25.024/25.024 NO 25.024/25.024 25.024/25.024 Problem 1: 25.024/25.024 25.024/25.024 25.024/25.024 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.024/25.024 Confluence Problem: 25.024/25.024 (VAR vNonEmpty:S x:S y:S z:S) 25.024/25.024 (STRATEGY CONTEXTSENSITIVE 25.024/25.024 (a) 25.024/25.024 (f 1 2) 25.024/25.024 (c) 25.024/25.024 (d) 25.024/25.024 (fSNonEmpty) 25.024/25.024 ) 25.024/25.024 (RULES 25.024/25.024 a -> c 25.024/25.024 a -> d 25.024/25.024 f(x:S,y:S) -> z:S | c ->* x:S, y:S ->* z:S 25.024/25.024 ) 25.024/25.024 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.024/25.024 25.024/25.024 25.024/25.024 Problem 1: 25.024/25.024 25.024/25.024 Inlining of Conditions Processor [STERN17]: 25.024/25.024 25.024/25.024 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.024/25.024 Confluence Problem: 25.024/25.024 (VAR vNonEmpty:S x:S y:S z:S) 25.024/25.024 (STRATEGY CONTEXTSENSITIVE 25.024/25.024 (a) 25.024/25.024 (f 1 2) 25.024/25.024 (c) 25.024/25.024 (d) 25.024/25.024 (fSNonEmpty) 25.024/25.024 ) 25.024/25.024 (RULES 25.024/25.024 a -> c 25.024/25.024 a -> d 25.024/25.024 f(x:S,y:S) -> y:S | c ->* x:S 25.024/25.024 ) 25.024/25.024 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.024/25.024 25.024/25.024 25.024/25.024 Problem 1: 25.024/25.024 25.024/25.024 Clean CTRS Processor: 25.024/25.024 25.024/25.024 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.024/25.024 Confluence Problem: 25.024/25.024 (VAR x:S y:S) 25.024/25.024 (STRATEGY CONTEXTSENSITIVE 25.024/25.024 (a) 25.024/25.024 (f 1 2) 25.024/25.024 (c) 25.024/25.024 (d) 25.024/25.024 (fSNonEmpty) 25.024/25.024 ) 25.024/25.024 (RULES 25.024/25.024 a -> c 25.024/25.024 a -> d 25.024/25.024 f(x:S,y:S) -> y:S | c ->* x:S 25.024/25.024 ) 25.024/25.024 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.024/25.024 25.024/25.024 CRule InfChecker Info: 25.024/25.024 a -> c 25.024/25.024 Rule remains 25.024/25.024 Proof: 25.024/25.024 NO_CONDS 25.024/25.024 25.024/25.024 CRule InfChecker Info: 25.024/25.024 a -> d 25.024/25.024 Rule remains 25.024/25.024 Proof: 25.024/25.024 NO_CONDS 25.024/25.024 25.024/25.024 CRule InfChecker Info: 25.024/25.024 f(x:S,y:S) -> y:S | c ->* x:S 25.024/25.024 Rule remains 25.024/25.024 Proof: 25.024/25.024 NO 25.024/25.024 25.024/25.024 Problem 1: 25.024/25.024 25.024/25.024 Infeasibility Problem: 25.024/25.024 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S y:S:S z:S:S) 25.024/25.024 (STRATEGY CONTEXTSENSITIVE 25.024/25.024 (a) 25.024/25.024 (f 1 2) 25.024/25.024 (c) 25.024/25.024 (d) 25.024/25.024 (fSNonEmpty) 25.024/25.024 ) 25.024/25.024 (RULES 25.024/25.024 a -> c 25.024/25.024 a -> d 25.024/25.024 f(x:S:S,y:S:S) -> y:S:S | c ->* x:S:S 25.024/25.024 )] 25.024/25.024 25.024/25.024 Infeasibility Conditions: 25.024/25.024 c ->* x:S:S 25.024/25.024 25.024/25.024 Problem 1: 25.024/25.024 25.024/25.024 Obtaining a proof using Prover9: 25.024/25.024 25.024/25.024 -> Prover9 Output: 25.024/25.024 ============================== Prover9 =============================== 25.024/25.024 Prover9 (64) version 2009-11A, November 2009. 25.024/25.024 Process 49413 was started by ubuntu on ubuntu, 25.024/25.024 Wed Jul 14 10:50:08 2021 25.024/25.024 The command was "./prover9 -f /tmp/prover949404-0.in". 25.024/25.024 ============================== end of head =========================== 25.024/25.024 25.024/25.024 ============================== INPUT ================================= 25.024/25.024 25.024/25.024 % Reading from file /tmp/prover949404-0.in 25.024/25.024 25.024/25.024 assign(max_seconds,20). 25.024/25.024 25.024/25.024 formulas(assumptions). 25.024/25.024 ->*_s0(x,x) # label(reflexivity). 25.024/25.024 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.024/25.024 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 25.024/25.024 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 25.024/25.024 ->_s0(a,c) # label(replacement). 25.024/25.024 ->_s0(a,d) # label(replacement). 25.024/25.024 ->*_s0(c,x2) -> ->_s0(f(x2,x3),x3) # label(replacement). 25.024/25.024 end_of_list. 25.024/25.024 25.024/25.024 formulas(goals). 25.024/25.024 (exists x2 ->*_s0(c,x2)) # label(goal). 25.024/25.024 end_of_list. 25.024/25.024 25.024/25.024 ============================== end of input ========================== 25.024/25.024 25.024/25.024 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.024/25.024 25.024/25.024 % Formulas that are not ordinary clauses: 25.024/25.024 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.024/25.024 2 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.024/25.024 3 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.024/25.024 4 ->*_s0(c,x2) -> ->_s0(f(x2,x3),x3) # label(replacement) # label(non_clause). [assumption]. 25.024/25.024 5 (exists x2 ->*_s0(c,x2)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.024/25.024 25.024/25.024 ============================== end of process non-clausal formulas === 25.024/25.024 25.024/25.024 ============================== PROCESS INITIAL CLAUSES =============== 25.024/25.024 25.024/25.024 % Clauses before input processing: 25.024/25.024 25.024/25.024 formulas(usable). 25.024/25.024 end_of_list. 25.024/25.024 25.024/25.024 formulas(sos). 25.024/25.024 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.024/25.024 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.024/25.024 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.024/25.024 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.024/25.024 ->_s0(a,c) # label(replacement). [assumption]. 25.024/25.024 ->_s0(a,d) # label(replacement). [assumption]. 25.024/25.024 -->*_s0(c,x) | ->_s0(f(x,y),y) # label(replacement). [clausify(4)]. 25.024/25.024 -->*_s0(c,x) # label(goal). [deny(5)]. 25.024/25.024 end_of_list. 25.024/25.024 25.024/25.024 formulas(demodulators). 25.024/25.024 end_of_list. 25.024/25.024 25.024/25.024 ============================== PREDICATE ELIMINATION ================= 25.024/25.024 25.024/25.024 No predicates eliminated. 25.024/25.024 25.024/25.024 ============================== end predicate elimination ============= 25.024/25.024 25.024/25.024 Auto_denials: 25.024/25.024 % copying label goal to answer in negative clause 25.024/25.024 25.024/25.024 Term ordering decisions: 25.024/25.024 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.024/25.024 Function symbol precedence: function_order([ a, c, d, f ]). 25.024/25.024 After inverse_order: (no changes). 25.024/25.024 Unfolding symbols: (none). 25.024/25.024 25.024/25.024 Auto_inference settings: 25.024/25.024 % set(neg_binary_resolution). % (HNE depth_diff=-3) 25.024/25.024 % clear(ordered_res). % (HNE depth_diff=-3) 25.024/25.024 % set(ur_resolution). % (HNE depth_diff=-3) 25.024/25.024 % set(ur_resolution) -> set(pos_ur_resolution). 25.024/25.024 % set(ur_resolution) -> set(neg_ur_resolution). 25.024/25.024 25.024/25.024 Auto_process settings: (no changes). 25.024/25.024 25.024/25.024 kept: 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.024/25.024 kept: 7 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.024/25.024 kept: 8 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.024/25.024 kept: 9 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.024/25.024 kept: 10 ->_s0(a,c) # label(replacement). [assumption]. 25.024/25.024 kept: 11 ->_s0(a,d) # label(replacement). [assumption]. 25.024/25.024 kept: 12 -->*_s0(c,x) | ->_s0(f(x,y),y) # label(replacement). [clausify(4)]. 25.024/25.024 kept: 13 -->*_s0(c,x) # label(goal) # answer(goal). [deny(5)]. 25.024/25.024 25.024/25.024 ============================== PROOF ================================= 25.024/25.024 25.024/25.024 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.024/25.024 % Length of proof is 4. 25.024/25.024 % Level of proof is 2. 25.024/25.024 % Maximum clause weight is 3.000. 25.024/25.024 % Given clauses 0. 25.024/25.024 25.024/25.024 5 (exists x2 ->*_s0(c,x2)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.024/25.024 6 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.024/25.024 13 -->*_s0(c,x) # label(goal) # answer(goal). [deny(5)]. 25.024/25.024 14 $F # answer(goal). [resolve(13,a,6,a)]. 25.024/25.024 25.024/25.024 ============================== end of proof ========================== 25.024/25.024 25.024/25.024 ============================== STATISTICS ============================ 25.024/25.024 25.024/25.024 Given=0. Generated=8. Kept=8. proofs=1. 25.024/25.024 Usable=0. Sos=0. Demods=0. Limbo=7, Disabled=8. Hints=0. 25.024/25.024 Kept_by_rule=0, Deleted_by_rule=0. 25.024/25.024 Forward_subsumed=0. Back_subsumed=0. 25.024/25.024 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.024/25.024 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.024/25.024 Demod_attempts=0. Demod_rewrites=0. 25.024/25.024 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.024/25.024 Nonunit_fsub_feature_tests=1. Nonunit_bsub_feature_tests=0. 25.024/25.024 Megabytes=0.04. 25.024/25.024 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.024/25.024 25.024/25.024 ============================== end of statistics ===================== 25.024/25.024 25.024/25.024 ============================== end of search ========================= 25.024/25.024 25.024/25.024 THEOREM PROVED 25.024/25.024 25.024/25.024 Exiting with 1 proof. 25.024/25.024 25.024/25.024 Process 49413 exit (max_proofs) Wed Jul 14 10:50:08 2021 25.024/25.024 25.024/25.024 25.024/25.024 The problem is feasible. 25.024/25.024 25.024/25.024 25.024/25.024 Problem 1: 25.024/25.024 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.024/25.024 Confluence Problem: 25.024/25.024 (VAR x:S y:S) 25.024/25.024 (STRATEGY CONTEXTSENSITIVE 25.024/25.024 (a) 25.024/25.024 (f 1 2) 25.024/25.024 (c) 25.024/25.024 (d) 25.024/25.024 (fSNonEmpty) 25.024/25.024 ) 25.024/25.024 (RULES 25.024/25.024 a -> c 25.024/25.024 a -> d 25.024/25.024 f(x:S,y:S) -> y:S | c ->* x:S 25.024/25.024 ) 25.024/25.024 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.024/25.024 25.024/25.024 Critical Pairs Processor: 25.024/25.024 -> Rules: 25.024/25.024 a -> c 25.024/25.024 a -> d 25.024/25.024 f(x:S,y:S) -> y:S | c ->* x:S 25.024/25.024 -> Vars: 25.024/25.024 "x", "y" 25.024/25.024 -> FVars: 25.024/25.024 "x3", "x4" 25.024/25.024 -> PVars: 25.024/25.024 "x": ["x3"], "y": ["x4"] 25.024/25.024 25.024/25.024 -> Rlps: 25.024/25.024 crule: a -> c, id: 1, possubterms: a-> [] 25.024/25.024 crule: a -> d, id: 2, possubterms: a-> [] 25.024/25.024 crule: f(x3:S,x4:S) -> x4:S | c ->* x3:S, id: 3, possubterms: f(x3:S,x4:S)-> [] 25.024/25.024 25.024/25.024 -> Unifications: 25.024/25.024 R2 unifies with R1 at p: [], l: a, lp: a, conds: {}, sig: {}, l': a, r: d, r': c 25.024/25.024 25.024/25.024 -> Critical pairs info: 25.024/25.024 => Not trivial, Overlay, N1 25.024/25.024 25.024/25.024 -> Problem conclusions: 25.024/25.024 Left linear, Right linear, Linear 25.024/25.024 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 25.024/25.024 CTRS Type: 1 25.024/25.024 Deterministic, Strongly deterministic 25.024/25.024 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.024/25.024 Maybe right-stable CTRS, Overlay CTRS 25.024/25.024 Maybe normal CTRS, Maybe almost normal CTRS 25.024/25.024 Maybe terminating CTRS, Maybe joinable CCPs 25.024/25.024 Maybe level confluent 25.024/25.024 Maybe confluent 25.024/25.024 25.024/25.024 Problem 1: 25.024/25.024 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.024/25.024 Confluence Problem: 25.024/25.024 (VAR x:S y:S) 25.024/25.024 (STRATEGY CONTEXTSENSITIVE 25.024/25.024 (a) 25.024/25.024 (f 1 2) 25.024/25.024 (c) 25.024/25.024 (d) 25.024/25.024 (fSNonEmpty) 25.024/25.024 ) 25.024/25.024 (RULES 25.024/25.024 a -> c 25.024/25.024 a -> d 25.024/25.024 f(x:S,y:S) -> y:S | c ->* x:S 25.024/25.024 ) 25.024/25.024 Critical Pairs: 25.024/25.024 => Not trivial, Overlay, N1 25.024/25.024 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.024/25.024 25.024/25.024 Conditional Critical Pairs Distributor Processor 25.024/25.024 25.024/25.024 Problem 1: 25.024/25.024 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.024/25.024 Confluence Problem: 25.024/25.024 (VAR x:S y:S) 25.024/25.024 (STRATEGY CONTEXTSENSITIVE 25.024/25.024 (a) 25.024/25.024 (f 1 2) 25.024/25.024 (c) 25.024/25.024 (d) 25.024/25.024 (fSNonEmpty) 25.024/25.024 ) 25.024/25.024 (RULES 25.024/25.024 a -> c 25.024/25.024 a -> d 25.024/25.024 f(x:S,y:S) -> y:S | c ->* x:S 25.024/25.024 ) 25.024/25.024 Critical Pairs: 25.024/25.024 => Not trivial, Overlay, N1 25.024/25.024 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.024/25.024 25.024/25.024 Infeasible Convergence No Conditions CCP Processor: 25.024/25.024 Infeasible convergence? 25.024/25.024 YES 25.024/25.024 25.024/25.024 Problem 1: 25.024/25.024 25.024/25.024 Infeasibility Problem: 25.024/25.024 [(VAR vNonEmpty:S x:S y:S z:S) 25.024/25.024 (STRATEGY CONTEXTSENSITIVE 25.024/25.024 (a) 25.024/25.024 (f 1 2) 25.024/25.024 (c) 25.024/25.024 (d) 25.024/25.024 (fSNonEmpty) 25.024/25.024 ) 25.024/25.024 (RULES 25.024/25.024 a -> c 25.024/25.024 a -> d 25.024/25.024 f(x:S,y:S) -> y:S | c ->* x:S 25.024/25.024 )] 25.024/25.024 25.024/25.024 Infeasibility Conditions: 25.024/25.024 c ->* z:S, d ->* z:S 25.024/25.024 25.024/25.024 Problem 1: 25.024/25.024 25.024/25.024 Obtaining a model using Mace4: 25.024/25.024 25.024/25.024 -> Usable Rules: 25.024/25.024 Empty 25.024/25.024 25.024/25.024 -> Mace4 Output: 25.024/25.024 ============================== Mace4 ================================= 25.024/25.024 Mace4 (64) version 2009-11A, November 2009. 25.024/25.024 Process 49453 was started by ubuntu on ubuntu, 25.024/25.024 Wed Jul 14 10:50:08 2021 25.024/25.024 The command was "./mace4 -c -f /tmp/mace449430-2.in". 25.024/25.024 ============================== end of head =========================== 25.024/25.024 25.024/25.024 ============================== INPUT ================================= 25.024/25.024 25.024/25.024 % Reading from file /tmp/mace449430-2.in 25.024/25.024 25.024/25.024 assign(max_seconds,20). 25.024/25.024 25.024/25.024 formulas(assumptions). 25.024/25.024 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). 25.024/25.024 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). 25.024/25.024 ->*(x,x) # label(reflexivity). 25.024/25.024 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 25.024/25.024 end_of_list. 25.024/25.024 25.024/25.024 formulas(goals). 25.024/25.024 (exists x3 (->*(c,x3) & ->*(d,x3))) # label(goal). 25.024/25.024 end_of_list. 25.024/25.024 25.024/25.024 ============================== end of input ========================== 25.024/25.024 25.024/25.024 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.024/25.024 25.024/25.024 % Formulas that are not ordinary clauses: 25.024/25.024 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.024/25.024 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.024/25.024 3 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.024/25.024 4 (exists x3 (->*(c,x3) & ->*(d,x3))) # label(goal) # label(non_clause) # label(goal). [goal]. 25.024/25.024 25.024/25.024 ============================== end of process non-clausal formulas === 25.024/25.024 25.024/25.024 ============================== CLAUSES FOR SEARCH ==================== 25.024/25.024 25.024/25.024 formulas(mace4_clauses). 25.024/25.024 -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). 25.024/25.024 -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). 25.024/25.024 ->*(x,x) # label(reflexivity). 25.024/25.024 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 25.024/25.024 -->*(c,x) | -->*(d,x) # label(goal). 25.024/25.024 end_of_list. 25.024/25.024 25.024/25.024 ============================== end of clauses for search ============= 25.024/25.024 25.024/25.024 % There are no natural numbers in the input. 25.024/25.024 25.024/25.024 ============================== DOMAIN SIZE 2 ========================= 25.024/25.024 25.024/25.024 ============================== MODEL ================================= 25.024/25.024 25.024/25.024 interpretation( 2, [number=1, seconds=0], [ 25.024/25.024 25.024/25.024 function(c, [ 0 ]), 25.024/25.024 25.024/25.024 function(d, [ 1 ]), 25.024/25.024 25.024/25.024 function(f(_,_), [ 25.024/25.024 0, 0, 25.024/25.024 0, 0 ]), 25.024/25.024 25.024/25.024 relation(->*(_,_), [ 25.024/25.024 1, 0, 25.024/25.024 0, 1 ]), 25.024/25.024 25.024/25.024 relation(->(_,_), [ 25.024/25.024 0, 0, 25.024/25.024 0, 0 ]) 25.024/25.024 ]). 25.024/25.024 25.024/25.024 ============================== end of model ========================== 25.024/25.024 25.024/25.024 ============================== STATISTICS ============================ 25.024/25.024 25.024/25.024 For domain size 2. 25.024/25.024 25.024/25.024 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 25.024/25.024 Ground clauses: seen=28, kept=24. 25.024/25.024 Selections=6, assignments=6, propagations=8, current_models=1. 25.024/25.024 Rewrite_terms=36, rewrite_bools=33, indexes=4. 25.024/25.024 Rules_from_neg_clauses=1, cross_offs=1. 25.024/25.024 25.024/25.024 ============================== end of statistics ===================== 25.024/25.024 25.024/25.024 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.024/25.024 25.024/25.024 Exiting with 1 model. 25.024/25.024 25.024/25.024 Process 49453 exit (max_models) Wed Jul 14 10:50:08 2021 25.024/25.024 The process finished Wed Jul 14 10:50:08 2021 25.024/25.024 25.024/25.024 25.024/25.024 Mace4 cooked interpretation: 25.024/25.024 25.024/25.024 25.024/25.024 25.024/25.024 The problem is infeasible. 25.024/25.024 25.024/25.024 25.024/25.024 The problem is not joinable. 25.024/25.024 23.21user 1.36system 0:25.24elapsed 97%CPU (0avgtext+0avgdata 254364maxresident)k 25.024/25.024 0inputs+0outputs (0major+261986minor)pagefaults 0swaps