# Theory Float

Up to index of Isabelle/HOL/HOL-Complex

theory Float
imports Real
uses [Float.ML]
begin

    (*  Title: HOL/Real/Float.thy
        ID:    $Id: Float.thy,v 1.2 2005/07/19 15:29:27 wenzelm Exp $
        Author: Steven Obua
    *)

    theory Float imports Real begin

    constdefs
      pow2 :: "int => real"
      "pow2 a == if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a))))"
      float :: "int * int => real"
      "float x == (real (fst x)) * (pow2 (snd x))"

    lemma pow2_0[simp]: "pow2 0 = 1"
    by (simp add: pow2_def)

    lemma pow2_1[simp]: "pow2 1 = 2"
    by (simp add: pow2_def)

    lemma pow2_neg: "pow2 x = inverse (pow2 (-x))"
    by (simp add: pow2_def)

    lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)"
    proof -
      have h: "! n. nat (2 + int n) - Suc 0 = nat (1 + int n)" by arith
      have g: "! a b. a - -1 = a + (1::int)" by arith
      have pos: "! n. pow2 (int n + 1) = 2 * pow2 (int n)"
        apply (auto, induct_tac n)
        apply (simp_all add: pow2_def)
        apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if])
        apply (auto simp add: h)
        apply arith
        done
      show ?thesis
      proof (induct a)
        case (1 n)
        from pos show ?case by (simp add: ring_eq_simps)
      next
        case (2 n)
        show ?case
          apply (auto)
          apply (subst pow2_neg[of "- int n"])
          apply (subst pow2_neg[of "-1 - int n"])
          apply (auto simp add: g pos)
          done
      qed
    qed

    lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)"
    proof (induct b)
      case (1 n)
      show ?case
      proof (induct n)
        case 0
        show ?case by simp
      next
        case (Suc m)
        show ?case by (auto simp add: ring_eq_simps pow2_add1 prems)
      qed
    next
      case (2 n)
      show ?case
      proof (induct n)
        case 0
        show ?case
          apply (auto)
          apply (subst pow2_neg[of "a + -1"])
          apply (subst pow2_neg[of "-1"])
          apply (simp)
          apply (insert pow2_add1[of "-a"])
          apply (simp add: ring_eq_simps)
          apply (subst pow2_neg[of "-a"])
          apply (simp)
          done
        case (Suc m)
        have a: "int m - (a + -2) =  1 + (int m - a + 1)" by arith
        have b: "int m - -2 = 1 + (int m + 1)" by arith
        show ?case
          apply (auto)
          apply (subst pow2_neg[of "a + (-2 - int m)"])
          apply (subst pow2_neg[of "-2 - int m"])
          apply (auto simp add: ring_eq_simps)
          apply (subst a)
          apply (subst b)
          apply (simp only: pow2_add1)
          apply (subst pow2_neg[of "int m - a + 1"])
          apply (subst pow2_neg[of "int m + 1"])
          apply auto
          apply (insert prems)
          apply (auto simp add: ring_eq_simps)
          done
      qed
    qed

    lemma "float (a, e) + float (b, e) = float (a + b, e)"
    by (simp add: float_def ring_eq_simps)

    constdefs
      int_of_real :: "real => int"
      "int_of_real x == SOME y. real y = x"
      real_is_int :: "real => bool"
      "real_is_int x == ? (u::int). x = real u"

    lemma real_is_int_def2: "real_is_int x = (x = real (int_of_real x))"
    by (auto simp add: real_is_int_def int_of_real_def)

    lemma float_transfer: "real_is_int ((real a)*(pow2 c)) ==> float (a, b) = float (int_of_real ((real a)*(pow2 c)), b - c)"
    by (simp add: float_def real_is_int_def2 pow2_add[symmetric])

    lemma pow2_int: "pow2 (int c) = (2::real)^c"
    by (simp add: pow2_def)

    lemma float_transfer_nat: "float (a, b) = float (a * 2^c, b - int c)"
    by (simp add: float_def pow2_int[symmetric] pow2_add[symmetric])

    lemma real_is_int_real[simp]: "real_is_int (real (x::int))"
    by (auto simp add: real_is_int_def int_of_real_def)

    lemma int_of_real_real[simp]: "int_of_real (real x) = x"
    by (simp add: int_of_real_def)

    lemma real_int_of_real[simp]: "real_is_int x ==> real (int_of_real x) = x"
    by (auto simp add: int_of_real_def real_is_int_def)

    lemma real_is_int_add_int_of_real: "real_is_int a ==> real_is_int b ==> (int_of_real (a+b)) = (int_of_real a) + (int_of_real b)"
    by (auto simp add: int_of_real_def real_is_int_def)

    lemma real_is_int_add[simp]: "real_is_int a ==> real_is_int b ==> real_is_int (a+b)"
    apply (subst real_is_int_def2)
    apply (simp add: real_is_int_add_int_of_real real_int_of_real)
    done

    lemma int_of_real_sub: "real_is_int a ==> real_is_int b ==> (int_of_real (a-b)) = (int_of_real a) - (int_of_real b)"
    by (auto simp add: int_of_real_def real_is_int_def)

    lemma real_is_int_sub[simp]: "real_is_int a ==> real_is_int b ==> real_is_int (a-b)"
    apply (subst real_is_int_def2)
    apply (simp add: int_of_real_sub real_int_of_real)
    done

    lemma real_is_int_rep: "real_is_int x ==> ?! (a::int). real a = x"
    by (auto simp add: real_is_int_def)

    lemma int_of_real_mult:
      assumes "real_is_int a" "real_is_int b"
      shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)"
    proof -
      from prems have a: "?! (a'::int). real a' = a" by (rule_tac real_is_int_rep, auto)
      from prems have b: "?! (b'::int). real b' = b" by (rule_tac real_is_int_rep, auto)
      from a obtain a'::int where a':"a = real a'" by auto
      from b obtain b'::int where b':"b = real b'" by auto
      have r: "real a' * real b' = real (a' * b')" by auto
      show ?thesis
        apply (simp add: a' b')
        apply (subst r)
        apply (simp only: int_of_real_real)
        done
    qed

    lemma real_is_int_mult[simp]: "real_is_int a ==> real_is_int b ==> real_is_int (a*b)"
    apply (subst real_is_int_def2)
    apply (simp add: int_of_real_mult)
    done

    lemma real_is_int_0[simp]: "real_is_int (0::real)"
    by (simp add: real_is_int_def int_of_real_def)

    lemma real_is_int_1[simp]: "real_is_int (1::real)"
    proof -
      have "real_is_int (1::real) = real_is_int(real (1::int))" by auto
      also have "… = True" by (simp only: real_is_int_real)
      ultimately show ?thesis by auto
    qed

    lemma real_is_int_n1: "real_is_int (-1::real)"
    proof -
      have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto
      also have "… = True" by (simp only: real_is_int_real)
      ultimately show ?thesis by auto
    qed

    lemma real_is_int_number_of[simp]: "real_is_int ((number_of::bin=>real) x)"
    proof -
      have neg1: "real_is_int (-1::real)"
      proof -
        have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto
        also have "… = True" by (simp only: real_is_int_real)
        ultimately show ?thesis by auto
      qed

      {
        fix x::int
        have "!! y. real_is_int ((number_of::bin=>real) (Abs_Bin x))"
          apply (simp add: number_of_eq)
          apply (subst Abs_Bin_inverse)
          apply (simp add: Bin_def)
          apply (induct x)
          apply (induct_tac n)
          apply (simp)
          apply (simp)
          apply (induct_tac n)
          apply (simp add: neg1)
        proof -
          fix n :: nat
          assume rn: "(real_is_int (of_int (- (int (Suc n)))))"
          have s: "-(int (Suc (Suc n))) = -1 + - (int (Suc n))" by simp
          show "real_is_int (of_int (- (int (Suc (Suc n)))))"
            apply (simp only: s of_int_add)
            apply (rule real_is_int_add)
            apply (simp add: neg1)
            apply (simp only: rn)
            done
        qed
      }
      note Abs_Bin = this
      {
        fix x :: bin
        have "? u. x = Abs_Bin u"
          apply (rule exI[where x = "Rep_Bin x"])
          apply (simp add: Rep_Bin_inverse)
          done
      }
      then obtain u::int where "x = Abs_Bin u" by auto
      with Abs_Bin show ?thesis by auto
    qed

    lemma int_of_real_0[simp]: "int_of_real (0::real) = (0::int)"
    by (simp add: int_of_real_def)

    lemma int_of_real_1[simp]: "int_of_real (1::real) = (1::int)"
    proof -
      have 1: "(1::real) = real (1::int)" by auto
      show ?thesis by (simp only: 1 int_of_real_real)
    qed

    lemma int_of_real_number_of[simp]: "int_of_real (number_of b) = number_of b"
    proof -
      have "real_is_int (number_of b)" by simp
      then have uu: "?! u::int. number_of b = real u" by (auto simp add: real_is_int_rep)
      then obtain u::int where u:"number_of b = real u" by auto
      have "number_of b = real ((number_of b)::int)"
        by (simp add: number_of_eq real_of_int_def)
      have ub: "number_of b = real ((number_of b)::int)"
        by (simp add: number_of_eq real_of_int_def)
      from uu u ub have unb: "u = number_of b"
        by blast
      have "int_of_real (number_of b) = u" by (simp add: u)
      with unb show ?thesis by simp
    qed

    lemma float_transfer_even: "even a ==> float (a, b) = float (a div 2, b+1)"
      apply (subst float_transfer[where a="a" and b="b" and c="-1", simplified])
      apply (simp_all add: pow2_def even_def real_is_int_def ring_eq_simps)
      apply (auto)
    proof -
      fix q::int
      have a:"b - (-1::int) = (1::int) + b" by arith
      show "(float (q, (b - (-1::int)))) = (float (q, ((1::int) + b)))"
        by (simp add: a)
    qed

    consts
      norm_float :: "int*int => int*int"

    lemma int_div_zdiv: "int (a div b) = (int a) div (int b)"
    apply (subst split_div, auto)
    apply (subst split_zdiv, auto)
    apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
    apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
    done

    lemma int_mod_zmod: "int (a mod b) = (int a) mod (int b)"
    apply (subst split_mod, auto)
    apply (subst split_zmod, auto)
    apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia in IntDiv.unique_remainder)
    apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
    done

    lemma abs_div_2_less: "a ≠ 0 ==> a ≠ -1 ==> abs((a::int) div 2) < abs a"
    by arith

    lemma terminating_norm_float: "∀a. (a::int) ≠ 0 ∧ even a --> a ≠ 0 ∧ ¦a div 2¦ < ¦a¦"
    apply (auto)
    apply (rule abs_div_2_less)
    apply (auto)
    done

    ML {* simp_depth_limit := 2 *}
    recdef norm_float "measure (% (a,b). nat (abs a))"
      "norm_float (a,b) = (if (a ≠ 0) & (even a) then norm_float (a div 2, b+1) else (if a=0 then (0,0) else (a,b)))"
    (hints simp: terminating_norm_float)
    ML {* simp_depth_limit := 1000 *}

    lemma norm_float: "float x = float (norm_float x)"
    proof -
      {
        fix a b :: int
        have norm_float_pair: "float (a,b) = float (norm_float (a,b))"
        proof (induct a b rule: norm_float.induct)
          case (1 u v)
          show ?case
          proof cases
            assume u: "u ≠ 0 ∧ even u"
            with prems have ind: "float (u div 2, v + 1) = float (norm_float (u div 2, v + 1))" by auto
            with u have "float (u,v) = float (u div 2, v+1)" by (simp add: float_transfer_even)
            then show ?thesis
              apply (subst norm_float.simps)
              apply (simp add: ind)
              done
          next
            assume "~(u ≠ 0 ∧ even u)"
            then show ?thesis
              by (simp add: prems float_def)
          qed
        qed
      }
      note helper = this
      have "? a b. x = (a,b)" by auto
      then obtain a b where "x = (a, b)" by blast
      then show ?thesis by (simp only: helper)
    qed

    lemma pow2_int: "pow2 (int n) = 2^n"
      by (simp add: pow2_def)

    lemma float_add:
      "float (a1, e1) + float (a2, e2) =
      (if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1)
      else float (a1*2^(nat (e1-e2))+a2, e2))"
      apply (simp add: float_def ring_eq_simps)
      apply (auto simp add: pow2_int[symmetric] pow2_add[symmetric])
      done

    lemma float_mult:
      "float (a1, e1) * float (a2, e2) =
      (float (a1 * a2, e1 + e2))"
      by (simp add: float_def pow2_add)

    lemma float_minus:
      "- (float (a,b)) = float (-a, b)"
      by (simp add: float_def)

    lemma zero_less_pow2:
      "0 < pow2 x"
    proof -
      {
        fix y
        have "0 <= y ==> 0 < pow2 y"
          by (induct y, induct_tac n, simp_all add: pow2_add)
      }
      note helper=this
      show ?thesis
        apply (case_tac "0 <= x")
        apply (simp add: helper)
        apply (subst pow2_neg)
        apply (simp add: helper)
        done
    qed

    lemma zero_le_float:
      "(0 <= float (a,b)) = (0 <= a)"
      apply (auto simp add: float_def)
      apply (auto simp add: zero_le_mult_iff zero_less_pow2)
      apply (insert zero_less_pow2[of b])
      apply (simp_all)
      done

    lemma float_le_zero:
      "(float (a,b) <= 0) = (a <= 0)"
      apply (auto simp add: float_def)
      apply (auto simp add: mult_le_0_iff)
      apply (insert zero_less_pow2[of b])
      apply auto
      done

    lemma float_abs:
      "abs (float (a,b)) = (if 0 <= a then (float (a,b)) else (float (-a,b)))"
      apply (auto simp add: abs_if)
      apply (simp_all add: zero_le_float[symmetric, of a b] float_minus)
      done

    lemma float_zero:
      "float (0, b) = 0"
      by (simp add: float_def)

    lemma float_pprt:
      "pprt (float (a, b)) = (if 0 <= a then (float (a,b)) else (float (0, b)))"
      by (auto simp add: zero_le_float float_le_zero float_zero)

    lemma float_nprt:
      "nprt (float (a, b)) = (if 0 <= a then (float (0,b)) else (float (a, b)))"
      by (auto simp add: zero_le_float float_le_zero float_zero)

    lemma norm_0_1: "(0::_::number_ring) = Numeral0 & (1::_::number_ring) = Numeral1"
      by auto

    lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)"
      by simp

    lemma add_right_zero: "a + 0 = (a::'a::comm_monoid_add)"
      by simp

    lemma mult_left_one: "1 * a = (a::'a::semiring_1)"
      by simp

    lemma mult_right_one: "a * 1 = (a::'a::semiring_1)"
      by simp

    lemma int_pow_0: "(a::int)^(Numeral0) = 1"
      by simp

    lemma int_pow_1: "(a::int)^(Numeral1) = a"
      by simp

    lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0"
      by simp

    lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1"
      by simp

    lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0"
      by simp

    lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1"
      by simp

    lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1"
      by simp

    lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1"
    proof -
      have 1:"((-1)::nat) = 0"
        by simp
      show ?thesis by (simp add: 1)
    qed

    lemma fst_cong: "a=a' ==> fst (a,b) = fst (a',b)"
      by simp

    lemma snd_cong: "b=b' ==> snd (a,b) = snd (a,b')"
      by simp

    lemma lift_bool: "x ==> x=True"
      by simp

    lemma nlift_bool: "~x ==> x=False"
      by simp

    lemma not_false_eq_true: "(~ False) = True" by simp

    lemma not_true_eq_false: "(~ True) = False" by simp

    lemmas binarith =
      Pls_0_eq Min_1_eq
      bin_pred_Pls bin_pred_Min bin_pred_1 bin_pred_0
      bin_succ_Pls bin_succ_Min bin_succ_1 bin_succ_0
      bin_add_Pls bin_add_Min bin_add_BIT_0 bin_add_BIT_10
      bin_add_BIT_11 bin_minus_Pls bin_minus_Min bin_minus_1
      bin_minus_0 bin_mult_Pls bin_mult_Min bin_mult_1 bin_mult_0
      bin_add_Pls_right bin_add_Min_right

    lemma int_eq_number_of_eq: "(((number_of v)::int)=(number_of w)) = iszero ((number_of (bin_add v (bin_minus w)))::int)"
      by simp

    lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)"
      by (simp only: iszero_number_of_Pls)

    lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))"
      by simp

    lemma int_iszero_number_of_0: "iszero ((number_of (w BIT bit.B0))::int) = iszero ((number_of w)::int)"
      by simp

    lemma int_iszero_number_of_1: "¬ iszero ((number_of (w BIT bit.B1))::int)"
      by simp

    lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (bin_add x (bin_minus y)))::int)"
      by simp

    lemma int_not_neg_number_of_Pls: "¬ (neg (Numeral0::int))"
      by simp

    lemma int_neg_number_of_Min: "neg (-1::int)"
      by simp

    lemma int_neg_number_of_BIT: "neg ((number_of (w BIT x))::int) = neg ((number_of w)::int)"
      by simp

    lemma int_le_number_of_eq: "(((number_of x)::int) ≤ number_of y) = (¬ neg ((number_of (bin_add y (bin_minus x)))::int))"
      by simp

    lemmas intarithrel =
      int_eq_number_of_eq
      lift_bool[OF int_iszero_number_of_Pls] nlift_bool[OF int_nonzero_number_of_Min] int_iszero_number_of_0
      lift_bool[OF int_iszero_number_of_1] int_less_number_of_eq_neg nlift_bool[OF int_not_neg_number_of_Pls] lift_bool[OF int_neg_number_of_Min]
      int_neg_number_of_BIT int_le_number_of_eq

    lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (bin_add v w)"
      by simp

    lemma int_number_of_diff_sym: "((number_of v)::int) - number_of w = number_of (bin_add v (bin_minus w))"
      by simp

    lemma int_number_of_mult_sym: "((number_of v)::int) * number_of w = number_of (bin_mult v w)"
      by simp

    lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (bin_minus v)"
      by simp

    lemmas intarith = int_number_of_add_sym int_number_of_minus_sym int_number_of_diff_sym int_number_of_mult_sym

    lemmas natarith = add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of

    lemmas powerarith = nat_number_of zpower_number_of_even
      zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring]
      zpower_Pls zpower_Min

    lemmas floatarith[simplified norm_0_1] = float_add float_mult float_minus float_abs zero_le_float float_pprt float_nprt

    (* for use with the compute oracle *)
    lemmas arith = binarith intarith intarithrel natarith powerarith floatarith not_false_eq_true not_true_eq_false

    end

lemma pow2_0:

      pow2 0 = 1

lemma pow2_1:

      pow2 1 = 2

lemma pow2_neg:

      pow2 x = inverse (pow2 (- x))

lemma pow2_add1:

      pow2 (1 + a) = 2 * pow2 a

lemma pow2_add:

      pow2 (a + b) = pow2 a * pow2 b

lemma

      float (a, e) + float (b, e) = float (a + b, e)

lemma real_is_int_def2:

      real_is_int x = (x = real (int_of_real x))

lemma float_transfer:

      real_is_int (real a * pow2 c)
      ==> float (a, b) = float (int_of_real (real a * pow2 c), b - c)

lemma pow2_int:

      pow2 (int c) = 2 ^ c

lemma float_transfer_nat:

      float (a, b) = float (a * 2 ^ c, b - int c)

lemma real_is_int_real:

      real_is_int (real x)

lemma int_of_real_real:

      int_of_real (real x) = x

lemma real_int_of_real:

      real_is_int x ==> real (int_of_real x) = x

lemma real_is_int_add_int_of_real:

      [| real_is_int a; real_is_int b |]
      ==> int_of_real (a + b) = int_of_real a + int_of_real b

lemma real_is_int_add:

      [| real_is_int a; real_is_int b |] ==> real_is_int (a + b)

lemma int_of_real_sub:

      [| real_is_int a; real_is_int b |]
      ==> int_of_real (a - b) = int_of_real a - int_of_real b

lemma real_is_int_sub:

      [| real_is_int a; real_is_int b |] ==> real_is_int (a - b)

lemma real_is_int_rep:

      real_is_int x ==> ∃!a. real a = x

lemma int_of_real_mult:

      [| real_is_int a; real_is_int b |]
      ==> int_of_real (a * b) = int_of_real a * int_of_real b

lemma real_is_int_mult:

      [| real_is_int a; real_is_int b |] ==> real_is_int (a * b)

lemma real_is_int_0:

      real_is_int 0

lemma real_is_int_1:

      real_is_int 1

lemma real_is_int_n1:

      real_is_int -1

lemma real_is_int_number_of:

      real_is_int (number_of x)

lemma int_of_real_0:

      int_of_real 0 = 0

lemma int_of_real_1:

      int_of_real 1 = 1

lemma int_of_real_number_of:

      int_of_real (number_of b) = number_of b

lemma float_transfer_even:

      even a ==> float (a, b) = float (a div 2, b + 1)

lemma int_div_zdiv:

      int (a div b) = int a div int b

lemma int_mod_zmod:

      int (a mod b) = int a mod int b

lemma abs_div_2_less:

      [| a ≠ 0; a ≠ -1 |] ==> ¦a div 2¦ < ¦a¦

lemma terminating_norm_float:

      ∀a. a ≠ 0 ∧ even a --> a ≠ 0 ∧ ¦a div 2¦ < ¦a¦

lemma norm_float:

      float x = float (norm_float x)

lemma pow2_int:

      pow2 (int n) = 2 ^ n

lemma float_add:

      float (a1.0, e1.0) + float (a2.0, e2.0) =
      (if e1.0 ≤ e2.0 then float (a1.0 + a2.0 * 2 ^ nat (e2.0 - e1.0), e1.0)
       else float (a1.0 * 2 ^ nat (e1.0 - e2.0) + a2.0, e2.0))

lemma float_mult:

      float (a1.0, e1.0) * float (a2.0, e2.0) = float (a1.0 * a2.0, e1.0 + e2.0)

lemma float_minus:

      - float (a, b) = float (- a, b)

lemma zero_less_pow2:

      0 < pow2 x

lemma zero_le_float:

      (0 ≤ float (a, b)) = (0 ≤ a)

lemma float_le_zero:

      (float (a, b) ≤ 0) = (a ≤ 0)

lemma float_abs:

      ¦float (a, b)¦ = (if 0 ≤ a then float (a, b) else float (- a, b))

lemma float_zero:

      float (0, b) = 0

lemma float_pprt:

      pprt (float (a, b)) = (if 0 ≤ a then float (a, b) else float (0, b))

lemma float_nprt:

      nprt (float (a, b)) = (if 0 ≤ a then float (0, b) else float (a, b))

lemma norm_0_1:

      (0::'a) = Numeral0 ∧ (1::'b) = Numeral1

lemma add_left_zero:

      (0::'a) + a = a

lemma add_right_zero:

      a + (0::'a) = a

lemma mult_left_one:

      (1::'a) * a = a

lemma mult_right_one:

      a * (1::'a) = a

lemma int_pow_0:

      a ^ Numeral0 = 1

lemma int_pow_1:

      a ^ Numeral1 = a

lemma zero_eq_Numeral0_nring:

      (0::'a) = Numeral0

lemma one_eq_Numeral1_nring:

      (1::'a) = Numeral1

lemma zero_eq_Numeral0_nat:

      0 = Numeral0

lemma one_eq_Numeral1_nat:

      1 = Numeral1

lemma zpower_Pls:

      z ^ Numeral0 = Numeral1

lemma zpower_Min:

      z ^ -1 = Numeral1

lemma fst_cong:

      a = a' ==> fst (a, b) = fst (a', b)

lemma snd_cong:

      b = b' ==> snd (a, b) = snd (a, b')

lemma lift_bool:

      x ==> x = True

lemma nlift_bool:

      ¬ x ==> x = False

lemma not_false_eq_true:

      (¬ False) = True

lemma not_true_eq_false:

      (¬ True) = False

lemmas binarith:

      Numeral.Pls BIT bit.B0 = Numeral.Pls

      Numeral.Min BIT bit.B1 = Numeral.Min

      bin_pred Numeral.Pls = Numeral.Min

      bin_pred Numeral.Min = Numeral.Min BIT bit.B0

      bin_pred (w BIT bit.B1) = w BIT bit.B0

      bin_pred (w BIT bit.B0) = bin_pred w BIT bit.B1

      bin_succ Numeral.Pls = Numeral.Pls BIT bit.B1

      bin_succ Numeral.Min = Numeral.Pls

      bin_succ (w BIT bit.B1) = bin_succ w BIT bit.B0

      bin_succ (w BIT bit.B0) = w BIT bit.B1

      bin_add Numeral.Pls w = w

      bin_add Numeral.Min w = bin_pred w

      bin_add (v BIT bit.B0) (w BIT y) = bin_add v w BIT y

      bin_add (v BIT bit.B1) (w BIT bit.B0) = bin_add v w BIT bit.B1

      bin_add (v BIT bit.B1) (w BIT bit.B1) = bin_add v (bin_succ w) BIT bit.B0

      bin_minus Numeral.Pls = Numeral.Pls

      bin_minus Numeral.Min = Numeral.Pls BIT bit.B1

      bin_minus (w BIT bit.B1) = bin_pred (bin_minus w) BIT bit.B1

      bin_minus (w BIT bit.B0) = bin_minus w BIT bit.B0

      bin_mult Numeral.Pls w = Numeral.Pls

      bin_mult Numeral.Min w = bin_minus w

      bin_mult (v BIT bit.B1) w = bin_add (bin_mult v w BIT bit.B0) w

      bin_mult (v BIT bit.B0) w = bin_mult v w BIT bit.B0

      bin_add w Numeral.Pls = w

      bin_add w Numeral.Min = bin_pred w

lemmas binarith:

      Numeral.Pls BIT bit.B0 = Numeral.Pls

      Numeral.Min BIT bit.B1 = Numeral.Min

      bin_pred Numeral.Pls = Numeral.Min

      bin_pred Numeral.Min = Numeral.Min BIT bit.B0

      bin_pred (w BIT bit.B1) = w BIT bit.B0

      bin_pred (w BIT bit.B0) = bin_pred w BIT bit.B1

      bin_succ Numeral.Pls = Numeral.Pls BIT bit.B1

      bin_succ Numeral.Min = Numeral.Pls

      bin_succ (w BIT bit.B1) = bin_succ w BIT bit.B0

      bin_succ (w BIT bit.B0) = w BIT bit.B1

      bin_add Numeral.Pls w = w

      bin_add Numeral.Min w = bin_pred w

      bin_add (v BIT bit.B0) (w BIT y) = bin_add v w BIT y

      bin_add (v BIT bit.B1) (w BIT bit.B0) = bin_add v w BIT bit.B1

      bin_add (v BIT bit.B1) (w BIT bit.B1) = bin_add v (bin_succ w) BIT bit.B0

      bin_minus Numeral.Pls = Numeral.Pls

      bin_minus Numeral.Min = Numeral.Pls BIT bit.B1

      bin_minus (w BIT bit.B1) = bin_pred (bin_minus w) BIT bit.B1

      bin_minus (w BIT bit.B0) = bin_minus w BIT bit.B0

      bin_mult Numeral.Pls w = Numeral.Pls

      bin_mult Numeral.Min w = bin_minus w

      bin_mult (v BIT bit.B1) w = bin_add (bin_mult v w BIT bit.B0) w

      bin_mult (v BIT bit.B0) w = bin_mult v w BIT bit.B0

      bin_add w Numeral.Pls = w

      bin_add w Numeral.Min = bin_pred w

lemma int_eq_number_of_eq:

      (number_of v = number_of w) = iszero (number_of (bin_add v (bin_minus w)))

lemma int_iszero_number_of_Pls:

      iszero Numeral0

lemma int_nonzero_number_of_Min:

      ¬ iszero -1

lemma int_iszero_number_of_0:

      iszero (number_of (w BIT bit.B0)) = iszero (number_of w)

lemma int_iszero_number_of_1:

      ¬ iszero (number_of (w BIT bit.B1))

lemma int_less_number_of_eq_neg:

      (number_of x < number_of y) = neg (number_of (bin_add x (bin_minus y)))

lemma int_not_neg_number_of_Pls:

      ¬ neg Numeral0

lemma int_neg_number_of_Min:

      neg -1

lemma int_neg_number_of_BIT:

      neg (number_of (w BIT x)) = neg (number_of w)

lemma int_le_number_of_eq:

      (number_of x ≤ number_of y) = (¬ neg (number_of (bin_add y (bin_minus x))))

lemmas intarithrel:

      (number_of v = number_of w) = iszero (number_of (bin_add v (bin_minus w)))

      iszero Numeral0 = True

      iszero -1 = False

      iszero (number_of (w BIT bit.B0)) = iszero (number_of w)

      (¬ iszero (number_of (w1 BIT bit.B1))) = True

      (number_of x < number_of y) = neg (number_of (bin_add x (bin_minus y)))

      neg Numeral0 = False

      neg -1 = True

      neg (number_of (w BIT x)) = neg (number_of w)

      (number_of x ≤ number_of y) = (¬ neg (number_of (bin_add y (bin_minus x))))

lemmas intarithrel:

      (number_of v = number_of w) = iszero (number_of (bin_add v (bin_minus w)))

      iszero Numeral0 = True

      iszero -1 = False

      iszero (number_of (w BIT bit.B0)) = iszero (number_of w)

      (¬ iszero (number_of (w1 BIT bit.B1))) = True

      (number_of x < number_of y) = neg (number_of (bin_add x (bin_minus y)))

      neg Numeral0 = False

      neg -1 = True

      neg (number_of (w BIT x)) = neg (number_of w)

      (number_of x ≤ number_of y) = (¬ neg (number_of (bin_add y (bin_minus x))))

lemma int_number_of_add_sym:

      number_of v + number_of w = number_of (bin_add v w)

lemma int_number_of_diff_sym:

      number_of v - number_of w = number_of (bin_add v (bin_minus w))

lemma int_number_of_mult_sym:

      number_of v * number_of w = number_of (bin_mult v w)

lemma int_number_of_minus_sym:

      - number_of v = number_of (bin_minus v)

lemmas intarith:

      number_of v + number_of w = number_of (bin_add v w)

      - number_of v = number_of (bin_minus v)

      number_of v - number_of w = number_of (bin_add v (bin_minus w))

      number_of v * number_of w = number_of (bin_mult v w)

lemmas intarith:

      number_of v + number_of w = number_of (bin_add v w)

      - number_of v = number_of (bin_minus v)

      number_of v - number_of w = number_of (bin_add v (bin_minus w))

      number_of v * number_of w = number_of (bin_mult v w)

lemmas natarith:

      number_of v + number_of v' =
      (if neg (number_of v) then number_of v'
       else if neg (number_of v') then number_of v else number_of (bin_add v v'))

      number_of v - number_of v' =
      (if neg (number_of v') then number_of v
       else let d = number_of (bin_add v (bin_minus v'))
            in if neg d then 0 else nat d)

      number_of v * number_of v' =
      (if neg (number_of v) then 0 else number_of (bin_mult v v'))

      (number_of v = number_of v') =
      (if neg (number_of v) then iszero (number_of v') ∨ neg (number_of v')
       else if neg (number_of v') then iszero (number_of v)
            else iszero (number_of (bin_add v (bin_minus v'))))

      (number_of v < number_of v') =
      (if neg (number_of v) then neg (number_of (bin_minus v'))
       else neg (number_of (bin_add v (bin_minus v'))))

lemmas natarith:

      number_of v + number_of v' =
      (if neg (number_of v) then number_of v'
       else if neg (number_of v') then number_of v else number_of (bin_add v v'))

      number_of v - number_of v' =
      (if neg (number_of v') then number_of v
       else let d = number_of (bin_add v (bin_minus v'))
            in if neg d then 0 else nat d)

      number_of v * number_of v' =
      (if neg (number_of v) then 0 else number_of (bin_mult v v'))

      (number_of v = number_of v') =
      (if neg (number_of v) then iszero (number_of v') ∨ neg (number_of v')
       else if neg (number_of v') then iszero (number_of v)
            else iszero (number_of (bin_add v (bin_minus v'))))

      (number_of v < number_of v') =
      (if neg (number_of v) then neg (number_of (bin_minus v'))
       else neg (number_of (bin_add v (bin_minus v'))))

lemmas powerarith:

      nat (number_of w) = number_of w

      z ^ number_of (w BIT bit.B0) = (let w = z ^ number_of w in w * w)

      z ^ number_of (w BIT bit.B1) =
      (if Numeral0 ≤ number_of w then let w = z ^ number_of w in z * w * w
       else Numeral1)

      z ^ Numeral0 = Numeral1

      z ^ -1 = Numeral1

lemmas powerarith:

      nat (number_of w) = number_of w

      z ^ number_of (w BIT bit.B0) = (let w = z ^ number_of w in w * w)

      z ^ number_of (w BIT bit.B1) =
      (if Numeral0 ≤ number_of w then let w = z ^ number_of w in z * w * w
       else Numeral1)

      z ^ Numeral0 = Numeral1

      z ^ -1 = Numeral1

lemmas floatarith:

      float (a1.0, e1.0) + float (a2.0, e2.0) =
      (if e1.0 ≤ e2.0 then float (a1.0 + a2.0 * 2 ^ nat (e2.0 - e1.0), e1.0)
       else float (a1.0 * 2 ^ nat (e1.0 - e2.0) + a2.0, e2.0))

      float (a1.0, e1.0) * float (a2.0, e2.0) = float (a1.0 * a2.0, e1.0 + e2.0)

      - float (a, b) = float (- a, b)

      ¦float (a, b)¦ = (if Numeral0 ≤ a then float (a, b) else float (- a, b))

      (Numeral0 ≤ float (a, b)) = (Numeral0 ≤ a)

      pprt (float (a, b)) =
      (if Numeral0 ≤ a then float (a, b) else float (Numeral0, b))

      nprt (float (a, b)) =
      (if Numeral0 ≤ a then float (Numeral0, b) else float (a, b))

lemmas floatarith:

      float (a1.0, e1.0) + float (a2.0, e2.0) =
      (if e1.0 ≤ e2.0 then float (a1.0 + a2.0 * 2 ^ nat (e2.0 - e1.0), e1.0)
       else float (a1.0 * 2 ^ nat (e1.0 - e2.0) + a2.0, e2.0))

      float (a1.0, e1.0) * float (a2.0, e2.0) = float (a1.0 * a2.0, e1.0 + e2.0)

      - float (a, b) = float (- a, b)

      ¦float (a, b)¦ = (if Numeral0 ≤ a then float (a, b) else float (- a, b))

      (Numeral0 ≤ float (a, b)) = (Numeral0 ≤ a)

      pprt (float (a, b)) =
      (if Numeral0 ≤ a then float (a, b) else float (Numeral0, b))

      nprt (float (a, b)) =
      (if Numeral0 ≤ a then float (Numeral0, b) else float (a, b))

lemmas arith:

      Numeral.Pls BIT bit.B0 = Numeral.Pls

      Numeral.Min BIT bit.B1 = Numeral.Min

      bin_pred Numeral.Pls = Numeral.Min

      bin_pred Numeral.Min = Numeral.Min BIT bit.B0

      bin_pred (w BIT bit.B1) = w BIT bit.B0

      bin_pred (w BIT bit.B0) = bin_pred w BIT bit.B1

      bin_succ Numeral.Pls = Numeral.Pls BIT bit.B1

      bin_succ Numeral.Min = Numeral.Pls

      bin_succ (w BIT bit.B1) = bin_succ w BIT bit.B0

      bin_succ (w BIT bit.B0) = w BIT bit.B1

      bin_add Numeral.Pls w = w

      bin_add Numeral.Min w = bin_pred w

      bin_add (v BIT bit.B0) (w BIT y) = bin_add v w BIT y

      bin_add (v BIT bit.B1) (w BIT bit.B0) = bin_add v w BIT bit.B1

      bin_add (v BIT bit.B1) (w BIT bit.B1) = bin_add v (bin_succ w) BIT bit.B0

      bin_minus Numeral.Pls = Numeral.Pls

      bin_minus Numeral.Min = Numeral.Pls BIT bit.B1

      bin_minus (w BIT bit.B1) = bin_pred (bin_minus w) BIT bit.B1

      bin_minus (w BIT bit.B0) = bin_minus w BIT bit.B0

      bin_mult Numeral.Pls w = Numeral.Pls

      bin_mult Numeral.Min w = bin_minus w

      bin_mult (v BIT bit.B1) w = bin_add (bin_mult v w BIT bit.B0) w

      bin_mult (v BIT bit.B0) w = bin_mult v w BIT bit.B0

      bin_add w Numeral.Pls = w

      bin_add w Numeral.Min = bin_pred w

      number_of v + number_of w = number_of (bin_add v w)

      - number_of v = number_of (bin_minus v)

      number_of v - number_of w = number_of (bin_add v (bin_minus w))

      number_of v * number_of w = number_of (bin_mult v w)

      (number_of v = number_of w) = iszero (number_of (bin_add v (bin_minus w)))

      iszero Numeral0 = True

      iszero -1 = False

      iszero (number_of (w BIT bit.B0)) = iszero (number_of w)

      (¬ iszero (number_of (w1 BIT bit.B1))) = True

      (number_of x < number_of y) = neg (number_of (bin_add x (bin_minus y)))

      neg Numeral0 = False

      neg -1 = True

      neg (number_of (w BIT x)) = neg (number_of w)

      (number_of x ≤ number_of y) = (¬ neg (number_of (bin_add y (bin_minus x))))

      number_of v + number_of v' =
      (if neg (number_of v) then number_of v'
       else if neg (number_of v') then number_of v else number_of (bin_add v v'))

      number_of v - number_of v' =
      (if neg (number_of v') then number_of v
       else let d = number_of (bin_add v (bin_minus v'))
            in if neg d then 0 else nat d)

      number_of v * number_of v' =
      (if neg (number_of v) then 0 else number_of (bin_mult v v'))

      (number_of v = number_of v') =
      (if neg (number_of v) then iszero (number_of v') ∨ neg (number_of v')
       else if neg (number_of v') then iszero (number_of v)
            else iszero (number_of (bin_add v (bin_minus v'))))

      (number_of v < number_of v') =
      (if neg (number_of v) then neg (number_of (bin_minus v'))
       else neg (number_of (bin_add v (bin_minus v'))))

      nat (number_of w) = number_of w

      z ^ number_of (w BIT bit.B0) = (let w = z ^ number_of w in w * w)

      z ^ number_of (w BIT bit.B1) =
      (if Numeral0 ≤ number_of w then let w = z ^ number_of w in z * w * w
       else Numeral1)

      z ^ Numeral0 = Numeral1

      z ^ -1 = Numeral1

      float (a1.0, e1.0) + float (a2.0, e2.0) =
      (if e1.0 ≤ e2.0 then float (a1.0 + a2.0 * 2 ^ nat (e2.0 - e1.0), e1.0)
       else float (a1.0 * 2 ^ nat (e1.0 - e2.0) + a2.0, e2.0))

      float (a1.0, e1.0) * float (a2.0, e2.0) = float (a1.0 * a2.0, e1.0 + e2.0)

      - float (a, b) = float (- a, b)

      ¦float (a, b)¦ = (if Numeral0 ≤ a then float (a, b) else float (- a, b))

      (Numeral0 ≤ float (a, b)) = (Numeral0 ≤ a)

      pprt (float (a, b)) =
      (if Numeral0 ≤ a then float (a, b) else float (Numeral0, b))

      nprt (float (a, b)) =
      (if Numeral0 ≤ a then float (Numeral0, b) else float (a, b))

      (¬ False) = True

      (¬ True) = False

lemmas arith:

      Numeral.Pls BIT bit.B0 = Numeral.Pls

      Numeral.Min BIT bit.B1 = Numeral.Min

      bin_pred Numeral.Pls = Numeral.Min

      bin_pred Numeral.Min = Numeral.Min BIT bit.B0

      bin_pred (w BIT bit.B1) = w BIT bit.B0

      bin_pred (w BIT bit.B0) = bin_pred w BIT bit.B1

      bin_succ Numeral.Pls = Numeral.Pls BIT bit.B1

      bin_succ Numeral.Min = Numeral.Pls

      bin_succ (w BIT bit.B1) = bin_succ w BIT bit.B0

      bin_succ (w BIT bit.B0) = w BIT bit.B1

      bin_add Numeral.Pls w = w

      bin_add Numeral.Min w = bin_pred w

      bin_add (v BIT bit.B0) (w BIT y) = bin_add v w BIT y

      bin_add (v BIT bit.B1) (w BIT bit.B0) = bin_add v w BIT bit.B1

      bin_add (v BIT bit.B1) (w BIT bit.B1) = bin_add v (bin_succ w) BIT bit.B0

      bin_minus Numeral.Pls = Numeral.Pls

      bin_minus Numeral.Min = Numeral.Pls BIT bit.B1

      bin_minus (w BIT bit.B1) = bin_pred (bin_minus w) BIT bit.B1

      bin_minus (w BIT bit.B0) = bin_minus w BIT bit.B0

      bin_mult Numeral.Pls w = Numeral.Pls

      bin_mult Numeral.Min w = bin_minus w

      bin_mult (v BIT bit.B1) w = bin_add (bin_mult v w BIT bit.B0) w

      bin_mult (v BIT bit.B0) w = bin_mult v w BIT bit.B0

      bin_add w Numeral.Pls = w

      bin_add w Numeral.Min = bin_pred w

      number_of v + number_of w = number_of (bin_add v w)

      - number_of v = number_of (bin_minus v)

      number_of v - number_of w = number_of (bin_add v (bin_minus w))

      number_of v * number_of w = number_of (bin_mult v w)

      (number_of v = number_of w) = iszero (number_of (bin_add v (bin_minus w)))

      iszero Numeral0 = True

      iszero -1 = False

      iszero (number_of (w BIT bit.B0)) = iszero (number_of w)

      (¬ iszero (number_of (w1 BIT bit.B1))) = True

      (number_of x < number_of y) = neg (number_of (bin_add x (bin_minus y)))

      neg Numeral0 = False

      neg -1 = True

      neg (number_of (w BIT x)) = neg (number_of w)

      (number_of x ≤ number_of y) = (¬ neg (number_of (bin_add y (bin_minus x))))

      number_of v + number_of v' =
      (if neg (number_of v) then number_of v'
       else if neg (number_of v') then number_of v else number_of (bin_add v v'))

      number_of v - number_of v' =
      (if neg (number_of v') then number_of v
       else let d = number_of (bin_add v (bin_minus v'))
            in if neg d then 0 else nat d)

      number_of v * number_of v' =
      (if neg (number_of v) then 0 else number_of (bin_mult v v'))

      (number_of v = number_of v') =
      (if neg (number_of v) then iszero (number_of v') ∨ neg (number_of v')
       else if neg (number_of v') then iszero (number_of v)
            else iszero (number_of (bin_add v (bin_minus v'))))

      (number_of v < number_of v') =
      (if neg (number_of v) then neg (number_of (bin_minus v'))
       else neg (number_of (bin_add v (bin_minus v'))))

      nat (number_of w) = number_of w

      z ^ number_of (w BIT bit.B0) = (let w = z ^ number_of w in w * w)

      z ^ number_of (w BIT bit.B1) =
      (if Numeral0 ≤ number_of w then let w = z ^ number_of w in z * w * w
       else Numeral1)

      z ^ Numeral0 = Numeral1

      z ^ -1 = Numeral1

      float (a1.0, e1.0) + float (a2.0, e2.0) =
      (if e1.0 ≤ e2.0 then float (a1.0 + a2.0 * 2 ^ nat (e2.0 - e1.0), e1.0)
       else float (a1.0 * 2 ^ nat (e1.0 - e2.0) + a2.0, e2.0))

      float (a1.0, e1.0) * float (a2.0, e2.0) = float (a1.0 * a2.0, e1.0 + e2.0)

      - float (a, b) = float (- a, b)

      ¦float (a, b)¦ = (if Numeral0 ≤ a then float (a, b) else float (- a, b))

      (Numeral0 ≤ float (a, b)) = (Numeral0 ≤ a)

      pprt (float (a, b)) =
      (if Numeral0 ≤ a then float (a, b) else float (Numeral0, b))

      nprt (float (a, b)) =
      (if Numeral0 ≤ a then float (Numeral0, b) else float (a, b))

      (¬ False) = True

      (¬ True) = False