0.012/0.012 NO 0.012/0.012 0.012/0.012 Problem 1: 0.012/0.012 0.012/0.012 0.012/0.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.012/0.012 Confluence Problem: 0.012/0.012 (VAR vNonEmpty x) 0.012/0.012 (REPLACEMENT-MAP 0.012/0.012 (a) 0.012/0.012 (b) 0.012/0.012 (f 1) 0.012/0.012 (g 1, 2) 0.012/0.012 (h 1, 2) 0.012/0.012 (c) 0.012/0.012 (d) 0.012/0.012 (fSNonEmpty) 0.012/0.012 ) 0.012/0.012 (RULES 0.012/0.012 a -> c 0.012/0.012 a -> d 0.012/0.012 b -> c 0.012/0.012 b -> d 0.012/0.012 f(x) -> x | x ->* c 0.012/0.012 g(x,x) -> h(x,x) 0.012/0.012 h(x,f(x)) -> x 0.012/0.012 ) 0.012/0.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.012/0.012 0.012/0.012 0.012/0.012 Problem 1: 0.012/0.012 0.012/0.012 Inlining of Conditions Processor [STERN17]: 0.012/0.012 0.012/0.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.012/0.012 Confluence Problem: 0.012/0.012 (VAR vNonEmpty x) 0.012/0.012 (REPLACEMENT-MAP 0.012/0.012 (a) 0.012/0.012 (b) 0.012/0.012 (f 1) 0.012/0.012 (g 1, 2) 0.012/0.012 (h 1, 2) 0.012/0.012 (c) 0.012/0.012 (d) 0.012/0.012 (fSNonEmpty) 0.012/0.012 ) 0.012/0.012 (RULES 0.012/0.012 a -> c 0.012/0.012 a -> d 0.012/0.012 b -> c 0.012/0.012 b -> d 0.012/0.012 f(x) -> x | x ->* c 0.012/0.012 g(x,x) -> h(x,x) 0.012/0.012 h(x,f(x)) -> x 0.012/0.012 ) 0.012/0.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.012/0.012 0.012/0.012 0.012/0.012 Problem 1: 0.012/0.012 0.012/0.012 Clean CTRS Processor: 0.012/0.012 0.012/0.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.012/0.012 Confluence Problem: 0.012/0.012 (VAR x) 0.012/0.012 (REPLACEMENT-MAP 0.012/0.012 (a) 0.012/0.012 (b) 0.012/0.012 (f 1) 0.012/0.012 (g 1, 2) 0.012/0.012 (h 1, 2) 0.012/0.012 (c) 0.012/0.012 (d) 0.012/0.012 (fSNonEmpty) 0.012/0.012 ) 0.012/0.012 (RULES 0.012/0.012 a -> c 0.012/0.012 a -> d 0.012/0.012 b -> c 0.012/0.012 b -> d 0.012/0.012 f(x) -> x | x ->* c 0.012/0.012 g(x,x) -> h(x,x) 0.012/0.012 h(x,f(x)) -> x 0.012/0.012 ) 0.012/0.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.012/0.012 0.012/0.012 CRule InfChecker Info: 0.012/0.012 a -> c 0.012/0.012 Rule remains 0.012/0.012 Proof: 0.012/0.012 NO_CONDS 0.012/0.012 0.012/0.012 CRule InfChecker Info: 0.012/0.012 a -> d 0.012/0.012 Rule remains 0.012/0.012 Proof: 0.012/0.012 NO_CONDS 0.012/0.012 0.012/0.012 CRule InfChecker Info: 0.012/0.012 b -> c 0.012/0.012 Rule remains 0.012/0.012 Proof: 0.012/0.012 NO_CONDS 0.012/0.012 0.012/0.012 CRule InfChecker Info: 0.012/0.012 b -> d 0.012/0.012 Rule remains 0.012/0.012 Proof: 0.012/0.012 NO_CONDS 0.012/0.012 0.012/0.012 CRule InfChecker Info: 0.012/0.012 f(x) -> x | x ->* c 0.012/0.012 Rule remains 0.012/0.012 Proof: 0.012/0.012 NO 0.012/0.012 0.012/0.012 Problem 1: 0.012/0.012 0.012/0.012 Infeasibility Problem: 0.012/0.012 [(VAR vNonEmpty x vNonEmpty x) 0.012/0.012 (STRATEGY CONTEXTSENSITIVE 0.012/0.012 (a) 0.012/0.012 (b) 0.012/0.012 (f 1) 0.012/0.012 (g 1 2) 0.012/0.012 (h 1 2) 0.012/0.012 (c) 0.012/0.012 (d) 0.012/0.012 (fSNonEmpty) 0.012/0.012 ) 0.012/0.012 (RULES 0.012/0.012 a -> c 0.012/0.012 a -> d 0.012/0.012 b -> c 0.012/0.012 b -> d 0.012/0.012 f(x) -> x | x ->* c 0.012/0.012 g(x,x) -> h(x,x) 0.012/0.012 h(x,f(x)) -> x 0.012/0.012 )] 0.012/0.012 0.012/0.012 Infeasibility Conditions: 0.012/0.012 x ->* c 0.012/0.012 0.012/0.012 Problem 1: 0.012/0.012 0.012/0.012 Obtaining a proof using Prover9: 0.012/0.012 0.012/0.012 -> Prover9 Output: 0.012/0.012 ============================== Prover9 =============================== 0.012/0.012 Prover9 (64) version 2009-11A, November 2009. 0.012/0.012 Process 61818 was started by ubuntu on ubuntu, 0.012/0.012 Wed Mar 9 09:49:39 2022 0.012/0.012 The command was "./prover9 -f /tmp/prover961809-0.in". 0.012/0.012 ============================== end of head =========================== 0.012/0.012 0.012/0.012 ============================== INPUT ================================= 0.012/0.012 0.012/0.012 % Reading from file /tmp/prover961809-0.in 0.012/0.012 0.012/0.012 assign(max_seconds,20). 0.012/0.012 0.012/0.012 formulas(assumptions). 0.012/0.012 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 0.012/0.012 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence). 0.012/0.012 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence). 0.012/0.012 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence). 0.012/0.012 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence). 0.012/0.012 ->_s0(a,c) # label(replacement). 0.012/0.012 ->_s0(a,d) # label(replacement). 0.012/0.012 ->_s0(b,c) # label(replacement). 0.012/0.012 ->_s0(b,d) # label(replacement). 0.012/0.012 ->*_s0(x1,c) -> ->_s0(f(x1),x1) # label(replacement). 0.012/0.012 ->_s0(g(x1,x1),h(x1,x1)) # label(replacement). 0.012/0.012 ->_s0(h(x1,f(x1)),x1) # label(replacement). 0.012/0.012 ->*_s0(x,x) # label(reflexivity). 0.012/0.012 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 0.012/0.012 end_of_list. 0.012/0.012 0.012/0.012 formulas(goals). 0.012/0.012 (exists x3 ->*_s0(x3,c)) # label(goal). 0.012/0.012 end_of_list. 0.012/0.012 0.012/0.012 ============================== end of input ========================== 0.012/0.012 0.012/0.012 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.012/0.012 0.012/0.012 % Formulas that are not ordinary clauses: 0.012/0.012 1 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 0.012/0.012 2 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.012/0.012 3 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.012/0.012 4 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.012/0.012 5 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.012/0.012 6 ->*_s0(x1,c) -> ->_s0(f(x1),x1) # label(replacement) # label(non_clause). [assumption]. 0.012/0.012 7 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 0.012/0.012 8 (exists x3 ->*_s0(x3,c)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.012/0.012 0.012/0.012 ============================== end of process non-clausal formulas === 0.012/0.012 0.012/0.012 ============================== PROCESS INITIAL CLAUSES =============== 0.012/0.012 0.012/0.012 % Clauses before input processing: 0.012/0.012 0.012/0.012 formulas(usable). 0.012/0.012 end_of_list. 0.012/0.012 0.012/0.012 formulas(sos). 0.012/0.012 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 0.012/0.012 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(2)]. 0.012/0.012 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(3)]. 0.012/0.012 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(4)]. 0.012/0.012 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(5)]. 0.012/0.012 ->_s0(a,c) # label(replacement). [assumption]. 0.012/0.012 ->_s0(a,d) # label(replacement). [assumption]. 0.012/0.012 ->_s0(b,c) # label(replacement). [assumption]. 0.012/0.012 ->_s0(b,d) # label(replacement). [assumption]. 0.012/0.012 -->*_s0(x,c) | ->_s0(f(x),x) # label(replacement). [clausify(6)]. 0.012/0.012 ->_s0(g(x,x),h(x,x)) # label(replacement). [assumption]. 0.012/0.012 ->_s0(h(x,f(x)),x) # label(replacement). [assumption]. 0.012/0.012 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.012/0.012 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 0.012/0.012 -->*_s0(x,c) # label(goal). [deny(8)]. 0.012/0.012 end_of_list. 0.012/0.012 0.012/0.012 formulas(demodulators). 0.012/0.012 end_of_list. 0.012/0.012 0.012/0.012 ============================== PREDICATE ELIMINATION ================= 0.012/0.012 0.012/0.012 No predicates eliminated. 0.012/0.012 0.012/0.012 ============================== end predicate elimination ============= 0.012/0.012 0.012/0.012 Auto_denials: 0.012/0.012 % copying label goal to answer in negative clause 0.012/0.012 0.012/0.012 Term ordering decisions: 0.012/0.012 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 0.012/0.012 Function symbol precedence: function_order([ c, a, b, d, h, g, f ]). 0.012/0.012 After inverse_order: (no changes). 0.012/0.012 Unfolding symbols: (none). 0.012/0.012 0.012/0.012 Auto_inference settings: 0.012/0.012 % set(neg_binary_resolution). % (HNE depth_diff=-6) 0.012/0.012 % clear(ordered_res). % (HNE depth_diff=-6) 0.012/0.012 % set(ur_resolution). % (HNE depth_diff=-6) 0.012/0.012 % set(ur_resolution) -> set(pos_ur_resolution). 0.012/0.012 % set(ur_resolution) -> set(neg_ur_resolution). 0.012/0.012 0.012/0.012 Auto_process settings: (no changes). 0.012/0.012 0.012/0.012 kept: 9 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 0.012/0.012 kept: 10 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(2)]. 0.012/0.012 kept: 11 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(3)]. 0.012/0.012 kept: 12 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(4)]. 0.012/0.012 kept: 13 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(5)]. 0.012/0.012 kept: 14 ->_s0(a,c) # label(replacement). [assumption]. 0.012/0.012 kept: 15 ->_s0(a,d) # label(replacement). [assumption]. 0.012/0.012 kept: 16 ->_s0(b,c) # label(replacement). [assumption]. 0.012/0.012 kept: 17 ->_s0(b,d) # label(replacement). [assumption]. 0.012/0.012 kept: 18 -->*_s0(x,c) | ->_s0(f(x),x) # label(replacement). [clausify(6)]. 0.012/0.012 kept: 19 ->_s0(g(x,x),h(x,x)) # label(replacement). [assumption]. 0.012/0.012 kept: 20 ->_s0(h(x,f(x)),x) # label(replacement). [assumption]. 0.012/0.012 kept: 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.012/0.012 kept: 22 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(7)]. 0.012/0.012 kept: 23 -->*_s0(x,c) # label(goal) # answer(goal). [deny(8)]. 0.012/0.012 0.012/0.012 ============================== PROOF ================================= 0.012/0.012 0.012/0.012 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 0.012/0.012 % Length of proof is 4. 0.012/0.012 % Level of proof is 2. 0.012/0.012 % Maximum clause weight is 3.000. 0.012/0.012 % Given clauses 0. 0.012/0.012 0.012/0.012 8 (exists x3 ->*_s0(x3,c)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.012/0.012 21 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.012/0.012 23 -->*_s0(x,c) # label(goal) # answer(goal). [deny(8)]. 0.012/0.012 24 $F # answer(goal). [resolve(23,a,21,a)]. 0.012/0.012 0.012/0.012 ============================== end of proof ========================== 0.012/0.012 0.012/0.012 ============================== STATISTICS ============================ 0.012/0.012 0.012/0.012 Given=0. Generated=15. Kept=15. proofs=1. 0.012/0.012 Usable=0. Sos=0. Demods=0. Limbo=14, Disabled=15. Hints=0. 0.012/0.012 Kept_by_rule=0, Deleted_by_rule=0. 0.012/0.012 Forward_subsumed=0. Back_subsumed=0. 0.012/0.012 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 0.012/0.012 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 0.012/0.012 Demod_attempts=0. Demod_rewrites=0. 0.012/0.012 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 0.012/0.012 Nonunit_fsub_feature_tests=2. Nonunit_bsub_feature_tests=0. 0.012/0.012 Megabytes=0.05. 0.012/0.012 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.012/0.012 0.012/0.012 ============================== end of statistics ===================== 0.012/0.012 0.012/0.012 ============================== end of search ========================= 0.012/0.012 0.012/0.012 THEOREM PROVED 0.012/0.012 0.012/0.012 Exiting with 1 proof. 0.012/0.012 0.012/0.012 Process 61818 exit (max_proofs) Wed Mar 9 09:49:39 2022 0.012/0.012 0.012/0.012 0.012/0.012 The problem is feasible. 0.012/0.012 0.012/0.012 0.012/0.012 CRule InfChecker Info: 0.012/0.012 g(x,x) -> h(x,x) 0.012/0.012 Rule remains 0.012/0.012 Proof: 0.012/0.012 NO_CONDS 0.012/0.012 0.012/0.012 CRule InfChecker Info: 0.012/0.012 h(x,f(x)) -> x 0.012/0.012 Rule remains 0.012/0.012 Proof: 0.012/0.012 NO_CONDS 0.012/0.012 0.012/0.012 Problem 1: 0.012/0.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.012/0.012 Confluence Problem: 0.012/0.012 (VAR x) 0.012/0.012 (REPLACEMENT-MAP 0.012/0.012 (a) 0.012/0.012 (b) 0.012/0.012 (f 1) 0.012/0.012 (g 1, 2) 0.012/0.012 (h 1, 2) 0.012/0.012 (c) 0.012/0.012 (d) 0.012/0.012 (fSNonEmpty) 0.012/0.012 ) 0.012/0.012 (RULES 0.012/0.012 a -> c 0.012/0.012 a -> d 0.012/0.012 b -> c 0.012/0.012 b -> d 0.012/0.012 f(x) -> x | x ->* c 0.012/0.012 g(x,x) -> h(x,x) 0.012/0.012 h(x,f(x)) -> x 0.012/0.012 ) 0.012/0.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.012/0.012 0.012/0.012 Critical Pairs Processor: 0.012/0.012 -> Rules: 0.012/0.012 a -> c 0.012/0.012 a -> d 0.012/0.012 b -> c 0.012/0.012 b -> d 0.012/0.012 f(x) -> x | x ->* c 0.012/0.012 g(x,x) -> h(x,x) 0.012/0.012 h(x,f(x)) -> x 0.012/0.012 -> Vars: 0.012/0.012 "x" 0.012/0.012 0.012/0.012 -> Rlps: 0.012/0.012 crule: a -> c, id: 1, possubterms: a-> [] 0.012/0.012 crule: a -> d, id: 2, possubterms: a-> [] 0.012/0.012 crule: b -> c, id: 3, possubterms: b-> [] 0.012/0.012 crule: b -> d, id: 4, possubterms: b-> [] 0.012/0.012 crule: f(x) -> x | x ->* c, id: 5, possubterms: f(x)-> [] 0.012/0.012 crule: g(x,x) -> h(x,x), id: 6, possubterms: g(x,x)-> [] 0.012/0.012 crule: h(x,f(x)) -> x, id: 7, possubterms: h(x,f(x))-> [], f(x)-> [2] 0.012/0.012 0.012/0.012 -> Unifications: 0.012/0.012 R2 unifies with R1 at p: [], l: a, lp: a, conds: {}, sig: {}, l': a, r: d, r': c 0.012/0.012 R4 unifies with R3 at p: [], l: b, lp: b, conds: {}, sig: {}, l': b, r: d, r': c 0.012/0.012 R7 unifies with R5 at p: [2], l: h(x,f(x)), lp: f(x), conds: {x' ->* c}, sig: {x -> x'}, l': f(x'), r: x, r': x' 0.012/0.012 0.012/0.012 -> Critical pairs info: 0.012/0.012 => Not trivial, Overlay, NW2, N1 0.012/0.012 | x' ->* c => Not trivial, Not overlay, NW2, N2 0.012/0.012 0.012/0.012 -> Problem conclusions: 0.012/0.012 Not left linear, Not right linear, Not linear 0.012/0.012 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.012/0.012 CTRS Type: 1 0.012/0.012 Deterministic, Strongly deterministic 0.012/0.012 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.012/0.012 Maybe right-stable CTRS, Not overlay CTRS 0.012/0.012 Maybe normal CTRS, Maybe almost normal CTRS 0.012/0.012 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 0.012/0.012 Maybe level confluent 0.012/0.012 Maybe confluent 0.012/0.012 0.012/0.012 Problem 1: 0.012/0.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.012/0.012 Confluence Problem: 0.012/0.012 (VAR x x') 0.012/0.012 (REPLACEMENT-MAP 0.012/0.012 (a) 0.012/0.012 (b) 0.012/0.012 (f 1) 0.012/0.012 (g 1, 2) 0.012/0.012 (h 1, 2) 0.012/0.012 (c) 0.012/0.012 (d) 0.012/0.012 (fSNonEmpty) 0.012/0.012 ) 0.012/0.012 (RULES 0.012/0.012 a -> c 0.012/0.012 a -> d 0.012/0.012 b -> c 0.012/0.012 b -> d 0.012/0.012 f(x) -> x | x ->* c 0.012/0.012 g(x,x) -> h(x,x) 0.012/0.012 h(x,f(x)) -> x 0.012/0.012 ) 0.012/0.012 Critical Pairs: 0.012/0.012 => Not trivial, Overlay, NW2, N1 0.012/0.012 | x' ->* c => Not trivial, Not overlay, NW2, N2 0.012/0.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.012/0.012 0.012/0.012 Conditional Critical Pairs Distributor Processor 0.012/0.012 0.012/0.012 Problem 1: 0.012/0.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.012/0.012 Confluence Problem: 0.012/0.012 (VAR x x') 0.012/0.012 (REPLACEMENT-MAP 0.012/0.012 (a) 0.012/0.012 (b) 0.012/0.012 (f 1) 0.012/0.012 (g 1, 2) 0.012/0.012 (h 1, 2) 0.012/0.012 (c) 0.012/0.012 (d) 0.012/0.012 (fSNonEmpty) 0.012/0.012 ) 0.012/0.012 (RULES 0.012/0.012 a -> c 0.012/0.012 a -> d 0.012/0.012 b -> c 0.012/0.012 b -> d 0.012/0.012 f(x) -> x | x ->* c 0.012/0.012 g(x,x) -> h(x,x) 0.012/0.012 h(x,f(x)) -> x 0.012/0.012 ) 0.012/0.012 Critical Pairs: 0.012/0.012 => Not trivial, Overlay, NW2, N1 0.012/0.012 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.012/0.012 0.012/0.012 InfChecker Different Ground and Normal No Conditions CCP Terms Processor: 0.012/0.012 Proof: 0.012/0.012 YES 0.012/0.012 0.012/0.012 Problem 1: 0.012/0.012 0.012/0.012 Infeasibility Problem: 0.012/0.012 [(VAR vNonEmpty x x1 vNonEmpty y) 0.012/0.012 (STRATEGY CONTEXTSENSITIVE 0.012/0.012 (a) 0.012/0.012 (b) 0.012/0.012 (f 1) 0.012/0.012 (g 1 2) 0.012/0.012 (h 1 2) 0.012/0.012 (c) 0.012/0.012 (d) 0.012/0.012 (fSNonEmpty) 0.012/0.012 ) 0.012/0.012 (RULES 0.012/0.012 a -> c 0.012/0.012 a -> d 0.012/0.012 b -> c 0.012/0.012 b -> d 0.012/0.012 f(x) -> x | x ->* c 0.012/0.012 g(x,x) -> h(x,x) 0.012/0.012 h(x,f(x)) -> x 0.012/0.012 )] 0.012/0.012 0.012/0.012 Infeasibility Conditions: 0.012/0.012 c -> y 0.012/0.012 0.012/0.012 Problem 1: 0.012/0.012 0.012/0.012 Obtaining a model using Mace4: 0.012/0.012 0.012/0.012 -> Usable Rules: 0.012/0.012 Empty 0.012/0.012 0.012/0.012 -> Mace4 Output: 0.012/0.012 ============================== Mace4 ================================= 0.012/0.012 Mace4 (64) version 2009-11A, November 2009. 0.012/0.012 Process 61853 was started by ubuntu on ubuntu, 0.012/0.012 Wed Mar 9 09:49:39 2022 0.012/0.012 The command was "./mace4 -c -f /tmp/mace461834-2.in". 0.012/0.012 ============================== end of head =========================== 0.012/0.012 0.012/0.012 ============================== INPUT ================================= 0.012/0.012 0.012/0.012 % Reading from file /tmp/mace461834-2.in 0.012/0.012 0.012/0.012 assign(max_seconds,6). 0.012/0.012 0.012/0.012 formulas(assumptions). 0.012/0.012 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 0.012/0.012 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence). 0.012/0.012 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence). 0.012/0.012 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence). 0.012/0.012 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence). 0.012/0.012 end_of_list. 0.012/0.012 0.012/0.012 formulas(goals). 0.012/0.012 (exists x4 ->(c,x4)) # label(goal). 0.012/0.012 end_of_list. 0.012/0.012 0.012/0.012 ============================== end of input ========================== 0.012/0.012 0.012/0.012 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.012/0.012 0.012/0.012 % Formulas that are not ordinary clauses: 0.012/0.012 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 0.012/0.012 2 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.012/0.012 3 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.012/0.012 4 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.012/0.012 5 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.012/0.012 6 (exists x4 ->(c,x4)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.012/0.012 0.012/0.012 ============================== end of process non-clausal formulas === 0.012/0.012 0.012/0.012 ============================== CLAUSES FOR SEARCH ==================== 0.012/0.012 0.012/0.012 formulas(mace4_clauses). 0.012/0.012 -->(x,y) | ->(f(x),f(y)) # label(congruence). 0.012/0.012 -->(x,y) | ->(g(x,z),g(y,z)) # label(congruence). 0.012/0.012 -->(x,y) | ->(g(z,x),g(z,y)) # label(congruence). 0.012/0.012 -->(x,y) | ->(h(x,z),h(y,z)) # label(congruence). 0.012/0.012 -->(x,y) | ->(h(z,x),h(z,y)) # label(congruence). 0.012/0.012 -->(c,x) # label(goal). 0.012/0.012 end_of_list. 0.012/0.012 0.012/0.012 ============================== end of clauses for search ============= 0.012/0.012 0.012/0.012 % There are no natural numbers in the input. 0.012/0.012 0.012/0.012 ============================== DOMAIN SIZE 2 ========================= 0.012/0.012 0.012/0.012 ============================== MODEL ================================= 0.012/0.012 0.012/0.012 interpretation( 2, [number=1, seconds=0], [ 0.012/0.012 0.012/0.012 function(c, [ 0 ]), 0.012/0.012 0.012/0.012 function(f(_), [ 0, 0 ]), 0.012/0.012 0.012/0.012 function(g(_,_), [ 0.012/0.012 0, 0, 0.012/0.012 0, 0 ]), 0.012/0.012 0.012/0.012 function(h(_,_), [ 0.012/0.012 0, 0, 0.012/0.012 0, 0 ]), 0.012/0.012 0.012/0.012 relation(->(_,_), [ 0.012/0.012 0, 0, 0.012/0.012 0, 0 ]) 0.012/0.012 ]). 0.012/0.012 0.012/0.012 ============================== end of model ========================== 0.012/0.012 0.012/0.012 ============================== STATISTICS ============================ 0.012/0.012 0.012/0.012 For domain size 2. 0.012/0.012 0.012/0.012 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 0.012/0.012 Ground clauses: seen=38, kept=38. 0.012/0.012 Selections=11, assignments=11, propagations=4, current_models=1. 0.012/0.012 Rewrite_terms=74, rewrite_bools=40, indexes=2. 0.012/0.012 Rules_from_neg_clauses=0, cross_offs=0. 0.012/0.012 0.012/0.012 ============================== end of statistics ===================== 0.012/0.012 0.012/0.012 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.012/0.012 0.012/0.012 Exiting with 1 model. 0.012/0.012 0.012/0.012 Process 61853 exit (max_models) Wed Mar 9 09:49:39 2022 0.012/0.012 The process finished Wed Mar 9 09:49:39 2022 0.012/0.012 0.012/0.012 0.012/0.012 Mace4 cooked interpretation: 0.012/0.012 0.012/0.012 0.012/0.012 0.012/0.012 The problem is infeasible. 0.012/0.012 YES 0.012/0.012 0.012/0.012 Problem 1: 0.012/0.012 0.012/0.012 Infeasibility Problem: 0.012/0.012 [(VAR vNonEmpty x x1 vNonEmpty x) 0.012/0.012 (STRATEGY CONTEXTSENSITIVE 0.012/0.012 (a) 0.012/0.012 (b) 0.012/0.012 (f 1) 0.012/0.012 (g 1 2) 0.012/0.012 (h 1 2) 0.012/0.012 (c) 0.012/0.012 (d) 0.012/0.012 (fSNonEmpty) 0.012/0.012 ) 0.012/0.012 (RULES 0.012/0.012 a -> c 0.012/0.012 a -> d 0.012/0.012 b -> c 0.012/0.012 b -> d 0.012/0.012 f(x) -> x | x ->* c 0.012/0.012 g(x,x) -> h(x,x) 0.012/0.012 h(x,f(x)) -> x 0.012/0.012 )] 0.012/0.012 0.012/0.012 Infeasibility Conditions: 0.012/0.012 d -> x 0.012/0.012 0.012/0.012 Problem 1: 0.012/0.012 0.012/0.012 Obtaining a model using Mace4: 0.012/0.012 0.012/0.012 -> Usable Rules: 0.012/0.012 Empty 0.012/0.012 0.012/0.012 -> Mace4 Output: 0.012/0.012 ============================== Mace4 ================================= 0.012/0.012 Mace4 (64) version 2009-11A, November 2009. 0.012/0.012 Process 61894 was started by ubuntu on ubuntu, 0.012/0.012 Wed Mar 9 09:49:39 2022 0.012/0.012 The command was "./mace4 -c -f /tmp/mace461882-2.in". 0.012/0.012 ============================== end of head =========================== 0.012/0.012 0.012/0.012 ============================== INPUT ================================= 0.012/0.012 0.012/0.012 % Reading from file /tmp/mace461882-2.in 0.012/0.012 0.012/0.012 assign(max_seconds,6). 0.012/0.012 0.012/0.012 formulas(assumptions). 0.012/0.012 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 0.012/0.012 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence). 0.012/0.012 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence). 0.012/0.012 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence). 0.012/0.012 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence). 0.012/0.012 end_of_list. 0.012/0.012 0.012/0.012 formulas(goals). 0.012/0.012 (exists x4 ->(d,x4)) # label(goal). 0.012/0.012 end_of_list. 0.012/0.012 0.012/0.012 ============================== end of input ========================== 0.012/0.012 0.012/0.012 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.012/0.012 0.012/0.012 % Formulas that are not ordinary clauses: 0.012/0.012 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 0.012/0.012 2 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.012/0.012 3 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.012/0.012 4 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.012/0.012 5 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.012/0.012 6 (exists x4 ->(d,x4)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.012/0.012 0.012/0.012 ============================== end of process non-clausal formulas === 0.012/0.012 0.012/0.012 ============================== CLAUSES FOR SEARCH ==================== 0.012/0.012 0.012/0.012 formulas(mace4_clauses). 0.012/0.012 -->(x,y) | ->(f(x),f(y)) # label(congruence). 0.012/0.012 -->(x,y) | ->(g(x,z),g(y,z)) # label(congruence). 0.012/0.012 -->(x,y) | ->(g(z,x),g(z,y)) # label(congruence). 0.012/0.012 -->(x,y) | ->(h(x,z),h(y,z)) # label(congruence). 0.012/0.012 -->(x,y) | ->(h(z,x),h(z,y)) # label(congruence). 0.012/0.012 -->(d,x) # label(goal). 0.012/0.012 end_of_list. 0.012/0.012 0.012/0.012 ============================== end of clauses for search ============= 0.012/0.012 0.012/0.012 % There are no natural numbers in the input. 0.012/0.012 0.012/0.012 ============================== DOMAIN SIZE 2 ========================= 0.012/0.012 0.012/0.012 ============================== MODEL ================================= 0.012/0.012 0.012/0.012 interpretation( 2, [number=1, seconds=0], [ 0.012/0.012 0.012/0.012 function(d, [ 0 ]), 0.012/0.012 0.012/0.012 function(f(_), [ 0, 0 ]), 0.012/0.012 0.012/0.012 function(g(_,_), [ 0.012/0.012 0, 0, 0.012/0.012 0, 0 ]), 0.012/0.012 0.012/0.012 function(h(_,_), [ 0.012/0.012 0, 0, 0.012/0.012 0, 0 ]), 0.012/0.012 0.012/0.012 relation(->(_,_), [ 0.012/0.012 0, 0, 0.012/0.012 0, 0 ]) 0.012/0.012 ]). 0.012/0.012 0.012/0.012 ============================== end of model ========================== 0.012/0.012 0.012/0.012 ============================== STATISTICS ============================ 0.012/0.012 0.012/0.012 For domain size 2. 0.012/0.012 0.012/0.012 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 0.012/0.012 Ground clauses: seen=38, kept=38. 0.012/0.012 Selections=11, assignments=11, propagations=4, current_models=1. 0.012/0.012 Rewrite_terms=74, rewrite_bools=40, indexes=2. 0.012/0.012 Rules_from_neg_clauses=0, cross_offs=0. 0.012/0.012 0.012/0.012 ============================== end of statistics ===================== 0.012/0.012 0.012/0.012 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.012/0.012 0.012/0.012 Exiting with 1 model. 0.012/0.012 0.012/0.012 Process 61894 exit (max_models) Wed Mar 9 09:49:39 2022 0.012/0.012 The process finished Wed Mar 9 09:49:39 2022 0.012/0.012 0.012/0.012 0.012/0.012 Mace4 cooked interpretation: 0.012/0.012 0.012/0.012 0.012/0.012 0.012/0.012 The problem is infeasible. 0.012/0.012 0.012/0.012 0.012/0.012 The problem is not joinable. 0.012/0.012 0.08user 0.02system 0:00.12elapsed 90%CPU (0avgtext+0avgdata 12048maxresident)k 0.012/0.012 8inputs+0outputs (0major+9517minor)pagefaults 0swaps