25.03/25.03 NO 25.03/25.03 25.03/25.03 Problem 1: 25.03/25.03 25.03/25.03 25.03/25.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.03/25.03 Confluence Problem: 25.03/25.03 (VAR vNonEmpty:S x:S) 25.03/25.03 (STRATEGY CONTEXTSENSITIVE 25.03/25.03 (f 1) 25.03/25.03 (g 1) 25.03/25.03 (a) 25.03/25.03 (b) 25.03/25.03 (fSNonEmpty) 25.03/25.03 ) 25.03/25.03 (RULES 25.03/25.03 f(x:S) -> b 25.03/25.03 g(x:S) -> f(x:S) 25.03/25.03 g(x:S) -> a | f(x:S) ->* b 25.03/25.03 ) 25.03/25.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.03/25.03 25.03/25.03 25.03/25.03 Problem 1: 25.03/25.03 25.03/25.03 Inlining of Conditions Processor [STERN17]: 25.03/25.03 25.03/25.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.03/25.03 Confluence Problem: 25.03/25.03 (VAR vNonEmpty:S x:S) 25.03/25.03 (STRATEGY CONTEXTSENSITIVE 25.03/25.03 (f 1) 25.03/25.03 (g 1) 25.03/25.03 (a) 25.03/25.03 (b) 25.03/25.03 (fSNonEmpty) 25.03/25.03 ) 25.03/25.03 (RULES 25.03/25.03 f(x:S) -> b 25.03/25.03 g(x:S) -> f(x:S) 25.03/25.03 g(x:S) -> a | f(x:S) ->* b 25.03/25.03 ) 25.03/25.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.03/25.03 25.03/25.03 25.03/25.03 Problem 1: 25.03/25.03 25.03/25.03 Clean CTRS Processor: 25.03/25.03 25.03/25.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.03/25.03 Confluence Problem: 25.03/25.03 (VAR x:S) 25.03/25.03 (STRATEGY CONTEXTSENSITIVE 25.03/25.03 (f 1) 25.03/25.03 (g 1) 25.03/25.03 (a) 25.03/25.03 (b) 25.03/25.03 (fSNonEmpty) 25.03/25.03 ) 25.03/25.03 (RULES 25.03/25.03 f(x:S) -> b 25.03/25.03 g(x:S) -> f(x:S) 25.03/25.03 g(x:S) -> a | f(x:S) ->* b 25.03/25.03 ) 25.03/25.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.03/25.03 25.03/25.03 CRule InfChecker Info: 25.03/25.03 f(x:S) -> b 25.03/25.03 Rule remains 25.03/25.03 Proof: 25.03/25.03 NO_CONDS 25.03/25.03 25.03/25.03 CRule InfChecker Info: 25.03/25.03 g(x:S) -> f(x:S) 25.03/25.03 Rule remains 25.03/25.03 Proof: 25.03/25.03 NO_CONDS 25.03/25.03 25.03/25.03 CRule InfChecker Info: 25.03/25.03 g(x:S) -> a | f(x:S) ->* b 25.03/25.03 Rule remains 25.03/25.03 Proof: 25.03/25.03 NO 25.03/25.03 25.03/25.03 Problem 1: 25.03/25.03 25.03/25.03 Infeasibility Problem: 25.03/25.03 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 25.03/25.03 (STRATEGY CONTEXTSENSITIVE 25.03/25.03 (f 1) 25.03/25.03 (g 1) 25.03/25.03 (a) 25.03/25.03 (b) 25.03/25.03 (fSNonEmpty) 25.03/25.03 ) 25.03/25.03 (RULES 25.03/25.03 f(x:S:S) -> b 25.03/25.03 g(x:S:S) -> f(x:S:S) 25.03/25.03 g(x:S:S) -> a | f(x:S:S) ->* b 25.03/25.03 )] 25.03/25.03 25.03/25.03 Infeasibility Conditions: 25.03/25.03 f(x:S:S) ->* b 25.03/25.03 25.03/25.03 Problem 1: 25.03/25.03 25.03/25.03 Obtaining a proof using Prover9: 25.03/25.03 25.03/25.03 -> Prover9 Output: 25.03/25.03 ============================== Prover9 =============================== 25.03/25.03 Prover9 (64) version 2009-11A, November 2009. 25.03/25.03 Process 33321 was started by ubuntu on ubuntu, 25.03/25.03 Wed Jul 14 09:48:07 2021 25.03/25.03 The command was "./prover9 -f /tmp/prover933312-0.in". 25.03/25.03 ============================== end of head =========================== 25.03/25.03 25.03/25.03 ============================== INPUT ================================= 25.03/25.03 25.03/25.03 % Reading from file /tmp/prover933312-0.in 25.03/25.03 25.03/25.03 assign(max_seconds,20). 25.03/25.03 25.03/25.03 formulas(assumptions). 25.03/25.03 ->*_s0(x,x) # label(reflexivity). 25.03/25.03 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.03/25.03 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 25.03/25.03 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 25.03/25.03 ->_s0(f(x2),b) # label(replacement). 25.03/25.03 ->_s0(g(x2),f(x2)) # label(replacement). 25.03/25.03 ->*_s0(f(x2),b) -> ->_s0(g(x2),a) # label(replacement). 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 formulas(goals). 25.03/25.03 (exists x2 ->*_s0(f(x2),b)) # label(goal). 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 ============================== end of input ========================== 25.03/25.03 25.03/25.03 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.03/25.03 25.03/25.03 % Formulas that are not ordinary clauses: 25.03/25.03 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.03/25.03 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.03/25.03 3 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.03/25.03 4 ->*_s0(f(x2),b) -> ->_s0(g(x2),a) # label(replacement) # label(non_clause). [assumption]. 25.03/25.03 5 (exists x2 ->*_s0(f(x2),b)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.03/25.03 25.03/25.03 ============================== end of process non-clausal formulas === 25.03/25.03 25.03/25.03 ============================== PROCESS INITIAL CLAUSES =============== 25.03/25.03 25.03/25.03 % Clauses before input processing: 25.03/25.03 25.03/25.03 formulas(usable). 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 formulas(sos). 25.03/25.03 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.03/25.03 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.03/25.03 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.03/25.03 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.03/25.03 ->_s0(f(x),b) # label(replacement). [assumption]. 25.03/25.03 ->_s0(g(x),f(x)) # label(replacement). [assumption]. 25.03/25.03 -->*_s0(f(x),b) | ->_s0(g(x),a) # label(replacement). [clausify(4)]. 25.03/25.03 -->*_s0(f(x),b) # label(goal). [deny(5)]. 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 formulas(demodulators). 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 ============================== PREDICATE ELIMINATION ================= 25.03/25.03 25.03/25.03 No predicates eliminated. 25.03/25.03 25.03/25.03 ============================== end predicate elimination ============= 25.03/25.03 25.03/25.03 Auto_denials: 25.03/25.03 % copying label goal to answer in negative clause 25.03/25.03 25.03/25.03 Term ordering decisions: 25.03/25.03 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.03/25.03 Function symbol precedence: function_order([ b, a, f, g ]). 25.03/25.03 After inverse_order: (no changes). 25.03/25.03 Unfolding symbols: (none). 25.03/25.03 25.03/25.03 Auto_inference settings: 25.03/25.03 % set(neg_binary_resolution). % (HNE depth_diff=-2) 25.03/25.03 % clear(ordered_res). % (HNE depth_diff=-2) 25.03/25.03 % set(ur_resolution). % (HNE depth_diff=-2) 25.03/25.03 % set(ur_resolution) -> set(pos_ur_resolution). 25.03/25.03 % set(ur_resolution) -> set(neg_ur_resolution). 25.03/25.03 25.03/25.03 Auto_process settings: (no changes). 25.03/25.03 25.03/25.03 kept: 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.03/25.03 kept: 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.03/25.03 kept: 8 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.03/25.03 kept: 9 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.03/25.03 kept: 10 ->_s0(f(x),b) # label(replacement). [assumption]. 25.03/25.03 kept: 11 ->_s0(g(x),f(x)) # label(replacement). [assumption]. 25.03/25.03 kept: 12 -->*_s0(f(x),b) | ->_s0(g(x),a) # label(replacement). [clausify(4)]. 25.03/25.03 kept: 13 -->*_s0(f(x),b) # label(goal) # answer(goal). [deny(5)]. 25.03/25.03 25.03/25.03 ============================== end of process initial clauses ======== 25.03/25.03 25.03/25.03 ============================== CLAUSES FOR SEARCH ==================== 25.03/25.03 25.03/25.03 % Clauses after input processing: 25.03/25.03 25.03/25.03 formulas(usable). 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 formulas(sos). 25.03/25.03 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.03/25.03 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.03/25.03 8 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.03/25.03 9 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.03/25.03 10 ->_s0(f(x),b) # label(replacement). [assumption]. 25.03/25.03 11 ->_s0(g(x),f(x)) # label(replacement). [assumption]. 25.03/25.03 13 -->*_s0(f(x),b) # label(goal) # answer(goal). [deny(5)]. 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 formulas(demodulators). 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 ============================== end of clauses for search ============= 25.03/25.03 25.03/25.03 ============================== SEARCH ================================ 25.03/25.03 25.03/25.03 % Starting search at 0.00 seconds. 25.03/25.03 25.03/25.03 given #1 (I,wt=3): 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.03/25.03 25.03/25.03 given #2 (I,wt=9): 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.03/25.03 25.03/25.03 given #3 (I,wt=8): 8 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.03/25.03 25.03/25.03 given #4 (I,wt=8): 9 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.03/25.03 25.03/25.03 given #5 (I,wt=4): 10 ->_s0(f(x),b) # label(replacement). [assumption]. 25.03/25.03 25.03/25.03 ============================== PROOF ================================= 25.03/25.03 25.03/25.03 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.03/25.03 % Length of proof is 8. 25.03/25.03 % Level of proof is 3. 25.03/25.03 % Maximum clause weight is 9.000. 25.03/25.03 % Given clauses 5. 25.03/25.03 25.03/25.03 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.03/25.03 5 (exists x2 ->*_s0(f(x2),b)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.03/25.03 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.03/25.03 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.03/25.03 10 ->_s0(f(x),b) # label(replacement). [assumption]. 25.03/25.03 13 -->*_s0(f(x),b) # label(goal) # answer(goal). [deny(5)]. 25.03/25.03 16 ->*_s0(f(x),b). [ur(7,a,10,a,b,6,a)]. 25.03/25.03 17 $F # answer(goal). [resolve(16,a,13,a)]. 25.03/25.03 25.03/25.03 ============================== end of proof ========================== 25.03/25.03 25.03/25.03 ============================== STATISTICS ============================ 25.03/25.03 25.03/25.03 Given=5. Generated=11. Kept=11. proofs=1. 25.03/25.03 Usable=5. Sos=2. Demods=0. Limbo=2, Disabled=9. Hints=0. 25.03/25.03 Kept_by_rule=0, Deleted_by_rule=0. 25.03/25.03 Forward_subsumed=0. Back_subsumed=1. 25.03/25.03 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.03/25.03 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.03/25.03 Demod_attempts=0. Demod_rewrites=0. 25.03/25.03 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.03/25.03 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=6. 25.03/25.03 Megabytes=0.05. 25.03/25.03 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.03/25.03 25.03/25.03 ============================== end of statistics ===================== 25.03/25.03 25.03/25.03 ============================== end of search ========================= 25.03/25.03 25.03/25.03 THEOREM PROVED 25.03/25.03 25.03/25.03 Exiting with 1 proof. 25.03/25.03 25.03/25.03 Process 33321 exit (max_proofs) Wed Jul 14 09:48:07 2021 25.03/25.03 25.03/25.03 25.03/25.03 The problem is feasible. 25.03/25.03 25.03/25.03 25.03/25.03 Problem 1: 25.03/25.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.03/25.03 Confluence Problem: 25.03/25.03 (VAR x:S) 25.03/25.03 (STRATEGY CONTEXTSENSITIVE 25.03/25.03 (f 1) 25.03/25.03 (g 1) 25.03/25.03 (a) 25.03/25.03 (b) 25.03/25.03 (fSNonEmpty) 25.03/25.03 ) 25.03/25.03 (RULES 25.03/25.03 f(x:S) -> b 25.03/25.03 g(x:S) -> f(x:S) 25.03/25.03 g(x:S) -> a | f(x:S) ->* b 25.03/25.03 ) 25.03/25.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.03/25.03 25.03/25.03 Critical Pairs Processor: 25.03/25.03 -> Rules: 25.03/25.03 f(x:S) -> b 25.03/25.03 g(x:S) -> f(x:S) 25.03/25.03 g(x:S) -> a | f(x:S) ->* b 25.03/25.03 -> Vars: 25.03/25.03 "x", "x", "x" 25.03/25.03 -> FVars: 25.03/25.03 "x2", "x3", "x4" 25.03/25.03 -> PVars: 25.03/25.03 "x": ["x2", "x3", "x4"] 25.03/25.03 25.03/25.03 -> Rlps: 25.03/25.03 crule: f(x2:S) -> b, id: 1, possubterms: f(x2:S)-> [] 25.03/25.03 crule: g(x3:S) -> f(x3:S), id: 2, possubterms: g(x3:S)-> [] 25.03/25.03 crule: g(x4:S) -> a | f(x4:S) ->* b, id: 3, possubterms: g(x4:S)-> [] 25.03/25.03 25.03/25.03 -> Unifications: 25.03/25.03 R3 unifies with R2 at p: [], l: g(x4:S), lp: g(x4:S), conds: {f(x4:S) ->* b}, sig: {x4:S -> x:S}, l': g(x:S), r: a, r': f(x:S) 25.03/25.03 25.03/25.03 -> Critical pairs info: 25.03/25.03 | f(x:S) ->* b => Not trivial, Overlay, N1 25.03/25.03 25.03/25.03 -> Problem conclusions: 25.03/25.03 Left linear, Right linear, Linear 25.03/25.03 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 25.03/25.03 CTRS Type: 1 25.03/25.03 Deterministic, Strongly deterministic 25.03/25.03 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.03/25.03 Maybe right-stable CTRS, Overlay CTRS 25.03/25.03 Maybe normal CTRS, Maybe almost normal CTRS 25.03/25.03 Maybe terminating CTRS, Maybe joinable CCPs 25.03/25.03 Maybe level confluent 25.03/25.03 Maybe confluent 25.03/25.03 25.03/25.03 Problem 1: 25.03/25.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.03/25.03 Confluence Problem: 25.03/25.03 (VAR x:S) 25.03/25.03 (STRATEGY CONTEXTSENSITIVE 25.03/25.03 (f 1) 25.03/25.03 (g 1) 25.03/25.03 (a) 25.03/25.03 (b) 25.03/25.03 (fSNonEmpty) 25.03/25.03 ) 25.03/25.03 (RULES 25.03/25.03 f(x:S) -> b 25.03/25.03 g(x:S) -> f(x:S) 25.03/25.03 g(x:S) -> a | f(x:S) ->* b 25.03/25.03 ) 25.03/25.03 Critical Pairs: 25.03/25.03 | f(x:S) ->* b => Not trivial, Overlay, N1 25.03/25.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.03/25.03 25.03/25.03 Conditional Critical Pairs Distributor Processor 25.03/25.03 25.03/25.03 Problem 1: 25.03/25.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.03/25.03 Confluence Problem: 25.03/25.03 (VAR x:S) 25.03/25.03 (STRATEGY CONTEXTSENSITIVE 25.03/25.03 (f 1) 25.03/25.03 (g 1) 25.03/25.03 (a) 25.03/25.03 (b) 25.03/25.03 (fSNonEmpty) 25.03/25.03 ) 25.03/25.03 (RULES 25.03/25.03 f(x:S) -> b 25.03/25.03 g(x:S) -> f(x:S) 25.03/25.03 g(x:S) -> a | f(x:S) ->* b 25.03/25.03 ) 25.03/25.03 Critical Pairs: 25.03/25.03 | f(x:S) ->* b => Not trivial, Overlay, N1 25.03/25.03 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.03/25.03 25.03/25.03 Infeasible Convergence Constantized CCP Processor: 25.03/25.03 25.03/25.03 Infeasible constantized conditions? 25.03/25.03 NO 25.03/25.03 25.03/25.03 Problem 1: 25.03/25.03 25.03/25.03 Infeasibility Problem: 25.03/25.03 [(VAR vNonEmpty:S x:S) 25.03/25.03 (STRATEGY CONTEXTSENSITIVE 25.03/25.03 (f 1) 25.03/25.03 (g 1) 25.03/25.03 (a) 25.03/25.03 (b) 25.03/25.03 (cx) 25.03/25.03 (fSNonEmpty) 25.03/25.03 ) 25.03/25.03 (RULES 25.03/25.03 f(x:S) -> b 25.03/25.03 g(x:S) -> f(x:S) 25.03/25.03 g(x:S) -> a | f(x:S) ->* b 25.03/25.03 )] 25.03/25.03 25.03/25.03 Infeasibility Conditions: 25.03/25.03 f(cx) ->* b 25.03/25.03 25.03/25.03 Problem 1: 25.03/25.03 25.03/25.03 Obtaining a proof using Prover9: 25.03/25.03 25.03/25.03 -> Prover9 Output: 25.03/25.03 ============================== Prover9 =============================== 25.03/25.03 Prover9 (64) version 2009-11A, November 2009. 25.03/25.03 Process 33354 was started by ubuntu on ubuntu, 25.03/25.03 Wed Jul 14 09:48:07 2021 25.03/25.03 The command was "./prover9 -f /tmp/prover933337-0.in". 25.03/25.03 ============================== end of head =========================== 25.03/25.03 25.03/25.03 ============================== INPUT ================================= 25.03/25.03 25.03/25.03 % Reading from file /tmp/prover933337-0.in 25.03/25.03 25.03/25.03 assign(max_seconds,20). 25.03/25.03 25.03/25.03 formulas(assumptions). 25.03/25.03 ->*_s0(x,x) # label(reflexivity). 25.03/25.03 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.03/25.03 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 25.03/25.03 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence). 25.03/25.03 ->_s0(f(x1),b) # label(replacement). 25.03/25.03 ->_s0(g(x1),f(x1)) # label(replacement). 25.03/25.03 ->*_s0(f(x1),b) -> ->_s0(g(x1),a) # label(replacement). 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 formulas(goals). 25.03/25.03 ->*_s0(f(cx),b) # label(goal). 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 ============================== end of input ========================== 25.03/25.03 25.03/25.03 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.03/25.03 25.03/25.03 % Formulas that are not ordinary clauses: 25.03/25.03 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.03/25.03 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.03/25.03 3 ->_s0(x1,y) -> ->_s0(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.03/25.03 4 ->*_s0(f(x1),b) -> ->_s0(g(x1),a) # label(replacement) # label(non_clause). [assumption]. 25.03/25.03 5 ->*_s0(f(cx),b) # label(goal) # label(non_clause) # label(goal). [goal]. 25.03/25.03 25.03/25.03 ============================== end of process non-clausal formulas === 25.03/25.03 25.03/25.03 ============================== PROCESS INITIAL CLAUSES =============== 25.03/25.03 25.03/25.03 % Clauses before input processing: 25.03/25.03 25.03/25.03 formulas(usable). 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 formulas(sos). 25.03/25.03 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.03/25.03 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.03/25.03 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.03/25.03 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.03/25.03 ->_s0(f(x),b) # label(replacement). [assumption]. 25.03/25.03 ->_s0(g(x),f(x)) # label(replacement). [assumption]. 25.03/25.03 -->*_s0(f(x),b) | ->_s0(g(x),a) # label(replacement). [clausify(4)]. 25.03/25.03 -->*_s0(f(cx),b) # label(goal). [deny(5)]. 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 formulas(demodulators). 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 ============================== PREDICATE ELIMINATION ================= 25.03/25.03 25.03/25.03 No predicates eliminated. 25.03/25.03 25.03/25.03 ============================== end predicate elimination ============= 25.03/25.03 25.03/25.03 Auto_denials: 25.03/25.03 % copying label goal to answer in negative clause 25.03/25.03 25.03/25.03 Term ordering decisions: 25.03/25.03 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.03/25.03 Function symbol precedence: function_order([ b, a, cx, f, g ]). 25.03/25.03 After inverse_order: (no changes). 25.03/25.03 Unfolding symbols: (none). 25.03/25.03 25.03/25.03 Auto_inference settings: 25.03/25.03 % set(neg_binary_resolution). % (HNE depth_diff=-2) 25.03/25.03 % clear(ordered_res). % (HNE depth_diff=-2) 25.03/25.03 % set(ur_resolution). % (HNE depth_diff=-2) 25.03/25.03 % set(ur_resolution) -> set(pos_ur_resolution). 25.03/25.03 % set(ur_resolution) -> set(neg_ur_resolution). 25.03/25.03 25.03/25.03 Auto_process settings: (no changes). 25.03/25.03 25.03/25.03 kept: 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.03/25.03 kept: 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.03/25.03 kept: 8 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.03/25.03 kept: 9 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.03/25.03 kept: 10 ->_s0(f(x),b) # label(replacement). [assumption]. 25.03/25.03 kept: 11 ->_s0(g(x),f(x)) # label(replacement). [assumption]. 25.03/25.03 kept: 12 -->*_s0(f(x),b) | ->_s0(g(x),a) # label(replacement). [clausify(4)]. 25.03/25.03 kept: 13 -->*_s0(f(cx),b) # label(goal) # answer(goal). [deny(5)]. 25.03/25.03 25.03/25.03 ============================== end of process initial clauses ======== 25.03/25.03 25.03/25.03 ============================== CLAUSES FOR SEARCH ==================== 25.03/25.03 25.03/25.03 % Clauses after input processing: 25.03/25.03 25.03/25.03 formulas(usable). 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 formulas(sos). 25.03/25.03 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.03/25.03 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.03/25.03 8 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.03/25.03 9 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.03/25.03 10 ->_s0(f(x),b) # label(replacement). [assumption]. 25.03/25.03 11 ->_s0(g(x),f(x)) # label(replacement). [assumption]. 25.03/25.03 12 -->*_s0(f(x),b) | ->_s0(g(x),a) # label(replacement). [clausify(4)]. 25.03/25.03 13 -->*_s0(f(cx),b) # label(goal) # answer(goal). [deny(5)]. 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 formulas(demodulators). 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 ============================== end of clauses for search ============= 25.03/25.03 25.03/25.03 ============================== SEARCH ================================ 25.03/25.03 25.03/25.03 % Starting search at 0.00 seconds. 25.03/25.03 25.03/25.03 given #1 (I,wt=3): 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.03/25.03 25.03/25.03 given #2 (I,wt=9): 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.03/25.03 25.03/25.03 given #3 (I,wt=8): 8 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.03/25.03 25.03/25.03 given #4 (I,wt=8): 9 -->_s0(x,y) | ->_s0(g(x),g(y)) # label(congruence). [clausify(3)]. 25.03/25.03 25.03/25.03 given #5 (I,wt=4): 10 ->_s0(f(x),b) # label(replacement). [assumption]. 25.03/25.03 25.03/25.03 ============================== PROOF ================================= 25.03/25.03 25.03/25.03 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.03/25.03 % Length of proof is 8. 25.03/25.03 % Level of proof is 3. 25.03/25.03 % Maximum clause weight is 9.000. 25.03/25.03 % Given clauses 5. 25.03/25.03 25.03/25.03 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.03/25.03 5 ->*_s0(f(cx),b) # label(goal) # label(non_clause) # label(goal). [goal]. 25.03/25.03 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.03/25.03 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.03/25.03 10 ->_s0(f(x),b) # label(replacement). [assumption]. 25.03/25.03 13 -->*_s0(f(cx),b) # label(goal) # answer(goal). [deny(5)]. 25.03/25.03 16 ->*_s0(f(x),b). [ur(7,a,10,a,b,6,a)]. 25.03/25.03 17 $F # answer(goal). [resolve(16,a,13,a)]. 25.03/25.03 25.03/25.03 ============================== end of proof ========================== 25.03/25.03 25.03/25.03 ============================== STATISTICS ============================ 25.03/25.03 25.03/25.03 Given=5. Generated=11. Kept=11. proofs=1. 25.03/25.03 Usable=5. Sos=3. Demods=0. Limbo=2, Disabled=8. Hints=0. 25.03/25.03 Kept_by_rule=0, Deleted_by_rule=0. 25.03/25.03 Forward_subsumed=0. Back_subsumed=0. 25.03/25.03 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.03/25.03 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.03/25.03 Demod_attempts=0. Demod_rewrites=0. 25.03/25.03 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.03/25.03 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=5. 25.03/25.03 Megabytes=0.05. 25.03/25.03 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.03/25.03 25.03/25.03 ============================== end of statistics ===================== 25.03/25.03 25.03/25.03 ============================== end of search ========================= 25.03/25.03 25.03/25.03 THEOREM PROVED 25.03/25.03 25.03/25.03 Exiting with 1 proof. 25.03/25.03 25.03/25.03 Process 33354 exit (max_proofs) Wed Jul 14 09:48:07 2021 25.03/25.03 25.03/25.03 25.03/25.03 The problem is feasible. 25.03/25.03 25.03/25.03 Infeasible constantized CCP convergence? 25.03/25.03 YES 25.03/25.03 25.03/25.03 Problem 1: 25.03/25.03 25.03/25.03 Infeasibility Problem: 25.03/25.03 [(VAR vNonEmpty:S x:S z:S) 25.03/25.03 (STRATEGY CONTEXTSENSITIVE 25.03/25.03 (f 1) 25.03/25.03 (g 1) 25.03/25.03 (a) 25.03/25.03 (b) 25.03/25.03 (cx) 25.03/25.03 (fSNonEmpty) 25.03/25.03 ) 25.03/25.03 (RULES 25.03/25.03 f(x:S) -> b 25.03/25.03 g(x:S) -> f(x:S) 25.03/25.03 g(x:S) -> a | f(x:S) ->* b 25.03/25.03 )] 25.03/25.03 25.03/25.03 Infeasibility Conditions: 25.03/25.03 f(cx) ->* z:S, a ->* z:S 25.03/25.03 25.03/25.03 Problem 1: 25.03/25.03 25.03/25.03 Obtaining a model using Mace4: 25.03/25.03 25.03/25.03 -> Usable Rules: 25.03/25.03 f(x:S) -> b 25.03/25.03 25.03/25.03 -> Mace4 Output: 25.03/25.03 ============================== Mace4 ================================= 25.03/25.03 Mace4 (64) version 2009-11A, November 2009. 25.03/25.03 Process 33397 was started by ubuntu on ubuntu, 25.03/25.03 Wed Jul 14 09:48:07 2021 25.03/25.03 The command was "./mace4 -c -f /tmp/mace433376-2.in". 25.03/25.03 ============================== end of head =========================== 25.03/25.03 25.03/25.03 ============================== INPUT ================================= 25.03/25.03 25.03/25.03 % Reading from file /tmp/mace433376-2.in 25.03/25.03 25.03/25.03 assign(max_seconds,20). 25.03/25.03 25.03/25.03 formulas(assumptions). 25.03/25.03 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 25.03/25.03 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence). 25.03/25.03 ->(f(x1),b) # label(replacement). 25.03/25.03 ->*(x,x) # label(reflexivity). 25.03/25.03 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 formulas(goals). 25.03/25.03 (exists x2 (->*(f(cx),x2) & ->*(a,x2))) # label(goal). 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 ============================== end of input ========================== 25.03/25.03 25.03/25.03 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.03/25.03 25.03/25.03 % Formulas that are not ordinary clauses: 25.03/25.03 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.03/25.03 2 ->(x1,y) -> ->(g(x1),g(y)) # label(congruence) # label(non_clause). [assumption]. 25.03/25.03 3 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.03/25.03 4 (exists x2 (->*(f(cx),x2) & ->*(a,x2))) # label(goal) # label(non_clause) # label(goal). [goal]. 25.03/25.03 25.03/25.03 ============================== end of process non-clausal formulas === 25.03/25.03 25.03/25.03 ============================== CLAUSES FOR SEARCH ==================== 25.03/25.03 25.03/25.03 formulas(mace4_clauses). 25.03/25.03 -->(x,y) | ->(f(x),f(y)) # label(congruence). 25.03/25.03 -->(x,y) | ->(g(x),g(y)) # label(congruence). 25.03/25.03 ->(f(x),b) # label(replacement). 25.03/25.03 ->*(x,x) # label(reflexivity). 25.03/25.03 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 25.03/25.03 -->*(f(cx),x) | -->*(a,x) # label(goal). 25.03/25.03 end_of_list. 25.03/25.03 25.03/25.03 ============================== end of clauses for search ============= 25.03/25.03 25.03/25.03 % There are no natural numbers in the input. 25.03/25.03 25.03/25.03 ============================== DOMAIN SIZE 2 ========================= 25.03/25.03 25.03/25.03 ============================== MODEL ================================= 25.03/25.03 25.03/25.03 interpretation( 2, [number=1, seconds=0], [ 25.03/25.03 25.03/25.03 function(a, [ 0 ]), 25.03/25.03 25.03/25.03 function(b, [ 1 ]), 25.03/25.03 25.03/25.03 function(cx, [ 0 ]), 25.03/25.03 25.03/25.03 function(f(_), [ 1, 1 ]), 25.03/25.03 25.03/25.03 function(g(_), [ 0, 1 ]), 25.03/25.03 25.03/25.03 relation(->*(_,_), [ 25.03/25.03 1, 0, 25.03/25.03 0, 1 ]), 25.03/25.03 25.03/25.03 relation(->(_,_), [ 25.03/25.03 0, 0, 25.03/25.03 0, 1 ]) 25.03/25.03 ]). 25.03/25.03 25.03/25.03 ============================== end of model ========================== 25.03/25.03 25.03/25.03 ============================== STATISTICS ============================ 25.03/25.03 25.03/25.03 For domain size 2. 25.03/25.03 25.03/25.03 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 25.03/25.03 Ground clauses: seen=22, kept=18. 25.03/25.03 Selections=8, assignments=12, propagations=16, current_models=1. 25.03/25.03 Rewrite_terms=49, rewrite_bools=36, indexes=16. 25.03/25.03 Rules_from_neg_clauses=3, cross_offs=3. 25.03/25.03 25.03/25.03 ============================== end of statistics ===================== 25.03/25.03 25.03/25.03 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.03/25.03 25.03/25.03 Exiting with 1 model. 25.03/25.03 25.03/25.03 Process 33397 exit (max_models) Wed Jul 14 09:48:07 2021 25.03/25.03 The process finished Wed Jul 14 09:48:07 2021 25.03/25.03 25.03/25.03 25.03/25.03 Mace4 cooked interpretation: 25.03/25.03 25.03/25.03 25.03/25.03 25.03/25.03 The problem is infeasible. 25.03/25.03 25.03/25.03 25.03/25.03 The problem is not joinable. 25.03/25.03 25.37user 0.79system 0:25.30elapsed 103%CPU (0avgtext+0avgdata 209032maxresident)k 25.03/25.03 0inputs+0outputs (0major+133131minor)pagefaults 0swaps