--- a/src/HOL/Library/Float.thy Sat Oct 17 01:05:59 2009 +0200
+++ b/src/HOL/Library/Float.thy Sat Oct 17 14:43:18 2009 +0200
@@ -243,7 +243,7 @@
fix x y z :: real
assume "y \<noteq> 0"
then have "(x * inverse y = z) = (x = z * y)"
- by auto
+ by auto
}
note inverse = this
have eq': "real a * (pow2 (b - b')) = real a'"
@@ -551,9 +551,9 @@
} moreover
{ have "x + 1 \<le> x - x mod 2 + 2"
proof -
- have "x mod 2 < 2" using `0 < x` by auto
- hence "x < x - x mod 2 + 2" unfolding algebra_simps by auto
- thus ?thesis by auto
+ have "x mod 2 < 2" using `0 < x` by auto
+ hence "x < x - x mod 2 + 2" unfolding algebra_simps by auto
+ thus ?thesis by auto
qed
also have "x - x mod 2 + 2 = (x div 2 + 1) * 2" unfolding algebra_simps using `0 < x` zdiv_zmod_equality2[of x 2 0] by auto
also have "\<dots> \<le> 2^nat (bitlen (x div 2)) * 2" using hyp[OF `0 < x div 2`, THEN conjunct2] by (rule mult_right_mono, auto)
@@ -839,7 +839,7 @@
have "0 < ?X div y"
proof -
have "2^nat (bitlen x - 1) \<le> y" and "y < 2^nat (bitlen y)"
- using bitlen_bounds[OF `0 < x`, THEN conjunct1] bitlen_bounds[OF `0 < y`, THEN conjunct2] `x < y` by auto
+ using bitlen_bounds[OF `0 < x`, THEN conjunct1] bitlen_bounds[OF `0 < y`, THEN conjunct2] `x < y` by auto
hence "(2::int)^nat (bitlen x - 1) < 2^nat (bitlen y)" by (rule order_le_less_trans)
hence "bitlen x \<le> bitlen y" by auto
hence len_less: "nat (bitlen x - 1) \<le> nat (int (n - 1) + bitlen y)" by auto
@@ -847,16 +847,16 @@
have "x \<noteq> 0" and "y \<noteq> 0" using `0 < x` `0 < y` by auto
have exp_eq: "nat (int (n - 1) + bitlen y) - nat (bitlen x - 1) = ?l"
- using `bitlen x \<le> bitlen y` bitlen_ge1[OF `x \<noteq> 0`] bitlen_ge1[OF `y \<noteq> 0`] `0 < n` by auto
+ using `bitlen x \<le> bitlen y` bitlen_ge1[OF `x \<noteq> 0`] bitlen_ge1[OF `y \<noteq> 0`] `0 < n` by auto
have "y * 2^nat (bitlen x - 1) \<le> y * x"
- using bitlen_bounds[OF `0 < x`, THEN conjunct1] `0 < y`[THEN less_imp_le] by (rule mult_left_mono)
+ using bitlen_bounds[OF `0 < x`, THEN conjunct1] `0 < y`[THEN less_imp_le] by (rule mult_left_mono)
also have "\<dots> \<le> 2^nat (bitlen y) * x" using bitlen_bounds[OF `0 < y`, THEN conjunct2, THEN less_imp_le] `0 \<le> x` by (rule mult_right_mono)
also have "\<dots> \<le> x * 2^nat (int (n - 1) + bitlen y)" unfolding mult_commute[of x] by (rule mult_right_mono, auto simp add: `0 \<le> x`)
finally have "real y * 2^nat (bitlen x - 1) * inverse (2^nat (bitlen x - 1)) \<le> real x * 2^nat (int (n - 1) + bitlen y) * inverse (2^nat (bitlen x - 1))"
- unfolding real_of_int_le_iff[symmetric] by auto
+ unfolding real_of_int_le_iff[symmetric] by auto
hence "real y \<le> real x * (2^nat (int (n - 1) + bitlen y) / (2^nat (bitlen x - 1)))"
- unfolding real_mult_assoc real_divide_def by auto
+ unfolding real_mult_assoc real_divide_def by auto
also have "\<dots> = real x * (2^(nat (int (n - 1) + bitlen y) - nat (bitlen x - 1)))" using power_diff[of "2::real", OF _ len_less] by auto
finally have "y \<le> x * 2^?l" unfolding exp_eq unfolding real_of_int_le_iff[symmetric] by auto
thus ?thesis using zdiv_greater_zero[OF `0 < y`] by auto
@@ -1194,19 +1194,19 @@
case True
have m: "m = m div ?p * ?p + 0" unfolding True[symmetric] using zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right, symmetric] .
have "real (Float m e) = real (Float (m div ?p) (e + ?d))" unfolding real_of_float_simp arg_cong[OF m, of real]
- by (auto simp add: pow2_add `0 < ?d` pow_d)
+ by (auto simp add: pow2_add `0 < ?d` pow_d)
thus ?thesis
- unfolding Float round_up.simps Let_def if_P[OF `m mod ?p = 0`] if_P[OF `0 < ?d`]
- by auto
+ unfolding Float round_up.simps Let_def if_P[OF `m mod ?p = 0`] if_P[OF `0 < ?d`]
+ by auto
next
case False
have "m = m div ?p * ?p + m mod ?p" unfolding zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right] ..
also have "\<dots> \<le> (m div ?p + 1) * ?p" unfolding left_distrib zmult_1 by (rule add_left_mono, rule pos_mod_bound[OF `0 < ?p`, THEN less_imp_le])
finally have "real (Float m e) \<le> real (Float (m div ?p + 1) (e + ?d))" unfolding real_of_float_simp add_commute[of e]
- unfolding pow2_add mult_assoc[symmetric] real_of_int_le_iff[of m, symmetric]
- by (auto intro!: mult_mono simp add: pow2_add `0 < ?d` pow_d)
+ unfolding pow2_add mult_assoc[symmetric] real_of_int_le_iff[of m, symmetric]
+ by (auto intro!: mult_mono simp add: pow2_add `0 < ?d` pow_d)
thus ?thesis
- unfolding Float round_up.simps Let_def if_not_P[OF `\<not> m mod ?p = 0`] if_P[OF `0 < ?d`] .
+ unfolding Float round_up.simps Let_def if_not_P[OF `\<not> m mod ?p = 0`] if_P[OF `0 < ?d`] .
qed
next
case False
@@ -1264,7 +1264,7 @@
have "real (m div 2^(nat ?l)) * pow2 ?l \<le> real m"
proof -
have "real (m div 2^(nat ?l)) * pow2 ?l = real (2^(nat ?l) * (m div 2^(nat ?l)))" unfolding real_of_int_mult real_of_int_power[symmetric] real_number_of unfolding pow2_int[symmetric]
- using `?l > 0` by auto
+ using `?l > 0` by auto
also have "\<dots> \<le> real (2^(nat ?l) * (m div 2^(nat ?l)) + m mod 2^(nat ?l))" unfolding real_of_int_add by auto
also have "\<dots> = real m" unfolding zmod_zdiv_equality[symmetric] ..
finally show ?thesis by auto