# HG changeset patch # User obua # Date 1183995520 -7200 # Node ID 9c486517354a0ac4c343cfd9153c3f6459d4e2e2 # Parent 84b5c89b8b49f1ec31079f5d605f82beee330240 added computing oracle support for HOL and numerals diff -r 84b5c89b8b49 -r 9c486517354a src/HOL/Tools/ComputeHOL.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Tools/ComputeHOL.ML Mon Jul 09 17:38:40 2007 +0200 @@ -0,0 +1,54 @@ +signature ComputeHOL = +sig + val prep_thms : thm list -> thm list + val to_meta_eq : thm -> thm + val to_hol_eq : thm -> thm + val symmetric : thm -> thm + val trans : thm -> thm -> thm +end + +structure ComputeHOL : ComputeHOL = +struct + +fun all_prems_conv ci ct = Conv.prems_conv (Logic.count_prems (term_of ct)) ci ct + +local +fun lhs_of eq = fst (Thm.dest_equals (cprop_of eq)); +in +fun rewrite_conv [] ct = raise CTERM ("rewrite_conv", []) + | rewrite_conv (eq :: eqs) ct = + Thm.instantiate (Thm.match (lhs_of eq, ct)) eq + handle Pattern.MATCH => rewrite_conv eqs ct; +end + +val convert_conditions = Conv.fconv_rule (all_prems_conv (fn _ => Conv.else_conv (rewrite_conv [@{thm "Trueprop_eq_eq"}], Conv.all_conv))) + +val eq_th = @{thm "HOL.eq_reflection"} +val meta_eq_trivial = @{thm "ComputeHOL.meta_eq_trivial"} +val bool_to_true = @{thm "ComputeHOL.bool_to_true"} + +fun to_meta_eq th = eq_th OF [th] handle THM _ => meta_eq_trivial OF [th] handle THM _ => bool_to_true OF [th] + +fun to_hol_eq th = @{thm "meta_eq_imp_eq"} OF [th] handle THM _ => @{thm "eq_trivial"} OF [th] + +fun prep_thms ths = map (convert_conditions o to_meta_eq) ths + +local + val sym_HOL = @{thm "HOL.sym"} + val sym_Pure = @{thm "ProtoPure.symmetric"} +in + fun symmetric th = ((sym_HOL OF [th]) handle THM _ => (sym_Pure OF [th])) +end + +local + val trans_HOL = @{thm "HOL.trans"} + val trans_HOL_1 = @{thm "ComputeHOL.transmeta_1"} + val trans_HOL_2 = @{thm "ComputeHOL.transmeta_2"} + val trans_HOL_3 = @{thm "ComputeHOL.transmeta_3"} + fun tr [] th1 th2 = trans_HOL OF [th1, th2] + | tr (t::ts) th1 th2 = (t OF [th1, th2] handle THM _ => tr ts th1 th2) +in + fun trans th1 th2 = tr [trans_HOL, trans_HOL_1, trans_HOL_2, trans_HOL_3] th1 th2 +end + +end \ No newline at end of file diff -r 84b5c89b8b49 -r 9c486517354a src/HOL/Tools/ComputeHOL.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Tools/ComputeHOL.thy Mon Jul 09 17:38:40 2007 +0200 @@ -0,0 +1,141 @@ +theory ComputeHOL +imports Main "~~/src/Tools/Compute_Oracle/Compute_Oracle" +begin + +lemma Trueprop_eq_eq: "Trueprop X == (X == True)" by (simp add: atomize_eq) +lemma meta_eq_trivial: "x == y \ x == y" by simp +lemma meta_eq_imp_eq: "x == y \ x = y" by auto +lemma eq_trivial: "x = y \ x = y" by auto +lemma bool_to_true: "x :: bool \ x == True" by simp +lemma transmeta_1: "x = y \ y == z \ x = z" by simp +lemma transmeta_2: "x == y \ y = z \ x = z" by simp +lemma transmeta_3: "x == y \ y == z \ x = z" by simp + + +(**** compute_if ****) + +lemma If_True: "If True = (\ x y. x)" by ((rule ext)+,auto) +lemma If_False: "If False = (\ x y. y)" by ((rule ext)+, auto) + +lemmas compute_if = If_True If_False + +(**** compute_bool ****) + +lemma bool1: "(\ True) = False" by blast +lemma bool2: "(\ False) = True" by blast +lemma bool3: "(P \ True) = P" by blast +lemma bool4: "(True \ P) = P" by blast +lemma bool5: "(P \ False) = False" by blast +lemma bool6: "(False \ P) = False" by blast +lemma bool7: "(P \ True) = True" by blast +lemma bool8: "(True \ P) = True" by blast +lemma bool9: "(P \ False) = P" by blast +lemma bool10: "(False \ P) = P" by blast +lemma bool11: "(True \ P) = P" by blast +lemma bool12: "(P \ True) = True" by blast +lemma bool13: "(True \ P) = P" by blast +lemma bool14: "(P \ False) = (\ P)" by blast +lemma bool15: "(False \ P) = True" by blast +lemma bool16: "(False = False) = True" by blast +lemma bool17: "(True = True) = True" by blast +lemma bool18: "(False = True) = False" by blast +lemma bool19: "(True = False) = False" by blast + +lemmas compute_bool = bool1 bool2 bool3 bool4 bool5 bool6 bool7 bool8 bool9 bool10 bool11 bool12 bool13 bool14 bool15 bool16 bool17 bool18 bool19 + + +(*** compute_pair ***) + +lemma compute_fst: "fst (x,y) = x" by simp +lemma compute_snd: "snd (x,y) = y" by simp +lemma compute_pair_eq: "((a, b) = (c, d)) = (a = c \ b = d)" by auto + +lemma prod_case_simp: "prod_case f (x,y) = f x y" by simp + +lemmas compute_pair = compute_fst compute_snd compute_pair_eq prod_case_simp + +(*** compute_option ***) + +lemma compute_the: "the (Some x) = x" by simp +lemma compute_None_Some_eq: "(None = Some x) = False" by auto +lemma compute_Some_None_eq: "(Some x = None) = False" by auto +lemma compute_None_None_eq: "(None = None) = True" by auto +lemma compute_Some_Some_eq: "(Some x = Some y) = (x = y)" by auto + +definition + option_case_compute :: "'b option \ 'a \ ('b \ 'a) \ 'a" +where + "option_case_compute opt a f = option_case a f opt" + +lemma option_case_compute: "option_case = (\ a f opt. option_case_compute opt a f)" + by (simp add: option_case_compute_def) + +lemma option_case_compute_None: "option_case_compute None = (\ a f. a)" + apply (rule ext)+ + apply (simp add: option_case_compute_def) + done + +lemma option_case_compute_Some: "option_case_compute (Some x) = (\ a f. f x)" + apply (rule ext)+ + apply (simp add: option_case_compute_def) + done + +lemmas compute_option_case = option_case_compute option_case_compute_None option_case_compute_Some + +lemmas compute_option = compute_the compute_None_Some_eq compute_Some_None_eq compute_None_None_eq compute_Some_Some_eq compute_option_case + +(**** compute_list_length ****) + +lemma length_cons:"length (x#xs) = 1 + (length xs)" + by simp + +lemma length_nil: "length [] = 0" + by simp + +lemmas compute_list_length = length_nil length_cons + +(*** compute_list_case ***) + +definition + list_case_compute :: "'b list \ 'a \ ('b \ 'b list \ 'a) \ 'a" +where + "list_case_compute l a f = list_case a f l" + +lemma list_case_compute: "list_case = (\ (a::'a) f (l::'b list). list_case_compute l a f)" + apply (rule ext)+ + apply (simp add: list_case_compute_def) + done + +lemma list_case_compute_empty: "list_case_compute ([]::'b list) = (\ (a::'a) f. a)" + apply (rule ext)+ + apply (simp add: list_case_compute_def) + done + +lemma list_case_compute_cons: "list_case_compute (u#v) = (\ (a::'a) f. (f (u::'b) v))" + apply (rule ext)+ + apply (simp add: list_case_compute_def) + done + +lemmas compute_list_case = list_case_compute list_case_compute_empty list_case_compute_cons + +(*** compute_list_nth ***) +(* Of course, you will need computation with nats for this to work \ *) + +lemma compute_list_nth: "((x#xs) ! n) = (if n = 0 then x else (xs ! (n - 1)))" + by (cases n, auto) + +(*** compute_list ***) + +lemmas compute_list = compute_list_case compute_list_length compute_list_nth + +(*** compute_let ***) + +lemmas compute_let = Let_def + +(***********************) +(* Everything together *) +(***********************) + +lemmas compute_hol = compute_if compute_bool compute_pair compute_option compute_list compute_let + +end diff -r 84b5c89b8b49 -r 9c486517354a src/HOL/Tools/ComputeNumeral.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Tools/ComputeNumeral.thy Mon Jul 09 17:38:40 2007 +0200 @@ -0,0 +1,250 @@ +theory ComputeNumeral +imports ComputeHOL Float +begin + +(* normalization of bit strings *) +lemmas bitnorm = Pls_0_eq Min_1_eq + +(* neg for bit strings *) +lemma neg1: "neg Numeral.Pls = False" by (simp add: Numeral.Pls_def) +lemma neg2: "neg Numeral.Min = True" apply (subst Numeral.Min_def) by auto +lemma neg3: "neg (x BIT Numeral.B0) = neg x" apply (simp add: neg_def) apply (subst Bit_def) by auto +lemma neg4: "neg (x BIT Numeral.B1) = neg x" apply (simp add: neg_def) apply (subst Bit_def) by auto +lemmas bitneg = neg1 neg2 neg3 neg4 + +(* iszero for bit strings *) +lemma iszero1: "iszero Numeral.Pls = True" by (simp add: Numeral.Pls_def iszero_def) +lemma iszero2: "iszero Numeral.Min = False" apply (subst Numeral.Min_def) apply (subst iszero_def) by simp +lemma iszero3: "iszero (x BIT Numeral.B0) = iszero x" apply (subst Numeral.Bit_def) apply (subst iszero_def)+ by auto +lemma iszero4: "iszero (x BIT Numeral.B1) = False" apply (subst Numeral.Bit_def) apply (subst iszero_def)+ apply simp by arith +lemmas bitiszero = iszero1 iszero2 iszero3 iszero4 + +(* lezero for bit strings *) +constdefs + "lezero x == (x \ 0)" +lemma lezero1: "lezero Numeral.Pls = True" unfolding Numeral.Pls_def lezero_def by auto +lemma lezero2: "lezero Numeral.Min = True" unfolding Numeral.Min_def lezero_def by auto +lemma lezero3: "lezero (x BIT Numeral.B0) = lezero x" unfolding Numeral.Bit_def lezero_def by auto +lemma lezero4: "lezero (x BIT Numeral.B1) = neg x" unfolding Numeral.Bit_def lezero_def neg_def by auto +lemmas bitlezero = lezero1 lezero2 lezero3 lezero4 + +(* equality for bit strings *) +lemma biteq1: "(Numeral.Pls = Numeral.Pls) = True" by auto +lemma biteq2: "(Numeral.Min = Numeral.Min) = True" by auto +lemma biteq3: "(Numeral.Pls = Numeral.Min) = False" unfolding Pls_def Min_def by auto +lemma biteq4: "(Numeral.Min = Numeral.Pls) = False" unfolding Pls_def Min_def by auto +lemma biteq5: "(x BIT Numeral.B0 = y BIT Numeral.B0) = (x = y)" unfolding Bit_def by auto +lemma biteq6: "(x BIT Numeral.B1 = y BIT Numeral.B1) = (x = y)" unfolding Bit_def by auto +lemma biteq7: "(x BIT Numeral.B0 = y BIT Numeral.B1) = False" unfolding Bit_def by (simp, arith) +lemma biteq8: "(x BIT Numeral.B1 = y BIT Numeral.B0) = False" unfolding Bit_def by (simp, arith) +lemma biteq9: "(Numeral.Pls = x BIT Numeral.B0) = (Numeral.Pls = x)" unfolding Bit_def Pls_def by auto +lemma biteq10: "(Numeral.Pls = x BIT Numeral.B1) = False" unfolding Bit_def Pls_def by (simp, arith) +lemma biteq11: "(Numeral.Min = x BIT Numeral.B0) = False" unfolding Bit_def Min_def by (simp, arith) +lemma biteq12: "(Numeral.Min = x BIT Numeral.B1) = (Numeral.Min = x)" unfolding Bit_def Min_def by auto +lemma biteq13: "(x BIT Numeral.B0 = Numeral.Pls) = (x = Numeral.Pls)" unfolding Bit_def Pls_def by auto +lemma biteq14: "(x BIT Numeral.B1 = Numeral.Pls) = False" unfolding Bit_def Pls_def by (simp, arith) +lemma biteq15: "(x BIT Numeral.B0 = Numeral.Min) = False" unfolding Bit_def Pls_def Min_def by (simp, arith) +lemma biteq16: "(x BIT Numeral.B1 = Numeral.Min) = (x = Numeral.Min)" unfolding Bit_def Min_def by (simp, arith) +lemmas biteq = biteq1 biteq2 biteq3 biteq4 biteq5 biteq6 biteq7 biteq8 biteq9 biteq10 biteq11 biteq12 biteq13 biteq14 biteq15 biteq16 + +(* x < y for bit strings *) +lemma bitless1: "(Numeral.Pls < Numeral.Min) = False" unfolding Pls_def Min_def by auto +lemma bitless2: "(Numeral.Pls < Numeral.Pls) = False" by auto +lemma bitless3: "(Numeral.Min < Numeral.Pls) = True" unfolding Pls_def Min_def by auto +lemma bitless4: "(Numeral.Min < Numeral.Min) = False" unfolding Pls_def Min_def by auto +lemma bitless5: "(x BIT Numeral.B0 < y BIT Numeral.B0) = (x < y)" unfolding Bit_def by auto +lemma bitless6: "(x BIT Numeral.B1 < y BIT Numeral.B1) = (x < y)" unfolding Bit_def by auto +lemma bitless7: "(x BIT Numeral.B0 < y BIT Numeral.B1) = (x \ y)" unfolding Bit_def by auto +lemma bitless8: "(x BIT Numeral.B1 < y BIT Numeral.B0) = (x < y)" unfolding Bit_def by auto +lemma bitless9: "(Numeral.Pls < x BIT Numeral.B0) = (Numeral.Pls < x)" unfolding Bit_def Pls_def by auto +lemma bitless10: "(Numeral.Pls < x BIT Numeral.B1) = (Numeral.Pls \ x)" unfolding Bit_def Pls_def by auto +lemma bitless11: "(Numeral.Min < x BIT Numeral.B0) = (Numeral.Pls \ x)" unfolding Bit_def Pls_def Min_def by auto +lemma bitless12: "(Numeral.Min < x BIT Numeral.B1) = (Numeral.Min < x)" unfolding Bit_def Min_def by auto +lemma bitless13: "(x BIT Numeral.B0 < Numeral.Pls) = (x < Numeral.Pls)" unfolding Bit_def Pls_def by auto +lemma bitless14: "(x BIT Numeral.B1 < Numeral.Pls) = (x < Numeral.Pls)" unfolding Bit_def Pls_def by auto +lemma bitless15: "(x BIT Numeral.B0 < Numeral.Min) = (x < Numeral.Pls)" unfolding Bit_def Pls_def Min_def by auto +lemma bitless16: "(x BIT Numeral.B1 < Numeral.Min) = (x < Numeral.Min)" unfolding Bit_def Min_def by auto +lemmas bitless = bitless1 bitless2 bitless3 bitless4 bitless5 bitless6 bitless7 bitless8 + bitless9 bitless10 bitless11 bitless12 bitless13 bitless14 bitless15 bitless16 + +(* x \ y for bit strings *) +lemma bitle1: "(Numeral.Pls \ Numeral.Min) = False" unfolding Pls_def Min_def by auto +lemma bitle2: "(Numeral.Pls \ Numeral.Pls) = True" by auto +lemma bitle3: "(Numeral.Min \ Numeral.Pls) = True" unfolding Pls_def Min_def by auto +lemma bitle4: "(Numeral.Min \ Numeral.Min) = True" unfolding Pls_def Min_def by auto +lemma bitle5: "(x BIT Numeral.B0 \ y BIT Numeral.B0) = (x \ y)" unfolding Bit_def by auto +lemma bitle6: "(x BIT Numeral.B1 \ y BIT Numeral.B1) = (x \ y)" unfolding Bit_def by auto +lemma bitle7: "(x BIT Numeral.B0 \ y BIT Numeral.B1) = (x \ y)" unfolding Bit_def by auto +lemma bitle8: "(x BIT Numeral.B1 \ y BIT Numeral.B0) = (x < y)" unfolding Bit_def by auto +lemma bitle9: "(Numeral.Pls \ x BIT Numeral.B0) = (Numeral.Pls \ x)" unfolding Bit_def Pls_def by auto +lemma bitle10: "(Numeral.Pls \ x BIT Numeral.B1) = (Numeral.Pls \ x)" unfolding Bit_def Pls_def by auto +lemma bitle11: "(Numeral.Min \ x BIT Numeral.B0) = (Numeral.Pls \ x)" unfolding Bit_def Pls_def Min_def by auto +lemma bitle12: "(Numeral.Min \ x BIT Numeral.B1) = (Numeral.Min \ x)" unfolding Bit_def Min_def by auto +lemma bitle13: "(x BIT Numeral.B0 \ Numeral.Pls) = (x \ Numeral.Pls)" unfolding Bit_def Pls_def by auto +lemma bitle14: "(x BIT Numeral.B1 \ Numeral.Pls) = (x < Numeral.Pls)" unfolding Bit_def Pls_def by auto +lemma bitle15: "(x BIT Numeral.B0 \ Numeral.Min) = (x < Numeral.Pls)" unfolding Bit_def Pls_def Min_def by auto +lemma bitle16: "(x BIT Numeral.B1 \ Numeral.Min) = (x \ Numeral.Min)" unfolding Bit_def Min_def by auto +lemmas bitle = bitle1 bitle2 bitle3 bitle4 bitle5 bitle6 bitle7 bitle8 + bitle9 bitle10 bitle11 bitle12 bitle13 bitle14 bitle15 bitle16 + +(* succ for bit strings *) +lemmas bitsucc = succ_Pls succ_Min succ_1 succ_0 + +(* pred for bit strings *) +lemmas bitpred = pred_Pls pred_Min pred_1 pred_0 + +(* unary minus for bit strings *) +lemmas bituminus = minus_Pls minus_Min minus_1 minus_0 + +(* addition for bit strings *) +lemmas bitadd = add_Pls add_Pls_right add_Min add_Min_right add_BIT_11 add_BIT_10 add_BIT_0[where b="Numeral.B0"] add_BIT_0[where b="Numeral.B1"] + +(* multiplication for bit strings *) +lemma mult_Pls_right: "x * Numeral.Pls = Numeral.Pls" by (simp add: Pls_def) +lemma mult_Min_right: "x * Numeral.Min = - x" by (subst mult_commute, simp add: mult_Min) +lemma multb0x: "(x BIT Numeral.B0) * y = (x * y) BIT Numeral.B0" unfolding Bit_def by simp +lemma multxb0: "x * (y BIT Numeral.B0) = (x * y) BIT Numeral.B0" unfolding Bit_def by simp +lemma multb1: "(x BIT Numeral.B1) * (y BIT Numeral.B1) = (((x * y) BIT Numeral.B0) + x + y) BIT Numeral.B1" + unfolding Bit_def by (simp add: ring_simps) +lemmas bitmul = mult_Pls mult_Min mult_Pls_right mult_Min_right multb0x multxb0 multb1 + +lemmas bitarith = bitnorm bitiszero bitneg bitlezero biteq bitless bitle bitsucc bitpred bituminus bitadd bitmul + +constdefs + "nat_norm_number_of (x::nat) == x" + +lemma nat_norm_number_of: "nat_norm_number_of (number_of w) = (if lezero w then 0 else number_of w)" + apply (simp add: nat_norm_number_of_def) + unfolding lezero_def iszero_def neg_def + apply (simp add: number_of_is_id) + done + +(* Normalization of nat literals *) +lemma natnorm0: "(0::nat) = number_of (Numeral.Pls)" by auto +lemma natnorm1: "(1 :: nat) = number_of (Numeral.Pls BIT Numeral.B1)" by auto +lemmas natnorm = natnorm0 natnorm1 nat_norm_number_of + +(* Suc *) +lemma natsuc: "Suc (number_of x) = (if neg x then 1 else number_of (Numeral.succ x))" by (auto simp add: number_of_is_id) + +(* Addition for nat *) +lemma natadd: "number_of x + ((number_of y)::nat) = (if neg x then (number_of y) else (if neg y then number_of x else (number_of (x + y))))" + by (auto simp add: number_of_is_id) + +(* Subtraction for nat *) +lemma natsub: "(number_of x) - ((number_of y)::nat) = + (if neg x then 0 else (if neg y then number_of x else (nat_norm_number_of (number_of (x + (- y))))))" + unfolding nat_norm_number_of + by (auto simp add: number_of_is_id neg_def lezero_def iszero_def Let_def nat_number_of_def) + +(* Multiplication for nat *) +lemma natmul: "(number_of x) * ((number_of y)::nat) = + (if neg x then 0 else (if neg y then 0 else number_of (x * y)))" + apply (auto simp add: number_of_is_id neg_def iszero_def) + apply (case_tac "x > 0") + apply auto + apply (simp add: mult_strict_left_mono[where a=y and b=0 and c=x, simplified]) + done + +lemma nateq: "(((number_of x)::nat) = (number_of y)) = ((lezero x \ lezero y) \ (x = y))" + by (auto simp add: iszero_def lezero_def neg_def number_of_is_id) + +lemma natless: "(((number_of x)::nat) < (number_of y)) = ((x < y) \ (\ (lezero y)))" + by (auto simp add: number_of_is_id neg_def lezero_def) + +lemma natle: "(((number_of x)::nat) \ (number_of y)) = (y < x \ lezero x)" + by (auto simp add: number_of_is_id lezero_def nat_number_of_def) + +fun natfac :: "nat \ nat" +where + "natfac n = (if n = 0 then 1 else n * (natfac (n - 1)))" + +lemmas compute_natarith = bitarith natnorm natsuc natadd natsub natmul nateq natless natle natfac.simps + +lemma number_eq: "(((number_of x)::'a::{number_ring, ordered_idom}) = (number_of y)) = (x = y)" + unfolding number_of_eq + apply simp + done + +lemma number_le: "(((number_of x)::'a::{number_ring, ordered_idom}) \ (number_of y)) = (x \ y)" + unfolding number_of_eq + apply simp + done + +lemma number_less: "(((number_of x)::'a::{number_ring, ordered_idom}) < (number_of y)) = (x < y)" + unfolding number_of_eq + apply simp + done + +lemma number_diff: "((number_of x)::'a::{number_ring, ordered_idom}) - number_of y = number_of (x + (- y))" + apply (subst diff_number_of_eq) + apply simp + done + +lemmas number_norm = number_of_Pls[symmetric] numeral_1_eq_1[symmetric] + +lemmas compute_numberarith = number_of_minus[symmetric] number_of_add[symmetric] number_diff number_of_mult[symmetric] number_norm number_eq number_le number_less + +lemma compute_real_of_nat_number_of: "real ((number_of v)::nat) = (if neg v then 0 else number_of v)" + by (simp only: real_of_nat_number_of number_of_is_id) + +lemma compute_nat_of_int_number_of: "nat ((number_of v)::int) = (number_of v)" + by simp + +lemmas compute_num_conversions = compute_real_of_nat_number_of compute_nat_of_int_number_of real_number_of + +lemmas zpowerarith = zpower_number_of_even + zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring] + zpower_Pls zpower_Min + +(* div, mod *) + +lemma adjust: "adjust b (q, r) = (if 0 \ r - b then (2 * q + 1, r - b) else (2 * q, r))" + by (auto simp only: adjust_def) + +lemma negateSnd: "negateSnd (q, r) = (q, -r)" + by (auto simp only: negateSnd_def) + +lemma divAlg: "divAlg (a, b) = (if 0\a then + if 0\b then posDivAlg a b + else if a=0 then (0, 0) + else negateSnd (negDivAlg (-a) (-b)) + else + if 0