--- a/src/HOL/Library/Float.thy Fri Apr 20 10:47:04 2012 +0200
+++ b/src/HOL/Library/Float.thy Fri Apr 20 11:14:39 2012 +0200
@@ -218,14 +218,14 @@
lemma transfer_numeral [transfer_rule]:
"fun_rel (op =) cr_float (numeral :: _ \<Rightarrow> real) (numeral :: _ \<Rightarrow> float)"
- unfolding fun_rel_def cr_float_def by (simp add: real_of_float_def[symmetric])
+ unfolding fun_rel_def cr_float_def by simp
lemma float_neg_numeral[simp]: "real (neg_numeral x :: float) = neg_numeral x"
by (simp add: minus_numeral[symmetric] del: minus_numeral)
lemma transfer_neg_numeral [transfer_rule]:
"fun_rel (op =) cr_float (neg_numeral :: _ \<Rightarrow> real) (neg_numeral :: _ \<Rightarrow> float)"
- unfolding fun_rel_def cr_float_def by (simp add: real_of_float_def[symmetric])
+ unfolding fun_rel_def cr_float_def by simp
lemma
shows float_of_numeral[simp]: "numeral k = float_of (numeral k)"
@@ -366,10 +366,6 @@
subsection {* Compute arithmetic operations *}
-lemma real_of_float_Float[code]: "real_of_float (Float m e) =
- (if e \<ge> 0 then m * 2 ^ nat e else m * inverse (2 ^ nat (- e)))"
-by (auto simp add: powr_realpow[symmetric] powr_minus real_of_float_def[symmetric])
-
lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f"
unfolding real_of_float_eq mantissa_exponent[of f] by simp
@@ -415,11 +411,13 @@
unfolding real_of_int_inject by auto
qed
-lemma compute_zero[code_unfold, code]: "0 = Float 0 0"
+lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0"
by transfer simp
+hide_fact (open) compute_float_zero
-lemma compute_one[code_unfold, code]: "1 = Float 1 0"
+lemma compute_float_one[code_unfold, code]: "1 = Float 1 0"
by transfer simp
+hide_fact (open) compute_float_one
definition normfloat :: "float \<Rightarrow> float" where
[simp]: "normfloat x = x"
@@ -429,56 +427,71 @@
else if m = 0 then 0 else Float m e)"
unfolding normfloat_def
by transfer (auto simp add: powr_add zmod_eq_0_iff)
+hide_fact (open) compute_normfloat
lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k"
by transfer simp
+hide_fact (open) compute_float_numeral
lemma compute_float_neg_numeral[code_abbrev]: "Float (neg_numeral k) 0 = neg_numeral k"
by transfer simp
+hide_fact (open) compute_float_neg_numeral
lemma compute_float_uminus[code]: "- Float m1 e1 = Float (- m1) e1"
by transfer simp
+hide_fact (open) compute_float_uminus
lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)"
by transfer (simp add: field_simps powr_add)
+hide_fact (open) compute_float_times
lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 =
(if e1 \<le> e2 then Float (m1 + m2 * 2^nat (e2 - e1)) e1
else Float (m2 + m1 * 2^nat (e1 - e2)) e2)"
by transfer (simp add: field_simps powr_realpow[symmetric] powr_divide2[symmetric])
+hide_fact (open) compute_float_plus
lemma compute_float_minus[code]: fixes f g::float shows "f - g = f + (-g)"
by simp
+hide_fact (open) compute_float_minus
lemma compute_float_sgn[code]: "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then -1 else 0)"
by transfer (simp add: sgn_times)
+hide_fact (open) compute_float_sgn
lift_definition is_float_pos :: "float \<Rightarrow> bool" is "op < 0 :: real \<Rightarrow> bool" ..
lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m"
by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0])
+hide_fact (open) compute_is_float_pos
lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b - a)"
by transfer (simp add: field_simps)
+hide_fact (open) compute_float_less
lift_definition is_float_nonneg :: "float \<Rightarrow> bool" is "op \<le> 0 :: real \<Rightarrow> bool" ..
lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m"
by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0])
+hide_fact (open) compute_is_float_nonneg
lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b - a)"
by transfer (simp add: field_simps)
+hide_fact (open) compute_float_le
lift_definition is_float_zero :: "float \<Rightarrow> bool" is "op = 0 :: real \<Rightarrow> bool" by simp
lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m"
by transfer (auto simp add: is_float_zero_def)
+hide_fact (open) compute_is_float_zero
lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e"
by transfer (simp add: abs_mult)
+hide_fact (open) compute_float_abs
lemma compute_float_eq[code]: "equal_class.equal f g = is_float_zero (f - g)"
by transfer simp
+hide_fact (open) compute_float_eq
subsection {* Rounding Real numbers *}
@@ -581,6 +594,7 @@
by transfer
(auto simp add: round_down_def field_simps powr_add powr_minus inverse_eq_divide)
qed
+hide_fact (open) compute_float_down
lemma ceil_divide_floor_conv:
assumes "b \<noteq> 0"
@@ -643,6 +657,7 @@
by transfer
(simp add: round_up_def floor_divide_eq_div field_simps powr_add powr_minus inverse_eq_divide)
qed
+hide_fact (open) compute_float_up
lemmas real_of_ints =
real_of_int_zero
@@ -777,6 +792,7 @@
unfolding bitlen_def
by (auto simp: pos_imp_zdiv_pos_iff not_le)
qed
+hide_fact (open) compute_bitlen
lemma float_gt1_scale: assumes "1 \<le> Float m e"
shows "0 \<le> e + (bitlen m - 1)"
@@ -853,6 +869,7 @@
by transfer
(simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide field_simps Let_def
del: two_powr_minus_int_float)
+hide_fact (open) compute_lapprox_posrat
lift_definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
is "\<lambda>prec (x::nat) (y::nat). round_up (rat_precision prec x y) (x / y)" by simp
@@ -900,6 +917,7 @@
(simp add: round_up_def ceil_divide_floor_conv floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0)
qed
qed
+hide_fact (open) compute_rapprox_posrat
lemma rat_precision_pos:
assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
@@ -959,6 +977,7 @@
then - (rapprox_posrat prec (nat (-x)) (nat y))
else lapprox_posrat prec (nat (-x)) (nat (-y))))"
by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)
+hide_fact (open) compute_lapprox_rat
lift_definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
"\<lambda>prec (x::int) (y::int). round_up (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp
@@ -973,6 +992,7 @@
then - (lapprox_posrat prec (nat (-x)) (nat y))
else rapprox_posrat prec (nat (-x)) (nat (-y))))"
by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)
+hide_fact (open) compute_rapprox_rat
subsection {* Division *}
@@ -994,6 +1014,7 @@
using not_0
by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_down_shift, simp add: field_simps)
qed (transfer, auto)
+hide_fact (open) compute_float_divl
lift_definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is
"\<lambda>(prec::nat) a b. round_up (prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor> - \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)" by simp
@@ -1013,6 +1034,7 @@
using not_0
by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_up_shift, simp add: field_simps)
qed (transfer, auto)
+hide_fact (open) compute_float_divr
subsection {* Lemmas needed by Approximate *}
@@ -1208,34 +1230,28 @@
"float_round_down prec (Float m e) = (let d = bitlen (abs m) - int prec in
if 0 < d then let P = 2^nat d ; n = m div P in Float n (e + d)
else Float m e)"
- using compute_float_down[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
+ using Float.compute_float_down[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
by transfer (simp add: field_simps abs_mult log_mult bitlen_def cong del: if_weak_cong)
+hide_fact (open) compute_float_round_down
lemma compute_float_round_up[code]:
"float_round_up prec (Float m e) = (let d = (bitlen (abs m) - int prec) in
if 0 < d then let P = 2^nat d ; n = m div P ; r = m mod P
in Float (n + (if r = 0 then 0 else 1)) (e + d)
else Float m e)"
- using compute_float_up[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
+ using Float.compute_float_up[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
unfolding Let_def
by transfer (simp add: field_simps abs_mult log_mult bitlen_def cong del: if_weak_cong)
+hide_fact (open) compute_float_round_up
lemma Float_le_zero_iff: "Float a b \<le> 0 \<longleftrightarrow> a \<le> 0"
apply (auto simp: zero_float_def mult_le_0_iff)
using powr_gt_zero[of 2 b] by simp
-(* TODO: how to use as code equation? -> pprt_float?! *)
-lemma compute_pprt[code]: "pprt (Float a e) = (if a <= 0 then 0 else (Float a e))"
-unfolding pprt_def sup_float_def max_def Float_le_zero_iff ..
-
-(* TODO: how to use as code equation? *)
-lemma compute_nprt[code]: "nprt (Float a e) = (if a <= 0 then (Float a e) else 0)"
-unfolding nprt_def inf_float_def min_def Float_le_zero_iff ..
-
-lemma of_float_pprt[simp]: fixes a::float shows "real (pprt a) = pprt (real a)"
+lemma real_of_float_pprt[simp]: fixes a::float shows "real (pprt a) = pprt (real a)"
unfolding pprt_def sup_float_def max_def sup_real_def by auto
-lemma of_float_nprt[simp]: fixes a::float shows "real (nprt a) = nprt (real a)"
+lemma real_of_float_nprt[simp]: fixes a::float shows "real (nprt a) = nprt (real a)"
unfolding nprt_def inf_float_def min_def inf_real_def by auto
lift_definition int_floor_fl :: "float \<Rightarrow> int" is floor by simp
@@ -1243,12 +1259,14 @@
lemma compute_int_floor_fl[code]:
"int_floor_fl (Float m e) = (if 0 \<le> e then m * 2 ^ nat e else m div (2 ^ (nat (-e))))"
by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
+hide_fact (open) compute_int_floor_fl
lift_definition floor_fl :: "float \<Rightarrow> float" is "\<lambda>x. real (floor x)" by simp
lemma compute_floor_fl[code]:
"floor_fl (Float m e) = (if 0 \<le> e then Float m e else Float (m div (2 ^ (nat (-e)))) 0)"
by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
+hide_fact (open) compute_floor_fl
lemma floor_fl: "real (floor_fl x) \<le> real x" by transfer simp