============================== Prover9 =============================== Prover9 (64) version Aug-2007, Aug 2007. Process 32400 was started by veroff on io, Mon Jan 21 11:14:49 2008 The command was "prover9 -f 3.4.in". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file 3.4.in clear(auto). % clear(auto) -> clear(auto_inference). % clear(auto_inference) -> clear(predicate_elim). % clear(auto_inference) -> assign(eq_defs, pass). % clear(auto) -> clear(auto_limits). % clear(auto_limits) -> assign(max_weight, 2147483647). % clear(auto_limits) -> assign(sos_limit, -1). % clear(auto) -> clear(auto_denials). % clear(auto) -> clear(auto_process). op(400,infix,[^,v,"->","=>",*]). op(200,prefix,[~,-]). lex([0,1,*,^,=>,->,v,~,-]). assign(order,kbo). set(lex_order_vars). set(paramodulation). % set(paramodulation) -> set(back_demod). set(para_units_only). % set(para_units_only) -> assign(para_lit_limit, 1). clear(back_demod). set(hyper_resolution). % set(hyper_resolution) -> set(pos_hyper_resolution). set(auto_denials). set(restrict_denials). formulas(assumptions). (x v y) v z = x v (y v z) # label("(D1)"). (x ^ y) ^ z = x ^ (y ^ z) # label("(D2)"). x v y = y v x # label("(D3)"). x ^ y = y ^ x # label("(D4)"). x v (x ^ y) = x # label("(D5)"). x ^ (x v y) = x # label("(D6)"). (x v y) ^ (x v z) = x v (y ^ z) # label("(D7)"). (x ^ y) v (x ^ z) = x ^ (y v z) # label("(D8)"). x v 0 = x # label("(D9)"). x ^ 1 = x # label("(D10)"). x ^ 0 = 0 # label("(2.1)"). ~ x v ~ y = ~ (x ^ y) # label("(DM1)"). ~ x ^ ~ y = ~ (x v y) # label("(DM2)"). ~ ~ x = x # label("(DN)"). ~ 1 = 0 # label("(2.2)"). (x ^ ~ x) ^ (y v ~ y) = x ^ ~ x # label("(N1)"). x -> x = 1 # label("(N2)"). (x -> y) ^ (~ x v y) = ~ x v y # label("(N3)"). x ^ (~ x v y) = x ^ (x -> y) # label("(N4)"). (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). (x ^ y) -> z = x -> (y -> z) # label("(N6)"). -x = x -> 0 # label("(N7)"). x * y = ~ (x -> ~ y) v ~ (y -> ~ x) # label("(* def)"). x => y = (x -> y) ^ (~ y -> ~ x) # label("(=> def)"). x -> 1 = 1 # label("(3.1)"). 1 -> x = x # label("(3.2)"). (x -> y) -> (x -> z) = x -> (y -> z) # label("(3.3)"). x -> (y -> z) = y -> (x -> z) # label("(3.4)"). x -> (x -> y) = x -> y # label("(3.5)"). (x -> y) ^ (z -> y) = (x v z) -> y # label("(3.6)"). x ^ ((x => y) => y) = x # label("(3.7)"). x -> ~ (x -> y) = x -> ~ y # label("(3.8)"). ~ (x -> y) -> z = x -> (~ y -> z) # label("(3.9)"). 1 => x = x # label("(BCK2)"). x => 1 = 1 # label("(BCK3)"). x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). end_of_list. formulas(assumptions). (x ^ y) -> y = 1 # label("(3.10)"). (x ^ y) => y = 1 # label("(3.11)"). ~ x => ~ y = y => x # label("(3.12)"). (x => y) ^ y = y # label("(3.13)"). x => (x ^ y) = x => y # label("(3.14)"). x => (x => y) = x -> y # label("(3.15)"). (x => y) => ((z => x) => (z => y)) = 1 # label("(BCK1')"). end_of_list. formulas(goals). x => ((x => y) => y) = 1 # label("(BCK5)"). end_of_list. formulas(hints). x => ((x => y) => y) = 1. end_of_list. ============================== end of input ========================== ============================== PROCESS NON-CLAUSAL FORMULAS ========== % Formulas that are not ordinary clauses: 1 x => ((x => y) => y) = 1 # label("(BCK5)") # label(non_clause) # label(goal). [goal]. ============================== end of process non-clausal formulas === ============================== PROCESS INITIAL CLAUSES =============== % Clauses before input processing: formulas(usable). end_of_list. formulas(sos). (x v y) v z = x v (y v z) # label("(D1)"). [assumption]. (x ^ y) ^ z = x ^ (y ^ z) # label("(D2)"). [assumption]. x v y = y v x # label("(D3)"). [assumption]. x ^ y = y ^ x # label("(D4)"). [assumption]. x v (x ^ y) = x # label("(D5)"). [assumption]. x ^ (x v y) = x # label("(D6)"). [assumption]. (x v y) ^ (x v z) = x v (y ^ z) # label("(D7)"). [assumption]. (x ^ y) v (x ^ z) = x ^ (y v z) # label("(D8)"). [assumption]. x v 0 = x # label("(D9)"). [assumption]. x ^ 1 = x # label("(D10)"). [assumption]. x ^ 0 = 0 # label("(2.1)"). [assumption]. ~ x v ~ y = ~ (x ^ y) # label("(DM1)"). [assumption]. ~ x ^ ~ y = ~ (x v y) # label("(DM2)"). [assumption]. ~ ~ x = x # label("(DN)"). [assumption]. ~ 1 = 0 # label("(2.2)"). [assumption]. (x ^ ~ x) ^ (y v ~ y) = x ^ ~ x # label("(N1)"). [assumption]. x -> x = 1 # label("(N2)"). [assumption]. (x -> y) ^ (~ x v y) = ~ x v y # label("(N3)"). [assumption]. x ^ (~ x v y) = x ^ (x -> y) # label("(N4)"). [assumption]. (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). [assumption]. (x ^ y) -> z = x -> (y -> z) # label("(N6)"). [assumption]. -x = x -> 0 # label("(N7)"). [assumption]. x * y = ~ (x -> ~ y) v ~ (y -> ~ x) # label("(* def)"). [assumption]. x => y = (x -> y) ^ (~ y -> ~ x) # label("(=> def)"). [assumption]. x -> 1 = 1 # label("(3.1)"). [assumption]. 1 -> x = x # label("(3.2)"). [assumption]. (x -> y) -> (x -> z) = x -> (y -> z) # label("(3.3)"). [assumption]. x -> (y -> z) = y -> (x -> z) # label("(3.4)"). [assumption]. x -> (x -> y) = x -> y # label("(3.5)"). [assumption]. (x -> y) ^ (z -> y) = (x v z) -> y # label("(3.6)"). [assumption]. x ^ ((x => y) => y) = x # label("(3.7)"). [assumption]. x -> ~ (x -> y) = x -> ~ y # label("(3.8)"). [assumption]. ~ (x -> y) -> z = x -> (~ y -> z) # label("(3.9)"). [assumption]. 1 => x = x # label("(BCK2)"). [assumption]. x => 1 = 1 # label("(BCK3)"). [assumption]. x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). [assumption]. (x ^ y) -> y = 1 # label("(3.10)"). [assumption]. (x ^ y) => y = 1 # label("(3.11)"). [assumption]. ~ x => ~ y = y => x # label("(3.12)"). [assumption]. (x => y) ^ y = y # label("(3.13)"). [assumption]. x => (x ^ y) = x => y # label("(3.14)"). [assumption]. x => (x => y) = x -> y # label("(3.15)"). [assumption]. (x => y) => ((z => x) => (z => y)) = 1 # label("(BCK1')"). [assumption]. c1 => ((c1 => c2) => c2) != 1 # label("(BCK5)"). [deny(1)]. end_of_list. formulas(demodulators). end_of_list. % 1 hints input. Auto_denials: % copying label "(BCK5)" to answer in negative clause Term ordering decisions: Function symbol KB weights: 0=1. 1=1. c1=1. c2=1. *=1. ^=1. =>=1. ->=1. v=1. ~=1. -=1. Predicate symbol precedence: predicate_order([ = ]). Function symbol precedence: function_order([ 0, 1, c1, c2, *, ^, =>, ->, v, ~, - ]). Skipping inverse_order, because there is a function_order (lex) command. % Operation v is commutative; C redundancy checks enabled. % Operation ^ is commutative; C redundancy checks enabled. restricted denial: (wt=9): 48 c1 => ((c1 => c2) => c2) != 1 # label("(BCK5)") # answer("(BCK5)"). [deny(1)]. ============================== end of process initial clauses ======== ============================== CLAUSES FOR SEARCH ==================== % Clauses after input processing: formulas(usable). 48 c1 => ((c1 => c2) => c2) != 1 # label("(BCK5)") # answer("(BCK5)"). [deny(1)]. end_of_list. formulas(sos). 2 (x v y) v z = x v (y v z) # label("(D1)"). [assumption]. 3 (x ^ y) ^ z = x ^ (y ^ z) # label("(D2)"). [assumption]. 4 x v y = y v x # label("(D3)"). [assumption]. 5 x ^ y = y ^ x # label("(D4)"). [assumption]. 6 x v (x ^ y) = x # label("(D5)"). [assumption]. 7 x ^ (x v y) = x # label("(D6)"). [assumption]. 8 (x v y) ^ (x v z) = x v (y ^ z) # label("(D7)"). [assumption]. 9 (x ^ y) v (x ^ z) = x ^ (y v z) # label("(D8)"). [assumption]. 10 x v 0 = x # label("(D9)"). [assumption]. 11 x ^ 1 = x # label("(D10)"). [assumption]. 12 x ^ 0 = 0 # label("(2.1)"). [assumption]. 13 ~ x v ~ y = ~ (x ^ y) # label("(DM1)"). [assumption]. 14 ~ x ^ ~ y = ~ (x v y) # label("(DM2)"). [assumption]. 15 ~ ~ x = x # label("(DN)"). [assumption]. 16 ~ 1 = 0 # label("(2.2)"). [assumption]. 17 (x ^ ~ x) ^ (y v ~ y) = x ^ ~ x # label("(N1)"). [assumption]. 18 x -> x = 1 # label("(N2)"). [assumption]. 19 (x -> y) ^ (~ x v y) = ~ x v y # label("(N3)"). [assumption]. 20 x ^ (~ x v y) = x ^ (x -> y) # label("(N4)"). [assumption]. 21 (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). [assumption]. 22 (x ^ y) -> z = x -> (y -> z) # label("(N6)"). [assumption]. 24 x -> 0 = -x. [copy(23),flip(a)]. 26 ~ (x -> ~ y) v ~ (y -> ~ x) = x * y. [copy(25),flip(a)]. 28 (x -> y) ^ (~ y -> ~ x) = x => y. [copy(27),flip(a)]. 29 x -> 1 = 1 # label("(3.1)"). [assumption]. 30 1 -> x = x # label("(3.2)"). [assumption]. 31 (x -> y) -> (x -> z) = x -> (y -> z) # label("(3.3)"). [assumption]. 32 x -> (y -> z) = y -> (x -> z) # label("(3.4)"). [assumption]. 33 x -> (x -> y) = x -> y # label("(3.5)"). [assumption]. 34 (x -> y) ^ (z -> y) = (x v z) -> y # label("(3.6)"). [assumption]. 35 x ^ ((x => y) => y) = x # label("(3.7)"). [assumption]. 36 x -> ~ (x -> y) = x -> ~ y # label("(3.8)"). [assumption]. 37 ~ (x -> y) -> z = x -> (~ y -> z) # label("(3.9)"). [assumption]. 38 1 => x = x # label("(BCK2)"). [assumption]. 39 x => 1 = 1 # label("(BCK3)"). [assumption]. 40 x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). [assumption]. 41 (x ^ y) -> y = 1 # label("(3.10)"). [assumption]. 42 (x ^ y) => y = 1 # label("(3.11)"). [assumption]. 43 ~ x => ~ y = y => x # label("(3.12)"). [assumption]. 44 (x => y) ^ y = y # label("(3.13)"). [assumption]. 45 x => (x ^ y) = x => y # label("(3.14)"). [assumption]. 46 x => (x => y) = x -> y # label("(3.15)"). [assumption]. 47 (x => y) => ((z => x) => (z => y)) = 1 # label("(BCK1')"). [assumption]. end_of_list. formulas(demodulators). end_of_list. % 1 hints processed (0 are redundant). ============================== end of clauses for search ============= ============================== SEARCH ================================ % Starting search at 0.01 seconds. given #1 (I,wt=11): 2 (x v y) v z = x v (y v z) # label("(D1)"). [assumption]. given #2 (I,wt=11): 3 (x ^ y) ^ z = x ^ (y ^ z) # label("(D2)"). [assumption]. given #3 (I,wt=7): 4 x v y = y v x # label("(D3)"). [assumption]. given #4 (I,wt=7): 5 x ^ y = y ^ x # label("(D4)"). [assumption]. given #5 (I,wt=7): 6 x v (x ^ y) = x # label("(D5)"). [assumption]. given #6 (I,wt=7): 7 x ^ (x v y) = x # label("(D6)"). [assumption]. given #7 (I,wt=13): 8 (x v y) ^ (x v z) = x v (y ^ z) # label("(D7)"). [assumption]. given #8 (I,wt=13): 9 (x ^ y) v (x ^ z) = x ^ (y v z) # label("(D8)"). [assumption]. given #9 (I,wt=5): 10 x v 0 = x # label("(D9)"). [assumption]. given #10 (I,wt=5): 11 x ^ 1 = x # label("(D10)"). [assumption]. given #11 (I,wt=5): 12 x ^ 0 = 0 # label("(2.1)"). [assumption]. given #12 (I,wt=10): 13 ~ x v ~ y = ~ (x ^ y) # label("(DM1)"). [assumption]. given #13 (I,wt=10): 14 ~ x ^ ~ y = ~ (x v y) # label("(DM2)"). [assumption]. given #14 (I,wt=5): 15 ~ ~ x = x # label("(DN)"). [assumption]. given #15 (I,wt=4): 16 ~ 1 = 0 # label("(2.2)"). [assumption]. given #16 (I,wt=14): 17 (x ^ ~ x) ^ (y v ~ y) = x ^ ~ x # label("(N1)"). [assumption]. given #17 (I,wt=5): 18 x -> x = 1 # label("(N2)"). [assumption]. given #18 (I,wt=13): 19 (x -> y) ^ (~ x v y) = ~ x v y # label("(N3)"). [assumption]. given #19 (I,wt=12): 20 x ^ (~ x v y) = x ^ (x -> y) # label("(N4)"). [assumption]. given #20 (I,wt=13): 21 (x -> y) ^ (x -> z) = x -> (y ^ z) # label("(N5)"). [assumption]. given #21 (I,wt=11): 22 (x ^ y) -> z = x -> (y -> z) # label("(N6)"). [assumption]. given #22 (I,wt=6): 24 x -> 0 = -x. [copy(23),flip(a)]. given #23 (I,wt=15): 26 ~ (x -> ~ y) v ~ (y -> ~ x) = x * y. [copy(25),flip(a)]. given #24 (I,wt=13): 28 (x -> y) ^ (~ y -> ~ x) = x => y. [copy(27),flip(a)]. given #25 (I,wt=5): 29 x -> 1 = 1 # label("(3.1)"). [assumption]. given #26 (I,wt=5): 30 1 -> x = x # label("(3.2)"). [assumption]. given #27 (I,wt=13): 31 (x -> y) -> (x -> z) = x -> (y -> z) # label("(3.3)"). [assumption]. given #28 (I,wt=11): 32 x -> (y -> z) = y -> (x -> z) # label("(3.4)"). [assumption]. given #29 (I,wt=9): 33 x -> (x -> y) = x -> y # label("(3.5)"). [assumption]. given #30 (I,wt=13): 34 (x -> y) ^ (z -> y) = (x v z) -> y # label("(3.6)"). [assumption]. given #31 (I,wt=9): 35 x ^ ((x => y) => y) = x # label("(3.7)"). [assumption]. given #32 (I,wt=11): 36 x -> ~ (x -> y) = x -> ~ y # label("(3.8)"). [assumption]. given #33 (I,wt=13): 37 ~ (x -> y) -> z = x -> (~ y -> z) # label("(3.9)"). [assumption]. given #34 (I,wt=5): 38 1 => x = x # label("(BCK2)"). [assumption]. given #35 (I,wt=5): 39 x => 1 = 1 # label("(BCK3)"). [assumption]. given #36 (I,wt=13): 40 x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). [assumption]. given #37 (I,wt=7): 41 (x ^ y) -> y = 1 # label("(3.10)"). [assumption]. given #38 (I,wt=7): 42 (x ^ y) => y = 1 # label("(3.11)"). [assumption]. -------- Proof 1 -------- "(BCK5)". ============================== PROOF ================================= % Proof 1 at 0.02 (+ 0.01) seconds: "(BCK5)". % Length of proof is 6. % Level of proof is 2. % Maximum clause weight is 9. % Given clauses 38. 1 x => ((x => y) => y) = 1 # label("(BCK5)") # label(non_clause) # label(goal). [goal]. 35 x ^ ((x => y) => y) = x # label("(3.7)"). [assumption]. 42 (x ^ y) => y = 1 # label("(3.11)"). [assumption]. 48 c1 => ((c1 => c2) => c2) != 1 # label("(BCK5)") # answer("(BCK5)"). [deny(1)]. 589 x => ((x => y) => y) = 1. [para(35(a,1),42(a,1,1))]. 590 $F # answer("(BCK5)"). [resolve(589,a,48,a)]. ============================== end of proof ========================== ============================== STATISTICS ============================ Given=38. Generated=746. Kept=585. proofs=1. Usable=39. Sos=530. Demods=0. Limbo=14, Disabled=45. Hints=1. Weight_deleted=0. Literals_deleted=0. Forward_subsumed=161. Back_subsumed=1. Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0. New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0. Demod_attempts=0. Demod_rewrites=0. Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0. Nonunit_fsub_feature_tests=0. Nonunit_bsub_feature_tests=1. Megabytes=0.82. User_CPU=0.02, System_CPU=0.01, Wall_clock=0. ============================== end of statistics ===================== ============================== end of search ========================= THEOREM PROVED THEOREM PROVED Exiting with 1 proof. ------ process 32400 exit (max_proofs) ------  Process 32400 exit (max_proofs) Mon Jan 21 11:14:49 2008