25.056/25.056 NO 25.056/25.056 25.056/25.056 Problem 1: 25.056/25.056 25.056/25.056 25.056/25.056 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.056/25.056 Confluence Problem: 25.056/25.056 (VAR vNonEmpty:S x:S y:S) 25.056/25.056 (STRATEGY CONTEXTSENSITIVE 25.056/25.056 (a) 25.056/25.056 (b) 25.056/25.056 (f 1 2) 25.056/25.056 (g 1 2) 25.056/25.056 (s 1) 25.056/25.056 (c) 25.056/25.056 (d) 25.056/25.056 (fSNonEmpty) 25.056/25.056 (h 1 2) 25.056/25.056 (k) 25.056/25.056 (l) 25.056/25.056 (t 1) 25.056/25.056 ) 25.056/25.056 (RULES 25.056/25.056 a -> c 25.056/25.056 a -> d 25.056/25.056 b -> c 25.056/25.056 b -> d 25.056/25.056 f(x:S,y:S) -> y:S | s(x:S) ->* t(y:S) 25.056/25.056 g(x:S,x:S) -> h(x:S,x:S) 25.056/25.056 s(k) -> t(a) 25.056/25.056 s(l) -> t(a) 25.056/25.056 ) 25.056/25.056 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.056/25.056 25.056/25.056 25.056/25.056 Problem 1: 25.056/25.056 25.056/25.056 Inlining of Conditions Processor [STERN17]: 25.056/25.056 25.056/25.056 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.056/25.056 Confluence Problem: 25.056/25.056 (VAR vNonEmpty:S x:S y:S) 25.056/25.056 (STRATEGY CONTEXTSENSITIVE 25.056/25.056 (a) 25.056/25.056 (b) 25.056/25.056 (f 1 2) 25.056/25.056 (g 1 2) 25.056/25.056 (s 1) 25.056/25.056 (c) 25.056/25.056 (d) 25.056/25.056 (fSNonEmpty) 25.056/25.056 (h 1 2) 25.056/25.056 (k) 25.056/25.056 (l) 25.056/25.056 (t 1) 25.056/25.056 ) 25.056/25.056 (RULES 25.056/25.056 a -> c 25.056/25.056 a -> d 25.056/25.056 b -> c 25.056/25.056 b -> d 25.056/25.056 f(x:S,y:S) -> y:S | s(x:S) ->* t(y:S) 25.056/25.056 g(x:S,x:S) -> h(x:S,x:S) 25.056/25.056 s(k) -> t(a) 25.056/25.056 s(l) -> t(a) 25.056/25.056 ) 25.056/25.056 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.056/25.056 25.056/25.056 25.056/25.056 Problem 1: 25.056/25.056 25.056/25.056 Clean CTRS Processor: 25.056/25.056 25.056/25.056 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.056/25.056 Confluence Problem: 25.056/25.056 (VAR x:S y:S) 25.056/25.056 (STRATEGY CONTEXTSENSITIVE 25.056/25.056 (a) 25.056/25.056 (b) 25.056/25.056 (f 1 2) 25.056/25.056 (g 1 2) 25.056/25.056 (s 1) 25.056/25.056 (c) 25.056/25.056 (d) 25.056/25.056 (fSNonEmpty) 25.056/25.056 (h 1 2) 25.056/25.056 (k) 25.056/25.056 (l) 25.056/25.056 (t 1) 25.056/25.056 ) 25.056/25.056 (RULES 25.056/25.056 a -> c 25.056/25.056 a -> d 25.056/25.056 b -> c 25.056/25.056 b -> d 25.056/25.056 f(x:S,y:S) -> y:S | s(x:S) ->* t(y:S) 25.056/25.056 g(x:S,x:S) -> h(x:S,x:S) 25.056/25.056 s(k) -> t(a) 25.056/25.056 s(l) -> t(a) 25.056/25.056 ) 25.056/25.056 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.056/25.056 25.056/25.056 CRule InfChecker Info: 25.056/25.056 a -> c 25.056/25.056 Rule remains 25.056/25.056 Proof: 25.056/25.056 NO_CONDS 25.056/25.056 25.056/25.056 CRule InfChecker Info: 25.056/25.056 a -> d 25.056/25.056 Rule remains 25.056/25.056 Proof: 25.056/25.056 NO_CONDS 25.056/25.056 25.056/25.056 CRule InfChecker Info: 25.056/25.056 b -> c 25.056/25.056 Rule remains 25.056/25.056 Proof: 25.056/25.056 NO_CONDS 25.056/25.056 25.056/25.056 CRule InfChecker Info: 25.056/25.056 b -> d 25.056/25.056 Rule remains 25.056/25.056 Proof: 25.056/25.056 NO_CONDS 25.056/25.056 25.056/25.056 CRule InfChecker Info: 25.056/25.056 f(x:S,y:S) -> y:S | s(x:S) ->* t(y:S) 25.056/25.056 Rule remains 25.056/25.056 Proof: 25.056/25.056 NO 25.056/25.056 25.056/25.056 Problem 1: 25.056/25.056 25.056/25.056 Infeasibility Problem: 25.056/25.056 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S y:S:S) 25.056/25.056 (STRATEGY CONTEXTSENSITIVE 25.056/25.056 (a) 25.056/25.056 (b) 25.056/25.056 (f 1 2) 25.056/25.056 (g 1 2) 25.056/25.056 (s 1) 25.056/25.056 (c) 25.056/25.056 (d) 25.056/25.056 (fSNonEmpty) 25.056/25.056 (h 1 2) 25.056/25.056 (k) 25.056/25.056 (l) 25.056/25.056 (t 1) 25.056/25.056 ) 25.056/25.056 (RULES 25.056/25.056 a -> c 25.056/25.056 a -> d 25.056/25.056 b -> c 25.056/25.056 b -> d 25.056/25.056 f(x:S:S,y:S:S) -> y:S:S | s(x:S:S) ->* t(y:S:S) 25.056/25.056 g(x:S:S,x:S:S) -> h(x:S:S,x:S:S) 25.056/25.056 s(k) -> t(a) 25.056/25.056 s(l) -> t(a) 25.056/25.056 )] 25.056/25.056 25.056/25.056 Infeasibility Conditions: 25.056/25.056 s(x:S:S) ->* t(y:S:S) 25.056/25.056 25.056/25.056 Problem 1: 25.056/25.056 25.056/25.056 Obtaining a proof using Prover9: 25.056/25.056 25.056/25.056 -> Prover9 Output: 25.056/25.056 ============================== Prover9 =============================== 25.056/25.056 Prover9 (64) version 2009-11A, November 2009. 25.056/25.056 Process 110220 was started by ubuntu on ubuntu, 25.056/25.056 Fri Jul 16 15:06:30 2021 25.056/25.056 The command was "./prover9 -f /tmp/prover9110213-0.in". 25.056/25.056 ============================== end of head =========================== 25.056/25.056 25.056/25.056 ============================== INPUT ================================= 25.056/25.056 25.056/25.056 % Reading from file /tmp/prover9110213-0.in 25.056/25.056 25.056/25.056 assign(max_seconds,20). 25.056/25.056 25.056/25.056 formulas(assumptions). 25.056/25.056 ->*_s0(x,x) # label(reflexivity). 25.056/25.056 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.056/25.056 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence). 25.056/25.056 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence). 25.056/25.056 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence). 25.056/25.056 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence). 25.056/25.056 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 25.056/25.056 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence). 25.056/25.056 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence). 25.056/25.056 ->_s0(x1,y) -> ->_s0(t(x1),t(y)) # label(congruence). 25.056/25.056 ->_s0(a,c) # label(replacement). 25.056/25.056 ->_s0(a,d) # label(replacement). 25.056/25.056 ->_s0(b,c) # label(replacement). 25.056/25.056 ->_s0(b,d) # label(replacement). 25.056/25.056 ->*_s0(s(x2),t(x3)) -> ->_s0(f(x2,x3),x3) # label(replacement). 25.056/25.056 ->_s0(g(x2,x2),h(x2,x2)) # label(replacement). 25.056/25.056 ->_s0(s(k),t(a)) # label(replacement). 25.056/25.056 ->_s0(s(l),t(a)) # label(replacement). 25.056/25.056 end_of_list. 25.056/25.056 25.056/25.056 formulas(goals). 25.056/25.056 (exists x2 exists x3 ->*_s0(s(x2),t(x3))) # label(goal). 25.056/25.056 end_of_list. 25.056/25.056 25.056/25.056 ============================== end of input ========================== 25.056/25.056 25.056/25.056 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.056/25.056 25.056/25.056 % Formulas that are not ordinary clauses: 25.056/25.056 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.056/25.056 2 ->_s0(x1,y) -> ->_s0(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 3 ->_s0(x2,y) -> ->_s0(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 4 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 5 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 6 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 7 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 8 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 9 ->_s0(x1,y) -> ->_s0(t(x1),t(y)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 10 ->*_s0(s(x2),t(x3)) -> ->_s0(f(x2,x3),x3) # label(replacement) # label(non_clause). [assumption]. 25.056/25.056 11 (exists x2 exists x3 ->*_s0(s(x2),t(x3))) # label(goal) # label(non_clause) # label(goal). [goal]. 25.056/25.056 25.056/25.056 ============================== end of process non-clausal formulas === 25.056/25.056 25.056/25.056 ============================== PROCESS INITIAL CLAUSES =============== 25.056/25.056 25.056/25.056 % Clauses before input processing: 25.056/25.056 25.056/25.056 formulas(usable). 25.056/25.056 end_of_list. 25.056/25.056 25.056/25.056 formulas(sos). 25.056/25.056 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.056/25.056 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.056/25.056 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.056/25.056 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.056/25.056 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(4)]. 25.056/25.056 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(5)]. 25.056/25.056 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.056/25.056 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(7)]. 25.056/25.056 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(8)]. 25.056/25.056 -->_s0(x,y) | ->_s0(t(x),t(y)) # label(congruence). [clausify(9)]. 25.056/25.056 ->_s0(a,c) # label(replacement). [assumption]. 25.056/25.056 ->_s0(a,d) # label(replacement). [assumption]. 25.056/25.056 ->_s0(b,c) # label(replacement). [assumption]. 25.056/25.056 ->_s0(b,d) # label(replacement). [assumption]. 25.056/25.056 -->*_s0(s(x),t(y)) | ->_s0(f(x,y),y) # label(replacement). [clausify(10)]. 25.056/25.056 ->_s0(g(x,x),h(x,x)) # label(replacement). [assumption]. 25.056/25.056 ->_s0(s(k),t(a)) # label(replacement). [assumption]. 25.056/25.056 ->_s0(s(l),t(a)) # label(replacement). [assumption]. 25.056/25.056 -->*_s0(s(x),t(y)) # label(goal). [deny(11)]. 25.056/25.056 end_of_list. 25.056/25.056 25.056/25.056 formulas(demodulators). 25.056/25.056 end_of_list. 25.056/25.056 25.056/25.056 ============================== PREDICATE ELIMINATION ================= 25.056/25.056 25.056/25.056 No predicates eliminated. 25.056/25.056 25.056/25.056 ============================== end predicate elimination ============= 25.056/25.056 25.056/25.056 Auto_denials: 25.056/25.056 % copying label goal to answer in negative clause 25.056/25.056 25.056/25.056 Term ordering decisions: 25.056/25.056 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.056/25.056 Function symbol precedence: function_order([ a, b, c, d, k, l, h, f, g, t, s ]). 25.056/25.056 After inverse_order: (no changes). 25.056/25.056 Unfolding symbols: (none). 25.056/25.056 25.056/25.056 Auto_inference settings: 25.056/25.056 % set(neg_binary_resolution). % (HNE depth_diff=-8) 25.056/25.056 % clear(ordered_res). % (HNE depth_diff=-8) 25.056/25.056 % set(ur_resolution). % (HNE depth_diff=-8) 25.056/25.056 % set(ur_resolution) -> set(pos_ur_resolution). 25.056/25.056 % set(ur_resolution) -> set(neg_ur_resolution). 25.056/25.056 25.056/25.056 Auto_process settings: (no changes). 25.056/25.056 25.056/25.056 kept: 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.056/25.056 kept: 13 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.056/25.056 kept: 14 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.056/25.056 kept: 15 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.056/25.056 kept: 16 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(4)]. 25.056/25.056 kept: 17 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(5)]. 25.056/25.056 kept: 18 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.056/25.056 kept: 19 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(7)]. 25.056/25.056 kept: 20 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(8)]. 25.056/25.056 kept: 21 -->_s0(x,y) | ->_s0(t(x),t(y)) # label(congruence). [clausify(9)]. 25.056/25.056 kept: 22 ->_s0(a,c) # label(replacement). [assumption]. 25.056/25.056 kept: 23 ->_s0(a,d) # label(replacement). [assumption]. 25.056/25.056 kept: 24 ->_s0(b,c) # label(replacement). [assumption]. 25.056/25.056 kept: 25 ->_s0(b,d) # label(replacement). [assumption]. 25.056/25.056 kept: 26 -->*_s0(s(x),t(y)) | ->_s0(f(x,y),y) # label(replacement). [clausify(10)]. 25.056/25.056 kept: 27 ->_s0(g(x,x),h(x,x)) # label(replacement). [assumption]. 25.056/25.056 kept: 28 ->_s0(s(k),t(a)) # label(replacement). [assumption]. 25.056/25.056 kept: 29 ->_s0(s(l),t(a)) # label(replacement). [assumption]. 25.056/25.056 kept: 30 -->*_s0(s(x),t(y)) # label(goal) # answer(goal). [deny(11)]. 25.056/25.056 25.056/25.056 ============================== end of process initial clauses ======== 25.056/25.056 25.056/25.056 ============================== CLAUSES FOR SEARCH ==================== 25.056/25.056 25.056/25.056 % Clauses after input processing: 25.056/25.056 25.056/25.056 formulas(usable). 25.056/25.056 end_of_list. 25.056/25.056 25.056/25.056 formulas(sos). 25.056/25.056 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.056/25.056 13 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.056/25.056 14 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.056/25.056 15 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.056/25.056 16 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(4)]. 25.056/25.056 17 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(5)]. 25.056/25.056 18 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.056/25.056 19 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(7)]. 25.056/25.056 20 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(8)]. 25.056/25.056 21 -->_s0(x,y) | ->_s0(t(x),t(y)) # label(congruence). [clausify(9)]. 25.056/25.056 22 ->_s0(a,c) # label(replacement). [assumption]. 25.056/25.056 23 ->_s0(a,d) # label(replacement). [assumption]. 25.056/25.056 24 ->_s0(b,c) # label(replacement). [assumption]. 25.056/25.056 25 ->_s0(b,d) # label(replacement). [assumption]. 25.056/25.056 27 ->_s0(g(x,x),h(x,x)) # label(replacement). [assumption]. 25.056/25.056 28 ->_s0(s(k),t(a)) # label(replacement). [assumption]. 25.056/25.056 29 ->_s0(s(l),t(a)) # label(replacement). [assumption]. 25.056/25.056 30 -->*_s0(s(x),t(y)) # label(goal) # answer(goal). [deny(11)]. 25.056/25.056 end_of_list. 25.056/25.056 25.056/25.056 formulas(demodulators). 25.056/25.056 end_of_list. 25.056/25.056 25.056/25.056 ============================== end of clauses for search ============= 25.056/25.056 25.056/25.056 ============================== SEARCH ================================ 25.056/25.056 25.056/25.056 % Starting search at 0.00 seconds. 25.056/25.056 25.056/25.056 given #1 (I,wt=3): 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.056/25.056 25.056/25.056 given #2 (I,wt=9): 13 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.056/25.056 25.056/25.056 given #3 (I,wt=10): 14 -->_s0(x,y) | ->_s0(f(x,z),f(y,z)) # label(congruence). [clausify(2)]. 25.056/25.056 25.056/25.056 given #4 (I,wt=10): 15 -->_s0(x,y) | ->_s0(f(z,x),f(z,y)) # label(congruence). [clausify(3)]. 25.056/25.056 25.056/25.056 given #5 (I,wt=10): 16 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(4)]. 25.056/25.056 25.056/25.056 given #6 (I,wt=10): 17 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(5)]. 25.056/25.056 25.056/25.056 given #7 (I,wt=8): 18 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(6)]. 25.056/25.056 25.056/25.056 given #8 (I,wt=10): 19 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(7)]. 25.056/25.056 25.056/25.056 given #9 (I,wt=10): 20 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(8)]. 25.056/25.056 25.056/25.056 given #10 (I,wt=8): 21 -->_s0(x,y) | ->_s0(t(x),t(y)) # label(congruence). [clausify(9)]. 25.056/25.056 25.056/25.056 given #11 (I,wt=3): 22 ->_s0(a,c) # label(replacement). [assumption]. 25.056/25.056 25.056/25.056 given #12 (I,wt=3): 23 ->_s0(a,d) # label(replacement). [assumption]. 25.056/25.056 25.056/25.056 given #13 (I,wt=3): 24 ->_s0(b,c) # label(replacement). [assumption]. 25.056/25.056 25.056/25.056 given #14 (I,wt=3): 25 ->_s0(b,d) # label(replacement). [assumption]. 25.056/25.056 25.056/25.056 given #15 (I,wt=7): 27 ->_s0(g(x,x),h(x,x)) # label(replacement). [assumption]. 25.056/25.056 25.056/25.056 given #16 (I,wt=5): 28 ->_s0(s(k),t(a)) # label(replacement). [assumption]. 25.056/25.056 25.056/25.056 ============================== PROOF ================================= 25.056/25.056 25.056/25.056 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.056/25.056 % Length of proof is 8. 25.056/25.056 % Level of proof is 3. 25.056/25.056 % Maximum clause weight is 9.000. 25.056/25.056 % Given clauses 16. 25.056/25.056 25.056/25.056 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.056/25.056 11 (exists x2 exists x3 ->*_s0(s(x2),t(x3))) # label(goal) # label(non_clause) # label(goal). [goal]. 25.056/25.056 12 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.056/25.056 13 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.056/25.056 28 ->_s0(s(k),t(a)) # label(replacement). [assumption]. 25.056/25.056 30 -->*_s0(s(x),t(y)) # label(goal) # answer(goal). [deny(11)]. 25.056/25.056 84 ->*_s0(s(k),t(a)). [ur(13,a,28,a,b,12,a)]. 25.056/25.056 85 $F # answer(goal). [resolve(84,a,30,a)]. 25.056/25.056 25.056/25.056 ============================== end of proof ========================== 25.056/25.056 25.056/25.056 ============================== STATISTICS ============================ 25.056/25.056 25.056/25.056 Given=16. Generated=73. Kept=73. proofs=1. 25.056/25.056 Usable=16. Sos=47. Demods=0. Limbo=8, Disabled=20. Hints=0. 25.056/25.056 Kept_by_rule=0, Deleted_by_rule=0. 25.056/25.056 Forward_subsumed=0. Back_subsumed=1. 25.056/25.056 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.056/25.056 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.056/25.056 Demod_attempts=0. Demod_rewrites=0. 25.056/25.056 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.056/25.056 Nonunit_fsub_feature_tests=3. Nonunit_bsub_feature_tests=18. 25.056/25.056 Megabytes=0.15. 25.056/25.056 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.056/25.056 25.056/25.056 ============================== end of statistics ===================== 25.056/25.056 25.056/25.056 ============================== end of search ========================= 25.056/25.056 25.056/25.056 THEOREM PROVED 25.056/25.056 25.056/25.056 Exiting with 1 proof. 25.056/25.056 25.056/25.056 Process 110220 exit (max_proofs) Fri Jul 16 15:06:30 2021 25.056/25.056 25.056/25.056 25.056/25.056 The problem is feasible. 25.056/25.056 25.056/25.056 25.056/25.056 CRule InfChecker Info: 25.056/25.056 g(x:S,x:S) -> h(x:S,x:S) 25.056/25.056 Rule remains 25.056/25.056 Proof: 25.056/25.056 NO_CONDS 25.056/25.056 25.056/25.056 CRule InfChecker Info: 25.056/25.056 s(k) -> t(a) 25.056/25.056 Rule remains 25.056/25.056 Proof: 25.056/25.056 NO_CONDS 25.056/25.056 25.056/25.056 CRule InfChecker Info: 25.056/25.056 s(l) -> t(a) 25.056/25.056 Rule remains 25.056/25.056 Proof: 25.056/25.056 NO_CONDS 25.056/25.056 25.056/25.056 Problem 1: 25.056/25.056 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.056/25.056 Confluence Problem: 25.056/25.056 (VAR x:S y:S) 25.056/25.056 (STRATEGY CONTEXTSENSITIVE 25.056/25.056 (a) 25.056/25.056 (b) 25.056/25.056 (f 1 2) 25.056/25.056 (g 1 2) 25.056/25.056 (s 1) 25.056/25.056 (c) 25.056/25.056 (d) 25.056/25.056 (fSNonEmpty) 25.056/25.056 (h 1 2) 25.056/25.056 (k) 25.056/25.056 (l) 25.056/25.056 (t 1) 25.056/25.056 ) 25.056/25.056 (RULES 25.056/25.056 a -> c 25.056/25.056 a -> d 25.056/25.056 b -> c 25.056/25.056 b -> d 25.056/25.056 f(x:S,y:S) -> y:S | s(x:S) ->* t(y:S) 25.056/25.056 g(x:S,x:S) -> h(x:S,x:S) 25.056/25.056 s(k) -> t(a) 25.056/25.056 s(l) -> t(a) 25.056/25.056 ) 25.056/25.056 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.056/25.056 25.056/25.056 Critical Pairs Processor: 25.056/25.056 -> Rules: 25.056/25.056 a -> c 25.056/25.056 a -> d 25.056/25.056 b -> c 25.056/25.056 b -> d 25.056/25.056 f(x:S,y:S) -> y:S | s(x:S) ->* t(y:S) 25.056/25.056 g(x:S,x:S) -> h(x:S,x:S) 25.056/25.056 s(k) -> t(a) 25.056/25.056 s(l) -> t(a) 25.056/25.056 -> Vars: 25.056/25.056 "x", "y", "x" 25.056/25.056 -> FVars: 25.056/25.056 "x3", "x4", "x5" 25.056/25.056 -> PVars: 25.056/25.056 "x": ["x3", "x5"], "y": ["x4"] 25.056/25.056 25.056/25.056 -> Rlps: 25.056/25.056 crule: a -> c, id: 1, possubterms: a-> [] 25.056/25.056 crule: a -> d, id: 2, possubterms: a-> [] 25.056/25.056 crule: b -> c, id: 3, possubterms: b-> [] 25.056/25.056 crule: b -> d, id: 4, possubterms: b-> [] 25.056/25.056 crule: f(x3:S,x4:S) -> x4:S | s(x3:S) ->* t(x4:S), id: 5, possubterms: f(x3:S,x4:S)-> [] 25.056/25.056 crule: g(x5:S,x5:S) -> h(x5:S,x5:S), id: 6, possubterms: g(x5:S,x5:S)-> [] 25.056/25.056 crule: s(k) -> t(a), id: 7, possubterms: s(k)-> [], k-> [1] 25.056/25.056 crule: s(l) -> t(a), id: 8, possubterms: s(l)-> [], l-> [1] 25.056/25.056 25.056/25.056 -> Unifications: 25.056/25.056 R2 unifies with R1 at p: [], l: a, lp: a, conds: {}, sig: {}, l': a, r: d, r': c 25.056/25.056 R4 unifies with R3 at p: [], l: b, lp: b, conds: {}, sig: {}, l': b, r: d, r': c 25.056/25.056 25.056/25.056 -> Critical pairs info: 25.056/25.056 => Not trivial, Overlay, N1 25.056/25.056 25.056/25.056 -> Problem conclusions: 25.056/25.056 Not left linear, Not right linear, Not linear 25.056/25.056 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 25.056/25.056 CTRS Type: 1 25.056/25.056 Deterministic, Strongly deterministic 25.056/25.056 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.056/25.056 Maybe right-stable CTRS, Overlay CTRS 25.056/25.056 Maybe normal CTRS, Maybe almost normal CTRS 25.056/25.056 Maybe terminating CTRS, Maybe joinable CCPs 25.056/25.056 Maybe level confluent 25.056/25.056 Maybe confluent 25.056/25.056 25.056/25.056 Problem 1: 25.056/25.056 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.056/25.056 Confluence Problem: 25.056/25.056 (VAR x:S y:S) 25.056/25.056 (STRATEGY CONTEXTSENSITIVE 25.056/25.056 (a) 25.056/25.056 (b) 25.056/25.056 (f 1 2) 25.056/25.056 (g 1 2) 25.056/25.056 (s 1) 25.056/25.056 (c) 25.056/25.056 (d) 25.056/25.056 (fSNonEmpty) 25.056/25.056 (h 1 2) 25.056/25.056 (k) 25.056/25.056 (l) 25.056/25.056 (t 1) 25.056/25.056 ) 25.056/25.056 (RULES 25.056/25.056 a -> c 25.056/25.056 a -> d 25.056/25.056 b -> c 25.056/25.056 b -> d 25.056/25.056 f(x:S,y:S) -> y:S | s(x:S) ->* t(y:S) 25.056/25.056 g(x:S,x:S) -> h(x:S,x:S) 25.056/25.056 s(k) -> t(a) 25.056/25.056 s(l) -> t(a) 25.056/25.056 ) 25.056/25.056 Critical Pairs: 25.056/25.056 => Not trivial, Overlay, N1 25.056/25.056 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.056/25.056 25.056/25.056 Conditional Critical Pairs Distributor Processor 25.056/25.056 25.056/25.056 Problem 1: 25.056/25.056 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.056/25.056 Confluence Problem: 25.056/25.056 (VAR x:S y:S) 25.056/25.056 (STRATEGY CONTEXTSENSITIVE 25.056/25.056 (a) 25.056/25.056 (b) 25.056/25.056 (f 1 2) 25.056/25.056 (g 1 2) 25.056/25.056 (s 1) 25.056/25.056 (c) 25.056/25.056 (d) 25.056/25.056 (fSNonEmpty) 25.056/25.056 (h 1 2) 25.056/25.056 (k) 25.056/25.056 (l) 25.056/25.056 (t 1) 25.056/25.056 ) 25.056/25.056 (RULES 25.056/25.056 a -> c 25.056/25.056 a -> d 25.056/25.056 b -> c 25.056/25.056 b -> d 25.056/25.056 f(x:S,y:S) -> y:S | s(x:S) ->* t(y:S) 25.056/25.056 g(x:S,x:S) -> h(x:S,x:S) 25.056/25.056 s(k) -> t(a) 25.056/25.056 s(l) -> t(a) 25.056/25.056 ) 25.056/25.056 Critical Pairs: 25.056/25.056 => Not trivial, Overlay, N1 25.056/25.056 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.056/25.056 25.056/25.056 Infeasible Convergence No Conditions CCP Processor: 25.056/25.056 Infeasible convergence? 25.056/25.056 YES 25.056/25.056 25.056/25.056 Problem 1: 25.056/25.056 25.056/25.056 Infeasibility Problem: 25.056/25.056 [(VAR vNonEmpty:S x:S y:S z:S) 25.056/25.056 (STRATEGY CONTEXTSENSITIVE 25.056/25.056 (a) 25.056/25.056 (b) 25.056/25.056 (f 1 2) 25.056/25.056 (g 1 2) 25.056/25.056 (s 1) 25.056/25.056 (c) 25.056/25.056 (d) 25.056/25.056 (fSNonEmpty) 25.056/25.056 (h 1 2) 25.056/25.056 (k) 25.056/25.056 (l) 25.056/25.056 (t 1) 25.056/25.056 ) 25.056/25.056 (RULES 25.056/25.056 a -> c 25.056/25.056 a -> d 25.056/25.056 b -> c 25.056/25.056 b -> d 25.056/25.056 f(x:S,y:S) -> y:S | s(x:S) ->* t(y:S) 25.056/25.056 g(x:S,x:S) -> h(x:S,x:S) 25.056/25.056 s(k) -> t(a) 25.056/25.056 s(l) -> t(a) 25.056/25.056 )] 25.056/25.056 25.056/25.056 Infeasibility Conditions: 25.056/25.056 c ->* z:S, d ->* z:S 25.056/25.056 25.056/25.056 Problem 1: 25.056/25.056 25.056/25.056 Obtaining a model using Mace4: 25.056/25.056 25.056/25.056 -> Usable Rules: 25.056/25.056 Empty 25.056/25.056 25.056/25.056 -> Mace4 Output: 25.056/25.056 ============================== Mace4 ================================= 25.056/25.056 Mace4 (64) version 2009-11A, November 2009. 25.056/25.056 Process 110266 was started by ubuntu on ubuntu, 25.056/25.056 Fri Jul 16 15:06:30 2021 25.056/25.056 The command was "./mace4 -c -f /tmp/mace4110239-2.in". 25.056/25.056 ============================== end of head =========================== 25.056/25.056 25.056/25.056 ============================== INPUT ================================= 25.056/25.056 25.056/25.056 % Reading from file /tmp/mace4110239-2.in 25.056/25.056 25.056/25.056 assign(max_seconds,10). 25.056/25.056 25.056/25.056 formulas(assumptions). 25.056/25.056 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence). 25.056/25.056 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence). 25.056/25.056 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence). 25.056/25.056 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence). 25.056/25.056 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 25.056/25.056 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence). 25.056/25.056 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence). 25.056/25.056 ->(x1,y) -> ->(t(x1),t(y)) # label(congruence). 25.056/25.056 ->*(x,x) # label(reflexivity). 25.056/25.056 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 25.056/25.056 end_of_list. 25.056/25.056 25.056/25.056 formulas(goals). 25.056/25.056 (exists x3 (->*(c,x3) & ->*(d,x3))) # label(goal). 25.056/25.056 end_of_list. 25.056/25.056 25.056/25.056 ============================== end of input ========================== 25.056/25.056 25.056/25.056 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.056/25.056 25.056/25.056 % Formulas that are not ordinary clauses: 25.056/25.056 1 ->(x1,y) -> ->(f(x1,x2),f(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 2 ->(x2,y) -> ->(f(x1,x2),f(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 3 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 4 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 5 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 6 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 7 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 8 ->(x1,y) -> ->(t(x1),t(y)) # label(congruence) # label(non_clause). [assumption]. 25.056/25.056 9 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.056/25.056 10 (exists x3 (->*(c,x3) & ->*(d,x3))) # label(goal) # label(non_clause) # label(goal). [goal]. 25.056/25.056 25.056/25.056 ============================== end of process non-clausal formulas === 25.056/25.056 25.056/25.056 ============================== CLAUSES FOR SEARCH ==================== 25.056/25.056 25.056/25.056 formulas(mace4_clauses). 25.056/25.056 -->(x,y) | ->(f(x,z),f(y,z)) # label(congruence). 25.056/25.056 -->(x,y) | ->(f(z,x),f(z,y)) # label(congruence). 25.056/25.056 -->(x,y) | ->(g(x,z),g(y,z)) # label(congruence). 25.056/25.056 -->(x,y) | ->(g(z,x),g(z,y)) # label(congruence). 25.056/25.056 -->(x,y) | ->(s(x),s(y)) # label(congruence). 25.056/25.056 -->(x,y) | ->(h(x,z),h(y,z)) # label(congruence). 25.056/25.056 -->(x,y) | ->(h(z,x),h(z,y)) # label(congruence). 25.056/25.056 -->(x,y) | ->(t(x),t(y)) # label(congruence). 25.056/25.056 ->*(x,x) # label(reflexivity). 25.056/25.056 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 25.056/25.056 -->*(c,x) | -->*(d,x) # label(goal). 25.056/25.056 end_of_list. 25.056/25.056 25.056/25.056 ============================== end of clauses for search ============= 25.056/25.056 25.056/25.056 % There are no natural numbers in the input. 25.056/25.056 25.056/25.056 ============================== DOMAIN SIZE 2 ========================= 25.056/25.056 25.056/25.056 ============================== MODEL ================================= 25.056/25.056 25.056/25.056 interpretation( 2, [number=1, seconds=0], [ 25.056/25.056 25.056/25.056 function(c, [ 0 ]), 25.056/25.056 25.056/25.056 function(d, [ 1 ]), 25.056/25.056 25.056/25.056 function(t(_), [ 0, 0 ]), 25.056/25.056 25.056/25.056 function(s(_), [ 0, 0 ]), 25.056/25.056 25.056/25.056 function(h(_,_), [ 25.056/25.056 0, 0, 25.056/25.056 0, 0 ]), 25.056/25.056 25.056/25.056 function(f(_,_), [ 25.056/25.056 0, 0, 25.056/25.056 0, 0 ]), 25.056/25.056 25.056/25.056 function(g(_,_), [ 25.056/25.056 0, 0, 25.056/25.056 0, 0 ]), 25.056/25.056 25.056/25.056 relation(->*(_,_), [ 25.056/25.056 1, 0, 25.056/25.056 0, 1 ]), 25.056/25.056 25.056/25.056 relation(->(_,_), [ 25.056/25.056 0, 0, 25.056/25.056 0, 0 ]) 25.056/25.056 ]). 25.056/25.056 25.056/25.056 ============================== end of model ========================== 25.056/25.056 25.056/25.056 ============================== STATISTICS ============================ 25.056/25.056 25.056/25.056 For domain size 2. 25.056/25.056 25.056/25.056 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 25.056/25.056 Ground clauses: seen=68, kept=64. 25.056/25.056 Selections=18, assignments=18, propagations=8, current_models=1. 25.056/25.056 Rewrite_terms=116, rewrite_bools=79, indexes=10. 25.056/25.056 Rules_from_neg_clauses=1, cross_offs=1. 25.056/25.056 25.056/25.056 ============================== end of statistics ===================== 25.056/25.056 25.056/25.056 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.056/25.056 25.056/25.056 Exiting with 1 model. 25.056/25.056 25.056/25.056 Process 110266 exit (max_models) Fri Jul 16 15:06:30 2021 25.056/25.056 The process finished Fri Jul 16 15:06:30 2021 25.056/25.056 25.056/25.056 25.056/25.056 Mace4 cooked interpretation: 25.056/25.056 25.056/25.056 25.056/25.056 25.056/25.056 The problem is infeasible. 25.056/25.056 25.056/25.056 25.056/25.056 The problem is not joinable. 25.056/25.056 28.32user 0.93system 0:25.56elapsed 114%CPU (0avgtext+0avgdata 223020maxresident)k 25.056/25.056 8inputs+0outputs (0major+115697minor)pagefaults 0swaps