25.047/25.047 NO 25.047/25.047 25.047/25.047 Problem 1: 25.047/25.047 25.047/25.047 25.047/25.047 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.047/25.047 Confluence Problem: 25.047/25.047 (VAR vNonEmpty:S x:S) 25.047/25.047 (STRATEGY CONTEXTSENSITIVE 25.047/25.047 (a) 25.047/25.047 (b) 25.047/25.047 (f 1) 25.047/25.047 (g 1 2) 25.047/25.047 (s 1) 25.047/25.047 (A) 25.047/25.047 (B) 25.047/25.047 (c) 25.047/25.047 (fSNonEmpty) 25.047/25.047 (h 1 2) 25.047/25.047 (t) 25.047/25.047 ) 25.047/25.047 (RULES 25.047/25.047 a -> c 25.047/25.047 b -> c 25.047/25.047 f(x:S) -> A | s(x:S) ->* t 25.047/25.047 f(x:S) -> B | s(x:S) ->* t 25.047/25.047 g(x:S,x:S) -> h(x:S,x:S) 25.047/25.047 s(a) -> t 25.047/25.047 s(b) -> t 25.047/25.047 ) 25.047/25.047 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.047/25.047 25.047/25.047 25.047/25.047 Problem 1: 25.047/25.047 25.047/25.047 Inlining of Conditions Processor [STERN17]: 25.047/25.047 25.047/25.047 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.047/25.047 Confluence Problem: 25.047/25.047 (VAR vNonEmpty:S x:S) 25.047/25.047 (STRATEGY CONTEXTSENSITIVE 25.047/25.047 (a) 25.047/25.047 (b) 25.047/25.047 (f 1) 25.047/25.047 (g 1 2) 25.047/25.047 (s 1) 25.047/25.047 (A) 25.047/25.047 (B) 25.047/25.047 (c) 25.047/25.047 (fSNonEmpty) 25.047/25.047 (h 1 2) 25.047/25.047 (t) 25.047/25.047 ) 25.047/25.047 (RULES 25.047/25.047 a -> c 25.047/25.047 b -> c 25.047/25.047 f(x:S) -> A | s(x:S) ->* t 25.047/25.047 f(x:S) -> B | s(x:S) ->* t 25.047/25.047 g(x:S,x:S) -> h(x:S,x:S) 25.047/25.047 s(a) -> t 25.047/25.047 s(b) -> t 25.047/25.047 ) 25.047/25.047 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.047/25.047 25.047/25.047 25.047/25.047 Problem 1: 25.047/25.047 25.047/25.047 Clean CTRS Processor: 25.047/25.047 25.047/25.047 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.047/25.047 Confluence Problem: 25.047/25.047 (VAR x:S) 25.047/25.047 (STRATEGY CONTEXTSENSITIVE 25.047/25.047 (a) 25.047/25.047 (b) 25.047/25.047 (f 1) 25.047/25.047 (g 1 2) 25.047/25.047 (s 1) 25.047/25.047 (A) 25.047/25.047 (B) 25.047/25.047 (c) 25.047/25.047 (fSNonEmpty) 25.047/25.047 (h 1 2) 25.047/25.047 (t) 25.047/25.047 ) 25.047/25.047 (RULES 25.047/25.047 a -> c 25.047/25.047 b -> c 25.047/25.047 f(x:S) -> A | s(x:S) ->* t 25.047/25.047 f(x:S) -> B | s(x:S) ->* t 25.047/25.047 g(x:S,x:S) -> h(x:S,x:S) 25.047/25.047 s(a) -> t 25.047/25.047 s(b) -> t 25.047/25.047 ) 25.047/25.047 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.047/25.047 25.047/25.047 CRule InfChecker Info: 25.047/25.047 a -> c 25.047/25.047 Rule remains 25.047/25.047 Proof: 25.047/25.047 NO_CONDS 25.047/25.047 25.047/25.047 CRule InfChecker Info: 25.047/25.047 b -> c 25.047/25.047 Rule remains 25.047/25.047 Proof: 25.047/25.047 NO_CONDS 25.047/25.047 25.047/25.047 CRule InfChecker Info: 25.047/25.047 f(x:S) -> A | s(x:S) ->* t 25.047/25.047 Rule remains 25.047/25.047 Proof: 25.047/25.047 NO 25.047/25.047 25.047/25.047 Problem 1: 25.047/25.047 25.047/25.047 Infeasibility Problem: 25.047/25.047 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 25.047/25.047 (STRATEGY CONTEXTSENSITIVE 25.047/25.047 (a) 25.047/25.047 (b) 25.047/25.047 (f 1) 25.047/25.047 (g 1 2) 25.047/25.047 (s 1) 25.047/25.047 (A) 25.047/25.047 (B) 25.047/25.047 (c) 25.047/25.047 (fSNonEmpty) 25.047/25.047 (h 1 2) 25.047/25.047 (t) 25.047/25.047 ) 25.047/25.047 (RULES 25.047/25.047 a -> c 25.047/25.047 b -> c 25.047/25.047 f(x:S:S) -> A | s(x:S:S) ->* t 25.047/25.047 f(x:S:S) -> B | s(x:S:S) ->* t 25.047/25.047 g(x:S:S,x:S:S) -> h(x:S:S,x:S:S) 25.047/25.047 s(a) -> t 25.047/25.047 s(b) -> t 25.047/25.047 )] 25.047/25.047 25.047/25.047 Infeasibility Conditions: 25.047/25.047 s(x:S:S) ->* t 25.047/25.047 25.047/25.047 Problem 1: 25.047/25.047 25.047/25.047 Obtaining a proof using Prover9: 25.047/25.047 25.047/25.047 -> Prover9 Output: 25.047/25.047 ============================== Prover9 =============================== 25.047/25.047 Prover9 (64) version 2009-11A, November 2009. 25.047/25.047 Process 50612 was started by ubuntu on ubuntu, 25.047/25.047 Wed Jul 14 10:55:50 2021 25.047/25.047 The command was "./prover9 -f /tmp/prover950603-0.in". 25.047/25.047 ============================== end of head =========================== 25.047/25.047 25.047/25.047 ============================== INPUT ================================= 25.047/25.047 25.047/25.047 % Reading from file /tmp/prover950603-0.in 25.047/25.047 25.047/25.047 assign(max_seconds,20). 25.047/25.047 25.047/25.047 formulas(assumptions). 25.047/25.047 ->*_s0(x,x) # label(reflexivity). 25.047/25.047 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.047/25.047 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 25.047/25.047 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence). 25.047/25.047 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence). 25.047/25.047 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 25.047/25.047 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence). 25.047/25.047 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence). 25.047/25.047 ->_s0(a,c) # label(replacement). 25.047/25.047 ->_s0(b,c) # label(replacement). 25.047/25.047 ->*_s0(s(x2),t) -> ->_s0(f(x2),A) # label(replacement). 25.047/25.047 ->*_s0(s(x2),t) -> ->_s0(f(x2),B) # label(replacement). 25.047/25.047 ->_s0(g(x2,x2),h(x2,x2)) # label(replacement). 25.047/25.047 ->_s0(s(a),t) # label(replacement). 25.047/25.047 ->_s0(s(b),t) # label(replacement). 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 formulas(goals). 25.047/25.047 (exists x2 ->*_s0(s(x2),t)) # label(goal). 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 ============================== end of input ========================== 25.047/25.047 25.047/25.047 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.047/25.047 25.047/25.047 % Formulas that are not ordinary clauses: 25.047/25.047 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.047/25.047 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 3 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 4 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 5 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 6 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 7 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 8 ->*_s0(s(x2),t) -> ->_s0(f(x2),A) # label(replacement) # label(non_clause). [assumption]. 25.047/25.047 9 ->*_s0(s(x2),t) -> ->_s0(f(x2),B) # label(replacement) # label(non_clause). [assumption]. 25.047/25.047 10 (exists x2 ->*_s0(s(x2),t)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.047/25.047 25.047/25.047 ============================== end of process non-clausal formulas === 25.047/25.047 25.047/25.047 ============================== PROCESS INITIAL CLAUSES =============== 25.047/25.047 25.047/25.047 % Clauses before input processing: 25.047/25.047 25.047/25.047 formulas(usable). 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 formulas(sos). 25.047/25.047 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.047/25.047 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.047/25.047 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.047/25.047 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 25.047/25.047 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 25.047/25.047 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 25.047/25.047 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 25.047/25.047 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 25.047/25.047 ->_s0(a,c) # label(replacement). [assumption]. 25.047/25.047 ->_s0(b,c) # label(replacement). [assumption]. 25.047/25.047 -->*_s0(s(x),t) | ->_s0(f(x),A) # label(replacement). [clausify(8)]. 25.047/25.047 -->*_s0(s(x),t) | ->_s0(f(x),B) # label(replacement). [clausify(9)]. 25.047/25.047 ->_s0(g(x,x),h(x,x)) # label(replacement). [assumption]. 25.047/25.047 ->_s0(s(a),t) # label(replacement). [assumption]. 25.047/25.047 ->_s0(s(b),t) # label(replacement). [assumption]. 25.047/25.047 -->*_s0(s(x),t) # label(goal). [deny(10)]. 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 formulas(demodulators). 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 ============================== PREDICATE ELIMINATION ================= 25.047/25.047 25.047/25.047 No predicates eliminated. 25.047/25.047 25.047/25.047 ============================== end predicate elimination ============= 25.047/25.047 25.047/25.047 Auto_denials: 25.047/25.047 % copying label goal to answer in negative clause 25.047/25.047 25.047/25.047 Term ordering decisions: 25.047/25.047 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.047/25.047 Function symbol precedence: function_order([ t, a, b, c, A, B, h, g, s, f ]). 25.047/25.047 After inverse_order: (no changes). 25.047/25.047 Unfolding symbols: (none). 25.047/25.047 25.047/25.047 Auto_inference settings: 25.047/25.047 % set(neg_binary_resolution). % (HNE depth_diff=-6) 25.047/25.047 % clear(ordered_res). % (HNE depth_diff=-6) 25.047/25.047 % set(ur_resolution). % (HNE depth_diff=-6) 25.047/25.047 % set(ur_resolution) -> set(pos_ur_resolution). 25.047/25.047 % set(ur_resolution) -> set(neg_ur_resolution). 25.047/25.047 25.047/25.047 Auto_process settings: (no changes). 25.047/25.047 25.047/25.047 kept: 11 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.047/25.047 kept: 12 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.047/25.047 kept: 13 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.047/25.047 kept: 14 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 25.047/25.047 kept: 15 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 25.047/25.047 kept: 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 25.047/25.047 kept: 17 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 25.047/25.047 kept: 18 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 25.047/25.047 kept: 19 ->_s0(a,c) # label(replacement). [assumption]. 25.047/25.047 kept: 20 ->_s0(b,c) # label(replacement). [assumption]. 25.047/25.047 kept: 21 -->*_s0(s(x),t) | ->_s0(f(x),A) # label(replacement). [clausify(8)]. 25.047/25.047 kept: 22 -->*_s0(s(x),t) | ->_s0(f(x),B) # label(replacement). [clausify(9)]. 25.047/25.047 kept: 23 ->_s0(g(x,x),h(x,x)) # label(replacement). [assumption]. 25.047/25.047 kept: 24 ->_s0(s(a),t) # label(replacement). [assumption]. 25.047/25.047 kept: 25 ->_s0(s(b),t) # label(replacement). [assumption]. 25.047/25.047 kept: 26 -->*_s0(s(x),t) # label(goal) # answer(goal). [deny(10)]. 25.047/25.047 25.047/25.047 ============================== end of process initial clauses ======== 25.047/25.047 25.047/25.047 ============================== CLAUSES FOR SEARCH ==================== 25.047/25.047 25.047/25.047 % Clauses after input processing: 25.047/25.047 25.047/25.047 formulas(usable). 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 formulas(sos). 25.047/25.047 11 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.047/25.047 12 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.047/25.047 13 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.047/25.047 14 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 25.047/25.047 15 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 25.047/25.047 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 25.047/25.047 17 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 25.047/25.047 18 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 25.047/25.047 19 ->_s0(a,c) # label(replacement). [assumption]. 25.047/25.047 20 ->_s0(b,c) # label(replacement). [assumption]. 25.047/25.047 23 ->_s0(g(x,x),h(x,x)) # label(replacement). [assumption]. 25.047/25.047 24 ->_s0(s(a),t) # label(replacement). [assumption]. 25.047/25.047 25 ->_s0(s(b),t) # label(replacement). [assumption]. 25.047/25.047 26 -->*_s0(s(x),t) # label(goal) # answer(goal). [deny(10)]. 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 formulas(demodulators). 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 ============================== end of clauses for search ============= 25.047/25.047 25.047/25.047 ============================== SEARCH ================================ 25.047/25.047 25.047/25.047 % Starting search at 0.00 seconds. 25.047/25.047 25.047/25.047 given #1 (I,wt=3): 11 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.047/25.047 25.047/25.047 given #2 (I,wt=9): 12 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.047/25.047 25.047/25.047 given #3 (I,wt=8): 13 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.047/25.047 25.047/25.047 given #4 (I,wt=10): 14 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 25.047/25.047 25.047/25.047 given #5 (I,wt=10): 15 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 25.047/25.047 25.047/25.047 given #6 (I,wt=8): 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 25.047/25.047 25.047/25.047 given #7 (I,wt=10): 17 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 25.047/25.047 25.047/25.047 given #8 (I,wt=10): 18 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 25.047/25.047 25.047/25.047 given #9 (I,wt=3): 19 ->_s0(a,c) # label(replacement). [assumption]. 25.047/25.047 25.047/25.047 given #10 (I,wt=3): 20 ->_s0(b,c) # label(replacement). [assumption]. 25.047/25.047 25.047/25.047 given #11 (I,wt=7): 23 ->_s0(g(x,x),h(x,x)) # label(replacement). [assumption]. 25.047/25.047 25.047/25.047 given #12 (I,wt=4): 24 ->_s0(s(a),t) # label(replacement). [assumption]. 25.047/25.047 25.047/25.047 ============================== PROOF ================================= 25.047/25.047 25.047/25.047 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.047/25.047 % Length of proof is 8. 25.047/25.047 % Level of proof is 3. 25.047/25.047 % Maximum clause weight is 9.000. 25.047/25.047 % Given clauses 12. 25.047/25.047 25.047/25.047 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.047/25.047 10 (exists x2 ->*_s0(s(x2),t)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.047/25.047 11 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.047/25.047 12 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.047/25.047 24 ->_s0(s(a),t) # label(replacement). [assumption]. 25.047/25.047 26 -->*_s0(s(x),t) # label(goal) # answer(goal). [deny(10)]. 25.047/25.047 54 ->*_s0(s(a),t). [ur(12,a,24,a,b,11,a)]. 25.047/25.047 55 $F # answer(goal). [resolve(54,a,26,a)]. 25.047/25.047 25.047/25.047 ============================== end of proof ========================== 25.047/25.047 25.047/25.047 ============================== STATISTICS ============================ 25.047/25.047 25.047/25.047 Given=12. Generated=44. Kept=44. proofs=1. 25.047/25.047 Usable=12. Sos=23. Demods=0. Limbo=6, Disabled=18. Hints=0. 25.047/25.047 Kept_by_rule=0, Deleted_by_rule=0. 25.047/25.047 Forward_subsumed=0. Back_subsumed=2. 25.047/25.047 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.047/25.047 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.047/25.047 Demod_attempts=0. Demod_rewrites=0. 25.047/25.047 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.047/25.047 Nonunit_fsub_feature_tests=2. Nonunit_bsub_feature_tests=16. 25.047/25.047 Megabytes=0.11. 25.047/25.047 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.047/25.047 25.047/25.047 ============================== end of statistics ===================== 25.047/25.047 25.047/25.047 ============================== end of search ========================= 25.047/25.047 25.047/25.047 THEOREM PROVED 25.047/25.047 25.047/25.047 Exiting with 1 proof. 25.047/25.047 25.047/25.047 Process 50612 exit (max_proofs) Wed Jul 14 10:55:50 2021 25.047/25.047 25.047/25.047 25.047/25.047 The problem is feasible. 25.047/25.047 25.047/25.047 25.047/25.047 CRule InfChecker Info: 25.047/25.047 f(x:S) -> B | s(x:S) ->* t 25.047/25.047 Rule remains 25.047/25.047 Proof: 25.047/25.047 NO 25.047/25.047 25.047/25.047 Problem 1: 25.047/25.047 25.047/25.047 Infeasibility Problem: 25.047/25.047 [(VAR vNonEmpty:S vNonEmpty:S:S x:S:S) 25.047/25.047 (STRATEGY CONTEXTSENSITIVE 25.047/25.047 (a) 25.047/25.047 (b) 25.047/25.047 (f 1) 25.047/25.047 (g 1 2) 25.047/25.047 (s 1) 25.047/25.047 (A) 25.047/25.047 (B) 25.047/25.047 (c) 25.047/25.047 (fSNonEmpty) 25.047/25.047 (h 1 2) 25.047/25.047 (t) 25.047/25.047 ) 25.047/25.047 (RULES 25.047/25.047 a -> c 25.047/25.047 b -> c 25.047/25.047 f(x:S:S) -> A | s(x:S:S) ->* t 25.047/25.047 f(x:S:S) -> B | s(x:S:S) ->* t 25.047/25.047 g(x:S:S,x:S:S) -> h(x:S:S,x:S:S) 25.047/25.047 s(a) -> t 25.047/25.047 s(b) -> t 25.047/25.047 )] 25.047/25.047 25.047/25.047 Infeasibility Conditions: 25.047/25.047 s(x:S:S) ->* t 25.047/25.047 25.047/25.047 Problem 1: 25.047/25.047 25.047/25.047 Obtaining a proof using Prover9: 25.047/25.047 25.047/25.047 -> Prover9 Output: 25.047/25.047 ============================== Prover9 =============================== 25.047/25.047 Prover9 (64) version 2009-11A, November 2009. 25.047/25.047 Process 50635 was started by ubuntu on ubuntu, 25.047/25.047 Wed Jul 14 10:55:50 2021 25.047/25.047 The command was "./prover9 -f /tmp/prover950625-0.in". 25.047/25.047 ============================== end of head =========================== 25.047/25.047 25.047/25.047 ============================== INPUT ================================= 25.047/25.047 25.047/25.047 % Reading from file /tmp/prover950625-0.in 25.047/25.047 25.047/25.047 assign(max_seconds,20). 25.047/25.047 25.047/25.047 formulas(assumptions). 25.047/25.047 ->*_s0(x,x) # label(reflexivity). 25.047/25.047 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity). 25.047/25.047 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence). 25.047/25.047 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence). 25.047/25.047 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence). 25.047/25.047 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence). 25.047/25.047 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence). 25.047/25.047 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence). 25.047/25.047 ->_s0(a,c) # label(replacement). 25.047/25.047 ->_s0(b,c) # label(replacement). 25.047/25.047 ->*_s0(s(x2),t) -> ->_s0(f(x2),A) # label(replacement). 25.047/25.047 ->*_s0(s(x2),t) -> ->_s0(f(x2),B) # label(replacement). 25.047/25.047 ->_s0(g(x2,x2),h(x2,x2)) # label(replacement). 25.047/25.047 ->_s0(s(a),t) # label(replacement). 25.047/25.047 ->_s0(s(b),t) # label(replacement). 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 formulas(goals). 25.047/25.047 (exists x2 ->*_s0(s(x2),t)) # label(goal). 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 ============================== end of input ========================== 25.047/25.047 25.047/25.047 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.047/25.047 25.047/25.047 % Formulas that are not ordinary clauses: 25.047/25.047 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.047/25.047 2 ->_s0(x1,y) -> ->_s0(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 3 ->_s0(x1,y) -> ->_s0(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 4 ->_s0(x2,y) -> ->_s0(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 5 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 6 ->_s0(x1,y) -> ->_s0(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 7 ->_s0(x2,y) -> ->_s0(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 8 ->*_s0(s(x2),t) -> ->_s0(f(x2),A) # label(replacement) # label(non_clause). [assumption]. 25.047/25.047 9 ->*_s0(s(x2),t) -> ->_s0(f(x2),B) # label(replacement) # label(non_clause). [assumption]. 25.047/25.047 10 (exists x2 ->*_s0(s(x2),t)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.047/25.047 25.047/25.047 ============================== end of process non-clausal formulas === 25.047/25.047 25.047/25.047 ============================== PROCESS INITIAL CLAUSES =============== 25.047/25.047 25.047/25.047 % Clauses before input processing: 25.047/25.047 25.047/25.047 formulas(usable). 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 formulas(sos). 25.047/25.047 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.047/25.047 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.047/25.047 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.047/25.047 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 25.047/25.047 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 25.047/25.047 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 25.047/25.047 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 25.047/25.047 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 25.047/25.047 ->_s0(a,c) # label(replacement). [assumption]. 25.047/25.047 ->_s0(b,c) # label(replacement). [assumption]. 25.047/25.047 -->*_s0(s(x),t) | ->_s0(f(x),A) # label(replacement). [clausify(8)]. 25.047/25.047 -->*_s0(s(x),t) | ->_s0(f(x),B) # label(replacement). [clausify(9)]. 25.047/25.047 ->_s0(g(x,x),h(x,x)) # label(replacement). [assumption]. 25.047/25.047 ->_s0(s(a),t) # label(replacement). [assumption]. 25.047/25.047 ->_s0(s(b),t) # label(replacement). [assumption]. 25.047/25.047 -->*_s0(s(x),t) # label(goal). [deny(10)]. 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 formulas(demodulators). 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 ============================== PREDICATE ELIMINATION ================= 25.047/25.047 25.047/25.047 No predicates eliminated. 25.047/25.047 25.047/25.047 ============================== end predicate elimination ============= 25.047/25.047 25.047/25.047 Auto_denials: 25.047/25.047 % copying label goal to answer in negative clause 25.047/25.047 25.047/25.047 Term ordering decisions: 25.047/25.047 Predicate symbol precedence: predicate_order([ ->_s0, ->*_s0 ]). 25.047/25.047 Function symbol precedence: function_order([ t, a, b, c, A, B, h, g, s, f ]). 25.047/25.047 After inverse_order: (no changes). 25.047/25.047 Unfolding symbols: (none). 25.047/25.047 25.047/25.047 Auto_inference settings: 25.047/25.047 % set(neg_binary_resolution). % (HNE depth_diff=-6) 25.047/25.047 % clear(ordered_res). % (HNE depth_diff=-6) 25.047/25.047 % set(ur_resolution). % (HNE depth_diff=-6) 25.047/25.047 % set(ur_resolution) -> set(pos_ur_resolution). 25.047/25.047 % set(ur_resolution) -> set(neg_ur_resolution). 25.047/25.047 25.047/25.047 Auto_process settings: (no changes). 25.047/25.047 25.047/25.047 kept: 11 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.047/25.047 kept: 12 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.047/25.047 kept: 13 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.047/25.047 kept: 14 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 25.047/25.047 kept: 15 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 25.047/25.047 kept: 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 25.047/25.047 kept: 17 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 25.047/25.047 kept: 18 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 25.047/25.047 kept: 19 ->_s0(a,c) # label(replacement). [assumption]. 25.047/25.047 kept: 20 ->_s0(b,c) # label(replacement). [assumption]. 25.047/25.047 kept: 21 -->*_s0(s(x),t) | ->_s0(f(x),A) # label(replacement). [clausify(8)]. 25.047/25.047 kept: 22 -->*_s0(s(x),t) | ->_s0(f(x),B) # label(replacement). [clausify(9)]. 25.047/25.047 kept: 23 ->_s0(g(x,x),h(x,x)) # label(replacement). [assumption]. 25.047/25.047 kept: 24 ->_s0(s(a),t) # label(replacement). [assumption]. 25.047/25.047 kept: 25 ->_s0(s(b),t) # label(replacement). [assumption]. 25.047/25.047 kept: 26 -->*_s0(s(x),t) # label(goal) # answer(goal). [deny(10)]. 25.047/25.047 25.047/25.047 ============================== end of process initial clauses ======== 25.047/25.047 25.047/25.047 ============================== CLAUSES FOR SEARCH ==================== 25.047/25.047 25.047/25.047 % Clauses after input processing: 25.047/25.047 25.047/25.047 formulas(usable). 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 formulas(sos). 25.047/25.047 11 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.047/25.047 12 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.047/25.047 13 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.047/25.047 14 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 25.047/25.047 15 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 25.047/25.047 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 25.047/25.047 17 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 25.047/25.047 18 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 25.047/25.047 19 ->_s0(a,c) # label(replacement). [assumption]. 25.047/25.047 20 ->_s0(b,c) # label(replacement). [assumption]. 25.047/25.047 23 ->_s0(g(x,x),h(x,x)) # label(replacement). [assumption]. 25.047/25.047 24 ->_s0(s(a),t) # label(replacement). [assumption]. 25.047/25.047 25 ->_s0(s(b),t) # label(replacement). [assumption]. 25.047/25.047 26 -->*_s0(s(x),t) # label(goal) # answer(goal). [deny(10)]. 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 formulas(demodulators). 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 ============================== end of clauses for search ============= 25.047/25.047 25.047/25.047 ============================== SEARCH ================================ 25.047/25.047 25.047/25.047 % Starting search at 0.00 seconds. 25.047/25.047 25.047/25.047 given #1 (I,wt=3): 11 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.047/25.047 25.047/25.047 given #2 (I,wt=9): 12 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.047/25.047 25.047/25.047 given #3 (I,wt=8): 13 -->_s0(x,y) | ->_s0(f(x),f(y)) # label(congruence). [clausify(2)]. 25.047/25.047 25.047/25.047 given #4 (I,wt=10): 14 -->_s0(x,y) | ->_s0(g(x,z),g(y,z)) # label(congruence). [clausify(3)]. 25.047/25.047 25.047/25.047 given #5 (I,wt=10): 15 -->_s0(x,y) | ->_s0(g(z,x),g(z,y)) # label(congruence). [clausify(4)]. 25.047/25.047 25.047/25.047 given #6 (I,wt=8): 16 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(5)]. 25.047/25.047 25.047/25.047 given #7 (I,wt=10): 17 -->_s0(x,y) | ->_s0(h(x,z),h(y,z)) # label(congruence). [clausify(6)]. 25.047/25.047 25.047/25.047 given #8 (I,wt=10): 18 -->_s0(x,y) | ->_s0(h(z,x),h(z,y)) # label(congruence). [clausify(7)]. 25.047/25.047 25.047/25.047 given #9 (I,wt=3): 19 ->_s0(a,c) # label(replacement). [assumption]. 25.047/25.047 25.047/25.047 given #10 (I,wt=3): 20 ->_s0(b,c) # label(replacement). [assumption]. 25.047/25.047 25.047/25.047 given #11 (I,wt=7): 23 ->_s0(g(x,x),h(x,x)) # label(replacement). [assumption]. 25.047/25.047 25.047/25.047 given #12 (I,wt=4): 24 ->_s0(s(a),t) # label(replacement). [assumption]. 25.047/25.047 25.047/25.047 ============================== PROOF ================================= 25.047/25.047 25.047/25.047 % Proof 1 at 0.00 (+ 0.00) seconds: goal. 25.047/25.047 % Length of proof is 8. 25.047/25.047 % Level of proof is 3. 25.047/25.047 % Maximum clause weight is 9.000. 25.047/25.047 % Given clauses 12. 25.047/25.047 25.047/25.047 1 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.047/25.047 10 (exists x2 ->*_s0(s(x2),t)) # label(goal) # label(non_clause) # label(goal). [goal]. 25.047/25.047 11 ->*_s0(x,x) # label(reflexivity). [assumption]. 25.047/25.047 12 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(1)]. 25.047/25.047 24 ->_s0(s(a),t) # label(replacement). [assumption]. 25.047/25.047 26 -->*_s0(s(x),t) # label(goal) # answer(goal). [deny(10)]. 25.047/25.047 54 ->*_s0(s(a),t). [ur(12,a,24,a,b,11,a)]. 25.047/25.047 55 $F # answer(goal). [resolve(54,a,26,a)]. 25.047/25.047 25.047/25.047 ============================== end of proof ========================== 25.047/25.047 25.047/25.047 ============================== STATISTICS ============================ 25.047/25.047 25.047/25.047 Given=12. Generated=44. Kept=44. proofs=1. 25.047/25.047 Usable=12. Sos=23. Demods=0. Limbo=6, Disabled=18. Hints=0. 25.047/25.047 Kept_by_rule=0, Deleted_by_rule=0. 25.047/25.047 Forward_subsumed=0. Back_subsumed=2. 25.047/25.047 Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. 25.047/25.047 New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. 25.047/25.047 Demod_attempts=0. Demod_rewrites=0. 25.047/25.047 Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. 25.047/25.047 Nonunit_fsub_feature_tests=2. Nonunit_bsub_feature_tests=16. 25.047/25.047 Megabytes=0.11. 25.047/25.047 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.047/25.047 25.047/25.047 ============================== end of statistics ===================== 25.047/25.047 25.047/25.047 ============================== end of search ========================= 25.047/25.047 25.047/25.047 THEOREM PROVED 25.047/25.047 25.047/25.047 Exiting with 1 proof. 25.047/25.047 25.047/25.047 Process 50635 exit (max_proofs) Wed Jul 14 10:55:50 2021 25.047/25.047 25.047/25.047 25.047/25.047 The problem is feasible. 25.047/25.047 25.047/25.047 25.047/25.047 CRule InfChecker Info: 25.047/25.047 g(x:S,x:S) -> h(x:S,x:S) 25.047/25.047 Rule remains 25.047/25.047 Proof: 25.047/25.047 NO_CONDS 25.047/25.047 25.047/25.047 CRule InfChecker Info: 25.047/25.047 s(a) -> t 25.047/25.047 Rule remains 25.047/25.047 Proof: 25.047/25.047 NO_CONDS 25.047/25.047 25.047/25.047 CRule InfChecker Info: 25.047/25.047 s(b) -> t 25.047/25.047 Rule remains 25.047/25.047 Proof: 25.047/25.047 NO_CONDS 25.047/25.047 25.047/25.047 Problem 1: 25.047/25.047 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.047/25.047 Confluence Problem: 25.047/25.047 (VAR x:S) 25.047/25.047 (STRATEGY CONTEXTSENSITIVE 25.047/25.047 (a) 25.047/25.047 (b) 25.047/25.047 (f 1) 25.047/25.047 (g 1 2) 25.047/25.047 (s 1) 25.047/25.047 (A) 25.047/25.047 (B) 25.047/25.047 (c) 25.047/25.047 (fSNonEmpty) 25.047/25.047 (h 1 2) 25.047/25.047 (t) 25.047/25.047 ) 25.047/25.047 (RULES 25.047/25.047 a -> c 25.047/25.047 b -> c 25.047/25.047 f(x:S) -> A | s(x:S) ->* t 25.047/25.047 f(x:S) -> B | s(x:S) ->* t 25.047/25.047 g(x:S,x:S) -> h(x:S,x:S) 25.047/25.047 s(a) -> t 25.047/25.047 s(b) -> t 25.047/25.047 ) 25.047/25.047 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.047/25.047 25.047/25.047 Critical Pairs Processor: 25.047/25.047 -> Rules: 25.047/25.047 a -> c 25.047/25.047 b -> c 25.047/25.047 f(x:S) -> A | s(x:S) ->* t 25.047/25.047 f(x:S) -> B | s(x:S) ->* t 25.047/25.047 g(x:S,x:S) -> h(x:S,x:S) 25.047/25.047 s(a) -> t 25.047/25.047 s(b) -> t 25.047/25.047 -> Vars: 25.047/25.047 "x", "x", "x" 25.047/25.047 -> FVars: 25.047/25.047 "x2", "x3", "x4" 25.047/25.047 -> PVars: 25.047/25.047 "x": ["x2", "x3", "x4"] 25.047/25.047 25.047/25.047 -> Rlps: 25.047/25.047 crule: a -> c, id: 1, possubterms: a-> [] 25.047/25.047 crule: b -> c, id: 2, possubterms: b-> [] 25.047/25.047 crule: f(x2:S) -> A | s(x2:S) ->* t, id: 3, possubterms: f(x2:S)-> [] 25.047/25.047 crule: f(x3:S) -> B | s(x3:S) ->* t, id: 4, possubterms: f(x3:S)-> [] 25.047/25.047 crule: g(x4:S,x4:S) -> h(x4:S,x4:S), id: 5, possubterms: g(x4:S,x4:S)-> [] 25.047/25.047 crule: s(a) -> t, id: 6, possubterms: s(a)-> [], a-> [1] 25.047/25.047 crule: s(b) -> t, id: 7, possubterms: s(b)-> [], b-> [1] 25.047/25.047 25.047/25.047 -> Unifications: 25.047/25.047 R4 unifies with R3 at p: [], l: f(x3:S), lp: f(x3:S), conds: {s(x3:S) ->* t, s(x:S) ->* t}, sig: {x3:S -> x:S}, l': f(x:S), r: B, r': A 25.047/25.047 R6 unifies with R1 at p: [1], l: s(a), lp: a, conds: {}, sig: {}, l': a, r: t, r': c 25.047/25.047 R7 unifies with R2 at p: [1], l: s(b), lp: b, conds: {}, sig: {}, l': b, r: t, r': c 25.047/25.047 25.047/25.047 -> Critical pairs info: 25.047/25.047 => Not trivial, Not overlay, N1 25.047/25.047 | s(x:S) ->* t => Not trivial, Overlay, N2 25.047/25.047 25.047/25.047 -> Problem conclusions: 25.047/25.047 Not left linear, Not right linear, Not linear 25.047/25.047 Not weakly orthogonal, Not almost orthogonal, Not orthogonal 25.047/25.047 CTRS Type: 1 25.047/25.047 Deterministic, Strongly deterministic 25.047/25.047 Oriented CTRS, Properly oriented CTRS, Not join CTRS 25.047/25.047 Maybe right-stable CTRS, Not overlay CTRS 25.047/25.047 Maybe normal CTRS, Maybe almost normal CTRS 25.047/25.047 Maybe terminating CTRS, Maybe joinable CCPs 25.047/25.047 Maybe level confluent 25.047/25.047 Maybe confluent 25.047/25.047 25.047/25.047 Problem 1: 25.047/25.047 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.047/25.047 Confluence Problem: 25.047/25.047 (VAR x:S) 25.047/25.047 (STRATEGY CONTEXTSENSITIVE 25.047/25.047 (a) 25.047/25.047 (b) 25.047/25.047 (f 1) 25.047/25.047 (g 1 2) 25.047/25.047 (s 1) 25.047/25.047 (A) 25.047/25.047 (B) 25.047/25.047 (c) 25.047/25.047 (fSNonEmpty) 25.047/25.047 (h 1 2) 25.047/25.047 (t) 25.047/25.047 ) 25.047/25.047 (RULES 25.047/25.047 a -> c 25.047/25.047 b -> c 25.047/25.047 f(x:S) -> A | s(x:S) ->* t 25.047/25.047 f(x:S) -> B | s(x:S) ->* t 25.047/25.047 g(x:S,x:S) -> h(x:S,x:S) 25.047/25.047 s(a) -> t 25.047/25.047 s(b) -> t 25.047/25.047 ) 25.047/25.047 Critical Pairs: 25.047/25.047 => Not trivial, Not overlay, N1 25.047/25.047 | s(x:S) ->* t => Not trivial, Overlay, N2 25.047/25.047 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.047/25.047 25.047/25.047 Conditional Critical Pairs Distributor Processor 25.047/25.047 25.047/25.047 Problem 1: 25.047/25.047 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.047/25.047 Confluence Problem: 25.047/25.047 (VAR x:S) 25.047/25.047 (STRATEGY CONTEXTSENSITIVE 25.047/25.047 (a) 25.047/25.047 (b) 25.047/25.047 (f 1) 25.047/25.047 (g 1 2) 25.047/25.047 (s 1) 25.047/25.047 (A) 25.047/25.047 (B) 25.047/25.047 (c) 25.047/25.047 (fSNonEmpty) 25.047/25.047 (h 1 2) 25.047/25.047 (t) 25.047/25.047 ) 25.047/25.047 (RULES 25.047/25.047 a -> c 25.047/25.047 b -> c 25.047/25.047 f(x:S) -> A | s(x:S) ->* t 25.047/25.047 f(x:S) -> B | s(x:S) ->* t 25.047/25.047 g(x:S,x:S) -> h(x:S,x:S) 25.047/25.047 s(a) -> t 25.047/25.047 s(b) -> t 25.047/25.047 ) 25.047/25.047 Critical Pairs: 25.047/25.047 => Not trivial, Not overlay, N1 25.047/25.047 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 25.047/25.047 25.047/25.047 Infeasible Convergence No Conditions CCP Processor: 25.047/25.047 Infeasible convergence? 25.047/25.047 YES 25.047/25.047 25.047/25.047 Problem 1: 25.047/25.047 25.047/25.047 Infeasibility Problem: 25.047/25.047 [(VAR vNonEmpty:S x:S z:S) 25.047/25.047 (STRATEGY CONTEXTSENSITIVE 25.047/25.047 (a) 25.047/25.047 (b) 25.047/25.047 (f 1) 25.047/25.047 (g 1 2) 25.047/25.047 (s 1) 25.047/25.047 (A) 25.047/25.047 (B) 25.047/25.047 (c) 25.047/25.047 (fSNonEmpty) 25.047/25.047 (h 1 2) 25.047/25.047 (t) 25.047/25.047 ) 25.047/25.047 (RULES 25.047/25.047 a -> c 25.047/25.047 b -> c 25.047/25.047 f(x:S) -> A | s(x:S) ->* t 25.047/25.047 f(x:S) -> B | s(x:S) ->* t 25.047/25.047 g(x:S,x:S) -> h(x:S,x:S) 25.047/25.047 s(a) -> t 25.047/25.047 s(b) -> t 25.047/25.047 )] 25.047/25.047 25.047/25.047 Infeasibility Conditions: 25.047/25.047 s(c) ->* z:S, t ->* z:S 25.047/25.047 25.047/25.047 Problem 1: 25.047/25.047 25.047/25.047 Obtaining a model using Mace4: 25.047/25.047 25.047/25.047 -> Usable Rules: 25.047/25.047 s(a) -> t 25.047/25.047 s(b) -> t 25.047/25.047 25.047/25.047 -> Mace4 Output: 25.047/25.047 ============================== Mace4 ================================= 25.047/25.047 Mace4 (64) version 2009-11A, November 2009. 25.047/25.047 Process 50683 was started by ubuntu on ubuntu, 25.047/25.047 Wed Jul 14 10:55:50 2021 25.047/25.047 The command was "./mace4 -c -f /tmp/mace450651-2.in". 25.047/25.047 ============================== end of head =========================== 25.047/25.047 25.047/25.047 ============================== INPUT ================================= 25.047/25.047 25.047/25.047 % Reading from file /tmp/mace450651-2.in 25.047/25.047 25.047/25.047 assign(max_seconds,20). 25.047/25.047 25.047/25.047 formulas(assumptions). 25.047/25.047 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence). 25.047/25.047 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence). 25.047/25.047 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence). 25.047/25.047 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence). 25.047/25.047 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence). 25.047/25.047 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence). 25.047/25.047 ->(s(a),t) # label(replacement). 25.047/25.047 ->(s(b),t) # label(replacement). 25.047/25.047 ->*(x,x) # label(reflexivity). 25.047/25.047 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity). 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 formulas(goals). 25.047/25.047 (exists x2 (->*(s(c),x2) & ->*(t,x2))) # label(goal). 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 ============================== end of input ========================== 25.047/25.047 25.047/25.047 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 25.047/25.047 25.047/25.047 % Formulas that are not ordinary clauses: 25.047/25.047 1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 2 ->(x1,y) -> ->(g(x1,x2),g(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 3 ->(x2,y) -> ->(g(x1,x2),g(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 4 ->(x1,y) -> ->(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 5 ->(x1,y) -> ->(h(x1,x2),h(y,x2)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 6 ->(x2,y) -> ->(h(x1,x2),h(x1,y)) # label(congruence) # label(non_clause). [assumption]. 25.047/25.047 7 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause). [assumption]. 25.047/25.047 8 (exists x2 (->*(s(c),x2) & ->*(t,x2))) # label(goal) # label(non_clause) # label(goal). [goal]. 25.047/25.047 25.047/25.047 ============================== end of process non-clausal formulas === 25.047/25.047 25.047/25.047 ============================== CLAUSES FOR SEARCH ==================== 25.047/25.047 25.047/25.047 formulas(mace4_clauses). 25.047/25.047 -->(x,y) | ->(f(x),f(y)) # label(congruence). 25.047/25.047 -->(x,y) | ->(g(x,z),g(y,z)) # label(congruence). 25.047/25.047 -->(x,y) | ->(g(z,x),g(z,y)) # label(congruence). 25.047/25.047 -->(x,y) | ->(s(x),s(y)) # label(congruence). 25.047/25.047 -->(x,y) | ->(h(x,z),h(y,z)) # label(congruence). 25.047/25.047 -->(x,y) | ->(h(z,x),h(z,y)) # label(congruence). 25.047/25.047 ->(s(a),t) # label(replacement). 25.047/25.047 ->(s(b),t) # label(replacement). 25.047/25.047 ->*(x,x) # label(reflexivity). 25.047/25.047 -->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity). 25.047/25.047 -->*(s(c),x) | -->*(t,x) # label(goal). 25.047/25.047 end_of_list. 25.047/25.047 25.047/25.047 ============================== end of clauses for search ============= 25.047/25.047 25.047/25.047 % There are no natural numbers in the input. 25.047/25.047 25.047/25.047 ============================== DOMAIN SIZE 2 ========================= 25.047/25.047 25.047/25.047 ============================== MODEL ================================= 25.047/25.047 25.047/25.047 interpretation( 2, [number=1, seconds=0], [ 25.047/25.047 25.047/25.047 function(t, [ 0 ]), 25.047/25.047 25.047/25.047 function(a, [ 0 ]), 25.047/25.047 25.047/25.047 function(b, [ 0 ]), 25.047/25.047 25.047/25.047 function(c, [ 1 ]), 25.047/25.047 25.047/25.047 function(f(_), [ 0, 0 ]), 25.047/25.047 25.047/25.047 function(s(_), [ 0, 1 ]), 25.047/25.047 25.047/25.047 function(h(_,_), [ 25.047/25.047 0, 0, 25.047/25.047 0, 0 ]), 25.047/25.047 25.047/25.047 function(g(_,_), [ 25.047/25.047 0, 0, 25.047/25.047 0, 0 ]), 25.047/25.047 25.047/25.047 relation(->*(_,_), [ 25.047/25.047 1, 0, 25.047/25.047 0, 1 ]), 25.047/25.047 25.047/25.047 relation(->(_,_), [ 25.047/25.047 1, 0, 25.047/25.047 0, 0 ]) 25.047/25.047 ]). 25.047/25.047 25.047/25.047 ============================== end of model ========================== 25.047/25.047 25.047/25.047 ============================== STATISTICS ============================ 25.047/25.047 25.047/25.047 For domain size 2. 25.047/25.047 25.047/25.047 Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds). 25.047/25.047 Ground clauses: seen=54, kept=50. 25.047/25.047 Selections=15, assignments=16, propagations=13, current_models=1. 25.047/25.047 Rewrite_terms=102, rewrite_bools=70, indexes=13. 25.047/25.047 Rules_from_neg_clauses=3, cross_offs=3. 25.047/25.047 25.047/25.047 ============================== end of statistics ===================== 25.047/25.047 25.047/25.047 User_CPU=0.00, System_CPU=0.00, Wall_clock=0. 25.047/25.047 25.047/25.047 Exiting with 1 model. 25.047/25.047 25.047/25.047 Process 50683 exit (max_models) Wed Jul 14 10:55:50 2021 25.047/25.047 The process finished Wed Jul 14 10:55:50 2021 25.047/25.047 25.047/25.047 25.047/25.047 Mace4 cooked interpretation: 25.047/25.047 25.047/25.047 25.047/25.047 25.047/25.047 The problem is infeasible. 25.047/25.047 25.047/25.047 25.047/25.047 The problem is not joinable. 25.047/25.047 19.35user 0.74system 0:25.47elapsed 78%CPU (0avgtext+0avgdata 116584maxresident)k 25.047/25.047 0inputs+0outputs (0major+61312minor)pagefaults 0swaps