25.012/25.012 NO 25.012/25.012 25.012/25.012 Problem 1: 25.012/25.012 25.012/25.012 25.012/25.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.012/25.012 Confluence Problem: 25.012/25.012 (VAR vNonEmpty:S x:S) 25.012/25.012 (STRATEGY CONTEXTSENSITIVE 25.012/25.012 (a) 25.012/25.012 (f 1) 25.012/25.012 (b) 25.012/25.012 (c) 25.012/25.012 (fSNonEmpty) 25.012/25.012 ) 25.012/25.012 (RULES 25.012/25.012 a -> b 25.012/25.012 a -> c 25.012/25.012 f(x:S) -> a | x:S ->* b 25.012/25.012 ) 25.012/25.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.012/25.012 25.012/25.012 25.012/25.012 Problem 1: 25.012/25.012 25.012/25.012 Inlining of Conditions Processor [STERN17]: 25.012/25.012 25.012/25.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.012/25.012 Confluence Problem: 25.012/25.012 (VAR vNonEmpty:S x:S) 25.012/25.012 (STRATEGY CONTEXTSENSITIVE 25.012/25.012 (a) 25.012/25.012 (f 1) 25.012/25.012 (b) 25.012/25.012 (c) 25.012/25.012 (fSNonEmpty) 25.012/25.012 ) 25.012/25.012 (RULES 25.012/25.012 a -> b 25.012/25.012 a -> c 25.012/25.012 f(x:S) -> a | x:S ->* b 25.012/25.012 ) 25.012/25.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.012/25.012 25.012/25.012 25.012/25.012 Problem 1: 25.012/25.012 25.012/25.012 Clean CTRS Processor: 25.012/25.012 25.012/25.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.012/25.012 Confluence Problem: 25.012/25.012 (VAR x:S) 25.012/25.012 (STRATEGY CONTEXTSENSITIVE 25.012/25.012 (a) 25.012/25.012 (f 1) 25.012/25.012 (b) 25.012/25.012 (c) 25.012/25.012 (fSNonEmpty) 25.012/25.012 ) 25.012/25.012 (RULES 25.012/25.012 a -> b 25.012/25.012 a -> c 25.012/25.012 f(x:S) -> a | x:S ->* b 25.012/25.012 ) 25.012/25.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.012/25.012 25.012/25.012 CRule InfChecker Info: 25.012/25.012 a -> b 25.012/25.012 Rule remains 25.012/25.012 Proof: 25.012/25.012 NO_CONDS 25.012/25.012 25.012/25.012 CRule InfChecker Info: 25.012/25.012 a -> c 25.012/25.012 Rule remains 25.012/25.012 Proof: 25.012/25.012 NO_CONDS 25.012/25.012 25.012/25.012 CRule InfChecker Info: 25.012/25.012 f(x:S) -> a | x:S ->* b 25.012/25.012 Rule remains 25.012/25.012 Proof: 25.012/25.012 NO 25.012/25.012 25.012/25.012 Problem 1: 25.012/25.012 25.012/25.012 Infeasibility Problem: 25.012/25.012 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 25.012/25.012 (STRATEGY CONTEXTSENSITIVE 25.012/25.012 (a) 25.012/25.012 (f 1) 25.012/25.012 (b) 25.012/25.012 (c) 25.012/25.012 (fSNonEmpty) 25.012/25.012 ) 25.012/25.012 (RULES 25.012/25.012 a -> b 25.012/25.012 a -> c 25.012/25.012 f(x:S:S) -> a | x:S:S ->* b 25.012/25.012 )] 25.012/25.012 25.012/25.012 Infeasibility Conditions: 25.012/25.012 x:S:S ->* b 25.012/25.012 25.012/25.012 Problem 1: 25.012/25.012 25.012/25.012 Obtaining a proof using Prover9: 25.012/25.012 25.012/25.012 -> Prover9 Output: 25.012/25.012 ============================== Prover9 =============================== 25.012/25.012 Prover9 (64) version 2009-11A, November 2009. 25.012/25.012 Process 46969 was started by ubuntu on ubuntu, 25.012/25.012 Wed Jul 14 10:40:21 2021 25.012/25.012 The command was "./prover9 -f /tmp/prover946960-0.in". 25.012/25.012 ============================== end of head =========================== 25.012/25.012 25.012/25.012 ============================== INPUT ================================= 25.012/25.012 25.012/25.012 % Reading from file /tmp/prover946960-0.in 25.012/25.012 25.012/25.012 assign(max_seconds,20). 25.012/25.012 25.012/25.012 formulas(assumptions). 25.012/25.012 ->*_s0(x,x) # label(reflexivity). 25.012/25.012 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.012/25.012 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 25.012/25.012 ->_s0(a,b) # label(replacement). 25.012/25.012 ->_s0(a,c) # label(replacement). 25.012/25.012 ->*_s0(x2,b) -> ->_s0(f(x2),a) # label(replacement). 25.012/25.012 end_of_list. 25.012/25.012 25.012/25.012 formulas(goals). 25.012/25.012 (exists x2 ->*_s0(x2,b)) # label(goal). 25.012/25.012 end_of_list. 25.012/25.012 25.012/25.012 ============================== end of input ========================== 25.012/25.012 25.012/25.012 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.012/25.012 25.012/25.012 % Formulas that are not ordinary clauses: 25.012/25.012 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.012/25.012 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.012/25.012 3 ->*_s0(x2,b) -> ->_s0(f(x2),a) # label(replacement) # label(non_clause). [assumption]. 25.012/25.012 4 (exists x2 ->*_s0(x2,b)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.012/25.012 25.012/25.012 ============================== end of process non-clausal formulas === 25.012/25.012 25.012/25.012 ============================== PROCESS INITIAL CLAUSES =============== 25.012/25.012 25.012/25.012 % Clauses before input processing: 25.012/25.012 25.012/25.012 formulas(usable). 25.012/25.012 end_of_list. 25.012/25.012 25.012/25.012 formulas(sos). 25.012/25.012 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.012/25.012 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.012/25.012 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.012/25.012 ->_s0(a,b) # label(replacement). [assumption]. 25.012/25.012 ->_s0(a,c) # label(replacement). [assumption]. 25.012/25.012 -->*_s0(x,b) | ->_s0(f(x),a) # label(replacement). [clausify(3)]. 25.012/25.012 -->*_s0(x,b) # label(goal). [deny(4)]. 25.012/25.012 end_of_list. 25.012/25.012 25.012/25.012 formulas(demodulators). 25.012/25.012 end_of_list. 25.012/25.012 25.012/25.012 ============================== PREDICATE ELIMINATION ================= 25.012/25.012 25.012/25.012 No predicates eliminated. 25.012/25.012 25.012/25.012 ============================== end predicate elimination ============= 25.012/25.012 25.012/25.012 Auto_denials: 25.012/25.012 % copying label goal to answer in negative clause 25.012/25.012 25.012/25.012 Term ordering decisions: 25.012/25.012 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.012/25.012 Function symbol precedence: function_order([ a, b, c, f ]). 25.012/25.012 After inverse_order: (no changes). 25.012/25.012 Unfolding symbols: (none). 25.012/25.012 25.012/25.012 Auto_inference settings: 25.012/25.012 % set(neg_binary_resolution). % (HNE depth_diff=-2) 25.012/25.012 % clear(ordered_res). % (HNE depth_diff=-2) 25.012/25.012 % set(ur_resolution). % (HNE depth_diff=-2) 25.012/25.012 % set(ur_resolution) -> set(pos_ur_resolution). 25.012/25.012 % set(ur_resolution) -> set(neg_ur_resolution). 25.012/25.012 25.012/25.012 Auto_process settings: (no changes). 25.012/25.012 25.012/25.012 kept: 5 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.012/25.012 kept: 6 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.012/25.012 kept: 7 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.012/25.012 kept: 8 ->_s0(a,b) # label(replacement). [assumption]. 25.012/25.012 kept: 9 ->_s0(a,c) # label(replacement). [assumption]. 25.012/25.012 kept: 10 -->*_s0(x,b) | ->_s0(f(x),a) # label(replacement). [clausify(3)]. 25.012/25.012 kept: 11 -->*_s0(x,b) # label(goal) # answer(goal). [deny(4)]. 25.012/25.012 25.012/25.012 ============================== PROOF ================================= 25.012/25.012 25.012/25.012 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.012/25.012 % Length of proof is 4. 25.012/25.012 % Level of proof is 2. 25.012/25.012 % Maximum clause weight is 3.000. 25.012/25.012 % Given clauses 0. 25.012/25.012 25.012/25.012 4 (exists x2 ->*_s0(x2,b)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.012/25.012 5 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.012/25.012 11 -->*_s0(x,b) # label(goal) # answer(goal). [deny(4)]. 25.012/25.012 12 $F # answer(goal). [resolve(11,a,5,a)]. 25.012/25.012 25.012/25.012 ============================== end of proof ========================== 25.012/25.012 25.012/25.012 ============================== STATISTICS ============================ 25.012/25.012 25.012/25.012 Given=0. Generated=7. Kept=7. proofs=1. 25.012/25.012 Usable=0. Sos=0. Demods=0. Limbo=6, Disabled=7. Hints=0. 25.012/25.012 Kept_by_rule=0, Deleted_by_rule=0. 25.012/25.012 Forward_subsumed=0. Back_subsumed=0. 25.012/25.012 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.012/25.012 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.012/25.012 Demod_attempts=0. Demod_rewrites=0. 25.012/25.012 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.012/25.012 Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=0. 25.012/25.012 Megabytes=0.04. 25.012/25.012 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.012/25.012 25.012/25.012 ============================== end of statistics ===================== 25.012/25.012 25.012/25.012 ============================== end of search ========================= 25.012/25.012 25.012/25.012 THEOREM PROVED 25.012/25.012 25.012/25.012 Exiting with 1 proof. 25.012/25.012 25.012/25.012 Process 46969 exit (max_proofs) Wed Jul 14 10:40:21 2021 25.012/25.012 25.012/25.012 25.012/25.012 The problem is feasible. 25.012/25.012 25.012/25.012 25.012/25.012 Problem 1: 25.012/25.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.012/25.012 Confluence Problem: 25.012/25.012 (VAR x:S) 25.012/25.012 (STRATEGY CONTEXTSENSITIVE 25.012/25.012 (a) 25.012/25.012 (f 1) 25.012/25.012 (b) 25.012/25.012 (c) 25.012/25.012 (fSNonEmpty) 25.012/25.012 ) 25.012/25.012 (RULES 25.012/25.012 a -> b 25.012/25.012 a -> c 25.012/25.012 f(x:S) -> a | x:S ->* b 25.012/25.012 ) 25.012/25.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.012/25.012 25.012/25.012 Critical Pairs Processor: 25.012/25.012 -> Rules: 25.012/25.012 a -> b 25.012/25.012 a -> c 25.012/25.012 f(x:S) -> a | x:S ->* b 25.012/25.012 -> Vars: 25.012/25.012 "x" 25.012/25.012 -> FVars: 25.012/25.012 "x2" 25.012/25.012 -> PVars: 25.012/25.012 "x": ["x2"] 25.012/25.012 25.012/25.012 -> Rlps: 25.012/25.012 crule: a -> b, id: 1, possubterms: a-> [] 25.012/25.012 crule: a -> c, id: 2, possubterms: a-> [] 25.012/25.012 crule: f(x2:S) -> a | x2:S ->* b, id: 3, possubterms: f(x2:S)-> [] 25.012/25.012 25.012/25.012 -> Unifications: 25.012/25.012 R2 unifies with R1 at p: [], l: a, lp: a, conds: {}, sig: {}, l': a, r: c, r': b 25.012/25.012 25.012/25.012 -> Critical pairs info: 25.012/25.012 => Not trivial, Overlay, N1 25.012/25.012 25.012/25.012 -> Problem conclusions: 25.012/25.012 Left linear, Right linear, Linear 25.012/25.012 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 25.012/25.012 CTRS Type: 1 25.012/25.012 Deterministic, Strongly deterministic 25.012/25.012 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.012/25.012 Maybe right-stable CTRS, Overlay CTRS 25.012/25.012 Maybe normal CTRS, Maybe almost normal CTRS 25.012/25.012 Maybe terminating CTRS, Maybe joinable CCPs 25.012/25.012 Maybe level confluent 25.012/25.012 Maybe confluent 25.012/25.012 25.012/25.012 Problem 1: 25.012/25.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.012/25.012 Confluence Problem: 25.012/25.012 (VAR x:S) 25.012/25.012 (STRATEGY CONTEXTSENSITIVE 25.012/25.012 (a) 25.012/25.012 (f 1) 25.012/25.012 (b) 25.012/25.012 (c) 25.012/25.012 (fSNonEmpty) 25.012/25.012 ) 25.012/25.012 (RULES 25.012/25.012 a -> b 25.012/25.012 a -> c 25.012/25.012 f(x:S) -> a | x:S ->* b 25.012/25.012 ) 25.012/25.012 Critical Pairs: 25.012/25.012 => Not trivial, Overlay, N1 25.012/25.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.012/25.012 25.012/25.012 Conditional Critical Pairs Distributor Processor 25.012/25.012 25.012/25.012 Problem 1: 25.012/25.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.012/25.012 Confluence Problem: 25.012/25.012 (VAR x:S) 25.012/25.012 (STRATEGY CONTEXTSENSITIVE 25.012/25.012 (a) 25.012/25.012 (f 1) 25.012/25.012 (b) 25.012/25.012 (c) 25.012/25.012 (fSNonEmpty) 25.012/25.012 ) 25.012/25.012 (RULES 25.012/25.012 a -> b 25.012/25.012 a -> c 25.012/25.012 f(x:S) -> a | x:S ->* b 25.012/25.012 ) 25.012/25.012 Critical Pairs: 25.012/25.012 => Not trivial, Overlay, N1 25.012/25.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.012/25.012 25.012/25.012 Infeasible Convergence No Conditions CCP Processor: 25.012/25.012 Infeasible convergence? 25.012/25.012 YES 25.012/25.012 25.012/25.012 Problem 1: 25.012/25.012 25.012/25.012 Infeasibility Problem: 25.012/25.012 [(VAR vNonEmpty:S x:S z:S) 25.012/25.012 (STRATEGY CONTEXTSENSITIVE 25.012/25.012 (a) 25.012/25.012 (f 1) 25.012/25.012 (b) 25.012/25.012 (c) 25.012/25.012 (fSNonEmpty) 25.012/25.012 ) 25.012/25.012 (RULES 25.012/25.012 a -> b 25.012/25.012 a -> c 25.012/25.012 f(x:S) -> a | x:S ->* b 25.012/25.012 )] 25.012/25.012 25.012/25.012 Infeasibility Conditions: 25.012/25.012 b ->* z:S, c ->* z:S 25.012/25.012 25.012/25.012 Problem 1: 25.012/25.012 25.012/25.012 Obtaining a model using Mace4: 25.012/25.012 25.012/25.012 -> Usable Rules: 25.012/25.012 Empty 25.012/25.012 25.012/25.012 -> Mace4 Output: 25.012/25.012 ============================== Mace4 ================================= 25.012/25.012 Mace4 (64) version 2009-11A, November 2009. 25.012/25.012 Process 47009 was started by ubuntu on ubuntu, 25.012/25.012 Wed Jul 14 10:40:21 2021 25.012/25.012 The command was "./mace4 -c -f /tmp/mace446986-2.in". 25.012/25.012 ============================== end of head =========================== 25.012/25.012 25.012/25.012 ============================== INPUT ================================= 25.012/25.012 25.012/25.012 % Reading from file /tmp/mace446986-2.in 25.012/25.012 25.012/25.012 assign(max_seconds,20). 25.012/25.012 25.012/25.012 formulas(assumptions). 25.012/25.012 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 25.012/25.012 ->*(x,x) # label(reflexivity). 25.012/25.012 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 25.012/25.012 end_of_list. 25.012/25.012 25.012/25.012 formulas(goals). 25.012/25.012 (exists x2 (->*(b,x2) & ->*(c,x2))) # label(goal). 25.012/25.012 end_of_list. 25.012/25.012 25.012/25.012 ============================== end of input ========================== 25.012/25.012 25.012/25.012 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.012/25.012 25.012/25.012 % Formulas that are not ordinary clauses: 25.012/25.012 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.012/25.012 2 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.012/25.012 3 (exists x2 (->*(b,x2) & ->*(c,x2))) # label(goal) # label(non_clause) # label(goal). [goal]. 25.012/25.012 25.012/25.012 ============================== end of process non-clausal formulas === 25.012/25.012 25.012/25.012 ============================== CLAUSES FOR SEARCH ==================== 25.012/25.012 25.012/25.012 formulas(mace4_clauses). 25.012/25.012 -->(x,y) | ->(f(x),f(y)) # label(congruence). 25.012/25.012 ->*(x,x) # label(reflexivity). 25.012/25.012 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 25.012/25.012 -->*(b,x) | -->*(c,x) # label(goal). 25.012/25.012 end_of_list. 25.012/25.012 25.012/25.012 ============================== end of clauses for search ============= 25.012/25.012 25.012/25.012 % There are no natural numbers in the input. 25.012/25.012 25.012/25.012 ============================== DOMAIN SIZE 2 ========================= 25.012/25.012 25.012/25.012 ============================== MODEL ================================= 25.012/25.012 25.012/25.012 interpretation( 2, [number=1, seconds=0], [ 25.012/25.012 25.012/25.012 function(b, [ 0 ]), 25.012/25.012 25.012/25.012 function(c, [ 1 ]), 25.012/25.012 25.012/25.012 function(f(_), [ 0, 0 ]), 25.012/25.012 25.012/25.012 relation(->*(_,_), [ 25.012/25.012 1, 0, 25.012/25.012 0, 1 ]), 25.012/25.012 25.012/25.012 relation(->(_,_), [ 25.012/25.012 0, 0, 25.012/25.012 0, 0 ]) 25.012/25.012 ]). 25.012/25.012 25.012/25.012 ============================== end of model ========================== 25.012/25.012 25.012/25.012 ============================== STATISTICS ============================ 25.012/25.012 25.012/25.012 For domain size 2. 25.012/25.012 25.012/25.012 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 25.012/25.012 Ground clauses: seen=16, kept=12. 25.012/25.012 Selections=4, assignments=4, propagations=8, current_models=1. 25.012/25.012 Rewrite_terms=12, rewrite_bools=20, indexes=3. 25.012/25.012 Rules_from_neg_clauses=1, cross_offs=1. 25.012/25.012 25.012/25.012 ============================== end of statistics ===================== 25.012/25.012 25.012/25.012 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.012/25.012 25.012/25.012 Exiting with 1 model. 25.012/25.012 25.012/25.012 Process 47009 exit (max_models) Wed Jul 14 10:40:21 2021 25.012/25.012 The process finished Wed Jul 14 10:40:21 2021 25.012/25.012 25.012/25.012 25.012/25.012 Mace4 cooked interpretation: 25.012/25.012 25.012/25.012 25.012/25.012 25.012/25.012 The problem is infeasible. 25.012/25.012 25.012/25.012 25.012/25.012 The problem is not joinable. 25.012/25.012 20.94user 0.53system 0:25.12elapsed 85%CPU (0avgtext+0avgdata 77056maxresident)k 25.012/25.012 0inputs+0outputs (0major+86748minor)pagefaults 0swaps