5.013/5.013 NO 5.013/5.013 5.013/5.013 Problem 1: 5.013/5.013 5.013/5.013 5.013/5.013 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5.013/5.013 Confluence Problem: 5.013/5.013 (VAR vNonEmpty x) 5.013/5.013 (REPLACEMENT-MAP 5.013/5.013 (A) 5.013/5.013 (a) 5.013/5.013 (b) 5.013/5.013 (c) 5.013/5.013 (d) 5.013/5.013 (f 1) 5.013/5.013 (g 1, 2, 3) 5.013/5.013 (h 1, 2) 5.013/5.013 (k) 5.013/5.013 (e) 5.013/5.013 (fSNonEmpty) 5.013/5.013 (l) 5.013/5.013 (m) 5.013/5.013 ) 5.013/5.013 (RULES 5.013/5.013 A -> h(f(a),f(b)) 5.013/5.013 a -> c 5.013/5.013 a -> d 5.013/5.013 b -> c 5.013/5.013 b -> d 5.013/5.013 c -> e 5.013/5.013 c -> l 5.013/5.013 d -> m 5.013/5.013 f(x) -> x | x ->* e 5.013/5.013 g(d,x,x) -> A 5.013/5.013 h(x,x) -> g(x,x,f(k)) 5.013/5.013 k -> l 5.013/5.013 k -> m 5.013/5.013 ) 5.013/5.013 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5.013/5.013 5.013/5.013 5.013/5.013 Problem 1: 5.013/5.013 5.013/5.013 Inlining of Conditions Processor [STERN17]: 5.013/5.013 5.013/5.013 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5.013/5.013 Confluence Problem: 5.013/5.013 (VAR vNonEmpty x) 5.013/5.013 (REPLACEMENT-MAP 5.013/5.013 (A) 5.013/5.013 (a) 5.013/5.013 (b) 5.013/5.013 (c) 5.013/5.013 (d) 5.013/5.013 (f 1) 5.013/5.013 (g 1, 2, 3) 5.013/5.013 (h 1, 2) 5.013/5.013 (k) 5.013/5.013 (e) 5.013/5.013 (fSNonEmpty) 5.013/5.013 (l) 5.013/5.013 (m) 5.013/5.013 ) 5.013/5.013 (RULES 5.013/5.013 A -> h(f(a),f(b)) 5.013/5.013 a -> c 5.013/5.013 a -> d 5.013/5.013 b -> c 5.013/5.013 b -> d 5.013/5.013 c -> e 5.013/5.013 c -> l 5.013/5.013 d -> m 5.013/5.013 f(x) -> x | x ->* e 5.013/5.013 g(d,x,x) -> A 5.013/5.013 h(x,x) -> g(x,x,f(k)) 5.013/5.013 k -> l 5.013/5.013 k -> m 5.013/5.013 ) 5.013/5.013 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5.013/5.013 5.013/5.013 5.013/5.013 Problem 1: 5.013/5.013 5.013/5.013 Clean CTRS Processor: 5.013/5.013 5.013/5.013 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5.013/5.013 Confluence Problem: 5.013/5.013 (VAR x) 5.013/5.013 (REPLACEMENT-MAP 5.013/5.013 (A) 5.013/5.013 (a) 5.013/5.013 (b) 5.013/5.013 (c) 5.013/5.013 (d) 5.013/5.013 (f 1) 5.013/5.013 (g 1, 2, 3) 5.013/5.013 (h 1, 2) 5.013/5.013 (k) 5.013/5.013 (e) 5.013/5.013 (fSNonEmpty) 5.013/5.013 (l) 5.013/5.013 (m) 5.013/5.013 ) 5.013/5.013 (RULES 5.013/5.013 A -> h(f(a),f(b)) 5.013/5.013 a -> c 5.013/5.013 a -> d 5.013/5.013 b -> c 5.013/5.013 b -> d 5.013/5.013 c -> e 5.013/5.013 c -> l 5.013/5.013 d -> m 5.013/5.013 f(x) -> x | x ->* e 5.013/5.013 g(d,x,x) -> A 5.013/5.013 h(x,x) -> g(x,x,f(k)) 5.013/5.013 k -> l 5.013/5.013 k -> m 5.013/5.013 ) 5.013/5.013 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5.013/5.013 5.013/5.013 CRule InfChecker Info: 5.013/5.013 A -> h(f(a),f(b)) 5.013/5.013 Rule remains 5.013/5.013 Proof: 5.013/5.013 NO_CONDS 5.013/5.013 5.013/5.013 CRule InfChecker Info: 5.013/5.013 a -> c 5.013/5.013 Rule remains 5.013/5.013 Proof: 5.013/5.013 NO_CONDS 5.013/5.013 5.013/5.013 CRule InfChecker Info: 5.013/5.013 a -> d 5.013/5.013 Rule remains 5.013/5.013 Proof: 5.013/5.013 NO_CONDS 5.013/5.013 5.013/5.013 CRule InfChecker Info: 5.013/5.013 b -> c 5.013/5.013 Rule remains 5.013/5.013 Proof: 5.013/5.013 NO_CONDS 5.013/5.013 5.013/5.013 CRule InfChecker Info: 5.013/5.013 b -> d 5.013/5.013 Rule remains 5.013/5.013 Proof: 5.013/5.013 NO_CONDS 5.013/5.013 5.013/5.013 CRule InfChecker Info: 5.013/5.013 c -> e 5.013/5.013 Rule remains 5.013/5.013 Proof: 5.013/5.013 NO_CONDS 5.013/5.013 5.013/5.013 CRule InfChecker Info: 5.013/5.013 c -> l 5.013/5.013 Rule remains 5.013/5.013 Proof: 5.013/5.013 NO_CONDS 5.013/5.013 5.013/5.013 CRule InfChecker Info: 5.013/5.013 d -> m 5.013/5.013 Rule remains 5.013/5.013 Proof: 5.013/5.013 NO_CONDS 5.013/5.013 5.013/5.013 CRule InfChecker Info: 5.013/5.013 f(x) -> x | x ->* e 5.013/5.013 Rule remains 5.013/5.013 Proof: 5.013/5.013 NO 5.013/5.013 5.013/5.013 Problem 1: 5.013/5.013 5.013/5.013 Infeasibility Problem: 5.013/5.013 [(VAR vNonEmpty x vNonEmpty x) 5.013/5.013 (STRATEGY CONTEXTSENSITIVE 5.013/5.013 (A) 5.013/5.013 (a) 5.013/5.013 (b) 5.013/5.013 (c) 5.013/5.013 (d) 5.013/5.013 (f 1) 5.013/5.013 (g 1 2 3) 5.013/5.013 (h 1 2) 5.013/5.013 (k) 5.013/5.013 (e) 5.013/5.013 (fSNonEmpty) 5.013/5.013 (l) 5.013/5.013 (m) 5.013/5.013 ) 5.013/5.013 (RULES 5.013/5.013 A -> h(f(a),f(b)) 5.013/5.013 a -> c 5.013/5.013 a -> d 5.013/5.013 b -> c 5.013/5.013 b -> d 5.013/5.013 c -> e 5.013/5.013 c -> l 5.013/5.013 d -> m 5.013/5.013 f(x) -> x | x ->* e 5.013/5.013 g(d,x,x) -> A 5.013/5.013 h(x,x) -> g(x,x,f(k)) 5.013/5.013 k -> l 5.013/5.013 k -> m 5.013/5.013 )] 5.013/5.013 5.013/5.013 Infeasibility Conditions: 5.013/5.013 x ->* e 5.013/5.013 5.013/5.013 Problem 1: 5.013/5.013 5.013/5.013 Obtaining a proof using Prover9: 5.013/5.013 5.013/5.013 -> Prover9 Output: 5.013/5.013 ============================== Prover9 =============================== 5.013/5.013 Prover9 (64) version 2009-11A, November 2009. 5.013/5.013 Process 58550 was started by ubuntu on ubuntu, 5.013/5.013 Wed Mar 9 09:37:06 2022 5.013/5.013 The command was "./prover9 -f /tmp/prover958543-0.in". 5.013/5.013 ============================== end of head =========================== 5.013/5.013 5.013/5.013 ============================== INPUT ================================= 5.013/5.013 5.013/5.013 % Reading from file /tmp/prover958543-0.in 5.013/5.013 5.013/5.013 assign(max_seconds,20). 5.013/5.013 5.013/5.013 formulas(assumptions). 5.013/5.013 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 5.013/5.013 ->_s0(x1,y) -> ->_s0(g(x1,x2,x3),g(y,x2,x3)) # label(congruence). 5.013/5.013 ->_s0(x2,y) -> ->_s0(g(x1,x2,x3),g(x1,y,x3)) # label(congruence). 5.013/5.013 ->_s0(x3,y) -> ->_s0(g(x1,x2,x3),g(x1,x2,y)) # label(congruence). 5.013/5.013 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence). 5.013/5.013 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence). 5.013/5.013 ->_s0(A,h(f(a),f(b))) # label(replacement). 5.013/5.013 ->_s0(a,c) # label(replacement). 5.013/5.013 ->_s0(a,d) # label(replacement). 5.013/5.013 ->_s0(b,c) # label(replacement). 5.013/5.013 ->_s0(b,d) # label(replacement). 5.013/5.013 ->_s0(c,e) # label(replacement). 5.013/5.013 ->_s0(c,l) # label(replacement). 5.013/5.013 ->_s0(d,m) # label(replacement). 5.013/5.013 ->*_s0(x1,e) -> ->_s0(f(x1),x1) # label(replacement). 5.013/5.013 ->_s0(g(d,x1,x1),A) # label(replacement). 5.013/5.013 ->_s0(h(x1,x1),g(x1,x1,f(k))) # label(replacement). 5.013/5.013 ->_s0(k,l) # label(replacement). 5.013/5.013 ->_s0(k,m) # label(replacement). 5.013/5.013 ->*_s0(x,x) # label(reflexivity). 5.013/5.013 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 5.013/5.013 end_of_list. 5.013/5.013 5.013/5.013 formulas(goals). 5.013/5.013 (exists x3 ->*_s0(x3,e)) # label(goal). 5.013/5.013 end_of_list. 5.013/5.013 5.013/5.013 ============================== end of input ========================== 5.013/5.013 5.013/5.013 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 5.013/5.013 5.013/5.013 % Formulas that are not ordinary clauses: 5.013/5.013 1 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 2 ->_s0(x1,y) -> ->_s0(g(x1,x2,x3),g(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 3 ->_s0(x2,y) -> ->_s0(g(x1,x2,x3),g(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 4 ->_s0(x3,y) -> ->_s0(g(x1,x2,x3),g(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 5 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 6 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 7 ->*_s0(x1,e) -> ->_s0(f(x1),x1) # label(replacement) # label(non_clause). [assumption]. 5.013/5.013 8 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 5.013/5.013 9 (exists x3 ->*_s0(x3,e)) # label(goal) # label(non_clause) # label(goal). [goal]. 5.013/5.013 5.013/5.013 ============================== end of process non-clausal formulas === 5.013/5.013 5.013/5.013 ============================== PROCESS INITIAL CLAUSES =============== 5.013/5.013 5.013/5.013 % Clauses before input processing: 5.013/5.013 5.013/5.013 formulas(usable). 5.013/5.013 end_of_list. 5.013/5.013 5.013/5.013 formulas(sos). 5.013/5.013 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 5.013/5.013 -->_s0(x,y) | ->_s0(g(x,z,u),g(y,z,u)) # label(congruence). [clausify(2)]. 5.013/5.013 -->_s0(x,y) | ->_s0(g(z,x,u),g(z,y,u)) # label(congruence). [clausify(3)]. 5.013/5.013 -->_s0(x,y) | ->_s0(g(z,u,x),g(z,u,y)) # label(congruence). [clausify(4)]. 5.013/5.013 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(5)]. 5.013/5.013 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(6)]. 5.013/5.013 ->_s0(A,h(f(a),f(b))) # label(replacement). [assumption]. 5.013/5.013 ->_s0(a,c) # label(replacement). [assumption]. 5.013/5.013 ->_s0(a,d) # label(replacement). [assumption]. 5.013/5.013 ->_s0(b,c) # label(replacement). [assumption]. 5.013/5.013 ->_s0(b,d) # label(replacement). [assumption]. 5.013/5.013 ->_s0(c,e) # label(replacement). [assumption]. 5.013/5.013 ->_s0(c,l) # label(replacement). [assumption]. 5.013/5.013 ->_s0(d,m) # label(replacement). [assumption]. 5.013/5.013 -->*_s0(x,e) | ->_s0(f(x),x) # label(replacement). [clausify(7)]. 5.013/5.013 ->_s0(g(d,x,x),A) # label(replacement). [assumption]. 5.013/5.013 ->_s0(h(x,x),g(x,x,f(k))) # label(replacement). [assumption]. 5.013/5.013 ->_s0(k,l) # label(replacement). [assumption]. 5.013/5.013 ->_s0(k,m) # label(replacement). [assumption]. 5.013/5.013 ->*_s0(x,x) # label(reflexivity). [assumption]. 5.013/5.013 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 5.013/5.013 -->*_s0(x,e) # label(goal). [deny(9)]. 5.013/5.013 end_of_list. 5.013/5.013 5.013/5.013 formulas(demodulators). 5.013/5.013 end_of_list. 5.013/5.013 5.013/5.013 ============================== PREDICATE ELIMINATION ================= 5.013/5.013 5.013/5.013 No predicates eliminated. 5.013/5.013 5.013/5.013 ============================== end predicate elimination ============= 5.013/5.013 5.013/5.013 Auto_denials: 5.013/5.013 % copying label goal to answer in negative clause 5.013/5.013 5.013/5.013 Term ordering decisions: 5.013/5.013 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 5.013/5.013 Function symbol precedence: function_order([ c, d, k, a, b, e, l, m, A, h, f, g ]). 5.013/5.013 After inverse_order: (no changes). 5.013/5.013 Unfolding symbols: (none). 5.013/5.013 5.013/5.013 Auto_inference settings: 5.013/5.013 % set(neg_binary_resolution). % (HNE depth_diff=-7) 5.013/5.013 % clear(ordered_res). % (HNE depth_diff=-7) 5.013/5.013 % set(ur_resolution). % (HNE depth_diff=-7) 5.013/5.013 % set(ur_resolution) -> set(pos_ur_resolution). 5.013/5.013 % set(ur_resolution) -> set(neg_ur_resolution). 5.013/5.013 5.013/5.013 Auto_process settings: (no changes). 5.013/5.013 5.013/5.013 kept: 10 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(1)]. 5.013/5.013 kept: 11 -->_s0(x,y) | ->_s0(g(x,z,u),g(y,z,u)) # label(congruence). [clausify(2)]. 5.013/5.013 kept: 12 -->_s0(x,y) | ->_s0(g(z,x,u),g(z,y,u)) # label(congruence). [clausify(3)]. 5.013/5.013 kept: 13 -->_s0(x,y) | ->_s0(g(z,u,x),g(z,u,y)) # label(congruence). [clausify(4)]. 5.013/5.013 kept: 14 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(5)]. 5.013/5.013 kept: 15 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(6)]. 5.013/5.013 kept: 16 ->_s0(A,h(f(a),f(b))) # label(replacement). [assumption]. 5.013/5.013 kept: 17 ->_s0(a,c) # label(replacement). [assumption]. 5.013/5.013 kept: 18 ->_s0(a,d) # label(replacement). [assumption]. 5.013/5.013 kept: 19 ->_s0(b,c) # label(replacement). [assumption]. 5.013/5.013 kept: 20 ->_s0(b,d) # label(replacement). [assumption]. 5.013/5.013 kept: 21 ->_s0(c,e) # label(replacement). [assumption]. 5.013/5.013 kept: 22 ->_s0(c,l) # label(replacement). [assumption]. 5.013/5.013 kept: 23 ->_s0(d,m) # label(replacement). [assumption]. 5.013/5.013 kept: 24 -->*_s0(x,e) | ->_s0(f(x),x) # label(replacement). [clausify(7)]. 5.013/5.013 kept: 25 ->_s0(g(d,x,x),A) # label(replacement). [assumption]. 5.013/5.013 kept: 26 ->_s0(h(x,x),g(x,x,f(k))) # label(replacement). [assumption]. 5.013/5.013 kept: 27 ->_s0(k,l) # label(replacement). [assumption]. 5.013/5.013 kept: 28 ->_s0(k,m) # label(replacement). [assumption]. 5.013/5.013 kept: 29 ->*_s0(x,x) # label(reflexivity). [assumption]. 5.013/5.013 kept: 30 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(8)]. 5.013/5.013 kept: 31 -->*_s0(x,e) # label(goal) # answer(goal). [deny(9)]. 5.013/5.013 5.013/5.013 ============================== PROOF ================================= 5.013/5.013 5.013/5.013 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 5.013/5.013 % Length of proof is 4. 5.013/5.013 % Level of proof is 2. 5.013/5.013 % Maximum clause weight is 3.000. 5.013/5.013 % Given clauses 0. 5.013/5.013 5.013/5.013 9 (exists x3 ->*_s0(x3,e)) # label(goal) # label(non_clause) # label(goal). [goal]. 5.013/5.013 29 ->*_s0(x,x) # label(reflexivity). [assumption]. 5.013/5.013 31 -->*_s0(x,e) # label(goal) # answer(goal). [deny(9)]. 5.013/5.013 32 $F # answer(goal). [resolve(31,a,29,a)]. 5.013/5.013 5.013/5.013 ============================== end of proof ========================== 5.013/5.013 5.013/5.013 ============================== STATISTICS ============================ 5.013/5.013 5.013/5.013 Given=0. Generated=22. Kept=22. proofs=1. 5.013/5.013 Usable=0. Sos=0. Demods=0. Limbo=21, Disabled=22. Hints=0. 5.013/5.013 Kept_by_rule=0, Deleted_by_rule=0. 5.013/5.013 Forward_subsumed=0. Back_subsumed=0. 5.013/5.013 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 5.013/5.013 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 5.013/5.013 Demod_attempts=0. Demod_rewrites=0. 5.013/5.013 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 5.013/5.013 Nonunit_fsub_feature_tests=4. Nonunit_bsub_feature_tests=0. 5.013/5.013 Megabytes=0.07. 5.013/5.013 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 5.013/5.013 5.013/5.013 ============================== end of statistics ===================== 5.013/5.013 5.013/5.013 ============================== end of search ========================= 5.013/5.013 5.013/5.013 THEOREM PROVED 5.013/5.013 5.013/5.013 Exiting with 1 proof. 5.013/5.013 5.013/5.013 Process 58550 exit (max_proofs) Wed Mar 9 09:37:06 2022 5.013/5.013 5.013/5.013 5.013/5.013 The problem is feasible. 5.013/5.013 5.013/5.013 5.013/5.013 CRule InfChecker Info: 5.013/5.013 g(d,x,x) -> A 5.013/5.013 Rule remains 5.013/5.013 Proof: 5.013/5.013 NO_CONDS 5.013/5.013 5.013/5.013 CRule InfChecker Info: 5.013/5.013 h(x,x) -> g(x,x,f(k)) 5.013/5.013 Rule remains 5.013/5.013 Proof: 5.013/5.013 NO_CONDS 5.013/5.013 5.013/5.013 CRule InfChecker Info: 5.013/5.013 k -> l 5.013/5.013 Rule remains 5.013/5.013 Proof: 5.013/5.013 NO_CONDS 5.013/5.013 5.013/5.013 CRule InfChecker Info: 5.013/5.013 k -> m 5.013/5.013 Rule remains 5.013/5.013 Proof: 5.013/5.013 NO_CONDS 5.013/5.013 5.013/5.013 Problem 1: 5.013/5.013 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5.013/5.013 Confluence Problem: 5.013/5.013 (VAR x) 5.013/5.013 (REPLACEMENT-MAP 5.013/5.013 (A) 5.013/5.013 (a) 5.013/5.013 (b) 5.013/5.013 (c) 5.013/5.013 (d) 5.013/5.013 (f 1) 5.013/5.013 (g 1, 2, 3) 5.013/5.013 (h 1, 2) 5.013/5.013 (k) 5.013/5.013 (e) 5.013/5.013 (fSNonEmpty) 5.013/5.013 (l) 5.013/5.013 (m) 5.013/5.013 ) 5.013/5.013 (RULES 5.013/5.013 A -> h(f(a),f(b)) 5.013/5.013 a -> c 5.013/5.013 a -> d 5.013/5.013 b -> c 5.013/5.013 b -> d 5.013/5.013 c -> e 5.013/5.013 c -> l 5.013/5.013 d -> m 5.013/5.013 f(x) -> x | x ->* e 5.013/5.013 g(d,x,x) -> A 5.013/5.013 h(x,x) -> g(x,x,f(k)) 5.013/5.013 k -> l 5.013/5.013 k -> m 5.013/5.013 ) 5.013/5.013 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5.013/5.013 5.013/5.013 Critical Pairs Processor: 5.013/5.013 -> Rules: 5.013/5.013 A -> h(f(a),f(b)) 5.013/5.013 a -> c 5.013/5.013 a -> d 5.013/5.013 b -> c 5.013/5.013 b -> d 5.013/5.013 c -> e 5.013/5.013 c -> l 5.013/5.013 d -> m 5.013/5.013 f(x) -> x | x ->* e 5.013/5.013 g(d,x,x) -> A 5.013/5.013 h(x,x) -> g(x,x,f(k)) 5.013/5.013 k -> l 5.013/5.013 k -> m 5.013/5.013 -> Vars: 5.013/5.013 "x" 5.013/5.013 5.013/5.013 -> Rlps: 5.013/5.013 crule: A -> h(f(a),f(b)), id: 1, possubterms: A-> [] 5.013/5.013 crule: a -> c, id: 2, possubterms: a-> [] 5.013/5.013 crule: a -> d, id: 3, possubterms: a-> [] 5.013/5.013 crule: b -> c, id: 4, possubterms: b-> [] 5.013/5.013 crule: b -> d, id: 5, possubterms: b-> [] 5.013/5.013 crule: c -> e, id: 6, possubterms: c-> [] 5.013/5.013 crule: c -> l, id: 7, possubterms: c-> [] 5.013/5.013 crule: d -> m, id: 8, possubterms: d-> [] 5.013/5.013 crule: f(x) -> x | x ->* e, id: 9, possubterms: f(x)-> [] 5.013/5.013 crule: g(d,x,x) -> A, id: 10, possubterms: g(d,x,x)-> [], d-> [1] 5.013/5.013 crule: h(x,x) -> g(x,x,f(k)), id: 11, possubterms: h(x,x)-> [] 5.013/5.013 crule: k -> l, id: 12, possubterms: k-> [] 5.013/5.013 crule: k -> m, id: 13, possubterms: k-> [] 5.013/5.013 5.013/5.013 -> Unifications: 5.013/5.013 R3 unifies with R2 at p: [], l: a, lp: a, conds: {}, sig: {}, l': a, r: d, r': c 5.013/5.013 R5 unifies with R4 at p: [], l: b, lp: b, conds: {}, sig: {}, l': b, r: d, r': c 5.013/5.013 R7 unifies with R6 at p: [], l: c, lp: c, conds: {}, sig: {}, l': c, r: l, r': e 5.013/5.013 R10 unifies with R8 at p: [1], l: g(d,x,x), lp: d, conds: {}, sig: {}, l': d, r: A, r': m 5.013/5.013 R13 unifies with R12 at p: [], l: k, lp: k, conds: {}, sig: {}, l': k, r: m, r': l 5.013/5.013 5.013/5.013 -> Critical pairs info: 5.013/5.013 => Not trivial, Overlay, NW2, N1 5.013/5.013 => Not trivial, Overlay, NW2, N2 5.013/5.013 => Not trivial, Not overlay, NW1, N3 5.013/5.013 => Not trivial, Overlay, NW0, N4 5.013/5.013 5.013/5.013 -> Problem conclusions: 5.013/5.013 Not left linear, Not right linear, Not linear 5.013/5.013 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 5.013/5.013 CTRS Type: 1 5.013/5.013 Deterministic, Strongly deterministic 5.013/5.013 Oriented CTRS, Properly oriented CTRS, Not join CTRS, Not semiequational CTRS 5.013/5.013 Maybe right-stable CTRS, Not overlay CTRS 5.013/5.013 Maybe normal CTRS, Maybe almost normal CTRS 5.013/5.013 Maybe terminating CTRS, Maybe operational terminating CTRS, Maybe joinable CCPs 5.013/5.013 Maybe level confluent 5.013/5.013 Maybe confluent 5.013/5.013 5.013/5.013 Problem 1: 5.013/5.013 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5.013/5.013 Confluence Problem: 5.013/5.013 (VAR x) 5.013/5.013 (REPLACEMENT-MAP 5.013/5.013 (A) 5.013/5.013 (a) 5.013/5.013 (b) 5.013/5.013 (c) 5.013/5.013 (d) 5.013/5.013 (f 1) 5.013/5.013 (g 1, 2, 3) 5.013/5.013 (h 1, 2) 5.013/5.013 (k) 5.013/5.013 (e) 5.013/5.013 (fSNonEmpty) 5.013/5.013 (l) 5.013/5.013 (m) 5.013/5.013 ) 5.013/5.013 (RULES 5.013/5.013 A -> h(f(a),f(b)) 5.013/5.013 a -> c 5.013/5.013 a -> d 5.013/5.013 b -> c 5.013/5.013 b -> d 5.013/5.013 c -> e 5.013/5.013 c -> l 5.013/5.013 d -> m 5.013/5.013 f(x) -> x | x ->* e 5.013/5.013 g(d,x,x) -> A 5.013/5.013 h(x,x) -> g(x,x,f(k)) 5.013/5.013 k -> l 5.013/5.013 k -> m 5.013/5.013 ) 5.013/5.013 Critical Pairs: 5.013/5.013 => Not trivial, Overlay, NW2, N1 5.013/5.013 => Not trivial, Overlay, NW2, N2 5.013/5.013 => Not trivial, Not overlay, NW1, N3 5.013/5.013 => Not trivial, Overlay, NW0, N4 5.013/5.013 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5.013/5.013 5.013/5.013 Conditional Critical Pairs Distributor Processor 5.013/5.013 5.013/5.013 Problem 1: 5.013/5.013 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5.013/5.013 Confluence Problem: 5.013/5.013 (VAR x) 5.013/5.013 (REPLACEMENT-MAP 5.013/5.013 (A) 5.013/5.013 (a) 5.013/5.013 (b) 5.013/5.013 (c) 5.013/5.013 (d) 5.013/5.013 (f 1) 5.013/5.013 (g 1, 2, 3) 5.013/5.013 (h 1, 2) 5.013/5.013 (k) 5.013/5.013 (e) 5.013/5.013 (fSNonEmpty) 5.013/5.013 (l) 5.013/5.013 (m) 5.013/5.013 ) 5.013/5.013 (RULES 5.013/5.013 A -> h(f(a),f(b)) 5.013/5.013 a -> c 5.013/5.013 a -> d 5.013/5.013 b -> c 5.013/5.013 b -> d 5.013/5.013 c -> e 5.013/5.013 c -> l 5.013/5.013 d -> m 5.013/5.013 f(x) -> x | x ->* e 5.013/5.013 g(d,x,x) -> A 5.013/5.013 h(x,x) -> g(x,x,f(k)) 5.013/5.013 k -> l 5.013/5.013 k -> m 5.013/5.013 ) 5.013/5.013 Critical Pairs: 5.013/5.013 => Not trivial, Overlay, NW2, N1 5.013/5.013 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 5.013/5.013 5.013/5.013 InfChecker Different Ground and Normal No Conditions CCP Terms Processor: 5.013/5.013 Proof: 5.013/5.013 YES 5.013/5.013 5.013/5.013 Problem 1: 5.013/5.013 5.013/5.013 Infeasibility Problem: 5.013/5.013 [(VAR vNonEmpty x vNonEmpty y) 5.013/5.013 (STRATEGY CONTEXTSENSITIVE 5.013/5.013 (A) 5.013/5.013 (a) 5.013/5.013 (b) 5.013/5.013 (c) 5.013/5.013 (d) 5.013/5.013 (f 1) 5.013/5.013 (g 1 2 3) 5.013/5.013 (h 1 2) 5.013/5.013 (k) 5.013/5.013 (e) 5.013/5.013 (fSNonEmpty) 5.013/5.013 (l) 5.013/5.013 (m) 5.013/5.013 ) 5.013/5.013 (RULES 5.013/5.013 A -> h(f(a),f(b)) 5.013/5.013 a -> c 5.013/5.013 a -> d 5.013/5.013 b -> c 5.013/5.013 b -> d 5.013/5.013 c -> e 5.013/5.013 c -> l 5.013/5.013 d -> m 5.013/5.013 f(x) -> x | x ->* e 5.013/5.013 g(d,x,x) -> A 5.013/5.013 h(x,x) -> g(x,x,f(k)) 5.013/5.013 k -> l 5.013/5.013 k -> m 5.013/5.013 )] 5.013/5.013 5.013/5.013 Infeasibility Conditions: 5.013/5.013 e -> y 5.013/5.013 5.013/5.013 Problem 1: 5.013/5.013 5.013/5.013 Obtaining a model using Mace4: 5.013/5.013 5.013/5.013 -> Usable Rules: 5.013/5.013 Empty 5.013/5.013 5.013/5.013 -> Mace4 Output: 5.013/5.013 ============================== Mace4 ================================= 5.013/5.013 Mace4 (64) version 2009-11A, November 2009. 5.013/5.013 Process 58594 was started by ubuntu on ubuntu, 5.013/5.013 Wed Mar 9 09:37:06 2022 5.013/5.013 The command was "./mace4 -c -f /tmp/mace458568-2.in". 5.013/5.013 ============================== end of head =========================== 5.013/5.013 5.013/5.013 ============================== INPUT ================================= 5.013/5.013 5.013/5.013 % Reading from file /tmp/mace458568-2.in 5.013/5.013 5.013/5.013 assign(max_seconds,6). 5.013/5.013 5.013/5.013 formulas(assumptions). 5.013/5.013 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 5.013/5.013 ->(x1,y) -> ->(g(x1,x2,x3),g(y,x2,x3)) # label(congruence). 5.013/5.013 ->(x2,y) -> ->(g(x1,x2,x3),g(x1,y,x3)) # label(congruence). 5.013/5.013 ->(x3,y) -> ->(g(x1,x2,x3),g(x1,x2,y)) # label(congruence). 5.013/5.013 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence). 5.013/5.013 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence). 5.013/5.013 end_of_list. 5.013/5.013 5.013/5.013 formulas(goals). 5.013/5.013 (exists x3 ->(e,x3)) # label(goal). 5.013/5.013 end_of_list. 5.013/5.013 5.013/5.013 ============================== end of input ========================== 5.013/5.013 5.013/5.013 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 5.013/5.013 5.013/5.013 % Formulas that are not ordinary clauses: 5.013/5.013 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 2 ->(x1,y) -> ->(g(x1,x2,x3),g(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 3 ->(x2,y) -> ->(g(x1,x2,x3),g(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 4 ->(x3,y) -> ->(g(x1,x2,x3),g(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 5 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 6 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 7 (exists x3 ->(e,x3)) # label(goal) # label(non_clause) # label(goal). [goal]. 5.013/5.013 5.013/5.013 ============================== end of process non-clausal formulas === 5.013/5.013 5.013/5.013 ============================== CLAUSES FOR SEARCH ==================== 5.013/5.013 5.013/5.013 formulas(mace4_clauses). 5.013/5.013 -->(x,y) | ->(f(x),f(y)) # label(congruence). 5.013/5.013 -->(x,y) | ->(g(x,z,u),g(y,z,u)) # label(congruence). 5.013/5.013 -->(x,y) | ->(g(z,x,u),g(z,y,u)) # label(congruence). 5.013/5.013 -->(x,y) | ->(g(z,u,x),g(z,u,y)) # label(congruence). 5.013/5.013 -->(x,y) | ->(h(x,z),h(y,z)) # label(congruence). 5.013/5.013 -->(x,y) | ->(h(z,x),h(z,y)) # label(congruence). 5.013/5.013 -->(e,x) # label(goal). 5.013/5.013 end_of_list. 5.013/5.013 5.013/5.013 ============================== end of clauses for search ============= 5.013/5.013 5.013/5.013 % There are no natural numbers in the input. 5.013/5.013 5.013/5.013 ============================== DOMAIN SIZE 2 ========================= 5.013/5.013 5.013/5.013 ============================== MODEL ================================= 5.013/5.013 5.013/5.013 interpretation( 2, [number=1, seconds=0], [ 5.013/5.013 5.013/5.013 function(e, [ 0 ]), 5.013/5.013 5.013/5.013 function(f(_), [ 0, 0 ]), 5.013/5.013 5.013/5.013 function(h(_,_), [ 5.013/5.013 0, 0, 5.013/5.013 0, 0 ]), 5.013/5.013 5.013/5.013 function(g(_,_,_), [ 5.013/5.013 0, 0, 5.013/5.013 0, 0, 5.013/5.013 0, 0, 5.013/5.013 0, 0 ]), 5.013/5.013 5.013/5.013 relation(->(_,_), [ 5.013/5.013 0, 0, 5.013/5.013 0, 0 ]) 5.013/5.013 ]). 5.013/5.013 5.013/5.013 ============================== end of model ========================== 5.013/5.013 5.013/5.013 ============================== STATISTICS ============================ 5.013/5.013 5.013/5.013 For domain size 2. 5.013/5.013 5.013/5.013 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 5.013/5.013 Ground clauses: seen=70, kept=70. 5.013/5.013 Selections=15, assignments=15, propagations=4, current_models=1. 5.013/5.013 Rewrite_terms=138, rewrite_bools=72, indexes=2. 5.013/5.013 Rules_from_neg_clauses=0, cross_offs=0. 5.013/5.013 5.013/5.013 ============================== end of statistics ===================== 5.013/5.013 5.013/5.013 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 5.013/5.013 5.013/5.013 Exiting with 1 model. 5.013/5.013 5.013/5.013 Process 58594 exit (max_models) Wed Mar 9 09:37:06 2022 5.013/5.013 The process finished Wed Mar 9 09:37:06 2022 5.013/5.013 5.013/5.013 5.013/5.013 Mace4 cooked interpretation: 5.013/5.013 5.013/5.013 5.013/5.013 5.013/5.013 The problem is infeasible. 5.013/5.013 YES 5.013/5.013 5.013/5.013 Problem 1: 5.013/5.013 5.013/5.013 Infeasibility Problem: 5.013/5.013 [(VAR vNonEmpty x vNonEmpty x) 5.013/5.013 (STRATEGY CONTEXTSENSITIVE 5.013/5.013 (A) 5.013/5.013 (a) 5.013/5.013 (b) 5.013/5.013 (c) 5.013/5.013 (d) 5.013/5.013 (f 1) 5.013/5.013 (g 1 2 3) 5.013/5.013 (h 1 2) 5.013/5.013 (k) 5.013/5.013 (e) 5.013/5.013 (fSNonEmpty) 5.013/5.013 (l) 5.013/5.013 (m) 5.013/5.013 ) 5.013/5.013 (RULES 5.013/5.013 A -> h(f(a),f(b)) 5.013/5.013 a -> c 5.013/5.013 a -> d 5.013/5.013 b -> c 5.013/5.013 b -> d 5.013/5.013 c -> e 5.013/5.013 c -> l 5.013/5.013 d -> m 5.013/5.013 f(x) -> x | x ->* e 5.013/5.013 g(d,x,x) -> A 5.013/5.013 h(x,x) -> g(x,x,f(k)) 5.013/5.013 k -> l 5.013/5.013 k -> m 5.013/5.013 )] 5.013/5.013 5.013/5.013 Infeasibility Conditions: 5.013/5.013 l -> x 5.013/5.013 5.013/5.013 Problem 1: 5.013/5.013 5.013/5.013 Obtaining a model using Mace4: 5.013/5.013 5.013/5.013 -> Usable Rules: 5.013/5.013 Empty 5.013/5.013 5.013/5.013 -> Mace4 Output: 5.013/5.013 ============================== Mace4 ================================= 5.013/5.013 Mace4 (64) version 2009-11A, November 2009. 5.013/5.013 Process 58625 was started by ubuntu on ubuntu, 5.013/5.013 Wed Mar 9 09:37:06 2022 5.013/5.013 The command was "./mace4 -c -f /tmp/mace458614-2.in". 5.013/5.013 ============================== end of head =========================== 5.013/5.013 5.013/5.013 ============================== INPUT ================================= 5.013/5.013 5.013/5.013 % Reading from file /tmp/mace458614-2.in 5.013/5.013 5.013/5.013 assign(max_seconds,6). 5.013/5.013 5.013/5.013 formulas(assumptions). 5.013/5.013 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 5.013/5.013 ->(x1,y) -> ->(g(x1,x2,x3),g(y,x2,x3)) # label(congruence). 5.013/5.013 ->(x2,y) -> ->(g(x1,x2,x3),g(x1,y,x3)) # label(congruence). 5.013/5.013 ->(x3,y) -> ->(g(x1,x2,x3),g(x1,x2,y)) # label(congruence). 5.013/5.013 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence). 5.013/5.013 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence). 5.013/5.013 end_of_list. 5.013/5.013 5.013/5.013 formulas(goals). 5.013/5.013 (exists x3 ->(l,x3)) # label(goal). 5.013/5.013 end_of_list. 5.013/5.013 5.013/5.013 ============================== end of input ========================== 5.013/5.013 5.013/5.013 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 5.013/5.013 5.013/5.013 % Formulas that are not ordinary clauses: 5.013/5.013 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 2 ->(x1,y) -> ->(g(x1,x2,x3),g(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 3 ->(x2,y) -> ->(g(x1,x2,x3),g(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 4 ->(x3,y) -> ->(g(x1,x2,x3),g(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 5 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 6 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 5.013/5.013 7 (exists x3 ->(l,x3)) # label(goal) # label(non_clause) # label(goal). [goal]. 5.013/5.013 5.013/5.013 ============================== end of process non-clausal formulas === 5.013/5.013 5.013/5.013 ============================== CLAUSES FOR SEARCH ==================== 5.013/5.013 5.013/5.013 formulas(mace4_clauses). 5.013/5.013 -->(x,y) | ->(f(x),f(y)) # label(congruence). 5.013/5.013 -->(x,y) | ->(g(x,z,u),g(y,z,u)) # label(congruence). 5.013/5.013 -->(x,y) | ->(g(z,x,u),g(z,y,u)) # label(congruence). 5.013/5.013 -->(x,y) | ->(g(z,u,x),g(z,u,y)) # label(congruence). 5.013/5.013 -->(x,y) | ->(h(x,z),h(y,z)) # label(congruence). 5.013/5.013 -->(x,y) | ->(h(z,x),h(z,y)) # label(congruence). 5.013/5.013 -->(l,x) # label(goal). 5.013/5.013 end_of_list. 5.013/5.013 5.013/5.013 ============================== end of clauses for search ============= 5.013/5.013 5.013/5.013 % There are no natural numbers in the input. 5.013/5.013 5.013/5.013 ============================== DOMAIN SIZE 2 ========================= 5.013/5.013 5.013/5.013 ============================== MODEL ================================= 5.013/5.013 5.013/5.013 interpretation( 2, [number=1, seconds=0], [ 5.013/5.013 5.013/5.013 function(l, [ 0 ]), 5.013/5.013 5.013/5.013 function(f(_), [ 0, 0 ]), 5.013/5.013 5.013/5.013 function(h(_,_), [ 5.013/5.013 0, 0, 5.013/5.013 0, 0 ]), 5.013/5.013 5.013/5.013 function(g(_,_,_), [ 5.013/5.013 0, 0, 5.013/5.013 0, 0, 5.013/5.013 0, 0, 5.013/5.013 0, 0 ]), 5.013/5.013 5.013/5.013 relation(->(_,_), [ 5.013/5.013 0, 0, 5.013/5.013 0, 0 ]) 5.013/5.013 ]). 5.013/5.013 5.013/5.013 ============================== end of model ========================== 5.013/5.013 5.013/5.013 ============================== STATISTICS ============================ 5.013/5.013 5.013/5.013 For domain size 2. 5.013/5.013 5.013/5.013 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 5.013/5.013 Ground clauses: seen=70, kept=70. 5.013/5.013 Selections=15, assignments=15, propagations=4, current_models=1. 5.013/5.013 Rewrite_terms=138, rewrite_bools=72, indexes=2. 5.013/5.013 Rules_from_neg_clauses=0, cross_offs=0. 5.013/5.013 5.013/5.013 ============================== end of statistics ===================== 5.013/5.013 5.013/5.013 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 5.013/5.013 5.013/5.013 Exiting with 1 model. 5.013/5.013 5.013/5.013 Process 58625 exit (max_models) Wed Mar 9 09:37:06 2022 5.013/5.013 The process finished Wed Mar 9 09:37:06 2022 5.013/5.013 5.013/5.013 5.013/5.013 Mace4 cooked interpretation: 5.013/5.013 5.013/5.013 5.013/5.013 5.013/5.013 The problem is infeasible. 5.013/5.013 5.013/5.013 5.013/5.013 The problem is not joinable. 5.013/5.013 0.91user 0.06system 0:05.13elapsed 19%CPU (0avgtext+0avgdata 19664maxresident)k 5.013/5.013 8inputs+0outputs (0major+33410minor)pagefaults 0swaps