20.066/20.066 NO 20.066/20.066 20.066/20.066 Problem 1: 20.066/20.066 20.066/20.066 20.066/20.066 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.066/20.066 Confluence Problem: 20.066/20.066 (VAR vNonEmpty x y) 20.066/20.066 (REPLACEMENT-MAP 20.066/20.066 (f 1, 2) 20.066/20.066 (a) 20.066/20.066 (b) 20.066/20.066 (fSNonEmpty) 20.066/20.066 ) 20.066/20.066 (RULES 20.066/20.066 f(x,x) -> b | f(x,x) ->* a, x ->* a 20.066/20.066 f(y,y) -> a 20.066/20.066 ) 20.066/20.066 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.066/20.066 20.066/20.066 20.066/20.066 Problem 1: 20.066/20.066 20.066/20.066 Inlining of Conditions Processor [STERN17]: 20.066/20.066 20.066/20.066 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.066/20.066 Confluence Problem: 20.066/20.066 (VAR vNonEmpty x y) 20.066/20.066 (REPLACEMENT-MAP 20.066/20.066 (f 1, 2) 20.066/20.066 (a) 20.066/20.066 (b) 20.066/20.066 (fSNonEmpty) 20.066/20.066 ) 20.066/20.066 (RULES 20.066/20.066 f(x,x) -> b | f(x,x) ->* a, x ->* a 20.066/20.066 f(y,y) -> a 20.066/20.066 ) 20.066/20.066 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.066/20.066 20.066/20.066 20.066/20.066 Problem 1: 20.066/20.066 20.066/20.066 Clean CTRS Processor: 20.066/20.066 20.066/20.066 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.066/20.066 Confluence Problem: 20.066/20.066 (VAR x y) 20.066/20.066 (REPLACEMENT-MAP 20.066/20.066 (f 1, 2) 20.066/20.066 (a) 20.066/20.066 (b) 20.066/20.066 (fSNonEmpty) 20.066/20.066 ) 20.066/20.066 (RULES 20.066/20.066 f(x,x) -> b | f(x,x) ->* a, x ->* a 20.066/20.066 f(y,y) -> a 20.066/20.066 ) 20.066/20.066 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.066/20.066 20.066/20.066 CRule InfChecker Info: 20.066/20.066 f(x,x) -> b | f(x,x) ->* a, x ->* a 20.066/20.066 Rule remains 20.066/20.066 Proof: 20.066/20.066 NO 20.066/20.066 20.066/20.066 Problem 1: 20.066/20.066 20.066/20.066 Infeasibility Problem: 20.066/20.066 [(VAR vNonEmpty x y vNonEmpty x) 20.066/20.066 (STRATEGY CONTEXTSENSITIVE 20.066/20.066 (f 1 2) 20.066/20.066 (a) 20.066/20.066 (b) 20.066/20.066 (fSNonEmpty) 20.066/20.066 ) 20.066/20.066 (RULES 20.066/20.066 f(x,x) -> b | f(x,x) ->* a, x ->* a 20.066/20.066 f(y,y) -> a 20.066/20.066 )] 20.066/20.066 20.066/20.066 Infeasibility Conditions: 20.066/20.066 f(x,x) ->* a, x ->* a 20.066/20.066 20.066/20.066 Problem 1: 20.066/20.066 20.066/20.066 Obtaining a proof using Prover9: 20.066/20.066 20.066/20.066 -> Prover9 Output: 20.066/20.066 ============================== Prover9 =============================== 20.066/20.066 Prover9 (64) version 2009-11A, November 2009. 20.066/20.066 Process 82196 was started by ubuntu on ubuntu, 20.066/20.066 Wed Mar 9 10:30:59 2022 20.066/20.066 The command was "./prover9 -f /tmp/prover982187-0.in". 20.066/20.066 ============================== end of head =========================== 20.066/20.066 20.066/20.066 ============================== INPUT ================================= 20.066/20.066 20.066/20.066 % Reading from file /tmp/prover982187-0.in 20.066/20.066 20.066/20.066 assign(max_seconds,20). 20.066/20.066 20.066/20.066 formulas(assumptions). 20.066/20.066 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 20.066/20.066 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 20.066/20.066 ->*_s0(f(x1,x1),a) & ->*_s0(x1,a) -> ->_s0(f(x1,x1),b) # label(replacement). 20.066/20.066 ->_s0(f(x2,x2),a) # label(replacement). 20.066/20.066 ->*_s0(x,x) # label(reflexivity). 20.066/20.066 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 formulas(goals). 20.066/20.066 (exists x4 (->*_s0(f(x4,x4),a) & ->*_s0(x4,a))) # label(goal). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 ============================== end of input ========================== 20.066/20.066 20.066/20.066 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.066/20.066 20.066/20.066 % Formulas that are not ordinary clauses: 20.066/20.066 1 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.066/20.066 2 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.066/20.066 3 ->*_s0(f(x1,x1),a) & ->*_s0(x1,a) -> ->_s0(f(x1,x1),b) # label(replacement) # label(non_clause). [assumption]. 20.066/20.066 4 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.066/20.066 5 (exists x4 (->*_s0(f(x4,x4),a) & ->*_s0(x4,a))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.066/20.066 20.066/20.066 ============================== end of process non-clausal formulas === 20.066/20.066 20.066/20.066 ============================== PROCESS INITIAL CLAUSES =============== 20.066/20.066 20.066/20.066 % Clauses before input processing: 20.066/20.066 20.066/20.066 formulas(usable). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 formulas(sos). 20.066/20.066 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.066/20.066 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.066/20.066 -->*_s0(f(x,x),a) | -->*_s0(x,a) | ->_s0(f(x,x),b) # label(replacement). [clausify(3)]. 20.066/20.066 ->_s0(f(x,x),a) # label(replacement). [assumption]. 20.066/20.066 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.066/20.066 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(4)]. 20.066/20.066 -->*_s0(f(x,x),a) | -->*_s0(x,a) # label(goal). [deny(5)]. 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 formulas(demodulators). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 ============================== PREDICATE ELIMINATION ================= 20.066/20.066 20.066/20.066 No predicates eliminated. 20.066/20.066 20.066/20.066 ============================== end predicate elimination ============= 20.066/20.066 20.066/20.066 Auto_denials: 20.066/20.066 % copying label goal to answer in negative clause 20.066/20.066 20.066/20.066 Term ordering decisions: 20.066/20.066 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.066/20.066 Function symbol precedence: function_order([ a, b, f ]). 20.066/20.066 After inverse_order: (no changes). 20.066/20.066 Unfolding symbols: (none). 20.066/20.066 20.066/20.066 Auto_inference settings: 20.066/20.066 % set(neg_binary_resolution). % (HNE depth_diff=-2) 20.066/20.066 % clear(ordered_res). % (HNE depth_diff=-2) 20.066/20.066 % set(ur_resolution). % (HNE depth_diff=-2) 20.066/20.066 % set(ur_resolution) -> set(pos_ur_resolution). 20.066/20.066 % set(ur_resolution) -> set(neg_ur_resolution). 20.066/20.066 20.066/20.066 Auto_process settings: 20.066/20.066 % set(unit_deletion). % (Horn set with negative nonunits) 20.066/20.066 20.066/20.066 kept: 6 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.066/20.066 kept: 7 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.066/20.066 kept: 8 -->*_s0(f(x,x),a) | -->*_s0(x,a) | ->_s0(f(x,x),b) # label(replacement). [clausify(3)]. 20.066/20.066 kept: 9 ->_s0(f(x,x),a) # label(replacement). [assumption]. 20.066/20.066 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.066/20.066 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(4)]. 20.066/20.066 kept: 12 -->*_s0(f(x,x),a) | -->*_s0(x,a) # label(goal) # answer(goal). [deny(5)]. 20.066/20.066 20.066/20.066 ============================== end of process initial clauses ======== 20.066/20.066 20.066/20.066 ============================== CLAUSES FOR SEARCH ==================== 20.066/20.066 20.066/20.066 % Clauses after input processing: 20.066/20.066 20.066/20.066 formulas(usable). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 formulas(sos). 20.066/20.066 6 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.066/20.066 7 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.066/20.066 9 ->_s0(f(x,x),a) # label(replacement). [assumption]. 20.066/20.066 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.066/20.066 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(4)]. 20.066/20.066 12 -->*_s0(f(x,x),a) | -->*_s0(x,a) # label(goal) # answer(goal). [deny(5)]. 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 formulas(demodulators). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 ============================== end of clauses for search ============= 20.066/20.066 20.066/20.066 ============================== SEARCH ================================ 20.066/20.066 20.066/20.066 % Starting search at 0.00 seconds. 20.066/20.066 20.066/20.066 given #1 (I,wt=10): 6 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.066/20.066 20.066/20.066 given #2 (I,wt=10): 7 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.066/20.066 20.066/20.066 given #3 (I,wt=5): 9 ->_s0(f(x,x),a) # label(replacement). [assumption]. 20.066/20.066 20.066/20.066 given #4 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.066/20.066 20.066/20.066 given #5 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(4)]. 20.066/20.066 20.066/20.066 ============================== PROOF ================================= 20.066/20.066 20.066/20.066 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.066/20.066 % Length of proof is 9. 20.066/20.066 % Level of proof is 4. 20.066/20.066 % Maximum clause weight is 9.000. 20.066/20.066 % Given clauses 5. 20.066/20.066 20.066/20.066 4 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.066/20.066 5 (exists x4 (->*_s0(f(x4,x4),a) & ->*_s0(x4,a))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.066/20.066 9 ->_s0(f(x,x),a) # label(replacement). [assumption]. 20.066/20.066 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.066/20.066 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(4)]. 20.066/20.066 12 -->*_s0(f(x,x),a) | -->*_s0(x,a) # label(goal) # answer(goal). [deny(5)]. 20.066/20.066 15 ->*_s0(f(x,x),a). [ur(11,a,9,a,b,10,a)]. 20.066/20.066 16 -->*_s0(x,a) # answer(goal). [back_unit_del(12),unit_del(a,15)]. 20.066/20.066 17 $F # answer(goal). [resolve(16,a,15,a)]. 20.066/20.066 20.066/20.066 ============================== end of proof ========================== 20.066/20.066 20.066/20.066 ============================== STATISTICS ============================ 20.066/20.066 20.066/20.066 Given=5. Generated=11. Kept=11. proofs=1. 20.066/20.066 Usable=5. Sos=2. Demods=0. Limbo=1, Disabled=9. Hints=0. 20.066/20.066 Kept_by_rule=0, Deleted_by_rule=0. 20.066/20.066 Forward_subsumed=0. Back_subsumed=1. 20.066/20.066 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.066/20.066 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=1. 20.066/20.066 Demod_attempts=0. Demod_rewrites=0. 20.066/20.066 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.066/20.066 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=9. 20.066/20.066 Megabytes=0.06. 20.066/20.066 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.066/20.066 20.066/20.066 ============================== end of statistics ===================== 20.066/20.066 20.066/20.066 ============================== end of search ========================= 20.066/20.066 20.066/20.066 THEOREM PROVED 20.066/20.066 20.066/20.066 Exiting with 1 proof. 20.066/20.066 20.066/20.066 Process 82196 exit (max_proofs) Wed Mar 9 10:30:59 2022 20.066/20.066 20.066/20.066 20.066/20.066 The problem is feasible. 20.066/20.066 20.066/20.066 20.066/20.066 CRule InfChecker Info: 20.066/20.066 f(y,y) -> a 20.066/20.066 Rule remains 20.066/20.066 Proof: 20.066/20.066 NO_CONDS 20.066/20.066 20.066/20.066 Problem 1: 20.066/20.066 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.066/20.066 Confluence Problem: 20.066/20.066 (VAR x y) 20.066/20.066 (REPLACEMENT-MAP 20.066/20.066 (f 1, 2) 20.066/20.066 (a) 20.066/20.066 (b) 20.066/20.066 (fSNonEmpty) 20.066/20.066 ) 20.066/20.066 (RULES 20.066/20.066 f(x,x) -> b | f(x,x) ->* a, x ->* a 20.066/20.066 f(y,y) -> a 20.066/20.066 ) 20.066/20.066 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.066/20.066 20.066/20.066 Critical Pairs Processor: 20.066/20.066 -> Rules: 20.066/20.066 f(x,x) -> b | f(x,x) ->* a, x ->* a 20.066/20.066 f(y,y) -> a 20.066/20.066 -> Vars: 20.066/20.066 "x", "y" 20.066/20.066 20.066/20.066 -> Rlps: 20.066/20.066 crule: f(x,x) -> b | f(x,x) ->* a, x ->* a, id: 1, possubterms: f(x,x)-> [] 20.066/20.066 crule: f(y,y) -> a, id: 2, possubterms: f(y,y)-> [] 20.066/20.066 20.066/20.066 -> Unifications: 20.066/20.066 R2 unifies with R1 at p: [], l: f(y,y), lp: f(y,y), conds: {f(x,x) ->* a, x ->* a}, sig: {y -> x}, l': f(x,x), r: a, r': b 20.066/20.066 20.066/20.066 -> Critical pairs info: 20.066/20.066 | f(x,x) ->* a, x ->* a => Not trivial, Overlay, NW2, N1 20.066/20.066 20.066/20.066 -> Problem conclusions: 20.066/20.066 Not left linear, Right linear, Not linear 20.066/20.066 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.066/20.066 CTRS Type: 1 20.066/20.066 Deterministic, Strongly deterministic 20.066/20.066 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.066/20.066 Maybe right-stable CTRS, Overlay CTRS 20.066/20.066 Maybe normal CTRS, Maybe almost normal CTRS 20.066/20.066 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.066/20.066 Maybe level confluent 20.066/20.066 Maybe confluent 20.066/20.066 20.066/20.066 Problem 1: 20.066/20.066 Clean Infeasible CCPs Processor: 20.066/20.066 Num of CPIs: 1 20.066/20.066 Timeout: 60 20.066/20.066 Timeout for each infeasibility problem: 60 s 20.066/20.066 | f(x,x) ->* a, x ->* a => Not trivial, Overlay, Feasible conditions, NW2, N1 20.066/20.066 (CONDITIONTYPE ORIENTED) 20.066/20.066 (VAR x y x) 20.066/20.066 (RULES 20.066/20.066 f(x,x) -> b | f(x,x) ->* a, x ->* a 20.066/20.066 f(y,y) -> a) 20.066/20.066 (VAR x5) 20.066/20.066 (CONDITION f(x5,x5) ->* a, x5 ->* a) 20.066/20.066 20.066/20.066 Proof: 20.066/20.066 NO 20.066/20.066 20.066/20.066 Problem 1: 20.066/20.066 20.066/20.066 Infeasibility Problem: 20.066/20.066 [(VAR vNonEmpty x y vNonEmpty x5) 20.066/20.066 (STRATEGY CONTEXTSENSITIVE 20.066/20.066 (f 1 2) 20.066/20.066 (a) 20.066/20.066 (b) 20.066/20.066 (fSNonEmpty) 20.066/20.066 ) 20.066/20.066 (RULES 20.066/20.066 f(x,x) -> b | f(x,x) ->* a, x ->* a 20.066/20.066 f(y,y) -> a 20.066/20.066 )] 20.066/20.066 20.066/20.066 Infeasibility Conditions: 20.066/20.066 f(x5,x5) ->* a, x5 ->* a 20.066/20.066 20.066/20.066 Problem 1: 20.066/20.066 20.066/20.066 Obtaining a proof using Prover9: 20.066/20.066 20.066/20.066 -> Prover9 Output: 20.066/20.066 ============================== Prover9 =============================== 20.066/20.066 Prover9 (64) version 2009-11A, November 2009. 20.066/20.066 Process 82228 was started by ubuntu on ubuntu, 20.066/20.066 Wed Mar 9 10:30:59 2022 20.066/20.066 The command was "./prover9 -f /tmp/prover982212-0.in". 20.066/20.066 ============================== end of head =========================== 20.066/20.066 20.066/20.066 ============================== INPUT ================================= 20.066/20.066 20.066/20.066 % Reading from file /tmp/prover982212-0.in 20.066/20.066 20.066/20.066 assign(max_seconds,20). 20.066/20.066 20.066/20.066 formulas(assumptions). 20.066/20.066 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 20.066/20.066 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 20.066/20.066 ->*_s0(f(x1,x1),a) & ->*_s0(x1,a) -> ->_s0(f(x1,x1),b) # label(replacement). 20.066/20.066 ->_s0(f(x2,x2),a) # label(replacement). 20.066/20.066 ->*_s0(x,x) # label(reflexivity). 20.066/20.066 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 formulas(goals). 20.066/20.066 (exists x4 (->*_s0(f(x4,x4),a) & ->*_s0(x4,a))) # label(goal). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 ============================== end of input ========================== 20.066/20.066 20.066/20.066 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.066/20.066 20.066/20.066 % Formulas that are not ordinary clauses: 20.066/20.066 1 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.066/20.066 2 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.066/20.066 3 ->*_s0(f(x1,x1),a) & ->*_s0(x1,a) -> ->_s0(f(x1,x1),b) # label(replacement) # label(non_clause). [assumption]. 20.066/20.066 4 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.066/20.066 5 (exists x4 (->*_s0(f(x4,x4),a) & ->*_s0(x4,a))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.066/20.066 20.066/20.066 ============================== end of process non-clausal formulas === 20.066/20.066 20.066/20.066 ============================== PROCESS INITIAL CLAUSES =============== 20.066/20.066 20.066/20.066 % Clauses before input processing: 20.066/20.066 20.066/20.066 formulas(usable). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 formulas(sos). 20.066/20.066 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.066/20.066 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.066/20.066 -->*_s0(f(x,x),a) | -->*_s0(x,a) | ->_s0(f(x,x),b) # label(replacement). [clausify(3)]. 20.066/20.066 ->_s0(f(x,x),a) # label(replacement). [assumption]. 20.066/20.066 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.066/20.066 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(4)]. 20.066/20.066 -->*_s0(f(x,x),a) | -->*_s0(x,a) # label(goal). [deny(5)]. 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 formulas(demodulators). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 ============================== PREDICATE ELIMINATION ================= 20.066/20.066 20.066/20.066 No predicates eliminated. 20.066/20.066 20.066/20.066 ============================== end predicate elimination ============= 20.066/20.066 20.066/20.066 Auto_denials: 20.066/20.066 % copying label goal to answer in negative clause 20.066/20.066 20.066/20.066 Term ordering decisions: 20.066/20.066 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 20.066/20.066 Function symbol precedence: function_order([ a, b, f ]). 20.066/20.066 After inverse_order: (no changes). 20.066/20.066 Unfolding symbols: (none). 20.066/20.066 20.066/20.066 Auto_inference settings: 20.066/20.066 % set(neg_binary_resolution). % (HNE depth_diff=-2) 20.066/20.066 % clear(ordered_res). % (HNE depth_diff=-2) 20.066/20.066 % set(ur_resolution). % (HNE depth_diff=-2) 20.066/20.066 % set(ur_resolution) -> set(pos_ur_resolution). 20.066/20.066 % set(ur_resolution) -> set(neg_ur_resolution). 20.066/20.066 20.066/20.066 Auto_process settings: 20.066/20.066 % set(unit_deletion). % (Horn set with negative nonunits) 20.066/20.066 20.066/20.066 kept: 6 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.066/20.066 kept: 7 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.066/20.066 kept: 8 -->*_s0(f(x,x),a) | -->*_s0(x,a) | ->_s0(f(x,x),b) # label(replacement). [clausify(3)]. 20.066/20.066 kept: 9 ->_s0(f(x,x),a) # label(replacement). [assumption]. 20.066/20.066 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.066/20.066 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(4)]. 20.066/20.066 kept: 12 -->*_s0(f(x,x),a) | -->*_s0(x,a) # label(goal) # answer(goal). [deny(5)]. 20.066/20.066 20.066/20.066 ============================== end of process initial clauses ======== 20.066/20.066 20.066/20.066 ============================== CLAUSES FOR SEARCH ==================== 20.066/20.066 20.066/20.066 % Clauses after input processing: 20.066/20.066 20.066/20.066 formulas(usable). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 formulas(sos). 20.066/20.066 6 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.066/20.066 7 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.066/20.066 9 ->_s0(f(x,x),a) # label(replacement). [assumption]. 20.066/20.066 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.066/20.066 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(4)]. 20.066/20.066 12 -->*_s0(f(x,x),a) | -->*_s0(x,a) # label(goal) # answer(goal). [deny(5)]. 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 formulas(demodulators). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 ============================== end of clauses for search ============= 20.066/20.066 20.066/20.066 ============================== SEARCH ================================ 20.066/20.066 20.066/20.066 % Starting search at 0.00 seconds. 20.066/20.066 20.066/20.066 given #1 (I,wt=10): 6 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(1)]. 20.066/20.066 20.066/20.066 given #2 (I,wt=10): 7 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(2)]. 20.066/20.066 20.066/20.066 given #3 (I,wt=5): 9 ->_s0(f(x,x),a) # label(replacement). [assumption]. 20.066/20.066 20.066/20.066 given #4 (I,wt=3): 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.066/20.066 20.066/20.066 given #5 (I,wt=9): 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(4)]. 20.066/20.066 20.066/20.066 ============================== PROOF ================================= 20.066/20.066 20.066/20.066 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 20.066/20.066 % Length of proof is 9. 20.066/20.066 % Level of proof is 4. 20.066/20.066 % Maximum clause weight is 9.000. 20.066/20.066 % Given clauses 5. 20.066/20.066 20.066/20.066 4 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 20.066/20.066 5 (exists x4 (->*_s0(f(x4,x4),a) & ->*_s0(x4,a))) # label(goal) # label(non_clause) # label(goal). [goal]. 20.066/20.066 9 ->_s0(f(x,x),a) # label(replacement). [assumption]. 20.066/20.066 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 20.066/20.066 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(4)]. 20.066/20.066 12 -->*_s0(f(x,x),a) | -->*_s0(x,a) # label(goal) # answer(goal). [deny(5)]. 20.066/20.066 15 ->*_s0(f(x,x),a). [ur(11,a,9,a,b,10,a)]. 20.066/20.066 16 -->*_s0(x,a) # answer(goal). [back_unit_del(12),unit_del(a,15)]. 20.066/20.066 17 $F # answer(goal). [resolve(16,a,15,a)]. 20.066/20.066 20.066/20.066 ============================== end of proof ========================== 20.066/20.066 20.066/20.066 ============================== STATISTICS ============================ 20.066/20.066 20.066/20.066 Given=5. Generated=11. Kept=11. proofs=1. 20.066/20.066 Usable=5. Sos=2. Demods=0. Limbo=1, Disabled=9. Hints=0. 20.066/20.066 Kept_by_rule=0, Deleted_by_rule=0. 20.066/20.066 Forward_subsumed=0. Back_subsumed=1. 20.066/20.066 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 20.066/20.066 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=1. 20.066/20.066 Demod_attempts=0. Demod_rewrites=0. 20.066/20.066 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 20.066/20.066 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=9. 20.066/20.066 Megabytes=0.06. 20.066/20.066 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.066/20.066 20.066/20.066 ============================== end of statistics ===================== 20.066/20.066 20.066/20.066 ============================== end of search ========================= 20.066/20.066 20.066/20.066 THEOREM PROVED 20.066/20.066 20.066/20.066 Exiting with 1 proof. 20.066/20.066 20.066/20.066 Process 82228 exit (max_proofs) Wed Mar 9 10:30:59 2022 20.066/20.066 20.066/20.066 20.066/20.066 The problem is feasible. 20.066/20.066 20.066/20.066 20.066/20.066 20.066/20.066 20.066/20.066 -> Problem conclusions: 20.066/20.066 Not left linear, Right linear, Not linear 20.066/20.066 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 20.066/20.066 CTRS Type: 1 20.066/20.066 Deterministic, Strongly deterministic 20.066/20.066 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 20.066/20.066 Maybe right-stable CTRS, Overlay CTRS 20.066/20.066 Maybe normal CTRS, Maybe almost normal CTRS 20.066/20.066 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 20.066/20.066 Maybe level confluent 20.066/20.066 Maybe confluent 20.066/20.066 20.066/20.066 Resulting CCPs: 20.066/20.066 | f(x,x) ->* a, x ->* a => Not trivial, Overlay, Feasible conditions, NW2, N1 20.066/20.066 20.066/20.066 Problem 1: 20.066/20.066 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.066/20.066 Confluence Problem: 20.066/20.066 (VAR x y x) 20.066/20.066 (REPLACEMENT-MAP 20.066/20.066 (f 1, 2) 20.066/20.066 (a) 20.066/20.066 (b) 20.066/20.066 (fSNonEmpty) 20.066/20.066 ) 20.066/20.066 (RULES 20.066/20.066 f(x,x) -> b | f(x,x) ->* a, x ->* a 20.066/20.066 f(y,y) -> a 20.066/20.066 ) 20.066/20.066 Critical Pairs: 20.066/20.066 | f(x,x) ->* a, x ->* a => Not trivial, Overlay, Feasible conditions, NW2, N1 20.066/20.066 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.066/20.066 20.066/20.066 Conditional Critical Pairs Distributor Processor 20.066/20.066 20.066/20.066 Problem 1: 20.066/20.066 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.066/20.066 Confluence Problem: 20.066/20.066 (VAR x y x) 20.066/20.066 (REPLACEMENT-MAP 20.066/20.066 (f 1, 2) 20.066/20.066 (a) 20.066/20.066 (b) 20.066/20.066 (fSNonEmpty) 20.066/20.066 ) 20.066/20.066 (RULES 20.066/20.066 f(x,x) -> b | f(x,x) ->* a, x ->* a 20.066/20.066 f(y,y) -> a 20.066/20.066 ) 20.066/20.066 Critical Pairs: 20.066/20.066 | f(x,x) ->* a, x ->* a => Not trivial, Overlay, Feasible conditions, NW2, N1 20.066/20.066 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 20.066/20.066 20.066/20.066 InfChecker Different Ground and Normal CCP Terms Processor: 20.066/20.066 Proof: 20.066/20.066 YES 20.066/20.066 20.066/20.066 Problem 1: 20.066/20.066 20.066/20.066 Infeasibility Problem: 20.066/20.066 [(VAR vNonEmpty x y vNonEmpty y) 20.066/20.066 (STRATEGY CONTEXTSENSITIVE 20.066/20.066 (f 1 2) 20.066/20.066 (a) 20.066/20.066 (b) 20.066/20.066 (fSNonEmpty) 20.066/20.066 ) 20.066/20.066 (RULES 20.066/20.066 f(x,x) -> b | f(x,x) ->* a, x ->* a 20.066/20.066 f(y,y) -> a 20.066/20.066 )] 20.066/20.066 20.066/20.066 Infeasibility Conditions: 20.066/20.066 b -> y 20.066/20.066 20.066/20.066 Problem 1: 20.066/20.066 20.066/20.066 Obtaining a model using Mace4: 20.066/20.066 20.066/20.066 -> Usable Rules: 20.066/20.066 Empty 20.066/20.066 20.066/20.066 -> Mace4 Output: 20.066/20.066 ============================== Mace4 ================================= 20.066/20.066 Mace4 (64) version 2009-11A, November 2009. 20.066/20.066 Process 82309 was started by ubuntu on ubuntu, 20.066/20.066 Wed Mar 9 10:31:19 2022 20.066/20.066 The command was "./mace4 -c -f /tmp/mace482266-2.in". 20.066/20.066 ============================== end of head =========================== 20.066/20.066 20.066/20.066 ============================== INPUT ================================= 20.066/20.066 20.066/20.066 % Reading from file /tmp/mace482266-2.in 20.066/20.066 20.066/20.066 assign(max_seconds,6). 20.066/20.066 20.066/20.066 formulas(assumptions). 20.066/20.066 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). 20.066/20.066 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 formulas(goals). 20.066/20.066 (exists x4 ->(b,x4)) # label(goal). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 ============================== end of input ========================== 20.066/20.066 20.066/20.066 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.066/20.066 20.066/20.066 % Formulas that are not ordinary clauses: 20.066/20.066 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.066/20.066 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.066/20.066 3 (exists x4 ->(b,x4)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.066/20.066 20.066/20.066 ============================== end of process non-clausal formulas === 20.066/20.066 20.066/20.066 ============================== CLAUSES FOR SEARCH ==================== 20.066/20.066 20.066/20.066 formulas(mace4_clauses). 20.066/20.066 -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). 20.066/20.066 -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). 20.066/20.066 -->(b,x) # label(goal). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 ============================== end of clauses for search ============= 20.066/20.066 20.066/20.066 % There are no natural numbers in the input. 20.066/20.066 20.066/20.066 ============================== DOMAIN SIZE 2 ========================= 20.066/20.066 20.066/20.066 ============================== MODEL ================================= 20.066/20.066 20.066/20.066 interpretation( 2, [number=1, seconds=0], [ 20.066/20.066 20.066/20.066 function(b, [ 0 ]), 20.066/20.066 20.066/20.066 function(f(_,_), [ 20.066/20.066 0, 0, 20.066/20.066 0, 0 ]), 20.066/20.066 20.066/20.066 relation(->(_,_), [ 20.066/20.066 0, 0, 20.066/20.066 0, 0 ]) 20.066/20.066 ]). 20.066/20.066 20.066/20.066 ============================== end of model ========================== 20.066/20.066 20.066/20.066 ============================== STATISTICS ============================ 20.066/20.066 20.066/20.066 For domain size 2. 20.066/20.066 20.066/20.066 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.066/20.066 Ground clauses: seen=18, kept=18. 20.066/20.066 Selections=5, assignments=5, propagations=4, current_models=1. 20.066/20.066 Rewrite_terms=34, rewrite_bools=20, indexes=2. 20.066/20.066 Rules_from_neg_clauses=0, cross_offs=0. 20.066/20.066 20.066/20.066 ============================== end of statistics ===================== 20.066/20.066 20.066/20.066 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.066/20.066 20.066/20.066 Exiting with 1 model. 20.066/20.066 20.066/20.066 Process 82309 exit (max_models) Wed Mar 9 10:31:19 2022 20.066/20.066 The process finished Wed Mar 9 10:31:19 2022 20.066/20.066 20.066/20.066 20.066/20.066 Mace4 cooked interpretation: 20.066/20.066 20.066/20.066 20.066/20.066 20.066/20.066 The problem is infeasible. 20.066/20.066 YES 20.066/20.066 20.066/20.066 Problem 1: 20.066/20.066 20.066/20.066 Infeasibility Problem: 20.066/20.066 [(VAR vNonEmpty x y vNonEmpty x) 20.066/20.066 (STRATEGY CONTEXTSENSITIVE 20.066/20.066 (f 1 2) 20.066/20.066 (a) 20.066/20.066 (b) 20.066/20.066 (fSNonEmpty) 20.066/20.066 ) 20.066/20.066 (RULES 20.066/20.066 f(x,x) -> b | f(x,x) ->* a, x ->* a 20.066/20.066 f(y,y) -> a 20.066/20.066 )] 20.066/20.066 20.066/20.066 Infeasibility Conditions: 20.066/20.066 a -> x 20.066/20.066 20.066/20.066 Problem 1: 20.066/20.066 20.066/20.066 Obtaining a model using Mace4: 20.066/20.066 20.066/20.066 -> Usable Rules: 20.066/20.066 Empty 20.066/20.066 20.066/20.066 -> Mace4 Output: 20.066/20.066 ============================== Mace4 ================================= 20.066/20.066 Mace4 (64) version 2009-11A, November 2009. 20.066/20.066 Process 82351 was started by ubuntu on ubuntu, 20.066/20.066 Wed Mar 9 10:31:19 2022 20.066/20.066 The command was "./mace4 -c -f /tmp/mace482337-2.in". 20.066/20.066 ============================== end of head =========================== 20.066/20.066 20.066/20.066 ============================== INPUT ================================= 20.066/20.066 20.066/20.066 % Reading from file /tmp/mace482337-2.in 20.066/20.066 20.066/20.066 assign(max_seconds,6). 20.066/20.066 20.066/20.066 formulas(assumptions). 20.066/20.066 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). 20.066/20.066 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 formulas(goals). 20.066/20.066 (exists x4 ->(a,x4)) # label(goal). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 ============================== end of input ========================== 20.066/20.066 20.066/20.066 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 20.066/20.066 20.066/20.066 % Formulas that are not ordinary clauses: 20.066/20.066 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 20.066/20.066 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 20.066/20.066 3 (exists x4 ->(a,x4)) # label(goal) # label(non_clause) # label(goal). [goal]. 20.066/20.066 20.066/20.066 ============================== end of process non-clausal formulas === 20.066/20.066 20.066/20.066 ============================== CLAUSES FOR SEARCH ==================== 20.066/20.066 20.066/20.066 formulas(mace4_clauses). 20.066/20.066 -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). 20.066/20.066 -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). 20.066/20.066 -->(a,x) # label(goal). 20.066/20.066 end_of_list. 20.066/20.066 20.066/20.066 ============================== end of clauses for search ============= 20.066/20.066 20.066/20.066 % There are no natural numbers in the input. 20.066/20.066 20.066/20.066 ============================== DOMAIN SIZE 2 ========================= 20.066/20.066 20.066/20.066 ============================== MODEL ================================= 20.066/20.066 20.066/20.066 interpretation( 2, [number=1, seconds=0], [ 20.066/20.066 20.066/20.066 function(a, [ 0 ]), 20.066/20.066 20.066/20.066 function(f(_,_), [ 20.066/20.066 0, 0, 20.066/20.066 0, 0 ]), 20.066/20.066 20.066/20.066 relation(->(_,_), [ 20.066/20.066 0, 0, 20.066/20.066 0, 0 ]) 20.066/20.066 ]). 20.066/20.066 20.066/20.066 ============================== end of model ========================== 20.066/20.066 20.066/20.066 ============================== STATISTICS ============================ 20.066/20.066 20.066/20.066 For domain size 2. 20.066/20.066 20.066/20.066 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 20.066/20.066 Ground clauses: seen=18, kept=18. 20.066/20.066 Selections=5, assignments=5, propagations=4, current_models=1. 20.066/20.066 Rewrite_terms=34, rewrite_bools=20, indexes=2. 20.066/20.066 Rules_from_neg_clauses=0, cross_offs=0. 20.066/20.066 20.066/20.066 ============================== end of statistics ===================== 20.066/20.066 20.066/20.066 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 20.066/20.066 20.066/20.066 Exiting with 1 model. 20.066/20.066 20.066/20.066 Process 82351 exit (max_models) Wed Mar 9 10:31:19 2022 20.066/20.066 The process finished Wed Mar 9 10:31:19 2022 20.066/20.066 20.066/20.066 20.066/20.066 Mace4 cooked interpretation: 20.066/20.066 20.066/20.066 20.066/20.066 20.066/20.066 The problem is infeasible. 20.066/20.066 20.066/20.066 20.066/20.066 The problem is not joinable. 20.066/20.066 47.99user 2.67system 0:20.66elapsed 245%CPU (0avgtext+0avgdata 2938496maxresident)k 20.066/20.066 8inputs+0outputs (0major+774151minor)pagefaults 0swaps