25.022/25.022 NO 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 Confluence Problem: 25.022/25.022 (VAR vNonEmpty:S x:S y:S z:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (a) 25.022/25.022 (f 1 2) 25.022/25.022 (c) 25.022/25.022 (d) 25.022/25.022 (fSNonEmpty) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 a -> c 25.022/25.022 a -> d 25.022/25.022 f(x:S,y:S) -> z:S | c ->* x:S, y:S ->* z:S 25.022/25.022 ) 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 Inlining of Conditions Processor [STERN17]: 25.022/25.022 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 Confluence Problem: 25.022/25.022 (VAR vNonEmpty:S x:S y:S z:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (a) 25.022/25.022 (f 1 2) 25.022/25.022 (c) 25.022/25.022 (d) 25.022/25.022 (fSNonEmpty) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 a -> c 25.022/25.022 a -> d 25.022/25.022 f(x:S,y:S) -> y:S | c ->* x:S 25.022/25.022 ) 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 Clean CTRS Processor: 25.022/25.022 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 Confluence Problem: 25.022/25.022 (VAR x:S y:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (a) 25.022/25.022 (f 1 2) 25.022/25.022 (c) 25.022/25.022 (d) 25.022/25.022 (fSNonEmpty) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 a -> c 25.022/25.022 a -> d 25.022/25.022 f(x:S,y:S) -> y:S | c ->* x:S 25.022/25.022 ) 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 25.022/25.022 CRule InfChecker Info: 25.022/25.022 a -> c 25.022/25.022 Rule remains 25.022/25.022 Proof: 25.022/25.022 NO_CONDS 25.022/25.022 25.022/25.022 CRule InfChecker Info: 25.022/25.022 a -> d 25.022/25.022 Rule remains 25.022/25.022 Proof: 25.022/25.022 NO_CONDS 25.022/25.022 25.022/25.022 CRule InfChecker Info: 25.022/25.022 f(x:S,y:S) -> y:S | c ->* x:S 25.022/25.022 Rule remains 25.022/25.022 Proof: 25.022/25.022 NO 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 Infeasibility Problem: 25.022/25.022 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S y:S:S z:S:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (a) 25.022/25.022 (f 1 2) 25.022/25.022 (c) 25.022/25.022 (d) 25.022/25.022 (fSNonEmpty) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 a -> c 25.022/25.022 a -> d 25.022/25.022 f(x:S:S,y:S:S) -> y:S:S | c ->* x:S:S 25.022/25.022 )] 25.022/25.022 25.022/25.022 Infeasibility Conditions: 25.022/25.022 c ->* x:S:S 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 Obtaining a proof using Prover9: 25.022/25.022 25.022/25.022 -> Prover9 Output: 25.022/25.022 ============================== Prover9 =============================== 25.022/25.022 Prover9 (64) version 2009-11A, November 2009. 25.022/25.022 Process 110442 was started by ubuntu on ubuntu, 25.022/25.022 Fri Jul 16 15:07:16 2021 25.022/25.022 The command was "./prover9 -f /tmp/prover9110431-0.in". 25.022/25.022 ============================== end of head =========================== 25.022/25.022 25.022/25.022 ============================== INPUT ================================= 25.022/25.022 25.022/25.022 % Reading from file /tmp/prover9110431-0.in 25.022/25.022 25.022/25.022 assign(max_seconds,20). 25.022/25.022 25.022/25.022 formulas(assumptions). 25.022/25.022 ->*_s0(x,x) # label(reflexivity). 25.022/25.022 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.022/25.022 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 25.022/25.022 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 25.022/25.022 ->_s0(a,c) # label(replacement). 25.022/25.022 ->_s0(a,d) # label(replacement). 25.022/25.022 ->*_s0(c,x2) -> ->_s0(f(x2,x3),x3) # label(replacement). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 formulas(goals). 25.022/25.022 (exists x2 ->*_s0(c,x2)) # label(goal). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 ============================== end of input ========================== 25.022/25.022 25.022/25.022 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.022/25.022 25.022/25.022 % Formulas that are not ordinary clauses: 25.022/25.022 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.022/25.022 2 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.022/25.022 3 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.022/25.022 4 ->*_s0(c,x2) -> ->_s0(f(x2,x3),x3) # label(replacement) # label(non_clause). [assumption]. 25.022/25.022 5 (exists x2 ->*_s0(c,x2)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.022/25.022 25.022/25.022 ============================== end of process non-clausal formulas === 25.022/25.022 25.022/25.022 ============================== PROCESS INITIAL CLAUSES =============== 25.022/25.022 25.022/25.022 % Clauses before input processing: 25.022/25.022 25.022/25.022 formulas(usable). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 formulas(sos). 25.022/25.022 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.022/25.022 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.022/25.022 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.022/25.022 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.022/25.022 ->_s0(a,c) # label(replacement). [assumption]. 25.022/25.022 ->_s0(a,d) # label(replacement). [assumption]. 25.022/25.022 -->*_s0(c,x) | ->_s0(f(x,y),y) # label(replacement). [clausify(4)]. 25.022/25.022 -->*_s0(c,x) # label(goal). [deny(5)]. 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 formulas(demodulators). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 ============================== PREDICATE ELIMINATION ================= 25.022/25.022 25.022/25.022 No predicates eliminated. 25.022/25.022 25.022/25.022 ============================== end predicate elimination ============= 25.022/25.022 25.022/25.022 Auto_denials: 25.022/25.022 % copying label goal to answer in negative clause 25.022/25.022 25.022/25.022 Term ordering decisions: 25.022/25.022 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.022/25.022 Function symbol precedence: function_order([ a, c, d, f ]). 25.022/25.022 After inverse_order: (no changes). 25.022/25.022 Unfolding symbols: (none). 25.022/25.022 25.022/25.022 Auto_inference settings: 25.022/25.022 % set(neg_binary_resolution). % (HNE depth_diff=-3) 25.022/25.022 % clear(ordered_res). % (HNE depth_diff=-3) 25.022/25.022 % set(ur_resolution). % (HNE depth_diff=-3) 25.022/25.022 % set(ur_resolution) -> set(pos_ur_resolution). 25.022/25.022 % set(ur_resolution) -> set(neg_ur_resolution). 25.022/25.022 25.022/25.022 Auto_process settings: (no changes). 25.022/25.022 25.022/25.022 kept: 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.022/25.022 kept: 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.022/25.022 kept: 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.022/25.022 kept: 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.022/25.022 kept: 10 ->_s0(a,c) # label(replacement). [assumption]. 25.022/25.022 kept: 11 ->_s0(a,d) # label(replacement). [assumption]. 25.022/25.022 kept: 12 -->*_s0(c,x) | ->_s0(f(x,y),y) # label(replacement). [clausify(4)]. 25.022/25.022 kept: 13 -->*_s0(c,x) # label(goal) # answer(goal). [deny(5)]. 25.022/25.022 25.022/25.022 ============================== PROOF ================================= 25.022/25.022 25.022/25.022 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.022/25.022 % Length of proof is 4. 25.022/25.022 % Level of proof is 2. 25.022/25.022 % Maximum clause weight is 3.000. 25.022/25.022 % Given clauses 0. 25.022/25.022 25.022/25.022 5 (exists x2 ->*_s0(c,x2)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.022/25.022 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.022/25.022 13 -->*_s0(c,x) # label(goal) # answer(goal). [deny(5)]. 25.022/25.022 14 $F # answer(goal). [resolve(13,a,6,a)]. 25.022/25.022 25.022/25.022 ============================== end of proof ========================== 25.022/25.022 25.022/25.022 ============================== STATISTICS ============================ 25.022/25.022 25.022/25.022 Given=0. Generated=8. Kept=8. proofs=1. 25.022/25.022 Usable=0. Sos=0. Demods=0. Limbo=7, Disabled=8. Hints=0. 25.022/25.022 Kept_by_rule=0, Deleted_by_rule=0. 25.022/25.022 Forward_subsumed=0. Back_subsumed=0. 25.022/25.022 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.022/25.022 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.022/25.022 Demod_attempts=0. Demod_rewrites=0. 25.022/25.022 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.022/25.022 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=0. 25.022/25.022 Megabytes=0.04. 25.022/25.022 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.022/25.022 25.022/25.022 ============================== end of statistics ===================== 25.022/25.022 25.022/25.022 ============================== end of search ========================= 25.022/25.022 25.022/25.022 THEOREM PROVED 25.022/25.022 25.022/25.022 Exiting with 1 proof. 25.022/25.022 25.022/25.022 Process 110442 exit (max_proofs) Fri Jul 16 15:07:16 2021 25.022/25.022 25.022/25.022 25.022/25.022 The problem is feasible. 25.022/25.022 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 Confluence Problem: 25.022/25.022 (VAR x:S y:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (a) 25.022/25.022 (f 1 2) 25.022/25.022 (c) 25.022/25.022 (d) 25.022/25.022 (fSNonEmpty) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 a -> c 25.022/25.022 a -> d 25.022/25.022 f(x:S,y:S) -> y:S | c ->* x:S 25.022/25.022 ) 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 25.022/25.022 Critical Pairs Processor: 25.022/25.022 -> Rules: 25.022/25.022 a -> c 25.022/25.022 a -> d 25.022/25.022 f(x:S,y:S) -> y:S | c ->* x:S 25.022/25.022 -> Vars: 25.022/25.022 "x", "y" 25.022/25.022 -> FVars: 25.022/25.022 "x3", "x4" 25.022/25.022 -> PVars: 25.022/25.022 "x": ["x3"], "y": ["x4"] 25.022/25.022 25.022/25.022 -> Rlps: 25.022/25.022 crule: a -> c, id: 1, possubterms: a-> [] 25.022/25.022 crule: a -> d, id: 2, possubterms: a-> [] 25.022/25.022 crule: f(x3:S,x4:S) -> x4:S | c ->* x3:S, id: 3, possubterms: f(x3:S,x4:S)-> [] 25.022/25.022 25.022/25.022 -> Unifications: 25.022/25.022 R2 unifies with R1 at p: [], l: a, lp: a, conds: {}, sig: {}, l': a, r: d, r': c 25.022/25.022 25.022/25.022 -> Critical pairs info: 25.022/25.022 => Not trivial, Overlay, N1 25.022/25.022 25.022/25.022 -> Problem conclusions: 25.022/25.022 Left linear, Right linear, Linear 25.022/25.022 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 25.022/25.022 CTRS Type: 1 25.022/25.022 Deterministic, Strongly deterministic 25.022/25.022 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.022/25.022 Maybe right-stable CTRS, Overlay CTRS 25.022/25.022 Maybe normal CTRS, Maybe almost normal CTRS 25.022/25.022 Maybe terminating CTRS, Maybe joinable CCPs 25.022/25.022 Maybe level confluent 25.022/25.022 Maybe confluent 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 Confluence Problem: 25.022/25.022 (VAR x:S y:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (a) 25.022/25.022 (f 1 2) 25.022/25.022 (c) 25.022/25.022 (d) 25.022/25.022 (fSNonEmpty) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 a -> c 25.022/25.022 a -> d 25.022/25.022 f(x:S,y:S) -> y:S | c ->* x:S 25.022/25.022 ) 25.022/25.022 Critical Pairs: 25.022/25.022 => Not trivial, Overlay, N1 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 25.022/25.022 Conditional Critical Pairs Distributor Processor 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 Confluence Problem: 25.022/25.022 (VAR x:S y:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (a) 25.022/25.022 (f 1 2) 25.022/25.022 (c) 25.022/25.022 (d) 25.022/25.022 (fSNonEmpty) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 a -> c 25.022/25.022 a -> d 25.022/25.022 f(x:S,y:S) -> y:S | c ->* x:S 25.022/25.022 ) 25.022/25.022 Critical Pairs: 25.022/25.022 => Not trivial, Overlay, N1 25.022/25.022 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.022/25.022 25.022/25.022 Infeasible Convergence No Conditions CCP Processor: 25.022/25.022 Infeasible convergence? 25.022/25.022 YES 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 Infeasibility Problem: 25.022/25.022 [(VAR vNonEmpty:S x:S y:S z:S) 25.022/25.022 (STRATEGY CONTEXTSENSITIVE 25.022/25.022 (a) 25.022/25.022 (f 1 2) 25.022/25.022 (c) 25.022/25.022 (d) 25.022/25.022 (fSNonEmpty) 25.022/25.022 ) 25.022/25.022 (RULES 25.022/25.022 a -> c 25.022/25.022 a -> d 25.022/25.022 f(x:S,y:S) -> y:S | c ->* x:S 25.022/25.022 )] 25.022/25.022 25.022/25.022 Infeasibility Conditions: 25.022/25.022 c ->* z:S, d ->* z:S 25.022/25.022 25.022/25.022 Problem 1: 25.022/25.022 25.022/25.022 Obtaining a model using Mace4: 25.022/25.022 25.022/25.022 -> Usable Rules: 25.022/25.022 Empty 25.022/25.022 25.022/25.022 -> Mace4 Output: 25.022/25.022 ============================== Mace4 ================================= 25.022/25.022 Mace4 (64) version 2009-11A, November 2009. 25.022/25.022 Process 110480 was started by ubuntu on ubuntu, 25.022/25.022 Fri Jul 16 15:07:16 2021 25.022/25.022 The command was "./mace4 -c -f /tmp/mace4110456-2.in". 25.022/25.022 ============================== end of head =========================== 25.022/25.022 25.022/25.022 ============================== INPUT ================================= 25.022/25.022 25.022/25.022 % Reading from file /tmp/mace4110456-2.in 25.022/25.022 25.022/25.022 assign(max_seconds,10). 25.022/25.022 25.022/25.022 formulas(assumptions). 25.022/25.022 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). 25.022/25.022 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). 25.022/25.022 ->*(x,x) # label(reflexivity). 25.022/25.022 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 formulas(goals). 25.022/25.022 (exists x3 (->*(c,x3) & ->*(d,x3))) # label(goal). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 ============================== end of input ========================== 25.022/25.022 25.022/25.022 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.022/25.022 25.022/25.022 % Formulas that are not ordinary clauses: 25.022/25.022 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.022/25.022 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.022/25.022 3 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.022/25.022 4 (exists x3 (->*(c,x3) & ->*(d,x3))) # label(goal) # label(non_clause) # label(goal). [goal]. 25.022/25.022 25.022/25.022 ============================== end of process non-clausal formulas === 25.022/25.022 25.022/25.022 ============================== CLAUSES FOR SEARCH ==================== 25.022/25.022 25.022/25.022 formulas(mace4_clauses). 25.022/25.022 -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). 25.022/25.022 -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). 25.022/25.022 ->*(x,x) # label(reflexivity). 25.022/25.022 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 25.022/25.022 -->*(c,x) | -->*(d,x) # label(goal). 25.022/25.022 end_of_list. 25.022/25.022 25.022/25.022 ============================== end of clauses for search ============= 25.022/25.022 25.022/25.022 % There are no natural numbers in the input. 25.022/25.022 25.022/25.022 ============================== DOMAIN SIZE 2 ========================= 25.022/25.022 25.022/25.022 ============================== MODEL ================================= 25.022/25.022 25.022/25.022 interpretation( 2, [number=1, seconds=0], [ 25.022/25.022 25.022/25.022 function(c, [ 0 ]), 25.022/25.022 25.022/25.022 function(d, [ 1 ]), 25.022/25.022 25.022/25.022 function(f(_,_), [ 25.022/25.022 0, 0, 25.022/25.022 0, 0 ]), 25.022/25.022 25.022/25.022 relation(->*(_,_), [ 25.022/25.022 1, 0, 25.022/25.022 0, 1 ]), 25.022/25.022 25.022/25.022 relation(->(_,_), [ 25.022/25.022 0, 0, 25.022/25.022 0, 0 ]) 25.022/25.022 ]). 25.022/25.022 25.022/25.022 ============================== end of model ========================== 25.022/25.022 25.022/25.022 ============================== STATISTICS ============================ 25.022/25.022 25.022/25.022 For domain size 2. 25.022/25.022 25.022/25.022 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 25.022/25.022 Ground clauses: seen=28, kept=24. 25.022/25.022 Selections=6, assignments=6, propagations=8, current_models=1. 25.022/25.022 Rewrite_terms=36, rewrite_bools=33, indexes=4. 25.022/25.022 Rules_from_neg_clauses=1, cross_offs=1. 25.022/25.022 25.022/25.022 ============================== end of statistics ===================== 25.022/25.022 25.022/25.022 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.022/25.022 25.022/25.022 Exiting with 1 model. 25.022/25.022 25.022/25.022 Process 110480 exit (max_models) Fri Jul 16 15:07:16 2021 25.022/25.022 The process finished Fri Jul 16 15:07:16 2021 25.022/25.022 25.022/25.022 25.022/25.022 Mace4 cooked interpretation: 25.022/25.022 25.022/25.022 25.022/25.022 25.022/25.022 The problem is infeasible. 25.022/25.022 25.022/25.022 25.022/25.022 The problem is not joinable. 25.022/25.022 22.70user 1.33system 0:25.22elapsed 95%CPU (0avgtext+0avgdata 253832maxresident)k 25.022/25.022 8inputs+0outputs (0major+262207minor)pagefaults 0swaps