0.015/0.015 NO 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 Confluence Problem: 0.015/0.015 (VAR vNonEmpty x) 0.015/0.015 (REPLACEMENT-MAP 0.015/0.015 (a) 0.015/0.015 (b) 0.015/0.015 (c) 0.015/0.015 (d) 0.015/0.015 (f 1) 0.015/0.015 (g 1, 2, 3) 0.015/0.015 (h 1, 2) 0.015/0.015 (k) 0.015/0.015 (A) 0.015/0.015 (e) 0.015/0.015 (fSNonEmpty) 0.015/0.015 (l) 0.015/0.015 (m) 0.015/0.015 ) 0.015/0.015 (RULES 0.015/0.015 a -> c 0.015/0.015 a -> d 0.015/0.015 b -> c 0.015/0.015 b -> d 0.015/0.015 c -> e 0.015/0.015 c -> l 0.015/0.015 d -> m 0.015/0.015 f(x) -> x | x ->* e 0.015/0.015 g(d,x,x) -> A 0.015/0.015 h(x,x) -> g(x,x,f(k)) 0.015/0.015 k -> l 0.015/0.015 k -> m 0.015/0.015 ) 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 Inlining of Conditions Processor [STERN17]: 0.015/0.015 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 Confluence Problem: 0.015/0.015 (VAR vNonEmpty x) 0.015/0.015 (REPLACEMENT-MAP 0.015/0.015 (a) 0.015/0.015 (b) 0.015/0.015 (c) 0.015/0.015 (d) 0.015/0.015 (f 1) 0.015/0.015 (g 1, 2, 3) 0.015/0.015 (h 1, 2) 0.015/0.015 (k) 0.015/0.015 (A) 0.015/0.015 (e) 0.015/0.015 (fSNonEmpty) 0.015/0.015 (l) 0.015/0.015 (m) 0.015/0.015 ) 0.015/0.015 (RULES 0.015/0.015 a -> c 0.015/0.015 a -> d 0.015/0.015 b -> c 0.015/0.015 b -> d 0.015/0.015 c -> e 0.015/0.015 c -> l 0.015/0.015 d -> m 0.015/0.015 f(x) -> x | x ->* e 0.015/0.015 g(d,x,x) -> A 0.015/0.015 h(x,x) -> g(x,x,f(k)) 0.015/0.015 k -> l 0.015/0.015 k -> m 0.015/0.015 ) 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 Clean CTRS Processor: 0.015/0.015 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 Confluence Problem: 0.015/0.015 (VAR x) 0.015/0.015 (REPLACEMENT-MAP 0.015/0.015 (a) 0.015/0.015 (b) 0.015/0.015 (c) 0.015/0.015 (d) 0.015/0.015 (f 1) 0.015/0.015 (g 1, 2, 3) 0.015/0.015 (h 1, 2) 0.015/0.015 (k) 0.015/0.015 (A) 0.015/0.015 (e) 0.015/0.015 (fSNonEmpty) 0.015/0.015 (l) 0.015/0.015 (m) 0.015/0.015 ) 0.015/0.015 (RULES 0.015/0.015 a -> c 0.015/0.015 a -> d 0.015/0.015 b -> c 0.015/0.015 b -> d 0.015/0.015 c -> e 0.015/0.015 c -> l 0.015/0.015 d -> m 0.015/0.015 f(x) -> x | x ->* e 0.015/0.015 g(d,x,x) -> A 0.015/0.015 h(x,x) -> g(x,x,f(k)) 0.015/0.015 k -> l 0.015/0.015 k -> m 0.015/0.015 ) 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 0.015/0.015 CRule InfChecker Info: 0.015/0.015 a -> c 0.015/0.015 Rule remains 0.015/0.015 Proof: 0.015/0.015 NO_CONDS 0.015/0.015 0.015/0.015 CRule InfChecker Info: 0.015/0.015 a -> d 0.015/0.015 Rule remains 0.015/0.015 Proof: 0.015/0.015 NO_CONDS 0.015/0.015 0.015/0.015 CRule InfChecker Info: 0.015/0.015 b -> c 0.015/0.015 Rule remains 0.015/0.015 Proof: 0.015/0.015 NO_CONDS 0.015/0.015 0.015/0.015 CRule InfChecker Info: 0.015/0.015 b -> d 0.015/0.015 Rule remains 0.015/0.015 Proof: 0.015/0.015 NO_CONDS 0.015/0.015 0.015/0.015 CRule InfChecker Info: 0.015/0.015 c -> e 0.015/0.015 Rule remains 0.015/0.015 Proof: 0.015/0.015 NO_CONDS 0.015/0.015 0.015/0.015 CRule InfChecker Info: 0.015/0.015 c -> l 0.015/0.015 Rule remains 0.015/0.015 Proof: 0.015/0.015 NO_CONDS 0.015/0.015 0.015/0.015 CRule InfChecker Info: 0.015/0.015 d -> m 0.015/0.015 Rule remains 0.015/0.015 Proof: 0.015/0.015 NO_CONDS 0.015/0.015 0.015/0.015 CRule InfChecker Info: 0.015/0.015 f(x) -> x | x ->* e 0.015/0.015 Rule remains 0.015/0.015 Proof: 0.015/0.015 NO 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 Infeasibility Problem: 0.015/0.015 [(VAR vNonEmpty x vNonEmpty x) 0.015/0.015 (STRATEGY CONTEXTSENSITIVE 0.015/0.015 (a) 0.015/0.015 (b) 0.015/0.015 (c) 0.015/0.015 (d) 0.015/0.015 (f 1) 0.015/0.015 (g 1 2 3) 0.015/0.015 (h 1 2) 0.015/0.015 (k) 0.015/0.015 (A) 0.015/0.015 (e) 0.015/0.015 (fSNonEmpty) 0.015/0.015 (l) 0.015/0.015 (m) 0.015/0.015 ) 0.015/0.015 (RULES 0.015/0.015 a -> c 0.015/0.015 a -> d 0.015/0.015 b -> c 0.015/0.015 b -> d 0.015/0.015 c -> e 0.015/0.015 c -> l 0.015/0.015 d -> m 0.015/0.015 f(x) -> x | x ->* e 0.015/0.015 g(d,x,x) -> A 0.015/0.015 h(x,x) -> g(x,x,f(k)) 0.015/0.015 k -> l 0.015/0.015 k -> m 0.015/0.015 )] 0.015/0.015 0.015/0.015 Infeasibility Conditions: 0.015/0.015 x ->* e 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 Obtaining a proof using Prover9: 0.015/0.015 0.015/0.015 -> Prover9 Output: 0.015/0.015 ============================== Prover9 =============================== 0.015/0.015 Prover9 (64) version 2009-11A, November 2009. 0.015/0.015 Process 11057 was started by ubuntu on ubuntu, 0.015/0.015 Wed Mar 9 09:15:55 2022 0.015/0.015 The command was "./prover9 -f /tmp/prover911050-0.in". 0.015/0.015 ============================== end of head =========================== 0.015/0.015 0.015/0.015 ============================== INPUT ================================= 0.015/0.015 0.015/0.015 % Reading from file /tmp/prover911050-0.in 0.015/0.015 0.015/0.015 assign(max_seconds,20). 0.015/0.015 0.015/0.015 formulas(assumptions). 0.015/0.015 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 0.015/0.015 ->_s0(x1,y) -> ->_s0(g(x1,x2,x3),g(y,x2,x3)) # label(congruence). 0.015/0.015 ->_s0(x2,y) -> ->_s0(g(x1,x2,x3),g(x1,y,x3)) # label(congruence). 0.015/0.015 ->_s0(x3,y) -> ->_s0(g(x1,x2,x3),g(x1,x2,y)) # label(congruence). 0.015/0.015 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence). 0.015/0.015 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence). 0.015/0.015 ->_s0(a,c) # label(replacement). 0.015/0.015 ->_s0(a,d) # label(replacement). 0.015/0.015 ->_s0(b,c) # label(replacement). 0.015/0.015 ->_s0(b,d) # label(replacement). 0.015/0.015 ->_s0(c,e) # label(replacement). 0.015/0.015 ->_s0(c,l) # label(replacement). 0.015/0.015 ->_s0(d,m) # label(replacement). 0.015/0.015 ->*_s0(x1,e) -> ->_s0(f(x1),x1) # label(replacement). 0.015/0.015 ->_s0(g(d,x1,x1),A) # label(replacement). 0.015/0.015 ->_s0(h(x1,x1),g(x1,x1,f(k))) # label(replacement). 0.015/0.015 ->_s0(k,l) # label(replacement). 0.015/0.015 ->_s0(k,m) # label(replacement). 0.015/0.015 ->*_s0(x,x) # label(reflexivity). 0.015/0.015 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 formulas(goals). 0.015/0.015 (exists x3 ->*_s0(x3,e)) # label(goal). 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 ============================== end of input ========================== 0.015/0.015 0.015/0.015 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.015/0.015 0.015/0.015 % Formulas that are not ordinary clauses: 0.015/0.015 1 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 2 ->_s0(x1,y) -> ->_s0(g(x1,x2,x3),g(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 3 ->_s0(x2,y) -> ->_s0(g(x1,x2,x3),g(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 4 ->_s0(x3,y) -> ->_s0(g(x1,x2,x3),g(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 5 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 6 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 7 ->*_s0(x1,e) -> ->_s0(f(x1),x1) # label(replacement) # label(non_clause). [assumption]. 0.015/0.015 8 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 0.015/0.015 9 (exists x3 ->*_s0(x3,e)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.015/0.015 0.015/0.015 ============================== end of process non-clausal formulas === 0.015/0.015 0.015/0.015 ============================== PROCESS INITIAL CLAUSES =============== 0.015/0.015 0.015/0.015 % Clauses before input processing: 0.015/0.015 0.015/0.015 formulas(usable). 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 formulas(sos). 0.015/0.015 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 0.015/0.015 -->_s0(x,y) | ->_s0(g(x,z,u),g(y,z,u)) # label(congruence). [clausify(2)]. 0.015/0.015 -->_s0(x,y) | ->_s0(g(z,x,u),g(z,y,u)) # label(congruence). [clausify(3)]. 0.015/0.015 -->_s0(x,y) | ->_s0(g(z,u,x),g(z,u,y)) # label(congruence). [clausify(4)]. 0.015/0.015 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(5)]. 0.015/0.015 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(6)]. 0.015/0.015 ->_s0(a,c) # label(replacement). [assumption]. 0.015/0.015 ->_s0(a,d) # label(replacement). [assumption]. 0.015/0.015 ->_s0(b,c) # label(replacement). [assumption]. 0.015/0.015 ->_s0(b,d) # label(replacement). [assumption]. 0.015/0.015 ->_s0(c,e) # label(replacement). [assumption]. 0.015/0.015 ->_s0(c,l) # label(replacement). [assumption]. 0.015/0.015 ->_s0(d,m) # label(replacement). [assumption]. 0.015/0.015 -->*_s0(x,e) | ->_s0(f(x),x) # label(replacement). [clausify(7)]. 0.015/0.015 ->_s0(g(d,x,x),A) # label(replacement). [assumption]. 0.015/0.015 ->_s0(h(x,x),g(x,x,f(k))) # label(replacement). [assumption]. 0.015/0.015 ->_s0(k,l) # label(replacement). [assumption]. 0.015/0.015 ->_s0(k,m) # label(replacement). [assumption]. 0.015/0.015 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.015/0.015 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 0.015/0.015 -->*_s0(x,e) # label(goal). [deny(9)]. 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 formulas(demodulators). 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 ============================== PREDICATE ELIMINATION ================= 0.015/0.015 0.015/0.015 No predicates eliminated. 0.015/0.015 0.015/0.015 ============================== end predicate elimination ============= 0.015/0.015 0.015/0.015 Auto_denials: 0.015/0.015 % copying label goal to answer in negative clause 0.015/0.015 0.015/0.015 Term ordering decisions: 0.015/0.015 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 0.015/0.015 Function symbol precedence: function_order([ c, d, k, e, l, m, a, b, A, h, f, g ]). 0.015/0.015 After inverse_order: (no changes). 0.015/0.015 Unfolding symbols: (none). 0.015/0.015 0.015/0.015 Auto_inference settings: 0.015/0.015 % set(neg_binary_resolution). % (HNE depth_diff=-7) 0.015/0.015 % clear(ordered_res). % (HNE depth_diff=-7) 0.015/0.015 % set(ur_resolution). % (HNE depth_diff=-7) 0.015/0.015 % set(ur_resolution) -> set(pos_ur_resolution). 0.015/0.015 % set(ur_resolution) -> set(neg_ur_resolution). 0.015/0.015 0.015/0.015 Auto_process settings: (no changes). 0.015/0.015 0.015/0.015 kept: 10 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 0.015/0.015 kept: 11 -->_s0(x,y) | ->_s0(g(x,z,u),g(y,z,u)) # label(congruence). [clausify(2)]. 0.015/0.015 kept: 12 -->_s0(x,y) | ->_s0(g(z,x,u),g(z,y,u)) # label(congruence). [clausify(3)]. 0.015/0.015 kept: 13 -->_s0(x,y) | ->_s0(g(z,u,x),g(z,u,y)) # label(congruence). [clausify(4)]. 0.015/0.015 kept: 14 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(5)]. 0.015/0.015 kept: 15 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(6)]. 0.015/0.015 kept: 16 ->_s0(a,c) # label(replacement). [assumption]. 0.015/0.015 kept: 17 ->_s0(a,d) # label(replacement). [assumption]. 0.015/0.015 kept: 18 ->_s0(b,c) # label(replacement). [assumption]. 0.015/0.015 kept: 19 ->_s0(b,d) # label(replacement). [assumption]. 0.015/0.015 kept: 20 ->_s0(c,e) # label(replacement). [assumption]. 0.015/0.015 kept: 21 ->_s0(c,l) # label(replacement). [assumption]. 0.015/0.015 kept: 22 ->_s0(d,m) # label(replacement). [assumption]. 0.015/0.015 kept: 23 -->*_s0(x,e) | ->_s0(f(x),x) # label(replacement). [clausify(7)]. 0.015/0.015 kept: 24 ->_s0(g(d,x,x),A) # label(replacement). [assumption]. 0.015/0.015 kept: 25 ->_s0(h(x,x),g(x,x,f(k))) # label(replacement). [assumption]. 0.015/0.015 kept: 26 ->_s0(k,l) # label(replacement). [assumption]. 0.015/0.015 kept: 27 ->_s0(k,m) # label(replacement). [assumption]. 0.015/0.015 kept: 28 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.015/0.015 kept: 29 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 0.015/0.015 kept: 30 -->*_s0(x,e) # label(goal) # answer(goal). [deny(9)]. 0.015/0.015 0.015/0.015 ============================== PROOF ================================= 0.015/0.015 0.015/0.015 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 0.015/0.015 % Length of proof is 4. 0.015/0.015 % Level of proof is 2. 0.015/0.015 % Maximum clause weight is 3.000. 0.015/0.015 % Given clauses 0. 0.015/0.015 0.015/0.015 9 (exists x3 ->*_s0(x3,e)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.015/0.015 28 ->*_s0(x,x) # label(reflexivity). [assumption]. 0.015/0.015 30 -->*_s0(x,e) # label(goal) # answer(goal). [deny(9)]. 0.015/0.015 31 $F # answer(goal). [resolve(30,a,28,a)]. 0.015/0.015 0.015/0.015 ============================== end of proof ========================== 0.015/0.015 0.015/0.015 ============================== STATISTICS ============================ 0.015/0.015 0.015/0.015 Given=0. Generated=21. Kept=21. proofs=1. 0.015/0.015 Usable=0. Sos=0. Demods=0. Limbo=20, Disabled=21. Hints=0. 0.015/0.015 Kept_by_rule=0, Deleted_by_rule=0. 0.015/0.015 Forward_subsumed=0. Back_subsumed=0. 0.015/0.015 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 0.015/0.015 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 0.015/0.015 Demod_attempts=0. Demod_rewrites=0. 0.015/0.015 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 0.015/0.015 Nonunit_fsub_feature_tests=4. Nonunit_bsub_feature_tests=0. 0.015/0.015 Megabytes=0.07. 0.015/0.015 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.015/0.015 0.015/0.015 ============================== end of statistics ===================== 0.015/0.015 0.015/0.015 ============================== end of search ========================= 0.015/0.015 0.015/0.015 THEOREM PROVED 0.015/0.015 0.015/0.015 Exiting with 1 proof. 0.015/0.015 0.015/0.015 Process 11057 exit (max_proofs) Wed Mar 9 09:15:55 2022 0.015/0.015 0.015/0.015 0.015/0.015 The problem is feasible. 0.015/0.015 0.015/0.015 0.015/0.015 CRule InfChecker Info: 0.015/0.015 g(d,x,x) -> A 0.015/0.015 Rule remains 0.015/0.015 Proof: 0.015/0.015 NO_CONDS 0.015/0.015 0.015/0.015 CRule InfChecker Info: 0.015/0.015 h(x,x) -> g(x,x,f(k)) 0.015/0.015 Rule remains 0.015/0.015 Proof: 0.015/0.015 NO_CONDS 0.015/0.015 0.015/0.015 CRule InfChecker Info: 0.015/0.015 k -> l 0.015/0.015 Rule remains 0.015/0.015 Proof: 0.015/0.015 NO_CONDS 0.015/0.015 0.015/0.015 CRule InfChecker Info: 0.015/0.015 k -> m 0.015/0.015 Rule remains 0.015/0.015 Proof: 0.015/0.015 NO_CONDS 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 Confluence Problem: 0.015/0.015 (VAR x) 0.015/0.015 (REPLACEMENT-MAP 0.015/0.015 (a) 0.015/0.015 (b) 0.015/0.015 (c) 0.015/0.015 (d) 0.015/0.015 (f 1) 0.015/0.015 (g 1, 2, 3) 0.015/0.015 (h 1, 2) 0.015/0.015 (k) 0.015/0.015 (A) 0.015/0.015 (e) 0.015/0.015 (fSNonEmpty) 0.015/0.015 (l) 0.015/0.015 (m) 0.015/0.015 ) 0.015/0.015 (RULES 0.015/0.015 a -> c 0.015/0.015 a -> d 0.015/0.015 b -> c 0.015/0.015 b -> d 0.015/0.015 c -> e 0.015/0.015 c -> l 0.015/0.015 d -> m 0.015/0.015 f(x) -> x | x ->* e 0.015/0.015 g(d,x,x) -> A 0.015/0.015 h(x,x) -> g(x,x,f(k)) 0.015/0.015 k -> l 0.015/0.015 k -> m 0.015/0.015 ) 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 0.015/0.015 Critical Pairs Processor: 0.015/0.015 -> Rules: 0.015/0.015 a -> c 0.015/0.015 a -> d 0.015/0.015 b -> c 0.015/0.015 b -> d 0.015/0.015 c -> e 0.015/0.015 c -> l 0.015/0.015 d -> m 0.015/0.015 f(x) -> x | x ->* e 0.015/0.015 g(d,x,x) -> A 0.015/0.015 h(x,x) -> g(x,x,f(k)) 0.015/0.015 k -> l 0.015/0.015 k -> m 0.015/0.015 -> Vars: 0.015/0.015 "x" 0.015/0.015 0.015/0.015 -> Rlps: 0.015/0.015 crule: a -> c, id: 1, possubterms: a-> [] 0.015/0.015 crule: a -> d, id: 2, possubterms: a-> [] 0.015/0.015 crule: b -> c, id: 3, possubterms: b-> [] 0.015/0.015 crule: b -> d, id: 4, possubterms: b-> [] 0.015/0.015 crule: c -> e, id: 5, possubterms: c-> [] 0.015/0.015 crule: c -> l, id: 6, possubterms: c-> [] 0.015/0.015 crule: d -> m, id: 7, possubterms: d-> [] 0.015/0.015 crule: f(x) -> x | x ->* e, id: 8, possubterms: f(x)-> [] 0.015/0.015 crule: g(d,x,x) -> A, id: 9, possubterms: g(d,x,x)-> [], d-> [1] 0.015/0.015 crule: h(x,x) -> g(x,x,f(k)), id: 10, possubterms: h(x,x)-> [] 0.015/0.015 crule: k -> l, id: 11, possubterms: k-> [] 0.015/0.015 crule: k -> m, id: 12, possubterms: k-> [] 0.015/0.015 0.015/0.015 -> Unifications: 0.015/0.015 R2 unifies with R1 at p: [], l: a, lp: a, conds: {}, sig: {}, l': a, r: d, r': c 0.015/0.015 R4 unifies with R3 at p: [], l: b, lp: b, conds: {}, sig: {}, l': b, r: d, r': c 0.015/0.015 R6 unifies with R5 at p: [], l: c, lp: c, conds: {}, sig: {}, l': c, r: l, r': e 0.015/0.015 R9 unifies with R7 at p: [1], l: g(d,x,x), lp: d, conds: {}, sig: {}, l': d, r: A, r': m 0.015/0.015 R12 unifies with R11 at p: [], l: k, lp: k, conds: {}, sig: {}, l': k, r: m, r': l 0.015/0.015 0.015/0.015 -> Critical pairs info: 0.015/0.015 => Not trivial, Overlay, NW2, N1 0.015/0.015 => Not trivial, Overlay, NW2, N2 0.015/0.015 => Not trivial, Not overlay, NW2, N3 0.015/0.015 => Not trivial, Overlay, NW0, N4 0.015/0.015 0.015/0.015 -> Problem conclusions: 0.015/0.015 Not left linear, Not right linear, Not linear 0.015/0.015 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 0.015/0.015 CTRS Type: 1 0.015/0.015 Deterministic, Strongly deterministic 0.015/0.015 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 0.015/0.015 Maybe right-stable CTRS, Not overlay CTRS 0.015/0.015 Maybe normal CTRS, Maybe almost normal CTRS 0.015/0.015 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 0.015/0.015 Maybe level confluent 0.015/0.015 Maybe confluent 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 Confluence Problem: 0.015/0.015 (VAR x) 0.015/0.015 (REPLACEMENT-MAP 0.015/0.015 (a) 0.015/0.015 (b) 0.015/0.015 (c) 0.015/0.015 (d) 0.015/0.015 (f 1) 0.015/0.015 (g 1, 2, 3) 0.015/0.015 (h 1, 2) 0.015/0.015 (k) 0.015/0.015 (A) 0.015/0.015 (e) 0.015/0.015 (fSNonEmpty) 0.015/0.015 (l) 0.015/0.015 (m) 0.015/0.015 ) 0.015/0.015 (RULES 0.015/0.015 a -> c 0.015/0.015 a -> d 0.015/0.015 b -> c 0.015/0.015 b -> d 0.015/0.015 c -> e 0.015/0.015 c -> l 0.015/0.015 d -> m 0.015/0.015 f(x) -> x | x ->* e 0.015/0.015 g(d,x,x) -> A 0.015/0.015 h(x,x) -> g(x,x,f(k)) 0.015/0.015 k -> l 0.015/0.015 k -> m 0.015/0.015 ) 0.015/0.015 Critical Pairs: 0.015/0.015 => Not trivial, Overlay, NW2, N1 0.015/0.015 => Not trivial, Overlay, NW2, N2 0.015/0.015 => Not trivial, Not overlay, NW2, N3 0.015/0.015 => Not trivial, Overlay, NW0, N4 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 0.015/0.015 Conditional Critical Pairs Distributor Processor 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 Confluence Problem: 0.015/0.015 (VAR x) 0.015/0.015 (REPLACEMENT-MAP 0.015/0.015 (a) 0.015/0.015 (b) 0.015/0.015 (c) 0.015/0.015 (d) 0.015/0.015 (f 1) 0.015/0.015 (g 1, 2, 3) 0.015/0.015 (h 1, 2) 0.015/0.015 (k) 0.015/0.015 (A) 0.015/0.015 (e) 0.015/0.015 (fSNonEmpty) 0.015/0.015 (l) 0.015/0.015 (m) 0.015/0.015 ) 0.015/0.015 (RULES 0.015/0.015 a -> c 0.015/0.015 a -> d 0.015/0.015 b -> c 0.015/0.015 b -> d 0.015/0.015 c -> e 0.015/0.015 c -> l 0.015/0.015 d -> m 0.015/0.015 f(x) -> x | x ->* e 0.015/0.015 g(d,x,x) -> A 0.015/0.015 h(x,x) -> g(x,x,f(k)) 0.015/0.015 k -> l 0.015/0.015 k -> m 0.015/0.015 ) 0.015/0.015 Critical Pairs: 0.015/0.015 => Not trivial, Overlay, NW2, N1 0.015/0.015 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 0.015/0.015 0.015/0.015 InfChecker Different Ground and Normal No Conditions CCP Terms Processor: 0.015/0.015 Proof: 0.015/0.015 YES 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 Infeasibility Problem: 0.015/0.015 [(VAR vNonEmpty x vNonEmpty y) 0.015/0.015 (STRATEGY CONTEXTSENSITIVE 0.015/0.015 (a) 0.015/0.015 (b) 0.015/0.015 (c) 0.015/0.015 (d) 0.015/0.015 (f 1) 0.015/0.015 (g 1 2 3) 0.015/0.015 (h 1 2) 0.015/0.015 (k) 0.015/0.015 (A) 0.015/0.015 (e) 0.015/0.015 (fSNonEmpty) 0.015/0.015 (l) 0.015/0.015 (m) 0.015/0.015 ) 0.015/0.015 (RULES 0.015/0.015 a -> c 0.015/0.015 a -> d 0.015/0.015 b -> c 0.015/0.015 b -> d 0.015/0.015 c -> e 0.015/0.015 c -> l 0.015/0.015 d -> m 0.015/0.015 f(x) -> x | x ->* e 0.015/0.015 g(d,x,x) -> A 0.015/0.015 h(x,x) -> g(x,x,f(k)) 0.015/0.015 k -> l 0.015/0.015 k -> m 0.015/0.015 )] 0.015/0.015 0.015/0.015 Infeasibility Conditions: 0.015/0.015 e -> y 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 Obtaining a model using Mace4: 0.015/0.015 0.015/0.015 -> Usable Rules: 0.015/0.015 Empty 0.015/0.015 0.015/0.015 -> Mace4 Output: 0.015/0.015 ============================== Mace4 ================================= 0.015/0.015 Mace4 (64) version 2009-11A, November 2009. 0.015/0.015 Process 11107 was started by ubuntu on ubuntu, 0.015/0.015 Wed Mar 9 09:15:55 2022 0.015/0.015 The command was "./mace4 -c -f /tmp/mace411076-2.in". 0.015/0.015 ============================== end of head =========================== 0.015/0.015 0.015/0.015 ============================== INPUT ================================= 0.015/0.015 0.015/0.015 % Reading from file /tmp/mace411076-2.in 0.015/0.015 0.015/0.015 assign(max_seconds,6). 0.015/0.015 0.015/0.015 formulas(assumptions). 0.015/0.015 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 0.015/0.015 ->(x1,y) -> ->(g(x1,x2,x3),g(y,x2,x3)) # label(congruence). 0.015/0.015 ->(x2,y) -> ->(g(x1,x2,x3),g(x1,y,x3)) # label(congruence). 0.015/0.015 ->(x3,y) -> ->(g(x1,x2,x3),g(x1,x2,y)) # label(congruence). 0.015/0.015 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence). 0.015/0.015 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence). 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 formulas(goals). 0.015/0.015 (exists x3 ->(e,x3)) # label(goal). 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 ============================== end of input ========================== 0.015/0.015 0.015/0.015 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.015/0.015 0.015/0.015 % Formulas that are not ordinary clauses: 0.015/0.015 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 2 ->(x1,y) -> ->(g(x1,x2,x3),g(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 3 ->(x2,y) -> ->(g(x1,x2,x3),g(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 4 ->(x3,y) -> ->(g(x1,x2,x3),g(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 5 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 6 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 7 (exists x3 ->(e,x3)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.015/0.015 0.015/0.015 ============================== end of process non-clausal formulas === 0.015/0.015 0.015/0.015 ============================== CLAUSES FOR SEARCH ==================== 0.015/0.015 0.015/0.015 formulas(mace4_clauses). 0.015/0.015 -->(x,y) | ->(f(x),f(y)) # label(congruence). 0.015/0.015 -->(x,y) | ->(g(x,z,u),g(y,z,u)) # label(congruence). 0.015/0.015 -->(x,y) | ->(g(z,x,u),g(z,y,u)) # label(congruence). 0.015/0.015 -->(x,y) | ->(g(z,u,x),g(z,u,y)) # label(congruence). 0.015/0.015 -->(x,y) | ->(h(x,z),h(y,z)) # label(congruence). 0.015/0.015 -->(x,y) | ->(h(z,x),h(z,y)) # label(congruence). 0.015/0.015 -->(e,x) # label(goal). 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 ============================== end of clauses for search ============= 0.015/0.015 0.015/0.015 % There are no natural numbers in the input. 0.015/0.015 0.015/0.015 ============================== DOMAIN SIZE 2 ========================= 0.015/0.015 0.015/0.015 ============================== MODEL ================================= 0.015/0.015 0.015/0.015 interpretation( 2, [number=1, seconds=0], [ 0.015/0.015 0.015/0.015 function(e, [ 0 ]), 0.015/0.015 0.015/0.015 function(f(_), [ 0, 0 ]), 0.015/0.015 0.015/0.015 function(h(_,_), [ 0.015/0.015 0, 0, 0.015/0.015 0, 0 ]), 0.015/0.015 0.015/0.015 function(g(_,_,_), [ 0.015/0.015 0, 0, 0.015/0.015 0, 0, 0.015/0.015 0, 0, 0.015/0.015 0, 0 ]), 0.015/0.015 0.015/0.015 relation(->(_,_), [ 0.015/0.015 0, 0, 0.015/0.015 0, 0 ]) 0.015/0.015 ]). 0.015/0.015 0.015/0.015 ============================== end of model ========================== 0.015/0.015 0.015/0.015 ============================== STATISTICS ============================ 0.015/0.015 0.015/0.015 For domain size 2. 0.015/0.015 0.015/0.015 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 0.015/0.015 Ground clauses: seen=70, kept=70. 0.015/0.015 Selections=15, assignments=15, propagations=4, current_models=1. 0.015/0.015 Rewrite_terms=138, rewrite_bools=72, indexes=2. 0.015/0.015 Rules_from_neg_clauses=0, cross_offs=0. 0.015/0.015 0.015/0.015 ============================== end of statistics ===================== 0.015/0.015 0.015/0.015 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.015/0.015 0.015/0.015 Exiting with 1 model. 0.015/0.015 0.015/0.015 Process 11107 exit (max_models) Wed Mar 9 09:15:55 2022 0.015/0.015 The process finished Wed Mar 9 09:15:55 2022 0.015/0.015 0.015/0.015 0.015/0.015 Mace4 cooked interpretation: 0.015/0.015 0.015/0.015 0.015/0.015 0.015/0.015 The problem is infeasible. 0.015/0.015 YES 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 Infeasibility Problem: 0.015/0.015 [(VAR vNonEmpty x vNonEmpty x) 0.015/0.015 (STRATEGY CONTEXTSENSITIVE 0.015/0.015 (a) 0.015/0.015 (b) 0.015/0.015 (c) 0.015/0.015 (d) 0.015/0.015 (f 1) 0.015/0.015 (g 1 2 3) 0.015/0.015 (h 1 2) 0.015/0.015 (k) 0.015/0.015 (A) 0.015/0.015 (e) 0.015/0.015 (fSNonEmpty) 0.015/0.015 (l) 0.015/0.015 (m) 0.015/0.015 ) 0.015/0.015 (RULES 0.015/0.015 a -> c 0.015/0.015 a -> d 0.015/0.015 b -> c 0.015/0.015 b -> d 0.015/0.015 c -> e 0.015/0.015 c -> l 0.015/0.015 d -> m 0.015/0.015 f(x) -> x | x ->* e 0.015/0.015 g(d,x,x) -> A 0.015/0.015 h(x,x) -> g(x,x,f(k)) 0.015/0.015 k -> l 0.015/0.015 k -> m 0.015/0.015 )] 0.015/0.015 0.015/0.015 Infeasibility Conditions: 0.015/0.015 l -> x 0.015/0.015 0.015/0.015 Problem 1: 0.015/0.015 0.015/0.015 Obtaining a model using Mace4: 0.015/0.015 0.015/0.015 -> Usable Rules: 0.015/0.015 Empty 0.015/0.015 0.015/0.015 -> Mace4 Output: 0.015/0.015 ============================== Mace4 ================================= 0.015/0.015 Mace4 (64) version 2009-11A, November 2009. 0.015/0.015 Process 11125 was started by ubuntu on ubuntu, 0.015/0.015 Wed Mar 9 09:15:55 2022 0.015/0.015 The command was "./mace4 -c -f /tmp/mace411115-2.in". 0.015/0.015 ============================== end of head =========================== 0.015/0.015 0.015/0.015 ============================== INPUT ================================= 0.015/0.015 0.015/0.015 % Reading from file /tmp/mace411115-2.in 0.015/0.015 0.015/0.015 assign(max_seconds,6). 0.015/0.015 0.015/0.015 formulas(assumptions). 0.015/0.015 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 0.015/0.015 ->(x1,y) -> ->(g(x1,x2,x3),g(y,x2,x3)) # label(congruence). 0.015/0.015 ->(x2,y) -> ->(g(x1,x2,x3),g(x1,y,x3)) # label(congruence). 0.015/0.015 ->(x3,y) -> ->(g(x1,x2,x3),g(x1,x2,y)) # label(congruence). 0.015/0.015 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence). 0.015/0.015 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence). 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 formulas(goals). 0.015/0.015 (exists x3 ->(l,x3)) # label(goal). 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 ============================== end of input ========================== 0.015/0.015 0.015/0.015 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 0.015/0.015 0.015/0.015 % Formulas that are not ordinary clauses: 0.015/0.015 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 2 ->(x1,y) -> ->(g(x1,x2,x3),g(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 3 ->(x2,y) -> ->(g(x1,x2,x3),g(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 4 ->(x3,y) -> ->(g(x1,x2,x3),g(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 5 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 6 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 0.015/0.015 7 (exists x3 ->(l,x3)) # label(goal) # label(non_clause) # label(goal). [goal]. 0.015/0.015 0.015/0.015 ============================== end of process non-clausal formulas === 0.015/0.015 0.015/0.015 ============================== CLAUSES FOR SEARCH ==================== 0.015/0.015 0.015/0.015 formulas(mace4_clauses). 0.015/0.015 -->(x,y) | ->(f(x),f(y)) # label(congruence). 0.015/0.015 -->(x,y) | ->(g(x,z,u),g(y,z,u)) # label(congruence). 0.015/0.015 -->(x,y) | ->(g(z,x,u),g(z,y,u)) # label(congruence). 0.015/0.015 -->(x,y) | ->(g(z,u,x),g(z,u,y)) # label(congruence). 0.015/0.015 -->(x,y) | ->(h(x,z),h(y,z)) # label(congruence). 0.015/0.015 -->(x,y) | ->(h(z,x),h(z,y)) # label(congruence). 0.015/0.015 -->(l,x) # label(goal). 0.015/0.015 end_of_list. 0.015/0.015 0.015/0.015 ============================== end of clauses for search ============= 0.015/0.015 0.015/0.015 % There are no natural numbers in the input. 0.015/0.015 0.015/0.015 ============================== DOMAIN SIZE 2 ========================= 0.015/0.015 0.015/0.015 ============================== MODEL ================================= 0.015/0.015 0.015/0.015 interpretation( 2, [number=1, seconds=0], [ 0.015/0.015 0.015/0.015 function(l, [ 0 ]), 0.015/0.015 0.015/0.015 function(f(_), [ 0, 0 ]), 0.015/0.015 0.015/0.015 function(h(_,_), [ 0.015/0.015 0, 0, 0.015/0.015 0, 0 ]), 0.015/0.015 0.015/0.015 function(g(_,_,_), [ 0.015/0.015 0, 0, 0.015/0.015 0, 0, 0.015/0.015 0, 0, 0.015/0.015 0, 0 ]), 0.015/0.015 0.015/0.015 relation(->(_,_), [ 0.015/0.015 0, 0, 0.015/0.015 0, 0 ]) 0.015/0.015 ]). 0.015/0.015 0.015/0.015 ============================== end of model ========================== 0.015/0.015 0.015/0.015 ============================== STATISTICS ============================ 0.015/0.015 0.015/0.015 For domain size 2. 0.015/0.015 0.015/0.015 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 0.015/0.015 Ground clauses: seen=70, kept=70. 0.015/0.015 Selections=15, assignments=15, propagations=4, current_models=1. 0.015/0.015 Rewrite_terms=138, rewrite_bools=72, indexes=2. 0.015/0.015 Rules_from_neg_clauses=0, cross_offs=0. 0.015/0.015 0.015/0.015 ============================== end of statistics ===================== 0.015/0.015 0.015/0.015 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 0.015/0.015 0.015/0.015 Exiting with 1 model. 0.015/0.015 0.015/0.015 Process 11125 exit (max_models) Wed Mar 9 09:15:55 2022 0.015/0.015 The process finished Wed Mar 9 09:15:55 2022 0.015/0.015 0.015/0.015 0.015/0.015 Mace4 cooked interpretation: 0.015/0.015 0.015/0.015 0.015/0.015 0.015/0.015 The problem is infeasible. 0.015/0.015 0.015/0.015 0.015/0.015 The problem is not joinable. 0.015/0.015 0.08user 0.04system 0:00.15elapsed 84%CPU (0avgtext+0avgdata 11992maxresident)k 0.015/0.015 8inputs+0outputs (0major+9662minor)pagefaults 0swaps