30.097/30.097 NO 30.097/30.097 30.097/30.097 Problem 1: 30.097/30.097 30.097/30.097 30.097/30.097 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 30.097/30.097 Confluence Problem: 30.097/30.097 (VAR vNonEmpty:S x:S) 30.097/30.097 (STRATEGY CONTEXTSENSITIVE 30.097/30.097 (A) 30.097/30.097 (a) 30.097/30.097 (b) 30.097/30.097 (c) 30.097/30.097 (d) 30.097/30.097 (f 1) 30.097/30.097 (g 1 2 3) 30.097/30.097 (h 1 2) 30.097/30.097 (k) 30.097/30.097 (e) 30.097/30.097 (fSNonEmpty) 30.097/30.097 (l) 30.097/30.097 (m) 30.097/30.097 ) 30.097/30.097 (RULES 30.097/30.097 A -> h(f(a),f(b)) 30.097/30.097 a -> c 30.097/30.097 a -> d 30.097/30.097 b -> c 30.097/30.097 b -> d 30.097/30.097 c -> e 30.097/30.097 c -> l 30.097/30.097 d -> m 30.097/30.097 f(x:S) -> x:S | x:S ->* e 30.097/30.097 g(d,x:S,x:S) -> A 30.097/30.097 h(x:S,x:S) -> g(x:S,x:S,f(k)) 30.097/30.097 k -> l 30.097/30.097 k -> m 30.097/30.097 ) 30.097/30.097 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 30.097/30.097 30.097/30.097 30.097/30.097 Problem 1: 30.097/30.097 30.097/30.097 Inlining of Conditions Processor [STERN17]: 30.097/30.097 30.097/30.097 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 30.097/30.097 Confluence Problem: 30.097/30.097 (VAR vNonEmpty:S x:S) 30.097/30.097 (STRATEGY CONTEXTSENSITIVE 30.097/30.097 (A) 30.097/30.097 (a) 30.097/30.097 (b) 30.097/30.097 (c) 30.097/30.097 (d) 30.097/30.097 (f 1) 30.097/30.097 (g 1 2 3) 30.097/30.097 (h 1 2) 30.097/30.097 (k) 30.097/30.097 (e) 30.097/30.097 (fSNonEmpty) 30.097/30.097 (l) 30.097/30.097 (m) 30.097/30.097 ) 30.097/30.097 (RULES 30.097/30.097 A -> h(f(a),f(b)) 30.097/30.097 a -> c 30.097/30.097 a -> d 30.097/30.097 b -> c 30.097/30.097 b -> d 30.097/30.097 c -> e 30.097/30.097 c -> l 30.097/30.097 d -> m 30.097/30.097 f(x:S) -> x:S | x:S ->* e 30.097/30.097 g(d,x:S,x:S) -> A 30.097/30.097 h(x:S,x:S) -> g(x:S,x:S,f(k)) 30.097/30.097 k -> l 30.097/30.097 k -> m 30.097/30.097 ) 30.097/30.097 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 30.097/30.097 30.097/30.097 30.097/30.097 Problem 1: 30.097/30.097 30.097/30.097 Clean CTRS Processor: 30.097/30.097 30.097/30.097 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 30.097/30.097 Confluence Problem: 30.097/30.097 (VAR x:S) 30.097/30.097 (STRATEGY CONTEXTSENSITIVE 30.097/30.097 (A) 30.097/30.097 (a) 30.097/30.097 (b) 30.097/30.097 (c) 30.097/30.097 (d) 30.097/30.097 (f 1) 30.097/30.097 (g 1 2 3) 30.097/30.097 (h 1 2) 30.097/30.097 (k) 30.097/30.097 (e) 30.097/30.097 (fSNonEmpty) 30.097/30.097 (l) 30.097/30.097 (m) 30.097/30.097 ) 30.097/30.097 (RULES 30.097/30.097 A -> h(f(a),f(b)) 30.097/30.097 a -> c 30.097/30.097 a -> d 30.097/30.097 b -> c 30.097/30.097 b -> d 30.097/30.097 c -> e 30.097/30.097 c -> l 30.097/30.097 d -> m 30.097/30.097 f(x:S) -> x:S | x:S ->* e 30.097/30.097 g(d,x:S,x:S) -> A 30.097/30.097 h(x:S,x:S) -> g(x:S,x:S,f(k)) 30.097/30.097 k -> l 30.097/30.097 k -> m 30.097/30.097 ) 30.097/30.097 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 30.097/30.097 30.097/30.097 CRule InfChecker Info: 30.097/30.097 A -> h(f(a),f(b)) 30.097/30.097 Rule remains 30.097/30.097 Proof: 30.097/30.097 NO_CONDS 30.097/30.097 30.097/30.097 CRule InfChecker Info: 30.097/30.097 a -> c 30.097/30.097 Rule remains 30.097/30.097 Proof: 30.097/30.097 NO_CONDS 30.097/30.097 30.097/30.097 CRule InfChecker Info: 30.097/30.097 a -> d 30.097/30.097 Rule remains 30.097/30.097 Proof: 30.097/30.097 NO_CONDS 30.097/30.097 30.097/30.097 CRule InfChecker Info: 30.097/30.097 b -> c 30.097/30.097 Rule remains 30.097/30.097 Proof: 30.097/30.097 NO_CONDS 30.097/30.097 30.097/30.097 CRule InfChecker Info: 30.097/30.097 b -> d 30.097/30.097 Rule remains 30.097/30.097 Proof: 30.097/30.097 NO_CONDS 30.097/30.097 30.097/30.097 CRule InfChecker Info: 30.097/30.097 c -> e 30.097/30.097 Rule remains 30.097/30.097 Proof: 30.097/30.097 NO_CONDS 30.097/30.097 30.097/30.097 CRule InfChecker Info: 30.097/30.097 c -> l 30.097/30.097 Rule remains 30.097/30.097 Proof: 30.097/30.097 NO_CONDS 30.097/30.097 30.097/30.097 CRule InfChecker Info: 30.097/30.097 d -> m 30.097/30.097 Rule remains 30.097/30.097 Proof: 30.097/30.097 NO_CONDS 30.097/30.097 30.097/30.097 CRule InfChecker Info: 30.097/30.097 f(x:S) -> x:S | x:S ->* e 30.097/30.097 Rule remains 30.097/30.097 Proof: 30.097/30.097 NO 30.097/30.097 30.097/30.097 Problem 1: 30.097/30.097 30.097/30.097 Infeasibility Problem: 30.097/30.097 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 30.097/30.097 (STRATEGY CONTEXTSENSITIVE 30.097/30.097 (A) 30.097/30.097 (a) 30.097/30.097 (b) 30.097/30.097 (c) 30.097/30.097 (d) 30.097/30.097 (f 1) 30.097/30.097 (g 1 2 3) 30.097/30.097 (h 1 2) 30.097/30.097 (k) 30.097/30.097 (e) 30.097/30.097 (fSNonEmpty) 30.097/30.097 (l) 30.097/30.097 (m) 30.097/30.097 ) 30.097/30.097 (RULES 30.097/30.097 A -> h(f(a),f(b)) 30.097/30.097 a -> c 30.097/30.097 a -> d 30.097/30.097 b -> c 30.097/30.097 b -> d 30.097/30.097 c -> e 30.097/30.097 c -> l 30.097/30.097 d -> m 30.097/30.097 f(x:S:S) -> x:S:S | x:S:S ->* e 30.097/30.097 g(d,x:S:S,x:S:S) -> A 30.097/30.097 h(x:S:S,x:S:S) -> g(x:S:S,x:S:S,f(k)) 30.097/30.097 k -> l 30.097/30.097 k -> m 30.097/30.097 )] 30.097/30.097 30.097/30.097 Infeasibility Conditions: 30.097/30.097 x:S:S ->* e 30.097/30.097 30.097/30.097 Problem 1: 30.097/30.097 30.097/30.097 Obtaining a proof using Prover9: 30.097/30.097 30.097/30.097 -> Prover9 Output: 30.097/30.097 ============================== Prover9 =============================== 30.097/30.097 Prover9 (64) version 2009-11A, November 2009. 30.097/30.097 Process 108361 was started by ubuntu on ubuntu, 30.097/30.097 Fri Jul 16 15:00:14 2021 30.097/30.097 The command was "./prover9 -f /tmp/prover9108353-0.in". 30.097/30.097 ============================== end of head =========================== 30.097/30.097 30.097/30.097 ============================== INPUT ================================= 30.097/30.097 30.097/30.097 % Reading from file /tmp/prover9108353-0.in 30.097/30.097 30.097/30.097 assign(max_seconds,20). 30.097/30.097 30.097/30.097 formulas(assumptions). 30.097/30.097 ->*_s0(x,x) # label(reflexivity). 30.097/30.097 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 30.097/30.097 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 30.097/30.097 ->_s0(x1,y) -> ->_s0(g(x1,x2,x3),g(y,x2,x3)) # label(congruence). 30.097/30.097 ->_s0(x2,y) -> ->_s0(g(x1,x2,x3),g(x1,y,x3)) # label(congruence). 30.097/30.097 ->_s0(x3,y) -> ->_s0(g(x1,x2,x3),g(x1,x2,y)) # label(congruence). 30.097/30.097 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence). 30.097/30.097 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence). 30.097/30.097 ->_s0(A,h(f(a),f(b))) # label(replacement). 30.097/30.097 ->_s0(a,c) # label(replacement). 30.097/30.097 ->_s0(a,d) # label(replacement). 30.097/30.097 ->_s0(b,c) # label(replacement). 30.097/30.097 ->_s0(b,d) # label(replacement). 30.097/30.097 ->_s0(c,e) # label(replacement). 30.097/30.097 ->_s0(c,l) # label(replacement). 30.097/30.097 ->_s0(d,m) # label(replacement). 30.097/30.097 ->*_s0(x2,e) -> ->_s0(f(x2),x2) # label(replacement). 30.097/30.097 ->_s0(g(d,x2,x2),A) # label(replacement). 30.097/30.097 ->_s0(h(x2,x2),g(x2,x2,f(k))) # label(replacement). 30.097/30.097 ->_s0(k,l) # label(replacement). 30.097/30.097 ->_s0(k,m) # label(replacement). 30.097/30.097 end_of_list. 30.097/30.097 30.097/30.097 formulas(goals). 30.097/30.097 (exists x2 ->*_s0(x2,e)) # label(goal). 30.097/30.097 end_of_list. 30.097/30.097 30.097/30.097 ============================== end of input ========================== 30.097/30.097 30.097/30.097 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 30.097/30.097 30.097/30.097 % Formulas that are not ordinary clauses: 30.097/30.097 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 30.097/30.097 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 30.097/30.097 3 ->_s0(x1,y) -> ->_s0(g(x1,x2,x3),g(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 30.097/30.097 4 ->_s0(x2,y) -> ->_s0(g(x1,x2,x3),g(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 30.097/30.097 5 ->_s0(x3,y) -> ->_s0(g(x1,x2,x3),g(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 30.097/30.097 6 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 30.097/30.097 7 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 30.097/30.097 8 ->*_s0(x2,e) -> ->_s0(f(x2),x2) # label(replacement) # label(non_clause). [assumption]. 30.097/30.097 9 (exists x2 ->*_s0(x2,e)) # label(goal) # label(non_clause) # label(goal). [goal]. 30.097/30.097 30.097/30.097 ============================== end of process non-clausal formulas === 30.097/30.097 30.097/30.097 ============================== PROCESS INITIAL CLAUSES =============== 30.097/30.097 30.097/30.097 % Clauses before input processing: 30.097/30.097 30.097/30.097 formulas(usable). 30.097/30.097 end_of_list. 30.097/30.097 30.097/30.097 formulas(sos). 30.097/30.097 ->*_s0(x,x) # label(reflexivity). [assumption]. 30.097/30.097 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 30.097/30.097 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 30.097/30.097 -->_s0(x,y) | ->_s0(g(x,z,u),g(y,z,u)) # label(congruence). [clausify(3)]. 30.097/30.097 -->_s0(x,y) | ->_s0(g(z,x,u),g(z,y,u)) # label(congruence). [clausify(4)]. 30.097/30.097 -->_s0(x,y) | ->_s0(g(z,u,x),g(z,u,y)) # label(congruence). [clausify(5)]. 30.097/30.097 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 30.097/30.097 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 30.097/30.097 ->_s0(A,h(f(a),f(b))) # label(replacement). [assumption]. 30.097/30.097 ->_s0(a,c) # label(replacement). [assumption]. 30.097/30.097 ->_s0(a,d) # label(replacement). [assumption]. 30.097/30.097 ->_s0(b,c) # label(replacement). [assumption]. 30.097/30.097 ->_s0(b,d) # label(replacement). [assumption]. 30.097/30.097 ->_s0(c,e) # label(replacement). [assumption]. 30.097/30.097 ->_s0(c,l) # label(replacement). [assumption]. 30.097/30.097 ->_s0(d,m) # label(replacement). [assumption]. 30.097/30.097 -->*_s0(x,e) | ->_s0(f(x),x) # label(replacement). [clausify(8)]. 30.097/30.097 ->_s0(g(d,x,x),A) # label(replacement). [assumption]. 30.097/30.097 ->_s0(h(x,x),g(x,x,f(k))) # label(replacement). [assumption]. 30.097/30.097 ->_s0(k,l) # label(replacement). [assumption]. 30.097/30.097 ->_s0(k,m) # label(replacement). [assumption]. 30.097/30.097 -->*_s0(x,e) # label(goal). [deny(9)]. 30.097/30.097 end_of_list. 30.097/30.097 30.097/30.097 formulas(demodulators). 30.097/30.097 end_of_list. 30.097/30.097 30.097/30.097 ============================== PREDICATE ELIMINATION ================= 30.097/30.097 30.097/30.097 No predicates eliminated. 30.097/30.097 30.097/30.097 ============================== end predicate elimination ============= 30.097/30.097 30.097/30.097 Auto_denials: 30.097/30.097 % copying label goal to answer in negative clause 30.097/30.097 30.097/30.097 Term ordering decisions: 30.097/30.097 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 30.097/30.097 Function symbol precedence: function_order([ c, d, k, a, b, e, l, m, A, h, f, g ]). 30.097/30.097 After inverse_order: (no changes). 30.097/30.097 Unfolding symbols: (none). 30.097/30.097 30.097/30.097 Auto_inference settings: 30.097/30.097 % set(neg_binary_resolution). % (HNE depth_diff=-7) 30.097/30.097 % clear(ordered_res). % (HNE depth_diff=-7) 30.097/30.097 % set(ur_resolution). % (HNE depth_diff=-7) 30.097/30.097 % set(ur_resolution) -> set(pos_ur_resolution). 30.097/30.097 % set(ur_resolution) -> set(neg_ur_resolution). 30.097/30.097 30.097/30.097 Auto_process settings: (no changes). 30.097/30.097 30.097/30.097 kept: 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 30.097/30.097 kept: 11 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 30.097/30.097 kept: 12 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 30.097/30.097 kept: 13 -->_s0(x,y) | ->_s0(g(x,z,u),g(y,z,u)) # label(congruence). [clausify(3)]. 30.097/30.097 kept: 14 -->_s0(x,y) | ->_s0(g(z,x,u),g(z,y,u)) # label(congruence). [clausify(4)]. 30.097/30.097 kept: 15 -->_s0(x,y) | ->_s0(g(z,u,x),g(z,u,y)) # label(congruence). [clausify(5)]. 30.097/30.097 kept: 16 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 30.097/30.097 kept: 17 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 30.097/30.097 kept: 18 ->_s0(A,h(f(a),f(b))) # label(replacement). [assumption]. 30.097/30.097 kept: 19 ->_s0(a,c) # label(replacement). [assumption]. 30.097/30.097 kept: 20 ->_s0(a,d) # label(replacement). [assumption]. 30.097/30.097 kept: 21 ->_s0(b,c) # label(replacement). [assumption]. 30.097/30.097 kept: 22 ->_s0(b,d) # label(replacement). [assumption]. 30.097/30.097 kept: 23 ->_s0(c,e) # label(replacement). [assumption]. 30.097/30.097 kept: 24 ->_s0(c,l) # label(replacement). [assumption]. 30.097/30.097 kept: 25 ->_s0(d,m) # label(replacement). [assumption]. 30.097/30.097 kept: 26 -->*_s0(x,e) | ->_s0(f(x),x) # label(replacement). [clausify(8)]. 30.097/30.097 kept: 27 ->_s0(g(d,x,x),A) # label(replacement). [assumption]. 30.097/30.097 kept: 28 ->_s0(h(x,x),g(x,x,f(k))) # label(replacement). [assumption]. 30.097/30.097 kept: 29 ->_s0(k,l) # label(replacement). [assumption]. 30.097/30.097 kept: 30 ->_s0(k,m) # label(replacement). [assumption]. 30.097/30.097 kept: 31 -->*_s0(x,e) # label(goal) # answer(goal). [deny(9)]. 30.097/30.097 30.097/30.097 ============================== PROOF ================================= 30.097/30.097 30.097/30.097 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 30.097/30.097 % Length of proof is 4. 30.097/30.097 % Level of proof is 2. 30.097/30.097 % Maximum clause weight is 3.000. 30.097/30.097 % Given clauses 0. 30.097/30.097 30.097/30.097 9 (exists x2 ->*_s0(x2,e)) # label(goal) # label(non_clause) # label(goal). [goal]. 30.097/30.097 10 ->*_s0(x,x) # label(reflexivity). [assumption]. 30.097/30.097 31 -->*_s0(x,e) # label(goal) # answer(goal). [deny(9)]. 30.097/30.097 32 $F # answer(goal). [resolve(31,a,10,a)]. 30.097/30.097 30.097/30.097 ============================== end of proof ========================== 30.097/30.097 30.097/30.097 ============================== STATISTICS ============================ 30.097/30.097 30.097/30.097 Given=0. Generated=22. Kept=22. proofs=1. 30.097/30.097 Usable=0. Sos=0. Demods=0. Limbo=21, Disabled=22. Hints=0. 30.097/30.097 Kept_by_rule=0, Deleted_by_rule=0. 30.097/30.097 Forward_subsumed=0. Back_subsumed=0. 30.097/30.097 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 30.097/30.097 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 30.097/30.097 Demod_attempts=0. Demod_rewrites=0. 30.097/30.097 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 30.097/30.097 Nonunit_fsub_feature_tests=4. Nonunit_bsub_feature_tests=0. 30.097/30.097 Megabytes=0.07. 30.097/30.097 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 30.097/30.097 30.097/30.097 ============================== end of statistics ===================== 30.097/30.097 30.097/30.097 ============================== end of search ========================= 30.097/30.097 30.097/30.097 THEOREM PROVED 30.097/30.097 30.097/30.097 Exiting with 1 proof. 30.097/30.097 30.097/30.097 Process 108361 exit (max_proofs) Fri Jul 16 15:00:14 2021 30.097/30.097 30.097/30.097 30.097/30.097 The problem is feasible. 30.097/30.097 30.097/30.097 30.097/30.097 CRule InfChecker Info: 30.097/30.097 g(d,x:S,x:S) -> A 30.097/30.097 Rule remains 30.097/30.097 Proof: 30.097/30.097 NO_CONDS 30.097/30.097 30.097/30.097 CRule InfChecker Info: 30.097/30.097 h(x:S,x:S) -> g(x:S,x:S,f(k)) 30.097/30.097 Rule remains 30.097/30.097 Proof: 30.097/30.097 NO_CONDS 30.097/30.097 30.097/30.097 CRule InfChecker Info: 30.097/30.097 k -> l 30.097/30.097 Rule remains 30.097/30.097 Proof: 30.097/30.097 NO_CONDS 30.097/30.097 30.097/30.097 CRule InfChecker Info: 30.097/30.097 k -> m 30.097/30.097 Rule remains 30.097/30.097 Proof: 30.097/30.097 NO_CONDS 30.097/30.097 30.097/30.097 Problem 1: 30.097/30.097 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 30.097/30.097 Confluence Problem: 30.097/30.097 (VAR x:S) 30.097/30.097 (STRATEGY CONTEXTSENSITIVE 30.097/30.097 (A) 30.097/30.097 (a) 30.097/30.097 (b) 30.097/30.097 (c) 30.097/30.097 (d) 30.097/30.097 (f 1) 30.097/30.097 (g 1 2 3) 30.097/30.097 (h 1 2) 30.097/30.097 (k) 30.097/30.097 (e) 30.097/30.097 (fSNonEmpty) 30.097/30.097 (l) 30.097/30.097 (m) 30.097/30.097 ) 30.097/30.097 (RULES 30.097/30.097 A -> h(f(a),f(b)) 30.097/30.097 a -> c 30.097/30.097 a -> d 30.097/30.097 b -> c 30.097/30.097 b -> d 30.097/30.097 c -> e 30.097/30.097 c -> l 30.097/30.097 d -> m 30.097/30.097 f(x:S) -> x:S | x:S ->* e 30.097/30.097 g(d,x:S,x:S) -> A 30.097/30.097 h(x:S,x:S) -> g(x:S,x:S,f(k)) 30.097/30.097 k -> l 30.097/30.097 k -> m 30.097/30.097 ) 30.097/30.097 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 30.097/30.097 30.097/30.097 Critical Pairs Processor: 30.097/30.097 -> Rules: 30.097/30.097 A -> h(f(a),f(b)) 30.097/30.097 a -> c 30.097/30.097 a -> d 30.097/30.097 b -> c 30.097/30.097 b -> d 30.097/30.097 c -> e 30.097/30.097 c -> l 30.097/30.097 d -> m 30.097/30.097 f(x:S) -> x:S | x:S ->* e 30.097/30.097 g(d,x:S,x:S) -> A 30.097/30.097 h(x:S,x:S) -> g(x:S,x:S,f(k)) 30.097/30.097 k -> l 30.097/30.097 k -> m 30.097/30.097 -> Vars: 30.097/30.097 "x", "x", "x" 30.097/30.097 -> FVars: 30.097/30.097 "x2", "x3", "x4" 30.097/30.097 -> PVars: 30.097/30.097 "x": ["x2", "x3", "x4"] 30.097/30.097 30.097/30.097 -> Rlps: 30.097/30.097 crule: A -> h(f(a),f(b)), id: 1, possubterms: A-> [] 30.097/30.097 crule: a -> c, id: 2, possubterms: a-> [] 30.097/30.097 crule: a -> d, id: 3, possubterms: a-> [] 30.097/30.097 crule: b -> c, id: 4, possubterms: b-> [] 30.097/30.097 crule: b -> d, id: 5, possubterms: b-> [] 30.097/30.097 crule: c -> e, id: 6, possubterms: c-> [] 30.097/30.097 crule: c -> l, id: 7, possubterms: c-> [] 30.097/30.097 crule: d -> m, id: 8, possubterms: d-> [] 30.097/30.097 crule: f(x2:S) -> x2:S | x2:S ->* e, id: 9, possubterms: f(x2:S)-> [] 30.097/30.097 crule: g(d,x3:S,x3:S) -> A, id: 10, possubterms: g(d,x3:S,x3:S)-> [], d-> [1] 30.097/30.097 crule: h(x4:S,x4:S) -> g(x4:S,x4:S,f(k)), id: 11, possubterms: h(x4:S,x4:S)-> [] 30.097/30.097 crule: k -> l, id: 12, possubterms: k-> [] 30.097/30.097 crule: k -> m, id: 13, possubterms: k-> [] 30.097/30.097 30.097/30.097 -> Unifications: 30.097/30.097 R3 unifies with R2 at p: [], l: a, lp: a, conds: {}, sig: {}, l': a, r: d, r': c 30.097/30.097 R5 unifies with R4 at p: [], l: b, lp: b, conds: {}, sig: {}, l': b, r: d, r': c 30.097/30.097 R7 unifies with R6 at p: [], l: c, lp: c, conds: {}, sig: {}, l': c, r: l, r': e 30.097/30.097 R10 unifies with R8 at p: [1], l: g(d,x3:S,x3:S), lp: d, conds: {}, sig: {}, l': d, r: A, r': m 30.097/30.097 R13 unifies with R12 at p: [], l: k, lp: k, conds: {}, sig: {}, l': k, r: m, r': l 30.097/30.097 30.097/30.097 -> Critical pairs info: 30.097/30.097 => Not trivial, Overlay, N1 30.097/30.097 => Not trivial, Overlay, N2 30.097/30.097 => Not trivial, Overlay, N3 30.097/30.097 => Not trivial, Not overlay, N4 30.097/30.097 30.097/30.097 -> Problem conclusions: 30.097/30.097 Not left linear, Not right linear, Not linear 30.097/30.097 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 30.097/30.097 CTRS Type: 1 30.097/30.097 Deterministic, Strongly deterministic 30.097/30.097 Oriented CTRS, Properly oriented CTRS, Not join CTRS 30.097/30.097 Maybe right-stable CTRS, Not overlay CTRS 30.097/30.097 Maybe normal CTRS, Maybe almost normal CTRS 30.097/30.097 Maybe terminating CTRS, Maybe joinable CCPs 30.097/30.097 Maybe level confluent 30.097/30.097 Maybe confluent 30.097/30.097 30.097/30.097 Problem 1: 30.097/30.097 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 30.097/30.097 Confluence Problem: 30.097/30.097 (VAR x:S) 30.097/30.097 (STRATEGY CONTEXTSENSITIVE 30.097/30.097 (A) 30.097/30.097 (a) 30.097/30.097 (b) 30.097/30.097 (c) 30.097/30.097 (d) 30.097/30.097 (f 1) 30.097/30.097 (g 1 2 3) 30.097/30.097 (h 1 2) 30.097/30.097 (k) 30.097/30.097 (e) 30.097/30.097 (fSNonEmpty) 30.097/30.097 (l) 30.097/30.097 (m) 30.097/30.097 ) 30.097/30.097 (RULES 30.097/30.097 A -> h(f(a),f(b)) 30.097/30.097 a -> c 30.097/30.097 a -> d 30.097/30.097 b -> c 30.097/30.097 b -> d 30.097/30.097 c -> e 30.097/30.097 c -> l 30.097/30.097 d -> m 30.097/30.097 f(x:S) -> x:S | x:S ->* e 30.097/30.097 g(d,x:S,x:S) -> A 30.097/30.097 h(x:S,x:S) -> g(x:S,x:S,f(k)) 30.097/30.097 k -> l 30.097/30.097 k -> m 30.097/30.097 ) 30.097/30.097 Critical Pairs: 30.097/30.097 => Not trivial, Overlay, N1 30.097/30.097 => Not trivial, Overlay, N2 30.097/30.097 => Not trivial, Overlay, N3 30.097/30.097 => Not trivial, Not overlay, N4 30.097/30.097 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 30.097/30.097 30.097/30.097 Conditional Critical Pairs Distributor Processor 30.097/30.097 30.097/30.097 Problem 1: 30.097/30.097 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 30.097/30.097 Confluence Problem: 30.097/30.097 (VAR x:S) 30.097/30.097 (STRATEGY CONTEXTSENSITIVE 30.097/30.097 (A) 30.097/30.097 (a) 30.097/30.097 (b) 30.097/30.097 (c) 30.097/30.097 (d) 30.097/30.097 (f 1) 30.097/30.097 (g 1 2 3) 30.097/30.097 (h 1 2) 30.097/30.097 (k) 30.097/30.097 (e) 30.097/30.097 (fSNonEmpty) 30.097/30.097 (l) 30.097/30.097 (m) 30.097/30.097 ) 30.097/30.097 (RULES 30.097/30.097 A -> h(f(a),f(b)) 30.097/30.097 a -> c 30.097/30.097 a -> d 30.097/30.097 b -> c 30.097/30.097 b -> d 30.097/30.097 c -> e 30.097/30.097 c -> l 30.097/30.097 d -> m 30.097/30.097 f(x:S) -> x:S | x:S ->* e 30.097/30.097 g(d,x:S,x:S) -> A 30.097/30.097 h(x:S,x:S) -> g(x:S,x:S,f(k)) 30.097/30.097 k -> l 30.097/30.097 k -> m 30.097/30.097 ) 30.097/30.097 Critical Pairs: 30.097/30.097 => Not trivial, Overlay, N1 30.097/30.097 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 30.097/30.097 30.097/30.097 Infeasible Convergence No Conditions CCP Processor: 30.097/30.097 Infeasible convergence? 30.097/30.097 YES 30.097/30.097 30.097/30.097 Problem 1: 30.097/30.097 30.097/30.097 Infeasibility Problem: 30.097/30.097 [(VAR vNonEmpty:S x:S z:S) 30.097/30.097 (STRATEGY CONTEXTSENSITIVE 30.097/30.097 (A) 30.097/30.097 (a) 30.097/30.097 (b) 30.097/30.097 (c) 30.097/30.097 (d) 30.097/30.097 (f 1) 30.097/30.097 (g 1 2 3) 30.097/30.097 (h 1 2) 30.097/30.097 (k) 30.097/30.097 (e) 30.097/30.097 (fSNonEmpty) 30.097/30.097 (l) 30.097/30.097 (m) 30.097/30.097 ) 30.097/30.097 (RULES 30.097/30.097 A -> h(f(a),f(b)) 30.097/30.097 a -> c 30.097/30.097 a -> d 30.097/30.097 b -> c 30.097/30.097 b -> d 30.097/30.097 c -> e 30.097/30.097 c -> l 30.097/30.097 d -> m 30.097/30.097 f(x:S) -> x:S | x:S ->* e 30.097/30.097 g(d,x:S,x:S) -> A 30.097/30.097 h(x:S,x:S) -> g(x:S,x:S,f(k)) 30.097/30.097 k -> l 30.097/30.097 k -> m 30.097/30.097 )] 30.097/30.097 30.097/30.097 Infeasibility Conditions: 30.097/30.097 l ->* z:S, m ->* z:S 30.097/30.097 30.097/30.097 Problem 1: 30.097/30.097 30.097/30.097 Obtaining a model using Mace4: 30.097/30.097 30.097/30.097 -> Usable Rules: 30.097/30.097 Empty 30.097/30.097 30.097/30.097 -> Mace4 Output: 30.097/30.097 ============================== Mace4 ================================= 30.097/30.097 Mace4 (64) version 2009-11A, November 2009. 30.097/30.097 Process 108407 was started by ubuntu on ubuntu, 30.097/30.097 Fri Jul 16 15:00:14 2021 30.097/30.097 The command was "./mace4 -c -f /tmp/mace4108379-2.in". 30.097/30.097 ============================== end of head =========================== 30.097/30.097 30.097/30.097 ============================== INPUT ================================= 30.097/30.097 30.097/30.097 % Reading from file /tmp/mace4108379-2.in 30.097/30.097 30.097/30.097 assign(max_seconds,10). 30.097/30.097 30.097/30.097 formulas(assumptions). 30.097/30.097 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 30.097/30.097 ->(x1,y) -> ->(g(x1,x2,x3),g(y,x2,x3)) # label(congruence). 30.097/30.097 ->(x2,y) -> ->(g(x1,x2,x3),g(x1,y,x3)) # label(congruence). 30.097/30.097 ->(x3,y) -> ->(g(x1,x2,x3),g(x1,x2,y)) # label(congruence). 30.097/30.097 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence). 30.097/30.097 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence). 30.097/30.097 ->*(x,x) # label(reflexivity). 30.097/30.097 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 30.097/30.097 end_of_list. 30.097/30.097 30.097/30.097 formulas(goals). 30.097/30.097 (exists x2 (->*(l,x2) & ->*(m,x2))) # label(goal). 30.097/30.097 end_of_list. 30.097/30.097 30.097/30.097 ============================== end of input ========================== 30.097/30.097 30.097/30.097 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 30.097/30.097 30.097/30.097 % Formulas that are not ordinary clauses: 30.097/30.097 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 30.097/30.097 2 ->(x1,y) -> ->(g(x1,x2,x3),g(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 30.097/30.097 3 ->(x2,y) -> ->(g(x1,x2,x3),g(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 30.097/30.097 4 ->(x3,y) -> ->(g(x1,x2,x3),g(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 30.097/30.097 5 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 30.097/30.097 6 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 30.097/30.097 7 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 30.097/30.097 8 (exists x2 (->*(l,x2) & ->*(m,x2))) # label(goal) # label(non_clause) # label(goal). [goal]. 30.097/30.097 30.097/30.097 ============================== end of process non-clausal formulas === 30.097/30.097 30.097/30.097 ============================== CLAUSES FOR SEARCH ==================== 30.097/30.097 30.097/30.097 formulas(mace4_clauses). 30.097/30.097 -->(x,y) | ->(f(x),f(y)) # label(congruence). 30.097/30.097 -->(x,y) | ->(g(x,z,u),g(y,z,u)) # label(congruence). 30.097/30.097 -->(x,y) | ->(g(z,x,u),g(z,y,u)) # label(congruence). 30.097/30.097 -->(x,y) | ->(g(z,u,x),g(z,u,y)) # label(congruence). 30.097/30.097 -->(x,y) | ->(h(x,z),h(y,z)) # label(congruence). 30.097/30.097 -->(x,y) | ->(h(z,x),h(z,y)) # label(congruence). 30.097/30.097 ->*(x,x) # label(reflexivity). 30.097/30.097 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 30.097/30.097 -->*(l,x) | -->*(m,x) # label(goal). 30.097/30.097 end_of_list. 30.097/30.097 30.097/30.097 ============================== end of clauses for search ============= 30.097/30.097 30.097/30.097 % There are no natural numbers in the input. 30.097/30.097 30.097/30.097 ============================== DOMAIN SIZE 2 ========================= 30.097/30.097 30.097/30.097 ============================== MODEL ================================= 30.097/30.097 30.097/30.097 interpretation( 2, [number=1, seconds=0], [ 30.097/30.097 30.097/30.097 function(l, [ 0 ]), 30.097/30.097 30.097/30.097 function(m, [ 1 ]), 30.097/30.097 30.097/30.097 function(f(_), [ 0, 0 ]), 30.097/30.097 30.097/30.097 function(h(_,_), [ 30.097/30.097 0, 0, 30.097/30.097 0, 0 ]), 30.097/30.097 30.097/30.097 function(g(_,_,_), [ 30.097/30.097 0, 0, 30.097/30.097 0, 0, 30.097/30.097 0, 0, 30.097/30.097 0, 0 ]), 30.097/30.097 30.097/30.097 relation(->*(_,_), [ 30.097/30.097 1, 0, 30.097/30.097 0, 1 ]), 30.097/30.097 30.097/30.097 relation(->(_,_), [ 30.097/30.097 0, 0, 30.097/30.097 0, 0 ]) 30.097/30.097 ]). 30.097/30.097 30.097/30.097 ============================== end of model ========================== 30.097/30.097 30.097/30.097 ============================== STATISTICS ============================ 30.097/30.097 30.097/30.097 For domain size 2. 30.097/30.097 30.097/30.097 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 30.097/30.097 Ground clauses: seen=80, kept=76. 30.097/30.097 Selections=16, assignments=16, propagations=8, current_models=1. 30.097/30.097 Rewrite_terms=140, rewrite_bools=89, indexes=8. 30.097/30.097 Rules_from_neg_clauses=1, cross_offs=1. 30.097/30.097 30.097/30.097 ============================== end of statistics ===================== 30.097/30.097 30.097/30.097 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 30.097/30.097 30.097/30.097 Exiting with 1 model. 30.097/30.097 30.097/30.097 Process 108407 exit (max_models) Fri Jul 16 15:00:14 2021 30.097/30.097 The process finished Fri Jul 16 15:00:14 2021 30.097/30.097 30.097/30.097 30.097/30.097 Mace4 cooked interpretation: 30.097/30.097 30.097/30.097 30.097/30.097 30.097/30.097 The problem is infeasible. 30.097/30.097 30.097/30.097 30.097/30.097 The problem is not joinable. 30.097/30.097 30.71user 1.76system 0:30.97elapsed 104%CPU (0avgtext+0avgdata 435076maxresident)k 30.097/30.097 8inputs+0outputs (0major+206508minor)pagefaults 0swaps