============================== Prover9 =============================== Prover9 (64) version Aug-2007, Aug 2007. Process 26278 was started by veroff on titan, Sat Jan 19 14:18:57 2008 The command was "prover9 -f 4.5.in". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file 4.5.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 => y) => ((y => z) => (x => z)) = 1 # label("(BCK1)"). 1 => x = x # label("(BCK2)"). x => 1 = 1 # label("(BCK3)"). x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). x * 1 = x # label("(M1)"). x * y = y * x # label("(M2)"). (x * y) * z = x * (y * z) # label("(M3)"). (x * y) => z = x => (y => z) # label("(P)"). x ^ 1 = x # label("(D10)"). x ^ ((x => y) => y) = x # label("(3.7)"). (x ^ y) => y = 1 # label("(3.11)"). x v 0 = x # label("(D9)"). x -> y = x => (x => y) # label("(-> def)"). ~ x = x => 0 # label("(~ def)"). -x = x => (x => 0) # label("(- def)"). x => (x => (x => y)) = x => (x => y) # label("(E_2)"). (x v y) ^ (x v z) = x v (y ^ z) # label("(D7)"). (x ^ y) v (x ^ z) = x ^ (y v z) # label("(D8)"). ~ ~ x = x # label("(DN)"). (x -> y) ^ (~ y -> ~ x) = x => y # label("(N)"). (x => y) => ((z => x) => (z => y)) = 1 # label("(BCK1')"). x => ((x => y) => y) = 1 # label("(BCK5)"). x => x = 1 # label("(3.16)"). x => (y => x) = 1 # label("(3.17)"). x => (y => z) = y => (x => z) # label("(3.18)"). (x * y) v (x * z) = x * (y v z) # label("(4.1)"). (x => y) ^ (x => z) = x => (y ^ z) # label("(4.2)"). (x * (x => y)) v y = y # label("(4.3)"). (x => y) ^ (z => y) = (x v z) => y # label("(4.4)"). ((x => y) * z) => (x => (y * z)) = 1 # label("(4.5)"). end_of_list. formulas(assumptions). (x => y) ^ y = y # label("(3.13)"). (x v y) => y = x => y # label("(4.6)"). x -> (y => z) = y => (x -> z) # label("(4.7)"). x ^ (~ x => y) = x # label("(4.8)"). x -> ((y v x) -> z) = x -> z # label("(4.9)"). x -> (~ x v y) = x -> y # label("(4.10)"). end_of_list. formulas(goals). x -> x = 1 # label("(N2)"). end_of_list. formulas(hints). 1 = x -> x. end_of_list. ============================== end of input ========================== ============================== PROCESS NON-CLAUSAL FORMULAS ========== % Formulas that are not ordinary clauses: 1 x -> x = 1 # label("(N2)") # 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 => y) => ((y => z) => (x => z)) = 1 # label("(BCK1)"). [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 * 1 = x # label("(M1)"). [assumption]. x * y = y * x # label("(M2)"). [assumption]. (x * y) * z = x * (y * z) # label("(M3)"). [assumption]. (x * y) => z = x => (y => z) # label("(P)"). [assumption]. x ^ 1 = x # label("(D10)"). [assumption]. x ^ ((x => y) => y) = x # label("(3.7)"). [assumption]. (x ^ y) => y = 1 # label("(3.11)"). [assumption]. x v 0 = x # label("(D9)"). [assumption]. x -> y = x => (x => y) # label("(-> def)"). [assumption]. ~ x = x => 0 # label("(~ def)"). [assumption]. -x = x => (x => 0) # label("(- def)"). [assumption]. x => (x => (x => y)) = x => (x => y) # label("(E_2)"). [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 = x # label("(DN)"). [assumption]. (x -> y) ^ (~ y -> ~ x) = x => y # label("(N)"). [assumption]. (x => y) => ((z => x) => (z => y)) = 1 # label("(BCK1')"). [assumption]. x => ((x => y) => y) = 1 # label("(BCK5)"). [assumption]. x => x = 1 # label("(3.16)"). [assumption]. x => (y => x) = 1 # label("(3.17)"). [assumption]. x => (y => z) = y => (x => z) # label("(3.18)"). [assumption]. (x * y) v (x * z) = x * (y v z) # label("(4.1)"). [assumption]. (x => y) ^ (x => z) = x => (y ^ z) # label("(4.2)"). [assumption]. (x * (x => y)) v y = y # label("(4.3)"). [assumption]. (x => y) ^ (z => y) = (x v z) => y # label("(4.4)"). [assumption]. ((x => y) * z) => (x => (y * z)) = 1 # label("(4.5)"). [assumption]. (x => y) ^ y = y # label("(3.13)"). [assumption]. (x v y) => y = x => y # label("(4.6)"). [assumption]. x -> (y => z) = y => (x -> z) # label("(4.7)"). [assumption]. x ^ (~ x => y) = x # label("(4.8)"). [assumption]. x -> ((y v x) -> z) = x -> z # label("(4.9)"). [assumption]. x -> (~ x v y) = x -> y # label("(4.10)"). [assumption]. c1 -> c1 != 1 # label("(N2)"). [deny(1)]. end_of_list. formulas(demodulators). end_of_list. % 1 hints input. Auto_denials: % copying label "(N2)" to answer in negative clause Term ordering decisions: Function symbol KB weights: 0=1. 1=1. c1=1. *=1. ^=1. =>=1. ->=1. v=1. ~=1. -=1. Predicate symbol precedence: predicate_order([ = ]). Function symbol precedence: function_order([ 0, 1, c1, *, ^, =>, ->, 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. % Operation * is commutative; C redundancy checks enabled. restricted denial: (wt=5): 47 c1 -> c1 != 1 # label("(N2)") # answer("(N2)"). [deny(1)]. ============================== end of process initial clauses ======== ============================== CLAUSES FOR SEARCH ==================== % Clauses after input processing: formulas(usable). 47 c1 -> c1 != 1 # label("(N2)") # answer("(N2)"). [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 => y) => ((y => z) => (x => z)) = 1 # label("(BCK1)"). [assumption]. 9 1 => x = x # label("(BCK2)"). [assumption]. 10 x => 1 = 1 # label("(BCK3)"). [assumption]. 11 x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). [assumption]. 12 x * 1 = x # label("(M1)"). [assumption]. 13 x * y = y * x # label("(M2)"). [assumption]. 14 (x * y) * z = x * (y * z) # label("(M3)"). [assumption]. 15 (x * y) => z = x => (y => z) # label("(P)"). [assumption]. 16 x ^ 1 = x # label("(D10)"). [assumption]. 17 x ^ ((x => y) => y) = x # label("(3.7)"). [assumption]. 18 (x ^ y) => y = 1 # label("(3.11)"). [assumption]. 19 x v 0 = x # label("(D9)"). [assumption]. 21 x => (x => y) = x -> y. [copy(20),flip(a)]. 23 x => 0 = ~ x. [copy(22),flip(a)]. 25 x => (x => 0) = -x. [copy(24),flip(a)]. 26 x => (x => (x => y)) = x => (x => y) # label("(E_2)"). [assumption]. 27 (x v y) ^ (x v z) = x v (y ^ z) # label("(D7)"). [assumption]. 28 (x ^ y) v (x ^ z) = x ^ (y v z) # label("(D8)"). [assumption]. 29 ~ ~ x = x # label("(DN)"). [assumption]. 30 (x -> y) ^ (~ y -> ~ x) = x => y # label("(N)"). [assumption]. 31 (x => y) => ((z => x) => (z => y)) = 1 # label("(BCK1')"). [assumption]. 32 x => ((x => y) => y) = 1 # label("(BCK5)"). [assumption]. 33 x => x = 1 # label("(3.16)"). [assumption]. 34 x => (y => x) = 1 # label("(3.17)"). [assumption]. 35 x => (y => z) = y => (x => z) # label("(3.18)"). [assumption]. 36 (x * y) v (x * z) = x * (y v z) # label("(4.1)"). [assumption]. 37 (x => y) ^ (x => z) = x => (y ^ z) # label("(4.2)"). [assumption]. 38 (x * (x => y)) v y = y # label("(4.3)"). [assumption]. 39 (x => y) ^ (z => y) = (x v z) => y # label("(4.4)"). [assumption]. 40 ((x => y) * z) => (x => (y * z)) = 1 # label("(4.5)"). [assumption]. 41 (x => y) ^ y = y # label("(3.13)"). [assumption]. 42 (x v y) => y = x => y # label("(4.6)"). [assumption]. 43 x -> (y => z) = y => (x -> z) # label("(4.7)"). [assumption]. 44 x ^ (~ x => y) = x # label("(4.8)"). [assumption]. 45 x -> ((y v x) -> z) = x -> z # label("(4.9)"). [assumption]. 46 x -> (~ x v y) = x -> y # label("(4.10)"). [assumption]. end_of_list. formulas(demodulators). end_of_list. % 1 hints processed (0 are redundant). ============================== end of clauses for search ============= ============================== SEARCH ================================ % Starting search at 0.00 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 => y) => ((y => z) => (x => z)) = 1 # label("(BCK1)"). [assumption]. given #8 (I,wt=5): 9 1 => x = x # label("(BCK2)"). [assumption]. given #9 (I,wt=5): 10 x => 1 = 1 # label("(BCK3)"). [assumption]. given #10 (I,wt=13): 11 x => y != 1 | y => x != 1 | x = y # label("(BCK4)"). [assumption]. given #11 (I,wt=5): 12 x * 1 = x # label("(M1)"). [assumption]. given #12 (I,wt=7): 13 x * y = y * x # label("(M2)"). [assumption]. given #13 (I,wt=11): 14 (x * y) * z = x * (y * z) # label("(M3)"). [assumption]. given #14 (I,wt=11): 15 (x * y) => z = x => (y => z) # label("(P)"). [assumption]. given #15 (I,wt=5): 16 x ^ 1 = x # label("(D10)"). [assumption]. given #16 (I,wt=9): 17 x ^ ((x => y) => y) = x # label("(3.7)"). [assumption]. given #17 (I,wt=7): 18 (x ^ y) => y = 1 # label("(3.11)"). [assumption]. given #18 (I,wt=5): 19 x v 0 = x # label("(D9)"). [assumption]. given #19 (I,wt=9): 21 x => (x => y) = x -> y. [copy(20),flip(a)]. given #20 (I,wt=6): 23 x => 0 = ~ x. [copy(22),flip(a)]. given #21 (I,wt=8): 25 x => (x => 0) = -x. [copy(24),flip(a)]. given #22 (I,wt=13): 26 x => (x => (x => y)) = x => (x => y) # label("(E_2)"). [assumption]. given #23 (I,wt=13): 27 (x v y) ^ (x v z) = x v (y ^ z) # label("(D7)"). [assumption]. given #24 (I,wt=13): 28 (x ^ y) v (x ^ z) = x ^ (y v z) # label("(D8)"). [assumption]. given #25 (I,wt=5): 29 ~ ~ x = x # label("(DN)"). [assumption]. given #26 (I,wt=13): 30 (x -> y) ^ (~ y -> ~ x) = x => y # label("(N)"). [assumption]. given #27 (I,wt=13): 31 (x => y) => ((z => x) => (z => y)) = 1 # label("(BCK1')"). [assumption]. given #28 (I,wt=9): 32 x => ((x => y) => y) = 1 # label("(BCK5)"). [assumption]. given #29 (I,wt=5): 33 x => x = 1 # label("(3.16)"). [assumption]. given #30 (I,wt=7): 34 x => (y => x) = 1 # label("(3.17)"). [assumption]. -------- Proof 1 -------- "(N2)". ============================== PROOF ================================= % Proof 1 at 0.01 (+ 0.00) seconds: "(N2)". % Length of proof is 7. % Level of proof is 3. % Maximum clause weight is 9. % Given clauses 30. 1 x -> x = 1 # label("(N2)") # label(non_clause) # label(goal). [goal]. 20 x -> y = x => (x => y) # label("(-> def)"). [assumption]. 21 x => (x => y) = x -> y. [copy(20),flip(a)]. 34 x => (y => x) = 1 # label("(3.17)"). [assumption]. 47 c1 -> c1 != 1 # label("(N2)") # answer("(N2)"). [deny(1)]. 344 x -> x = 1. [para(34(a,1),21(a,1)),flip(a)]. 345 $F # answer("(N2)"). [resolve(344,a,47,a)]. ============================== end of proof ========================== ============================== STATISTICS ============================ Given=30. Generated=489. Kept=340. proofs=1. Usable=31. Sos=296. Demods=0. Limbo=8, Disabled=47. Hints=1. Weight_deleted=0. Literals_deleted=0. Forward_subsumed=149. Back_subsumed=4. 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=0.46. User_CPU=0.01, System_CPU=0.00, Wall_clock=0. ============================== end of statistics ===================== ============================== end of search ========================= THEOREM PROVED THEOREM PROVED Exiting with 1 proof. ------ process 26278 exit (max_proofs) ------  Process 26278 exit (max_proofs) Sat Jan 19 14:18:57 2008