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theory Float(* 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