src/HOL/Tools/ComputeNumeral.thy
author wenzelm
Sun, 01 Mar 2009 23:36:12 +0100
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permissions -rw-r--r--
use long names for old-style fold combinators;
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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