25.025/25.025 NO 25.025/25.025 25.025/25.025 Problem 1: 25.025/25.025 25.025/25.025 25.025/25.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.025/25.025 Confluence Problem: 25.025/25.025 (VAR vNonEmpty:S x:S y:S) 25.025/25.025 (STRATEGY CONTEXTSENSITIVE 25.025/25.025 (f 1 2) 25.025/25.025 (b) 25.025/25.025 (fSNonEmpty) 25.025/25.025 ) 25.025/25.025 (RULES 25.025/25.025 f(x:S,x:S) -> x:S | f(x:S,x:S) ->* b 25.025/25.025 f(y:S,y:S) -> b 25.025/25.025 ) 25.025/25.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.025/25.025 25.025/25.025 25.025/25.025 Problem 1: 25.025/25.025 25.025/25.025 Inlining of Conditions Processor [STERN17]: 25.025/25.025 25.025/25.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.025/25.025 Confluence Problem: 25.025/25.025 (VAR vNonEmpty:S x:S y:S) 25.025/25.025 (STRATEGY CONTEXTSENSITIVE 25.025/25.025 (f 1 2) 25.025/25.025 (b) 25.025/25.025 (fSNonEmpty) 25.025/25.025 ) 25.025/25.025 (RULES 25.025/25.025 f(x:S,x:S) -> x:S | f(x:S,x:S) ->* b 25.025/25.025 f(y:S,y:S) -> b 25.025/25.025 ) 25.025/25.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.025/25.025 25.025/25.025 25.025/25.025 Problem 1: 25.025/25.025 25.025/25.025 Clean CTRS Processor: 25.025/25.025 25.025/25.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.025/25.025 Confluence Problem: 25.025/25.025 (VAR x:S y:S) 25.025/25.025 (STRATEGY CONTEXTSENSITIVE 25.025/25.025 (f 1 2) 25.025/25.025 (b) 25.025/25.025 (fSNonEmpty) 25.025/25.025 ) 25.025/25.025 (RULES 25.025/25.025 f(x:S,x:S) -> x:S | f(x:S,x:S) ->* b 25.025/25.025 f(y:S,y:S) -> b 25.025/25.025 ) 25.025/25.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.025/25.025 25.025/25.025 CRule InfChecker Info: 25.025/25.025 f(x:S,x:S) -> x:S | f(x:S,x:S) ->* b 25.025/25.025 Rule remains 25.025/25.025 Proof: 25.025/25.025 NO 25.025/25.025 25.025/25.025 Problem 1: 25.025/25.025 25.025/25.025 Infeasibility Problem: 25.025/25.025 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S y:S:S) 25.025/25.025 (STRATEGY CONTEXTSENSITIVE 25.025/25.025 (f 1 2) 25.025/25.025 (b) 25.025/25.025 (fSNonEmpty) 25.025/25.025 ) 25.025/25.025 (RULES 25.025/25.025 f(x:S:S,x:S:S) -> x:S:S | f(x:S:S,x:S:S) ->* b 25.025/25.025 f(y:S:S,y:S:S) -> b 25.025/25.025 )] 25.025/25.025 25.025/25.025 Infeasibility Conditions: 25.025/25.025 f(x:S:S,x:S:S) ->* b 25.025/25.025 25.025/25.025 Problem 1: 25.025/25.025 25.025/25.025 Obtaining a proof using Prover9: 25.025/25.025 25.025/25.025 -> Prover9 Output: 25.025/25.025 ============================== Prover9 =============================== 25.025/25.025 Prover9 (64) version 2009-11A, November 2009. 25.025/25.025 Process 33179 was started by ubuntu on ubuntu, 25.025/25.025 Wed Jul 14 09:47:41 2021 25.025/25.025 The command was "./prover9 -f /tmp/prover933170-0.in". 25.025/25.025 ============================== end of head =========================== 25.025/25.025 25.025/25.025 ============================== INPUT ================================= 25.025/25.025 25.025/25.025 % Reading from file /tmp/prover933170-0.in 25.025/25.025 25.025/25.025 assign(max_seconds,20). 25.025/25.025 25.025/25.025 formulas(assumptions). 25.025/25.025 ->*_s0(x,x) # label(reflexivity). 25.025/25.025 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.025/25.025 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 25.025/25.025 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 25.025/25.025 ->*_s0(f(x2,x2),b) -> ->_s0(f(x2,x2),x2) # label(replacement). 25.025/25.025 ->_s0(f(x3,x3),b) # label(replacement). 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 formulas(goals). 25.025/25.025 (exists x2 ->*_s0(f(x2,x2),b)) # label(goal). 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 ============================== end of input ========================== 25.025/25.025 25.025/25.025 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.025/25.025 25.025/25.025 % Formulas that are not ordinary clauses: 25.025/25.025 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.025/25.025 2 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.025/25.025 3 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.025/25.025 4 ->*_s0(f(x2,x2),b) -> ->_s0(f(x2,x2),x2) # label(replacement) # label(non_clause). [assumption]. 25.025/25.025 5 (exists x2 ->*_s0(f(x2,x2),b)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.025/25.025 25.025/25.025 ============================== end of process non-clausal formulas === 25.025/25.025 25.025/25.025 ============================== PROCESS INITIAL CLAUSES =============== 25.025/25.025 25.025/25.025 % Clauses before input processing: 25.025/25.025 25.025/25.025 formulas(usable). 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 formulas(sos). 25.025/25.025 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.025/25.025 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.025/25.025 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.025/25.025 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.025/25.025 -->*_s0(f(x,x),b) | ->_s0(f(x,x),x) # label(replacement). [clausify(4)]. 25.025/25.025 ->_s0(f(x,x),b) # label(replacement). [assumption]. 25.025/25.025 -->*_s0(f(x,x),b) # label(goal). [deny(5)]. 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 formulas(demodulators). 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 ============================== PREDICATE ELIMINATION ================= 25.025/25.025 25.025/25.025 No predicates eliminated. 25.025/25.025 25.025/25.025 ============================== end predicate elimination ============= 25.025/25.025 25.025/25.025 Auto_denials: 25.025/25.025 % copying label goal to answer in negative clause 25.025/25.025 25.025/25.025 Term ordering decisions: 25.025/25.025 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.025/25.025 Function symbol precedence: function_order([ b, f ]). 25.025/25.025 After inverse_order: (no changes). 25.025/25.025 Unfolding symbols: (none). 25.025/25.025 25.025/25.025 Auto_inference settings: 25.025/25.025 % set(neg_binary_resolution). % (HNE depth_diff=-2) 25.025/25.025 % clear(ordered_res). % (HNE depth_diff=-2) 25.025/25.025 % set(ur_resolution). % (HNE depth_diff=-2) 25.025/25.025 % set(ur_resolution) -> set(pos_ur_resolution). 25.025/25.025 % set(ur_resolution) -> set(neg_ur_resolution). 25.025/25.025 25.025/25.025 Auto_process settings: (no changes). 25.025/25.025 25.025/25.025 kept: 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.025/25.025 kept: 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.025/25.025 kept: 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.025/25.025 kept: 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.025/25.025 kept: 10 -->*_s0(f(x,x),b) | ->_s0(f(x,x),x) # label(replacement). [clausify(4)]. 25.025/25.025 kept: 11 ->_s0(f(x,x),b) # label(replacement). [assumption]. 25.025/25.025 kept: 12 -->*_s0(f(x,x),b) # label(goal) # answer(goal). [deny(5)]. 25.025/25.025 25.025/25.025 ============================== end of process initial clauses ======== 25.025/25.025 25.025/25.025 ============================== CLAUSES FOR SEARCH ==================== 25.025/25.025 25.025/25.025 % Clauses after input processing: 25.025/25.025 25.025/25.025 formulas(usable). 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 formulas(sos). 25.025/25.025 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.025/25.025 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.025/25.025 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.025/25.025 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.025/25.025 11 ->_s0(f(x,x),b) # label(replacement). [assumption]. 25.025/25.025 12 -->*_s0(f(x,x),b) # label(goal) # answer(goal). [deny(5)]. 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 formulas(demodulators). 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 ============================== end of clauses for search ============= 25.025/25.025 25.025/25.025 ============================== SEARCH ================================ 25.025/25.025 25.025/25.025 % Starting search at 0.00 seconds. 25.025/25.025 25.025/25.025 given #1 (I,wt=3): 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.025/25.025 25.025/25.025 given #2 (I,wt=9): 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.025/25.025 25.025/25.025 given #3 (I,wt=10): 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.025/25.025 25.025/25.025 given #4 (I,wt=10): 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.025/25.025 25.025/25.025 given #5 (I,wt=5): 11 ->_s0(f(x,x),b) # label(replacement). [assumption]. 25.025/25.025 25.025/25.025 ============================== PROOF ================================= 25.025/25.025 25.025/25.025 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.025/25.025 % Length of proof is 8. 25.025/25.025 % Level of proof is 3. 25.025/25.025 % Maximum clause weight is 9.000. 25.025/25.025 % Given clauses 5. 25.025/25.025 25.025/25.025 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.025/25.025 5 (exists x2 ->*_s0(f(x2,x2),b)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.025/25.025 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.025/25.025 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.025/25.025 11 ->_s0(f(x,x),b) # label(replacement). [assumption]. 25.025/25.025 12 -->*_s0(f(x,x),b) # label(goal) # answer(goal). [deny(5)]. 25.025/25.025 15 ->*_s0(f(x,x),b). [ur(7,a,11,a,b,6,a)]. 25.025/25.025 16 $F # answer(goal). [resolve(15,a,12,a)]. 25.025/25.025 25.025/25.025 ============================== end of proof ========================== 25.025/25.025 25.025/25.025 ============================== STATISTICS ============================ 25.025/25.025 25.025/25.025 Given=5. Generated=10. Kept=10. proofs=1. 25.025/25.025 Usable=5. Sos=1. Demods=0. Limbo=2, Disabled=8. Hints=0. 25.025/25.025 Kept_by_rule=0, Deleted_by_rule=0. 25.025/25.025 Forward_subsumed=0. Back_subsumed=1. 25.025/25.025 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.025/25.025 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.025/25.025 Demod_attempts=0. Demod_rewrites=0. 25.025/25.025 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.025/25.025 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=8. 25.025/25.025 Megabytes=0.05. 25.025/25.025 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.025/25.025 25.025/25.025 ============================== end of statistics ===================== 25.025/25.025 25.025/25.025 ============================== end of search ========================= 25.025/25.025 25.025/25.025 THEOREM PROVED 25.025/25.025 25.025/25.025 Exiting with 1 proof. 25.025/25.025 25.025/25.025 Process 33179 exit (max_proofs) Wed Jul 14 09:47:41 2021 25.025/25.025 25.025/25.025 25.025/25.025 The problem is feasible. 25.025/25.025 25.025/25.025 25.025/25.025 CRule InfChecker Info: 25.025/25.025 f(y:S,y:S) -> b 25.025/25.025 Rule remains 25.025/25.025 Proof: 25.025/25.025 NO_CONDS 25.025/25.025 25.025/25.025 Problem 1: 25.025/25.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.025/25.025 Confluence Problem: 25.025/25.025 (VAR x:S y:S) 25.025/25.025 (STRATEGY CONTEXTSENSITIVE 25.025/25.025 (f 1 2) 25.025/25.025 (b) 25.025/25.025 (fSNonEmpty) 25.025/25.025 ) 25.025/25.025 (RULES 25.025/25.025 f(x:S,x:S) -> x:S | f(x:S,x:S) ->* b 25.025/25.025 f(y:S,y:S) -> b 25.025/25.025 ) 25.025/25.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.025/25.025 25.025/25.025 Critical Pairs Processor: 25.025/25.025 -> Rules: 25.025/25.025 f(x:S,x:S) -> x:S | f(x:S,x:S) ->* b 25.025/25.025 f(y:S,y:S) -> b 25.025/25.025 -> Vars: 25.025/25.025 "x", "y" 25.025/25.025 -> FVars: 25.025/25.025 "x3", "x4" 25.025/25.025 -> PVars: 25.025/25.025 "x": ["x3"], "y": ["x4"] 25.025/25.025 25.025/25.025 -> Rlps: 25.025/25.025 crule: f(x3:S,x3:S) -> x3:S | f(x3:S,x3:S) ->* b, id: 1, possubterms: f(x3:S,x3:S)-> [] 25.025/25.025 crule: f(x4:S,x4:S) -> b, id: 2, possubterms: f(x4:S,x4:S)-> [] 25.025/25.025 25.025/25.025 -> Unifications: 25.025/25.025 R2 unifies with R1 at p: [], l: f(x4:S,x4:S), lp: f(x4:S,x4:S), conds: {f(x:S,x:S) ->* b}, sig: {x4:S -> x:S}, l': f(x:S,x:S), r: b, r': x:S 25.025/25.025 25.025/25.025 -> Critical pairs info: 25.025/25.025 | f(x:S,x:S) ->* b => Not trivial, Overlay, N1 25.025/25.025 25.025/25.025 -> Problem conclusions: 25.025/25.025 Not left linear, Right linear, Not linear 25.025/25.025 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 25.025/25.025 CTRS Type: 1 25.025/25.025 Deterministic, Strongly deterministic 25.025/25.025 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.025/25.025 Maybe right-stable CTRS, Overlay CTRS 25.025/25.025 Maybe normal CTRS, Maybe almost normal CTRS 25.025/25.025 Maybe terminating CTRS, Maybe joinable CCPs 25.025/25.025 Maybe level confluent 25.025/25.025 Maybe confluent 25.025/25.025 25.025/25.025 Problem 1: 25.025/25.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.025/25.025 Confluence Problem: 25.025/25.025 (VAR x:S y:S) 25.025/25.025 (STRATEGY CONTEXTSENSITIVE 25.025/25.025 (f 1 2) 25.025/25.025 (b) 25.025/25.025 (fSNonEmpty) 25.025/25.025 ) 25.025/25.025 (RULES 25.025/25.025 f(x:S,x:S) -> x:S | f(x:S,x:S) ->* b 25.025/25.025 f(y:S,y:S) -> b 25.025/25.025 ) 25.025/25.025 Critical Pairs: 25.025/25.025 | f(x:S,x:S) ->* b => Not trivial, Overlay, N1 25.025/25.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.025/25.025 25.025/25.025 Conditional Critical Pairs Distributor Processor 25.025/25.025 25.025/25.025 Problem 1: 25.025/25.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.025/25.025 Confluence Problem: 25.025/25.025 (VAR x:S y:S) 25.025/25.025 (STRATEGY CONTEXTSENSITIVE 25.025/25.025 (f 1 2) 25.025/25.025 (b) 25.025/25.025 (fSNonEmpty) 25.025/25.025 ) 25.025/25.025 (RULES 25.025/25.025 f(x:S,x:S) -> x:S | f(x:S,x:S) ->* b 25.025/25.025 f(y:S,y:S) -> b 25.025/25.025 ) 25.025/25.025 Critical Pairs: 25.025/25.025 | f(x:S,x:S) ->* b => Not trivial, Overlay, N1 25.025/25.025 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.025/25.025 25.025/25.025 Infeasible Convergence Constantized CCP Processor: 25.025/25.025 25.025/25.025 Infeasible constantized conditions? 25.025/25.025 NO 25.025/25.025 25.025/25.025 Problem 1: 25.025/25.025 25.025/25.025 Infeasibility Problem: 25.025/25.025 [(VAR vNonEmpty:S x:S y:S) 25.025/25.025 (STRATEGY CONTEXTSENSITIVE 25.025/25.025 (f 1 2) 25.025/25.025 (b) 25.025/25.025 (cx) 25.025/25.025 (fSNonEmpty) 25.025/25.025 ) 25.025/25.025 (RULES 25.025/25.025 f(x:S,x:S) -> x:S | f(x:S,x:S) ->* b 25.025/25.025 f(y:S,y:S) -> b 25.025/25.025 )] 25.025/25.025 25.025/25.025 Infeasibility Conditions: 25.025/25.025 f(cx,cx) ->* b 25.025/25.025 25.025/25.025 Problem 1: 25.025/25.025 25.025/25.025 Obtaining a proof using Prover9: 25.025/25.025 25.025/25.025 -> Prover9 Output: 25.025/25.025 ============================== Prover9 =============================== 25.025/25.025 Prover9 (64) version 2009-11A, November 2009. 25.025/25.025 Process 33213 was started by ubuntu on ubuntu, 25.025/25.025 Wed Jul 14 09:47:41 2021 25.025/25.025 The command was "./prover9 -f /tmp/prover933195-0.in". 25.025/25.025 ============================== end of head =========================== 25.025/25.025 25.025/25.025 ============================== INPUT ================================= 25.025/25.025 25.025/25.025 % Reading from file /tmp/prover933195-0.in 25.025/25.025 25.025/25.025 assign(max_seconds,20). 25.025/25.025 25.025/25.025 formulas(assumptions). 25.025/25.025 ->*_s0(x,x) # label(reflexivity). 25.025/25.025 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.025/25.025 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 25.025/25.025 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 25.025/25.025 ->*_s0(f(x1,x1),b) -> ->_s0(f(x1,x1),x1) # label(replacement). 25.025/25.025 ->_s0(f(x2,x2),b) # label(replacement). 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 formulas(goals). 25.025/25.025 ->*_s0(f(cx,cx),b) # label(goal). 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 ============================== end of input ========================== 25.025/25.025 25.025/25.025 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.025/25.025 25.025/25.025 % Formulas that are not ordinary clauses: 25.025/25.025 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.025/25.025 2 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.025/25.025 3 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.025/25.025 4 ->*_s0(f(x1,x1),b) -> ->_s0(f(x1,x1),x1) # label(replacement) # label(non_clause). [assumption]. 25.025/25.025 5 ->*_s0(f(cx,cx),b) # label(goal) # label(non_clause) # label(goal). [goal]. 25.025/25.025 25.025/25.025 ============================== end of process non-clausal formulas === 25.025/25.025 25.025/25.025 ============================== PROCESS INITIAL CLAUSES =============== 25.025/25.025 25.025/25.025 % Clauses before input processing: 25.025/25.025 25.025/25.025 formulas(usable). 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 formulas(sos). 25.025/25.025 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.025/25.025 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.025/25.025 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.025/25.025 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.025/25.025 -->*_s0(f(x,x),b) | ->_s0(f(x,x),x) # label(replacement). [clausify(4)]. 25.025/25.025 ->_s0(f(x,x),b) # label(replacement). [assumption]. 25.025/25.025 -->*_s0(f(cx,cx),b) # label(goal). [deny(5)]. 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 formulas(demodulators). 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 ============================== PREDICATE ELIMINATION ================= 25.025/25.025 25.025/25.025 No predicates eliminated. 25.025/25.025 25.025/25.025 ============================== end predicate elimination ============= 25.025/25.025 25.025/25.025 Auto_denials: 25.025/25.025 % copying label goal to answer in negative clause 25.025/25.025 25.025/25.025 Term ordering decisions: 25.025/25.025 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.025/25.025 Function symbol precedence: function_order([ b, cx, f ]). 25.025/25.025 After inverse_order: (no changes). 25.025/25.025 Unfolding symbols: (none). 25.025/25.025 25.025/25.025 Auto_inference settings: 25.025/25.025 % set(neg_binary_resolution). % (HNE depth_diff=-2) 25.025/25.025 % clear(ordered_res). % (HNE depth_diff=-2) 25.025/25.025 % set(ur_resolution). % (HNE depth_diff=-2) 25.025/25.025 % set(ur_resolution) -> set(pos_ur_resolution). 25.025/25.025 % set(ur_resolution) -> set(neg_ur_resolution). 25.025/25.025 25.025/25.025 Auto_process settings: (no changes). 25.025/25.025 25.025/25.025 kept: 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.025/25.025 kept: 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.025/25.025 kept: 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.025/25.025 kept: 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.025/25.025 kept: 10 -->*_s0(f(x,x),b) | ->_s0(f(x,x),x) # label(replacement). [clausify(4)]. 25.025/25.025 kept: 11 ->_s0(f(x,x),b) # label(replacement). [assumption]. 25.025/25.025 kept: 12 -->*_s0(f(cx,cx),b) # label(goal) # answer(goal). [deny(5)]. 25.025/25.025 25.025/25.025 ============================== end of process initial clauses ======== 25.025/25.025 25.025/25.025 ============================== CLAUSES FOR SEARCH ==================== 25.025/25.025 25.025/25.025 % Clauses after input processing: 25.025/25.025 25.025/25.025 formulas(usable). 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 formulas(sos). 25.025/25.025 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.025/25.025 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.025/25.025 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.025/25.025 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.025/25.025 10 -->*_s0(f(x,x),b) | ->_s0(f(x,x),x) # label(replacement). [clausify(4)]. 25.025/25.025 11 ->_s0(f(x,x),b) # label(replacement). [assumption]. 25.025/25.025 12 -->*_s0(f(cx,cx),b) # label(goal) # answer(goal). [deny(5)]. 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 formulas(demodulators). 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 ============================== end of clauses for search ============= 25.025/25.025 25.025/25.025 ============================== SEARCH ================================ 25.025/25.025 25.025/25.025 % Starting search at 0.00 seconds. 25.025/25.025 25.025/25.025 given #1 (I,wt=3): 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.025/25.025 25.025/25.025 given #2 (I,wt=9): 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.025/25.025 25.025/25.025 given #3 (I,wt=10): 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.025/25.025 25.025/25.025 given #4 (I,wt=10): 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.025/25.025 25.025/25.025 given #5 (I,wt=10): 10 -->*_s0(f(x,x),b) | ->_s0(f(x,x),x) # label(replacement). [clausify(4)]. 25.025/25.025 25.025/25.025 given #6 (I,wt=5): 11 ->_s0(f(x,x),b) # label(replacement). [assumption]. 25.025/25.025 25.025/25.025 ============================== PROOF ================================= 25.025/25.025 25.025/25.025 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.025/25.025 % Length of proof is 8. 25.025/25.025 % Level of proof is 3. 25.025/25.025 % Maximum clause weight is 9.000. 25.025/25.025 % Given clauses 6. 25.025/25.025 25.025/25.025 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.025/25.025 5 ->*_s0(f(cx,cx),b) # label(goal) # label(non_clause) # label(goal). [goal]. 25.025/25.025 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.025/25.025 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.025/25.025 11 ->_s0(f(x,x),b) # label(replacement). [assumption]. 25.025/25.025 12 -->*_s0(f(cx,cx),b) # label(goal) # answer(goal). [deny(5)]. 25.025/25.025 15 ->*_s0(f(x,x),b). [ur(7,a,11,a,b,6,a)]. 25.025/25.025 16 $F # answer(goal). [resolve(15,a,12,a)]. 25.025/25.025 25.025/25.025 ============================== end of proof ========================== 25.025/25.025 25.025/25.025 ============================== STATISTICS ============================ 25.025/25.025 25.025/25.025 Given=6. Generated=10. Kept=10. proofs=1. 25.025/25.025 Usable=6. Sos=1. Demods=0. Limbo=2, Disabled=7. Hints=0. 25.025/25.025 Kept_by_rule=0, Deleted_by_rule=0. 25.025/25.025 Forward_subsumed=0. Back_subsumed=0. 25.025/25.025 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.025/25.025 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.025/25.025 Demod_attempts=0. Demod_rewrites=0. 25.025/25.025 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.025/25.025 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=7. 25.025/25.025 Megabytes=0.05. 25.025/25.025 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.025/25.025 25.025/25.025 ============================== end of statistics ===================== 25.025/25.025 25.025/25.025 ============================== end of search ========================= 25.025/25.025 25.025/25.025 THEOREM PROVED 25.025/25.025 25.025/25.025 Exiting with 1 proof. 25.025/25.025 25.025/25.025 Process 33213 exit (max_proofs) Wed Jul 14 09:47:41 2021 25.025/25.025 25.025/25.025 25.025/25.025 The problem is feasible. 25.025/25.025 25.025/25.025 Infeasible constantized CCP convergence? 25.025/25.025 YES 25.025/25.025 25.025/25.025 Problem 1: 25.025/25.025 25.025/25.025 Infeasibility Problem: 25.025/25.025 [(VAR vNonEmpty:S x:S y:S z:S) 25.025/25.025 (STRATEGY CONTEXTSENSITIVE 25.025/25.025 (f 1 2) 25.025/25.025 (b) 25.025/25.025 (cx) 25.025/25.025 (fSNonEmpty) 25.025/25.025 ) 25.025/25.025 (RULES 25.025/25.025 f(x:S,x:S) -> x:S | f(x:S,x:S) ->* b 25.025/25.025 f(y:S,y:S) -> b 25.025/25.025 )] 25.025/25.025 25.025/25.025 Infeasibility Conditions: 25.025/25.025 cx ->* z:S, b ->* z:S 25.025/25.025 25.025/25.025 Problem 1: 25.025/25.025 25.025/25.025 Obtaining a model using Mace4: 25.025/25.025 25.025/25.025 -> Usable Rules: 25.025/25.025 Empty 25.025/25.025 25.025/25.025 -> Mace4 Output: 25.025/25.025 ============================== Mace4 ================================= 25.025/25.025 Mace4 (64) version 2009-11A, November 2009. 25.025/25.025 Process 33251 was started by ubuntu on ubuntu, 25.025/25.025 Wed Jul 14 09:47:41 2021 25.025/25.025 The command was "./mace4 -c -f /tmp/mace433237-2.in". 25.025/25.025 ============================== end of head =========================== 25.025/25.025 25.025/25.025 ============================== INPUT ================================= 25.025/25.025 25.025/25.025 % Reading from file /tmp/mace433237-2.in 25.025/25.025 25.025/25.025 assign(max_seconds,20). 25.025/25.025 25.025/25.025 formulas(assumptions). 25.025/25.025 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). 25.025/25.025 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). 25.025/25.025 ->*(x,x) # label(reflexivity). 25.025/25.025 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 formulas(goals). 25.025/25.025 (exists x3 (->*(cx,x3) & ->*(b,x3))) # label(goal). 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 ============================== end of input ========================== 25.025/25.025 25.025/25.025 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.025/25.025 25.025/25.025 % Formulas that are not ordinary clauses: 25.025/25.025 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.025/25.025 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.025/25.025 3 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.025/25.025 4 (exists x3 (->*(cx,x3) & ->*(b,x3))) # label(goal) # label(non_clause) # label(goal). [goal]. 25.025/25.025 25.025/25.025 ============================== end of process non-clausal formulas === 25.025/25.025 25.025/25.025 ============================== CLAUSES FOR SEARCH ==================== 25.025/25.025 25.025/25.025 formulas(mace4_clauses). 25.025/25.025 -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). 25.025/25.025 -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). 25.025/25.025 ->*(x,x) # label(reflexivity). 25.025/25.025 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 25.025/25.025 -->*(cx,x) | -->*(b,x) # label(goal). 25.025/25.025 end_of_list. 25.025/25.025 25.025/25.025 ============================== end of clauses for search ============= 25.025/25.025 25.025/25.025 % There are no natural numbers in the input. 25.025/25.025 25.025/25.025 ============================== DOMAIN SIZE 2 ========================= 25.025/25.025 25.025/25.025 ============================== MODEL ================================= 25.025/25.025 25.025/25.025 interpretation( 2, [number=1, seconds=0], [ 25.025/25.025 25.025/25.025 function(b, [ 0 ]), 25.025/25.025 25.025/25.025 function(cx, [ 1 ]), 25.025/25.025 25.025/25.025 function(f(_,_), [ 25.025/25.025 0, 0, 25.025/25.025 0, 0 ]), 25.025/25.025 25.025/25.025 relation(->*(_,_), [ 25.025/25.025 1, 0, 25.025/25.025 0, 1 ]), 25.025/25.025 25.025/25.025 relation(->(_,_), [ 25.025/25.025 0, 0, 25.025/25.025 0, 0 ]) 25.025/25.025 ]). 25.025/25.025 25.025/25.025 ============================== end of model ========================== 25.025/25.025 25.025/25.025 ============================== STATISTICS ============================ 25.025/25.025 25.025/25.025 For domain size 2. 25.025/25.025 25.025/25.025 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 25.025/25.025 Ground clauses: seen=28, kept=24. 25.025/25.025 Selections=6, assignments=6, propagations=8, current_models=1. 25.025/25.025 Rewrite_terms=36, rewrite_bools=33, indexes=4. 25.025/25.025 Rules_from_neg_clauses=1, cross_offs=1. 25.025/25.025 25.025/25.025 ============================== end of statistics ===================== 25.025/25.025 25.025/25.025 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.025/25.025 25.025/25.025 Exiting with 1 model. 25.025/25.025 25.025/25.025 Process 33251 exit (max_models) Wed Jul 14 09:47:41 2021 25.025/25.025 The process finished Wed Jul 14 09:47:41 2021 25.025/25.025 25.025/25.025 25.025/25.025 Mace4 cooked interpretation: 25.025/25.025 25.025/25.025 25.025/25.025 25.025/25.025 The problem is infeasible. 25.025/25.025 25.025/25.025 25.025/25.025 The problem is not joinable. 25.025/25.025 23.00user 1.17system 0:25.25elapsed 95%CPU (0avgtext+0avgdata 243568maxresident)k 25.025/25.025 0inputs+0outputs (0major+144768minor)pagefaults 0swaps