author  wenzelm 
Sun, 01 Mar 2009 23:36:12 +0100  
changeset 30190  479806475f3c 
parent 29804  e15b74577368 
permissions  rwrr 
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theory ComputeNumeral 
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imports ComputeHOL ComputeFloat 
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begin 
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(* normalization of bit strings *) 

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lemmas bitnorm = normalize_bin_simps 
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(* neg for bit strings *) 

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lemma neg1: "neg Int.Pls = False" by (simp add: Int.Pls_def) 
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lemma neg2: "neg Int.Min = True" apply (subst Int.Min_def) by auto 
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lemma neg3: "neg (Int.Bit0 x) = neg x" apply (simp add: neg_def) apply (subst Bit0_def) by auto 
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lemma neg4: "neg (Int.Bit1 x) = neg x" apply (simp add: neg_def) apply (subst Bit1_def) by auto 
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lemmas bitneg = neg1 neg2 neg3 neg4 
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(* iszero for bit strings *) 

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lemma iszero1: "iszero Int.Pls = True" by (simp add: Int.Pls_def iszero_def) 
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lemma iszero2: "iszero Int.Min = False" apply (subst Int.Min_def) apply (subst iszero_def) by simp 
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lemma iszero3: "iszero (Int.Bit0 x) = iszero x" apply (subst Int.Bit0_def) apply (subst iszero_def)+ by auto 
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lemma iszero4: "iszero (Int.Bit1 x) = False" apply (subst Int.Bit1_def) apply (subst iszero_def)+ apply simp by arith 
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lemmas bitiszero = iszero1 iszero2 iszero3 iszero4 
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(* lezero for bit strings *) 

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constdefs 

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"lezero x == (x \<le> 0)" 

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lemma lezero1: "lezero Int.Pls = True" unfolding Int.Pls_def lezero_def by auto 
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lemma lezero2: "lezero Int.Min = True" unfolding Int.Min_def lezero_def by auto 
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lemma lezero3: "lezero (Int.Bit0 x) = lezero x" unfolding Int.Bit0_def lezero_def by auto 
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lemma lezero4: "lezero (Int.Bit1 x) = neg x" unfolding Int.Bit1_def lezero_def neg_def by auto 
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lemmas bitlezero = lezero1 lezero2 lezero3 lezero4 
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(* equality for bit strings *) 

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lemmas biteq = eq_bin_simps 
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(* x < y for bit strings *) 

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lemmas bitless = less_bin_simps 
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(* x \<le> y for bit strings *) 

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lemmas bitle = le_bin_simps 
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(* succ for bit strings *) 

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lemmas bitsucc = succ_bin_simps 
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(* pred for bit strings *) 

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lemmas bitpred = pred_bin_simps 
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(* unary minus for bit strings *) 

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lemmas bituminus = minus_bin_simps 
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(* addition for bit strings *) 

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lemmas bitadd = add_bin_simps 
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(* multiplication for bit strings *) 

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lemma mult_Pls_right: "x * Int.Pls = Int.Pls" by (simp add: Pls_def) 
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lemma mult_Min_right: "x * Int.Min =  x" by (subst mult_commute, simp add: mult_Min) 
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lemma multb0x: "(Int.Bit0 x) * y = Int.Bit0 (x * y)" by (rule mult_Bit0) 
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lemma multxb0: "x * (Int.Bit0 y) = Int.Bit0 (x * y)" unfolding Bit0_def by simp 
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lemma multb1: "(Int.Bit1 x) * (Int.Bit1 y) = Int.Bit1 (Int.Bit0 (x * y) + x + y)" 
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unfolding Bit0_def Bit1_def by (simp add: algebra_simps) 
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lemmas bitmul = mult_Pls mult_Min mult_Pls_right mult_Min_right multb0x multxb0 multb1 
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lemmas bitarith = bitnorm bitiszero bitneg bitlezero biteq bitless bitle bitsucc bitpred bituminus bitadd bitmul 

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constdefs 

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"nat_norm_number_of (x::nat) == x" 

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lemma nat_norm_number_of: "nat_norm_number_of (number_of w) = (if lezero w then 0 else number_of w)" 

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apply (simp add: nat_norm_number_of_def) 

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unfolding lezero_def iszero_def neg_def 

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apply (simp add: numeral_simps) 
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done 
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(* Normalization of nat literals *) 

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lemma natnorm0: "(0::nat) = number_of (Int.Pls)" by auto 
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lemma natnorm1: "(1 :: nat) = number_of (Int.Bit1 Int.Pls)" by auto 
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lemmas natnorm = natnorm0 natnorm1 nat_norm_number_of 
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(* Suc *) 

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lemma natsuc: "Suc (number_of x) = (if neg x then 1 else number_of (Int.succ x))" by (auto simp add: number_of_is_id) 
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(* Addition for nat *) 

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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))))" 

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unfolding nat_number_of_def number_of_is_id neg_def 
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by auto 

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(* Subtraction for nat *) 

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lemma natsub: "(number_of x)  ((number_of y)::nat) = 

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(if neg x then 0 else (if neg y then number_of x else (nat_norm_number_of (number_of (x + ( y))))))" 

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unfolding nat_norm_number_of 

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by (auto simp add: number_of_is_id neg_def lezero_def iszero_def Let_def nat_number_of_def) 

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(* Multiplication for nat *) 

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lemma natmul: "(number_of x) * ((number_of y)::nat) = 

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(if neg x then 0 else (if neg y then 0 else number_of (x * y)))" 

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unfolding nat_number_of_def number_of_is_id neg_def 
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by (simp add: nat_mult_distrib) 

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lemma nateq: "(((number_of x)::nat) = (number_of y)) = ((lezero x \<and> lezero y) \<or> (x = y))" 

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by (auto simp add: iszero_def lezero_def neg_def number_of_is_id) 

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lemma natless: "(((number_of x)::nat) < (number_of y)) = ((x < y) \<and> (\<not> (lezero y)))" 

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by (simp add: lezero_def numeral_simps not_le) 
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lemma natle: "(((number_of x)::nat) \<le> (number_of y)) = (y < x \<longrightarrow> lezero x)" 

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by (auto simp add: number_of_is_id lezero_def nat_number_of_def) 

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fun natfac :: "nat \<Rightarrow> nat" 

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where 

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"natfac n = (if n = 0 then 1 else n * (natfac (n  1)))" 

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lemmas compute_natarith = bitarith natnorm natsuc natadd natsub natmul nateq natless natle natfac.simps 

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lemma number_eq: "(((number_of x)::'a::{number_ring, ordered_idom}) = (number_of y)) = (x = y)" 

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unfolding number_of_eq 

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apply simp 

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done 

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lemma number_le: "(((number_of x)::'a::{number_ring, ordered_idom}) \<le> (number_of y)) = (x \<le> y)" 

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unfolding number_of_eq 

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apply simp 

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done 

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lemma number_less: "(((number_of x)::'a::{number_ring, ordered_idom}) < (number_of y)) = (x < y)" 

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unfolding number_of_eq 

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apply simp 

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done 

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lemma number_diff: "((number_of x)::'a::{number_ring, ordered_idom})  number_of y = number_of (x + ( y))" 

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apply (subst diff_number_of_eq) 

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apply simp 

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done 

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lemmas number_norm = number_of_Pls[symmetric] numeral_1_eq_1[symmetric] 

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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 

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lemma compute_real_of_nat_number_of: "real ((number_of v)::nat) = (if neg v then 0 else number_of v)" 

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by (simp only: real_of_nat_number_of number_of_is_id) 

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lemma compute_nat_of_int_number_of: "nat ((number_of v)::int) = (number_of v)" 

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by simp 

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lemmas compute_num_conversions = compute_real_of_nat_number_of compute_nat_of_int_number_of real_number_of 

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lemmas zpowerarith = zpower_number_of_even 

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zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring] 

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zpower_Pls zpower_Min 

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(* div, mod *) 

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lemma adjust: "adjust b (q, r) = (if 0 \<le> r  b then (2 * q + 1, r  b) else (2 * q, r))" 

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by (auto simp only: adjust_def) 

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lemma negateSnd: "negateSnd (q, r) = (q, r)" 

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by (simp add: negateSnd_def) 
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lemma divmod: "IntDiv.divmod a b = (if 0\<le>a then 
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if 0\<le>b then posDivAlg a b 
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else if a=0 then (0, 0) 

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else negateSnd (negDivAlg (a) (b)) 

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else 

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if 0<b then negDivAlg a b 

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else negateSnd (posDivAlg (a) (b)))" 

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by (auto simp only: IntDiv.divmod_def) 
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lemmas compute_div_mod = div_def mod_def divmod adjust negateSnd posDivAlg.simps negDivAlg.simps 
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(* collecting all the theorems *) 

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lemma even_Pls: "even (Int.Pls) = True" 
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apply (unfold Pls_def even_def) 
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by simp 

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lemma even_Min: "even (Int.Min) = False" 
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apply (unfold Min_def even_def) 
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by simp 

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lemma even_B0: "even (Int.Bit0 x) = True" 
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apply (unfold Bit0_def) 
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by simp 
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lemma even_B1: "even (Int.Bit1 x) = False" 
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apply (unfold Bit1_def) 
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by simp 
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lemma even_number_of: "even ((number_of w)::int) = even w" 

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by (simp only: number_of_is_id) 

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lemmas compute_even = even_Pls even_Min even_B0 even_B1 even_number_of 

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lemmas compute_numeral = compute_if compute_let compute_pair compute_bool 

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compute_natarith compute_numberarith max_def min_def compute_num_conversions zpowerarith compute_div_mod compute_even 

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end 