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 e1.0e2.0 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 e1.0e2.0 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 e1.0e2.0 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 e1.0e2.0 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 e1.0e2.0 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