25.081/25.081 NO 25.081/25.081 25.081/25.081 Problem 1: 25.081/25.081 25.081/25.081 25.081/25.081 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.081/25.081 Confluence Problem: 25.081/25.081 (VAR vNonEmpty:S x:S y:S) 25.081/25.081 (STRATEGY CONTEXTSENSITIVE 25.081/25.081 (a) 25.081/25.081 (b) 25.081/25.081 (c) 25.081/25.081 (d) 25.081/25.081 (f 1) 25.081/25.081 (g 1 2 3) 25.081/25.081 (h 1 2) 25.081/25.081 (k) 25.081/25.081 (A) 25.081/25.081 (e) 25.081/25.081 (fSNonEmpty) 25.081/25.081 (l) 25.081/25.081 (m) 25.081/25.081 ) 25.081/25.081 (RULES 25.081/25.081 a -> c 25.081/25.081 a -> d 25.081/25.081 b -> c 25.081/25.081 b -> d 25.081/25.081 c -> e 25.081/25.081 c -> l 25.081/25.081 d -> m 25.081/25.081 f(x:S) -> x:S | x:S ->* e 25.081/25.081 g(d,x:S,y:S) -> A | y:S ->* x:S 25.081/25.081 h(x:S,y:S) -> g(x:S,y:S,f(k)) | y:S ->* x:S 25.081/25.081 k -> l 25.081/25.081 k -> m 25.081/25.081 ) 25.081/25.081 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.081/25.081 25.081/25.081 25.081/25.081 Problem 1: 25.081/25.081 25.081/25.081 Inlining of Conditions Processor [STERN17]: 25.081/25.081 25.081/25.081 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.081/25.081 Confluence Problem: 25.081/25.081 (VAR vNonEmpty:S x:S y:S) 25.081/25.081 (STRATEGY CONTEXTSENSITIVE 25.081/25.081 (a) 25.081/25.081 (b) 25.081/25.081 (c) 25.081/25.081 (d) 25.081/25.081 (f 1) 25.081/25.081 (g 1 2 3) 25.081/25.081 (h 1 2) 25.081/25.081 (k) 25.081/25.081 (A) 25.081/25.081 (e) 25.081/25.081 (fSNonEmpty) 25.081/25.081 (l) 25.081/25.081 (m) 25.081/25.081 ) 25.081/25.081 (RULES 25.081/25.081 a -> c 25.081/25.081 a -> d 25.081/25.081 b -> c 25.081/25.081 b -> d 25.081/25.081 c -> e 25.081/25.081 c -> l 25.081/25.081 d -> m 25.081/25.081 f(x:S) -> x:S | x:S ->* e 25.081/25.081 g(d,x:S,y:S) -> A | y:S ->* x:S 25.081/25.081 h(x:S,y:S) -> g(x:S,y:S,f(k)) | y:S ->* x:S 25.081/25.081 k -> l 25.081/25.081 k -> m 25.081/25.081 ) 25.081/25.081 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.081/25.081 25.081/25.081 25.081/25.081 Problem 1: 25.081/25.081 25.081/25.081 Clean CTRS Processor: 25.081/25.081 25.081/25.081 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.081/25.081 Confluence Problem: 25.081/25.081 (VAR x:S y:S) 25.081/25.081 (STRATEGY CONTEXTSENSITIVE 25.081/25.081 (a) 25.081/25.081 (b) 25.081/25.081 (c) 25.081/25.081 (d) 25.081/25.081 (f 1) 25.081/25.081 (g 1 2 3) 25.081/25.081 (h 1 2) 25.081/25.081 (k) 25.081/25.081 (A) 25.081/25.081 (e) 25.081/25.081 (fSNonEmpty) 25.081/25.081 (l) 25.081/25.081 (m) 25.081/25.081 ) 25.081/25.081 (RULES 25.081/25.081 a -> c 25.081/25.081 a -> d 25.081/25.081 b -> c 25.081/25.081 b -> d 25.081/25.081 c -> e 25.081/25.081 c -> l 25.081/25.081 d -> m 25.081/25.081 f(x:S) -> x:S | x:S ->* e 25.081/25.081 g(d,x:S,y:S) -> A | y:S ->* x:S 25.081/25.081 h(x:S,y:S) -> g(x:S,y:S,f(k)) | y:S ->* x:S 25.081/25.081 k -> l 25.081/25.081 k -> m 25.081/25.081 ) 25.081/25.081 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.081/25.081 25.081/25.081 CRule InfChecker Info: 25.081/25.081 a -> c 25.081/25.081 Rule remains 25.081/25.081 Proof: 25.081/25.081 NO_CONDS 25.081/25.081 25.081/25.081 CRule InfChecker Info: 25.081/25.081 a -> d 25.081/25.081 Rule remains 25.081/25.081 Proof: 25.081/25.081 NO_CONDS 25.081/25.081 25.081/25.081 CRule InfChecker Info: 25.081/25.081 b -> c 25.081/25.081 Rule remains 25.081/25.081 Proof: 25.081/25.081 NO_CONDS 25.081/25.081 25.081/25.081 CRule InfChecker Info: 25.081/25.081 b -> d 25.081/25.081 Rule remains 25.081/25.081 Proof: 25.081/25.081 NO_CONDS 25.081/25.081 25.081/25.081 CRule InfChecker Info: 25.081/25.081 c -> e 25.081/25.081 Rule remains 25.081/25.081 Proof: 25.081/25.081 NO_CONDS 25.081/25.081 25.081/25.081 CRule InfChecker Info: 25.081/25.081 c -> l 25.081/25.081 Rule remains 25.081/25.081 Proof: 25.081/25.081 NO_CONDS 25.081/25.081 25.081/25.081 CRule InfChecker Info: 25.081/25.081 d -> m 25.081/25.081 Rule remains 25.081/25.081 Proof: 25.081/25.081 NO_CONDS 25.081/25.081 25.081/25.081 CRule InfChecker Info: 25.081/25.081 f(x:S) -> x:S | x:S ->* e 25.081/25.081 Rule remains 25.081/25.081 Proof: 25.081/25.081 NO 25.081/25.081 25.081/25.081 Problem 1: 25.081/25.081 25.081/25.081 Infeasibility Problem: 25.081/25.081 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S y:S:S) 25.081/25.081 (STRATEGY CONTEXTSENSITIVE 25.081/25.081 (a) 25.081/25.081 (b) 25.081/25.081 (c) 25.081/25.081 (d) 25.081/25.081 (f 1) 25.081/25.081 (g 1 2 3) 25.081/25.081 (h 1 2) 25.081/25.081 (k) 25.081/25.081 (A) 25.081/25.081 (e) 25.081/25.081 (fSNonEmpty) 25.081/25.081 (l) 25.081/25.081 (m) 25.081/25.081 ) 25.081/25.081 (RULES 25.081/25.081 a -> c 25.081/25.081 a -> d 25.081/25.081 b -> c 25.081/25.081 b -> d 25.081/25.081 c -> e 25.081/25.081 c -> l 25.081/25.081 d -> m 25.081/25.081 f(x:S:S) -> x:S:S | x:S:S ->* e 25.081/25.081 g(d,x:S:S,y:S:S) -> A | y:S:S ->* x:S:S 25.081/25.081 h(x:S:S,y:S:S) -> g(x:S:S,y:S:S,f(k)) | y:S:S ->* x:S:S 25.081/25.081 k -> l 25.081/25.081 k -> m 25.081/25.081 )] 25.081/25.081 25.081/25.081 Infeasibility Conditions: 25.081/25.081 x:S:S ->* e 25.081/25.081 25.081/25.081 Problem 1: 25.081/25.081 25.081/25.081 Obtaining a proof using Prover9: 25.081/25.081 25.081/25.081 -> Prover9 Output: 25.081/25.081 ============================== Prover9 =============================== 25.081/25.081 Prover9 (64) version 2009-11A, November 2009. 25.081/25.081 Process 39817 was started by ubuntu on ubuntu, 25.081/25.081 Wed Jul 14 10:13:42 2021 25.081/25.081 The command was "./prover9 -f /tmp/prover939810-0.in". 25.081/25.081 ============================== end of head =========================== 25.081/25.081 25.081/25.081 ============================== INPUT ================================= 25.081/25.081 25.081/25.081 % Reading from file /tmp/prover939810-0.in 25.081/25.081 25.081/25.081 assign(max_seconds,20). 25.081/25.081 25.081/25.081 formulas(assumptions). 25.081/25.081 ->*_s0(x,x) # label(reflexivity). 25.081/25.081 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.081/25.081 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 25.081/25.081 ->_s0(x1,y) -> ->_s0(g(x1,x2,x3),g(y,x2,x3)) # label(congruence). 25.081/25.081 ->_s0(x2,y) -> ->_s0(g(x1,x2,x3),g(x1,y,x3)) # label(congruence). 25.081/25.081 ->_s0(x3,y) -> ->_s0(g(x1,x2,x3),g(x1,x2,y)) # label(congruence). 25.081/25.081 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence). 25.081/25.081 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence). 25.081/25.081 ->_s0(a,c) # label(replacement). 25.081/25.081 ->_s0(a,d) # label(replacement). 25.081/25.081 ->_s0(b,c) # label(replacement). 25.081/25.081 ->_s0(b,d) # label(replacement). 25.081/25.081 ->_s0(c,e) # label(replacement). 25.081/25.081 ->_s0(c,l) # label(replacement). 25.081/25.081 ->_s0(d,m) # label(replacement). 25.081/25.081 ->*_s0(x2,e) -> ->_s0(f(x2),x2) # label(replacement). 25.081/25.081 ->*_s0(x3,x2) -> ->_s0(g(d,x2,x3),A) # label(replacement). 25.081/25.081 ->*_s0(x3,x2) -> ->_s0(h(x2,x3),g(x2,x3,f(k))) # label(replacement). 25.081/25.081 ->_s0(k,l) # label(replacement). 25.081/25.081 ->_s0(k,m) # label(replacement). 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 formulas(goals). 25.081/25.081 (exists x2 ->*_s0(x2,e)) # label(goal). 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 ============================== end of input ========================== 25.081/25.081 25.081/25.081 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.081/25.081 25.081/25.081 % Formulas that are not ordinary clauses: 25.081/25.081 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.081/25.081 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 3 ->_s0(x1,y) -> ->_s0(g(x1,x2,x3),g(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 4 ->_s0(x2,y) -> ->_s0(g(x1,x2,x3),g(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 5 ->_s0(x3,y) -> ->_s0(g(x1,x2,x3),g(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 6 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 7 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 8 ->*_s0(x2,e) -> ->_s0(f(x2),x2) # label(replacement) # label(non_clause). [assumption]. 25.081/25.081 9 ->*_s0(x3,x2) -> ->_s0(g(d,x2,x3),A) # label(replacement) # label(non_clause). [assumption]. 25.081/25.081 10 ->*_s0(x3,x2) -> ->_s0(h(x2,x3),g(x2,x3,f(k))) # label(replacement) # label(non_clause). [assumption]. 25.081/25.081 11 (exists x2 ->*_s0(x2,e)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.081/25.081 25.081/25.081 ============================== end of process non-clausal formulas === 25.081/25.081 25.081/25.081 ============================== PROCESS INITIAL CLAUSES =============== 25.081/25.081 25.081/25.081 % Clauses before input processing: 25.081/25.081 25.081/25.081 formulas(usable). 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 formulas(sos). 25.081/25.081 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.081/25.081 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.081/25.081 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.081/25.081 -->_s0(x,y) | ->_s0(g(x,z,u),g(y,z,u)) # label(congruence). [clausify(3)]. 25.081/25.081 -->_s0(x,y) | ->_s0(g(z,x,u),g(z,y,u)) # label(congruence). [clausify(4)]. 25.081/25.081 -->_s0(x,y) | ->_s0(g(z,u,x),g(z,u,y)) # label(congruence). [clausify(5)]. 25.081/25.081 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 25.081/25.081 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 25.081/25.081 ->_s0(a,c) # label(replacement). [assumption]. 25.081/25.081 ->_s0(a,d) # label(replacement). [assumption]. 25.081/25.081 ->_s0(b,c) # label(replacement). [assumption]. 25.081/25.081 ->_s0(b,d) # label(replacement). [assumption]. 25.081/25.081 ->_s0(c,e) # label(replacement). [assumption]. 25.081/25.081 ->_s0(c,l) # label(replacement). [assumption]. 25.081/25.081 ->_s0(d,m) # label(replacement). [assumption]. 25.081/25.081 -->*_s0(x,e) | ->_s0(f(x),x) # label(replacement). [clausify(8)]. 25.081/25.081 -->*_s0(x,y) | ->_s0(g(d,y,x),A) # label(replacement). [clausify(9)]. 25.081/25.081 -->*_s0(x,y) | ->_s0(h(y,x),g(y,x,f(k))) # label(replacement). [clausify(10)]. 25.081/25.081 ->_s0(k,l) # label(replacement). [assumption]. 25.081/25.081 ->_s0(k,m) # label(replacement). [assumption]. 25.081/25.081 -->*_s0(x,e) # label(goal). [deny(11)]. 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 formulas(demodulators). 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 ============================== PREDICATE ELIMINATION ================= 25.081/25.081 25.081/25.081 No predicates eliminated. 25.081/25.081 25.081/25.081 ============================== end predicate elimination ============= 25.081/25.081 25.081/25.081 Auto_denials: 25.081/25.081 % copying label goal to answer in negative clause 25.081/25.081 25.081/25.081 Term ordering decisions: 25.081/25.081 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.081/25.081 Function symbol precedence: function_order([ c, d, k, e, l, m, a, b, A, h, f, g ]). 25.081/25.081 After inverse_order: (no changes). 25.081/25.081 Unfolding symbols: (none). 25.081/25.081 25.081/25.081 Auto_inference settings: 25.081/25.081 % set(neg_binary_resolution). % (HNE depth_diff=-10) 25.081/25.081 % clear(ordered_res). % (HNE depth_diff=-10) 25.081/25.081 % set(ur_resolution). % (HNE depth_diff=-10) 25.081/25.081 % set(ur_resolution) -> set(pos_ur_resolution). 25.081/25.081 % set(ur_resolution) -> set(neg_ur_resolution). 25.081/25.081 25.081/25.081 Auto_process settings: (no changes). 25.081/25.081 25.081/25.081 kept: 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.081/25.081 kept: 13 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.081/25.081 kept: 14 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.081/25.081 kept: 15 -->_s0(x,y) | ->_s0(g(x,z,u),g(y,z,u)) # label(congruence). [clausify(3)]. 25.081/25.081 kept: 16 -->_s0(x,y) | ->_s0(g(z,x,u),g(z,y,u)) # label(congruence). [clausify(4)]. 25.081/25.081 kept: 17 -->_s0(x,y) | ->_s0(g(z,u,x),g(z,u,y)) # label(congruence). [clausify(5)]. 25.081/25.081 kept: 18 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 25.081/25.081 kept: 19 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 25.081/25.081 kept: 20 ->_s0(a,c) # label(replacement). [assumption]. 25.081/25.081 kept: 21 ->_s0(a,d) # label(replacement). [assumption]. 25.081/25.081 kept: 22 ->_s0(b,c) # label(replacement). [assumption]. 25.081/25.081 kept: 23 ->_s0(b,d) # label(replacement). [assumption]. 25.081/25.081 kept: 24 ->_s0(c,e) # label(replacement). [assumption]. 25.081/25.081 kept: 25 ->_s0(c,l) # label(replacement). [assumption]. 25.081/25.081 kept: 26 ->_s0(d,m) # label(replacement). [assumption]. 25.081/25.081 kept: 27 -->*_s0(x,e) | ->_s0(f(x),x) # label(replacement). [clausify(8)]. 25.081/25.081 kept: 28 -->*_s0(x,y) | ->_s0(g(d,y,x),A) # label(replacement). [clausify(9)]. 25.081/25.081 kept: 29 -->*_s0(x,y) | ->_s0(h(y,x),g(y,x,f(k))) # label(replacement). [clausify(10)]. 25.081/25.081 kept: 30 ->_s0(k,l) # label(replacement). [assumption]. 25.081/25.081 kept: 31 ->_s0(k,m) # label(replacement). [assumption]. 25.081/25.081 kept: 32 -->*_s0(x,e) # label(goal) # answer(goal). [deny(11)]. 25.081/25.081 25.081/25.081 ============================== PROOF ================================= 25.081/25.081 25.081/25.081 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.081/25.081 % Length of proof is 4. 25.081/25.081 % Level of proof is 2. 25.081/25.081 % Maximum clause weight is 3.000. 25.081/25.081 % Given clauses 0. 25.081/25.081 25.081/25.081 11 (exists x2 ->*_s0(x2,e)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.081/25.081 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.081/25.081 32 -->*_s0(x,e) # label(goal) # answer(goal). [deny(11)]. 25.081/25.081 33 $F # answer(goal). [resolve(32,a,12,a)]. 25.081/25.081 25.081/25.081 ============================== end of proof ========================== 25.081/25.081 25.081/25.081 ============================== STATISTICS ============================ 25.081/25.081 25.081/25.081 Given=0. Generated=21. Kept=21. proofs=1. 25.081/25.081 Usable=0. Sos=0. Demods=0. Limbo=20, Disabled=21. Hints=0. 25.081/25.081 Kept_by_rule=0, Deleted_by_rule=0. 25.081/25.081 Forward_subsumed=0. Back_subsumed=0. 25.081/25.081 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.081/25.081 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.081/25.081 Demod_attempts=0. Demod_rewrites=0. 25.081/25.081 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.081/25.081 Nonunit_fsub_feature_tests=4. Nonunit_bsub_feature_tests=0. 25.081/25.081 Megabytes=0.07. 25.081/25.081 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.081/25.081 25.081/25.081 ============================== end of statistics ===================== 25.081/25.081 25.081/25.081 ============================== end of search ========================= 25.081/25.081 25.081/25.081 THEOREM PROVED 25.081/25.081 25.081/25.081 Exiting with 1 proof. 25.081/25.081 25.081/25.081 Process 39817 exit (max_proofs) Wed Jul 14 10:13:42 2021 25.081/25.081 25.081/25.081 25.081/25.081 The problem is feasible. 25.081/25.081 25.081/25.081 25.081/25.081 CRule InfChecker Info: 25.081/25.081 g(d,x:S,y:S) -> A | y:S ->* x:S 25.081/25.081 Rule remains 25.081/25.081 Proof: 25.081/25.081 NO 25.081/25.081 25.081/25.081 Problem 1: 25.081/25.081 25.081/25.081 Infeasibility Problem: 25.081/25.081 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S y:S:S) 25.081/25.081 (STRATEGY CONTEXTSENSITIVE 25.081/25.081 (a) 25.081/25.081 (b) 25.081/25.081 (c) 25.081/25.081 (d) 25.081/25.081 (f 1) 25.081/25.081 (g 1 2 3) 25.081/25.081 (h 1 2) 25.081/25.081 (k) 25.081/25.081 (A) 25.081/25.081 (e) 25.081/25.081 (fSNonEmpty) 25.081/25.081 (l) 25.081/25.081 (m) 25.081/25.081 ) 25.081/25.081 (RULES 25.081/25.081 a -> c 25.081/25.081 a -> d 25.081/25.081 b -> c 25.081/25.081 b -> d 25.081/25.081 c -> e 25.081/25.081 c -> l 25.081/25.081 d -> m 25.081/25.081 f(x:S:S) -> x:S:S | x:S:S ->* e 25.081/25.081 g(d,x:S:S,y:S:S) -> A | y:S:S ->* x:S:S 25.081/25.081 h(x:S:S,y:S:S) -> g(x:S:S,y:S:S,f(k)) | y:S:S ->* x:S:S 25.081/25.081 k -> l 25.081/25.081 k -> m 25.081/25.081 )] 25.081/25.081 25.081/25.081 Infeasibility Conditions: 25.081/25.081 y:S:S ->* x:S:S 25.081/25.081 25.081/25.081 Problem 1: 25.081/25.081 25.081/25.081 Obtaining a proof using Prover9: 25.081/25.081 25.081/25.081 -> Prover9 Output: 25.081/25.081 ============================== Prover9 =============================== 25.081/25.081 Prover9 (64) version 2009-11A, November 2009. 25.081/25.081 Process 39840 was started by ubuntu on ubuntu, 25.081/25.081 Wed Jul 14 10:13:42 2021 25.081/25.081 The command was "./prover9 -f /tmp/prover939832-0.in". 25.081/25.081 ============================== end of head =========================== 25.081/25.081 25.081/25.081 ============================== INPUT ================================= 25.081/25.081 25.081/25.081 % Reading from file /tmp/prover939832-0.in 25.081/25.081 25.081/25.081 assign(max_seconds,20). 25.081/25.081 25.081/25.081 formulas(assumptions). 25.081/25.081 ->*_s0(x,x) # label(reflexivity). 25.081/25.081 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.081/25.081 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 25.081/25.081 ->_s0(x1,y) -> ->_s0(g(x1,x2,x3),g(y,x2,x3)) # label(congruence). 25.081/25.081 ->_s0(x2,y) -> ->_s0(g(x1,x2,x3),g(x1,y,x3)) # label(congruence). 25.081/25.081 ->_s0(x3,y) -> ->_s0(g(x1,x2,x3),g(x1,x2,y)) # label(congruence). 25.081/25.081 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence). 25.081/25.081 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence). 25.081/25.081 ->_s0(a,c) # label(replacement). 25.081/25.081 ->_s0(a,d) # label(replacement). 25.081/25.081 ->_s0(b,c) # label(replacement). 25.081/25.081 ->_s0(b,d) # label(replacement). 25.081/25.081 ->_s0(c,e) # label(replacement). 25.081/25.081 ->_s0(c,l) # label(replacement). 25.081/25.081 ->_s0(d,m) # label(replacement). 25.081/25.081 ->*_s0(x2,e) -> ->_s0(f(x2),x2) # label(replacement). 25.081/25.081 ->*_s0(x3,x2) -> ->_s0(g(d,x2,x3),A) # label(replacement). 25.081/25.081 ->*_s0(x3,x2) -> ->_s0(h(x2,x3),g(x2,x3,f(k))) # label(replacement). 25.081/25.081 ->_s0(k,l) # label(replacement). 25.081/25.081 ->_s0(k,m) # label(replacement). 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 formulas(goals). 25.081/25.081 (exists x2 exists x3 ->*_s0(x3,x2)) # label(goal). 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 ============================== end of input ========================== 25.081/25.081 25.081/25.081 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.081/25.081 25.081/25.081 % Formulas that are not ordinary clauses: 25.081/25.081 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.081/25.081 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 3 ->_s0(x1,y) -> ->_s0(g(x1,x2,x3),g(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 4 ->_s0(x2,y) -> ->_s0(g(x1,x2,x3),g(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 5 ->_s0(x3,y) -> ->_s0(g(x1,x2,x3),g(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 6 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 7 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 8 ->*_s0(x2,e) -> ->_s0(f(x2),x2) # label(replacement) # label(non_clause). [assumption]. 25.081/25.081 9 ->*_s0(x3,x2) -> ->_s0(g(d,x2,x3),A) # label(replacement) # label(non_clause). [assumption]. 25.081/25.081 10 ->*_s0(x3,x2) -> ->_s0(h(x2,x3),g(x2,x3,f(k))) # label(replacement) # label(non_clause). [assumption]. 25.081/25.081 11 (exists x2 exists x3 ->*_s0(x3,x2)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.081/25.081 25.081/25.081 ============================== end of process non-clausal formulas === 25.081/25.081 25.081/25.081 ============================== PROCESS INITIAL CLAUSES =============== 25.081/25.081 25.081/25.081 % Clauses before input processing: 25.081/25.081 25.081/25.081 formulas(usable). 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 formulas(sos). 25.081/25.081 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.081/25.081 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.081/25.081 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.081/25.081 -->_s0(x,y) | ->_s0(g(x,z,u),g(y,z,u)) # label(congruence). [clausify(3)]. 25.081/25.081 -->_s0(x,y) | ->_s0(g(z,x,u),g(z,y,u)) # label(congruence). [clausify(4)]. 25.081/25.081 -->_s0(x,y) | ->_s0(g(z,u,x),g(z,u,y)) # label(congruence). [clausify(5)]. 25.081/25.081 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 25.081/25.081 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 25.081/25.081 ->_s0(a,c) # label(replacement). [assumption]. 25.081/25.081 ->_s0(a,d) # label(replacement). [assumption]. 25.081/25.081 ->_s0(b,c) # label(replacement). [assumption]. 25.081/25.081 ->_s0(b,d) # label(replacement). [assumption]. 25.081/25.081 ->_s0(c,e) # label(replacement). [assumption]. 25.081/25.081 ->_s0(c,l) # label(replacement). [assumption]. 25.081/25.081 ->_s0(d,m) # label(replacement). [assumption]. 25.081/25.081 -->*_s0(x,e) | ->_s0(f(x),x) # label(replacement). [clausify(8)]. 25.081/25.081 -->*_s0(x,y) | ->_s0(g(d,y,x),A) # label(replacement). [clausify(9)]. 25.081/25.081 -->*_s0(x,y) | ->_s0(h(y,x),g(y,x,f(k))) # label(replacement). [clausify(10)]. 25.081/25.081 ->_s0(k,l) # label(replacement). [assumption]. 25.081/25.081 ->_s0(k,m) # label(replacement). [assumption]. 25.081/25.081 -->*_s0(x,y) # label(goal). [deny(11)]. 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 formulas(demodulators). 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 ============================== PREDICATE ELIMINATION ================= 25.081/25.081 25.081/25.081 No predicates eliminated. 25.081/25.081 25.081/25.081 ============================== end predicate elimination ============= 25.081/25.081 25.081/25.081 Auto_denials: 25.081/25.081 % copying label goal to answer in negative clause 25.081/25.081 25.081/25.081 Term ordering decisions: 25.081/25.081 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.081/25.081 Function symbol precedence: function_order([ c, d, k, e, l, m, a, b, A, h, f, g ]). 25.081/25.081 After inverse_order: (no changes). 25.081/25.081 Unfolding symbols: (none). 25.081/25.081 25.081/25.081 Auto_inference settings: 25.081/25.081 % set(neg_binary_resolution). % (HNE depth_diff=-10) 25.081/25.081 % clear(ordered_res). % (HNE depth_diff=-10) 25.081/25.081 % set(ur_resolution). % (HNE depth_diff=-10) 25.081/25.081 % set(ur_resolution) -> set(pos_ur_resolution). 25.081/25.081 % set(ur_resolution) -> set(neg_ur_resolution). 25.081/25.081 25.081/25.081 Auto_process settings: (no changes). 25.081/25.081 25.081/25.081 kept: 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.081/25.081 kept: 13 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.081/25.081 kept: 14 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.081/25.081 kept: 15 -->_s0(x,y) | ->_s0(g(x,z,u),g(y,z,u)) # label(congruence). [clausify(3)]. 25.081/25.081 kept: 16 -->_s0(x,y) | ->_s0(g(z,x,u),g(z,y,u)) # label(congruence). [clausify(4)]. 25.081/25.081 kept: 17 -->_s0(x,y) | ->_s0(g(z,u,x),g(z,u,y)) # label(congruence). [clausify(5)]. 25.081/25.081 kept: 18 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 25.081/25.081 kept: 19 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 25.081/25.081 kept: 20 ->_s0(a,c) # label(replacement). [assumption]. 25.081/25.081 kept: 21 ->_s0(a,d) # label(replacement). [assumption]. 25.081/25.081 kept: 22 ->_s0(b,c) # label(replacement). [assumption]. 25.081/25.081 kept: 23 ->_s0(b,d) # label(replacement). [assumption]. 25.081/25.081 kept: 24 ->_s0(c,e) # label(replacement). [assumption]. 25.081/25.081 kept: 25 ->_s0(c,l) # label(replacement). [assumption]. 25.081/25.081 kept: 26 ->_s0(d,m) # label(replacement). [assumption]. 25.081/25.081 kept: 27 -->*_s0(x,e) | ->_s0(f(x),x) # label(replacement). [clausify(8)]. 25.081/25.081 kept: 28 -->*_s0(x,y) | ->_s0(g(d,y,x),A) # label(replacement). [clausify(9)]. 25.081/25.081 kept: 29 -->*_s0(x,y) | ->_s0(h(y,x),g(y,x,f(k))) # label(replacement). [clausify(10)]. 25.081/25.081 kept: 30 ->_s0(k,l) # label(replacement). [assumption]. 25.081/25.081 kept: 31 ->_s0(k,m) # label(replacement). [assumption]. 25.081/25.081 kept: 32 -->*_s0(x,y) # label(goal) # answer(goal). [deny(11)]. 25.081/25.081 25.081/25.081 ============================== PROOF ================================= 25.081/25.081 25.081/25.081 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.081/25.081 % Length of proof is 4. 25.081/25.081 % Level of proof is 2. 25.081/25.081 % Maximum clause weight is 3.000. 25.081/25.081 % Given clauses 0. 25.081/25.081 25.081/25.081 11 (exists x2 exists x3 ->*_s0(x3,x2)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.081/25.081 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.081/25.081 32 -->*_s0(x,y) # label(goal) # answer(goal). [deny(11)]. 25.081/25.081 33 $F # answer(goal). [resolve(32,a,12,a)]. 25.081/25.081 25.081/25.081 ============================== end of proof ========================== 25.081/25.081 25.081/25.081 ============================== STATISTICS ============================ 25.081/25.081 25.081/25.081 Given=0. Generated=21. Kept=21. proofs=1. 25.081/25.081 Usable=0. Sos=0. Demods=0. Limbo=20, Disabled=21. Hints=0. 25.081/25.081 Kept_by_rule=0, Deleted_by_rule=0. 25.081/25.081 Forward_subsumed=0. Back_subsumed=0. 25.081/25.081 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.081/25.081 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.081/25.081 Demod_attempts=0. Demod_rewrites=0. 25.081/25.081 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.081/25.081 Nonunit_fsub_feature_tests=4. Nonunit_bsub_feature_tests=0. 25.081/25.081 Megabytes=0.07. 25.081/25.081 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.081/25.081 25.081/25.081 ============================== end of statistics ===================== 25.081/25.081 25.081/25.081 ============================== end of search ========================= 25.081/25.081 25.081/25.081 THEOREM PROVED 25.081/25.081 25.081/25.081 Exiting with 1 proof. 25.081/25.081 25.081/25.081 Process 39840 exit (max_proofs) Wed Jul 14 10:13:42 2021 25.081/25.081 25.081/25.081 25.081/25.081 The problem is feasible. 25.081/25.081 25.081/25.081 25.081/25.081 CRule InfChecker Info: 25.081/25.081 h(x:S,y:S) -> g(x:S,y:S,f(k)) | y:S ->* x:S 25.081/25.081 Rule remains 25.081/25.081 Proof: 25.081/25.081 NO 25.081/25.081 25.081/25.081 Problem 1: 25.081/25.081 25.081/25.081 Infeasibility Problem: 25.081/25.081 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S y:S:S) 25.081/25.081 (STRATEGY CONTEXTSENSITIVE 25.081/25.081 (a) 25.081/25.081 (b) 25.081/25.081 (c) 25.081/25.081 (d) 25.081/25.081 (f 1) 25.081/25.081 (g 1 2 3) 25.081/25.081 (h 1 2) 25.081/25.081 (k) 25.081/25.081 (A) 25.081/25.081 (e) 25.081/25.081 (fSNonEmpty) 25.081/25.081 (l) 25.081/25.081 (m) 25.081/25.081 ) 25.081/25.081 (RULES 25.081/25.081 a -> c 25.081/25.081 a -> d 25.081/25.081 b -> c 25.081/25.081 b -> d 25.081/25.081 c -> e 25.081/25.081 c -> l 25.081/25.081 d -> m 25.081/25.081 f(x:S:S) -> x:S:S | x:S:S ->* e 25.081/25.081 g(d,x:S:S,y:S:S) -> A | y:S:S ->* x:S:S 25.081/25.081 h(x:S:S,y:S:S) -> g(x:S:S,y:S:S,f(k)) | y:S:S ->* x:S:S 25.081/25.081 k -> l 25.081/25.081 k -> m 25.081/25.081 )] 25.081/25.081 25.081/25.081 Infeasibility Conditions: 25.081/25.081 y:S:S ->* x:S:S 25.081/25.081 25.081/25.081 Problem 1: 25.081/25.081 25.081/25.081 Obtaining a proof using Prover9: 25.081/25.081 25.081/25.081 -> Prover9 Output: 25.081/25.081 ============================== Prover9 =============================== 25.081/25.081 Prover9 (64) version 2009-11A, November 2009. 25.081/25.081 Process 39861 was started by ubuntu on ubuntu, 25.081/25.081 Wed Jul 14 10:13:42 2021 25.081/25.081 The command was "./prover9 -f /tmp/prover939854-0.in". 25.081/25.081 ============================== end of head =========================== 25.081/25.081 25.081/25.081 ============================== INPUT ================================= 25.081/25.081 25.081/25.081 % Reading from file /tmp/prover939854-0.in 25.081/25.081 25.081/25.081 assign(max_seconds,20). 25.081/25.081 25.081/25.081 formulas(assumptions). 25.081/25.081 ->*_s0(x,x) # label(reflexivity). 25.081/25.081 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.081/25.081 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 25.081/25.081 ->_s0(x1,y) -> ->_s0(g(x1,x2,x3),g(y,x2,x3)) # label(congruence). 25.081/25.081 ->_s0(x2,y) -> ->_s0(g(x1,x2,x3),g(x1,y,x3)) # label(congruence). 25.081/25.081 ->_s0(x3,y) -> ->_s0(g(x1,x2,x3),g(x1,x2,y)) # label(congruence). 25.081/25.081 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence). 25.081/25.081 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence). 25.081/25.081 ->_s0(a,c) # label(replacement). 25.081/25.081 ->_s0(a,d) # label(replacement). 25.081/25.081 ->_s0(b,c) # label(replacement). 25.081/25.081 ->_s0(b,d) # label(replacement). 25.081/25.081 ->_s0(c,e) # label(replacement). 25.081/25.081 ->_s0(c,l) # label(replacement). 25.081/25.081 ->_s0(d,m) # label(replacement). 25.081/25.081 ->*_s0(x2,e) -> ->_s0(f(x2),x2) # label(replacement). 25.081/25.081 ->*_s0(x3,x2) -> ->_s0(g(d,x2,x3),A) # label(replacement). 25.081/25.081 ->*_s0(x3,x2) -> ->_s0(h(x2,x3),g(x2,x3,f(k))) # label(replacement). 25.081/25.081 ->_s0(k,l) # label(replacement). 25.081/25.081 ->_s0(k,m) # label(replacement). 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 formulas(goals). 25.081/25.081 (exists x2 exists x3 ->*_s0(x3,x2)) # label(goal). 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 ============================== end of input ========================== 25.081/25.081 25.081/25.081 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.081/25.081 25.081/25.081 % Formulas that are not ordinary clauses: 25.081/25.081 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.081/25.081 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 3 ->_s0(x1,y) -> ->_s0(g(x1,x2,x3),g(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 4 ->_s0(x2,y) -> ->_s0(g(x1,x2,x3),g(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 5 ->_s0(x3,y) -> ->_s0(g(x1,x2,x3),g(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 6 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 7 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 8 ->*_s0(x2,e) -> ->_s0(f(x2),x2) # label(replacement) # label(non_clause). [assumption]. 25.081/25.081 9 ->*_s0(x3,x2) -> ->_s0(g(d,x2,x3),A) # label(replacement) # label(non_clause). [assumption]. 25.081/25.081 10 ->*_s0(x3,x2) -> ->_s0(h(x2,x3),g(x2,x3,f(k))) # label(replacement) # label(non_clause). [assumption]. 25.081/25.081 11 (exists x2 exists x3 ->*_s0(x3,x2)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.081/25.081 25.081/25.081 ============================== end of process non-clausal formulas === 25.081/25.081 25.081/25.081 ============================== PROCESS INITIAL CLAUSES =============== 25.081/25.081 25.081/25.081 % Clauses before input processing: 25.081/25.081 25.081/25.081 formulas(usable). 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 formulas(sos). 25.081/25.081 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.081/25.081 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.081/25.081 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.081/25.081 -->_s0(x,y) | ->_s0(g(x,z,u),g(y,z,u)) # label(congruence). [clausify(3)]. 25.081/25.081 -->_s0(x,y) | ->_s0(g(z,x,u),g(z,y,u)) # label(congruence). [clausify(4)]. 25.081/25.081 -->_s0(x,y) | ->_s0(g(z,u,x),g(z,u,y)) # label(congruence). [clausify(5)]. 25.081/25.081 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 25.081/25.081 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 25.081/25.081 ->_s0(a,c) # label(replacement). [assumption]. 25.081/25.081 ->_s0(a,d) # label(replacement). [assumption]. 25.081/25.081 ->_s0(b,c) # label(replacement). [assumption]. 25.081/25.081 ->_s0(b,d) # label(replacement). [assumption]. 25.081/25.081 ->_s0(c,e) # label(replacement). [assumption]. 25.081/25.081 ->_s0(c,l) # label(replacement). [assumption]. 25.081/25.081 ->_s0(d,m) # label(replacement). [assumption]. 25.081/25.081 -->*_s0(x,e) | ->_s0(f(x),x) # label(replacement). [clausify(8)]. 25.081/25.081 -->*_s0(x,y) | ->_s0(g(d,y,x),A) # label(replacement). [clausify(9)]. 25.081/25.081 -->*_s0(x,y) | ->_s0(h(y,x),g(y,x,f(k))) # label(replacement). [clausify(10)]. 25.081/25.081 ->_s0(k,l) # label(replacement). [assumption]. 25.081/25.081 ->_s0(k,m) # label(replacement). [assumption]. 25.081/25.081 -->*_s0(x,y) # label(goal). [deny(11)]. 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 formulas(demodulators). 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 ============================== PREDICATE ELIMINATION ================= 25.081/25.081 25.081/25.081 No predicates eliminated. 25.081/25.081 25.081/25.081 ============================== end predicate elimination ============= 25.081/25.081 25.081/25.081 Auto_denials: 25.081/25.081 % copying label goal to answer in negative clause 25.081/25.081 25.081/25.081 Term ordering decisions: 25.081/25.081 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.081/25.081 Function symbol precedence: function_order([ c, d, k, e, l, m, a, b, A, h, f, g ]). 25.081/25.081 After inverse_order: (no changes). 25.081/25.081 Unfolding symbols: (none). 25.081/25.081 25.081/25.081 Auto_inference settings: 25.081/25.081 % set(neg_binary_resolution). % (HNE depth_diff=-10) 25.081/25.081 % clear(ordered_res). % (HNE depth_diff=-10) 25.081/25.081 % set(ur_resolution). % (HNE depth_diff=-10) 25.081/25.081 % set(ur_resolution) -> set(pos_ur_resolution). 25.081/25.081 % set(ur_resolution) -> set(neg_ur_resolution). 25.081/25.081 25.081/25.081 Auto_process settings: (no changes). 25.081/25.081 25.081/25.081 kept: 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.081/25.081 kept: 13 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.081/25.081 kept: 14 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.081/25.081 kept: 15 -->_s0(x,y) | ->_s0(g(x,z,u),g(y,z,u)) # label(congruence). [clausify(3)]. 25.081/25.081 kept: 16 -->_s0(x,y) | ->_s0(g(z,x,u),g(z,y,u)) # label(congruence). [clausify(4)]. 25.081/25.081 kept: 17 -->_s0(x,y) | ->_s0(g(z,u,x),g(z,u,y)) # label(congruence). [clausify(5)]. 25.081/25.081 kept: 18 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 25.081/25.081 kept: 19 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 25.081/25.081 kept: 20 ->_s0(a,c) # label(replacement). [assumption]. 25.081/25.081 kept: 21 ->_s0(a,d) # label(replacement). [assumption]. 25.081/25.081 kept: 22 ->_s0(b,c) # label(replacement). [assumption]. 25.081/25.081 kept: 23 ->_s0(b,d) # label(replacement). [assumption]. 25.081/25.081 kept: 24 ->_s0(c,e) # label(replacement). [assumption]. 25.081/25.081 kept: 25 ->_s0(c,l) # label(replacement). [assumption]. 25.081/25.081 kept: 26 ->_s0(d,m) # label(replacement). [assumption]. 25.081/25.081 kept: 27 -->*_s0(x,e) | ->_s0(f(x),x) # label(replacement). [clausify(8)]. 25.081/25.081 kept: 28 -->*_s0(x,y) | ->_s0(g(d,y,x),A) # label(replacement). [clausify(9)]. 25.081/25.081 kept: 29 -->*_s0(x,y) | ->_s0(h(y,x),g(y,x,f(k))) # label(replacement). [clausify(10)]. 25.081/25.081 kept: 30 ->_s0(k,l) # label(replacement). [assumption]. 25.081/25.081 kept: 31 ->_s0(k,m) # label(replacement). [assumption]. 25.081/25.081 kept: 32 -->*_s0(x,y) # label(goal) # answer(goal). [deny(11)]. 25.081/25.081 25.081/25.081 ============================== PROOF ================================= 25.081/25.081 25.081/25.081 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.081/25.081 % Length of proof is 4. 25.081/25.081 % Level of proof is 2. 25.081/25.081 % Maximum clause weight is 3.000. 25.081/25.081 % Given clauses 0. 25.081/25.081 25.081/25.081 11 (exists x2 exists x3 ->*_s0(x3,x2)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.081/25.081 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.081/25.081 32 -->*_s0(x,y) # label(goal) # answer(goal). [deny(11)]. 25.081/25.081 33 $F # answer(goal). [resolve(32,a,12,a)]. 25.081/25.081 25.081/25.081 ============================== end of proof ========================== 25.081/25.081 25.081/25.081 ============================== STATISTICS ============================ 25.081/25.081 25.081/25.081 Given=0. Generated=21. Kept=21. proofs=1. 25.081/25.081 Usable=0. Sos=0. Demods=0. Limbo=20, Disabled=21. Hints=0. 25.081/25.081 Kept_by_rule=0, Deleted_by_rule=0. 25.081/25.081 Forward_subsumed=0. Back_subsumed=0. 25.081/25.081 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.081/25.081 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.081/25.081 Demod_attempts=0. Demod_rewrites=0. 25.081/25.081 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.081/25.081 Nonunit_fsub_feature_tests=4. Nonunit_bsub_feature_tests=0. 25.081/25.081 Megabytes=0.07. 25.081/25.081 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.081/25.081 25.081/25.081 ============================== end of statistics ===================== 25.081/25.081 25.081/25.081 ============================== end of search ========================= 25.081/25.081 25.081/25.081 THEOREM PROVED 25.081/25.081 25.081/25.081 Exiting with 1 proof. 25.081/25.081 25.081/25.081 Process 39861 exit (max_proofs) Wed Jul 14 10:13:42 2021 25.081/25.081 25.081/25.081 25.081/25.081 The problem is feasible. 25.081/25.081 25.081/25.081 25.081/25.081 CRule InfChecker Info: 25.081/25.081 k -> l 25.081/25.081 Rule remains 25.081/25.081 Proof: 25.081/25.081 NO_CONDS 25.081/25.081 25.081/25.081 CRule InfChecker Info: 25.081/25.081 k -> m 25.081/25.081 Rule remains 25.081/25.081 Proof: 25.081/25.081 NO_CONDS 25.081/25.081 25.081/25.081 Problem 1: 25.081/25.081 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.081/25.081 Confluence Problem: 25.081/25.081 (VAR x:S y:S) 25.081/25.081 (STRATEGY CONTEXTSENSITIVE 25.081/25.081 (a) 25.081/25.081 (b) 25.081/25.081 (c) 25.081/25.081 (d) 25.081/25.081 (f 1) 25.081/25.081 (g 1 2 3) 25.081/25.081 (h 1 2) 25.081/25.081 (k) 25.081/25.081 (A) 25.081/25.081 (e) 25.081/25.081 (fSNonEmpty) 25.081/25.081 (l) 25.081/25.081 (m) 25.081/25.081 ) 25.081/25.081 (RULES 25.081/25.081 a -> c 25.081/25.081 a -> d 25.081/25.081 b -> c 25.081/25.081 b -> d 25.081/25.081 c -> e 25.081/25.081 c -> l 25.081/25.081 d -> m 25.081/25.081 f(x:S) -> x:S | x:S ->* e 25.081/25.081 g(d,x:S,y:S) -> A | y:S ->* x:S 25.081/25.081 h(x:S,y:S) -> g(x:S,y:S,f(k)) | y:S ->* x:S 25.081/25.081 k -> l 25.081/25.081 k -> m 25.081/25.081 ) 25.081/25.081 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.081/25.081 25.081/25.081 Critical Pairs Processor: 25.081/25.081 -> Rules: 25.081/25.081 a -> c 25.081/25.081 a -> d 25.081/25.081 b -> c 25.081/25.081 b -> d 25.081/25.081 c -> e 25.081/25.081 c -> l 25.081/25.081 d -> m 25.081/25.081 f(x:S) -> x:S | x:S ->* e 25.081/25.081 g(d,x:S,y:S) -> A | y:S ->* x:S 25.081/25.081 h(x:S,y:S) -> g(x:S,y:S,f(k)) | y:S ->* x:S 25.081/25.081 k -> l 25.081/25.081 k -> m 25.081/25.081 -> Vars: 25.081/25.081 "x", "x", "y", "x", "y" 25.081/25.081 -> FVars: 25.081/25.081 "x3", "x4", "x5", "x6", "x7" 25.081/25.081 -> PVars: 25.081/25.081 "x": ["x3", "x4", "x6"], "y": ["x5", "x7"] 25.081/25.081 25.081/25.081 -> Rlps: 25.081/25.081 crule: a -> c, id: 1, possubterms: a-> [] 25.081/25.081 crule: a -> d, id: 2, possubterms: a-> [] 25.081/25.081 crule: b -> c, id: 3, possubterms: b-> [] 25.081/25.081 crule: b -> d, id: 4, possubterms: b-> [] 25.081/25.081 crule: c -> e, id: 5, possubterms: c-> [] 25.081/25.081 crule: c -> l, id: 6, possubterms: c-> [] 25.081/25.081 crule: d -> m, id: 7, possubterms: d-> [] 25.081/25.081 crule: f(x3:S) -> x3:S | x3:S ->* e, id: 8, possubterms: f(x3:S)-> [] 25.081/25.081 crule: g(d,x4:S,x5:S) -> A | x5:S ->* x4:S, id: 9, possubterms: g(d,x4:S,x5:S)-> [], d-> [1] 25.081/25.081 crule: h(x6:S,x7:S) -> g(x6:S,x7:S,f(k)) | x7:S ->* x6:S, id: 10, possubterms: h(x6:S,x7:S)-> [] 25.081/25.081 crule: k -> l, id: 11, possubterms: k-> [] 25.081/25.081 crule: k -> m, id: 12, possubterms: k-> [] 25.081/25.081 25.081/25.081 -> Unifications: 25.081/25.081 R2 unifies with R1 at p: [], l: a, lp: a, conds: {}, sig: {}, l': a, r: d, r': c 25.081/25.081 R4 unifies with R3 at p: [], l: b, lp: b, conds: {}, sig: {}, l': b, r: d, r': c 25.081/25.081 R6 unifies with R5 at p: [], l: c, lp: c, conds: {}, sig: {}, l': c, r: l, r': e 25.081/25.081 R9 unifies with R7 at p: [1], l: g(d,x4:S,x5:S), lp: d, conds: {x5:S ->* x4:S}, sig: {}, l': d, r: A, r': m 25.081/25.081 R12 unifies with R11 at p: [], l: k, lp: k, conds: {}, sig: {}, l': k, r: m, r': l 25.081/25.081 25.081/25.081 -> Critical pairs info: 25.081/25.081 => Not trivial, Overlay, N1 25.081/25.081 => Not trivial, Overlay, N2 25.081/25.081 => Not trivial, Overlay, N3 25.081/25.081 | y:S ->* x:S => Not trivial, Not overlay, N4 25.081/25.081 25.081/25.081 -> Problem conclusions: 25.081/25.081 Left linear, Right linear, Linear 25.081/25.081 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 25.081/25.081 CTRS Type: 1 25.081/25.081 Deterministic, Maybe strongly deterministic 25.081/25.081 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.081/25.081 Maybe right-stable CTRS, Not overlay CTRS 25.081/25.081 Maybe normal CTRS, Maybe almost normal CTRS 25.081/25.081 Maybe terminating CTRS, Maybe joinable CCPs 25.081/25.081 Maybe level confluent 25.081/25.081 Maybe confluent 25.081/25.081 25.081/25.081 Problem 1: 25.081/25.081 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.081/25.081 Confluence Problem: 25.081/25.081 (VAR x:S y:S) 25.081/25.081 (STRATEGY CONTEXTSENSITIVE 25.081/25.081 (a) 25.081/25.081 (b) 25.081/25.081 (c) 25.081/25.081 (d) 25.081/25.081 (f 1) 25.081/25.081 (g 1 2 3) 25.081/25.081 (h 1 2) 25.081/25.081 (k) 25.081/25.081 (A) 25.081/25.081 (e) 25.081/25.081 (fSNonEmpty) 25.081/25.081 (l) 25.081/25.081 (m) 25.081/25.081 ) 25.081/25.081 (RULES 25.081/25.081 a -> c 25.081/25.081 a -> d 25.081/25.081 b -> c 25.081/25.081 b -> d 25.081/25.081 c -> e 25.081/25.081 c -> l 25.081/25.081 d -> m 25.081/25.081 f(x:S) -> x:S | x:S ->* e 25.081/25.081 g(d,x:S,y:S) -> A | y:S ->* x:S 25.081/25.081 h(x:S,y:S) -> g(x:S,y:S,f(k)) | y:S ->* x:S 25.081/25.081 k -> l 25.081/25.081 k -> m 25.081/25.081 ) 25.081/25.081 Critical Pairs: 25.081/25.081 => Not trivial, Overlay, N1 25.081/25.081 => Not trivial, Overlay, N2 25.081/25.081 => Not trivial, Overlay, N3 25.081/25.081 | y:S ->* x:S => Not trivial, Not overlay, N4 25.081/25.081 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.081/25.081 25.081/25.081 Conditional Critical Pairs Distributor Processor 25.081/25.081 25.081/25.081 Problem 1: 25.081/25.081 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.081/25.081 Confluence Problem: 25.081/25.081 (VAR x:S y:S) 25.081/25.081 (STRATEGY CONTEXTSENSITIVE 25.081/25.081 (a) 25.081/25.081 (b) 25.081/25.081 (c) 25.081/25.081 (d) 25.081/25.081 (f 1) 25.081/25.081 (g 1 2 3) 25.081/25.081 (h 1 2) 25.081/25.081 (k) 25.081/25.081 (A) 25.081/25.081 (e) 25.081/25.081 (fSNonEmpty) 25.081/25.081 (l) 25.081/25.081 (m) 25.081/25.081 ) 25.081/25.081 (RULES 25.081/25.081 a -> c 25.081/25.081 a -> d 25.081/25.081 b -> c 25.081/25.081 b -> d 25.081/25.081 c -> e 25.081/25.081 c -> l 25.081/25.081 d -> m 25.081/25.081 f(x:S) -> x:S | x:S ->* e 25.081/25.081 g(d,x:S,y:S) -> A | y:S ->* x:S 25.081/25.081 h(x:S,y:S) -> g(x:S,y:S,f(k)) | y:S ->* x:S 25.081/25.081 k -> l 25.081/25.081 k -> m 25.081/25.081 ) 25.081/25.081 Critical Pairs: 25.081/25.081 => Not trivial, Overlay, N1 25.081/25.081 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.081/25.081 25.081/25.081 Infeasible Convergence No Conditions CCP Processor: 25.081/25.081 Infeasible convergence? 25.081/25.081 YES 25.081/25.081 25.081/25.081 Problem 1: 25.081/25.081 25.081/25.081 Infeasibility Problem: 25.081/25.081 [(VAR vNonEmpty:S x:S y:S z:S) 25.081/25.081 (STRATEGY CONTEXTSENSITIVE 25.081/25.081 (a) 25.081/25.081 (b) 25.081/25.081 (c) 25.081/25.081 (d) 25.081/25.081 (f 1) 25.081/25.081 (g 1 2 3) 25.081/25.081 (h 1 2) 25.081/25.081 (k) 25.081/25.081 (A) 25.081/25.081 (e) 25.081/25.081 (fSNonEmpty) 25.081/25.081 (l) 25.081/25.081 (m) 25.081/25.081 ) 25.081/25.081 (RULES 25.081/25.081 a -> c 25.081/25.081 a -> d 25.081/25.081 b -> c 25.081/25.081 b -> d 25.081/25.081 c -> e 25.081/25.081 c -> l 25.081/25.081 d -> m 25.081/25.081 f(x:S) -> x:S | x:S ->* e 25.081/25.081 g(d,x:S,y:S) -> A | y:S ->* x:S 25.081/25.081 h(x:S,y:S) -> g(x:S,y:S,f(k)) | y:S ->* x:S 25.081/25.081 k -> l 25.081/25.081 k -> m 25.081/25.081 )] 25.081/25.081 25.081/25.081 Infeasibility Conditions: 25.081/25.081 l ->* z:S, m ->* z:S 25.081/25.081 25.081/25.081 Problem 1: 25.081/25.081 25.081/25.081 Obtaining a model using Mace4: 25.081/25.081 25.081/25.081 -> Usable Rules: 25.081/25.081 Empty 25.081/25.081 25.081/25.081 -> Mace4 Output: 25.081/25.081 ============================== Mace4 ================================= 25.081/25.081 Mace4 (64) version 2009-11A, November 2009. 25.081/25.081 Process 39910 was started by ubuntu on ubuntu, 25.081/25.081 Wed Jul 14 10:13:42 2021 25.081/25.081 The command was "./mace4 -c -f /tmp/mace439882-2.in". 25.081/25.081 ============================== end of head =========================== 25.081/25.081 25.081/25.081 ============================== INPUT ================================= 25.081/25.081 25.081/25.081 % Reading from file /tmp/mace439882-2.in 25.081/25.081 25.081/25.081 assign(max_seconds,20). 25.081/25.081 25.081/25.081 formulas(assumptions). 25.081/25.081 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 25.081/25.081 ->(x1,y) -> ->(g(x1,x2,x3),g(y,x2,x3)) # label(congruence). 25.081/25.081 ->(x2,y) -> ->(g(x1,x2,x3),g(x1,y,x3)) # label(congruence). 25.081/25.081 ->(x3,y) -> ->(g(x1,x2,x3),g(x1,x2,y)) # label(congruence). 25.081/25.081 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence). 25.081/25.081 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence). 25.081/25.081 ->*(x,x) # label(reflexivity). 25.081/25.081 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 formulas(goals). 25.081/25.081 (exists x3 (->*(l,x3) & ->*(m,x3))) # label(goal). 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 ============================== end of input ========================== 25.081/25.081 25.081/25.081 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.081/25.081 25.081/25.081 % Formulas that are not ordinary clauses: 25.081/25.081 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 2 ->(x1,y) -> ->(g(x1,x2,x3),g(y,x2,x3)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 3 ->(x2,y) -> ->(g(x1,x2,x3),g(x1,y,x3)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 4 ->(x3,y) -> ->(g(x1,x2,x3),g(x1,x2,y)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 5 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 6 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.081/25.081 7 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.081/25.081 8 (exists x3 (->*(l,x3) & ->*(m,x3))) # label(goal) # label(non_clause) # label(goal). [goal]. 25.081/25.081 25.081/25.081 ============================== end of process non-clausal formulas === 25.081/25.081 25.081/25.081 ============================== CLAUSES FOR SEARCH ==================== 25.081/25.081 25.081/25.081 formulas(mace4_clauses). 25.081/25.081 -->(x,y) | ->(f(x),f(y)) # label(congruence). 25.081/25.081 -->(x,y) | ->(g(x,z,u),g(y,z,u)) # label(congruence). 25.081/25.081 -->(x,y) | ->(g(z,x,u),g(z,y,u)) # label(congruence). 25.081/25.081 -->(x,y) | ->(g(z,u,x),g(z,u,y)) # label(congruence). 25.081/25.081 -->(x,y) | ->(h(x,z),h(y,z)) # label(congruence). 25.081/25.081 -->(x,y) | ->(h(z,x),h(z,y)) # label(congruence). 25.081/25.081 ->*(x,x) # label(reflexivity). 25.081/25.081 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 25.081/25.081 -->*(l,x) | -->*(m,x) # label(goal). 25.081/25.081 end_of_list. 25.081/25.081 25.081/25.081 ============================== end of clauses for search ============= 25.081/25.081 25.081/25.081 % There are no natural numbers in the input. 25.081/25.081 25.081/25.081 ============================== DOMAIN SIZE 2 ========================= 25.081/25.081 25.081/25.081 ============================== MODEL ================================= 25.081/25.081 25.081/25.081 interpretation( 2, [number=1, seconds=0], [ 25.081/25.081 25.081/25.081 function(l, [ 0 ]), 25.081/25.081 25.081/25.081 function(m, [ 1 ]), 25.081/25.081 25.081/25.081 function(f(_), [ 0, 0 ]), 25.081/25.081 25.081/25.081 function(h(_,_), [ 25.081/25.081 0, 0, 25.081/25.081 0, 0 ]), 25.081/25.081 25.081/25.081 function(g(_,_,_), [ 25.081/25.081 0, 0, 25.081/25.081 0, 0, 25.081/25.081 0, 0, 25.081/25.081 0, 0 ]), 25.081/25.081 25.081/25.081 relation(->*(_,_), [ 25.081/25.081 1, 0, 25.081/25.081 0, 1 ]), 25.081/25.081 25.081/25.081 relation(->(_,_), [ 25.081/25.081 0, 0, 25.081/25.081 0, 0 ]) 25.081/25.081 ]). 25.081/25.081 25.081/25.081 ============================== end of model ========================== 25.081/25.081 25.081/25.081 ============================== STATISTICS ============================ 25.081/25.081 25.081/25.081 For domain size 2. 25.081/25.081 25.081/25.081 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 25.081/25.081 Ground clauses: seen=80, kept=76. 25.081/25.081 Selections=16, assignments=16, propagations=8, current_models=1. 25.081/25.081 Rewrite_terms=140, rewrite_bools=89, indexes=8. 25.081/25.081 Rules_from_neg_clauses=1, cross_offs=1. 25.081/25.081 25.081/25.081 ============================== end of statistics ===================== 25.081/25.081 25.081/25.081 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.081/25.081 25.081/25.081 Exiting with 1 model. 25.081/25.081 25.081/25.081 Process 39910 exit (max_models) Wed Jul 14 10:13:42 2021 25.081/25.081 The process finished Wed Jul 14 10:13:42 2021 25.081/25.081 25.081/25.081 25.081/25.081 Mace4 cooked interpretation: 25.081/25.081 25.081/25.081 25.081/25.081 25.081/25.081 The problem is infeasible. 25.081/25.081 25.081/25.081 25.081/25.081 The problem is not joinable. 25.081/25.081 29.11user 1.41system 0:25.81elapsed 118%CPU (0avgtext+0avgdata 116632maxresident)k 25.081/25.081 0inputs+0outputs (0major+95630minor)pagefaults 0swaps