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theory ComputeFloat(* Title: HOL/Tools/ComputeFloat.thy Author: Steven Obua *) header {* Floating Point Representation of the Reals *} theory ComputeFloat imports Complex_Main uses "~~/src/Tools/float.ML" ("~~/src/HOL/Tools/float_arith.ML") begin definition pow2 :: "int => real" where "pow2 a = (if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a)))))" definition float :: "int * int => real" where "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]) by (auto simp add: h) show ?thesis proof (induct a) case (1 n) from pos show ?case by (simp add: algebra_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: algebra_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: algebra_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: algebra_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: algebra_simps) done qed qed lemma "float (a, e) + float (b, e) = float (a + b, e)" by (simp add: float_def algebra_simps) definition int_of_real :: "real => int" where "int_of_real x = (SOME y. real y = x)" definition real_is_int :: "real => bool" where "real_is_int x = (EX (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^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 :: int => 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 "real_is_int ((number_of :: int => real) x)" unfolding number_of_eq 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 :: int have "? u. x = u" apply (rule exI[where x = "x"]) apply (simp) done } then obtain u::int where "x = 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 algebra_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 lemma int_div_zdiv: "int (a div b) = (int a) div (int b)" by (rule zdiv_int) lemma int_mod_zmod: "int (a mod b) = (int a) mod (int b)" by (rule zmod_int) lemma abs_div_2_less: "a ≠ 0 ==> a ≠ -1 ==> abs((a::int) div 2) < abs a" by arith function norm_float :: "int => int => int × int" where "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))" by auto termination by (relation "measure (nat o abs o fst)") (auto intro: abs_div_2_less) lemma norm_float: "float x = float (split 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 add: helper) qed lemma float_add_l0: "float (0, e) + x = x" by (simp add: float_def) lemma float_add_r0: "x + float (0, e) = x" by (simp add: float_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 algebra_simps) apply (auto simp add: pow2_int[symmetric] pow2_add[symmetric]) done lemma float_add_assoc1: "(x + float (y1, e1)) + float (y2, e2) = (float (y1, e1) + float (y2, e2)) + x" by simp lemma float_add_assoc2: "(float (y1, e1) + x) + float (y2, e2) = (float (y1, e1) + float (y2, e2)) + x" by simp lemma float_add_assoc3: "float (y1, e1) + (x + float (y2, e2)) = (float (y1, e1) + float (y2, e2)) + x" by simp lemma float_add_assoc4: "float (y1, e1) + (float (y2, e2) + x) = (float (y1, e1) + float (y2, e2)) + x" by simp lemma float_mult_l0: "float (0, e) * x = float (0, 0)" by (simp add: float_def) lemma float_mult_r0: "x * float (0, e) = float (0, 0)" by (simp add: float_def) definition lbound :: "real => real" where "lbound x = min 0 x" definition ubound :: "real => real" where "ubound x = max 0 x" lemma lbound: "lbound x ≤ x" by (simp add: lbound_def) lemma ubound: "x ≤ ubound x" by (simp add: ubound_def) 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 pprt_lbound: "pprt (lbound x) = float (0, 0)" apply (simp add: float_def) apply (rule pprt_eq_0) apply (simp add: lbound_def) done lemma nprt_ubound: "nprt (ubound x) = float (0, 0)" apply (simp add: float_def) apply (rule nprt_eq_0) apply (simp add: ubound_def) done 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 = normalize_bin_simps pred_bin_simps succ_bin_simps add_bin_simps minus_bin_simps mult_bin_simps lemma int_eq_number_of_eq: "(((number_of v)::int)=(number_of w)) = iszero ((number_of (v + uminus w))::int)" by (rule eq_number_of_eq) 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_Bit0: "iszero ((number_of (Int.Bit0 w))::int) = iszero ((number_of w)::int)" by simp lemma int_iszero_number_of_Bit1: "¬ iszero ((number_of (Int.Bit1 w))::int)" by simp lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (x + (uminus y)))::int)" unfolding neg_def number_of_is_id 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_Bit0: "neg ((number_of (Int.Bit0 w))::int) = neg ((number_of w)::int)" by simp lemma int_neg_number_of_Bit1: "neg ((number_of (Int.Bit1 w))::int) = neg ((number_of w)::int)" by simp lemma int_le_number_of_eq: "(((number_of x)::int) ≤ number_of y) = (¬ neg ((number_of (y + (uminus x)))::int))" unfolding neg_def number_of_is_id by (simp add: not_less) 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_Bit0 lift_bool[OF int_iszero_number_of_Bit1] 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_Bit0 int_neg_number_of_Bit1 int_le_number_of_eq lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (v + w)" by simp lemma int_number_of_diff_sym: "((number_of v)::int) - number_of w = number_of (v + (uminus w))" by simp lemma int_number_of_mult_sym: "((number_of v)::int) * number_of w = number_of (v * w)" by simp lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (uminus 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_add_l0 float_add_r0 float_mult float_mult_l0 float_mult_r0 float_minus float_abs zero_le_float float_pprt float_nprt pprt_lbound nprt_ubound (* for use with the compute oracle *) lemmas arith = binarith intarith intarithrel natarith powerarith floatarith not_false_eq_true not_true_eq_false use "~~/src/HOL/Tools/float_arith.ML" end