, Almost normal CTRS 0.007/0.007 Maybe terminating CTRS, Maybe joinable CCPs 0.007/0.007 Maybe level confluent 0.007/0.007 Maybe confluent 0.007/0.007 0.007/0.007 Problem 1: 0.007/0.007 Underlying TRS Termination Processor: 0.007/0.007 0.007/0.007 Resulting Underlying TRS: 0.007/0.007 (VAR vNonEmpty:S x:S y:S) 0.007/0.007 (STRATEGY CONTEXTSENSITIVE 0.007/0.007 (a) 0.007/0.007 (f 1 2) 0.007/0.007 (b) 0.007/0.007 ) 0.007/0.007 (RULES 0.007/0.007 a -> b 0.007/0.007 f(c,x:S) -> a 0.007/0.007 f(y:S,y:S) -> b 0.007/0.007 ) 0.007/0.007 Underlying TRS terminating? 0.007/0.007 YES 0.007/0.007 0.007/0.007 Problem 1: 0.007/0.007 0.007/0.007 (VAR vu95NonEmpty:S vNonEmptyu58S:S xu58S:S yu58S:S) 0.007/0.007 (RULES 0.007/0.007 a -> b 0.007/0.007 f(c,xu58S:S) -> a 0.007/0.007 f(yu58S:S,yu58S:S) -> b 0.007/0.007 ) 0.007/0.007 0.007/0.007 Problem 1: 0.007/0.007 0.007/0.007 Dependency Pairs Processor: 0.007/0.007 -> Pairs: 0.007/0.007 F(c,xu58S:S) -> A 0.007/0.007 -> Rules: 0.007/0.007 a -> b 0.007/0.007 f(c,xu58S:S) -> a 0.007/0.007 f(yu58S:S,yu58S:S) -> b 0.007/0.007 0.007/0.007 Problem 1: 0.007/0.007 0.007/0.007 SCC Processor: 0.007/0.007 -> Pairs: 0.007/0.007 F(c,xu58S:S) -> A 0.007/0.007 -> Rules: 0.007/0.007 a -> b 0.007/0.007 f(c,xu58S:S) -> a 0.007/0.007 f(yu58S:S,yu58S:S) -> b 0.007/0.007 ->Strongly Connected Components: 0.007/0.007 There is no strongly connected component 0.007/0.007 0.007/0.007 The problem is finite. 0.007/0.007 0.007/0.007 0.007/0.007 -> Problem conclusions: 0.007/0.007 Not left linear, Right linear, Not linear 0.007/0.007 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.007/0.007 CTRS Type: 1 0.007/0.007 Deterministic, Strongly deterministic 0.007/0.007 Not oriented CTRS, Properly oriented CTRS, Join CTRS 0.007/0.007 Maybe right-stable CTRS, Overlay CTRS 0.007/0.007 Normal CTRS, Almost normal CTRS 0.007/0.007 Terminating CTRS, Maybe joinable CCPs 0.007/0.007 Maybe level confluent 0.007/0.007 Maybe confluent 0.007/0.007 0.007/0.007 Problem 1: 0.007/0.007 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.007/0.007 Confluence Problem: 0.007/0.007 (VAR vNonEmpty:S x:S y:S) 0.007/0.007 (STRATEGY CONTEXTSENSITIVE 0.007/0.007 (a) 0.007/0.007 (f 1 2) 0.007/0.007 (b) 0.007/0.007 (c) 0.007/0.007 (fSNonEmpty) 0.007/0.007 ) 0.007/0.007 (RULES 0.007/0.007 a -> b 0.007/0.007 f(c,x:S) -> a | f(x:S,a) ->*<- f(b,b) 0.007/0.007 f(y:S,y:S) -> b 0.007/0.007 ) 0.007/0.007 Critical Pairs: 0.007/0.007 | f(c,a) ->*<- f(b,b) => Not trivial, Overlay, N1 0.007/0.007 Terminating:YES 0.007/0.007 0.007/0.007 -> Problem conclusions: 0.007/0.007 Not left linear, Right linear, Not linear 0.007/0.007 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.007/0.007 CTRS Type: 1 0.007/0.007 Deterministic, Strongly deterministic 0.007/0.007 Not oriented CTRS, Properly oriented CTRS, Join CTRS 0.007/0.007 Maybe right-stable CTRS, Overlay CTRS 0.007/0.007 Normal CTRS, Almost normal CTRS 0.007/0.007 Terminating CTRS, Maybe joinable CCPs 0.007/0.007 Maybe level confluent 0.007/0.007 Maybe confluent 0.007/0.007 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.007/0.007 0.007/0.007 Right Stable Processor: 0.007/0.007 Not right-stable CTRS 0.007/0.007 Justification: 0.007/0.007 0.007/0.007 -> Term right-stability conditions: 0.007/0.007 Term: f(b,b) 0.007/0.007 Not right-stable term 0.007/0.007 Not a linear constructor form 0.007/0.007 Not a ground normal form 0.007/0.007 Right-stability condition achieved 0.007/0.007 InfChecker proof: 0.007/0.007 NO 0.007/0.007 0.007/0.007 Problem 1: 0.007/0.007 0.007/0.007 Infeasibility Problem: 0.007/0.007 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S y:S:S z:S) 0.007/0.007 (STRATEGY CONTEXTSENSITIVE 0.007/0.007 (a) 0.007/0.007 (f 1 2) 0.007/0.007 (b) 0.007/0.007 (c) 0.007/0.007 (fSNonEmpty) 0.007/0.007 ) 0.007/0.007 (RULES 0.007/0.007 a -> b 0.007/0.007 f(c,x:S:S) -> a | f(x:S:S,a) ->*<- f(b,b) 0.007/0.007 f(y:S:S,y:S:S) -> b 0.007/0.007 )] 0.007/0.007 0.007/0.007 Infeasibility Conditions: 0.007/0.007 f(b,b) -> z:S 0.007/0.007 0.007/0.007 Problem 1: 0.007/0.007 0.007/0.007 Obtaining a proof using Prover9: 0.007/0.007 0.007/0.007 -> Prover9 Output: 0.007/0.007 ============================== Prover9 =============================== 0.007/0.007 Prover9 (64) version 2009-11A, November 2009. 0.007/0.007 Process 29914 was started by ubuntu on ubuntu, 0.007/0.007 Wed Jul 14 09:34:59 2021 0.007/0.007 The command was "./prover9 -f /tmp/prover929905-0.in". 0.007/0.007 ============================== end of head =========================== 0.007/0.007 0.007/0.007 ============================== INPUT ================================= 0.007/0.007 0.007/0.007 % Reading from file /tmp/prover929905-0.in 0.007/0.007 0.007/0.007 assign(max_seconds,20). 0.007/0.007 0.007/0.007 formulas(assumptions). 0.007/0.007 ->*_s0(x,x) # label(reflexivity). 0.007/0.007 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 0.007/0.007 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 0.007/0.007 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 0.007/0.007 ->_s0(a,b) # label(replacement). 0.007/0.007 ->*<-_s0(f(x2,a),f(b,b)) -> ->_s0(f(c,x2),a) # label(replacement). 0.007/0.007 ->_s0(f(x3,x3),b) # label(replacement). 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 formulas(goals). 0.007/0.007 (exists x4 ->_s0(f(b,b),x4)) # label(goal). 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 ============================== end of input ========================== 0.007/0.007 0.007/0.007 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.007/0.007 0.007/0.007 % Formulas that are not ordinary clauses: 0.007/0.007 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 0.007/0.007 2 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.007/0.007 3 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.007/0.007 4 ->*<-_s0(f(x2,a),f(b,b)) -> ->_s0(f(c,x2),a) # label(replacement) # label(non_clause). [assumption]. 0.007/0.007 5 (exists x4 ->_s0(f(b,b),x4)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.007/0.007 0.007/0.007 ============================== end of process non-clausal formulas === 0.007/0.007 0.007/0.007 ============================== PROCESS INITIAL CLAUSES =============== 0.007/0.007 0.007/0.007 % Clauses before input processing: 0.007/0.007 0.007/0.007 formulas(usable). 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 formulas(sos). 0.007/0.007 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.007/0.007 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 0.007/0.007 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 0.007/0.007 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 0.007/0.007 ->_s0(a,b) # label(replacement). [assumption]. 0.007/0.007 -->*<-_s0(f(x,a),f(b,b)) | ->_s0(f(c,x),a) # label(replacement). [clausify(4)]. 0.007/0.007 ->_s0(f(x,x),b) # label(replacement). [assumption]. 0.007/0.007 -->_s0(f(b,b),x) # label(goal). [deny(5)]. 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 formulas(demodulators). 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 ============================== PREDICATE ELIMINATION ================= 0.007/0.007 0.007/0.007 Eliminating ->*<-_s0/2 0.007/0.007 0.007/0.007 ============================== end predicate elimination ============= 0.007/0.007 0.007/0.007 Auto_denials: 0.007/0.007 % copying label goal to answer in negative clause 0.007/0.007 0.007/0.007 Term ordering decisions: 0.007/0.007 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 0.007/0.007 Function symbol precedence: function_order([ b, a, f ]). 0.007/0.007 After inverse_order: (no changes). 0.007/0.007 Unfolding symbols: (none). 0.007/0.007 0.007/0.007 Auto_inference settings: 0.007/0.007 % set(neg_binary_resolution). % (HNE depth_diff=-2) 0.007/0.007 % clear(ordered_res). % (HNE depth_diff=-2) 0.007/0.007 % set(ur_resolution). % (HNE depth_diff=-2) 0.007/0.007 % set(ur_resolution) -> set(pos_ur_resolution). 0.007/0.007 % set(ur_resolution) -> set(neg_ur_resolution). 0.007/0.007 0.007/0.007 Auto_process settings: (no changes). 0.007/0.007 0.007/0.007 kept: 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.007/0.007 kept: 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 0.007/0.007 kept: 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 0.007/0.007 kept: 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 0.007/0.007 kept: 10 ->_s0(a,b) # label(replacement). [assumption]. 0.007/0.007 kept: 11 ->_s0(f(x,x),b) # label(replacement). [assumption]. 0.007/0.007 kept: 12 -->_s0(f(b,b),x) # label(goal) # answer(goal). [deny(5)]. 0.007/0.007 0.007/0.007 ============================== PROOF ================================= 0.007/0.007 0.007/0.007 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 0.007/0.007 % Length of proof is 4. 0.007/0.007 % Level of proof is 2. 0.007/0.007 % Maximum clause weight is 5.000. 0.007/0.007 % Given clauses 0. 0.007/0.007 0.007/0.007 5 (exists x4 ->_s0(f(b,b),x4)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.007/0.007 11 ->_s0(f(x,x),b) # label(replacement). [assumption]. 0.007/0.007 12 -->_s0(f(b,b),x) # label(goal) # answer(goal). [deny(5)]. 0.007/0.007 13 $F # answer(goal). [resolve(12,a,11,a)]. 0.007/0.007 0.007/0.007 ============================== end of proof ========================== 0.007/0.007 0.007/0.007 ============================== STATISTICS ============================ 0.007/0.007 0.007/0.007 Given=0. Generated=7. Kept=7. proofs=1. 0.007/0.007 Usable=0. Sos=0. Demods=0. Limbo=6, Disabled=8. Hints=0. 0.007/0.007 Kept_by_rule=0, Deleted_by_rule=0. 0.007/0.007 Forward_subsumed=0. Back_subsumed=0. 0.007/0.007 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 0.007/0.007 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 0.007/0.007 Demod_attempts=0. Demod_rewrites=0. 0.007/0.007 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 0.007/0.007 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=0. 0.007/0.007 Megabytes=0.04. 0.007/0.007 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.007/0.007 0.007/0.007 ============================== end of statistics ===================== 0.007/0.007 0.007/0.007 ============================== end of search ========================= 0.007/0.007 0.007/0.007 THEOREM PROVED 0.007/0.007 0.007/0.007 Exiting with 1 proof. 0.007/0.007 0.007/0.007 Process 29914 exit (max_proofs) Wed Jul 14 09:34:59 2021 0.007/0.007 0.007/0.007 0.007/0.007 The problem is feasible. 0.007/0.007 0.007/0.007 0.007/0.007 -> Problem conclusions: 0.007/0.007 Not left linear, Right linear, Not linear 0.007/0.007 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.007/0.007 CTRS Type: 1 0.007/0.007 Deterministic, Strongly deterministic 0.007/0.007 Not oriented CTRS, Properly oriented CTRS, Join CTRS 0.007/0.007 Not right-stable CTRS, Overlay CTRS 0.007/0.007 Normal CTRS, Almost normal CTRS 0.007/0.007 Terminating CTRS, Maybe joinable CCPs 0.007/0.007 Maybe level confluent 0.007/0.007 Maybe confluent 0.007/0.007 0.007/0.007 Problem 1: 0.007/0.007 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.007/0.007 Confluence Problem: 0.007/0.007 (VAR vNonEmpty:S x:S y:S) 0.007/0.007 (STRATEGY CONTEXTSENSITIVE 0.007/0.007 (a) 0.007/0.007 (f 1 2) 0.007/0.007 (b) 0.007/0.007 (c) 0.007/0.007 (fSNonEmpty) 0.007/0.007 ) 0.007/0.007 (RULES 0.007/0.007 a -> b 0.007/0.007 f(c,x:S) -> a | f(x:S,a) ->*<- f(b,b) 0.007/0.007 f(y:S,y:S) -> b 0.007/0.007 ) 0.007/0.007 Critical Pairs: 0.007/0.007 | f(c,a) ->*<- f(b,b) => Not trivial, Overlay, N1 0.007/0.007 Terminating:YES 0.007/0.007 0.007/0.007 -> Problem conclusions: 0.007/0.007 Not left linear, Right linear, Not linear 0.007/0.007 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.007/0.007 CTRS Type: 1 0.007/0.007 Deterministic, Strongly deterministic 0.007/0.007 Not oriented CTRS, Properly oriented CTRS, Join CTRS 0.007/0.007 Not right-stable CTRS, Overlay CTRS 0.007/0.007 Normal CTRS, Almost normal CTRS 0.007/0.007 Terminating CTRS, Maybe joinable CCPs 0.007/0.007 Maybe level confluent 0.007/0.007 Maybe confluent 0.007/0.007 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.007/0.007 0.007/0.007 Normal RConds Processor: 0.007/0.007 NO 0.007/0.007 0.007/0.007 Problem 1: 0.007/0.007 0.007/0.007 Infeasibility Problem: 0.007/0.007 [(VAR vNonEmpty:S x:S y:S x1:S) 0.007/0.007 (STRATEGY CONTEXTSENSITIVE 0.007/0.007 (a) 0.007/0.007 (f 1 2) 0.007/0.007 (b) 0.007/0.007 (c) 0.007/0.007 (fSNonEmpty) 0.007/0.007 ) 0.007/0.007 (RULES 0.007/0.007 a -> b 0.007/0.007 f(c,x:S) -> a | f(x:S,a) ->*<- f(b,b) 0.007/0.007 f(y:S,y:S) -> b 0.007/0.007 )] 0.007/0.007 0.007/0.007 Infeasibility Conditions: 0.007/0.007 f(b,b) -> x1:S 0.007/0.007 0.007/0.007 Problem 1: 0.007/0.007 0.007/0.007 Obtaining a proof using Prover9: 0.007/0.007 0.007/0.007 -> Prover9 Output: 0.007/0.007 ============================== Prover9 =============================== 0.007/0.007 Prover9 (64) version 2009-11A, November 2009. 0.007/0.007 Process 29936 was started by ubuntu on ubuntu, 0.007/0.007 Wed Jul 14 09:34:59 2021 0.007/0.007 The command was "./prover9 -f /tmp/prover929927-0.in". 0.007/0.007 ============================== end of head =========================== 0.007/0.007 0.007/0.007 ============================== INPUT ================================= 0.007/0.007 0.007/0.007 % Reading from file /tmp/prover929927-0.in 0.007/0.007 0.007/0.007 assign(max_seconds,20). 0.007/0.007 0.007/0.007 formulas(assumptions). 0.007/0.007 ->*_s0(x,x) # label(reflexivity). 0.007/0.007 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 0.007/0.007 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 0.007/0.007 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 0.007/0.007 ->_s0(a,b) # label(replacement). 0.007/0.007 ->*<-_s0(f(x1,a),f(b,b)) -> ->_s0(f(c,x1),a) # label(replacement). 0.007/0.007 ->_s0(f(x2,x2),b) # label(replacement). 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 formulas(goals). 0.007/0.007 (exists x3 ->_s0(f(b,b),x3)) # label(goal). 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 ============================== end of input ========================== 0.007/0.007 0.007/0.007 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.007/0.007 0.007/0.007 % Formulas that are not ordinary clauses: 0.007/0.007 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 0.007/0.007 2 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.007/0.007 3 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.007/0.007 4 ->*<-_s0(f(x1,a),f(b,b)) -> ->_s0(f(c,x1),a) # label(replacement) # label(non_clause). [assumption]. 0.007/0.007 5 (exists x3 ->_s0(f(b,b),x3)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.007/0.007 0.007/0.007 ============================== end of process non-clausal formulas === 0.007/0.007 0.007/0.007 ============================== PROCESS INITIAL CLAUSES =============== 0.007/0.007 0.007/0.007 % Clauses before input processing: 0.007/0.007 0.007/0.007 formulas(usable). 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 formulas(sos). 0.007/0.007 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.007/0.007 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 0.007/0.007 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 0.007/0.007 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 0.007/0.007 ->_s0(a,b) # label(replacement). [assumption]. 0.007/0.007 -->*<-_s0(f(x,a),f(b,b)) | ->_s0(f(c,x),a) # label(replacement). [clausify(4)]. 0.007/0.007 ->_s0(f(x,x),b) # label(replacement). [assumption]. 0.007/0.007 -->_s0(f(b,b),x) # label(goal). [deny(5)]. 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 formulas(demodulators). 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 ============================== PREDICATE ELIMINATION ================= 0.007/0.007 0.007/0.007 Eliminating ->*<-_s0/2 0.007/0.007 0.007/0.007 ============================== end predicate elimination ============= 0.007/0.007 0.007/0.007 Auto_denials: 0.007/0.007 % copying label goal to answer in negative clause 0.007/0.007 0.007/0.007 Term ordering decisions: 0.007/0.007 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 0.007/0.007 Function symbol precedence: function_order([ b, a, f ]). 0.007/0.007 After inverse_order: (no changes). 0.007/0.007 Unfolding symbols: (none). 0.007/0.007 0.007/0.007 Auto_inference settings: 0.007/0.007 % set(neg_binary_resolution). % (HNE depth_diff=-2) 0.007/0.007 % clear(ordered_res). % (HNE depth_diff=-2) 0.007/0.007 % set(ur_resolution). % (HNE depth_diff=-2) 0.007/0.007 % set(ur_resolution) -> set(pos_ur_resolution). 0.007/0.007 % set(ur_resolution) -> set(neg_ur_resolution). 0.007/0.007 0.007/0.007 Auto_process settings: (no changes). 0.007/0.007 0.007/0.007 kept: 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.007/0.007 kept: 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 0.007/0.007 kept: 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 0.007/0.007 kept: 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 0.007/0.007 kept: 10 ->_s0(a,b) # label(replacement). [assumption]. 0.007/0.007 kept: 11 ->_s0(f(x,x),b) # label(replacement). [assumption]. 0.007/0.007 kept: 12 -->_s0(f(b,b),x) # label(goal) # answer(goal). [deny(5)]. 0.007/0.007 0.007/0.007 ============================== PROOF ================================= 0.007/0.007 0.007/0.007 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 0.007/0.007 % Length of proof is 4. 0.007/0.007 % Level of proof is 2. 0.007/0.007 % Maximum clause weight is 5.000. 0.007/0.007 % Given clauses 0. 0.007/0.007 0.007/0.007 5 (exists x3 ->_s0(f(b,b),x3)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.007/0.007 11 ->_s0(f(x,x),b) # label(replacement). [assumption]. 0.007/0.007 12 -->_s0(f(b,b),x) # label(goal) # answer(goal). [deny(5)]. 0.007/0.007 13 $F # answer(goal). [resolve(12,a,11,a)]. 0.007/0.007 0.007/0.007 ============================== end of proof ========================== 0.007/0.007 0.007/0.007 ============================== STATISTICS ============================ 0.007/0.007 0.007/0.007 Given=0. Generated=7. Kept=7. proofs=1. 0.007/0.007 Usable=0. Sos=0. Demods=0. Limbo=6, Disabled=8. Hints=0. 0.007/0.007 Kept_by_rule=0, Deleted_by_rule=0. 0.007/0.007 Forward_subsumed=0. Back_subsumed=0. 0.007/0.007 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 0.007/0.007 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 0.007/0.007 Demod_attempts=0. Demod_rewrites=0. 0.007/0.007 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 0.007/0.007 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=0. 0.007/0.007 Megabytes=0.04. 0.007/0.007 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.007/0.007 0.007/0.007 ============================== end of statistics ===================== 0.007/0.007 0.007/0.007 ============================== end of search ========================= 0.007/0.007 0.007/0.007 THEOREM PROVED 0.007/0.007 0.007/0.007 Exiting with 1 proof. 0.007/0.007 0.007/0.007 Process 29936 exit (max_proofs) Wed Jul 14 09:34:59 2021 0.007/0.007 0.007/0.007 0.007/0.007 The problem is feasible. 0.007/0.007 0.007/0.007 0.007/0.007 -> Problem conclusions: 0.007/0.007 Not left linear, Right linear, Not linear 0.007/0.007 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.007/0.007 CTRS Type: 1 0.007/0.007 Deterministic, Strongly deterministic 0.007/0.007 Not oriented CTRS, Properly oriented CTRS, Join CTRS 0.007/0.007 Not right-stable CTRS, Overlay CTRS 0.007/0.007 Normal CTRS, Almost normal CTRS 0.007/0.007 Terminating CTRS, Maybe joinable CCPs 0.007/0.007 Maybe level confluent 0.007/0.007 Maybe confluent 0.007/0.007 0.007/0.007 Problem 1: 0.007/0.007 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.007/0.007 Confluence Problem: 0.007/0.007 (VAR vNonEmpty:S x:S y:S) 0.007/0.007 (STRATEGY CONTEXTSENSITIVE 0.007/0.007 (a) 0.007/0.007 (f 1 2) 0.007/0.007 (b) 0.007/0.007 (c) 0.007/0.007 (fSNonEmpty) 0.007/0.007 ) 0.007/0.007 (RULES 0.007/0.007 a -> b 0.007/0.007 f(c,x:S) -> a | f(x:S,a) ->*<- f(b,b) 0.007/0.007 f(y:S,y:S) -> b 0.007/0.007 ) 0.007/0.007 Critical Pairs: 0.007/0.007 | f(c,a) ->*<- f(b,b) => Not trivial, Overlay, N1 0.007/0.007 Terminating:YES 0.007/0.007 0.007/0.007 -> Problem conclusions: 0.007/0.007 Not left linear, Right linear, Not linear 0.007/0.007 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.007/0.007 CTRS Type: 1 0.007/0.007 Deterministic, Strongly deterministic 0.007/0.007 Not oriented CTRS, Properly oriented CTRS, Join CTRS 0.007/0.007 Not right-stable CTRS, Overlay CTRS 0.007/0.007 Normal CTRS, Almost normal CTRS 0.007/0.007 Terminating CTRS, Maybe joinable CCPs 0.007/0.007 Maybe level confluent 0.007/0.007 Maybe confluent 0.007/0.007 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.007/0.007 0.007/0.007 Conditional Critical Pairs Distributor Processor 0.007/0.007 Problem 1.1: 0.007/0.007 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.007/0.007 Confluence Problem: 0.007/0.007 (VAR vNonEmpty:S x:S y:S) 0.007/0.007 (STRATEGY CONTEXTSENSITIVE 0.007/0.007 (a) 0.007/0.007 (f 1 2) 0.007/0.007 (b) 0.007/0.007 (c) 0.007/0.007 (fSNonEmpty) 0.007/0.007 ) 0.007/0.007 (RULES 0.007/0.007 a -> b 0.007/0.007 f(c,x:S) -> a | f(x:S,a) ->*<- f(b,b) 0.007/0.007 f(y:S,y:S) -> b 0.007/0.007 ) 0.007/0.007 Critical Pairs: 0.007/0.007 | f(c,a) ->*<- f(b,b) => Not trivial, Overlay, N1 0.007/0.007 Terminating:YES 0.007/0.007 0.007/0.007 -> Problem conclusions: 0.007/0.007 Not left linear, Right linear, Not linear 0.007/0.007 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.007/0.007 CTRS Type: 1 0.007/0.007 Deterministic, Strongly deterministic 0.007/0.007 Not oriented CTRS, Properly oriented CTRS, Join CTRS 0.007/0.007 Not right-stable CTRS, Overlay CTRS 0.007/0.007 Normal CTRS, Almost normal CTRS 0.007/0.007 Terminating CTRS, Maybe joinable CCPs 0.007/0.007 Maybe level confluent 0.007/0.007 Maybe confluent 0.007/0.007 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.007/0.007 0.007/0.007 Prover9 CCP Convergence Checker Processor: 0.007/0.007 formulas(sos). 0.007/0.007 %Reflexivity 0.007/0.007 reds(x,x). 0.007/0.007 0.007/0.007 %Transitivity 0.007/0.007 (red(x,y) & reds(y,z)) -> reds(x,z). 0.007/0.007 0.007/0.007 %Congruence 0.007/0.007 0.007/0.007 red(x,y) -> red(f(x,z2),f(y,z2)). 0.007/0.007 red(x,y) -> red(f(z1,x),f(z1,y)). 0.007/0.007 0.007/0.007 0.007/0.007 %Replacement 0.007/0.007 red(a,b). 0.007/0.007 reds(f(x,a),f(b,b)) -> red(f(c,x),a). 0.007/0.007 red(f(y,y),b). 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 formulas(goals). 0.007/0.007 exists z (reds(a,z) & reds(b,z)). 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 Proof: 0.007/0.007 ============================== Prover9 =============================== 0.007/0.007 Prover9 (64) version 2009-11A, November 2009. 0.007/0.007 Process 29902 was started by ubuntu on ubuntu, 0.007/0.007 Wed Jul 14 09:34:59 2021 0.007/0.007 The command was "./prover9 -t 5 -f /tmp/prover929888-2.in". 0.007/0.007 ============================== end of head =========================== 0.007/0.007 0.007/0.007 ============================== INPUT ================================= 0.007/0.007 0.007/0.007 % Reading from file /tmp/prover929888-2.in 0.007/0.007 0.007/0.007 0.007/0.007 formulas(sos). 0.007/0.007 reds(x,x). 0.007/0.007 red(x,y) & reds(y,z) -> reds(x,z). 0.007/0.007 red(x,y) -> red(f(x,z2),f(y,z2)). 0.007/0.007 red(x,y) -> red(f(z1,x),f(z1,y)). 0.007/0.007 red(a,b). 0.007/0.007 reds(f(x,a),f(b,b)) -> red(f(c,x),a). 0.007/0.007 red(f(y,y),b). 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 formulas(goals). 0.007/0.007 (exists z (reds(a,z) & reds(b,z))). 0.007/0.007 end_of_list. 0.007/0.007 assign(max_megs,-1). 0.007/0.007 set(quiet). 0.007/0.007 0.007/0.007 ============================== end of input ========================== 0.007/0.007 0.007/0.007 % From the command line: assign(max_seconds, 5). 0.007/0.007 0.007/0.007 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.007/0.007 0.007/0.007 % Formulas that are not ordinary clauses: 0.007/0.007 1 red(x,y) & reds(y,z) -> reds(x,z) # label(non_clause). [assumption]. 0.007/0.007 2 red(x,y) -> red(f(x,z2),f(y,z2)) # label(non_clause). [assumption]. 0.007/0.007 3 red(x,y) -> red(f(z1,x),f(z1,y)) # label(non_clause). [assumption]. 0.007/0.007 4 reds(f(x,a),f(b,b)) -> red(f(c,x),a) # label(non_clause). [assumption]. 0.007/0.007 5 (exists z (reds(a,z) & reds(b,z))) # label(non_clause) # label(goal). [goal]. 0.007/0.007 0.007/0.007 ============================== end of process non-clausal formulas === 0.007/0.007 0.007/0.007 ============================== PROCESS INITIAL CLAUSES =============== 0.007/0.007 0.007/0.007 % Clauses before input processing: 0.007/0.007 0.007/0.007 formulas(usable). 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 formulas(sos). 0.007/0.007 reds(x,x). [assumption]. 0.007/0.007 -red(x,y) | -reds(y,z) | reds(x,z). [clausify(1)]. 0.007/0.007 -red(x,y) | red(f(x,z),f(y,z)). [clausify(2)]. 0.007/0.007 -red(x,y) | red(f(z,x),f(z,y)). [clausify(3)]. 0.007/0.007 red(a,b). [assumption]. 0.007/0.007 -reds(f(x,a),f(b,b)) | red(f(c,x),a). [clausify(4)]. 0.007/0.007 red(f(x,x),b). [assumption]. 0.007/0.007 -reds(a,x) | -reds(b,x). [deny(5)]. 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 formulas(demodulators). 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 ============================== PREDICATE ELIMINATION ================= 0.007/0.007 0.007/0.007 ============================== end predicate elimination ============= 0.007/0.007 0.007/0.007 Auto_denials: (no changes). 0.007/0.007 0.007/0.007 Term ordering decisions: 0.007/0.007 0.007/0.007 kept: 6 reds(x,x). [assumption]. 0.007/0.007 kept: 7 -red(x,y) | -reds(y,z) | reds(x,z). [clausify(1)]. 0.007/0.007 kept: 8 -red(x,y) | red(f(x,z),f(y,z)). [clausify(2)]. 0.007/0.007 kept: 9 -red(x,y) | red(f(z,x),f(z,y)). [clausify(3)]. 0.007/0.007 kept: 10 red(a,b). [assumption]. 0.007/0.007 kept: 11 -reds(f(x,a),f(b,b)) | red(f(c,x),a). [clausify(4)]. 0.007/0.007 kept: 12 red(f(x,x),b). [assumption]. 0.007/0.007 kept: 13 -reds(a,x) | -reds(b,x). [deny(5)]. 0.007/0.007 0.007/0.007 ============================== end of process initial clauses ======== 0.007/0.007 0.007/0.007 ============================== CLAUSES FOR SEARCH ==================== 0.007/0.007 0.007/0.007 % Clauses after input processing: 0.007/0.007 0.007/0.007 formulas(usable). 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 formulas(sos). 0.007/0.007 6 reds(x,x). [assumption]. 0.007/0.007 7 -red(x,y) | -reds(y,z) | reds(x,z). [clausify(1)]. 0.007/0.007 8 -red(x,y) | red(f(x,z),f(y,z)). [clausify(2)]. 0.007/0.007 9 -red(x,y) | red(f(z,x),f(z,y)). [clausify(3)]. 0.007/0.007 10 red(a,b). [assumption]. 0.007/0.007 11 -reds(f(x,a),f(b,b)) | red(f(c,x),a). [clausify(4)]. 0.007/0.007 12 red(f(x,x),b). [assumption]. 0.007/0.007 13 -reds(a,x) | -reds(b,x). [deny(5)]. 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 formulas(demodulators). 0.007/0.007 end_of_list. 0.007/0.007 0.007/0.007 ============================== end of clauses for search ============= 0.007/0.007 0.007/0.007 ============================== SEARCH ================================ 0.007/0.007 0.007/0.007 % Starting search at 0.00 seconds. 0.007/0.007 0.007/0.007 given #1 (I,wt=3): 6 reds(x,x). [assumption]. 0.007/0.007 0.007/0.007 given #2 (I,wt=9): 7 -red(x,y) | -reds(y,z) | reds(x,z). [clausify(1)]. 0.007/0.007 0.007/0.007 given #3 (I,wt=10): 8 -red(x,y) | red(f(x,z),f(y,z)). [clausify(2)]. 0.007/0.007 0.007/0.007 given #4 (I,wt=10): 9 -red(x,y) | red(f(z,x),f(z,y)). [clausify(3)]. 0.007/0.007 0.007/0.007 given #5 (I,wt=3): 10 red(a,b). [assumption]. 0.007/0.007 0.007/0.007 given #6 (I,wt=12): 11 -reds(f(x,a),f(b,b)) | red(f(c,x),a). [clausify(4)]. 0.007/0.007 0.007/0.007 given #7 (I,wt=5): 12 red(f(x,x),b). [assumption]. 0.007/0.007 0.007/0.007 given #8 (I,wt=6): 13 -reds(a,x) | -reds(b,x). [deny(5)]. 0.007/0.007 0.007/0.007 ============================== PROOF ================================= 0.007/0.007 0.007/0.007 % Proof 1 at 0.01 (+ 0.00) seconds. 0.007/0.007 % Length of proof is 8. 0.007/0.007 % Level of proof is 3. 0.007/0.007 % Maximum clause weight is 9.000. 0.007/0.007 % Given clauses 8. 0.007/0.007 0.007/0.007 1 red(x,y) & reds(y,z) -> reds(x,z) # label(non_clause). [assumption]. 0.007/0.007 5 (exists z (reds(a,z) & reds(b,z))) # label(non_clause) # label(goal). [goal]. 0.007/0.007 6 reds(x,x). [assumption]. 0.007/0.007 7 -red(x,y) | -reds(y,z) | reds(x,z). [clausify(1)]. 0.007/0.007 10 red(a,b). [assumption]. 0.007/0.007 13 -reds(a,x) | -reds(b,x). [deny(5)]. 0.007/0.007 16 reds(a,b). [ur(7,a,10,a,b,6,a)]. 0.007/0.007 23 $F. [resolve(13,b,6,a),unit_del(a,16)]. 0.007/0.007 0.007/0.007 ============================== end of proof ========================== 0.007/0.007 0.007/0.007 ============================== STATISTICS ============================ 0.007/0.007 0.007/0.007 Given=8. Generated=18. Kept=17. proofs=1. 0.007/0.007 Usable=8. Sos=6. Demods=0. Limbo=3, Disabled=8. Hints=0. 0.007/0.007 Kept_by_rule=0, Deleted_by_rule=0. 0.007/0.007 Forward_subsumed=0. Back_subsumed=0. 0.007/0.007 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 0.007/0.007 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 0.007/0.007 Demod_attempts=0. Demod_rewrites=0. 0.007/0.007 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 0.007/0.007 Nonunit_fsub_feature_tests=4. Nonunit_bsub_feature_tests=8. 0.007/0.007 Megabytes=0.06. 0.007/0.007 User_CPU=0.01, System_CPU=0.00, Wall_clock=0. 0.007/0.007 0.007/0.007 ============================== end of statistics ===================== 0.007/0.007 0.007/0.007 ============================== end of search ========================= 0.007/0.007 0.007/0.007 THEOREM PROVED 0.007/0.007 0.007/0.007 Exiting with 1 proof. 0.007/0.007 0.007/0.007 Process 29902 exit (max_proofs) Wed Jul 14 09:34:59 2021 0.007/0.007 0.007/0.007 0.007/0.007 The problem is joinable. 0.007/0.007 0.04user 0.00system 0:00.07elapsed 73%CPU (0avgtext+0avgdata 10660maxresident)k 0.007/0.007 0inputs+0outputs (0major+4849minor)pagefaults 0swaps