dedicated fact collections for algebraic simplification rules potentially splitting goals
--- a/NEWS Wed Oct 09 23:00:52 2019 +0200
+++ b/NEWS Wed Oct 09 14:51:54 2019 +0000
@@ -71,10 +71,11 @@
1): theory HOL/Library/Bit.thy has been renamed accordingly.
INCOMPATIBILITY.
-* Fact collection sign_simps contains only simplification rules for
-products being less / greater / equal to zero; combine with existing
-collections algebra_simps or field_simps to obtain reasonable
-simplification. INCOMPATIBILITY.
+* Fact collections algebra_split_simps and field_split_simps correspond
+to algebra_simps and field_simps but contain more aggressive rules
+potentially splitting goals; algebra_split_simps roughly replaces
+sign_simps and field_split_simps can be used instead of divide_simps.
+INCOMPATIBILITY.
* Theory HOL-Library.Monad_Syntax: infix operation "bind" (>>=)
associates to the left now as is customary.
--- a/src/HOL/Analysis/Abstract_Euclidean_Space.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Abstract_Euclidean_Space.thy Wed Oct 09 14:51:54 2019 +0000
@@ -270,7 +270,7 @@
done
next
have 1: "\<And>x i. \<lbrakk> i \<le> n; x i \<noteq> 0\<rbrakk> \<Longrightarrow> (\<Sum>i\<le>n. (x i / sqrt (\<Sum>j\<le>n. (x j)\<^sup>2))\<^sup>2) = 1"
- by (force simp: sum_nonneg sum_nonneg_eq_0_iff divide_simps simp flip: sum_divide_distrib)
+ by (force simp: sum_nonneg sum_nonneg_eq_0_iff field_split_simps simp flip: sum_divide_distrib)
have cm: "continuous_map ?Y (nsphere n) (\<lambda>x i. x i / sqrt (\<Sum>j\<le>n. (x j)\<^sup>2))"
unfolding Euclidean_space_def nsphere subtopology_subtopology continuous_map_in_subtopology
proof (intro continuous_intros conjI)
@@ -390,7 +390,7 @@
have 2: "(?g \<circ> ?h) (0, x) = f x" if "x \<in> topspace (nsphere p)" for x
using that f1 by simp
have 3: "(?g \<circ> ?h) (1, x) = (\<lambda>i. - a i)" for x
- using a by (force simp: divide_simps nsphere)
+ using a by (force simp: field_split_simps nsphere)
then show "homotopic_with (\<lambda>x. True) (nsphere p) (nsphere p) f (\<lambda>x. (\<lambda>i. - a i))"
by (force simp: homotopic_with intro: 1 2 3)
qed
--- a/src/HOL/Analysis/Arcwise_Connected.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Arcwise_Connected.thy Wed Oct 09 14:51:54 2019 +0000
@@ -211,7 +211,7 @@
proof
assume "of_nat m = (4 * of_nat k + 1) / (2 * 2^n :: 'a)"
then have "of_nat (m * (2 * 2^n)) = (of_nat (Suc (4 * k)) :: 'a)"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
then have "m * (2 * 2^n) = Suc (4 * k)"
using of_nat_eq_iff by blast
then have "odd (m * (2 * 2^n))"
@@ -224,7 +224,7 @@
proof
assume "of_nat m = (4 * of_nat k + 3) / (2 * 2^n :: 'a)"
then have "of_nat (m * (2 * 2^n)) = (of_nat (4 * k + 3) :: 'a)"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
then have "m * (2 * 2^n) = (4 * k) + 3"
using of_nat_eq_iff by blast
then have "odd (m * (2 * 2^n))"
@@ -520,7 +520,7 @@
lemma dyadics_in_open_unit_interval:
"{0<..<1} \<inter> (\<Union>k m. {real m / 2^k}) = (\<Union>k. \<Union>m \<in> {0<..<2^k}. {real m / 2^k})"
- by (auto simp: divide_simps)
+ by (auto simp: field_split_simps)
@@ -738,9 +738,9 @@
by simp
finally have "(real i * 2^n - real j * 2^m) / 2^m \<in> \<int>" .
with True Ints_abs show "\<bar>2^n\<bar> * \<bar>real i / 2^m - real j / 2^n\<bar> \<in> \<int>"
- by (fastforce simp: divide_simps)
+ by (fastforce simp: field_split_simps)
show "\<bar>\<bar>2^n\<bar> * \<bar>real i / 2^m - real j / 2^n\<bar>\<bar> < 1"
- using less.prems by (auto simp: divide_simps)
+ using less.prems by (auto simp: field_split_simps)
qed
then have "real i / 2^m = real j / 2^n"
by auto
@@ -770,7 +770,7 @@
proof (cases "real i / 2^m" "real j / 2^n" rule: linorder_cases)
case less
moreover have "real (4 * k + 1) / 2 ^ Suc n + 1 / (2 ^ Suc n) = real j / 2 ^ n"
- using k by (force simp: divide_simps)
+ using k by (force simp: field_split_simps)
moreover have "\<bar>real i / 2 ^ m - real j / 2 ^ n\<bar> < 2 / (2 ^ Suc n)"
using less.prems by simp
ultimately have closer: "\<bar>real i / 2 ^ m - real (4 * k + 1) / 2 ^ Suc n\<bar> < 1 / (2 ^ Suc n)"
@@ -778,7 +778,7 @@
have *: "a (real (4 * k + 1) / 2 ^ Suc n) \<le> c (real i / 2 ^ m) \<and>
c (real i / 2 ^ m) \<le> b (real (4 * k + 1) / 2 ^ Suc n)"
apply (rule less.IH [OF _ refl])
- using closer \<open>n < m\<close> \<open>d = m - n\<close> apply (auto simp: divide_simps \<open>n < m\<close> diff_less_mono2)
+ using closer \<open>n < m\<close> \<open>d = m - n\<close> apply (auto simp: field_split_simps \<open>n < m\<close> diff_less_mono2)
done
show ?thesis
using LR [of "c((2*k + 1) / 2^n)" "a((2*k + 1) / 2^n)" "b((2*k + 1) / 2^n)"]
@@ -791,7 +791,7 @@
next
case greater
moreover have "real (4 * k + 3) / 2 ^ Suc n - 1 / (2 ^ Suc n) = real j / 2 ^ n"
- using k by (force simp: divide_simps)
+ using k by (force simp: field_split_simps)
moreover have "\<bar>real i / 2 ^ m - real j / 2 ^ n\<bar> < 2 * 1 / (2 ^ Suc n)"
using less.prems by simp
ultimately have closer: "\<bar>real i / 2 ^ m - real (4 * k + 3) / 2 ^ Suc n\<bar> < 1 / (2 ^ Suc n)"
@@ -799,7 +799,7 @@
have *: "a (real (4 * k + 3) / 2 ^ Suc n) \<le> c (real i / 2 ^ m) \<and>
c (real i / 2 ^ m) \<le> b (real (4 * k + 3) / 2 ^ Suc n)"
apply (rule less.IH [OF _ refl])
- using closer \<open>n < m\<close> \<open>d = m - n\<close> apply (auto simp: divide_simps \<open>n < m\<close> diff_less_mono2)
+ using closer \<open>n < m\<close> \<open>d = m - n\<close> apply (auto simp: field_split_simps \<open>n < m\<close> diff_less_mono2)
done
show ?thesis
using LR [of "c((2*k + 1) / 2^n)" "a((2*k + 1) / 2^n)" "b((2*k + 1) / 2^n)"]
@@ -963,7 +963,7 @@
obtain n where n: "1/2^n < min d 1"
by (metis \<open>0 < d\<close> divide_less_eq_1 less_numeral_extra(1) min_def one_less_numeral_iff power_one_over real_arch_pow_inv semiring_norm(76) zero_less_numeral)
with gr0I have "n > 0"
- by (force simp: divide_simps)
+ by (force simp: field_split_simps)
show "\<exists>d>0. \<forall>x\<in>D01. \<forall>x'\<in>D01. dist x' x < d \<longrightarrow> dist (f (c x')) (f (c x)) < e"
proof (intro exI ballI impI conjI)
show "(0::real) < 1/2^n" by auto
@@ -1100,12 +1100,12 @@
by auto
show "\<bar>real i / 2 ^ m - real j / 2 ^ n\<bar> < 1/2 ^ n"
using clo less_j1 j_le
- apply (auto simp: le_max_iff_disj divide_simps dist_norm)
+ apply (auto simp: le_max_iff_disj field_split_simps dist_norm)
apply (auto simp: algebra_simps abs_if split: if_split_asm dest: 1 2)
done
show "\<bar>real k / 2 ^ p - real j / 2 ^ n\<bar> < 1/2 ^ n"
using clo less_j1 j_le
- apply (auto simp: le_max_iff_disj divide_simps dist_norm)
+ apply (auto simp: le_max_iff_disj field_split_simps dist_norm)
apply (auto simp: algebra_simps not_less abs_if split: if_split_asm dest: 3 2)
done
qed (use False in simp)
@@ -1392,7 +1392,7 @@
proof cases
case 1
with less.prems show ?thesis
- by (rule_tac x=1 in exI)+ (fastforce simp: divide_simps)
+ by (rule_tac x=1 in exI)+ (fastforce simp: field_split_simps)
next
case 2 show ?thesis
proof (cases m)
@@ -1415,7 +1415,7 @@
then show ?thesis
using less.IH [of "m'+p'" i m' k p'] less.prems \<open>m = Suc m'\<close> 2 Suc
apply atomize
- apply (force simp: divide_simps)
+ apply (force simp: field_split_simps)
done
qed
qed
@@ -1464,7 +1464,7 @@
have "c(i / 2^m) \<le> b(real(4 * q + 1) / 2 ^ (Suc n'))"
apply (rule ci_le_bj, force)
apply (rule * [OF less])
- using i_le_j clo_ij q apply (auto simp: divide_simps)
+ using i_le_j clo_ij q apply (auto simp: field_split_simps)
done
then show ?thesis
using alec [of "2*q+1" n'] cleb [of "2*q+1" n'] a_ge_0 [of "2*q+1" n'] b_le_1 [of "2*q+1" n'] b41 [of n' q] \<open>n' \<noteq> 0\<close>
@@ -1484,7 +1484,7 @@
have "a(real(4*q + 3) / 2 ^ (Suc n')) \<le> c(j / 2^n)"
apply (rule aj_le_ci, force)
apply (rule * [OF less])
- using j_le_j clo_jj q apply (auto simp: divide_simps)
+ using j_le_j clo_jj q apply (auto simp: field_split_simps)
done
then show ?thesis
using alec [of "2*q+1" n'] cleb [of "2*q+1" n'] a_ge_0 [of "2*q+1" n'] b_le_1 [of "2*q+1" n'] a43 [of n' q] \<open>n' \<noteq> 0\<close>
@@ -1589,7 +1589,7 @@
then show ?thesis by (metis that)
qed
have m_div: "0 < m / 2^n" "m / 2^n < 1"
- using m by (auto simp: divide_simps)
+ using m by (auto simp: field_split_simps)
have closure0m: "{0..m / 2^n} = closure ({0<..< m / 2^n} \<inter> (\<Union>k m. {real m / 2 ^ k}))"
by (subst closure_dyadic_rationals_in_convex_set_pos_1, simp_all add: not_le m)
have closurem1: "{m / 2^n .. 1} = closure ({m / 2^n <..< 1} \<inter> (\<Union>k m. {real m / 2 ^ k}))"
@@ -1611,7 +1611,7 @@
fix p q
assume p: "0 < real p / 2 ^ q" "real p / 2 ^ q < real m / 2 ^ n"
then have [simp]: "0 < p" "p < 2 ^ q"
- apply (simp add: divide_simps)
+ apply (simp add: field_split_simps)
apply (blast intro: p less_2I m_div less_trans)
done
have "f (c (real p / 2 ^ q)) \<in> f ` {0..c (real m / 2 ^ n)}"
--- a/src/HOL/Analysis/Ball_Volume.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Ball_Volume.thy Wed Oct 09 14:51:54 2019 +0000
@@ -46,7 +46,7 @@
(auto intro!: derivative_eq_intros continuous_intros simp: set_borel_measurable_def)
also have "\<dots> = (\<integral>\<^sup>+ x. 2 * ennreal ((1 - x\<^sup>2) powr (real n / 2) * indicator {0..1} x) \<partial>lborel)"
by (intro nn_integral_cong_AE AE_I[of _ _ "{0}"])
- (auto simp: indicator_def powr_minus powr_half_sqrt divide_simps ennreal_mult' mult_ac)
+ (auto simp: indicator_def powr_minus powr_half_sqrt field_split_simps ennreal_mult' mult_ac)
also have "\<dots> = (\<integral>\<^sup>+ x. ennreal ((1 - x\<^sup>2) powr (real n / 2) * indicator {0..1} x) \<partial>lborel) +
(\<integral>\<^sup>+ x. ennreal ((1 - x\<^sup>2) powr (real n / 2) * indicator {0..1} x) \<partial>lborel)"
(is "_ = ?I + _") by (simp add: mult_2 nn_integral_add)
--- a/src/HOL/Analysis/Cartesian_Euclidean_Space.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Cartesian_Euclidean_Space.thy Wed Oct 09 14:51:54 2019 +0000
@@ -289,7 +289,7 @@
apply (rule onorm_le_matrix_component)
using Bclo by (simp add: abs_minus_commute less_imp_le)
also have "\<dots> < e"
- using \<open>0 < e\<close> by (simp add: divide_simps)
+ using \<open>0 < e\<close> by (simp add: field_split_simps)
finally show "onorm ((*v) (A - B)) < e" .
qed (use B in auto)
qed
--- a/src/HOL/Analysis/Cauchy_Integral_Theorem.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Cauchy_Integral_Theorem.thy Wed Oct 09 14:51:54 2019 +0000
@@ -323,7 +323,7 @@
show "(g1 +++ g2 \<circ> (\<lambda>x. (x + 1) / 2)) x' = g2 x'" if "x' \<in> {0..1}" "dist x' x < dist ((x + 1) / 2) (1/2)" for x'
proof -
have [simp]: "(2*x'+2)/2 = x'+1"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
show ?thesis
using that by (auto simp: joinpaths_def)
qed
@@ -691,7 +691,7 @@
show "g1 +++ g2 \<circ> (\<lambda>x. (x + 1) / 2) differentiable at x"
using g12D that
apply (simp only: joinpaths_def)
- apply (drule_tac x= "(x+1) / 2" in bspec, force simp: divide_simps)
+ apply (drule_tac x= "(x+1) / 2" in bspec, force simp: field_split_simps)
apply (rule differentiable_chain_at derivative_intros | force)+
done
show "\<And>x'. dist x' x < dist ((x + 1) / 2) (1/2) \<Longrightarrow> (g1 +++ g2 \<circ> (\<lambda>x. (x + 1) / 2)) x' = g2 x'"
@@ -699,7 +699,7 @@
qed (use that in \<open>auto simp: dist_norm\<close>)
have [simp]: "vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at ((x+1)/2)) = 2 *\<^sub>R vector_derivative g2 (at x)"
if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S)" for x
- using that by (auto simp: vector_derivative_chain_at divide_simps g2D)
+ using that by (auto simp: vector_derivative_chain_at field_split_simps g2D)
have "continuous_on ({1/2..1} - insert (1/2) S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
using co12 by (rule continuous_on_subset) force
then have coDhalf: "continuous_on ({1/2..1} - insert (1/2) S) (\<lambda>x. vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x))"
@@ -1926,7 +1926,7 @@
{ assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}"
have **: "\<And>x. ((k - x) / k) *\<^sub>R a + (x / k) *\<^sub>R c = (1 - x) *\<^sub>R a + x *\<^sub>R b"
using False apply (simp add: c' algebra_simps)
- apply (simp add: real_vector.scale_left_distrib [symmetric] divide_simps)
+ apply (simp add: real_vector.scale_left_distrib [symmetric] field_split_simps)
done
have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}"
using k has_integral_affinity01 [OF *, of "inverse k" "0"]
@@ -2185,28 +2185,28 @@
case 1 then have "?\<Phi> a c' b'"
using assms
apply (clarsimp simp: c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
- apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
+ apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps)
done
then show ?thesis by blast
next
case 2 then have "?\<Phi> a' c' b"
using assms
apply (clarsimp simp: a'_def c'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
- apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
+ apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps)
done
then show ?thesis by blast
next
case 3 then have "?\<Phi> a' c b'"
using assms
apply (clarsimp simp: a'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
- apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
+ apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps)
done
then show ?thesis by blast
next
case 4 then have "?\<Phi> a' b' c'"
using assms
apply (clarsimp simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
- apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
+ apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps)
done
then show ?thesis by blast
qed
@@ -2576,7 +2576,7 @@
by (simp add: algebra_simps)
have "cmod y / (24 * C) \<le> cmod y / cmod (b - a) / 12"
using False \<open>C>0\<close> diff_2C [of b a] ynz
- by (auto simp: divide_simps hull_inc)
+ by (auto simp: field_split_simps hull_inc)
have less_C: "\<lbrakk>u \<in> convex hull {a, b, c}; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> x * cmod u < C" for x u
apply (cases "x=0", simp add: \<open>0<C\<close>)
using Cno [of u] mult_left_le_one_le [of "cmod u" x] le_less_trans norm_ge_zero by blast
@@ -2610,7 +2610,7 @@
by (auto simp: hull_inc intro: d1 interior_subset [THEN subsetD] linepath_in_convex_hull)
also have "\<dots> \<le> cmod y / cmod (v - u) / 12"
using False uv \<open>C>0\<close> diff_2C [of v u] ynz
- by (auto simp: divide_simps hull_inc)
+ by (auto simp: field_split_simps hull_inc)
finally have "cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) \<le> cmod y / cmod (v - u) / 12"
by simp
then have cmod_12_le: "cmod (v - u) * cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) * 12 \<le> cmod y"
@@ -3993,7 +3993,7 @@
using eq by (simp add: exp_eq_1 complex_is_Int_iff)
have "exp (contour_integral p (\<lambda>w. 1 / (w - z))) = (\<gamma> 1 - z) / (\<gamma> 0 - z)"
using p winding_number_exp_integral(2) [of p 0 1 z]
- apply (simp add: valid_path_def path_defs contour_integral_integral exp_minus divide_simps)
+ apply (simp add: valid_path_def path_defs contour_integral_integral exp_minus field_split_simps)
by (metis path_image_def pathstart_def pathstart_in_path_image)
then have "winding_number p z \<in> \<int> \<longleftrightarrow> pathfinish p = pathstart p"
using p wneq iff by (auto simp: path_defs)
@@ -4528,7 +4528,7 @@
define z' where "z' = z - (d / (2 * cmod a)) *\<^sub>R a"
have *: "a \<bullet> z' \<le> b - d / 3 * cmod a"
unfolding z'_def inner_mult_right' divide_inverse
- apply (simp add: divide_simps algebra_simps dot_square_norm power2_eq_square anz)
+ apply (simp add: field_split_simps algebra_simps dot_square_norm power2_eq_square anz)
apply (metis \<open>0 < d\<close> add_increasing azb less_eq_real_def mult_nonneg_nonneg mult_right_mono norm_ge_zero norm_numeral)
done
have "cmod (winding_number \<gamma> z' - winding_number \<gamma> z) < \<bar>Re (winding_number \<gamma> z)\<bar> - 1/2"
@@ -4542,7 +4542,7 @@
moreover have "\<bar>Re (winding_number \<gamma> z')\<bar> < 1/2"
apply (rule winding_number_lt_half [OF \<gamma> *])
using azb \<open>d>0\<close> pag
- apply (auto simp: add_strict_increasing anz divide_simps algebra_simps dest!: subsetD)
+ apply (auto simp: add_strict_increasing anz field_split_simps algebra_simps dest!: subsetD)
done
ultimately have False
by simp
@@ -4620,7 +4620,7 @@
show ?thesis
apply (rule has_contour_integral_eq)
using znotin has_contour_integral_add [OF has_contour_integral_lmul [OF has_contour_integral_winding_number [OF vpg znotin], of "f z"] *]
- apply (auto simp: mult_ac divide_simps)
+ apply (auto simp: ac_simps divide_simps)
done
qed
@@ -5009,7 +5009,7 @@
then have "z + of_real r * exp (\<i> * of_real z) \<in> (\<lambda>x. z + of_real r * exp (\<i> * linepath s t x)) ` {0..1}"
apply (rule_tac x="(z - s)/(t - s)" in image_eqI)
apply (simp add: linepath_def scaleR_conv_of_real divide_simps exp_eq)
- apply (auto simp: algebra_simps divide_simps)
+ apply (auto simp: field_split_simps)
done
}
ultimately show ?thesis
@@ -5148,7 +5148,7 @@
have *: "finite {x. cmod (complex_of_real (2 * real_of_int x * pi) * \<i>) \<le> b + cmod (Ln w)}"
apply (simp add: norm_mult finite_int_iff_bounded_le)
apply (rule_tac x="\<lfloor>(b + cmod (Ln w)) / (2*pi)\<rfloor>" in exI)
- apply (auto simp: divide_simps le_floor_iff)
+ apply (auto simp: field_split_simps le_floor_iff)
done
have [simp]: "\<And>P f. {z. P z \<and> (\<exists>n. z = f n)} = f ` {n. P (f n)}"
by blast
@@ -5274,7 +5274,8 @@
proof (rule ccontr)
assume "\<not> \<bar>s - t\<bar> \<le> 2 * pi"
then have *: "\<And>n. t - s \<noteq> of_int n * \<bar>s - t\<bar>"
- using False that [of "2*pi / \<bar>t - s\<bar>"] by (simp add: abs_minus_commute divide_simps)
+ using False that [of "2*pi / \<bar>t - s\<bar>"]
+ by (simp add: abs_minus_commute divide_simps)
show False
using * [of 1] * [of "-1"] by auto
qed
@@ -5292,7 +5293,7 @@
apply (subst abs_away)
apply (auto simp: 1)
apply (rule ccontr)
- apply (auto simp: 2 divide_simps abs_mult dest: of_int_leD)
+ apply (auto simp: 2 field_split_simps abs_mult dest: of_int_leD)
done
qed
@@ -5408,7 +5409,7 @@
apply (rule sincos_total_2pi [of "Re(w/of_real(norm w))" "Im(w/of_real(norm w))"])
apply (simp add: divide_simps \<open>w \<noteq> 0\<close> cmod_power2 [symmetric])
apply (rule_tac x="t / (2*pi)" in image_eqI)
- apply (simp add: divide_simps \<open>w \<noteq> 0\<close>)
+ apply (simp add: field_simps \<open>w \<noteq> 0\<close>)
using False **
apply (auto simp: w_def)
done
@@ -5565,7 +5566,7 @@
using eventually_happens [OF eventually_conj]
by (fastforce simp: contour_integrable_on path_image_def)
have Ble: "B * e / (\<bar>B\<bar> + 1) \<le> e"
- using \<open>0 \<le> B\<close> \<open>0 < e\<close> by (simp add: divide_simps)
+ using \<open>0 \<le> B\<close> \<open>0 < e\<close> by (simp add: field_split_simps)
have "\<exists>h. (\<forall>x\<in>{0..1}. cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - h x) \<le> e) \<and> h integrable_on {0..1}"
proof (intro exI conjI ballI)
show "cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - f a (\<gamma> x) * vector_derivative \<gamma> (at x)) \<le> e"
@@ -5693,7 +5694,8 @@
qed
{ fix a::real and b::real assume ab: "a > 0" "b > 0"
then have "k * (1 + real k) * (1 / a) \<le> k * (1 + real k) * (4 / b) \<longleftrightarrow> b \<le> 4 * a"
- by (subst mult_le_cancel_left_pos) (use \<open>k \<noteq> 0\<close> in \<open>auto simp: divide_simps\<close>)
+ by (subst mult_le_cancel_left_pos)
+ (use \<open>k \<noteq> 0\<close> in \<open>auto simp: divide_simps\<close>)
with ab have "real k * (1 + real k) / a \<le> (real k * 4 + real k * real k * 4) / b \<longleftrightarrow> b \<le> 4 * a"
by (simp add: field_simps)
} note canc = this
@@ -5706,14 +5708,14 @@
using lessd d x
by (auto simp add: dist_norm path_image_def ball_def not_less [symmetric] del: divide_const_simps)
then have "d \<le> cmod (x - v) * 2"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
then have dpow_le: "d ^ (k+2) \<le> (cmod (x - v) * 2) ^ (k+2)"
using \<open>0 < d\<close> order_less_imp_le power_mono by blast
have "x \<noteq> v" using that
using \<open>x \<in> path_image \<gamma>\<close> ball_divide_subset_numeral d by fastforce
then show ?thesis
using \<open>d > 0\<close> apply (simp add: ff_def norm_mult norm_divide norm_power dist_norm canc)
- using dpow_le apply (simp add: algebra_simps divide_simps mult_less_0_iff)
+ using dpow_le apply (simp add: algebra_simps field_split_simps mult_less_0_iff)
done
qed
have ub: "u \<in> ball w (d/2)"
@@ -6370,7 +6372,7 @@
show ?thes1 using *
using contour_integrable_on_def by blast
show ?thes2
- unfolding contour_integral_unique [OF *] by (simp add: divide_simps)
+ unfolding contour_integral_unique [OF *] by (simp add: field_split_simps)
qed
corollary Cauchy_contour_integral_circlepath:
@@ -6461,7 +6463,7 @@
also have "\<dots> \<le> e * norm (u - w)"
using r kle \<open>0 < e\<close> by (simp add: dist_commute dist_norm)
finally show ?thesis
- by (simp add: divide_simps norm_divide del: power_Suc)
+ by (simp add: field_split_simps norm_divide del: power_Suc)
qed
with \<open>0 < r\<close> show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>sphere z r.
norm ((\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k)) - f x / (x - w)) < e"
@@ -6527,7 +6529,7 @@
by (rule B) (use norm_triangle_ineq4 [of x z] in auto)
with \<open>R > 0\<close> fz show False
using has_contour_integral_bound_circlepath [OF *, of "norm(f z)/2/R"]
- by (auto simp: less_imp_le norm_mult norm_divide divide_simps)
+ by (auto simp: less_imp_le norm_mult norm_divide field_split_simps)
qed
proposition Liouville_weak:
@@ -6546,7 +6548,7 @@
have 2: "((\<lambda>x. 1 / f x) \<longlongrightarrow> 0) at_infinity"
apply (rule tendstoI [OF eventually_mono])
apply (rule_tac B="2/e" in unbounded)
- apply (simp add: dist_norm norm_divide divide_simps mult_ac)
+ apply (simp add: dist_norm norm_divide field_split_simps mult_ac)
done
have False
using Liouville_weak_0 [OF 1 2] f by simp
@@ -6679,7 +6681,7 @@
then have f': "f' n w = 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)"
using DERIV_unique [OF fnd] w by blast
show ?thesis
- by (simp add: f' Cauchy_contour_integral_circlepath_2 [OF g w] derg [OF w] divide_simps)
+ by (simp add: f' Cauchy_contour_integral_circlepath_2 [OF g w] derg [OF w] field_split_simps)
qed
define d where "d = (r - norm(w - z))^2"
have "d > 0"
@@ -6703,7 +6705,7 @@
fix e::real
assume "0 < e"
with \<open>r > 0\<close> show "\<forall>\<^sub>F n in F. \<forall>x\<in>sphere z r. dist (f n x / (x - w)\<^sup>2) (g x / (x - w)\<^sup>2) < e"
- apply (simp add: norm_divide divide_simps sphere_def dist_norm)
+ apply (simp add: norm_divide field_split_simps sphere_def dist_norm)
apply (rule eventually_mono [OF uniform_limitD [OF ulim], of "e*d"])
apply (simp add: \<open>0 < d\<close>)
apply (force simp: dist_norm dle intro: less_le_trans)
@@ -6949,7 +6951,7 @@
have inb: "z + complex_of_real ((dist z w + r) / 2) \<in> ball z r"
proof -
have wz: "cmod (w - z) < r" using w
- by (auto simp: divide_simps dist_norm norm_minus_commute)
+ by (auto simp: field_split_simps dist_norm norm_minus_commute)
then have "0 \<le> r"
by (meson less_eq_real_def norm_ge_zero order_trans)
show ?thesis
@@ -7145,7 +7147,7 @@
shows "(\<lambda>z. if z = a then deriv g a
else f z - g a/(z - a)) holomorphic_on S"
using pole_lemma [OF holg a]
- by (rule holomorphic_transform) (simp add: eq divide_simps)
+ by (rule holomorphic_transform) (simp add: eq field_split_simps)
lemma pole_lemma_open:
assumes "f holomorphic_on S" "open S"
@@ -7175,7 +7177,7 @@
and [simp]: "f a = deriv g a" "g a = 0"
shows "f holomorphic_on S"
using pole_theorem [OF holg a eq]
- by (rule holomorphic_transform) (auto simp: eq divide_simps)
+ by (rule holomorphic_transform) (auto simp: eq field_split_simps)
lemma pole_theorem_open_0:
assumes holg: "g holomorphic_on S" and S: "open S"
@@ -7183,7 +7185,7 @@
and [simp]: "f a = deriv g a" "g a = 0"
shows "f holomorphic_on S"
using pole_theorem_open [OF holg S eq]
- by (rule holomorphic_transform) (auto simp: eq divide_simps)
+ by (rule holomorphic_transform) (auto simp: eq field_split_simps)
lemma pole_theorem_analytic:
assumes g: "g analytic_on S"
@@ -7379,7 +7381,7 @@
then have "((\<lambda>x. f z * (1 / (x - z))) has_contour_integral 0) \<gamma>"
using has_contour_integral_lmul by fastforce
then have "((\<lambda>x. f z / (x - z)) has_contour_integral 0) \<gamma>"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
moreover have "((\<lambda>x. (f x - f z) / (x - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
using z
apply (auto simp: v_def)
@@ -7467,7 +7469,7 @@
apply (rule mult_mono)
using that D interior_subset apply blast
using \<open>L>0\<close> \<open>e>0\<close> \<open>D>0\<close> DL2
- apply (auto simp: norm_divide divide_simps algebra_simps)
+ apply (auto simp: norm_divide field_split_simps algebra_simps)
done
finally show ?thesis .
qed
@@ -7701,7 +7703,7 @@
then have "((\<lambda>w. f z * (1 / (w - z))) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
by (metis mult.commute has_contour_integral_lmul)
then have 1: "((\<lambda>w. f z / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
moreover have 2: "((\<lambda>w. (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
using U [OF z] pasz d_def by (force elim: has_contour_integral_eq [where g = "\<lambda>w. (f w - f z)/(w - z)"])
show ?thesis
--- a/src/HOL/Analysis/Change_Of_Vars.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Change_Of_Vars.thy Wed Oct 09 14:51:54 2019 +0000
@@ -540,9 +540,9 @@
then have "dist (f' x ((y - x) /\<^sub>R r)) ((f y - f x) /\<^sub>R r) = norm (f' x (y - x) /\<^sub>R r - (f y - f x) /\<^sub>R r)"
by (simp add: dist_norm)
also have "\<dots> = norm (f' x (y - x) - (f y - f x)) / r"
- using \<open>r > 0\<close> by (simp add: scale_right_diff_distrib [symmetric] divide_simps)
+ using \<open>r > 0\<close> by (simp add: divide_simps scale_right_diff_distrib [symmetric])
also have "\<dots> \<le> norm (f y - (f x + f' x (y - x))) / norm (y - x)"
- using that \<open>r > 0\<close> False by (simp add: algebra_simps divide_simps dist_norm norm_minus_commute mult_right_mono)
+ using that \<open>r > 0\<close> False by (simp add: algebra_simps field_split_simps dist_norm norm_minus_commute mult_right_mono)
also have "\<dots> < k"
using that \<open>0 < \<zeta>\<close> by (simp add: dist_commute r_def \<zeta> [OF \<open>y \<in> S\<close> False])
finally show "dist (f' x ((y - x) /\<^sub>R r)) ((f y - f x) /\<^sub>R r) < k" .
@@ -883,7 +883,7 @@
with ml have 0: "0 < - (m - ?\<mu> (f ` S))/2 / ?\<mu> S"
using that zero_less_measure_iff by force
then show ?thesis
- using * [OF 0] that by (auto simp: divide_simps m_def split: if_split_asm)
+ using * [OF 0] that by (auto simp: field_split_simps m_def split: if_split_asm)
qed
then show "?\<mu> (f ` S) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
by fastforce
@@ -991,7 +991,7 @@
fix k::real
assume "k \<in> \<int>" and k: "\<bar>k\<bar> \<le> 2 ^ (2*n)"
show "0 \<le> k/2^n * ?\<Omega> n k x"
- using f \<open>k \<in> \<int>\<close> apply (auto simp: indicator_def divide_simps Ints_def)
+ using f \<open>k \<in> \<int>\<close> apply (auto simp: indicator_def field_split_simps Ints_def)
apply (drule spec [where x=x])
using zero_le_power [of "2::real" n] mult_nonneg_nonneg [of "f x" "2^n"]
by linarith
@@ -1002,7 +1002,7 @@
(\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n).
k/2^n * (indicator {y. k/2^n \<le> f y \<and> f y < (k+1/2)/2^n} x +
indicator {y. (k+1/2)/2^n \<le> f y \<and> f y < (k+1)/2^n} x))"
- by (rule sum.cong [OF refl]) (simp add: indicator_def divide_simps)
+ by (rule sum.cong [OF refl]) (simp add: indicator_def field_split_simps)
also have "\<dots> = (\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * indicator {y. k/2^n \<le> f y \<and> f y < (k+1/2)/2^n} x) +
(\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n). k/2^n * indicator {y. (k+1/2)/2^n \<le> f y \<and> f y < (k+1)/2^n} x)"
by (simp add: comm_monoid_add_class.sum.distrib algebra_simps)
@@ -1021,7 +1021,7 @@
show "(\<Sum>k | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n).
2 * k/2 ^ Suc n * indicator {y. (2 * k+1)/2 ^ Suc n \<le> f y \<and> f y < (2 * k+1 + 1)/2 ^ Suc n} x)
\<le> sum ?h {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}"
- by (rule sum_mono) (simp add: indicator_def divide_simps)
+ by (rule sum_mono) (simp add: indicator_def field_split_simps)
next
have \<alpha>: "?a = (\<Sum>k \<in> (*)2 ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}.
k/2 ^ Suc n * indicator {y. k/2 ^ Suc n \<le> f y \<and> f y < (k+1)/2 ^ Suc n} x)"
@@ -1183,7 +1183,7 @@
ultimately have lim_dl: "((\<lambda>x. (\<bar>d \<bullet> x\<bar> - k)) \<circ> (?\<beta> \<circ> \<rho>)) \<longlonglongrightarrow> (\<bar>d \<bullet> l\<bar> - k)"
by (meson continuous_imp_tendsto)
have "\<forall>\<^sub>F i in sequentially. 0 \<le> ((\<lambda>x. \<bar>d \<bullet> x\<bar> - k) \<circ> ((\<lambda>n. \<alpha> n /\<^sub>R norm (\<alpha> n)) \<circ> \<rho>)) i"
- using \<alpha> kd by (auto simp: divide_simps)
+ using \<alpha> kd by (auto simp: field_split_simps)
then have "k \<le> \<bar>d \<bullet> l\<bar>"
using tendsto_lowerbound [OF lim_dl, of 0] by auto
moreover have "d \<bullet> l = 0"
@@ -1606,7 +1606,7 @@
also have "\<dots> = norm ((A i - A j) *v (\<gamma> p - x)) / norm (\<gamma> p - x)"
by (simp add: divide_inverse matrix_vector_mult_scaleR)
also have "\<dots> \<le> 2 / N"
- using no_le by (auto simp: divide_simps)
+ using no_le by (auto simp: field_split_simps)
finally show "norm (?V i p - ?V j p) \<le> 2 / N" .
qed
have "isCont (\<lambda>w. (norm(A i *v w - A j *v w) - 2 / N)) z"
@@ -1652,7 +1652,7 @@
show "linear ((*v) (matrix (f' x) - B))"
by (rule matrix_vector_mul_linear)
have "((\<lambda>y. ((f x + f' x (y - x)) - f y) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within S)"
- using tendsto_minus [OF lim_df] by (simp add: algebra_simps divide_simps)
+ using tendsto_minus [OF lim_df] by (simp add: algebra_simps field_split_simps)
then show "((\<lambda>y. (matrix (f' x) - B) *v (y - x) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within S)"
proof (rule Lim_transform)
have "((\<lambda>y. ((f y + B *v x - (f x + B *v y)) /\<^sub>R norm (y - x))) \<longlongrightarrow> 0) (at x within S)"
@@ -1704,7 +1704,7 @@
also have "\<dots> \<le> (e - 1 / (Suc (p + q))) * norm (y - x)"
proof (rule mult_right_mono)
have "1 / Suc (p + q) \<le> 1 / q"
- using \<open>q \<noteq> 0\<close> by (auto simp: divide_simps)
+ using \<open>q \<noteq> 0\<close> by (auto simp: field_split_simps)
also have "\<dots> < e/2"
using qe2 by auto
finally show "e / 2 \<le> e - 1 / real (Suc (p + q))"
@@ -1714,7 +1714,7 @@
by (simp add: algebra_simps)
qed
then show "inverse (norm (y - x)) * norm (f y + B *v x - (f x + B *v y)) < e"
- using \<open>y \<noteq> x\<close> by (simp add: divide_simps algebra_simps)
+ using \<open>y \<noteq> x\<close> by (simp add: field_split_simps algebra_simps)
qed
qed
then show "((\<lambda>y. (matrix (f' x) - B) *v (y - x) /\<^sub>R
@@ -1784,7 +1784,7 @@
by (rule Bo_e6)
finally have "norm ((?A - B) *v (y - x)) / norm (y - x) < e / 6" .
then show "norm ((?A - B) *v (y - x)) \<le> e * norm (y - x) / 6"
- by (simp add: divide_simps False)
+ by (simp add: field_split_simps False)
have "norm ((matrix (f' x) - B) *v (y - x) - ((?A - B) *v (y - x))) = norm ((\<chi> i j. if i = m \<and> j = n then e / 4 else 0) *v (y - x))"
by (simp add: algebra_simps)
also have "\<dots> = norm((e/4) *\<^sub>R (y - x)$n *\<^sub>R axis m (1::real))"
@@ -2192,7 +2192,7 @@
finally show ?thesis .
qed
finally have *: "1 / real (?m ^ ?m) * norm (v - u) ^ ?n \<le> ?\<mu> K"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
show ?thesis
using mult_left_mono [OF *, of "e / (2*c) ^ ?m"] \<open>c > 0\<close> \<open>e > 0\<close> by auto
qed
@@ -2235,7 +2235,7 @@
using \<open>c > 0\<close> by (simp add: content_cbox_if_cart)
finally have "sum ?\<mu> \<F> \<le> (2 ^ ?m * c ^ ?m)" .
then show ?thesis
- using \<open>e > 0\<close> \<open>c > 0\<close> by (auto simp: divide_simps mult_less_0_iff)
+ using \<open>e > 0\<close> \<open>c > 0\<close> by (auto simp: field_split_simps mult_less_0_iff)
qed
finally show ?thesis .
qed
@@ -2362,7 +2362,7 @@
show "norm (if x \<in> {t. h n (g t) = y} then ?D x else 0) \<le> ?D x *\<^sub>R f (g x) /\<^sub>R y"
if "x \<in> S" for x
using nonneg_h [of n x] \<open>y > 0\<close> nonneg_fg [of x] h_le_f [of x n] that
- by (auto simp: divide_simps ordered_semiring_class.mult_left_mono)
+ by (auto simp: divide_simps mult_left_mono)
qed (use S in auto)
then have int_det: "(\<lambda>t. \<bar>det (matrix (g' t))\<bar>) integrable_on ({t. h n (g t) = y} \<inter> S)"
using integrable_restrict_Int by force
--- a/src/HOL/Analysis/Complex_Analysis_Basics.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Complex_Analysis_Basics.thy Wed Oct 09 14:51:54 2019 +0000
@@ -401,13 +401,13 @@
"\<lbrakk>f field_differentiable at z; f z \<noteq> 0\<rbrakk>
\<Longrightarrow> deriv (\<lambda>w. inverse (f w)) z = - deriv f z / f z ^ 2"
unfolding DERIV_deriv_iff_field_differentiable[symmetric]
- by (safe intro!: DERIV_imp_deriv derivative_eq_intros) (auto simp: divide_simps power2_eq_square)
+ by (safe intro!: DERIV_imp_deriv derivative_eq_intros) (auto simp: field_split_simps power2_eq_square)
lemma deriv_divide [simp]:
"\<lbrakk>f field_differentiable at z; g field_differentiable at z; g z \<noteq> 0\<rbrakk>
\<Longrightarrow> deriv (\<lambda>w. f w / g w) z = (deriv f z * g z - f z * deriv g z) / g z ^ 2"
by (simp add: field_class.field_divide_inverse field_differentiable_inverse)
- (simp add: divide_simps power2_eq_square)
+ (simp add: field_split_simps power2_eq_square)
lemma deriv_cdivide_right:
"f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w / c) z = deriv f z / c"
--- a/src/HOL/Analysis/Complex_Transcendental.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Complex_Transcendental.thy Wed Oct 09 14:51:54 2019 +0000
@@ -332,11 +332,11 @@
proof -
have "cos (\<theta> j - \<Theta>) = ((r j)\<^sup>2 + R\<^sup>2 - (norm(?z j - ?Z))\<^sup>2) / (2 * R * r j)" for j
apply (subst cmod_diff_squared)
- using assms by (auto simp: divide_simps less_le)
+ using assms by (auto simp: field_split_simps less_le)
moreover have "(\<lambda>j. ((r j)\<^sup>2 + R\<^sup>2 - (norm(?z j - ?Z))\<^sup>2) / (2 * R * r j)) \<longlonglongrightarrow> ((R\<^sup>2 + R\<^sup>2 - (norm(?Z - ?Z))\<^sup>2) / (2 * R * R))"
by (intro L rR tendsto_intros) (use \<open>R > 0\<close> in force)
moreover have "((R\<^sup>2 + R\<^sup>2 - (norm(?Z - ?Z))\<^sup>2) / (2 * R * R)) = 1"
- using \<open>R > 0\<close> by (simp add: power2_eq_square divide_simps)
+ using \<open>R > 0\<close> by (simp add: power2_eq_square field_split_simps)
ultimately have "(\<lambda>j. cos (\<theta> j - \<Theta>)) \<longlonglongrightarrow> 1"
by auto
then show ?thesis
@@ -582,7 +582,7 @@
lemma sinh_complex:
fixes z :: complex
shows "(exp z - inverse (exp z)) / 2 = -\<i> * sin(\<i> * z)"
- by (simp add: sin_exp_eq divide_simps exp_minus)
+ by (simp add: sin_exp_eq field_split_simps exp_minus)
lemma sin_i_times:
fixes z :: complex
@@ -597,12 +597,12 @@
lemma cosh_complex:
fixes z :: complex
shows "(exp z + inverse (exp z)) / 2 = cos(\<i> * z)"
- by (simp add: cos_exp_eq divide_simps exp_minus exp_of_real)
+ by (simp add: cos_exp_eq field_split_simps exp_minus exp_of_real)
lemma cosh_real:
fixes x :: real
shows "of_real((exp x + inverse (exp x)) / 2) = cos(\<i> * of_real x)"
- by (simp add: cos_exp_eq divide_simps exp_minus exp_of_real)
+ by (simp add: cos_exp_eq field_split_simps exp_minus exp_of_real)
lemmas cos_i_times = cosh_complex [symmetric]
@@ -612,7 +612,7 @@
apply (simp add: cos_add cmod_power2 cos_of_real sin_of_real Complex_eq)
apply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide power_divide)
apply (simp only: left_diff_distrib [symmetric] power_mult_distrib sin_squared_eq)
- apply (simp add: power2_eq_square algebra_simps divide_simps)
+ apply (simp add: power2_eq_square algebra_simps field_split_simps)
done
lemma norm_sin_squared:
@@ -621,7 +621,7 @@
apply (simp add: sin_add cmod_power2 cos_of_real sin_of_real cos_double_cos exp_double Complex_eq)
apply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide power_divide)
apply (simp only: left_diff_distrib [symmetric] power_mult_distrib cos_squared_eq)
- apply (simp add: power2_eq_square algebra_simps divide_simps)
+ apply (simp add: power2_eq_square algebra_simps field_split_simps)
done
lemma exp_uminus_Im: "exp (- Im z) \<le> exp (cmod z)"
@@ -928,7 +928,7 @@
unfolding is_Arg_def
apply (rule complex_eqI)
using t False ReIm
- apply (auto simp: exp_Euler sin_of_real cos_of_real divide_simps)
+ apply (auto simp: exp_Euler sin_of_real cos_of_real field_split_simps)
done
show ?thesis
apply (simp add: Arg2pi_def False)
@@ -1055,7 +1055,7 @@
using assms Arg2pi_eq [of z] Arg2pi_eq [of w]
apply auto
apply (rule_tac x="norm w / norm z" in exI)
- apply (simp add: divide_simps)
+ apply (simp add: field_split_simps)
by (metis mult.commute mult.left_commute)
lemma Arg2pi_inverse_eq_0: "Arg2pi(inverse z) = 0 \<longleftrightarrow> Arg2pi z = 0"
@@ -1102,7 +1102,7 @@
by auto
lemma Arg2pi_cnj_eq_inverse: "z\<noteq>0 \<Longrightarrow> Arg2pi (cnj z) = Arg2pi (inverse z)"
- apply (simp add: Arg2pi_eq_iff divide_simps complex_norm_square [symmetric] mult.commute)
+ apply (simp add: Arg2pi_eq_iff field_split_simps complex_norm_square [symmetric] mult.commute)
by (metis of_real_power zero_less_norm_iff zero_less_power)
lemma Arg2pi_cnj: "Arg2pi(cnj z) = (if z \<in> \<real> then Arg2pi z else 2*pi - Arg2pi z)"
@@ -1166,10 +1166,10 @@
proof -
obtain \<psi> where z: "z / (cmod z) = Complex (cos \<psi>) (sin \<psi>)"
using complex_unimodular_polar [of "z / (norm z)"] assms
- by (auto simp: norm_divide divide_simps)
+ by (auto simp: norm_divide field_split_simps)
obtain \<phi> where \<phi>: "- pi < \<phi>" "\<phi> \<le> pi" "sin \<phi> = sin \<psi>" "cos \<phi> = cos \<psi>"
using sincos_principal_value [of "\<psi>"] assms
- by (auto simp: norm_divide divide_simps)
+ by (auto simp: norm_divide field_split_simps)
have "exp(ln z) = z & -pi < Im(ln z) & Im(ln z) \<le> pi" unfolding ln_complex_def
apply (rule theI [where a = "Complex (ln(norm z)) \<phi>"])
using z assms \<phi>
@@ -1437,7 +1437,7 @@
by (intro summable_norm)
(auto simp: norm_power norm_inverse norm_mult mult_ac simp del: of_nat_add of_nat_Suc)
also have "norm ((-z)^2 * (-z)^i) * inverse (of_nat (i+2)) \<le> norm ((-z)^2 * (-z)^i) * 1" for i
- by (intro mult_left_mono) (simp_all add: divide_simps)
+ by (intro mult_left_mono) (simp_all add: field_split_simps)
hence "(\<Sum>i. norm (-(z^2) * inverse (of_nat (i+2)) * (-z)^i))
\<le> (\<Sum>i. norm (-(z^2) * (-z)^i))"
using A assms
@@ -1907,14 +1907,14 @@
using assms Arg_eq [of z] Arg_eq [of w]
apply auto
apply (rule_tac x="norm w / norm z" in exI)
- apply (simp add: divide_simps)
+ apply (simp add: field_split_simps)
by (metis mult.commute mult.left_commute)
lemma Arg_inverse_eq_0: "Arg(inverse z) = 0 \<longleftrightarrow> Arg z = 0"
by (metis Arg_eq_0 Arg_inverse inverse_inverse_eq)
lemma Arg_cnj_eq_inverse: "z\<noteq>0 \<Longrightarrow> Arg (cnj z) = Arg (inverse z)"
- apply (simp add: Arg_eq_iff divide_simps complex_norm_square [symmetric] mult.commute)
+ apply (simp add: Arg_eq_iff field_split_simps complex_norm_square [symmetric] mult.commute)
by (metis of_real_power zero_less_norm_iff zero_less_power)
lemma Arg_cnj: "Arg(cnj z) = (if z \<in> \<real> then Arg z else - Arg z)"
@@ -2307,7 +2307,7 @@
by auto
then show ?thesis
unfolding dd'_def using gderiv that \<open>g z\<noteq>0\<close>
- by (auto intro!: derivative_eq_intros simp add:divide_simps power_add[symmetric])
+ by (auto intro!: derivative_eq_intros simp add:field_split_simps power_add[symmetric])
qed
finally have "((\<lambda>z. g z powr of_int n) has_field_derivative dd) (at z within s)" .
then show ?thesis unfolding dd_def by simp
@@ -2549,7 +2549,7 @@
using e by (auto simp: field_simps)
have "norm (Ln (of_nat n) / of_nat n powr s) < e" if "n \<ge> nat \<lceil>exp xo\<rceil>" for n
using e xo [of "ln n"] that
- apply (auto simp: norm_divide norm_powr_real divide_simps)
+ apply (auto simp: norm_divide norm_powr_real field_split_simps)
apply (metis exp_less_mono exp_ln not_le of_nat_0_less_iff)
done
then show "\<exists>no. \<forall>n\<ge>no. norm (Ln (of_nat n) / of_nat n powr s) < e"
@@ -2593,7 +2593,7 @@
using ln_272_gt_1
by (force intro: order_trans [of _ "ln (272/100)"])
then show "\<forall>\<^sub>F x in sequentially. cmod (1 / of_nat x powr s) \<le> cmod (Ln (of_nat x) / of_nat x powr s)"
- by (auto simp: norm_divide divide_simps eventually_sequentially)
+ by (auto simp: norm_divide field_split_simps eventually_sequentially)
show "(\<lambda>n. cmod (Ln (of_nat n) / of_nat n powr s)) \<longlonglongrightarrow> 0"
using lim_Ln_over_power [OF assms] by (metis tendsto_norm_zero_iff)
qed
@@ -2609,7 +2609,7 @@
done
lemma lim_1_over_Ln: "((\<lambda>n. 1 / Ln(of_nat n)) \<longlongrightarrow> 0) sequentially"
-proof (clarsimp simp add: lim_sequentially dist_norm norm_divide divide_simps)
+proof (clarsimp simp add: lim_sequentially dist_norm norm_divide field_split_simps)
fix r::real
assume "0 < r"
have ir: "inverse (exp (inverse r)) > 0"
@@ -2618,7 +2618,7 @@
using ex_less_of_nat_mult [of _ 1, OF ir]
by auto
then have "exp (inverse r) < of_nat n"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
then have "ln (exp (inverse r)) < ln (of_nat n)"
by (metis exp_gt_zero less_trans ln_exp ln_less_cancel_iff)
with \<open>0 < r\<close> have "1 < r * ln (real_of_nat n)"
@@ -2627,7 +2627,7 @@
using neq0_conv by fastforce
ultimately show "\<exists>no. \<forall>k. Ln (of_nat k) \<noteq> 0 \<longrightarrow> no \<le> k \<longrightarrow> 1 < r * cmod (Ln (of_nat k))"
using n \<open>0 < r\<close>
- by (rule_tac x=n in exI) (force simp: divide_simps intro: less_le_trans)
+ by (rule_tac x=n in exI) (force simp: field_split_simps intro: less_le_trans)
qed
lemma lim_1_over_ln: "((\<lambda>n. 1 / ln(real_of_nat n)) \<longlongrightarrow> 0) sequentially"
@@ -2650,7 +2650,7 @@
then have "ln 3 \<le> ln n" and ln0: "0 \<le> ln n"
by auto
with ln3_gt_1 have "1/ ln n \<le> 1"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
moreover have "ln (1 + 1 / real n) \<le> 1/n"
by (simp add: ln_add_one_self_le_self)
ultimately have "ln (1 + 1 / real n) * (1 / ln n) \<le> (1/n) * 1"
@@ -2662,7 +2662,7 @@
then show "(\<lambda>n. 1 + ln(1 + 1/n) / ln n) \<longlonglongrightarrow> 1"
by (metis (full_types) add.right_neutral tendsto_add_const_iff)
show "\<forall>\<^sub>F k in sequentially. 1 + ln (1 + 1 / k) / ln k = ln(Suc k) / ln k"
- by (simp add: divide_simps ln_div eventually_sequentiallyI [of 2])
+ by (simp add: field_split_simps ln_div eventually_sequentiallyI [of 2])
qed
lemma lim_ln_over_ln1: "(\<lambda>n. ln n / ln(Suc n)) \<longlonglongrightarrow> 1"
@@ -2721,7 +2721,7 @@
have z: "z \<noteq> 0"
using assms by auto
then have *: "inverse z = inverse (2*z) * 2"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
have [simp]: "exp (Ln z / 2) * inverse z = inverse (csqrt z)"
by (simp add: z field_simps csqrt_exp_Ln [symmetric]) (metis power2_csqrt power2_eq_square)
have "Im z = 0 \<Longrightarrow> 0 < Re z"
@@ -2904,7 +2904,7 @@
unfolding Arctan_def scaleR_conv_of_real
apply (intro derivative_eq_intros | simp add: nz0 *)+
using nz0 nz1 zz
- apply (simp add: algebra_simps divide_simps power2_eq_square)
+ apply (simp add: algebra_simps field_split_simps power2_eq_square)
apply algebra
done
qed
@@ -2954,12 +2954,12 @@
have "ereal (norm (h u n) / norm (h u (Suc n))) =
ereal (inverse (norm u)^2) * ereal (((2*Suc n+1) / (Suc n)) /
((2*Suc n-1) / (Suc n)))"
- by (simp add: h_def norm_mult norm_power norm_divide divide_simps
+ by (simp add: h_def norm_mult norm_power norm_divide field_split_simps
power2_eq_square eval_nat_numeral del: of_nat_add of_nat_Suc)
also have "of_nat (2*Suc n+1) / of_nat (Suc n) = (2::real) + inverse (real (Suc n))"
- by (auto simp: divide_simps simp del: of_nat_Suc) simp_all?
+ by (auto simp: field_split_simps simp del: of_nat_Suc) simp_all?
also have "of_nat (2*Suc n-1) / of_nat (Suc n) = (2::real) - inverse (real (Suc n))"
- by (auto simp: divide_simps simp del: of_nat_Suc) simp_all?
+ by (auto simp: field_split_simps simp del: of_nat_Suc) simp_all?
finally show "ereal (norm (h u n) / norm (h u (Suc n))) = ereal (inverse (norm u)^2) *
ereal ((2 + inverse (real (Suc n))) / (2 - inverse (real (Suc n))))" .
qed
@@ -2968,7 +2968,7 @@
finally have "liminf (\<lambda>n. ereal (cmod (h u n) / cmod (h u (Suc n)))) = inverse (norm u)^2"
by (intro lim_imp_Liminf) simp_all
moreover from power_strict_mono[OF that, of 2] u have "inverse (norm u)^2 > 1"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
ultimately have A: "liminf (\<lambda>n. ereal (cmod (h u n) / cmod (h u (Suc n)))) > 1" by simp
from u have "summable (h u)"
by (intro summable_norm_cancel[OF ratio_test_convergence[OF _ A]])
@@ -3049,7 +3049,7 @@
apply (rule norm_exp_imaginary)
apply (subst exp_Ln)
using ne apply (simp_all add: cmod_def complex_eq_iff)
- apply (auto simp: divide_simps)
+ apply (auto simp: field_split_simps)
apply algebra
done
then show ?thesis
@@ -3065,7 +3065,7 @@
done
next
have *: " (1 - \<i>*x) / (1 + \<i>*x) \<noteq> 0"
- by (simp add: divide_simps) ( simp add: complex_eq_iff)
+ by (simp add: field_split_simps) ( simp add: complex_eq_iff)
show "Re (Arctan (complex_of_real x)) < pi / 2"
using mpi_less_Im_Ln [OF *]
by (simp add: Arctan_def)
@@ -3143,7 +3143,7 @@
case False
then have *: "\<bar>arctan x\<bar> < pi / 2 - \<bar>arctan y\<bar>" using assms
apply (auto simp add: abs_arctan arctan_inverse [symmetric] arctan_less_iff)
- apply (simp add: divide_simps abs_mult)
+ apply (simp add: field_split_simps abs_mult)
done
show ?thesis
apply (rule arctan_add_raw)
@@ -3192,7 +3192,7 @@
have nonneg: "0 \<le> ?a n" for n
by (force intro!: divide_nonneg_nonneg mult_nonneg_nonneg zero_le_power assms)
have le: "?a (Suc n) \<le> ?a n" for n
- by (rule mult_mono[OF _ power_decreasing]) (auto simp: divide_simps assms less_imp_le)
+ by (rule mult_mono[OF _ power_decreasing]) (auto simp: field_split_simps assms less_imp_le)
from summable_Leibniz'(4)[of ?a, OF tendsto_zero nonneg le, of n]
summable_Leibniz'(2)[of ?a, OF tendsto_zero nonneg le, of n]
assms
@@ -3565,7 +3565,7 @@
have "(Im (Arccos w))\<^sup>2 \<le> (cmod (cos (Arccos w)))\<^sup>2 - (cos (Re (Arccos w)))\<^sup>2"
using norm_cos_squared [of "Arccos w"] real_le_abs_sinh [of "Im (Arccos w)"]
apply (simp only: abs_le_square_iff)
- apply (simp add: divide_simps)
+ apply (simp add: field_split_simps)
done
also have "... \<le> (cmod w)\<^sup>2"
by (auto simp: cmod_power2)
@@ -3974,7 +3974,7 @@
note * = this
show ?thesis
using assms
- by (simp add: exp_eq divide_simps mult_ac of_real_numeral *)
+ by (simp add: exp_eq field_split_simps mult_ac of_real_numeral *)
qed
corollary bij_betw_roots_unity:
@@ -4059,7 +4059,7 @@
then h (fst z, Arg2pi (snd z) / (2 * pi))
else h (fst z, 1 - Arg2pi (cnj (snd z)) / (2 * pi))"
have Arg2pi_eq: "1 - Arg2pi (cnj y) / (2 * pi) = Arg2pi y / (2 * pi) \<or> Arg2pi y = 0 \<and> Arg2pi (cnj y) = 0" if "cmod y = 1" for y
- using that Arg2pi_eq_0_pi Arg2pi_eq_pi by (force simp: Arg2pi_cnj divide_simps)
+ using that Arg2pi_eq_0_pi Arg2pi_eq_pi by (force simp: Arg2pi_cnj field_split_simps)
show ?thesis
proof (simp add: homotopic_with; intro conjI ballI exI)
show "continuous_on ({0..1} \<times> sphere 0 1) (\<lambda>w. h (fst w, Arg2pi (snd w) / (2 * pi)))"
@@ -4250,7 +4250,7 @@
next
case False
then have *: "(Arg2pi (exp (\<i>*(2* of_real pi* of_real x))) / (2*pi)) = x"
- using that by (auto simp: Arg2pi_exp divide_simps)
+ using that by (auto simp: Arg2pi_exp field_split_simps)
show ?thesis
by (rule_tac x="exp(\<i> * of_real(2*pi*x))" in bexI) (auto simp: *)
qed
--- a/src/HOL/Analysis/Conformal_Mappings.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Conformal_Mappings.thy Wed Oct 09 14:51:54 2019 +0000
@@ -39,7 +39,7 @@
define a where "a = (2 * pi)/(fact n)"
have "0 < a" by (simp add: a_def)
have "B0/r^(Suc n)*2 * pi * r = a*((fact n)*B0/r^n)"
- using \<open>0 < r\<close> by (simp add: a_def divide_simps)
+ using \<open>0 < r\<close> by (simp add: a_def field_split_simps)
have der_dif: "(deriv ^^ n) (\<lambda>w. f w - y) z = (deriv ^^ n) f z"
using \<open>0 < r\<close> \<open>0 < n\<close>
by (auto simp: higher_deriv_diff [OF holf holomorphic_on_const])
@@ -123,7 +123,7 @@
then have wge: "fact k * B / cmod ((deriv ^^ k) f 0) < norm w"
by (metis norm_of_real w_def)
then have "fact k * B / norm w < cmod ((deriv ^^ k) f 0)"
- using False by (simp add: divide_simps mult.commute split: if_split_asm)
+ using False by (simp add: field_split_simps mult.commute split: if_split_asm)
also have "... \<le> fact k * (B * norm w ^ n) / norm w ^ k"
apply (rule Cauchy_inequality)
using holf holomorphic_on_subset apply force
@@ -132,7 +132,7 @@
by (metis nof wgeA dist_0_norm dist_norm)
also have "... = fact k * (B * 1 / cmod w ^ (k-n))"
apply (simp only: mult_cancel_left times_divide_eq_right [symmetric])
- using \<open>k>n\<close> \<open>w \<noteq> 0\<close> \<open>0 < B\<close> apply (simp add: divide_simps semiring_normalization_rules)
+ using \<open>k>n\<close> \<open>w \<noteq> 0\<close> \<open>0 < B\<close> apply (simp add: field_split_simps semiring_normalization_rules)
done
also have "... = fact k * B / cmod w ^ (k-n)"
by simp
@@ -862,7 +862,7 @@
have h0: "(h has_field_derivative 0) (at \<xi>)"
apply (simp add: h_def has_field_derivative_iff)
apply (rule Lim_transform_within [OF that, of 1])
- apply (auto simp: divide_simps power2_eq_square)
+ apply (auto simp: field_split_simps power2_eq_square)
done
have holh: "h holomorphic_on S"
proof (simp add: holomorphic_on_def, clarify)
@@ -952,7 +952,7 @@
have "cmod (f z) \<le> cmod z ^ n / B" if "2/e \<le> cmod z" for z
proof -
have ize: "inverse z \<in> ball 0 e - {0}" using that \<open>0 < e\<close>
- by (auto simp: norm_divide divide_simps algebra_simps)
+ by (auto simp: norm_divide field_split_simps algebra_simps)
then have [simp]: "z \<noteq> 0" and izr: "inverse z \<in> ball 0 r - {0}" using \<open>e \<le> r\<close>
by auto
then have [simp]: "f z \<noteq> 0"
@@ -960,7 +960,7 @@
have [simp]: "f z = inverse (g (inverse z))"
using izr geq [of "inverse z"] by simp
show ?thesis using ize leg [of "inverse z"] \<open>0 < B\<close> \<open>0 < e\<close>
- by (simp add: divide_simps norm_divide algebra_simps)
+ by (simp add: field_split_simps norm_divide algebra_simps)
qed
then show ?thesis
apply (rule_tac a = "\<lambda>k. (deriv ^^ k) f 0 / (fact k)" and n=n in that)
@@ -1852,7 +1852,7 @@
apply (rule Le1) using r x \<open>0 < r\<close> by simp
also have "... \<le> norm (x *\<^sub>R z) / (r - norm z) * C"
using r x \<open>0 < r\<close>
- apply (simp add: divide_simps)
+ apply (simp add: field_split_simps)
by (simp add: \<open>0 < C\<close> mult.assoc mult_left_le_one_le ordered_comm_semiring_class.comm_mult_left_mono)
finally have "norm (deriv f (x *\<^sub>R z) - deriv f 0) * norm z \<le> norm (x *\<^sub>R z) / (r - norm z) * C * norm z"
by (rule mult_right_mono) simp
@@ -1994,7 +1994,7 @@
then have "norm (deriv f y) * abs (r - norm (y - a)) \<le> norm (deriv f p) * abs (r - norm (p - a))"
by (simp only: dist_norm g_def norm_mult norm_of_real)
with that \<open>norm (p - a) < r\<close> show ?thesis
- by (simp add: dist_norm divide_simps)
+ by (simp add: dist_norm field_split_simps)
qed
have le_norm_dfp: "r / (r - norm (p - a)) \<le> norm (deriv f p)"
using gen_le_dfp [of a] \<open>r > 0\<close> by auto
@@ -3044,7 +3044,7 @@
by (intro tendsto_mult_filterlim_at_infinity[of _ "f z"]
filterlim_compose[OF filterlim_inverse_at_infinity])+
(insert assms, auto simp: isCont_def)
- thus ?thesis by (simp add: divide_simps is_pole_def)
+ thus ?thesis by (simp add: field_split_simps is_pole_def)
qed
lemma is_pole_basic:
@@ -3526,7 +3526,7 @@
apply (elim filterlim_transform_within[OF _ _ \<open>r1>0\<close>],simp)
apply (subst fg_times,simp add:dist_commute)
apply (subst powr_of_int)
- using that by (auto simp add:divide_simps)
+ using that by (auto simp add:field_split_simps)
then show ?thesis unfolding not_essential_def fg_def by auto
qed
ultimately show ?thesis unfolding not_essential_def fg_def by fastforce
@@ -4573,7 +4573,7 @@
have "eventually (\<lambda>w. w \<noteq> z) (at z)"
by (auto simp: eventually_at_filter)
hence "eventually (\<lambda>w. ((f w - f z) / (w - z)) / ((g w - g z) / (w - z)) = f w / g w) (at z)"
- by eventually_elim (simp add: assms divide_simps)
+ by eventually_elim (simp add: assms field_split_simps)
moreover have "((\<lambda>w. ((f w - f z) / (w - z)) / ((g w - g z) / (w - z))) \<longlongrightarrow> f' / g') (at z)"
by (intro tendsto_divide has_field_derivativeD assms)
ultimately have "((\<lambda>w. f w / g w) \<longlongrightarrow> f' / g') (at z)"
@@ -4654,7 +4654,7 @@
show "(g has_field_derivative g') (at z)" by fact
qed (insert assms, auto)
then show "((\<lambda>w. (f w / g w) * (w - z)) \<longlongrightarrow> f z / g') (at z)"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
qed
qed
--- a/src/HOL/Analysis/Continuous_Extension.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Continuous_Extension.thy Wed Oct 09 14:51:54 2019 +0000
@@ -91,7 +91,7 @@
done
qed
moreover have "\<lbrakk>U \<in> \<C>; x \<in> S\<rbrakk> \<Longrightarrow> 0 \<le> F U x" for U x
- using nonneg [of x] by (simp add: F_def divide_simps setdist_pos_le)
+ using nonneg [of x] by (simp add: F_def field_split_simps setdist_pos_le)
ultimately show "\<And>U. U \<in> \<C> \<Longrightarrow> continuous_on S (F U) \<and> (\<forall>x\<in>S. 0 \<le> F U x)"
by metis
next
@@ -100,7 +100,7 @@
next
show "supp_sum (\<lambda>W. F W x) \<C> = 1" if "x \<in> S" for x
using that ss_pos [OF that]
- by (simp add: F_def divide_simps supp_sum_divide_distrib [symmetric])
+ by (simp add: F_def field_split_simps supp_sum_divide_distrib [symmetric])
next
show "\<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>. \<exists>x\<in>V. F U x \<noteq> 0}" if "x \<in> S" for x
using fin [OF that] that
@@ -157,7 +157,7 @@
done
also have "... \<longleftrightarrow> setdist {x} T = 0 \<and> setdist {x} S \<noteq> 0"
using sdpos that
- by (simp add: divide_simps) linarith
+ by (simp add: field_split_simps) linarith
also have "... \<longleftrightarrow> x \<in> T"
using \<open>S \<noteq> {}\<close> \<open>T \<noteq> {}\<close> \<open>S \<inter> T = {}\<close> that
by (force simp: S0 T0)
--- a/src/HOL/Analysis/Convex.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Convex.thy Wed Oct 09 14:51:54 2019 +0000
@@ -988,7 +988,7 @@
assume "u \<in> {0<..}" "v \<in> {0<..}" "u \<le> v"
with assms show "-inverse (u^2) \<le> -inverse (v^2)"
by (intro le_imp_neg_le le_imp_inverse_le power_mono) (simp_all)
-qed (insert assms, auto intro!: derivative_eq_intros simp: divide_simps power2_eq_square)
+qed (insert assms, auto intro!: derivative_eq_intros simp: field_split_simps power2_eq_square)
lemma convex_onD_Icc':
assumes "convex_on {x..y} f" "c \<in> {x..y}"
@@ -999,7 +999,7 @@
then have d: "d > 0"
by (simp add: d_def)
from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1"
- by (simp_all add: d_def divide_simps)
+ by (simp_all add: d_def field_split_simps)
have "f c = f (x + (c - x) * 1)"
by simp
also from less have "1 = ((y - x) / d)"
@@ -1022,7 +1022,7 @@
then have d: "d > 0"
by (simp add: d_def)
from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1"
- by (simp_all add: d_def divide_simps)
+ by (simp_all add: d_def field_split_simps)
have "f c = f (y - (y - c) * 1)"
by simp
also from less have "1 = ((y - x) / d)"
--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy Wed Oct 09 14:51:54 2019 +0000
@@ -1602,7 +1602,7 @@
by simp
also have "\<dots> < norm(x - closest_point S x)"
using clo_notx \<open>e > 0\<close>
- by (auto simp: mult_less_cancel_right2 divide_simps)
+ by (auto simp: mult_less_cancel_right2 field_split_simps)
finally have no_less: "norm (x - y) < norm (x - closest_point S x)" .
have "y \<in> affine hull S"
unfolding y_def
@@ -2394,7 +2394,7 @@
using \<open>0 < t\<close> \<open>2 < t\<close> and \<open>y \<in> S\<close> \<open>w \<in> S\<close>
by (auto simp:field_simps)
also have "\<dots> = (f w + t * f y) / (1 + t)"
- using \<open>t > 0\<close> by (auto simp: divide_simps)
+ using \<open>t > 0\<close> by (simp add: add_divide_distrib)
also have "\<dots> < e + f y"
using \<open>t > 0\<close> * \<open>e > 0\<close> by (auto simp: field_simps)
finally have "f x - f y < e" by auto
@@ -2504,7 +2504,7 @@
have "e \<le> e * real DIM('a)"
using e(2) real_of_nat_ge_one_iff by auto
then have "d \<le> e"
- by (simp add: d_def divide_simps)
+ by (simp add: d_def field_split_simps)
then have dsube: "cball x d \<subseteq> cball x e"
by (rule subset_cball)
have conv: "convex_on (cball x d) f"
--- a/src/HOL/Analysis/Derivative.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Derivative.thy Wed Oct 09 14:51:54 2019 +0000
@@ -2111,7 +2111,10 @@
let ?e = "\<lambda>i x. (inverse (1 + real i) * inverse (fact i) * (x - t) ^ i) *\<^sub>R (A * A ^ i)"
have "\<forall>\<^sub>F n in sequentially.
\<forall>x\<in>T \<inter> {t - 1<..<t + 1}. norm (?e n x) \<le> norm (A ^ (n + 1) /\<^sub>R fact (n + 1))"
- by (auto simp: divide_simps power_abs intro!: mult_left_le_one_le power_le_one eventuallyI)
+ apply (auto simp: algebra_split_simps intro!: eventuallyI)
+ apply (rule mult_left_mono)
+ apply (auto simp add: field_simps power_abs intro!: divide_right_mono power_le_one)
+ done
then have "uniform_limit (T \<inter> {t - 1<..<t + 1}) (\<lambda>n x. \<Sum>i<n. ?e i x) (\<lambda>x. \<Sum>i. ?e i x) sequentially"
by (rule Weierstrass_m_test_ev) (intro summable_ignore_initial_segment summable_norm_exp)
moreover
@@ -2156,7 +2159,7 @@
also have "(\<dots> - (x - t) *\<^sub>R A) /\<^sub>R norm (x - t) = (x - t) *\<^sub>R ((\<Sum>n. ?e n x) - A) /\<^sub>R norm (x - t)"
by (simp add: algebra_simps)
finally show ?case
- by (simp add: divide_simps)
+ by simp (simp add: field_simps)
qed
ultimately have "((\<lambda>y. (exp ((y - t) *\<^sub>R A) - 1 - (y - t) *\<^sub>R A) /\<^sub>R norm (y - t)) \<longlongrightarrow> 0) (at t within T)"
@@ -2165,7 +2168,7 @@
show "((\<lambda>y. (exp (y *\<^sub>R A) - exp (t *\<^sub>R A) - (y - t) *\<^sub>R (exp (t *\<^sub>R A) * A)) /\<^sub>R norm (y - t)) \<longlongrightarrow> 0)
(at t within T)"
by (rule Lim_transform_eventually)
- (auto simp: algebra_simps divide_simps exp_add_commuting[symmetric])
+ (auto simp: algebra_simps field_split_simps exp_add_commuting[symmetric])
qed (rule bounded_linear_scaleR_left)
lemma exp_times_scaleR_commute: "exp (t *\<^sub>R A) * A = A * exp (t *\<^sub>R A)"
@@ -2262,7 +2265,7 @@
using interior_subset[of A]
by (intro convex_onD_Icc' convex_on_subset[OF convex] connected_contains_Icc) auto
hence "f y - f c \<le> (f x - f c) / (x - c) * (y - c)" by simp
- thus "(f y - f c) / (y - c) \<le> (f x - f c) / (x - c)" using y xc by (simp add: divide_simps)
+ thus "(f y - f c) / (y - c) \<le> (f x - f c) / (x - c)" using y xc by (simp add: field_split_simps)
qed
hence "eventually (\<lambda>y. (f y - f c) / (y - c) \<le> (f x - f c) / (x - c)) (at c within ?A')"
by (blast intro: filter_leD at_le)
@@ -2286,7 +2289,7 @@
hence "f y - f c \<le> (f x - f c) * ((c - y) / (c - x))" by simp
also have "(c - y) / (c - x) = (y - c) / (x - c)" using y xc by (simp add: field_simps)
finally show "(f y - f c) / (y - c) \<ge> (f x - f c) / (x - c)" using y xc
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
qed
hence "eventually (\<lambda>y. (f y - f c) / (y - c) \<ge> (f x - f c) / (x - c)) (at c within ?A')"
by (blast intro: filter_leD at_le)
@@ -2418,8 +2421,8 @@
let ?le = "\<lambda>x'. norm (f x' y - f x y - (fx) (x' - x)) \<le> norm (x' - x) * e"
from fx[unfolded has_derivative_within, THEN conjunct2, THEN tendstoD, OF \<open>0 < e\<close>]
have "\<forall>\<^sub>F x' in at x within X. ?le x'"
- by eventually_elim
- (simp add: dist_norm divide_simps diff_diff_add ac_simps split: if_split_asm)
+ by eventually_elim (simp,
+ simp add: dist_norm field_split_simps split: if_split_asm)
then have "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. ?le x'"
by (rule eventually_at_Pair_within_TimesI1)
(simp add: blinfun.bilinear_simps)
@@ -2476,7 +2479,7 @@
using \<open>0 < e'\<close> nz
by (auto simp: e_def)
finally show "norm ((f x' y' - f x y - (fx (x' - x) + fy x y (y' - y))) /\<^sub>R norm ((x', y') - (x, y))) < e'"
- by (auto simp: divide_simps dist_norm mult.commute)
+ by (simp add: dist_norm) (auto simp add: field_split_simps)
qed
then show ?case
by eventually_elim (auto simp: dist_norm field_simps)
--- a/src/HOL/Analysis/Elementary_Metric_Spaces.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Elementary_Metric_Spaces.thy Wed Oct 09 14:51:54 2019 +0000
@@ -210,9 +210,9 @@
lemma ball_divide_subset: "d \<ge> 1 \<Longrightarrow> ball x (e/d) \<subseteq> ball x e"
apply (cases "e \<le> 0")
- apply (simp add: ball_empty divide_simps)
+ apply (simp add: ball_empty field_split_simps)
apply (rule subset_ball)
- apply (simp add: divide_simps)
+ apply (simp add: field_split_simps)
done
lemma ball_divide_subset_numeral: "ball x (e / numeral w) \<subseteq> ball x e"
@@ -220,7 +220,7 @@
lemma cball_divide_subset: "d \<ge> 1 \<Longrightarrow> cball x (e/d) \<subseteq> cball x e"
apply (cases "e < 0")
- apply (simp add: divide_simps)
+ apply (simp add: field_split_simps)
apply (rule subset_cball)
apply (metis div_by_1 frac_le not_le order_refl zero_less_one)
done
@@ -1335,7 +1335,7 @@
then have "diameter(closure(S)) - d / 2 < diameter(closure(S))"
by simp
have dd: "diameter (closure S) - d / 2 = (diameter(closure(S)) + diameter(S)) / 2"
- by (simp add: d_def divide_simps)
+ by (simp add: d_def field_split_simps)
have bocl: "bounded (closure S)"
using assms by blast
moreover have "0 \<le> diameter S"
@@ -2423,7 +2423,7 @@
then have "dist (f (r (max N1 N2))) x < 1 / real (Suc (r (max N1 N2)))"
by simp
also have "... \<le> 1 / real (Suc (max N1 N2))"
- apply (simp add: divide_simps del: max.bounded_iff)
+ apply (simp add: field_split_simps del: max.bounded_iff)
using \<open>strict_mono r\<close> seq_suble by blast
also have "... \<le> 1 / real (Suc N2)"
by (simp add: field_simps)
--- a/src/HOL/Analysis/Elementary_Normed_Spaces.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Elementary_Normed_Spaces.thy Wed Oct 09 14:51:54 2019 +0000
@@ -599,7 +599,7 @@
qed
show "\<forall>\<^sub>F x in net. dist (f x *\<^sub>R g x) 0 < \<epsilon>"
apply (rule eventually_mono [OF eventually_conj [OF tendstoD [OF f B] gB] ])
- apply (auto simp: \<open>0 < \<epsilon>\<close> divide_simps * split: if_split_asm)
+ apply (auto simp: \<open>0 < \<epsilon>\<close> field_split_simps * split: if_split_asm)
done
qed
@@ -1562,17 +1562,17 @@
proof
have "x \<in> (\<lambda>z. z /\<^sub>R (1 - norm z)) ` ball 0 1" for x::'a
apply (rule_tac x="x /\<^sub>R (1 + norm x)" in image_eqI)
- apply (auto simp: divide_simps)
+ apply (auto simp: field_split_simps)
using norm_ge_zero [of x] apply linarith+
done
then show "(\<lambda>z::'a. z /\<^sub>R (1 - norm z)) ` ?B = ?U"
by blast
have "x \<in> range (\<lambda>z. (1 / (1 + norm z)) *\<^sub>R z)" if "norm x < 1" for x::'a
apply (rule_tac x="x /\<^sub>R (1 - norm x)" in image_eqI)
- using that apply (auto simp: divide_simps)
+ using that apply (auto simp: field_split_simps)
done
then show "(\<lambda>z::'a. z /\<^sub>R (1 + norm z)) ` ?U = ?B"
- by (force simp: divide_simps dest: add_less_zeroD)
+ by (force simp: field_split_simps dest: add_less_zeroD)
show "continuous_on (ball 0 1) (\<lambda>z. z /\<^sub>R (1 - norm z))"
by (rule continuous_intros | force)+
show "continuous_on UNIV (\<lambda>z. z /\<^sub>R (1 + norm z))"
@@ -1581,9 +1581,9 @@
done
show "\<And>x. x \<in> ball 0 1 \<Longrightarrow>
x /\<^sub>R (1 - norm x) /\<^sub>R (1 + norm (x /\<^sub>R (1 - norm x))) = x"
- by (auto simp: divide_simps)
+ by (auto simp: field_split_simps)
show "\<And>y. y /\<^sub>R (1 + norm y) /\<^sub>R (1 - norm (y /\<^sub>R (1 + norm y))) = y"
- apply (auto simp: divide_simps)
+ apply (auto simp: field_split_simps)
apply (metis le_add_same_cancel1 norm_ge_zero not_le zero_less_one)
done
qed
--- a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy Wed Oct 09 14:51:54 2019 +0000
@@ -147,7 +147,7 @@
then have "x \<in> ball (T X k) (1 / (3 * Suc m))"
using in_s[OF \<open>(x, k) \<in> s\<close>] by (auto simp: C_def subset_eq dist_commute)
then have "ball x (1 / (3 * Suc m)) \<subseteq> ball (T X k) (1 / Suc m)"
- by (rule ball_trans) (auto simp: divide_simps)
+ by (rule ball_trans) (auto simp: field_split_simps)
with k in_s[OF \<open>(x, k) \<in> s\<close>] have "k \<subseteq> d (T X k)"
by (auto simp: C_def) }
then have "d fine ?p"
@@ -1423,7 +1423,7 @@
then have "negligible (cbox ?p ?q)"
by (meson \<open>cball a e \<subseteq> S\<close> neg negligible_subset)
with \<open>e > 0\<close> show False
- by (simp add: negligible_interval box_eq_empty algebra_simps divide_simps mult_le_0_iff)
+ by (simp add: negligible_interval box_eq_empty algebra_simps field_split_simps mult_le_0_iff)
qed
lemma negligible_convex_interior:
@@ -1661,7 +1661,7 @@
then have "S \<in> sets lebesgue"
by blast
have e22: "0 < e/2 / (2 * B * real DIM('M)) ^ DIM('N)"
- using \<open>0 < e\<close> \<open>0 < B\<close> by (simp add: divide_simps)
+ using \<open>0 < e\<close> \<open>0 < B\<close> by (simp add: field_split_simps)
obtain T where "open T" "S \<subseteq> T" "(T - S) \<in> lmeasurable"
"measure lebesgue (T - S) < e/2 / (2 * B * DIM('M)) ^ DIM('N)"
using sets_lebesgue_outer_open [OF \<open>S \<in> sets lebesgue\<close> e22]
@@ -3040,7 +3040,7 @@
have 1: "cbox a b - S \<in> lmeasurable"
by (simp add: fmeasurable.Diff that)
have 2: "0 < e / (1 + \<bar>m\<bar>)"
- using \<open>e > 0\<close> by (simp add: divide_simps abs_add_one_gt_zero)
+ using \<open>e > 0\<close> by (simp add: field_split_simps abs_add_one_gt_zero)
obtain \<D>
where "countable \<D>"
and cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b \<and> K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
@@ -3075,7 +3075,7 @@
fix e :: real
assume "e > 0"
have em: "0 < e / (1 + \<bar>m\<bar>)"
- using \<open>e > 0\<close> by (simp add: divide_simps abs_add_one_gt_zero)
+ using \<open>e > 0\<close> by (simp add: field_split_simps abs_add_one_gt_zero)
obtain \<D>
where "countable \<D>"
and cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b \<and> K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
@@ -3163,7 +3163,7 @@
by (metis (no_types) abs_minus_commute mult.commute right_diff_distrib' mm)
qed
also have "\<dots> < e"
- using \<open>e > 0\<close> by (auto simp: divide_simps)
+ using \<open>e > 0\<close> by (cases "m \<ge> 0") (simp_all add: field_simps)
finally have "\<bar>?\<mu> (f ` S \<inter> cbox a b) - m * ?\<mu> S\<bar> < e" .
with 1 show ?thesis by auto
qed
@@ -4138,7 +4138,7 @@
show "{m} \<subseteq> {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}"
using assms by auto
show "\<forall>i\<in>{k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)} - {m}. ?f i = 0"
- using assms by (auto simp: indicator_def Ints_def abs_le_iff divide_simps)
+ using assms by (auto simp: indicator_def Ints_def abs_le_iff field_split_simps)
qed
also have "\<dots> = m/2^n"
using assms by (auto simp: indicator_def not_less)
@@ -4196,7 +4196,7 @@
proof -
define m where "m \<equiv> floor(2^n * (f x))"
have "1 \<le> \<bar>2^n\<bar> * e"
- using n N2 \<open>e > 0\<close> less_eq_real_def less_le_trans by (fastforce simp add: divide_simps)
+ using n N2 \<open>e > 0\<close> less_eq_real_def less_le_trans by (fastforce simp add: field_split_simps)
then have *: "\<lbrakk>x \<le> y; y < x + 1\<rbrakk> \<Longrightarrow> abs(x - y) < \<bar>2^n\<bar> * e" for x y::real
by linarith
have "\<bar>2^n\<bar> * \<bar>m/2^n - f x\<bar> = \<bar>2^n * (m/2^n - f x)\<bar>"
@@ -4243,7 +4243,7 @@
by (metis abs_of_nonneg floor_mono le_floor_iff m_def of_int_0_le_iff power2_eq_square power_mult real_mult_le_cancel_iff1 zero_less_numeral mult.commute zero_less_power)
qed
then have "?g n x = m/2^n"
- by (rule indicator_sum_eq) (auto simp: m_def mult.commute divide_simps)
+ by (rule indicator_sum_eq) (auto simp add: m_def field_split_simps, linarith)
then have "norm (?g n x - f x) = norm (m/2^n - f x)"
by simp
also have "\<dots> < e"
@@ -4779,7 +4779,7 @@
proof -
have "1 / real (max i j) < \<delta>"
using j \<open>j \<noteq> 0\<close> \<open>0 < \<delta>\<close>
- by (auto simp: divide_simps max_mult_distrib_left of_nat_max)
+ by (auto simp: field_split_simps max_mult_distrib_left of_nat_max)
then have "norm (y - x) < \<delta>"
using less by linarith
with \<delta> \<open>y \<in> S\<close> have le: "norm (f y - f x - f' x (y - x)) \<le> B * norm (y - x) - norm (y - x)/i"
@@ -4789,7 +4789,7 @@
using norm_triangle_ineq3 [of "f - f'" y] by simp
show ?thesis
apply (rule * [OF le B])
- using \<open>i \<noteq> 0\<close> \<open>j \<noteq> 0\<close> by (simp add: divide_simps max_mult_distrib_left of_nat_max less_max_iff_disj)
+ using \<open>i \<noteq> 0\<close> \<open>j \<noteq> 0\<close> by (simp add: field_split_simps max_mult_distrib_left of_nat_max less_max_iff_disj)
qed
with \<open>x \<in> S\<close> \<open>i \<noteq> 0\<close> \<open>j \<noteq> 0\<close> show "\<exists>n\<in>{0<..}. x \<in> U n"
by (rule_tac x="max i j" in bexI) (auto simp: field_simps U_def less_max_iff_disj)
--- a/src/HOL/Analysis/Equivalence_Measurable_On_Borel.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Equivalence_Measurable_On_Borel.thy Wed Oct 09 14:51:54 2019 +0000
@@ -114,7 +114,7 @@
using \<D>lm less.prems(2) by auto
qed
then have "measure lebesgue (\<Union>\<D> - cbox a3 b3) / 3 ^ DIM('a) \<le> measure lebesgue (\<Union> \<D>')"
- using \<D>'m by (simp add: divide_simps)
+ using \<D>'m by (simp add: field_split_simps)
then show ?thesis
by (simp add: m3 field_simps)
qed
@@ -129,7 +129,7 @@
qed (simp add: \<D>lm \<open>D \<in> \<D>\<close>)
qed
finally show "measure lebesgue (\<Union>\<D>) / 3 ^ DIM('a) \<le> measure lebesgue (\<Union>(insert D \<D>'))"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
qed
qed
qed
@@ -421,7 +421,7 @@
unfolding fine_def \<Phi>_def using BOX_\<gamma> \<open>\<F> \<subseteq> \<D>\<close> \<open>\<G> \<subseteq> \<F>\<close> tag_in_E by blast
qed
finally show ?thesis
- using \<open>\<mu> > 0\<close> by (auto simp: divide_simps)
+ using \<open>\<mu> > 0\<close> by (auto simp: field_split_simps)
qed
finally show ?thesis .
qed
@@ -448,7 +448,7 @@
qed
ultimately
have "measure lebesgue (\<Union>\<F>) \<le> e/2"
- by (auto simp: divide_simps)
+ by (auto simp: field_split_simps)
then show "measure lebesgue (\<Union>\<D>) \<le> e"
using \<F> by linarith
qed
@@ -620,7 +620,7 @@
have no: "norm (y - x) < 1"
using that by (auto simp: dist_norm)
have le1: "inverse (1 + real n) \<le> 1"
- by (auto simp: divide_simps)
+ by (auto simp: field_split_simps)
have "norm (integral (cbox y (y + One /\<^sub>R real (Suc n))) f
- integral (cbox x (x + One /\<^sub>R real (Suc n))) f)
< e / (1 + real n) ^ DIM('a)"
@@ -742,7 +742,7 @@
show "continuous_on UNIV (g \<circ> (\<theta> n))" for n :: nat
unfolding \<theta>_def
apply (intro continuous_on_compose2 [OF contg] continuous_intros conth)
- apply (auto simp: aibi * mem_box less_max_iff_disj min_less_iff_disj algebra_simps divide_simps)
+ apply (auto simp: aibi * mem_box less_max_iff_disj min_less_iff_disj algebra_simps field_split_simps)
done
show "(\<lambda>n. (g \<circ> \<theta> n) x) \<longlonglongrightarrow> g (f x)"
if "x \<notin> N" for x
@@ -1278,7 +1278,7 @@
proof -
have "{a<..<b} = ball ((b+a) / 2) ((b-a) / 2)"
using assms
- by (auto simp: dist_real_def abs_if divide_simps split: if_split_asm)
+ by (auto simp: dist_real_def abs_if field_split_simps split: if_split_asm)
then show ?thesis
by (simp add: homeomorphic_ball_UNIV assms)
qed
--- a/src/HOL/Analysis/FPS_Convergence.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/FPS_Convergence.thy Wed Oct 09 14:51:54 2019 +0000
@@ -327,7 +327,7 @@
show ?case
proof (cases m)
case 0
- thus ?thesis using assms by (simp add: divide_simps norm_inverse norm_divide)
+ thus ?thesis using assms by (simp add: field_split_simps norm_inverse norm_divide)
next
case [simp]: (Suc n)
have "norm (natfun_inverse f (Suc n)) =
@@ -345,7 +345,7 @@
also have "\<dots> \<le> norm (fps_nth f i) * inverse (\<delta> ^ (Suc n - i))"
using 1 by (intro mult_left_mono less.IH) auto
also have "\<dots> = norm (fps_nth f i) / \<delta> ^ (Suc n - i)"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
finally show ?case .
qed
also have "\<dots> = (\<Sum>i = Suc 0..Suc n. norm (fps_nth f i) * \<delta> ^ i) / \<delta> ^ Suc n"
@@ -360,7 +360,7 @@
using \<open>\<delta> > 0\<close> by (intro sum_le_suminf ballI mult_nonneg_nonneg zero_le_power summable) auto
also have "\<dots> \<le> 1" by fact
finally show ?thesis using \<open>\<delta> > 0\<close>
- by (simp add: divide_right_mono divide_simps)
+ by (simp add: divide_right_mono field_split_simps)
qed
qed
@@ -457,7 +457,7 @@
have "(\<lambda>n. norm (c * z) ^ n /\<^sub>R fact n) sums exp (norm (c * z))"
by (rule exp_converges)
also have "(\<lambda>n. norm (c * z) ^ n /\<^sub>R fact n) = (\<lambda>n. norm (fps_nth (fps_exp c) n * z ^ n))"
- by (rule ext) (simp add: norm_divide norm_mult norm_power divide_simps power_mult_distrib)
+ by (rule ext) (simp add: norm_divide norm_mult norm_power field_split_simps power_mult_distrib)
finally have "summable \<dots>" by (simp add: sums_iff)
thus "summable (\<lambda>n. fps_nth (fps_exp c) n * z ^ n)"
by (rule summable_norm_cancel)
@@ -494,7 +494,7 @@
by (intro Suc.IH [symmetric]) auto
also have "\<dots> / of_nat (Suc n) = fps_nth f (Suc n)"
by (simp add: fps_deriv_def del: of_nat_Suc)
- finally show ?case by (simp add: divide_simps)
+ finally show ?case by (simp add: field_split_simps)
qed
theorem eval_fps_eqD:
@@ -613,7 +613,7 @@
lemma eval_fps_exp [simp]:
fixes c :: "'a :: {banach, real_normed_field}"
shows "eval_fps (fps_exp c) z = exp (c * z)" unfolding eval_fps_def exp_def
- by (simp add: eval_fps_def exp_def scaleR_conv_of_real divide_simps power_mult_distrib)
+ by (simp add: eval_fps_def exp_def scaleR_conv_of_real field_split_simps power_mult_distrib)
lemma
fixes f :: "complex fps" and r :: ereal
@@ -716,7 +716,7 @@
by (intro eval_fps_inverse[where r = r] less_le_trans[OF z] nz)
(auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
also have "f * inverse g = f / g" by fact
- finally show "eval_fps (f / g) z = eval_fps f z / eval_fps g z" by (simp add: divide_simps)
+ finally show "eval_fps (f / g) z = eval_fps f z / eval_fps g z" by (simp add: field_split_simps)
qed
lemma
@@ -1000,7 +1000,7 @@
by (subst eval_fps_mult) auto
also have "eval_fps (inverse F * F) z = 1"
using assms by (simp add: inverse_mult_eq_1)
- finally show ?case by (auto simp: divide_simps)
+ finally show ?case by (auto simp: field_split_simps)
qed
with radius show ?thesis by (auto simp: has_fps_expansion_def)
qed
--- a/src/HOL/Analysis/Function_Topology.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Function_Topology.thy Wed Oct 09 14:51:54 2019 +0000
@@ -1005,7 +1005,7 @@
also have "... \<le> (1/2)^(to_nat i) * e i"
unfolding m_def apply (rule Min_le) using \<open>finite I\<close> \<open>i \<in> I\<close> by auto
ultimately have "min (dist (x i) (y i)) 1 < e i"
- by (auto simp add: divide_simps)
+ by (auto simp add: field_split_simps)
then have "dist (x i) (y i) < e i"
using \<open>e i < 1\<close> by auto
then show "y i \<in> X i" using \<open>ball (x i) (e i) \<subseteq> X i\<close> by auto
@@ -1058,7 +1058,7 @@
have "dist x y \<le> 2 * Max {dist (x (from_nat n)) (y (from_nat n))|n. n \<le> N} + (1/2)^N"
by (rule dist_fun_le_dist_first_terms)
also have "... \<le> 2 * f + e / 8"
- apply (rule add_mono) using ** \<open>2^N > 8/e\<close> by(auto simp add: algebra_simps divide_simps)
+ apply (rule add_mono) using ** \<open>2^N > 8/e\<close> by(auto simp add: algebra_simps field_split_simps)
also have "... \<le> e/2 + e/8"
unfolding f_def by auto
also have "... < e"
@@ -1202,7 +1202,7 @@
also have "... < (1/2)^k * min e 1"
using C \<open>m \<ge> N\<close> \<open>n \<ge> N\<close> by auto
finally have "min (dist (u m i) (u n i)) 1 < min e 1"
- by (auto simp add: algebra_simps divide_simps)
+ by (auto simp add: algebra_simps field_split_simps)
then show "dist (u m i) (u n i) < e" by auto
qed
then show "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (u m i) (u n i) < e"
@@ -1242,7 +1242,7 @@
using dist_fun_le_dist_first_terms by auto
also have "... \<le> 2 * e/4 + e/4"
apply (rule add_mono)
- using ** \<open>2^N > 4/e\<close> less_imp_le by (auto simp add: algebra_simps divide_simps)
+ using ** \<open>2^N > 4/e\<close> less_imp_le by (auto simp add: algebra_simps field_split_simps)
also have "... < e" by auto
finally show ?thesis by simp
qed
--- a/src/HOL/Analysis/Further_Topology.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Further_Topology.thy Wed Oct 09 14:51:54 2019 +0000
@@ -3148,7 +3148,7 @@
using Suc_leI assms \<open>2 \<le> n\<close> complex_roots_unity [THEN eqset_imp_iff, of n "w/z"]
by force
have "cmod (w/z - 1) < 2 * sin (pi / real n)"
- using lt assms \<open>z \<noteq> 0\<close> by (simp add: divide_simps norm_divide)
+ using lt assms \<open>z \<noteq> 0\<close> by (simp add: field_split_simps norm_divide)
then have "cmod (exp (\<i> * of_real (2 * pi * j / n)) - 1) < 2 * sin (pi / real n)"
by (simp add: j field_simps)
then have "2 * \<bar>sin((2 * pi * j / n) / 2)\<bar> < 2 * sin (pi / real n)"
--- a/src/HOL/Analysis/Gamma_Function.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Gamma_Function.thy Wed Oct 09 14:51:54 2019 +0000
@@ -66,7 +66,7 @@
proof
assume "of_int m / (of_int n :: 'a) \<in> \<int>"
then obtain k where "of_int m / of_int n = (of_int k :: 'a)" by (elim Ints_cases)
- with assms have "of_int m = (of_int (k * n) :: 'a)" by (auto simp add: divide_simps)
+ with assms have "of_int m = (of_int (k * n) :: 'a)" by (auto simp add: field_split_simps)
hence "m = k * n" by (subst (asm) of_int_eq_iff)
hence "n dvd m" by simp
with assms(1) show False by contradiction
@@ -295,7 +295,7 @@
by (cases n, simp)
(auto simp add: Gamma_series_def Gamma_series'_def pochhammer_rec'
dest: pochhammer_eq_0_imp_nonpos_Int plus_of_nat_eq_0_imp)
- also from n have "\<dots> = z / of_nat n + 1" by (simp add: divide_simps)
+ also from n have "\<dots> = z / of_nat n + 1" by (simp add: field_split_simps)
finally show "z / of_nat n + 1 = Gamma_series' z n / Gamma_series z n" ..
qed
have "(\<lambda>x. z / of_nat x) \<longlonglongrightarrow> 0"
@@ -403,7 +403,7 @@
moreover have "N \<ge> 2" "N \<ge> M" unfolding N_def by simp_all
moreover have "(\<Sum>k=m..n. 1/(of_nat k)\<^sup>2) < e'" if "m \<ge> N" for m n
using M[OF order.trans[OF \<open>N \<ge> M\<close> that]] unfolding real_norm_def
- by (subst (asm) abs_of_nonneg) (auto intro: sum_nonneg simp: divide_simps)
+ by (subst (asm) abs_of_nonneg) (auto intro: sum_nonneg simp: field_split_simps)
moreover note calculation
} note N = this
@@ -437,7 +437,7 @@
by (intro sum_cong_Suc)
(simp_all del: of_nat_Suc add: field_simps Ln_of_nat Ln_of_nat_over_of_nat)
also have "of_nat (k - 1) / of_nat k = 1 - 1 / (of_nat k :: complex)" if "k \<in> {Suc m..n}" for k
- using that of_nat_eq_0_iff[of "Suc i" for i] by (cases k) (simp_all add: divide_simps)
+ using that of_nat_eq_0_iff[of "Suc i" for i] by (cases k) (simp_all add: field_split_simps)
hence "(\<Sum>k = Suc m..n. t * Ln (of_nat (k - 1) / of_nat k)) =
(\<Sum>k = Suc m..n. t * Ln (1 - 1 / of_nat k))" using mn N
by (intro sum.cong) simp_all
@@ -516,7 +516,7 @@
unfolding pochhammer_prod
by (simp add: prod.atLeast0_atMost_Suc atLeastLessThanSuc_atLeastAtMost)
also have "of_nat n powr z / (pochhammer z (Suc n) / fact n) = Gamma_series z n"
- unfolding Gamma_series_def using assms by (simp add: divide_simps powr_def)
+ unfolding Gamma_series_def using assms by (simp add: field_split_simps powr_def)
finally show ?thesis .
qed
@@ -582,9 +582,9 @@
hence t': "t \<noteq> -of_nat (Suc n)" by (intro notI) (simp del: of_nat_Suc)
with t_nz have "?f n t = 1 / (of_nat (Suc n) * (1 + of_nat (Suc n)/t))"
- by (simp add: divide_simps eq_neg_iff_add_eq_0 del: of_nat_Suc)
+ by (simp add: field_split_simps eq_neg_iff_add_eq_0 del: of_nat_Suc)
also have "norm \<dots> = inverse (of_nat (Suc n)) * inverse (norm (of_nat (Suc n)/t + 1))"
- by (simp add: norm_divide norm_mult divide_simps add_ac del: of_nat_Suc)
+ by (simp add: norm_divide norm_mult field_split_simps add_ac del: of_nat_Suc)
also {
from z t_nz ball[OF t] have "of_nat (Suc n) / (4 * norm z) \<le> of_nat (Suc n) / (2 * norm t)"
by (intro divide_left_mono mult_pos_pos) simp_all
@@ -592,11 +592,11 @@
using t_nz n by (simp add: field_simps norm_divide del: of_nat_Suc)
also have "\<dots> \<le> norm (of_nat (Suc n)/t + 1)" by (rule norm_diff_ineq)
finally have "inverse (norm (of_nat (Suc n)/t + 1)) \<le> 4 * norm z / of_nat (Suc n)"
- using z by (simp add: divide_simps norm_divide mult_ac del: of_nat_Suc)
+ using z by (simp add: field_split_simps norm_divide mult_ac del: of_nat_Suc)
}
also have "inverse (real_of_nat (Suc n)) * (4 * norm z / real_of_nat (Suc n)) =
4 * norm z * inverse (of_nat (Suc n)^2)"
- by (simp add: divide_simps power2_eq_square del: of_nat_Suc)
+ by (simp add: field_split_simps power2_eq_square del: of_nat_Suc)
finally show "norm (?f n t) \<le> 4 * norm z * inverse (of_nat (Suc n)^2)"
by (simp del: of_nat_Suc)
qed
@@ -1094,7 +1094,7 @@
have A: "eventually (\<lambda>n. rGamma_series 1 n = of_nat (Suc n) / of_nat n) sequentially"
using eventually_gt_at_top[of "0::nat"]
by (force elim!: eventually_mono simp: rGamma_series_def exp_of_real pochhammer_fact
- divide_simps pochhammer_rec' dest!: pochhammer_eq_0_imp_nonpos_Int)
+ field_split_simps pochhammer_rec' dest!: pochhammer_eq_0_imp_nonpos_Int)
have "rGamma_series 1 \<longlonglongrightarrow> 1" by (subst tendsto_cong[OF A]) (rule LIMSEQ_Suc_n_over_n)
moreover have "rGamma_series 1 \<longlonglongrightarrow> rGamma 1" by (rule tendsto_intros)
ultimately show ?thesis by (intro LIMSEQ_unique)
@@ -1111,7 +1111,7 @@
pochhammer z (Suc (Suc n)) / (fact n * exp (z * of_real (ln (of_nat n))))"
by (subst pochhammer_rec) (simp add: rGamma_series_def field_simps exp_add exp_of_real)
also from n have "\<dots> = ?f n * rGamma_series z n"
- by (subst pochhammer_rec') (simp_all add: divide_simps rGamma_series_def add_ac)
+ by (subst pochhammer_rec') (simp_all add: field_split_simps rGamma_series_def add_ac)
finally show "?f n * rGamma_series z n = z * rGamma_series (z + 1) n" ..
qed
moreover have "(\<lambda>n. ?f n * rGamma_series z n) \<longlonglongrightarrow> ((z+1) * 0 + 1) * rGamma z"
@@ -1315,7 +1315,7 @@
pochhammer z (Suc (Suc n)) / (fact n * exp (z * of_real (ln (of_nat n))))"
by (subst pochhammer_rec) (simp add: rGamma_series_def field_simps exp_add exp_of_real)
also from n have "\<dots> = ?f n * rGamma_series z n"
- by (subst pochhammer_rec') (simp_all add: divide_simps rGamma_series_def add_ac)
+ by (subst pochhammer_rec') (simp_all add: field_split_simps rGamma_series_def add_ac)
finally show "?f n * rGamma_series z n = z * rGamma_series (z + 1) n" ..
qed
moreover have "(\<lambda>n. ?f n * rGamma_series z n) \<longlonglongrightarrow> ((z+1) * 0 + 1) * rGamma z"
@@ -1374,7 +1374,7 @@
have A: "eventually (\<lambda>n. rGamma_series 1 n = of_nat (Suc n) / of_nat n) sequentially"
using eventually_gt_at_top[of "0::nat"]
by (force elim!: eventually_mono simp: rGamma_series_def exp_of_real pochhammer_fact
- divide_simps pochhammer_rec' dest!: pochhammer_eq_0_imp_nonpos_Int)
+ field_split_simps pochhammer_rec' dest!: pochhammer_eq_0_imp_nonpos_Int)
have "rGamma_series 1 \<longlonglongrightarrow> 1" by (subst tendsto_cong[OF A]) (rule LIMSEQ_Suc_n_over_n)
thus "rGamma 1 = (1 :: complex)" unfolding rGamma_complex_def by (rule limI)
qed
@@ -2042,7 +2042,7 @@
moreover from z have "pochhammer (z + 1/2) n \<noteq> 0" by (auto dest: pochhammer_eq_0_imp_nonpos_Int)
ultimately have "?powr 2 (2*z) * (Gamma_series' z n * Gamma_series' (z + 1/2) n) / Gamma_series' (2*z) (2*n) =
?f^2 / ?f' * of_nat (2^(2*n)) * (?powr n ((4*z + 1)/2) * ?powr n (-2*z))"
- using n unfolding A B by (simp add: divide_simps exp_minus)
+ using n unfolding A B by (simp add: field_split_simps exp_minus)
also have "?powr n ((4*z + 1)/2) * ?powr n (-2*z) = ?powr n (1/2)"
by (simp add: algebra_simps exp_add[symmetric] add_divide_distrib)
finally show "?g n = ?h n" by (simp only: mult_ac)
@@ -2062,7 +2062,7 @@
by (intro not_in_Ints_imp_not_in_nonpos_Ints) simp_all
with lim[of "1/2 :: 'a"] have "?h \<longlonglongrightarrow> 2 * Gamma (1/2 :: 'a)" by (simp add: exp_of_real)
from LIMSEQ_unique[OF this lim[OF assms]] z' show ?thesis
- by (simp add: divide_simps Gamma_eq_zero_iff ring_distribs exp_diff exp_of_real ac_simps)
+ by (simp add: field_split_simps Gamma_eq_zero_iff ring_distribs exp_diff exp_of_real ac_simps)
qed
theorem Gamma_reflection_complex:
@@ -2330,7 +2330,7 @@
show ?thesis
proof (cases "z \<in> \<int>")
case False
- with \<open>g z = pi\<close> show ?thesis by (auto simp: g_def divide_simps)
+ with \<open>g z = pi\<close> show ?thesis by (auto simp: g_def field_split_simps)
next
case True
then obtain n where n: "z = of_int n" by (elim Ints_cases)
@@ -2344,7 +2344,7 @@
lemma rGamma_reflection_complex:
"rGamma z * rGamma (1 - z :: complex) = sin (of_real pi * z) / of_real pi"
using Gamma_reflection_complex[of z]
- by (simp add: Gamma_def divide_simps split: if_split_asm)
+ by (simp add: Gamma_def field_split_simps split: if_split_asm)
lemma rGamma_reflection_complex':
"rGamma z * rGamma (- z :: complex) = -z * sin (of_real pi * z) / of_real pi"
@@ -2358,7 +2358,7 @@
lemma Gamma_reflection_complex':
"Gamma z * Gamma (- z :: complex) = - of_real pi / (z * sin (of_real pi * z))"
- using rGamma_reflection_complex'[of z] by (force simp add: Gamma_def divide_simps mult_ac)
+ using rGamma_reflection_complex'[of z] by (force simp add: Gamma_def field_split_simps mult_ac)
@@ -2403,7 +2403,7 @@
from eventually_at_ball'[OF zero_less_one, of "- of_nat n :: 'a" UNIV]
show "eventually (\<lambda>z. rGamma z / (z + of_nat n) = ?f z) (at (- of_nat n))"
by (subst pochhammer_rGamma[of _ "Suc n"])
- (auto elim!: eventually_mono simp: divide_simps pochhammer_rec' eq_neg_iff_add_eq_0)
+ (auto elim!: eventually_mono simp: field_split_simps pochhammer_rec' eq_neg_iff_add_eq_0)
have "isCont ?f (- of_nat n)" by (intro continuous_intros)
thus "?f \<midarrow> (- of_nat n) \<rightarrow> (- 1) ^ n * fact n" unfolding isCont_def
by (simp add: pochhammer_same)
@@ -2437,7 +2437,7 @@
from eventually_at_ball'[OF zero_less_one, of "- of_nat n :: 'a" UNIV]
show "eventually (\<lambda>z. Gamma z * (z + of_nat n) = inverse (rGamma z / (z + of_nat n)))
(at (- of_nat n))"
- by (auto elim!: eventually_mono simp: divide_simps rGamma_inverse_Gamma)
+ by (auto elim!: eventually_mono simp: field_split_simps rGamma_inverse_Gamma)
have "(\<lambda>z. inverse (rGamma z / (z + of_nat n))) \<midarrow> (- of_nat n) \<rightarrow>
inverse ((- 1) ^ n * fact n :: 'a)"
by (intro tendsto_intros rGamma_zeros) simp_all
@@ -2484,9 +2484,9 @@
also have "(\<Sum>k=1..n. ln (1 + 1 / of_nat k) :: real) = ln (\<Prod>k=1..n. 1 + 1 / real_of_nat k)"
by (subst ln_prod [symmetric]) (auto intro!: add_pos_nonneg)
also have "(\<Prod>k=1..n. 1 + 1 / of_nat k :: real) = (\<Prod>k=1..n. (of_nat k + 1) / of_nat k)"
- by (intro prod.cong) (simp_all add: divide_simps)
+ by (intro prod.cong) (simp_all add: field_split_simps)
also have "(\<Prod>k=1..n. (of_nat k + 1) / of_nat k :: real) = of_nat n + 1"
- by (induction n) (simp_all add: prod.nat_ivl_Suc' divide_simps)
+ by (induction n) (simp_all add: prod.nat_ivl_Suc' field_split_simps)
finally show ?thesis ..
qed
@@ -2500,7 +2500,7 @@
from z have z': "z \<noteq> 0" by auto
have "eventually (\<lambda>n. ?r' n = ?r n) sequentially"
- using z by (auto simp: divide_simps Gamma_series_def ring_distribs exp_diff ln_div add_ac
+ using z by (auto simp: field_split_simps Gamma_series_def ring_distribs exp_diff ln_div add_ac
intro: eventually_mono eventually_gt_at_top[of "0::nat"] dest: pochhammer_eq_0_imp_nonpos_Int)
moreover have "?r' \<longlonglongrightarrow> exp (z * of_real (ln 1))"
by (intro tendsto_intros LIMSEQ_Suc_n_over_n) simp_all
@@ -2572,7 +2572,7 @@
by (simp add: exp_add exp_sum exp_diff mult_ac Gamma_complex_altdef A)
from tendsto_mult[OF tendsto_const[of "exp (-euler_mascheroni * z) / z"] this] z
show "Gamma_series_Weierstrass z \<longlonglongrightarrow> Gamma z"
- by (simp add: exp_minus divide_simps Gamma_series_Weierstrass_def [abs_def])
+ by (simp add: exp_minus field_split_simps Gamma_series_Weierstrass_def [abs_def])
qed
lemma tendsto_complex_of_real_iff: "((\<lambda>x. complex_of_real (f x)) \<longlongrightarrow> of_real c) F = (f \<longlongrightarrow> c) F"
@@ -2616,7 +2616,7 @@
from False have "((z + of_nat n) gchoose n) = pochhammer z (Suc n) / z / fact n"
by (simp add: gbinomial_pochhammer' pochhammer_rec)
also have "pochhammer z (Suc n) / z / fact n * ?powr n (-z) = rGamma_series z n / z"
- by (simp add: rGamma_series_def divide_simps exp_minus)
+ by (simp add: rGamma_series_def field_split_simps exp_minus)
finally show "rGamma_series z n / z = ((z + of_nat n) gchoose n) * ?powr n (-z)" ..
qed
@@ -2644,7 +2644,7 @@
have "(\<lambda>n. of_nat ((k + n) choose n) / of_real (exp (of_nat k * ln (real_of_nat n))))
\<longlonglongrightarrow> 1 / Gamma (of_nat (Suc k) :: 'a)" (is "?f \<longlonglongrightarrow> _")
using Gamma_gbinomial[of "of_nat k :: 'a"]
- by (simp add: binomial_gbinomial add_ac Gamma_def divide_simps exp_of_real [symmetric] exp_minus)
+ by (simp add: binomial_gbinomial add_ac Gamma_def field_split_simps exp_of_real [symmetric] exp_minus)
also have "Gamma (of_nat (Suc k)) = fact k" by (simp add: Gamma_fact)
finally show "?f \<longlonglongrightarrow> 1 / fact k" .
@@ -2710,7 +2710,7 @@
have "(z gchoose n) = Gamma (z + 2) / (z + 1) / (fact n * Gamma (z - of_nat n + 1))"
by (subst gbinomial_Beta[OF assms]) (simp_all add: Beta_def Gamma_fact [symmetric] add_ac)
also from assms have "Gamma (z + 2) / (z + 1) = Gamma (z + 1)"
- using Gamma_plus1[of "z+1"] by (auto simp add: divide_simps mult_ac add_ac)
+ using Gamma_plus1[of "z+1"] by (auto simp add: field_split_simps mult_ac add_ac)
finally show ?thesis .
qed
@@ -2847,7 +2847,7 @@
using Suc.IH[of "z+1"] Suc.prems by (intro has_integral_mult_left) (simp_all add: add_ac pochhammer_rec)
also have "?A = (\<lambda>t. ?f t * ?g' t)" by (intro ext) (simp_all add: field_simps)
also have "?B = - (of_nat (Suc n) * fact n / pochhammer z (n+2))"
- by (simp add: divide_simps pochhammer_rec
+ by (simp add: field_split_simps pochhammer_rec
prod.shift_bounds_cl_Suc_ivl del: of_nat_Suc)
finally show "((\<lambda>t. ?f t * ?g' t) has_integral (?f 1 * ?g 1 - ?f 0 * ?g 0 - ?I)) {0..1}"
by simp
@@ -2957,7 +2957,7 @@
fix x :: real and n :: nat assume x: "x \<le> of_nat n"
have "(1 - complex_of_real x / of_nat n) = complex_of_real ((1 - x / of_nat n))" by simp
also have "norm \<dots> = \<bar>(1 - x / real n)\<bar>" by (subst norm_of_real) (rule refl)
- also from x have "\<dots> = (1 - x / real n)" by (intro abs_of_nonneg) (simp_all add: divide_simps)
+ also from x have "\<dots> = (1 - x / real n)" by (intro abs_of_nonneg) (simp_all add: field_split_simps)
finally have "cmod (1 - complex_of_real x / of_nat n) = 1 - x / real n" .
} note D = this
from D[of x k] x
@@ -3196,10 +3196,10 @@
(\<Prod>k=1..n. 1 - z^2 / of_nat k ^ 2)"
by (intro prod.cong) (simp_all add: power2_eq_square field_simps)
finally show "(- of_real pi * inverse z) * (?f z n * ?f (-z) n) = of_real pi * z * \<dots>"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
qed
also have "(- of_real pi * inverse z) * (rGamma z * rGamma (- z)) = sin (of_real pi * z)"
- by (subst rGamma_reflection_complex') (simp add: divide_simps)
+ by (subst rGamma_reflection_complex') (simp add: field_split_simps)
finally show ?thesis .
qed
@@ -3228,7 +3228,7 @@
by (simp add: prod_inversef [symmetric])
also have "(\<lambda>n. (\<Prod>k=1..n. inverse (1 - (1/2)\<^sup>2 / (real k)\<^sup>2))) =
(\<lambda>n. (\<Prod>k=1..n. (4*real k^2)/(4*real k^2 - 1)))"
- by (intro ext prod.cong refl) (simp add: divide_simps)
+ by (intro ext prod.cong refl) (simp add: field_split_simps)
finally show ?thesis .
qed
@@ -3295,7 +3295,7 @@
by (simp add: eval_nat_numeral)
from sums_divide[OF this, of "x^3 * pi"] x
have "(\<lambda>n. - (sin_coeff (n+3) * pi^(n+2) * x^n)) sums ((1 - sin (pi*x) / (pi*x)) / x^2)"
- by (simp add: divide_simps eval_nat_numeral power_mult_distrib mult_ac)
+ by (simp add: field_split_simps eval_nat_numeral power_mult_distrib mult_ac)
with x have "(\<lambda>n. - (sin_coeff (n+3) * pi^(n+2) * x^n)) sums (g x / x^2)"
by (simp add: g_def)
hence "f' x = g x / x^2" by (simp add: sums_iff f'_def)
@@ -3310,7 +3310,7 @@
assume x: "x \<noteq> 0"
from summable_divide[OF sums_summable[OF sums_split_initial_segment[OF
sin_converges[of "pi*x"]], of 3], of "-pi*x^3"] x
- show ?thesis by (simp add: mult_ac power_mult_distrib divide_simps eval_nat_numeral)
+ show ?thesis by (simp add: mult_ac power_mult_distrib field_split_simps eval_nat_numeral)
qed (simp only: summable_0_powser)
qed
hence "f' \<midarrow> 0 \<rightarrow> f' 0" by (simp add: isCont_def)
--- a/src/HOL/Analysis/Great_Picard.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Great_Picard.thy Wed Oct 09 14:51:54 2019 +0000
@@ -70,7 +70,7 @@
fix n::nat
assume "0 < n" "ln (n + sqrt ((real n)\<^sup>2 - 1)) / pi \<le> x"
then have "ln (n + sqrt ((real n)\<^sup>2 - 1)) \<le> x * pi"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
then have *: "exp (ln (n + sqrt ((real n)\<^sup>2 - 1))) \<le> exp (x * pi)"
by blast
have 0: "0 \<le> sqrt ((real n)\<^sup>2 - 1)"
@@ -130,10 +130,10 @@
by (auto simp: real_le_lsqrt power2_eq_square algebra_simps 1 **)
then
have "((1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) / (real n + sqrt ((real n)\<^sup>2 - 1))) \<le> 2"
- using Schottky_lemma1 \<open>0 < n\<close> by (simp add: divide_simps)
+ using Schottky_lemma1 \<open>0 < n\<close> by (simp add: field_split_simps)
then have "ln ((1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) / (real n + sqrt ((real n)\<^sup>2 - 1))) \<le> ln 2"
apply (subst ln_le_cancel_iff)
- using Schottky_lemma1 \<open>0 < n\<close> by auto (force simp: divide_simps)
+ using Schottky_lemma1 \<open>0 < n\<close> by auto (force simp: field_split_simps)
also have "... \<le> 3"
using ln_add_one_self_le_self [of 1] by auto
finally show ?thesis .
@@ -141,7 +141,7 @@
also have "... < pi"
using pi_approx by simp
finally show ?thesis
- by (simp add: a_def b_def divide_simps)
+ by (simp add: a_def b_def field_split_simps)
qed
ultimately have "\<bar>x - a\<bar> < 1/2 \<or> \<bar>x - b\<bar> < 1/2"
by (auto simp: abs_if)
@@ -255,8 +255,8 @@
also have "... \<le> 2 + cmod (f 0) * 3"
by simp
finally have "1 + norm(2 * f 0 - 1) / 3 \<le> (2 + norm(f 0) - 1) * 3"
- apply (simp add: divide_simps)
- using norm_ge_zero [of "2 * f 0 - 1"]
+ apply (simp add: field_split_simps)
+ using norm_ge_zero [of "f 0 * 2 - 1"]
by linarith
with nh0 have "norm(h 0) \<le> (2 + norm(f 0) - 1) * 3"
by linarith
@@ -349,7 +349,7 @@
using has_contour_integral_bound_linepath [OF int_g', of "12/(1 - t)"] assms
by simp
with \<open>cmod z \<le> t\<close> \<open>t < 1\<close> show ?thesis
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
qed
have fz: "f z = (1 + cos(of_real pi * h z)) / 2"
using h \<open>z \<in> ball 0 1\<close> by (auto simp: field_simps)
@@ -880,13 +880,13 @@
then have r4_le_xy: "r/4 \<le> cmod (x - y)"
using \<open>r > 0\<close> by simp
then have neq: "x \<noteq> y" "x \<noteq> z"
- using that \<open>r > 0\<close> by (auto simp: divide_simps norm_minus_commute)
+ using that \<open>r > 0\<close> by (auto simp: field_split_simps norm_minus_commute)
have leM: "cmod (\<F> n x) \<le> M"
by (simp add: M dist_commute dist_norm that)
have "cmod (\<F> n x / (x - y) - \<F> n x / (x - z)) = cmod (\<F> n x) * cmod (1 / (x - y) - 1 / (x - z))"
by (metis (no_types, lifting) divide_inverse mult.left_neutral norm_mult right_diff_distrib')
also have "... = cmod (\<F> n x) * cmod ((y - z) / ((x - y) * (x - z)))"
- using neq by (simp add: divide_simps)
+ using neq by (simp add: field_split_simps)
also have "... = cmod (\<F> n x) * (cmod (y - z) / (cmod(x - y) * (2/3 * r)))"
by (simp add: norm_mult norm_divide that)
also have "... \<le> M * (cmod (y - z) / (cmod(x - y) * (2/3 * r)))"
@@ -899,7 +899,7 @@
apply (simp add: field_simps mult_less_0_iff norm_minus_commute)
done
also have "... \<le> e/r"
- using \<open>e > 0\<close> \<open>r > 0\<close> r4_le_xy by (simp add: divide_simps)
+ using \<open>e > 0\<close> \<open>r > 0\<close> r4_le_xy by (simp add: field_split_simps)
finally show ?thesis by simp
qed
have "(2 * pi) * cmod (\<F> n y - \<F> n z) = cmod ((2 * pi) * \<i> * \<F> n y - (2 * pi) * \<i> * \<F> n z)"
@@ -908,7 +908,7 @@
apply (rule has_contour_integral_bound_circlepath [OF has_contour_integral_diff [OF y z], of "e/r"])
using \<open>e > 0\<close> \<open>r > 0\<close> le_er by auto
also have "... = (2 * pi) * e * ((2 / 3))"
- using \<open>r > 0\<close> by (simp add: divide_simps)
+ using \<open>r > 0\<close> by (simp add: field_split_simps)
finally have "cmod (\<F> n y - \<F> n z) \<le> e * (2 / 3)"
by simp
also have "... < e"
@@ -1142,7 +1142,7 @@
done
qed
with \<open>0 < r\<close> \<open>0 < m\<close> w w_inb show "deriv g w / g w = of_nat m / (w - z0) + deriv h w / h w"
- by (auto simp: geq divide_simps hnz)
+ by (auto simp: geq field_split_simps hnz)
qed
moreover
have "contour_integral (circlepath z0 (r/2)) (\<lambda>z. m / (z - z0) + deriv h z / h z) =
@@ -1268,7 +1268,7 @@
apply (rule subsetD [OF e])
using \<open>0 < e\<close> by (auto simp: dist_norm norm_mult)
have "cmod (h z) \<le> cmod (h (w + of_real e * (inverse e * (z - w))))"
- using \<open>0 < e\<close> by (simp add: divide_simps)
+ using \<open>0 < e\<close> by (simp add: field_split_simps)
also have "... \<le> exp (pi * exp (pi * (14 + 2 * r)))"
using r [OF \<open>h \<in> Y\<close>] Schottky [OF hol_h_o norm_le_r _ _ _ le12] non01 by auto
finally
@@ -1483,7 +1483,7 @@
have [simp]: "cmod (1 + of_nat n) = 1 + of_nat n" for n
by (metis norm_of_nat of_nat_Suc)
have *: "(\<lambda>x::complex. x / of_nat (Suc n)) ` (ball 0 1 - {0}) \<subseteq> ball 0 1 - {0}" for n
- by (auto simp: norm_divide divide_simps split: if_split_asm)
+ by (auto simp: norm_divide field_split_simps split: if_split_asm)
define h where "h \<equiv> \<lambda>n z::complex. f (z / (Suc n))"
have holh: "(h n) holomorphic_on ball 0 1 - {0}" for n
unfolding h_def
@@ -1606,7 +1606,7 @@
then have "B > 0"
by (metis \<open>0 < \<epsilon>\<close> dense leI order.asym vector_choose_size)
then have "inverse B > 0"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
then show ?thesis
apply (rule that [OF \<open>0 < \<epsilon>\<close> \<open>\<epsilon> < 1\<close>])
using \<epsilon> by auto
@@ -1632,7 +1632,7 @@
using \<open>0 < r\<close> \<open>a \<noteq> 0\<close> r
by (auto simp: dist_norm norm_mult subset_eq)
show "\<And>w. w \<in> ball 0 1 - {0} \<Longrightarrow> f (z + of_real r * w) / a \<noteq> 0 \<and> f (z + of_real r * w) / a \<noteq> 1"
- apply (simp add: divide_simps \<open>a \<noteq> 0\<close>)
+ apply (simp add: field_split_simps \<open>a \<noteq> 0\<close>)
apply (rule f0a)
using \<open>0 < r\<close> r by (auto simp: dist_norm norm_mult subset_eq)
qed
@@ -1652,14 +1652,14 @@
case 1
have "\<lbrakk>dist z w < e * r; w \<noteq> z\<rbrakk> \<Longrightarrow> cmod (f w) \<le> B * norm a" for w
using \<open>a \<noteq> 0\<close> \<open>0 < r\<close> 1 [of "(w - z) / r"]
- by (simp add: norm_divide dist_norm divide_simps)
+ by (simp add: norm_divide dist_norm field_split_simps)
then show ?thesis
by (force simp: intro!: boundedI)
next
case 2
have "\<lbrakk>dist z w < e * r; w \<noteq> z\<rbrakk> \<Longrightarrow> cmod (f w) \<ge> B * norm a" for w
using \<open>a \<noteq> 0\<close> \<open>0 < r\<close> 2 [of "(w - z) / r"]
- by (simp add: norm_divide dist_norm divide_simps)
+ by (simp add: norm_divide dist_norm field_split_simps)
then have "\<lbrakk>dist z w < e * r; w \<noteq> z\<rbrakk> \<Longrightarrow> inverse (cmod (f w)) \<le> inverse (B * norm a)" for w
by (metis \<open>0 < B\<close> \<open>a \<noteq> 0\<close> mult_pos_pos norm_inverse norm_inverse_le_norm zero_less_norm_iff)
then show ?thesis
@@ -1754,7 +1754,7 @@
apply (rule tendsto_eq_intros refl gz \<open>g z \<noteq> 0\<close>)+
by (simp add: True)
show "\<And>x. \<lbrakk>x \<in> ball z r; x \<noteq> z\<rbrakk> \<Longrightarrow> (1 + a * g x) / g x = f x"
- using fab fab zrM by (fastforce simp add: gf divide_simps)
+ using fab fab zrM by (fastforce simp add: gf field_split_simps)
qed (use \<open>0 < r\<close> in auto)
then show ?thesis
using that by blast
--- a/src/HOL/Analysis/Harmonic_Numbers.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Harmonic_Numbers.thy Wed Oct 09 14:51:54 2019 +0000
@@ -258,7 +258,7 @@
shows "ln (x + a) - ln x \<le> a * (inverse x + inverse (x + a))/2" (is "_ \<le> ?A")
proof -
define f' where "f' = (inverse (x + a) - inverse x)/a"
- have f'_nonpos: "f' \<le> 0" using assms by (simp add: f'_def divide_simps)
+ have f'_nonpos: "f' \<le> 0" using assms by (simp add: f'_def field_simps)
let ?f = "\<lambda>t. (t - x) * f' + inverse x"
let ?F = "\<lambda>t. (t - x)^2 * f' / 2 + t * inverse x"
have diff: "\<And>t. t \<in> {x..x+a} \<Longrightarrow> (?F has_vector_derivative ?f t) (at t within {x..x+a})"
@@ -271,14 +271,14 @@
intro!: derivative_eq_intros)
also have "?F (x+a) - ?F x = (a*2 + f'*a\<^sup>2*x) / (2*x)" using assms by (simp add: field_simps)
also have "f'*a^2 = - (a^2) / (x*(x + a))" using assms
- by (simp add: divide_simps f'_def power2_eq_square)
+ by (simp add: f'_def power2_eq_square) (simp add: field_simps)
also have "(a*2 + - a\<^sup>2/(x*(x+a))*x) / (2*x) = ?A" using assms
- by (simp add: divide_simps power2_eq_square) (simp add: algebra_simps)
+ by (simp add: power2_eq_square) (simp add: field_split_simps)
finally have int1: "((\<lambda>t. (t - x) * f' + inverse x) has_integral ?A) {x..x + a}" .
from assms have int2: "(inverse has_integral (ln (x + a) - ln x)) {x..x+a}"
by (intro fundamental_theorem_of_calculus)
- (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric] divide_simps
+ (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric] field_split_simps
intro!: derivative_eq_intros)
hence "ln (x + a) - ln x = integral {x..x+a} inverse" by (simp add: integral_unique)
also have ineq: "\<forall>xa\<in>{x..x + a}. inverse xa \<le> (xa - x) * f' + inverse x"
@@ -291,7 +291,7 @@
from convex_onD_Icc[OF this _ t] assms
have "?A \<le> (1 - (t - x) / a) * inverse x + (t - x) / a * inverse (x + a)" by simp
also have "\<dots> = (t - x) * f' + inverse x" using assms
- by (simp add: f'_def divide_simps) (simp add: f'_def field_simps)
+ by (simp add: f'_def divide_simps) (simp add: field_simps)
finally show "inverse t \<le> (t - x) * f' + inverse x" .
qed
hence "integral {x..x+a} inverse \<le> integral {x..x+a} ?f" using f'_nonpos assms
@@ -310,7 +310,7 @@
let ?F = "\<lambda>t. (t - m)^2 * f' / 2 + t / m"
from assms have "((\<lambda>t. (t - m) * f' + inverse m) has_integral (?F y - ?F x)) {x..y}"
by (intro fundamental_theorem_of_calculus)
- (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric] divide_simps
+ (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric] field_split_simps
intro!: derivative_eq_intros)
also from m have "?F y - ?F x = ((y - m)^2 - (x - m)^2) * f' / 2 + (y - x) / m"
by (simp add: field_simps)
@@ -320,7 +320,7 @@
from assms have int2: "(inverse has_integral (ln y - ln x)) {x..y}"
by (intro fundamental_theorem_of_calculus)
- (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric] divide_simps
+ (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric] field_split_simps
intro!: derivative_eq_intros)
hence "ln y - ln x = integral {x..y} inverse" by (simp add: integral_unique)
also have ineq: "\<forall>xa\<in>{x..y}. inverse xa \<ge> (xa - m) * f' + inverse m"
@@ -484,7 +484,7 @@
hence "inverse c * (ln x - (\<Sum>k<n. 2*y^(2*k+1)/of_nat (2*k+1))) \<le> (1 / (1-y^2) / of_nat (2*n+1))"
by simp
hence "(ln x - (\<Sum>k<n. 2*y^(2*k+1) / of_nat (2*k+1))) / c \<le> (1 / (1 - y^2) / of_nat (2*n+1))"
- by (auto simp add: divide_simps)
+ by (auto simp add: field_split_simps)
with c_pos have "ln x \<le> c / (1 - y^2) / of_nat (2*n+1) + approx"
by (subst (asm) pos_divide_le_eq) (simp_all add: mult_ac approx_def)
moreover {
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Wed Oct 09 14:51:54 2019 +0000
@@ -1848,7 +1848,7 @@
using g_near_f[OF x, of n] g_near_f[OF x, of m] by simp
also have "\<dots> \<le> 1 / (real M) + 1 / (real M)"
apply (rule add_mono)
- using \<open>M \<noteq> 0\<close> m n by (auto simp: divide_simps)
+ using \<open>M \<noteq> 0\<close> m n by (auto simp: field_split_simps)
also have "\<dots> = 2 / real M"
by auto
finally show "norm (g n x - g m x) \<le> 2 / real M"
@@ -1900,7 +1900,7 @@
then have norm_less: "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g (N1 + N2) x) - h (N1 + N2)) < e/3"
using g' by blast
have "content (cbox a b) < e/3 * (of_nat N2)"
- using \<open>N2 \<noteq> 0\<close> N2 using True by (auto simp: divide_simps)
+ using \<open>N2 \<noteq> 0\<close> N2 using True by (auto simp: field_split_simps)
moreover have "e/3 * of_nat N2 \<le> e/3 * (of_nat (N1 + N2) + 1)"
using \<open>e>0\<close> by auto
ultimately have "content (cbox a b) < e/3 * (of_nat (N1 + N2) + 1)"
@@ -2904,7 +2904,7 @@
"Dg n s = (if n < p then (-1)^n * (b - s)^(p - 1 - n) / fact (p - 1 - n) else 0)" for n s
have g0: "Dg 0 = g"
using \<open>p > 0\<close>
- by (auto simp add: Dg_def divide_simps g_def split: if_split_asm)
+ by (auto simp add: Dg_def field_split_simps g_def split: if_split_asm)
{
fix m
assume "p > Suc m"
@@ -2916,7 +2916,7 @@
have Dg: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
(Dg m has_vector_derivative Dg (Suc m) t) (at t within {a..b})"
unfolding Dg_def
- by (auto intro!: derivative_eq_intros simp: has_vector_derivative_def fact_eq divide_simps)
+ by (auto intro!: derivative_eq_intros simp: has_vector_derivative_def fact_eq field_split_simps)
let ?sum = "\<lambda>t. \<Sum>i<p. (- 1) ^ i *\<^sub>R Dg i t *\<^sub>R Df (p - Suc i) t"
from sum_prod_derivatives_has_vector_derivative[of _ Dg _ _ _ Df,
OF \<open>p > 0\<close> g0 Dg f0 Df]
@@ -6582,7 +6582,7 @@
integrable_continuous continuous_intros)
also have "\<dots> < e'"
using \<open>0 < e'\<close> \<open>e > 0\<close>
- by (auto simp: e_def divide_simps)
+ by (auto simp: e_def field_split_simps)
finally show "dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) < e'" .
qed
qed
@@ -6672,8 +6672,8 @@
also have "\<dots> < e' * norm (x - x0)"
using \<open>e' > 0\<close>
apply (intro mult_strict_right_mono[OF _ \<open>0 < norm (x - x0)\<close>])
- apply (auto simp: divide_simps e_def)
- by (metis \<open>0 < e\<close> e_def order.asym zero_less_divide_iff)
+ apply (simp add: divide_simps e_def not_less)
+ done
finally have "norm (?F x - ?F x0 - ?dF (x - x0)) < e' * norm (x - x0)" .
then show ?case
by (auto simp: divide_simps)
@@ -7263,10 +7263,10 @@
\<le> e * content (cbox w z) / content ?CBOX/2 * content (cbox u v)"
apply (rule integrable_bound [OF _ _ normint_wz])
using cbp \<open>0 < e/content ?CBOX\<close>
- apply (auto simp: divide_simps content_pos_le integrable_diff int_integrable integrable_const)
+ apply (auto simp: field_split_simps content_pos_le integrable_diff int_integrable integrable_const)
done
also have "... \<le> e * content (cbox (u,w) (v,z)) / content ?CBOX/2"
- by (simp add: content_Pair divide_simps)
+ by (simp add: content_Pair field_split_simps)
finally have 2: "norm (integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x,y))) -
integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (t1,t2))))
\<le> e * content (cbox (u,w) (v,z)) / content ?CBOX/2"
@@ -7561,7 +7561,7 @@
with assms have "a powr (e + 1) \<ge> n powr (e + 1)"
by (intro powr_mono2') simp_all
with assms show ?thesis by (auto simp: divide_simps F_def integral_f)
- qed (insert assms, simp add: integral_f F_def divide_simps)
+ qed (insert assms, simp add: integral_f F_def field_split_simps)
thus "bounded (range(\<lambda>k. integral {a..} (f k)))"
unfolding bounded_iff by (intro exI[of _ "-F a"]) auto
qed
@@ -7591,12 +7591,12 @@
note has_integral_powr_to_inf[OF this \<open>a > 0\<close>]
also have "- (a powr (real_of_int (- int n) + 1)) / (real_of_int (- int n) + 1) =
1 / (real (n - 1) * a powr (real (n - 1)))" using assms
- by (simp add: divide_simps powr_add [symmetric] of_nat_diff)
+ by (simp add: field_split_simps powr_add [symmetric] of_nat_diff)
also from assms have "a powr (real (n - 1)) = a ^ (n - 1)"
by (intro powr_realpow)
finally show ?thesis
by (rule has_integral_eq [rotated])
- (insert assms, simp_all add: powr_minus powr_realpow divide_simps)
+ (insert assms, simp_all add: powr_minus powr_realpow field_split_simps)
qed
subsubsection \<open>Adaption to ordered Euclidean spaces and the Cartesian Euclidean space\<close>
--- a/src/HOL/Analysis/Homeomorphism.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Homeomorphism.thy Wed Oct 09 14:51:54 2019 +0000
@@ -95,7 +95,7 @@
then have [simp]: "(norm x / norm y) *\<^sub>R y = x"
by (rule eqI) (simp add: \<open>y \<noteq> 0\<close>)
have no: "0 \<le> norm x / norm y" "norm x / norm y < 1"
- using * by (auto simp: divide_simps)
+ using * by (auto simp: field_split_simps)
then show "x = y"
using starI [OF \<open>y \<in> S\<close> no] xynot by auto
next
@@ -107,7 +107,7 @@
then have [simp]: "(norm y / norm x) *\<^sub>R x = y"
by (rule eqI) (simp add: \<open>x \<noteq> 0\<close>)
have no: "0 \<le> norm y / norm x" "norm y / norm x < 1"
- using * by (auto simp: divide_simps)
+ using * by (auto simp: field_split_simps)
then show "x = y"
using starI [OF \<open>x \<in> S\<close> no] xynot by auto
qed
@@ -133,7 +133,7 @@
show "x \<in> (\<lambda>x. x /\<^sub>R norm x) ` (S - rel_interior S)"
apply (rule_tac x="d *\<^sub>R x" in image_eqI)
using \<open>0 < d\<close>
- using dx \<open>closed S\<close> apply (auto simp: rel_frontier_def divide_simps nox)
+ using dx \<open>closed S\<close> apply (auto simp: rel_frontier_def field_split_simps nox)
done
qed
qed
@@ -265,7 +265,7 @@
show ?thesis
apply (rule_tac x="inverse(norm(surf (proj x))) *\<^sub>R x" in image_eqI)
apply (metis (no_types, hide_lams) mult.commute scaleproj abs_inverse abs_norm_cancel divide_inverse norm_scaleR nxx positive_imp_inverse_positive proj_scaleR psp sproj_nz zero_less_norm_iff)
- apply (auto simp: divide_simps nole affineI)
+ apply (auto simp: field_split_simps nole affineI)
done
qed
ultimately have im_cball: "(\<lambda>x. norm x *\<^sub>R surf (proj x)) ` ?CBALL = S"
@@ -503,24 +503,28 @@
define f where "f \<equiv> \<lambda>x. 2 *\<^sub>R b + (2 / (1 - b\<bullet>x)) *\<^sub>R (x - b)"
define g where "g \<equiv> \<lambda>y. b + (4 / (norm y ^ 2 + 4)) *\<^sub>R (y - 2 *\<^sub>R b)"
have [simp]: "\<And>x. \<lbrakk>x \<in> T; b\<bullet>x = 0\<rbrakk> \<Longrightarrow> f (g x) = x"
- unfolding f_def g_def by (simp add: algebra_simps divide_simps add_nonneg_eq_0_iff)
- have no: "\<And>x. \<lbrakk>norm x = 1; b\<bullet>x \<noteq> 1\<rbrakk> \<Longrightarrow> (norm (f x))\<^sup>2 = 4 * (1 + b\<bullet>x) / (1 - b\<bullet>x)"
- apply (simp add: dot_square_norm [symmetric])
- apply (simp add: f_def vector_add_divide_simps divide_simps norm_eq_1)
- apply (simp add: algebra_simps inner_commute)
+ unfolding f_def g_def by (simp add: algebra_simps field_split_simps add_nonneg_eq_0_iff)
+ have no: "(norm (f x))\<^sup>2 = 4 * (1 + b \<bullet> x) / (1 - b \<bullet> x)"
+ if "norm x = 1" and "b \<bullet> x \<noteq> 1" for x
+ using that
+ apply (simp flip: dot_square_norm add: norm_eq_1 nonzero_eq_divide_eq)
+ apply (simp add: f_def vector_add_divide_simps inner_simps)
+ apply (use sum_sqs_eq [of 1 "b \<bullet> x"]
+ in \<open>auto simp add: field_split_simps inner_commute\<close>)
done
have [simp]: "\<And>u::real. 8 + u * (u * 8) = u * 16 \<longleftrightarrow> u=1"
by algebra
have [simp]: "\<And>x. \<lbrakk>norm x = 1; b \<bullet> x \<noteq> 1\<rbrakk> \<Longrightarrow> g (f x) = x"
- unfolding g_def no by (auto simp: f_def divide_simps)
- have [simp]: "\<And>x. \<lbrakk>x \<in> T; b \<bullet> x = 0\<rbrakk> \<Longrightarrow> norm (g x) = 1"
- unfolding g_def
+ unfolding g_def no by (auto simp: f_def field_split_simps)
+ have [simp]: "norm (g x) = 1" if "x \<in> T" and "b \<bullet> x = 0" for x
+ using that
+ apply (simp only: g_def)
apply (rule power2_eq_imp_eq)
apply (simp_all add: dot_square_norm [symmetric] divide_simps vector_add_divide_simps)
apply (simp add: algebra_simps inner_commute)
done
- have [simp]: "\<And>x. \<lbrakk>x \<in> T; b \<bullet> x = 0\<rbrakk> \<Longrightarrow> b \<bullet> g x \<noteq> 1"
- unfolding g_def
+ have [simp]: "b \<bullet> g x \<noteq> 1" if "x \<in> T" and "b \<bullet> x = 0" for x
+ using that unfolding g_def
apply (simp_all add: dot_square_norm [symmetric] divide_simps vector_add_divide_simps add_nonneg_eq_0_iff)
apply (auto simp: algebra_simps)
done
@@ -547,7 +551,7 @@
apply (auto simp: power2_eq_square algebra_simps inner_commute)
done
have [simp]: "\<And>x. \<lbrakk>norm x = 1; b \<bullet> x \<noteq> 1\<rbrakk> \<Longrightarrow> b \<bullet> f x = 0"
- by (simp add: f_def algebra_simps divide_simps)
+ by (simp add: f_def algebra_simps field_split_simps)
have [simp]: "\<And>x. \<lbrakk>x \<in> T; norm x = 1; b \<bullet> x \<noteq> 1\<rbrakk> \<Longrightarrow> f x \<in> T"
unfolding f_def
by (blast intro: \<open>subspace T\<close> \<open>b \<in> T\<close> subspace_add subspace_mul subspace_diff)
@@ -1442,9 +1446,9 @@
obtain t where "t \<in> tk" and t: "{real n / real N .. (1 + real n) / real N} \<subseteq> K t"
proof (rule bexE [OF \<delta>])
show "{real n / real N .. (1 + real n) / real N} \<subseteq> {0..1}"
- using Suc.prems by (auto simp: divide_simps algebra_simps)
+ using Suc.prems by (auto simp: field_split_simps algebra_simps)
show diameter_less: "diameter {real n / real N .. (1 + real n) / real N} < \<delta>"
- using \<open>0 < \<delta>\<close> N by (auto simp: divide_simps algebra_simps)
+ using \<open>0 < \<delta>\<close> N by (auto simp: field_split_simps algebra_simps)
qed blast
have t01: "t \<in> {0..1}"
using \<open>t \<in> tk\<close> \<open>tk \<subseteq> {0..1}\<close> by blast
@@ -1454,7 +1458,7 @@
and homuu: "\<And>U. U \<in> \<V> \<Longrightarrow> \<exists>q. homeomorphism U (UU (X t)) p q"
using inUS [OF t01] UU by meson
have n_div_N_in: "real n / real N \<in> {real n / real N .. (1 + real n) / real N}"
- using N by (auto simp: divide_simps algebra_simps)
+ using N by (auto simp: field_split_simps algebra_simps)
with t have nN_in_kkt: "real n / real N \<in> K t"
by blast
have "k (real n / real N, y) \<in> C \<inter> p -` UU (X t)"
@@ -1466,7 +1470,7 @@
also have "... \<in> h ` (({0..1} \<inter> K t) \<times> (U \<inter> NN t))"
apply (rule imageI)
using \<open>y \<in> V\<close> t01 \<open>n \<le> N\<close>
- apply (simp add: nN_in_kkt \<open>y \<in> U\<close> inUS divide_simps)
+ apply (simp add: nN_in_kkt \<open>y \<in> U\<close> inUS field_split_simps)
done
also have "... \<subseteq> UU (X t)"
using him t01 by blast
@@ -1514,7 +1518,7 @@
then have "y \<in> Q'" "Q \<subseteq> (U \<inter> NN(t)) \<inter> N' \<inter> V"
by blast+
have neq: "{0..real n / real N} \<union> {real n / real N..(1 + real n) / real N} = {0..(1 + real n) / real N}"
- apply (auto simp: divide_simps)
+ apply (auto simp: field_split_simps)
by (metis mult_zero_left of_nat_0_le_iff of_nat_0_less_iff order_trans real_mult_le_cancel_iff1)
then have neqQ': "{0..real n / real N} \<times> Q' \<union> {real n / real N..(1 + real n) / real N} \<times> Q' = {0..(1 + real n) / real N} \<times> Q'"
by blast
--- a/src/HOL/Analysis/Homotopy.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Homotopy.thy Wed Oct 09 14:51:54 2019 +0000
@@ -4624,7 +4624,7 @@
using norm_triangle_ineq by blast
also have "\<dots> = norm y + (norm x - norm y) * (norm u / r)"
using yx \<open>r > 0\<close>
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
also have "\<dots> < norm y + (norm x - norm y) * 1"
apply (subst add_less_cancel_left)
apply (rule mult_strict_left_mono)
@@ -5039,7 +5039,7 @@
using assms by auto
have "f ` {a..b} = {c..d}"
unfolding f_def image_affinity_atLeastAtMost
- using assms sum_sqs_eq by (auto simp: divide_simps algebra_simps)
+ using assms sum_sqs_eq by (auto simp: field_split_simps algebra_simps)
then show "f ` cbox a b = cbox c d"
by auto
show "inj_on f (cbox a b)"
@@ -5051,7 +5051,7 @@
show "f a = c"
by (simp add: f_def)
show "f b = d"
- using assms sum_sqs_eq [of a b] by (auto simp: f_def divide_simps algebra_simps)
+ using assms sum_sqs_eq [of a b] by (auto simp: f_def field_split_simps algebra_simps)
qed
qed
@@ -5631,7 +5631,7 @@
also have "\<dots> < e"
apply (rule d [unfolded dist_norm])
using greater \<open>0 < d\<close> \<open>b1 \<in> Basis\<close> that
- by (auto simp: dist_norm divide_simps)
+ by (simp_all add: dist_norm) (simp add: field_simps)
finally show ?thesis .
qed
show ?thesis
--- a/src/HOL/Analysis/Improper_Integral.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Improper_Integral.thy Wed Oct 09 14:51:54 2019 +0000
@@ -725,7 +725,7 @@
case False
have prod_if: "(\<Prod>k\<in>Basis \<inter> - {i}. f k) = (\<Prod>k\<in>Basis. f k) / f i" if "f i \<noteq> (0::real)" for f
using that mk_disjoint_insert [OF i]
- apply (clarsimp simp add: divide_simps)
+ apply (clarsimp simp add: field_split_simps)
by (metis Int_insert_left_if0 finite_Basis finite_insert le_iff_inf mult.commute order_refl prod.insert subset_Compl_singleton)
have abc: "a \<bullet> i < c" "c < b \<bullet> i"
using False assms by auto
@@ -733,7 +733,7 @@
\<le> content(cbox a b') / (c - a \<bullet> i)"
"(\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<ge> c}))) K)
\<le> content(cbox a' b) / (b \<bullet> i - c)"
- using lec gec by (simp_all add: divide_simps mult.commute)
+ using lec gec by (simp_all add: field_split_simps mult.commute)
moreover
have "(\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))
\<le> (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c}))) K) +
@@ -780,7 +780,7 @@
\<le> 2 * content (cbox a b) / (b \<bullet> i - a \<bullet> i)"
by linarith
then show ?thesis
- using abc by (simp add: divide_simps mult.commute)
+ using abc by (simp add: field_split_simps mult.commute)
qed
qed
@@ -856,8 +856,8 @@
have "gauge (\<lambda>x. ball x
(\<epsilon> * (INF m\<in>Basis. b \<bullet> m - a \<bullet> m) / ((8 * norm (f x) + 8) * content (cbox a b))))"
using \<open>0 < content (cbox a b)\<close> \<open>0 < \<epsilon>\<close> a_less_b
- apply (auto simp: gauge_def divide_simps mult_less_0_iff zero_less_mult_iff add_nonneg_eq_0_iff finite_less_Inf_iff)
- apply (meson add_nonneg_nonneg mult_nonneg_nonneg norm_ge_zero not_less zero_le_numeral)
+ apply (auto simp: gauge_def field_split_simps mult_less_0_iff zero_less_mult_iff add_nonneg_eq_0_iff finite_less_Inf_iff)
+ apply (meson add_increasing measure_nonneg mult_nonneg_nonneg norm_ge_zero not_le zero_le_numeral)
done
then have "gauge \<gamma>"
unfolding \<gamma>_def using \<open>gauge \<gamma>0\<close> gauge_Int by auto
@@ -936,7 +936,7 @@
by simp
also have "... < \<epsilon> * (b \<bullet> i - a \<bullet> i) / dist u v / (4 * content (cbox a b))"
using duv dist_uv contab_gt0
- apply (simp add: divide_simps algebra_simps mult_less_0_iff zero_less_mult_iff split: if_split_asm)
+ apply (simp add: field_split_simps algebra_simps mult_less_0_iff zero_less_mult_iff split: if_split_asm)
by (meson add_nonneg_nonneg linorder_not_le measure_nonneg mult_nonneg_nonneg norm_ge_zero zero_le_numeral)
also have "... = \<epsilon> * (b \<bullet> i - a \<bullet> i) / norm (v - u) / (4 * content (cbox a b))"
by (simp add: dist_norm norm_minus_commute)
@@ -953,7 +953,7 @@
using mult_left_mono by fastforce
also have "... = \<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b) *
content K / ?\<Delta>"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
finally show ?thesis .
qed
qed
@@ -1533,7 +1533,7 @@
using n \<open>n \<noteq> 0\<close> emin [OF \<open>i \<in> Basis\<close>]
by (simp_all add: inverse_eq_divide)
show "1 / real (Suc m) \<le> 1 / real n"
- using \<open>n \<noteq> 0\<close> \<open>m \<ge> n\<close> by (simp add: divide_simps)
+ using \<open>n \<noteq> 0\<close> \<open>m \<ge> n\<close> by (simp add: field_split_simps)
qed
then show ?thesis by (simp add: mem_box)
qed
@@ -1684,7 +1684,7 @@
with N have "k > ((b - a) \<bullet> i) / ((x - a) \<bullet> i)"
using \<open>N \<le> k\<close> by linarith
with x that show ?thesis
- by (auto simp: mem_box algebra_simps divide_simps)
+ by (auto simp: mem_box algebra_simps field_split_simps)
qed
have bx_gt: "(b - x) \<bullet> i > ((b - a) \<bullet> i) / k" if "i \<in> Basis" for i
proof -
@@ -1696,7 +1696,7 @@
with N have "k > ((b - a) \<bullet> i) / ((b - x) \<bullet> i)"
using \<open>N \<le> k\<close> by linarith
with x that show ?thesis
- by (auto simp: mem_box algebra_simps divide_simps)
+ by (auto simp: mem_box algebra_simps field_split_simps)
qed
show "x \<in> cbox (a + (b - a) /\<^sub>R k) (b - (b - a) /\<^sub>R k)"
using that \<Delta> \<open>k > 0\<close>
@@ -1883,7 +1883,7 @@
then obtain m::nat where m: "floor(n * f x) = int m"
using nonneg_int_cases zero_le_floor by blast
then have kn: "real k / real n \<le> f x \<longleftrightarrow> k \<le> m" for k
- using \<open>n \<noteq> 0\<close> by (simp add: divide_simps mult.commute) linarith
+ using \<open>n \<noteq> 0\<close> by (simp add: field_split_simps mult.commute) linarith
then have "Suc n / real n \<le> f x \<longleftrightarrow> Suc n \<le> m"
by blast
have "real n * f x \<le> real n"
@@ -1894,7 +1894,7 @@
by (subst sum.inter_restrict) (auto simp: kn)
also have "\<dots> < inverse n"
using \<open>m \<le> n\<close> \<open>n \<noteq> 0\<close> m
- by (simp add: min_absorb2 divide_simps mult.commute) linarith
+ by (simp add: min_absorb2 field_split_simps mult.commute) linarith
finally show ?thesis .
qed
@@ -2156,7 +2156,7 @@
by (metis Suc_n_not_le_n non0)
qed (use x 01 non0 in auto)
also have "\<dots> \<le> inverse N"
- using seq_suble [OF \<sigma>] \<open>N \<noteq> 0\<close> non0 that by (auto intro: order_trans simp: divide_simps)
+ using seq_suble [OF \<sigma>] \<open>N \<noteq> 0\<close> non0 that by (auto intro: order_trans simp: field_split_simps)
finally show ?thesis
by linarith
qed
@@ -2222,7 +2222,7 @@
proof (subst has_integral_cong)
show "g x * f x - g a * f x = (g b - g a) * h x *\<^sub>R f x"
if "x \<in> {a..b}" for x
- using 1 that by (simp add: h_def divide_simps algebra_simps)
+ using 1 that by (simp add: h_def field_split_simps algebra_simps)
show "((\<lambda>x. (g b - g a) * h x *\<^sub>R f x) has_integral (g b - g a) * integral {c..b} f) {a..b}"
using has_integral_mult_right [OF c, of "g b - g a"] .
qed
--- a/src/HOL/Analysis/Infinite_Products.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Infinite_Products.thy Wed Oct 09 14:51:54 2019 +0000
@@ -1732,7 +1732,7 @@
have "norm (?f n) = cmod (z k) ^ n / cmod (1 + of_nat n)"
by (auto simp: norm_divide norm_mult norm_power)
also have "\<dots> \<le> cmod (z k) ^ n"
- by (auto simp: divide_simps mult_le_cancel_left1 in_Reals_norm)
+ by (auto simp: field_split_simps mult_le_cancel_left1 in_Reals_norm)
also have "\<dots> \<le> (1 / 2) ^ n"
using N [OF that] by (simp add: power_mono)
finally show "norm (?f n) \<le> (1 / 2) ^ n" .
--- a/src/HOL/Analysis/Lebesgue_Measure.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Lebesgue_Measure.thy Wed Oct 09 14:51:54 2019 +0000
@@ -1001,7 +1001,7 @@
using c by (auto simp: box_def T_def field_simps inner_simps divide_less_eq)
with le c show "emeasure ?D (box l u) = (\<Prod>b\<in>?B. (u - l) \<bullet> b)"
by (auto simp: emeasure_density emeasure_distr nn_integral_multc emeasure_lborel_box_eq inner_simps
- field_simps divide_simps ennreal_mult'[symmetric] prod_nonneg prod.distrib[symmetric]
+ field_simps field_split_simps ennreal_mult'[symmetric] prod_nonneg prod.distrib[symmetric]
intro!: prod.cong)
qed simp
--- a/src/HOL/Analysis/Linear_Algebra.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Linear_Algebra.thy Wed Oct 09 14:51:54 2019 +0000
@@ -1806,7 +1806,7 @@
show "pairwise orthogonal ?S"
using \<open>pairwise orthogonal S\<close> by (auto simp: pairwise_def orthogonal_def)
show "\<And>x. x \<in> (\<lambda>x. x /\<^sub>R norm x) ` S \<Longrightarrow> norm x = 1"
- using \<open>0 \<notin> S\<close> by (auto simp: divide_simps)
+ using \<open>0 \<notin> S\<close> by (auto simp: field_split_simps)
then show "independent ?S"
by (metis \<open>pairwise orthogonal ((\<lambda>x. x /\<^sub>R norm x) ` S)\<close> norm_zero pairwise_orthogonal_independent zero_neq_one)
have "inj_on (\<lambda>x. x /\<^sub>R norm x) S"
@@ -1932,7 +1932,7 @@
proof (rule eventually_mono)
show "norm (h (f x) - h l) < e" if "norm (f x - l) < e / B" for x
using that B
- apply (simp add: divide_simps)
+ apply (simp add: field_split_simps)
by (metis \<open>linear h\<close> le_less_trans linear_diff mult.commute)
qed
qed
--- a/src/HOL/Analysis/Lipschitz.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Lipschitz.thy Wed Oct 09 14:51:54 2019 +0000
@@ -595,7 +595,7 @@
have "?d \<le> e/2/(L + 1)" by simp
also have "(L + 1) * \<dots> \<le> e / 2"
using \<open>0 < e\<close> \<open>L \<ge> 0\<close>
- by (auto simp: divide_simps)
+ by (auto simp: field_split_simps)
finally have le1: "(L + 1) * dist y b < e / 2" using \<open>L \<ge> 0\<close> by simp
have "dist x a \<le> dist (x, y) (a, b)"
@@ -684,7 +684,7 @@
from this elim d[of "rx (ry (rt n))"]
have "\<dots> < dist (f (?t n) (?y n)) (f (?t n) (?x n)) / rx (ry (rt (n)))"
using lx'(2) ly'(2) lt'(2) \<open>0 < rx _\<close>
- by (auto simp add: divide_simps algebra_simps strict_mono_def)
+ by (auto simp add: field_split_simps algebra_simps strict_mono_def)
also have "\<dots> \<le> diameter ?S / n"
by (force intro!: \<open>0 < n\<close> strict_mono_def xy diameter_bounded_bound frac_le
compact_imp_bounded compact t
--- a/src/HOL/Analysis/Path_Connected.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Path_Connected.thy Wed Oct 09 14:51:54 2019 +0000
@@ -642,14 +642,14 @@
assume "g2 x = g2 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1"
then show "x = y"
using * [of "(x + 1) / 2" "(y + 1) / 2"]
- by (force simp: joinpaths_def split_ifs divide_simps)
+ by (force simp: joinpaths_def split_ifs field_split_simps)
qed
ultimately have "arc g2"
using assms by (simp add: arc_def)
have "g2 y = g1 0 \<or> g2 y = g1 1"
if "g1 x = g2 y" "0 \<le> x" "x \<le> 1" "0 \<le> y" "y \<le> 1" for x y
using * [of "x / 2" "(y + 1) / 2"] that
- by (auto simp: joinpaths_def split_ifs divide_simps)
+ by (auto simp: joinpaths_def split_ifs field_split_simps)
then have "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}"
by (fastforce simp: pathstart_def pathfinish_def path_image_def)
with \<open>arc g1\<close> \<open>arc g2\<close> show ?thesis using that by blast
@@ -674,7 +674,7 @@
using assms by (simp add: simple_path_def)
have False if "g1 0 = g2 u" "0 \<le> u" "u \<le> 1" for u
using * [of 0 "(u + 1) / 2"] that assms arc_distinct_ends [OF \<open>?lhs\<close>]
- by (auto simp: joinpaths_def pathstart_def pathfinish_def split_ifs divide_simps)
+ by (auto simp: joinpaths_def pathstart_def pathfinish_def split_ifs field_split_simps)
then have n1: "pathstart g1 \<notin> path_image g2"
unfolding pathstart_def path_image_def
using atLeastAtMost_iff by blast
@@ -848,9 +848,9 @@
{ fix x y
assume "x \<in> closed_segment u v" "y \<in> closed_segment u v" "g x = g y"
then have "x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u"
- using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
- by (auto simp: closed_segment_real_eq image_affinity_atLeastAtMost divide_simps
- split: if_split_asm)
+ using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
+ by (auto split: if_split_asm simp add: closed_segment_real_eq image_affinity_atLeastAtMost)
+ (simp_all add: field_split_simps)
} moreover
have "path(subpath u v g) \<and> u\<noteq>v"
using sim [of "1/3" "2/3"] p
@@ -884,9 +884,10 @@
{ fix x y
assume "x \<in> closed_segment u v" "y \<in> closed_segment u v" "g x = g y"
then have "x = y"
- using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
- by (force simp: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost divide_simps
- split: if_split_asm)
+ using sim [of "(x-u)/(v-u)" "(y-u)/(v-u)"] p
+ by (cases "v = u")
+ (simp_all split: if_split_asm add: inj_on_def closed_segment_real_eq image_affinity_atLeastAtMost,
+ simp add: field_simps)
} moreover
have "path(subpath u v g) \<and> u\<noteq>v"
using sim [of "1/3" "2/3"] p
@@ -937,7 +938,7 @@
done
lemma join_subpaths_middle: "subpath (0) ((1 / 2)) p +++ subpath ((1 / 2)) 1 p = p"
- by (rule ext) (simp add: joinpaths_def subpath_def divide_simps)
+ by (rule ext) (simp add: joinpaths_def subpath_def field_split_simps)
subsection\<^marker>\<open>tag unimportant\<close>\<open>There is a subpath to the frontier\<close>
@@ -2457,9 +2458,8 @@
{ fix u
assume u: "(1 - u) *\<^sub>R x + u *\<^sub>R (a + C *\<^sub>R (x - a)) \<in> s" and "0 \<le> u" "u \<le> 1"
have CC: "1 \<le> 1 + (C - 1) * u"
- using \<open>x \<noteq> a\<close> \<open>0 \<le> u\<close>
- apply (simp add: C_def divide_simps norm_minus_commute)
- using Bx by auto
+ using \<open>x \<noteq> a\<close> \<open>0 \<le> u\<close> Bx
+ by (auto simp add: C_def norm_minus_commute)
have *: "\<And>v. (1 - u) *\<^sub>R x + u *\<^sub>R (a + v *\<^sub>R (x - a)) = a + (1 + (v - 1) * u) *\<^sub>R (x - a)"
by (simp add: algebra_simps)
have "a + ((1 / (1 + C * u - u)) *\<^sub>R x + ((u / (1 + C * u - u)) *\<^sub>R a + (C * u / (1 + C * u - u)) *\<^sub>R x)) =
@@ -2492,9 +2492,8 @@
{ fix u
assume u: "(1 - u) *\<^sub>R y + u *\<^sub>R (a + D *\<^sub>R (y - a)) \<in> s" and "0 \<le> u" "u \<le> 1"
have DD: "1 \<le> 1 + (D - 1) * u"
- using \<open>y \<noteq> a\<close> \<open>0 \<le> u\<close>
- apply (simp add: D_def divide_simps norm_minus_commute)
- using By by auto
+ using \<open>y \<noteq> a\<close> \<open>0 \<le> u\<close> By
+ by (auto simp add: D_def norm_minus_commute)
have *: "\<And>v. (1 - u) *\<^sub>R y + u *\<^sub>R (a + v *\<^sub>R (y - a)) = a + (1 + (v - 1) * u) *\<^sub>R (y - a)"
by (simp add: algebra_simps)
have "a + ((1 / (1 + D * u - u)) *\<^sub>R y + ((u / (1 + D * u - u)) *\<^sub>R a + (D * u / (1 + D * u - u)) *\<^sub>R y)) =
@@ -3008,11 +3007,20 @@
case True with B Bno show ?thesis by force
next
case False
- with B C
have "closed_segment x (((B + C) / norm x) *\<^sub>R x) \<subseteq> - ball 0 B"
- apply (clarsimp simp add: closed_segment_def ball_def dist_norm real_vector_class.scaleR_add_left [symmetric] divide_simps)
- using segment_bound_lemma [of B "norm x" "B+C" ] Bno
- by (meson le_add_same_cancel1 less_eq_real_def not_le)
+ proof
+ fix w
+ assume "w \<in> closed_segment x (((B + C) / norm x) *\<^sub>R x)"
+ then obtain u where
+ w: "w = (1 - u + u * (B + C) / norm x) *\<^sub>R x" "0 \<le> u" "u \<le> 1"
+ by (auto simp add: closed_segment_def real_vector_class.scaleR_add_left [symmetric])
+ with False B C have "B \<le> (1 - u) * norm x + u * (B + C)"
+ using segment_bound_lemma [of B "norm x" "B + C" u] Bno
+ by simp
+ with False B C show "w \<in> - ball 0 B"
+ using distrib_right [of _ _ "norm x"]
+ by (simp add: ball_def w not_less)
+ qed
also have "... \<subseteq> -S"
by (simp add: B)
finally have "\<exists>T. connected T \<and> T \<subseteq> - S \<and> x \<in> T \<and> ((B + C) / norm x) *\<^sub>R x \<in> T"
@@ -3303,7 +3311,7 @@
by (metis mem_Collect_eq)
define C where "C = (B + 1 + norm z) / norm (z-a)"
have "C > 0"
- using \<open>0 < B\<close> zna by (simp add: C_def divide_simps add_strict_increasing)
+ using \<open>0 < B\<close> zna by (simp add: C_def field_split_simps add_strict_increasing)
have "\<bar>norm (z + C *\<^sub>R (z-a)) - norm (C *\<^sub>R (z-a))\<bar> \<le> norm z"
by (metis add_diff_cancel norm_triangle_ineq3)
moreover have "norm (C *\<^sub>R (z-a)) > norm z + B"
@@ -3314,10 +3322,10 @@
then have Cpos: "1 + u * C > 0"
by (meson \<open>0 < C\<close> add_pos_nonneg less_eq_real_def zero_le_mult_iff zero_less_one)
then have *: "(1 / (1 + u * C)) *\<^sub>R z + (u * C / (1 + u * C)) *\<^sub>R z = z"
- by (simp add: scaleR_add_left [symmetric] divide_simps)
+ by (simp add: scaleR_add_left [symmetric] field_split_simps)
then have False
using convexD_alt [OF s \<open>a \<in> s\<close> ins, of "1/(u*C + 1)"] \<open>C>0\<close> \<open>z \<notin> s\<close> Cpos u
- by (simp add: * divide_simps algebra_simps)
+ by (simp add: * field_split_simps algebra_simps)
} note contra = this
have "connected_component (- s) z (z + C *\<^sub>R (z-a))"
apply (rule connected_componentI [OF connected_segment [of z "z + C *\<^sub>R (z-a)"]])
@@ -3327,7 +3335,7 @@
done
then have False
using zna B [of "z + C *\<^sub>R (z-a)"] C
- by (auto simp: divide_simps max_mult_distrib_right)
+ by (auto simp: field_split_simps max_mult_distrib_right)
}
then show ?thesis
by (auto simp: outside_def z)
--- a/src/HOL/Analysis/Polytope.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Polytope.thy Wed Oct 09 14:51:54 2019 +0000
@@ -192,7 +192,7 @@
have nbc: "norm (b - c) + e > 0" using \<open>e > 0\<close>
by (metis add.commute le_less_trans less_add_same_cancel2 norm_ge_zero)
then have [simp]: "d \<noteq> c" using False scaleR_cancel_left [of "1 + (e / norm (b - c))" b c]
- by (simp add: algebra_simps d_def) (simp add: divide_simps)
+ by (simp add: algebra_simps d_def) (simp add: field_split_simps)
have [simp]: "((e - e * e / (e + norm (b - c))) / norm (b - c)) = (e / (e + norm (b - c)))"
using False nbc
by (simp add: divide_simps) (simp add: algebra_simps)
@@ -287,7 +287,7 @@
proof -
have "inverse(k) *\<^sub>R (x - y) = (1 - inverse k) *\<^sub>R inverse(1 - k) *\<^sub>R y + inverse(k) *\<^sub>R x"
using assms
- by (simp add: algebra_simps) (simp add: scaleR_left_distrib [symmetric] divide_simps)
+ by (simp add: algebra_simps) (simp add: scaleR_left_distrib [symmetric] field_split_simps)
then show ?thesis
using \<open>affine S\<close> xy by (auto simp: affine_alt)
qed
@@ -1316,7 +1316,7 @@
apply (rule_tac x="u/x" in exI)
apply (rule_tac x="v/x" in exI)
apply (rule_tac x="w/x" in exI)
- using x apply (auto simp: algebra_simps divide_simps)
+ using x apply (auto simp: algebra_simps field_split_simps)
done
ultimately show ?thesis by force
qed
@@ -1445,14 +1445,14 @@
have "x = (((1 - v) * ub) *\<^sub>R b + (v * uc) *\<^sub>R c) /\<^sub>R ux"
by (metis \<open>ux \<noteq> 0\<close> uxx mult.commute right_inverse scaleR_one scaleR_scaleR)
also have "... = (1 - v * uc / ux) *\<^sub>R b + (v * uc / ux) *\<^sub>R c"
- using \<open>ux \<noteq> 0\<close> equx apply (auto simp: algebra_simps divide_simps)
+ using \<open>ux \<noteq> 0\<close> equx apply (auto simp: algebra_simps field_split_simps)
by (metis add.commute add_diff_eq add_divide_distrib diff_add_cancel scaleR_add_left)
finally have "x = (1 - v * uc / ux) *\<^sub>R b + (v * uc / ux) *\<^sub>R c" .
then have "x \<in> open_segment b c"
apply (simp add: in_segment \<open>b \<noteq> c\<close>)
apply (rule_tac x="(v * uc) / ux" in exI)
using \<open>0 \<le> ux\<close> \<open>ux \<noteq> 0\<close> \<open>0 < uc\<close> \<open>0 < v\<close> \<open>0 < ub\<close> \<open>v < 1\<close> equx
- apply (force simp: algebra_simps divide_simps)
+ apply (force simp: algebra_simps field_split_simps)
done
then show ?thesis
by (rule face_ofD [OF F _ b c \<open>x \<in> F\<close>])
@@ -1841,7 +1841,7 @@
also have "... = \<xi> * norm (x - z)"
using \<open>e > 0\<close> by (simp add: \<xi>_def)
also have "... < e"
- using \<open>z \<noteq> x\<close> \<open>e > 0\<close> by (simp add: \<xi>_def min_def divide_simps norm_minus_commute)
+ using \<open>z \<noteq> x\<close> \<open>e > 0\<close> by (simp add: \<xi>_def min_def field_split_simps norm_minus_commute)
finally have lte: "norm (\<xi> *\<^sub>R x - \<xi> *\<^sub>R z) < e" .
have \<xi>_aff: "\<xi> *\<^sub>R z + (1 - \<xi>) *\<^sub>R x \<in> affine hull S"
by (metis \<open>x \<in> S\<close> add.commute affine_affine_hull diff_add_cancel hull_inc mem_affine zaff)
@@ -2034,7 +2034,7 @@
define w where "w = (1 - l) *\<^sub>R x + l *\<^sub>R z"
have "0 < l" "l < 1"
using able xltz \<open>b h < a h \<bullet> z\<close> \<open>h \<in> F\<close>
- by (auto simp: l_def divide_simps)
+ by (auto simp: l_def field_split_simps)
have awlt: "a i \<bullet> w < b i" if "i \<in> F" "i \<noteq> h" for i
proof -
have "(1 - l) * (a i \<bullet> x) < (1 - l) * b i"
@@ -2116,7 +2116,7 @@
done
then show ?thesis
using \<open>0 < inff\<close> awlt [OF that] mult_strict_left_mono
- by (fastforce simp add: algebra_simps divide_simps split: if_split_asm)
+ by (fastforce simp add: algebra_simps field_split_simps split: if_split_asm)
next
case False
with \<open>0 < inff\<close> have "inff * (a j \<bullet> y - a j \<bullet> w) \<le> 0"
@@ -3002,7 +3002,7 @@
apply (simp add: C_def)
apply (erule rev_finite_subset)
using \<open>0 < e\<close>
- apply (auto simp: divide_simps)
+ apply (auto simp: field_split_simps)
done
then have "finite I"
by (simp add: I_def)
--- a/src/HOL/Analysis/Radon_Nikodym.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Radon_Nikodym.thy Wed Oct 09 14:51:54 2019 +0000
@@ -89,7 +89,7 @@
show "summable (\<lambda>n. (\<integral>x. norm ((1 / 2) ^ Suc n * indicator (A n) x / (1 + Sigma_Algebra.measure M (A n))) \<partial>M))"
apply (rule summable_comparison_test'[of "\<lambda>n. (1/2)^(Suc n)" 0], auto)
using power_half_series summable_def apply auto[1]
- apply (auto simp add: divide_simps) using measure_nonneg[of M] not_less by fastforce
+ apply (auto simp add: field_split_simps) using measure_nonneg[of M] not_less by fastforce
qed
define f where "f = (\<lambda>x. if x \<in> space M then g x else 1)"
--- a/src/HOL/Analysis/Riemann_Mapping.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Riemann_Mapping.thy Wed Oct 09 14:51:54 2019 +0000
@@ -525,7 +525,7 @@
by (metis d1 dist_norm less_le_trans not_less not_less_iff_gr_or_eq segment_bound1 that(4))
qed (use \<open>e > 0\<close> in auto)
with \<open>e > 0\<close> have "inverse (norm (y - x)) * norm (z - f x * (y - x)) \<le> e/2"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
also have "... < e"
using \<open>e > 0\<close> by simp
finally show ?thesis
@@ -701,7 +701,7 @@
apply (rule 3)
unfolding norm_divide
using \<open>r > 0\<close> g_not_r [OF \<open>z \<in> S\<close>] g_not_r [OF \<open>a \<in> S\<close>]
- by (simp_all add: divide_simps dist_commute dist_norm)
+ by (simp_all add: field_split_simps dist_commute dist_norm)
then show "?f ` S \<subseteq> ball 0 1"
by auto
show "inj_on ?f S"
@@ -945,9 +945,9 @@
then obtain n where n: "1 / (1 - norm x) < of_nat n"
using reals_Archimedean2 by blast
with \<open>cmod x < 1\<close> gr0I have XX: "1 / of_nat n < 1 - norm x" and "n > 0"
- by (fastforce simp: divide_simps algebra_simps)+
+ by (fastforce simp: field_split_simps algebra_simps)+
have "f x \<in> f ` (D n)"
- using n \<open>cmod x < 1\<close> by (auto simp: divide_simps algebra_simps D_def)
+ using n \<open>cmod x < 1\<close> by (auto simp: field_split_simps algebra_simps D_def)
moreover have " f ` D n \<inter> closure (f ` A n) = {}"
proof -
have op_fDn: "open(f ` (D n))"
--- a/src/HOL/Analysis/Simplex_Content.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Simplex_Content.thy Wed Oct 09 14:51:54 2019 +0000
@@ -252,7 +252,7 @@
((a + b + c) - 2 * c)"
unfolding power2_eq_square by (simp add: s_def a_def b_def c_def algebra_simps)
also have "\<dots> = 16 * s * (s - a) * (s - b) * (s - c)"
- by (simp add: s_def divide_simps mult_ac)
+ by (simp add: s_def field_split_simps mult_ac)
finally have "content (convex hull {A, B, C}) ^ 2 = s * (s - a) * (s - b) * (s - c)"
by simp
also have "\<dots> = sqrt (s * (s - a) * (s - b) * (s - c)) ^ 2"
--- a/src/HOL/Analysis/Starlike.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Starlike.thy Wed Oct 09 14:51:54 2019 +0000
@@ -1221,7 +1221,7 @@
show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<in> interior S"
apply (clarsimp simp add: min_def a)
apply (rule mem_interior_closure_convex_shrink [OF \<open>convex S\<close> a x])
- using \<open>0 < e\<close> False apply (auto simp: divide_simps)
+ using \<open>0 < e\<close> False apply (auto simp: field_split_simps)
done
qed
qed
@@ -3433,7 +3433,7 @@
qed
qed
then show False
- using \<open>0 < e\<close> \<open>l \<noteq> 0\<close> by (simp add: d_def [symmetric] divide_simps)
+ using \<open>0 < e\<close> \<open>l \<noteq> 0\<close> by (simp add: d_def [symmetric] field_simps)
qed
moreover have "a + d *\<^sub>R l \<in> closure S"
proof (clarsimp simp: closure_approachable)
@@ -3444,7 +3444,7 @@
have "norm l * min d (\<eta> / (norm l * 2)) \<le> norm l * (\<eta> / (norm l * 2))"
by (metis min_def mult_left_mono norm_ge_zero order_refl)
also have "... < \<eta>"
- using \<open>l \<noteq> 0\<close> \<open>0 < \<eta>\<close> by (simp add: divide_simps)
+ using \<open>l \<noteq> 0\<close> \<open>0 < \<eta>\<close> by (simp add: field_simps)
finally have 2: "norm l * min d (\<eta> / (norm l * 2)) < \<eta>" .
show "\<exists>y\<in>S. dist y (a + d *\<^sub>R l) < \<eta>"
apply (rule_tac x="a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l" in bexI)
@@ -3498,7 +3498,7 @@
using xy
apply (auto simp: in_segment)
apply (rule_tac x="d" in exI)
- using \<open>0 < d\<close> that apply (auto simp: divide_simps algebra_simps)
+ using \<open>0 < d\<close> that apply (auto simp: algebra_simps)
done
ultimately have "1 \<le> d"
using df rel_frontier_def by fastforce
@@ -3507,7 +3507,7 @@
ultimately show "y \<in> closed_segment x (x + d *\<^sub>R (y - x))"
apply (auto simp: in_segment)
apply (rule_tac x="1/d" in exI)
- apply (auto simp: divide_simps algebra_simps)
+ apply (auto simp: algebra_simps)
done
next
show "open_segment x (x + d *\<^sub>R (y - x)) \<subseteq> rel_interior S"
@@ -5972,7 +5972,7 @@
ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> sphere 0 1"
by simp
have aa: "a /\<^sub>R (norm a) \<in> (\<Inter>c\<in>C. {x. 0 \<le> c \<bullet> x})"
- apply (clarsimp simp add: divide_simps)
+ apply (clarsimp simp add: field_split_simps)
using ab \<open>0 < b\<close>
by (metis hull_inc inner_commute less_eq_real_def less_trans)
show ?thesis
@@ -7446,7 +7446,7 @@
by (intro closure_minimal UN_mono ball_subset_cball order_refl cl)
also have "... \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})"
apply (intro UN_mono order_refl)
- apply (simp add: cball_subset_ball_iff divide_simps)
+ apply (simp add: cball_subset_ball_iff field_split_simps)
done
finally show "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
\<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})" .
@@ -7463,7 +7463,7 @@
obtain N2 where N2: "norm(x - a) \<le> real N2"
by (meson real_arch_simple)
have N12: "inverse((N1 + N2) + 1) \<le> inverse(N1)"
- using \<open>N1 > 0\<close> by (auto simp: divide_simps)
+ using \<open>N1 > 0\<close> by (auto simp: field_split_simps)
have "x \<noteq> y + z" if "y \<notin> S" "norm z < 1 / (1 + (real N1 + real N2))" for y z
proof -
have "e * real N1 < e * (1 + (real N1 + real N2))"
--- a/src/HOL/Analysis/Summation_Tests.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Summation_Tests.thy Wed Oct 09 14:51:54 2019 +0000
@@ -396,7 +396,7 @@
from this and \<open>summable f\<close> have "summable (\<lambda>n. p N * f N * inverse (p n))"
by (rule summable_comparison_test_ev)
from summable_mult[OF this, of "inverse (p N * f N)"] N[OF le_refl]
- have "summable (\<lambda>n. inverse (p n))" by (simp add: divide_simps)
+ have "summable (\<lambda>n. inverse (p n))" by (simp add: field_split_simps)
with divergent_p show False by contradiction
qed
@@ -757,7 +757,7 @@
fix r assume r: "0 < r" "ereal r < c"
let ?l = "liminf (\<lambda>n. ereal (norm (f n * of_real r ^ n) / norm (f (Suc n) * of_real r ^ Suc n)))"
have "?l = liminf (\<lambda>n. ereal (norm (f n) / (norm (f (Suc n)))) * ereal (inverse r))"
- using r by (simp add: norm_mult norm_power divide_simps)
+ using r by (simp add: norm_mult norm_power field_split_simps)
also from r have "\<dots> = liminf (\<lambda>n. ereal (norm (f n) / (norm (f (Suc n))))) * ereal (inverse r)"
by (intro Liminf_ereal_mult_right) simp_all
also have "liminf (\<lambda>n. ereal (norm (f n) / (norm (f (Suc n))))) = c"
@@ -771,7 +771,7 @@
fix r assume r: "0 < r" "ereal r > c"
let ?l = "limsup (\<lambda>n. ereal (norm (f n * of_real r ^ n) / norm (f (Suc n) * of_real r ^ Suc n)))"
have "?l = limsup (\<lambda>n. ereal (norm (f n) / (norm (f (Suc n)))) * ereal (inverse r))"
- using r by (simp add: norm_mult norm_power divide_simps)
+ using r by (simp add: norm_mult norm_power field_split_simps)
also from r have "\<dots> = limsup (\<lambda>n. ereal (norm (f n) / (norm (f (Suc n))))) * ereal (inverse r)"
by (intro Limsup_ereal_mult_right) simp_all
also have "limsup (\<lambda>n. ereal (norm (f n) / (norm (f (Suc n))))) = c"
--- a/src/HOL/Analysis/Tagged_Division.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Tagged_Division.thy Wed Oct 09 14:51:54 2019 +0000
@@ -9,8 +9,6 @@
imports Topology_Euclidean_Space
begin
-term comm_monoid
-
lemma sum_Sigma_product:
assumes "finite S"
and "\<And>i. i \<in> S \<Longrightarrow> finite (T i)"
@@ -2337,13 +2335,14 @@
show "\<Union>?D0 \<subseteq> cbox a b"
apply (simp add: UN_subset_iff)
apply (intro conjI allI ballI subset_box_imp)
- apply (simp add: divide_simps zero_le_mult_iff aibi)
- apply (force simp: aibi scaling_mono nat_less_real_le dest: PiE_mem)
+ apply (simp add: field_simps)
+ apply (auto intro: mult_right_mono aibi)
+ apply (force simp: aibi scaling_mono nat_less_real_le dest: PiE_mem intro: mult_right_mono)
done
next
show "\<And>K. K \<in> ?D0 \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
using \<open>box a b \<noteq> {}\<close>
- by (clarsimp simp: box_eq_empty) (fastforce simp add: divide_simps dest: PiE_mem)
+ by (clarsimp simp: box_eq_empty) (fastforce simp add: field_split_simps dest: PiE_mem)
next
have realff: "(real w) * 2^m < (real v) * 2^n \<longleftrightarrow> w * 2^m < v * 2^n" for m n v w
using of_nat_less_iff less_imp_of_nat_less by fastforce
@@ -2359,7 +2358,9 @@
apply (drule_tac x=i in bspec, assumption)
using \<open>m\<le>n\<close> realff [of _ _ "1+_"] realff [of "1+_"_ "1+_"]
apply (auto simp: divide_simps add.commute not_le nat_le_iff_add realff)
- apply (simp add: power_add, metis (no_types, hide_lams) mult_Suc mult_less_cancel2 not_less_eq mult.assoc)+
+ apply (simp_all add: power_add)
+ apply (metis (no_types, hide_lams) mult_Suc mult_less_cancel2 not_less_eq mult.assoc)
+ apply (metis (no_types, hide_lams) mult_Suc mult_less_cancel2 not_less_eq mult.assoc)
done
then show "?K0(m,v) \<subseteq> ?K0(n,w) \<or> ?K0(n,w) \<subseteq> ?K0(m,v) \<or> interior(?K0(m,v)) \<inter> interior(?K0(n,w)) = {}"
by meson
@@ -2387,7 +2388,7 @@
also have "... \<le> DIM('a) * (\<epsilon> / (2 * real DIM('a)))"
by (meson sum_bounded_above that)
also have "... = \<epsilon> / 2"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
also have "... < \<epsilon>"
by (simp add: \<open>0 < \<epsilon>\<close>)
finally show ?thesis .
@@ -2399,9 +2400,9 @@
using \<open>x \<in> S\<close> \<open>S \<subseteq> cbox a b\<close> by blast
obtain n where n: "norm (b - a) / 2^n < e"
using real_arch_pow_inv [of "e / norm(b - a)" "1/2"] normab \<open>0 < e\<close>
- by (auto simp: divide_simps)
+ by (auto simp: field_split_simps)
then have "norm (b - a) < e * 2^n"
- by (auto simp: divide_simps)
+ by (auto simp: field_split_simps)
then have bai: "b \<bullet> i - a \<bullet> i < e * 2 ^ n" if "i \<in> Basis" for i
proof -
have "b \<bullet> i - a \<bullet> i \<le> norm (b - a)"
@@ -2423,7 +2424,7 @@
x \<bullet> i \<le> a \<bullet> i + (real k + 1) * (b \<bullet> i - a \<bullet> i) / 2 ^ n)"
proof cases
case 1 then show ?thesis
- by (rule_tac x = "2^n - 1" in exI) (auto simp: algebra_simps divide_simps of_nat_diff \<open>i \<in> Basis\<close> aibi)
+ by (rule_tac x = "2^n - 1" in exI) (auto simp: algebra_simps field_split_simps of_nat_diff \<open>i \<in> Basis\<close> aibi)
next
case 2
then have abi_less: "a \<bullet> i < b \<bullet> i"
@@ -2433,20 +2434,20 @@
proof (intro exI conjI)
show "?k < 2 ^ n"
using aibi xab \<open>i \<in> Basis\<close>
- by (force simp: nat_less_iff floor_less_iff divide_simps 2 mem_box)
+ by (force simp: nat_less_iff floor_less_iff field_split_simps 2 mem_box)
next
have "a \<bullet> i + real ?k * (b \<bullet> i - a \<bullet> i) / 2 ^ n \<le>
a \<bullet> i + (2 ^ n * (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i)) * (b \<bullet> i - a \<bullet> i) / 2 ^ n"
apply (intro add_left_mono mult_right_mono divide_right_mono of_nat_floor)
using aibi [OF \<open>i \<in> Basis\<close>] xab 2
- apply (simp_all add: \<open>i \<in> Basis\<close> mem_box divide_simps)
+ apply (simp_all add: \<open>i \<in> Basis\<close> mem_box field_split_simps)
done
also have "... = x \<bullet> i"
- using abi_less by (simp add: divide_simps)
+ using abi_less by (simp add: field_split_simps)
finally show "a \<bullet> i + real ?k * (b \<bullet> i - a \<bullet> i) / 2 ^ n \<le> x \<bullet> i" .
next
have "x \<bullet> i \<le> a \<bullet> i + (2 ^ n * (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i)) * (b \<bullet> i - a \<bullet> i) / 2 ^ n"
- using abi_less by (simp add: divide_simps algebra_simps)
+ using abi_less by (simp add: field_split_simps algebra_simps)
also have "... \<le> a \<bullet> i + (real ?k + 1) * (b \<bullet> i - a \<bullet> i) / 2 ^ n"
apply (intro add_left_mono mult_right_mono divide_right_mono of_nat_floor)
using aibi [OF \<open>i \<in> Basis\<close>] xab
@@ -2467,7 +2468,8 @@
apply (drule bspec D, assumption)+
apply (erule order_trans)
apply (simp add: divide_simps)
- using bai by (force simp: algebra_simps)
+ using bai apply (force simp add: algebra_simps)
+ done
ultimately show ?thesis by auto
qed
qed
@@ -2486,12 +2488,12 @@
proof -
have dd: "w / m \<le> v / n \<and> (v+1) / n \<le> (w+1) / m
\<Longrightarrow> inverse n \<le> inverse m" for w m v n::real
- by (auto simp: divide_simps algebra_simps)
+ by (auto simp: field_split_simps algebra_simps)
have bjaj: "b \<bullet> j - a \<bullet> j > 0"
using \<open>j \<in> Basis\<close> \<open>box a b \<noteq> {}\<close> box_eq_empty(1) by fastforce
have "((g j) / 2 ^ m) * (b \<bullet> j - a \<bullet> j) \<le> ((f j) / 2 ^ n) * (b \<bullet> j - a \<bullet> j) \<and>
(((f j) + 1) / 2 ^ n) * (b \<bullet> j - a \<bullet> j) \<le> (((g j) + 1) / 2 ^ m) * (b \<bullet> j - a \<bullet> j)"
- using that \<open>j \<in> Basis\<close> by (simp add: subset_box algebra_simps divide_simps aibi)
+ using that \<open>j \<in> Basis\<close> by (simp add: subset_box algebra_simps field_split_simps aibi)
then have "((g j) / 2 ^ m) \<le> ((f j) / 2 ^ n) \<and>
((real(f j) + 1) / 2 ^ n) \<le> ((real(g j) + 1) / 2 ^ m)"
by (metis bjaj mult.commute of_nat_1 of_nat_add real_mult_le_cancel_iff2)
--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy Wed Oct 09 14:51:54 2019 +0000
@@ -66,7 +66,7 @@
have "sqrt ((representation B x b)\<^sup>2 * (norm b)\<^sup>2) \<le> sqrt (\<Sum>b\<in>B. (representation B x b)\<^sup>2 * (norm b)\<^sup>2)"
using \<open>b \<in> B\<close> \<open>finite B\<close> by (auto intro: member_le_sum)
then have "\<bar>representation B x b\<bar> \<le> (1 / norm b) * sqrt (\<Sum>b\<in>B. (representation B x b)\<^sup>2 * (norm b)\<^sup>2)"
- using \<open>b \<noteq> 0\<close> by (simp add: divide_simps real_sqrt_mult del: real_sqrt_le_iff)
+ using \<open>b \<noteq> 0\<close> by (simp add: field_split_simps real_sqrt_mult del: real_sqrt_le_iff)
ultimately show "\<bar>representation B x b\<bar> \<le> (1 / norm b) * norm x"
by simp
next
@@ -2028,7 +2028,7 @@
lemma diameter_closed_interval [simp]: "diameter {a..b} = (if b < a then 0 else b-a)"
proof -
have "{a .. b} = cball ((a+b)/2) ((b-a)/2)"
- by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)
+ by (auto simp: dist_norm abs_if field_split_simps split: if_split_asm)
then show ?thesis
by simp
qed
@@ -2036,7 +2036,7 @@
lemma diameter_open_interval [simp]: "diameter {a<..<b} = (if b < a then 0 else b-a)"
proof -
have "{a <..< b} = ball ((a+b)/2) ((b-a)/2)"
- by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)
+ by (auto simp: dist_norm abs_if field_split_simps split: if_split_asm)
then show ?thesis
by simp
qed
@@ -2068,7 +2068,7 @@
show "\<exists>e>0. \<forall>y. dist y (f x) < e \<longrightarrow> y \<in> f ` A"
proof (intro exI conjI)
show "\<delta> > 0"
- using \<open>e > 0\<close> \<open>B > 0\<close> by (simp add: \<delta>_def divide_simps)
+ using \<open>e > 0\<close> \<open>B > 0\<close> by (simp add: \<delta>_def field_split_simps)
have "y \<in> f ` A" if "dist y (f x) * (B * real DIM('b)) < e" for y
proof -
define u where "u \<equiv> y - f x"
@@ -2095,7 +2095,7 @@
qed
then show "\<forall>y. dist y (f x) < \<delta> \<longrightarrow> y \<in> f ` A"
using \<open>e > 0\<close> \<open>B > 0\<close>
- by (auto simp: \<delta>_def divide_simps mult_less_0_iff)
+ by (auto simp: \<delta>_def field_split_simps mult_less_0_iff)
qed
qed
--- a/src/HOL/Analysis/Uniform_Limit.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Uniform_Limit.thy Wed Oct 09 14:51:54 2019 +0000
@@ -596,7 +596,7 @@
using \<open>e > 0\<close> that b [OF \<open>y \<in> S\<close>] apply (simp add: power2_eq_square)
by (metis \<open>b > 0\<close> less_eq_real_def mult.left_commute mult_mono')
finally show ?thesis
- by (auto simp: dist_norm divide_simps norm_mult norm_divide)
+ by (auto simp: dist_norm field_split_simps norm_mult norm_divide)
qed
have "\<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (f n x) (l x) < b/2"
using uniform_limitD [OF f, of "b/2"] by (simp add: \<open>b > 0\<close>)
--- a/src/HOL/Analysis/Vitali_Covering_Theorem.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Vitali_Covering_Theorem.thy Wed Oct 09 14:51:54 2019 +0000
@@ -99,7 +99,7 @@
proof
assume "i \<in> C"
have "B/2 ^ Suc n \<le> B/2 ^ n"
- using \<open>B > 0\<close> by (simp add: divide_simps)
+ using \<open>B > 0\<close> by (simp add: field_split_simps)
also have "\<dots> \<le> r i"
using Ble \<open>i \<in> C\<close> by blast
finally show ?thesis .
@@ -224,7 +224,7 @@
then obtain n where "(1/2) ^ n < r i / B"
using \<open>B > 0\<close> assms real_arch_pow_inv by fastforce
then have B2: "B/2 ^ n < r i"
- using \<open>B > 0\<close> by (simp add: divide_simps)
+ using \<open>B > 0\<close> by (simp add: field_split_simps)
have "0 < r i" "r i \<le> B"
by (auto simp: \<open>i \<in> K\<close> assms)
show "\<exists>j. j \<in> (Union(range F)) \<and>
@@ -467,7 +467,7 @@
by simp
qed
ultimately show ?thesis
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
qed
have co: "countable (D - D1)"
by (simp add: \<open>countable D\<close>)
@@ -576,7 +576,7 @@
using ball_subset_ball_iff k by auto
finally show "ball x r \<subseteq> ball z 1" .
have "?\<epsilon> * ?\<mu> (ball x r) \<le> e * content (ball x r) / content (ball z 1)"
- using r \<open>e > 0\<close> by (simp add: ord_class.min_def divide_simps)
+ using r \<open>e > 0\<close> by (simp add: ord_class.min_def field_split_simps)
with mU show "?\<mu> U < e * content (ball x r) / content (ball z 1)"
by auto
qed (use r U x in auto)
--- a/src/HOL/Analysis/Weierstrass_Theorems.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Weierstrass_Theorems.thy Wed Oct 09 14:51:54 2019 +0000
@@ -110,12 +110,12 @@
then have "(\<Sum>k\<le>n. (k - n * x)\<^sup>2 * Bernstein n k x)/n^2 = x * (1 - x) / n"
by (simp add: power2_eq_square)
then show ?thesis
- using n by (simp add: sum_divide_distrib divide_simps mult.commute power2_commute)
+ using n by (simp add: sum_divide_distrib field_split_simps mult.commute power2_commute)
qed
{ fix k
assume k: "k \<le> n"
then have kn: "0 \<le> k / n" "k / n \<le> 1"
- by (auto simp: divide_simps)
+ by (auto simp: field_split_simps)
consider (lessd) "\<bar>x - k / n\<bar> < d" | (ged) "d \<le> \<bar>x - k / n\<bar>"
by linarith
then have "\<bar>(f x - f (k/n))\<bar> \<le> e/2 + 2 * M / d\<^sup>2 * (x - k/n)\<^sup>2"
@@ -158,7 +158,7 @@
also have "... < e"
apply (simp add: field_simps)
using \<open>d>0\<close> nbig e \<open>n>0\<close>
- apply (simp add: divide_simps algebra_simps)
+ apply (simp add: field_split_simps algebra_simps)
using ed0 by linarith
finally have "\<bar>f x - (\<Sum>k\<le>n. f (real k / real n) * Bernstein n k x)\<bar> < e" .
}
@@ -282,7 +282,7 @@
obtain i where i: "i \<in> subU" "t \<in> Uf i" using subU t by blast
show ?thesis
using subU i t
- apply (clarsimp simp: p_def divide_simps)
+ apply (clarsimp simp: p_def field_split_simps)
apply (rule sum_pos2 [OF \<open>finite subU\<close>])
using Uf t pf01 apply auto
apply (force elim!: subsetCE)
@@ -292,12 +292,12 @@
proof -
have "0 \<le> p x"
using subU cardp t
- apply (simp add: p_def divide_simps sum_nonneg)
+ apply (simp add: p_def field_split_simps sum_nonneg)
apply (rule sum_nonneg)
using pf01 by force
moreover have "p x \<le> 1"
using subU cardp t
- apply (simp add: p_def divide_simps sum_nonneg)
+ apply (simp add: p_def field_split_simps sum_nonneg)
apply (rule sum_bounded_above [where 'a=real and K=1, simplified])
using pf01 by force
ultimately show ?thesis
@@ -331,12 +331,12 @@
with \<delta>01 have "k \<le> (1+\<delta>)/\<delta>"
by (auto simp: algebra_simps add_divide_distrib)
also have "... < 2/\<delta>"
- using \<delta>01 by (auto simp: divide_simps)
+ using \<delta>01 by (auto simp: field_split_simps)
finally have k2\<delta>: "k < 2/\<delta>" .
have "1/\<delta> < k"
using \<delta>01 unfolding k_def by linarith
with \<delta>01 k2\<delta> have k\<delta>: "1 < k*\<delta>" "k*\<delta> < 2"
- by (auto simp: divide_simps)
+ by (auto simp: field_split_simps)
define q where [abs_def]: "q n t = (1 - p t ^ n) ^ (k^n)" for n t
have qR: "q n \<in> R" for n
by (simp add: q_def const diff power pR)
@@ -484,12 +484,12 @@
proof -
have "0 \<le> pff x"
using subA cardp t
- apply (simp add: pff_def divide_simps sum_nonneg)
+ apply (simp add: pff_def field_split_simps sum_nonneg)
apply (rule Groups_Big.linordered_semidom_class.prod_nonneg)
using ff by fastforce
moreover have "pff x \<le> 1"
using subA cardp t
- apply (simp add: pff_def divide_simps sum_nonneg)
+ apply (simp add: pff_def field_split_simps sum_nonneg)
apply (rule prod_mono [where g = "\<lambda>x. 1", simplified])
using ff by fastforce
ultimately show ?thesis
@@ -512,7 +512,7 @@
using ff [THEN conjunct2, THEN conjunct2, THEN conjunct1] v subA x
by auto
also have "... \<le> e"
- using cardp e by (simp add: divide_simps)
+ using cardp e by (simp add: field_split_simps)
finally have "pff x < e" .
}
then have "\<And>x. x \<in> A \<Longrightarrow> pff x < e"
@@ -532,7 +532,7 @@
apply (rule prod_mono_strict [where f = "\<lambda>x. 1 - e / card subA", simplified])
apply (simp_all add: subA(2))
apply (intro ballI conjI)
- using e apply (force simp: divide_simps)
+ using e apply (force simp: field_split_simps)
using ff [THEN conjunct2, THEN conjunct2, THEN conjunct2] subA B x
apply blast
done
--- a/src/HOL/Analysis/Winding_Numbers.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/Winding_Numbers.thy Wed Oct 09 14:51:54 2019 +0000
@@ -997,15 +997,17 @@
fix t :: "real"
assume t: "t \<in> {0..1}"
show "vector_derivative (exp \<circ> g) (at t) / exp (g t) = vector_derivative g (at t)"
- proof (simp add: divide_simps, rule vector_derivative_unique_at)
- show "(exp \<circ> g has_vector_derivative vector_derivative (exp \<circ> g) (at t)) (at t)"
+ proof -
+ have "(exp \<circ> g has_vector_derivative vector_derivative (exp \<circ> g) (at t)) (at t)"
by (meson DERIV_exp differentiable_def field_vector_diff_chain_at has_vector_derivative_def
has_vector_derivative_polynomial_function pfg vector_derivative_works)
- show "(exp \<circ> g has_vector_derivative vector_derivative g (at t) * exp (g t)) (at t)"
+ moreover have "(exp \<circ> g has_vector_derivative vector_derivative g (at t) * exp (g t)) (at t)"
apply (rule field_vector_diff_chain_at)
apply (metis has_vector_derivative_polynomial_function pfg vector_derivative_at)
using DERIV_exp has_field_derivative_def apply blast
done
+ ultimately show ?thesis
+ by (simp add: divide_simps, rule vector_derivative_unique_at)
qed
qed
also have "... = (pathfinish p - pathstart p) / (2 * of_real pi * \<i>)"
--- a/src/HOL/Analysis/ex/Approximations.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Analysis/ex/Approximations.thy Wed Oct 09 14:51:54 2019 +0000
@@ -114,7 +114,7 @@
from assms have "(\<lambda>k. (x^n / fact n) * (x / 2)^k) sums ((x^n / fact n) * (1 / (1 - x / 2)))"
by (intro sums_mult geometric_sums) simp_all
also from assms have "((x^n / fact n) * (1 / (1 - x / 2))) = (2 * x^n / (2 - x)) / fact n"
- by (auto simp: divide_simps)
+ by (auto simp: field_split_simps)
finally have "(\<lambda>k. (x^n / fact n) * (x / 2)^k) sums \<dots>" .
}
ultimately show "(exp x - approx) \<le> (2 * x^n / (2 - x)) / fact n"
@@ -130,7 +130,7 @@
proof -
from assms have "x^n / (2 - x) \<le> x^n / 1" by (intro frac_le) simp_all
hence "(2 * x^n / (2 - x)) / fact n \<le> 2 * x^n / fact n"
- using assms by (simp add: divide_simps)
+ using assms by (simp add: field_split_simps)
with exp_approx[of n x] assms
have "exp (x::real) - (\<Sum>k<n. x^k / fact k) \<in> {0..2 * x^n / fact n}" by simp
moreover have "(\<Sum>k\<le>n. x^k / fact k) = (\<Sum>k<n. x^k / fact k) + x ^ n / fact n"
@@ -152,7 +152,7 @@
proof -
from assms have "\<bar>exp x - (\<Sum>k\<le>n. x ^ k / fact k)\<bar> \<le> x ^ n / fact n"
by (rule exp_approx')
- also from assms have "\<dots> \<le> 1 / fact n" by (simp add: divide_simps power_le_one)
+ also from assms have "\<dots> \<le> 1 / fact n" by (simp add: field_split_simps power_le_one)
finally show ?thesis .
qed
@@ -169,7 +169,7 @@
lemma exp_1_approx:
"n > 0 \<Longrightarrow> \<bar>exp (1::real) - euler_approx n\<bar> \<le> 1 / fact n"
- using exp_approx''[of n 1] by (simp add: euler_approx_def divide_simps)
+ using exp_approx''[of n 1] by (simp add: euler_approx_def field_split_simps)
text \<open>
The following allows us to compute the numerator and the denominator of the result
@@ -282,7 +282,7 @@
also have "2 * y * (\<Sum>k<n. inverse (real (2 * k + 1)) * y\<^sup>2 ^ k) =
(\<Sum>k<n. 2 * y^(2*k+1) / (real (2 * k + 1)))"
by (subst sum_distrib_left, simp, subst power_mult)
- (simp_all add: divide_simps mult_ac power_mult)
+ (simp_all add: field_split_simps mult_ac power_mult)
finally show ?case by (simp only: d_def y_def approx_def)
qed
@@ -629,17 +629,17 @@
(is "_ \<in> {-?l..?u1 + ?u2}")
apply ((rule combine_bounds(1,2))+; (rule combine_bounds(3); (rule arctan_approx)?)?)
apply (simp_all add: n)
- apply (simp_all add: divide_simps)?
+ apply (simp_all add: field_split_simps)?
done
also {
have "?l \<le> (1/8) * (1/2)^(46*n)"
- unfolding power_mult by (intro mult_mono power_mono) (simp_all add: divide_simps)
+ unfolding power_mult by (intro mult_mono power_mono) (simp_all add: field_split_simps)
also have "\<dots> \<le> (1/2) ^ (46 * n - 1)"
- by (cases n; simp_all add: power_add divide_simps)
+ by (cases n; simp_all add: power_add field_split_simps)
finally have "?l \<le> (1/2) ^ (46 * n - 1)" .
moreover {
have "?u1 + ?u2 \<le> 4 * (1/2)^(48*n) + 1 * (1/2)^(46*n)"
- unfolding power_mult by (intro add_mono mult_mono power_mono) (simp_all add: divide_simps)
+ unfolding power_mult by (intro add_mono mult_mono power_mono) (simp_all add: field_split_simps)
also from \<open>n > 0\<close> have "4 * (1/2::real)^(48*n) \<le> (1/2)^(46*n)"
by (cases n) (simp_all add: field_simps power_add)
also from \<open>n > 0\<close> have "(1/2::real) ^ (46 * n) + 1 * (1 / 2) ^ (46 * n) = (1/2) ^ (46 * n - 1)"
@@ -650,7 +650,7 @@
by (subst atLeastatMost_subset_iff) simp_all
}
finally have "\<bar>pi - pi_approx2 n\<bar> \<le> ((1/2) ^ (46 * n - 1))" by auto
- thus ?thesis by (simp add: divide_simps)
+ thus ?thesis by (simp add: field_split_simps)
qed
lemma pi_approx2':
--- a/src/HOL/Complex.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Complex.thy Wed Oct 09 14:51:54 2019 +0000
@@ -92,8 +92,7 @@
by (simp add: divide_complex_def add_divide_distrib)
lemma Im_divide: "Im (x / y) = (Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
- unfolding divide_complex_def times_complex.sel inverse_complex.sel
- by (simp add: divide_simps)
+ by (simp add: divide_complex_def diff_divide_distrib)
lemma Complex_divide:
"(x / y) = Complex ((Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2))
@@ -184,10 +183,10 @@
by (simp add: Im_divide sqr_conv_mult)
lemma Re_divide_of_nat [simp]: "Re (z / of_nat n) = Re z / of_nat n"
- by (cases n) (simp_all add: Re_divide divide_simps power2_eq_square del: of_nat_Suc)
+ by (cases n) (simp_all add: Re_divide field_split_simps power2_eq_square del: of_nat_Suc)
lemma Im_divide_of_nat [simp]: "Im (z / of_nat n) = Im z / of_nat n"
- by (cases n) (simp_all add: Im_divide divide_simps power2_eq_square del: of_nat_Suc)
+ by (cases n) (simp_all add: Im_divide field_split_simps power2_eq_square del: of_nat_Suc)
lemma of_real_Re [simp]: "z \<in> \<real> \<Longrightarrow> of_real (Re z) = z"
by (auto simp: Reals_def)
@@ -397,7 +396,7 @@
done
lemma complex_unit_circle: "z \<noteq> 0 \<Longrightarrow> (Re z / cmod z)\<^sup>2 + (Im z / cmod z)\<^sup>2 = 1"
- by (simp add: norm_complex_def divide_simps complex_eq_iff)
+ by (simp add: norm_complex_def complex_eq_iff power2_eq_square add_divide_distrib [symmetric])
text \<open>Properties of complex signum.\<close>
@@ -1043,7 +1042,8 @@
then obtain m :: int where "2 * pi * (real k - real l) / real n = real_of_int m * 2 * pi"
by (subst (asm) cos_one_2pi_int) blast
hence "real_of_int (int k - int l) = real_of_int (m * int n)"
- unfolding of_int_diff of_int_mult using assms by (simp add: divide_simps)
+ unfolding of_int_diff of_int_mult using assms
+ by (simp add: nonzero_divide_eq_eq)
also note of_int_eq_iff
finally have *: "abs m * n = abs (int k - int l)" by (simp add: abs_mult)
also have "\<dots> < int n" using kl by linarith
--- a/src/HOL/Computational_Algebra/Computational_Algebra.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Computational_Algebra/Computational_Algebra.thy Wed Oct 09 14:51:54 2019 +0000
@@ -19,4 +19,3 @@
begin
end
-
--- a/src/HOL/Computational_Algebra/Field_as_Ring.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Computational_Algebra/Field_as_Ring.thy Wed Oct 09 14:51:54 2019 +0000
@@ -35,7 +35,7 @@
instance
by standard
- (simp_all add: dvd_field_iff divide_simps split: if_splits)
+ (simp_all add: dvd_field_iff field_split_simps split: if_splits)
end
@@ -66,7 +66,7 @@
instance
by standard
- (simp_all add: dvd_field_iff divide_simps split: if_splits)
+ (simp_all add: dvd_field_iff field_split_simps split: if_splits)
end
@@ -97,7 +97,7 @@
instance
by standard
- (simp_all add: dvd_field_iff divide_simps split: if_splits)
+ (simp_all add: dvd_field_iff field_split_simps split: if_splits)
end
--- a/src/HOL/Computational_Algebra/Formal_Laurent_Series.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Computational_Algebra/Formal_Laurent_Series.thy Wed Oct 09 14:51:54 2019 +0000
@@ -2832,26 +2832,25 @@
qed
lemma fls_lr_inverse_power_divring:
- fixes f :: "'a::division_ring fls"
- shows "fls_left_inverse (f ^ n) ((inverse (f $$ fls_subdegree f)) ^ n) =
- (fls_left_inverse f (inverse (f $$ fls_subdegree f))) ^ n"
- "fls_right_inverse (f ^ n) ((inverse (f $$ fls_subdegree f)) ^ n) =
- (fls_right_inverse f (inverse (f $$ fls_subdegree f))) ^ n"
+ "fls_left_inverse (f ^ n) ((inverse (f $$ fls_subdegree f)) ^ n) =
+ (fls_left_inverse f (inverse (f $$ fls_subdegree f))) ^ n" (is ?P)
+ and "fls_right_inverse (f ^ n) ((inverse (f $$ fls_subdegree f)) ^ n) =
+ (fls_right_inverse f (inverse (f $$ fls_subdegree f))) ^ n" (is ?Q)
+ for f :: "'a::division_ring fls"
proof -
- have
+ note fls_left_inverse_eq_inverse [of f] fls_right_inverse_eq_inverse[of f]
+ moreover have
"fls_right_inverse (f ^ n) ((inverse (f $$ fls_subdegree f)) ^ n) =
inverse f ^ n"
+ using fls_right_inverse_eq_inverse [of "f ^ n"]
+ by (simp add: fls_subdegree_pow power_inverse)
+ moreover have
"fls_left_inverse (f ^ n) ((inverse (f $$ fls_subdegree f)) ^ n) =
inverse f ^ n"
- using fls_left_inverse_eq_inverse[of "f^n"] fls_right_inverse_eq_inverse[of "f^n"]
- by (auto simp add: divide_simps fls_subdegree_pow)
- thus
- "fls_left_inverse (f ^ n) ((inverse (f $$ fls_subdegree f)) ^ n) =
- (fls_left_inverse f (inverse (f $$ fls_subdegree f))) ^ n"
- "fls_right_inverse (f ^ n) ((inverse (f $$ fls_subdegree f)) ^ n) =
- (fls_right_inverse f (inverse (f $$ fls_subdegree f))) ^ n"
- using fls_left_inverse_eq_inverse[of f] fls_right_inverse_eq_inverse[of f]
- by auto
+ using fls_left_inverse_eq_inverse [of "f ^ n"]
+ by (simp add: fls_subdegree_pow power_inverse)
+ ultimately show ?P and ?Q
+ by simp_all
qed
instance fls :: (field) field
--- a/src/HOL/Computational_Algebra/Formal_Power_Series.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Computational_Algebra/Formal_Power_Series.thy Wed Oct 09 14:51:54 2019 +0000
@@ -1489,7 +1489,7 @@
have "dist (?s n) a < r" if nn0: "n \<ge> n0" for n
proof -
from that have thnn0: "(1/2)^n \<le> (1/2 :: real)^n0"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
show ?thesis
proof (cases "?s n = a")
case True
@@ -1505,7 +1505,7 @@
then have "dist (?s n) a < (1/2)^n"
by (simp add: field_simps dist_fps_def)
also have "\<dots> \<le> (1/2)^n0"
- using nn0 by (simp add: divide_simps)
+ using nn0 by (simp add: field_split_simps)
also have "\<dots> < r"
using n0 by simp
finally show ?thesis .
--- a/src/HOL/Decision_Procs/Approximation_Bounds.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Decision_Procs/Approximation_Bounds.thy Wed Oct 09 14:51:54 2019 +0000
@@ -305,7 +305,7 @@
case (Float m e)
hence "0 < m"
using assms
- by (auto simp: sign_simps)
+ by (auto simp: algebra_split_simps)
hence "0 < sqrt m" by auto
have int_nat_bl: "(nat (bitlen m)) = bitlen m"
--- a/src/HOL/Fields.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Fields.thy Wed Oct 09 14:51:54 2019 +0000
@@ -87,7 +87,6 @@
lemmas [arith_split] = nat_diff_split split_min split_max
-
text\<open>Lemmas \<open>divide_simps\<close> move division to the outside and eliminates them on (in)equalities.\<close>
named_theorems divide_simps "rewrite rules to eliminate divisions"
@@ -198,7 +197,7 @@
lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
by (simp add: divide_inverse)
-lemma inverse_eq_divide [field_simps, divide_simps]: "inverse a = 1 / a"
+lemma inverse_eq_divide [field_simps, field_split_simps, divide_simps]: "inverse a = 1 / a"
by (simp add: divide_inverse)
lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
@@ -332,7 +331,7 @@
by (cases "b = 0 \<or> c = 0")
(auto simp: divide_inverse mult.assoc nonzero_inverse_mult_distrib)
-lemma add_divide_eq_if_simps [divide_simps]:
+lemma add_divide_eq_if_simps [field_split_simps, divide_simps]:
"a + b / z = (if z = 0 then a else (a * z + b) / z)"
"a / z + b = (if z = 0 then b else (a + b * z) / z)"
"- (a / z) + b = (if z = 0 then b else (-a + b * z) / z)"
@@ -342,7 +341,7 @@
by (simp_all add: add_divide_eq_iff divide_add_eq_iff diff_divide_eq_iff divide_diff_eq_iff
minus_divide_diff_eq_iff)
-lemma [divide_simps]:
+lemma [field_split_simps, divide_simps]:
shows divide_eq_eq: "b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)"
and eq_divide_eq: "a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)"
and minus_divide_eq_eq: "- (b / c) = a \<longleftrightarrow> (if c \<noteq> 0 then - b = a * c else a = 0)"
@@ -1096,7 +1095,7 @@
lemma inverse_le_1_iff: "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x"
by (simp add: not_less [symmetric] one_less_inverse_iff)
-lemma [divide_simps]:
+lemma [field_split_simps, divide_simps]:
shows le_divide_eq: "a \<le> b / c \<longleftrightarrow> (if 0 < c then a * c \<le> b else if c < 0 then b \<le> a * c else a \<le> 0)"
and divide_le_eq: "b / c \<le> a \<longleftrightarrow> (if 0 < c then b \<le> a * c else if c < 0 then a * c \<le> b else 0 \<le> a)"
and less_divide_eq: "a < b / c \<longleftrightarrow> (if 0 < c then a * c < b else if c < 0 then b < a * c else a < 0)"
@@ -1114,7 +1113,7 @@
and divide_less_0_iff: "a / b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
and zero_le_divide_iff: "0 \<le> a / b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
and divide_le_0_iff: "a / b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
- by (auto simp add: divide_simps)
+ by (auto simp add: field_split_simps)
text \<open>Division and the Number One\<close>
@@ -1193,19 +1192,19 @@
lemma divide_nonneg_nonneg [simp]:
"0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x / y"
- by (auto simp add: divide_simps)
+ by (auto simp add: field_split_simps)
lemma divide_nonpos_nonpos:
"x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> 0 \<le> x / y"
- by (auto simp add: divide_simps)
+ by (auto simp add: field_split_simps)
lemma divide_nonneg_nonpos:
"0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> x / y \<le> 0"
- by (auto simp add: divide_simps)
+ by (auto simp add: field_split_simps)
lemma divide_nonpos_nonneg:
"x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x / y \<le> 0"
- by (auto simp add: divide_simps)
+ by (auto simp add: field_split_simps)
text \<open>Conditional Simplification Rules: No Case Splits\<close>
--- a/src/HOL/Groups.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Groups.thy Wed Oct 09 14:51:54 2019 +0000
@@ -16,8 +16,9 @@
named_theorems ac_simps "associativity and commutativity simplification rules"
and algebra_simps "algebra simplification rules for rings"
+ and algebra_split_simps "algebra simplification rules for rings, with potential goal splitting"
and field_simps "algebra simplification rules for fields"
- and sign_simps "algebra simplification rules for comparision with zero"
+ and field_split_simps "algebra simplification rules for fields, with potential goal splitting"
text \<open>
The rewrites accumulated in \<open>algebra_simps\<close> deal with the classical
@@ -34,9 +35,9 @@
inequalities). Can be too aggressive and is therefore separate from the more
benign \<open>algebra_simps\<close>.
- Lemmas \<open>sign_simps\<close> is a first attempt to automate proofs
- of positivity/negativity needed for \<open>field_simps\<close>. Have not added \<open>sign_simps\<close> to \<open>field_simps\<close>
- because the former can lead to case explosions.
+ Collections \<open>algebra_split_simps\<close> and \<open>field_split_simps\<close>
+ correspond to \<open>algebra_simps\<close> and \<open>field_simps\<close>
+ but contain more aggresive rules that may lead to goal splitting.
\<close>
@@ -206,7 +207,8 @@
subsection \<open>Semigroups and Monoids\<close>
class semigroup_add = plus +
- assumes add_assoc [algebra_simps, field_simps]: "(a + b) + c = a + (b + c)"
+ assumes add_assoc [algebra_simps, algebra_split_simps, field_simps, field_split_simps]:
+ "(a + b) + c = a + (b + c)"
begin
sublocale add: semigroup plus
@@ -217,13 +219,14 @@
hide_fact add_assoc
class ab_semigroup_add = semigroup_add +
- assumes add_commute [algebra_simps, field_simps]: "a + b = b + a"
+ assumes add_commute [algebra_simps, algebra_split_simps, field_simps, field_split_simps]:
+ "a + b = b + a"
begin
sublocale add: abel_semigroup plus
by standard (fact add_commute)
-declare add.left_commute [algebra_simps, field_simps]
+declare add.left_commute [algebra_simps, algebra_split_simps, field_simps, field_split_simps]
lemmas add_ac = add.assoc add.commute add.left_commute
@@ -234,7 +237,8 @@
lemmas add_ac = add.assoc add.commute add.left_commute
class semigroup_mult = times +
- assumes mult_assoc [algebra_simps, field_simps]: "(a * b) * c = a * (b * c)"
+ assumes mult_assoc [algebra_simps, algebra_split_simps, field_simps, field_split_simps]:
+ "(a * b) * c = a * (b * c)"
begin
sublocale mult: semigroup times
@@ -245,13 +249,14 @@
hide_fact mult_assoc
class ab_semigroup_mult = semigroup_mult +
- assumes mult_commute [algebra_simps, field_simps]: "a * b = b * a"
+ assumes mult_commute [algebra_simps, algebra_split_simps, field_simps, field_split_simps]:
+ "a * b = b * a"
begin
sublocale mult: abel_semigroup times
by standard (fact mult_commute)
-declare mult.left_commute [algebra_simps, field_simps]
+declare mult.left_commute [algebra_simps, algebra_split_simps, field_simps, field_split_simps]
lemmas mult_ac = mult.assoc mult.commute mult.left_commute
@@ -326,7 +331,8 @@
class cancel_ab_semigroup_add = ab_semigroup_add + minus +
assumes add_diff_cancel_left' [simp]: "(a + b) - a = b"
- assumes diff_diff_add [algebra_simps, field_simps]: "a - b - c = a - (b + c)"
+ assumes diff_diff_add [algebra_simps, algebra_split_simps, field_simps, field_split_simps]:
+ "a - b - c = a - (b + c)"
begin
lemma add_diff_cancel_right' [simp]: "(a + b) - b = a"
@@ -547,19 +553,23 @@
lemma minus_diff_eq [simp]: "- (a - b) = b - a"
by (simp only: neg_eq_iff_add_eq_0 diff_conv_add_uminus add.assoc minus_add_cancel) simp
-lemma add_diff_eq [algebra_simps, field_simps]: "a + (b - c) = (a + b) - c"
+lemma add_diff_eq [algebra_simps, algebra_split_simps, field_simps, field_split_simps]:
+ "a + (b - c) = (a + b) - c"
by (simp only: diff_conv_add_uminus add.assoc)
lemma diff_add_eq_diff_diff_swap: "a - (b + c) = a - c - b"
by (simp only: diff_conv_add_uminus add.assoc minus_add)
-lemma diff_eq_eq [algebra_simps, field_simps]: "a - b = c \<longleftrightarrow> a = c + b"
+lemma diff_eq_eq [algebra_simps, algebra_split_simps, field_simps, field_split_simps]:
+ "a - b = c \<longleftrightarrow> a = c + b"
by auto
-lemma eq_diff_eq [algebra_simps, field_simps]: "a = c - b \<longleftrightarrow> a + b = c"
+lemma eq_diff_eq [algebra_simps, algebra_split_simps, field_simps, field_split_simps]:
+ "a = c - b \<longleftrightarrow> a + b = c"
by auto
-lemma diff_diff_eq2 [algebra_simps, field_simps]: "a - (b - c) = (a + c) - b"
+lemma diff_diff_eq2 [algebra_simps, algebra_split_simps, field_simps, field_split_simps]:
+ "a - (b - c) = (a + c) - b"
by (simp only: diff_conv_add_uminus add.assoc) simp
lemma diff_eq_diff_eq: "a - b = c - d \<Longrightarrow> a = b \<longleftrightarrow> c = d"
@@ -592,7 +602,8 @@
lemma minus_add_distrib [simp]: "- (a + b) = - a + - b"
by (simp add: algebra_simps)
-lemma diff_add_eq [algebra_simps, field_simps]: "(a - b) + c = (a + c) - b"
+lemma diff_add_eq [algebra_simps, algebra_split_simps, field_simps, field_split_simps]:
+ "(a - b) + c = (a + c) - b"
by (simp add: algebra_simps)
end
@@ -938,13 +949,15 @@
lemmas less_iff_diff_less_0 = diff_less_0_iff_less [symmetric]
-lemma diff_less_eq [algebra_simps, field_simps]: "a - b < c \<longleftrightarrow> a < c + b"
+lemma diff_less_eq [algebra_simps, algebra_split_simps, field_simps, field_split_simps]:
+ "a - b < c \<longleftrightarrow> a < c + b"
apply (subst less_iff_diff_less_0 [of a])
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
apply (simp add: algebra_simps)
done
-lemma less_diff_eq[algebra_simps, field_simps]: "a < c - b \<longleftrightarrow> a + b < c"
+lemma less_diff_eq [algebra_simps, algebra_split_simps, field_simps, field_split_simps]:
+ "a < c - b \<longleftrightarrow> a + b < c"
apply (subst less_iff_diff_less_0 [of "a + b"])
apply (subst less_iff_diff_less_0 [of a])
apply (simp add: algebra_simps)
@@ -953,10 +966,12 @@
lemma diff_gt_0_iff_gt [simp]: "a - b > 0 \<longleftrightarrow> a > b"
by (simp add: less_diff_eq)
-lemma diff_le_eq [algebra_simps, field_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
+lemma diff_le_eq [algebra_simps, algebra_split_simps, field_simps, field_split_simps]:
+ "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
by (auto simp add: le_less diff_less_eq )
-lemma le_diff_eq [algebra_simps, field_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
+lemma le_diff_eq [algebra_simps, algebra_split_simps, field_simps, field_split_simps]:
+ "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
by (auto simp add: le_less less_diff_eq)
lemma diff_le_0_iff_le [simp]: "a - b \<le> 0 \<longleftrightarrow> a \<le> b"
--- a/src/HOL/Homology/Simplices.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Homology/Simplices.thy Wed Oct 09 14:51:54 2019 +0000
@@ -1229,7 +1229,7 @@
by simp
with False show "(\<Sum>j\<le>p. x (Suc j) * inverse (1 - x 0)) = 1"
by (metis add_diff_cancel_left' diff_diff_eq2 diff_zero right_inverse sum_distrib_right)
- qed (use x01 x0 xle \<open>x 0 < 1\<close> in \<open>auto simp: divide_simps\<close>)
+ qed (use x01 x0 xle \<open>x 0 < 1\<close> in \<open>auto simp: field_split_simps\<close>)
then show "(\<lambda>i. inverse (1 - x 0) * (\<Sum>j\<le>p. l j i * x (Suc j))) \<in> S"
by (simp add: field_simps sum_divide_distrib)
qed (use x01 in auto)
@@ -1448,11 +1448,11 @@
have "(\<lambda>i. \<Sum>j\<le>Suc p. l j i * ((if j \<le> Suc p then (1 - u) * inverse (p + 2) + u * x j else 0))) \<in> S"
proof (rule l)
have i2p: "inverse (2 + real p) \<le> 1"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
show "(\<lambda>j. if j \<le> Suc p then (1 - u) * inverse (real (p + 2)) + u * x j else 0) \<in> standard_simplex (Suc p)"
using x \<open>0 \<le> u\<close> \<open>u \<le> 1\<close>
apply (simp add: sum.distrib standard_simplex_def i2p linepath_le_1 flip: sum_distrib_left del: sum.atMost_Suc)
- apply (simp add: divide_simps)
+ apply (simp add: field_split_simps)
done
qed
moreover have "(\<lambda>i. \<Sum>j\<le>Suc p. l j i * ((1 - u) * inverse (2 + real p) + u * x j))
@@ -1604,7 +1604,7 @@
done
show "\<bar>l i k - l i' k\<bar> / real (p + 2) \<le> (1 + real p) * d / real (p + 2) / real (card ({..Suc p} - {i}))"
using i i' lf [of i] lf [of i'] 2
- by (auto simp: divide_simps image_iff)
+ by (auto simp: image_iff divide_simps)
qed auto
finally have "(\<Sum>j\<le>Suc p. \<bar>l i k / (p + 2) - l j k / (p + 2)\<bar>) \<le> (1 + real p) * d / (p + 2)" .
then have "\<bar>\<Sum>j\<le>Suc p. l i k / (p + 2) - l j k / (p + 2)\<bar> \<le> (1 + real p) * d / (p + 2)"
@@ -1810,7 +1810,7 @@
\<in> standard_simplex (Suc p)"
by (simp add: simplicial_vertex_def standard_simplex_def del: sum.atMost_Suc)
have ss_Sp: "(\<lambda>i. (if i \<le> Suc p then 1 else 0) / (real p + 2)) \<in> standard_simplex (Suc p)"
- by (simp add: standard_simplex_def divide_simps)
+ by (simp add: standard_simplex_def field_split_simps)
obtain l where feq: "f = oriented_simplex (Suc p) l"
using one unfolding simplicial_simplex by blast
then have 3: "f (\<lambda>i. (\<Sum>j\<le>Suc p. simplicial_vertex j (restrict id (standard_simplex (Suc p))) i) / (real p + 2))
--- a/src/HOL/Library/Float.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Library/Float.thy Wed Oct 09 14:51:54 2019 +0000
@@ -492,13 +492,13 @@
qed
lemma mantissa_pos_iff: "0 < mantissa x \<longleftrightarrow> 0 < x"
- by (auto simp: mantissa_exponent sign_simps)
+ by (auto simp: mantissa_exponent algebra_split_simps)
lemma mantissa_nonneg_iff: "0 \<le> mantissa x \<longleftrightarrow> 0 \<le> x"
- by (auto simp: mantissa_exponent sign_simps)
+ by (auto simp: mantissa_exponent algebra_split_simps)
lemma mantissa_neg_iff: "0 > mantissa x \<longleftrightarrow> 0 > x"
- by (auto simp: mantissa_exponent sign_simps)
+ by (auto simp: mantissa_exponent algebra_split_simps)
lemma
fixes m e :: int
@@ -1139,7 +1139,7 @@
auto
intro!: floor_eq2
intro: powr_strict powr
- simp: powr_diff powr_add divide_simps algebra_simps)+
+ simp: powr_diff powr_add field_split_simps algebra_simps)+
finally
show ?thesis by simp
qed
--- a/src/HOL/Library/Landau_Symbols.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Library/Landau_Symbols.thy Wed Oct 09 14:51:54 2019 +0000
@@ -1833,7 +1833,7 @@
by (rule asymp_equivD)
from order_tendstoD(1)[OF this zero_less_one]
show ?th1 ?th2 ?th3
- by (eventually_elim; force simp: sgn_if divide_simps split: if_splits)+
+ by (eventually_elim; force simp: sgn_if field_split_simps split: if_splits)+
qed
lemma asymp_equiv_tendsto_transfer:
@@ -1900,7 +1900,7 @@
have ev: "eventually (\<lambda>x. g x = 0 \<longrightarrow> h x = 0) F" by eventually_elim auto
have "(\<lambda>x. f x + h x) \<sim>[F] g \<longleftrightarrow> ((\<lambda>x. ?T f g x + h x / g x) \<longlongrightarrow> 1) F"
unfolding asymp_equiv_def using ev
- by (intro tendsto_cong) (auto elim!: eventually_mono simp: divide_simps)
+ by (intro tendsto_cong) (auto elim!: eventually_mono simp: field_split_simps)
also have "\<dots> \<longleftrightarrow> ((\<lambda>x. ?T f g x + h x / g x) \<longlongrightarrow> 1 + 0) F" by simp
also have \<dots> by (intro tendsto_intros asymp_equivD assms smalloD_tendsto)
finally show "(\<lambda>x. f x + h x) \<sim>[F] g" .
@@ -2154,7 +2154,7 @@
proof eventually_elim
case (elim x)
from elim have "norm (f x - g x) \<le> norm (f x / g x - 1) * norm (g x)"
- by (subst norm_mult [symmetric]) (auto split: if_splits simp: divide_simps)
+ by (subst norm_mult [symmetric]) (auto split: if_splits simp add: algebra_simps)
also have "norm (f x / g x - 1) * norm (g x) \<le> c * norm (g x)" using elim
by (auto split: if_splits simp: mult_right_mono)
finally show ?case .
--- a/src/HOL/Library/Nonpos_Ints.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Library/Nonpos_Ints.thy Wed Oct 09 14:51:54 2019 +0000
@@ -265,7 +265,7 @@
apply (auto intro: nonpos_Reals_mult_I2)
apply (auto simp: nonpos_Reals_def)
apply (rule_tac x="r/n" in exI)
- apply (auto simp: divide_simps)
+ apply (auto simp: field_split_simps)
done
lemma nonpos_Reals_inverse_I:
@@ -288,7 +288,7 @@
apply (auto intro: nonpos_Reals_divide_I2)
apply (auto simp: nonpos_Reals_def)
apply (rule_tac x="r*n" in exI)
- apply (auto simp: divide_simps mult_le_0_iff)
+ apply (auto simp: field_split_simps mult_le_0_iff)
done
lemma nonpos_Reals_inverse_iff [simp]:
--- a/src/HOL/Limits.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Limits.thy Wed Oct 09 14:51:54 2019 +0000
@@ -285,7 +285,7 @@
with Bfun_const have "Bseq (\<lambda>x. inverse c * (c * f x))"
by (rule Bseq_mult)
with \<open>c \<noteq> 0\<close> show "Bseq f"
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
qed (intro Bseq_mult Bfun_const)
lemma Bseq_subseq: "Bseq f \<Longrightarrow> Bseq (\<lambda>x. f (g x))"
--- a/src/HOL/MacLaurin.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/MacLaurin.thy Wed Oct 09 14:51:54 2019 +0000
@@ -57,7 +57,7 @@
by (rule nonzero_divide_eq_eq[THEN iffD2]) auto
moreover
have "(n - m) * t ^ (n - Suc m) * B / fact (n - m) = B * (t ^ (n - Suc m) / fact (n - Suc m))"
- using \<open>0 < n - m\<close> by (simp add: divide_simps fact_reduce)
+ using \<open>0 < n - m\<close> by (simp add: field_split_simps fact_reduce)
ultimately show "DERIV (difg m) t :> difg (Suc m) t"
unfolding difg_def by (simp add: mult.commute)
qed
--- a/src/HOL/Modules.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Modules.thy Wed Oct 09 14:51:54 2019 +0000
@@ -48,8 +48,10 @@
spaces by replacing the scalar field by a scalar ring.\<close>
locale module =
fixes scale :: "'a::comm_ring_1 \<Rightarrow> 'b::ab_group_add \<Rightarrow> 'b" (infixr "*s" 75)
- assumes scale_right_distrib [algebra_simps]: "a *s (x + y) = a *s x + a *s y"
- and scale_left_distrib [algebra_simps]: "(a + b) *s x = a *s x + b *s x"
+ assumes scale_right_distrib [algebra_simps, algebra_split_simps]:
+ "a *s (x + y) = a *s x + a *s y"
+ and scale_left_distrib [algebra_simps, algebra_split_simps]:
+ "(a + b) *s x = a *s x + b *s x"
and scale_scale [simp]: "a *s (b *s x) = (a * b) *s x"
and scale_one [simp]: "1 *s x = x"
begin
@@ -59,7 +61,8 @@
lemma scale_zero_left [simp]: "0 *s x = 0"
and scale_minus_left [simp]: "(- a) *s x = - (a *s x)"
- and scale_left_diff_distrib [algebra_simps]: "(a - b) *s x = a *s x - b *s x"
+ and scale_left_diff_distrib [algebra_simps, algebra_split_simps]:
+ "(a - b) *s x = a *s x - b *s x"
and scale_sum_left: "(sum f A) *s x = (\<Sum>a\<in>A. (f a) *s x)"
proof -
interpret s: additive "\<lambda>a. a *s x"
@@ -72,7 +75,8 @@
lemma scale_zero_right [simp]: "a *s 0 = 0"
and scale_minus_right [simp]: "a *s (- x) = - (a *s x)"
- and scale_right_diff_distrib [algebra_simps]: "a *s (x - y) = a *s x - a *s y"
+ and scale_right_diff_distrib [algebra_simps, algebra_split_simps]:
+ "a *s (x - y) = a *s x - a *s y"
and scale_sum_right: "a *s (sum f A) = (\<Sum>x\<in>A. a *s (f x))"
proof -
interpret s: additive "\<lambda>x. a *s x"
@@ -93,7 +97,7 @@
context module
begin
-lemma [field_simps]:
+lemma [field_simps, field_split_simps]:
shows scale_left_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a + b) *s x = a *s x + b *s x"
and scale_right_distrib_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a *s (x + y) = a *s x + a *s y"
and scale_left_diff_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a - b) *s x = a *s x - b *s x"
--- a/src/HOL/Number_Theory/Fib.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Number_Theory/Fib.thy Wed Oct 09 14:51:54 2019 +0000
@@ -200,8 +200,10 @@
by (rule LIMSEQ_power_zero) (simp_all add: \<phi>_def \<psi>_def field_simps add_pos_pos)
then have "(\<lambda>n. 1 - (\<psi> / \<phi>) ^ n) \<longlonglongrightarrow> 1 - 0"
by (intro tendsto_diff tendsto_const)
- with * show ?thesis
- by (simp add: divide_simps fib_closed_form [folded \<phi>_def \<psi>_def])
+ with * have "(\<lambda>n. (\<phi> ^ n - \<psi> ^ n) / \<phi> ^ n) \<longlonglongrightarrow> 1"
+ by (simp add: field_simps)
+ then show ?thesis
+ by (simp add: fib_closed_form \<phi>_def \<psi>_def)
qed
--- a/src/HOL/Power.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Power.thy Wed Oct 09 14:51:54 2019 +0000
@@ -123,8 +123,9 @@
context comm_monoid_mult
begin
-lemma power_mult_distrib [field_simps]: "(a * b) ^ n = (a ^ n) * (b ^ n)"
- by (induct n) (simp_all add: ac_simps)
+lemma power_mult_distrib [algebra_simps, algebra_split_simps, field_simps, field_split_simps, divide_simps]:
+ "(a * b) ^ n = (a ^ n) * (b ^ n)"
+ by (induction n) (simp_all add: ac_simps)
end
@@ -378,7 +379,7 @@
begin
text \<open>Perhaps these should be simprules.\<close>
-lemma power_inverse [field_simps, divide_simps]: "inverse a ^ n = inverse (a ^ n)"
+lemma power_inverse [field_simps, field_split_simps, divide_simps]: "inverse a ^ n = inverse (a ^ n)"
proof (cases "a = 0")
case True
then show ?thesis by (simp add: power_0_left)
@@ -389,7 +390,7 @@
then show ?thesis by simp
qed
-lemma power_one_over [field_simps, divide_simps]: "(1 / a) ^ n = 1 / a ^ n"
+lemma power_one_over [field_simps, field_split_simps, divide_simps]: "(1 / a) ^ n = 1 / a ^ n"
using power_inverse [of a] by (simp add: divide_inverse)
end
@@ -397,7 +398,7 @@
context field
begin
-lemma power_divide [field_simps, divide_simps]: "(a / b) ^ n = a ^ n / b ^ n"
+lemma power_divide [field_simps, field_split_simps, divide_simps]: "(a / b) ^ n = a ^ n / b ^ n"
by (induct n) simp_all
end
--- a/src/HOL/Probability/Conditional_Expectation.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Probability/Conditional_Expectation.thy Wed Oct 09 14:51:54 2019 +0000
@@ -855,7 +855,7 @@
fixes c::real
assumes "integrable M f"
shows "AE x in M. real_cond_exp M F (\<lambda>x. f x / c) x = real_cond_exp M F f x / c"
-using real_cond_exp_cmult[of _ "1/c", OF assms] by (auto simp add: divide_simps)
+using real_cond_exp_cmult[of _ "1/c", OF assms] by (auto simp add: field_split_simps)
lemma real_cond_exp_diff [intro, simp]:
assumes [measurable]: "integrable M f" "integrable M g"
@@ -1140,7 +1140,7 @@
have "q x + phi x * (y-x) \<le> q y"
unfolding phi_def apply (rule convex_le_Inf_differential) using \<open>convex_on I q\<close> that \<open>interior I = I\<close> by auto
then have "phi x \<le> (q x - q y) / (x - y)"
- using that \<open>x < y\<close> by (auto simp add: divide_simps algebra_simps)
+ using that \<open>x < y\<close> by (auto simp add: field_split_simps algebra_simps)
moreover have "(q x - q y)/(x - y) \<le> phi y"
unfolding phi_def proof (rule cInf_greatest, auto)
fix t assume "t \<in> I" "y < t"
@@ -1172,7 +1172,7 @@
using X measurable_from_subalg[OF subalg] unfolding G_def g_def by auto
have int1: "integrable (restr_to_subalg M F) (\<lambda>x. g x * q (real_cond_exp M F X x))"
apply (rule Bochner_Integration.integrable_bound[of _ f], auto simp add: subalg \<open>integrable (restr_to_subalg M F) f\<close>)
- unfolding g_def by (auto simp add: divide_simps abs_mult algebra_simps)
+ unfolding g_def by (auto simp add: field_split_simps abs_mult algebra_simps)
have int2: "integrable M (\<lambda>x. G x * (X x - real_cond_exp M F X x))"
apply (rule Bochner_Integration.integrable_bound[of _ "\<lambda>x. \<bar>X x\<bar> + \<bar>real_cond_exp M F X x\<bar>"])
apply (auto intro!: Bochner_Integration.integrable_add integrable_abs real_cond_exp_int \<open>integrable M X\<close> AE_I2)
@@ -1208,7 +1208,7 @@
ultimately have "AE x in M. g x * real_cond_exp M F (\<lambda>x. q (X x)) x \<ge> g x * q (real_cond_exp M F X x)"
by auto
then show "AE x in M. real_cond_exp M F (\<lambda>x. q (X x)) x \<ge> q (real_cond_exp M F X x)"
- using g(1) by (auto simp add: divide_simps)
+ using g(1) by (auto simp add: field_split_simps)
qed
text \<open>Jensen's inequality does not imply that $q(E(X|F))$ is integrable, as it only proves an upper
--- a/src/HOL/Probability/Distributions.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Probability/Distributions.thy Wed Oct 09 14:51:54 2019 +0000
@@ -612,7 +612,7 @@
also have "\<dots> = (ln l - 1) / ln b"
by simp
finally show ?thesis
- by (simp add: log_def divide_simps ln_div)
+ by (simp add: log_def ln_div) (simp add: field_split_simps)
qed
subsection \<open>Uniform distribution\<close>
@@ -999,7 +999,7 @@
case (Suc k)
note step[OF this]
also have "(real (2 * k + 1 + 1) / (2::real) * ((fact k) / 2)) = (fact (Suc k)) / 2"
- by (simp add: field_simps of_nat_Suc divide_simps del: fact_Suc) (simp add: field_simps)
+ by (simp add: field_simps of_nat_Suc field_split_simps del: fact_Suc) (simp add: field_simps)
finally show ?case
by simp
qed (insert gaussian_moment_1, simp)
@@ -1230,7 +1230,7 @@
apply (intro ext nn_integral_cong)
apply (simp add: normal_density_def \<sigma>'_def[symmetric] \<tau>'_def[symmetric] sqrt mult_exp_exp)
apply (simp add: divide_simps power2_eq_square)
- apply (simp add: field_simps)
+ apply (simp add: algebra_simps)
done
also have "... =
@@ -1365,7 +1365,7 @@
by (intro arg_cong[where f="uminus"] Bochner_Integration.integral_cong)
(auto simp: normal_density_def field_simps ln_mult log_def ln_div ln_sqrt)
also have "\<dots> = - (\<integral>x. - (normal_density \<mu> \<sigma> x * (ln (2 * pi * \<sigma>\<^sup>2)) + (normal_density \<mu> \<sigma> x * (x - \<mu>)\<^sup>2) / \<sigma>\<^sup>2) / (2 * ln b) \<partial>lborel)"
- by (intro arg_cong[where f="uminus"] Bochner_Integration.integral_cong) (auto simp: divide_simps field_simps)
+ by (intro arg_cong[where f="uminus"] Bochner_Integration.integral_cong) (auto simp: field_split_simps field_simps)
also have "\<dots> = (\<integral>x. normal_density \<mu> \<sigma> x * (ln (2 * pi * \<sigma>\<^sup>2)) + (normal_density \<mu> \<sigma> x * (x - \<mu>)\<^sup>2) / \<sigma>\<^sup>2 \<partial>lborel) / (2 * ln b)"
by (simp del: minus_add_distrib)
also have "\<dots> = (ln (2 * pi * \<sigma>\<^sup>2) + 1) / (2 * ln b)"
--- a/src/HOL/Probability/Levy.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Probability/Levy.thy Wed Oct 09 14:51:54 2019 +0000
@@ -106,7 +106,7 @@
by (intro interval_lebesgue_integral_add(2) [symmetric] interval_integrable_mirror[THEN iffD2]) simp_all
also have "\<dots> = (CLBINT t=(0::real)..T. ((iexp(t * (x - a)) - iexp (-(t * (x - a)))) -
(iexp(t * (x - b)) - iexp (-(t * (x - b))))) / (\<i> * t))"
- using \<open>T \<ge> 0\<close> by (intro interval_integral_cong) (auto simp add: divide_simps)
+ using \<open>T \<ge> 0\<close> by (intro interval_integral_cong) (auto simp add: field_split_simps)
also have "\<dots> = (CLBINT t=(0::real)..T. complex_of_real(
2 * (sin (t * (x - a)) / t) - 2 * (sin (t * (x - b)) / t)))"
using \<open>T \<ge> 0\<close>
--- a/src/HOL/Probability/Probability_Mass_Function.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Probability/Probability_Mass_Function.thy Wed Oct 09 14:51:54 2019 +0000
@@ -1785,7 +1785,7 @@
assumes "A \<noteq> {}" "finite A" "\<And>x. x \<in> A \<Longrightarrow> finite (set_pmf (f x))"
shows "measure_pmf.expectation (pmf_of_set A \<bind> f) h =
(\<Sum>a\<in>A. measure_pmf.expectation (f a) h /\<^sub>R real (card A))"
- using assms by (subst pmf_expectation_bind[of A]) (auto simp: divide_simps)
+ using assms by (subst pmf_expectation_bind[of A]) (auto simp: field_split_simps)
lemma map_pmf_of_set:
assumes "finite A" "A \<noteq> {}"
@@ -1995,7 +1995,7 @@
by (simp add: indicator_def)
show "pmf (binomial_pmf (Suc n) p) k = pmf ?rhs k"
by (cases k; cases "k > n")
- (insert assms, auto simp: pmf_bind measure_pmf_single A divide_simps algebra_simps
+ (insert assms, auto simp: pmf_bind measure_pmf_single A field_split_simps algebra_simps
not_less less_eq_Suc_le [symmetric] power_diff')
qed
--- a/src/HOL/Probability/Random_Permutations.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Probability/Random_Permutations.thy Wed Oct 09 14:51:54 2019 +0000
@@ -215,7 +215,7 @@
also have "real (card A choose length xs) = fact (card A) / (fact ?n1 * fact (card A - ?n1))"
by (subst binomial_fact) (auto intro!: card_mono assms)
also have "\<dots> / fact (card A) = 1 / (fact ?n1 * fact ?n2)"
- by (simp add: divide_simps card_eq)
+ by (simp add: field_split_simps card_eq)
also have "\<dots> = pmf ?rhs (xs, ys)" using True assms by (simp add: pmf_pair)
finally show ?thesis .
next
--- a/src/HOL/Real.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Real.thy Wed Oct 09 14:51:54 2019 +0000
@@ -816,7 +816,7 @@
then have "\<exists>n. y < r * 2 ^ n"
by (metis divide_less_eq less_trans mult.commute of_nat_less_two_power that)
then show ?thesis
- by (simp add: divide_simps)
+ by (simp add: field_split_simps)
qed
have PA: "\<not> P (A n)" for n
by (induct n) (simp_all add: a)
--- a/src/HOL/Real_Asymp/Multiseries_Expansion.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Real_Asymp/Multiseries_Expansion.thy Wed Oct 09 14:51:54 2019 +0000
@@ -184,7 +184,7 @@
"convergent_powser' (msllist_of_msstream exp_series_stream) exp"
proof -
have "(\<lambda>n. lcoeff (msllist_of_msstream exp_series_stream) n * x ^ n) sums exp x" for x :: real
- using exp_converges[of x] by (simp add: mssnth_exp_series_stream divide_simps)
+ using exp_converges[of x] by (simp add: mssnth_exp_series_stream field_split_simps)
thus ?thesis by (auto intro: exI[of _ "1::real"] simp: convergent_powser'_def)
qed
@@ -617,7 +617,7 @@
show "\<exists>N. \<forall>n\<ge>N. norm (arctan_coeffs n * x ^ n) \<le> 1 * norm x ^ n"
unfolding norm_mult norm_power
by (intro exI[of _ "0::nat"] allI impI mult_right_mono)
- (simp_all add: arctan_coeffs_def divide_simps)
+ (simp_all add: arctan_coeffs_def field_split_simps)
from that show "summable (\<lambda>n. 1 * norm x ^ n)"
by (intro summable_mult summable_geometric) simp_all
qed
@@ -3334,7 +3334,7 @@
lemma expands_to_div':
assumes "basis_wf basis" "(f expands_to F) basis" "((\<lambda>x. inverse (g x)) expands_to G) basis"
shows "((\<lambda>x. f x / g x) expands_to F * G) basis"
- using expands_to_mult[OF assms] by (simp add: divide_simps)
+ using expands_to_mult[OF assms] by (simp add: field_split_simps)
lemma expands_to_basic:
assumes "basis_wf (b # basis)" "length basis = expansion_level TYPE('a::multiseries)"
@@ -4270,7 +4270,7 @@
from assms(2) have "eventually (\<lambda>x. b x > 0) at_top"
by (simp add: filterlim_at_top_dense basis_wf_Cons)
with assms(3) show "eventually (\<lambda>x. b x powr c * g x = g x / b x ^ n) at_top"
- by (auto elim!: eventually_mono simp: powr_realpow powr_minus divide_simps)
+ by (auto elim!: eventually_mono simp: powr_realpow powr_minus field_split_simps)
qed
lemma eval_monom_collapse: "c * eval_monom (c', es) basis x = eval_monom (c * c', es) basis x"
@@ -4492,7 +4492,7 @@
by (auto simp: compare_expansions_def trimmed_imp_dominant_term_nz split: cmp_result.splits)
have "(\<lambda>x. inverse c * (c' * f x)) \<sim>[at_top] (\<lambda>x. inverse c * (c * g x))"
by (rule asymp_equiv_mult[OF asymp_equiv_refl]) (rule compare_expansions_EQ[OF assms(1-6)])
- with assms(7) show ?thesis by (simp add: divide_simps)
+ with assms(7) show ?thesis by (simp add: field_split_simps)
qed
lemma compare_expansions_EQ_imp_bigtheta:
@@ -4573,7 +4573,7 @@
(auto intro: expands_to_hd'' simp: trimmed_ms_aux_MSLCons basis_wf_Cons)
have "(\<lambda>x. inverse c' * (c' * eval C x)) \<sim>[at_top] (\<lambda>x. inverse c' * (c * g x))"
by (rule asymp_equiv_mult) (insert *, simp_all)
- hence "eval C \<sim>[at_top] (\<lambda>x. c / c' * g x)" by (simp add: divide_simps)
+ hence "eval C \<sim>[at_top] (\<lambda>x. c / c' * g x)" by (simp add: field_split_simps)
finally show ?thesis .
qed
--- a/src/HOL/Real_Asymp/Multiseries_Expansion_Bounds.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Real_Asymp/Multiseries_Expansion_Bounds.thy Wed Oct 09 14:51:54 2019 +0000
@@ -411,7 +411,7 @@
(* TODO Move *)
lemma powr_mono': "a \<le> (b::real) \<Longrightarrow> x \<ge> 0 \<Longrightarrow> x \<le> 1 \<Longrightarrow> x powr b \<le> x powr a"
- using powr_mono[of "-b" "-a" "inverse x"] by (auto simp: powr_def ln_inverse ln_div divide_simps)
+ using powr_mono[of "-b" "-a" "inverse x"] by (auto simp: powr_def ln_inverse ln_div field_split_simps)
qualified lemma powr_left_bounds:
fixes f g :: "real \<Rightarrow> real"
--- a/src/HOL/Real_Asymp/Real_Asymp_Examples.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Real_Asymp/Real_Asymp_Examples.thy Wed Oct 09 14:51:54 2019 +0000
@@ -438,7 +438,7 @@
by (subst binomial_gbinomial)
(simp add: gbinomial_pochhammer' numeral_3_eq_3 pochhammer_Suc add_ac)
moreover have "\<dots> \<sim> (\<lambda>n. 3 / ln 2 * ln n)"
- by (real_asymp simp: divide_simps)
+ by (real_asymp simp: field_split_simps)
ultimately show ?thesis by simp
qed
@@ -487,7 +487,7 @@
by real_asymp
lemma "A > 1 \<Longrightarrow> (\<lambda>n. ((1 + 1 / sqrt A) / 2) powr real_of_int (2 ^ Suc n)) \<longlonglongrightarrow> 0"
- by (real_asymp simp: divide_simps add_pos_pos)
+ by (real_asymp simp: field_split_simps add_pos_pos)
lemma
assumes "c > (1 :: real)" "k \<noteq> 0"
--- a/src/HOL/Real_Vector_Spaces.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Real_Vector_Spaces.thy Wed Oct 09 14:51:54 2019 +0000
@@ -566,6 +566,30 @@
for a b :: "'a :: ordered_real_vector"
using that by (simp add: less_le_not_le neg_le_minus_divideR_eq neg_minus_divideR_le_eq)
+lemma [field_split_simps]:
+ "a = b /\<^sub>R c \<longleftrightarrow> (if c = 0 then a = 0 else c *\<^sub>R a = b)"
+ "b /\<^sub>R c = a \<longleftrightarrow> (if c = 0 then a = 0 else b = c *\<^sub>R a)"
+ "a + b /\<^sub>R c = (if c = 0 then a else (c *\<^sub>R a + b) /\<^sub>R c)"
+ "a /\<^sub>R c + b = (if c = 0 then b else (a + c *\<^sub>R b) /\<^sub>R c)"
+ "a - b /\<^sub>R c = (if c = 0 then a else (c *\<^sub>R a - b) /\<^sub>R c)"
+ "a /\<^sub>R c - b = (if c = 0 then - b else (a - c *\<^sub>R b) /\<^sub>R c)"
+ "- (a /\<^sub>R c) + b = (if c = 0 then b else (- a + c *\<^sub>R b) /\<^sub>R c)"
+ "- (a /\<^sub>R c) - b = (if c = 0 then - b else (- a - c *\<^sub>R b) /\<^sub>R c)"
+ for a b :: "'a :: real_vector"
+ by (auto simp add: field_simps)
+
+lemma [field_split_simps]:
+ "0 < c \<Longrightarrow> a \<le> b /\<^sub>R c \<longleftrightarrow> (if c > 0 then c *\<^sub>R a \<le> b else if c < 0 then b \<le> c *\<^sub>R a else a \<le> 0)"
+ "0 < c \<Longrightarrow> a < b /\<^sub>R c \<longleftrightarrow> (if c > 0 then c *\<^sub>R a < b else if c < 0 then b < c *\<^sub>R a else a < 0)"
+ "0 < c \<Longrightarrow> b /\<^sub>R c \<le> a \<longleftrightarrow> (if c > 0 then b \<le> c *\<^sub>R a else if c < 0 then c *\<^sub>R a \<le> b else a \<ge> 0)"
+ "0 < c \<Longrightarrow> b /\<^sub>R c < a \<longleftrightarrow> (if c > 0 then b < c *\<^sub>R a else if c < 0 then c *\<^sub>R a < b else a > 0)"
+ "0 < c \<Longrightarrow> a \<le> - (b /\<^sub>R c) \<longleftrightarrow> (if c > 0 then c *\<^sub>R a \<le> - b else if c < 0 then - b \<le> c *\<^sub>R a else a \<le> 0)"
+ "0 < c \<Longrightarrow> a < - (b /\<^sub>R c) \<longleftrightarrow> (if c > 0 then c *\<^sub>R a < - b else if c < 0 then - b < c *\<^sub>R a else a < 0)"
+ "0 < c \<Longrightarrow> - (b /\<^sub>R c) \<le> a \<longleftrightarrow> (if c > 0 then - b \<le> c *\<^sub>R a else if c < 0 then c *\<^sub>R a \<le> - b else a \<ge> 0)"
+ "0 < c \<Longrightarrow> - (b /\<^sub>R c) < a \<longleftrightarrow> (if c > 0 then - b < c *\<^sub>R a else if c < 0 then c *\<^sub>R a < - b else a > 0)"
+ for a b :: "'a :: ordered_real_vector"
+ by (clarsimp intro!: field_simps)+
+
lemma scaleR_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> a *\<^sub>R x"
for x :: "'a::ordered_real_vector"
using scaleR_left_mono [of 0 x a] by simp
@@ -1052,7 +1076,7 @@
shows "norm (z^m - w^m) \<le> m * norm (z - w)"
proof -
have "norm (z^m - w^m) = norm ((\<Prod> i < m. z) - (\<Prod> i < m. w))"
- by (simp add: prod_constant)
+ by simp
also have "\<dots> \<le> (\<Sum>i<m. norm (z - w))"
by (intro norm_prod_diff) (auto simp: assms)
also have "\<dots> = m * norm (z - w)"
@@ -2003,7 +2027,7 @@
have "inverse e < of_nat (nat \<lceil>inverse e + 1\<rceil>)" by linarith
also assume "n \<ge> nat \<lceil>inverse e + 1\<rceil>"
finally show "dist (1 / of_nat n :: 'a) 0 < e"
- using e by (simp add: divide_simps mult.commute norm_divide)
+ using e by (simp add: field_split_simps norm_divide)
qed
lemma (in metric_space) complete_def:
--- a/src/HOL/Rings.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Rings.thy Wed Oct 09 14:51:54 2019 +0000
@@ -16,8 +16,8 @@
subsection \<open>Semirings and rings\<close>
class semiring = ab_semigroup_add + semigroup_mult +
- assumes distrib_right [algebra_simps]: "(a + b) * c = a * c + b * c"
- assumes distrib_left [algebra_simps]: "a * (b + c) = a * b + a * c"
+ assumes distrib_right [algebra_simps, algebra_split_simps]: "(a + b) * c = a * c + b * c"
+ assumes distrib_left [algebra_simps, algebra_split_simps]: "a * (b + c) = a * b + a * c"
begin
text \<open>For the \<open>combine_numerals\<close> simproc\<close>
@@ -250,14 +250,16 @@
class comm_semiring_1_cancel =
comm_semiring + cancel_comm_monoid_add + zero_neq_one + comm_monoid_mult +
- assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
+ assumes right_diff_distrib' [algebra_simps, algebra_split_simps]:
+ "a * (b - c) = a * b - a * c"
begin
subclass semiring_1_cancel ..
subclass comm_semiring_0_cancel ..
subclass comm_semiring_1 ..
-lemma left_diff_distrib' [algebra_simps]: "(b - c) * a = b * a - c * a"
+lemma left_diff_distrib' [algebra_simps, algebra_split_simps]:
+ "(b - c) * a = b * a - c * a"
by (simp add: algebra_simps)
lemma dvd_add_times_triv_left_iff [simp]: "a dvd c * a + b \<longleftrightarrow> a dvd b"
@@ -331,10 +333,12 @@
lemma minus_mult_commute: "- a * b = a * - b"
by simp
-lemma right_diff_distrib [algebra_simps]: "a * (b - c) = a * b - a * c"
+lemma right_diff_distrib [algebra_simps, algebra_split_simps]:
+ "a * (b - c) = a * b - a * c"
using distrib_left [of a b "-c "] by simp
-lemma left_diff_distrib [algebra_simps]: "(a - b) * c = a * c - b * c"
+lemma left_diff_distrib [algebra_simps, algebra_split_simps]:
+ "(a - b) * c = a * c - b * c"
using distrib_right [of a "- b" c] by simp
lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
@@ -646,7 +650,7 @@
context semiring
begin
-lemma [field_simps]:
+lemma [field_simps, field_split_simps]:
shows distrib_left_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b + c) = a * b + a * c"
and distrib_right_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a + b) * c = a * c + b * c"
by (rule distrib_left distrib_right)+
@@ -656,7 +660,7 @@
context ring
begin
-lemma [field_simps]:
+lemma [field_simps, field_split_simps]:
shows left_diff_distrib_NO_MATCH: "NO_MATCH (x div y) c \<Longrightarrow> (a - b) * c = a * c - b * c"
and right_diff_distrib_NO_MATCH: "NO_MATCH (x div y) a \<Longrightarrow> a * (b - c) = a * b - a * c"
by (rule left_diff_distrib right_diff_distrib)+
@@ -2176,17 +2180,21 @@
by (simp add: neq_iff)
qed
-lemma zero_less_mult_iff [sign_simps]: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
+lemma zero_less_mult_iff [algebra_split_simps, field_split_simps]:
+ "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
(auto simp add: mult_neg_neg not_less le_less dest: zero_less_mult_pos zero_less_mult_pos2)
-lemma zero_le_mult_iff [sign_simps]: "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
+lemma zero_le_mult_iff [algebra_split_simps, field_split_simps]:
+ "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
-lemma mult_less_0_iff [sign_simps]: "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
+lemma mult_less_0_iff [algebra_split_simps, field_split_simps]:
+ "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
using zero_less_mult_iff [of "- a" b] by auto
-lemma mult_le_0_iff [sign_simps]: "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
+lemma mult_le_0_iff [algebra_split_simps, field_split_simps]:
+ "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
using zero_le_mult_iff [of "- a" b] by auto
text \<open>
--- a/src/HOL/Set_Interval.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Set_Interval.thy Wed Oct 09 14:51:54 2019 +0000
@@ -2247,7 +2247,7 @@
fixes x :: "'a::{comm_ring,division_ring}"
shows "(\<Sum>i\<le>n. x^i) = (if x = 1 then of_nat(n + 1) else (1 - x^Suc n) / (1 - x))"
using sum_gp_basic[of x n]
- by (simp add: mult.commute divide_simps)
+ by (simp add: mult.commute field_split_simps)
lemma sum_power_add:
fixes x :: "'a::{comm_ring,monoid_mult}"
@@ -2264,7 +2264,7 @@
lemma sum_gp_strict:
fixes x :: "'a::{comm_ring,division_ring}"
shows "(\<Sum>i<n. x^i) = (if x = 1 then of_nat n else (1 - x^n) / (1 - x))"
- by (induct n) (auto simp: algebra_simps divide_simps)
+ by (induct n) (auto simp: algebra_simps field_split_simps)
subsubsection \<open>The formulae for arithmetic sums\<close>
--- a/src/HOL/Transcendental.thy Wed Oct 09 23:00:52 2019 +0200
+++ b/src/HOL/Transcendental.thy Wed Oct 09 14:51:54 2019 +0000
@@ -102,7 +102,7 @@
also from k have "fact k = fact (Suc k) / (real k + 1)" by (simp add: field_simps)
also have "fact n / (fact (Suc k) / (real k + 1) * ((real n - real k) * fact (n - Suc k))) =
(n choose (Suc k)) * ((real k + 1) / (real n - real k))"
- using k by (simp add: divide_simps binomial_fact)
+ using k by (simp add: field_split_simps binomial_fact)
also from assms have "(real k + 1) / (real n - real k) < 1" by simp
finally show ?thesis using k by (simp add: mult_less_cancel_left)
qed
@@ -299,7 +299,7 @@
shows "(\<lambda>n. of_nat n * x ^ n) \<longlonglongrightarrow> 0"
proof -
have "norm x / (1 - norm x) \<ge> 0"
- using assms by (auto simp: divide_simps)
+ using assms by (auto simp: field_split_simps)
moreover obtain N where N: "norm x / (1 - norm x) < of_int N"
using ex_le_of_int by (meson ex_less_of_int)
ultimately have N0: "N>0"
@@ -326,7 +326,7 @@
fixes x :: "'a::{real_normed_field,banach}"
shows "1 < norm x \<Longrightarrow> ((\<lambda>n. of_nat n / x^n) \<longlongrightarrow> 0) sequentially"
using powser_times_n_limit_0 [of "inverse x"]
- by (simp add: norm_divide divide_simps)
+ by (simp add: norm_divide field_split_simps)
lemma sum_split_even_odd:
fixes f :: "nat \<Rightarrow> real"
@@ -955,7 +955,7 @@
by (intro termdiff_converges[OF norm] sums_summable[OF sums])
from norm have "eventually (\<lambda>z. z \<in> norm -` {..<K}) (nhds z)"
by (intro eventually_nhds_in_open open_vimage)
- (simp_all add: continuous_on_norm continuous_on_id)
+ (simp_all add: continuous_on_norm)
hence eq: "eventually (\<lambda>z. (\<Sum>n. c n * z^n) = f z) (nhds z)"
by eventually_elim (insert sums, simp add: sums_iff)
@@ -1534,12 +1534,14 @@
then show ?thesis by simp
next
case False
- then have [simp]: "x * of_nat n / (1 + of_nat n) / of_nat n = x / (1 + of_nat n)"
+ have [simp]: "1 + (of_nat n * of_nat n + of_nat n * 2) \<noteq> (0::'a)"
+ using of_nat_eq_iff [of "1 + n * n + n * 2" "0"]
+ by simp
+ from False have [simp]: "x * of_nat n / (1 + of_nat n) / of_nat n = x / (1 + of_nat n)"
by simp
have [simp]: "x / (1 + of_nat n) + x * of_nat n / (1 + of_nat n) = x"
- apply (simp add: divide_simps)
- using of_nat_eq_0_iff apply (fastforce simp: distrib_left)
- done
+ using of_nat_neq_0
+ by (auto simp add: field_split_simps)
show ?thesis
using Suc.IH [of "x * of_nat n / (1 + of_nat n)"] False
by (simp add: exp_add [symmetric])
@@ -3049,7 +3051,7 @@
have "((\<lambda>x. exp (f x * ln (g x))) has_derivative
(\<lambda>h. (g x powr f x) * (f' h * ln (g x) + g' h * f x / g x))) (at x within X)"
using pos
- by (auto intro!: derivative_eq_intros simp: divide_simps powr_def)
+ by (auto intro!: derivative_eq_intros simp: field_split_simps powr_def)
then show ?thesis
by (rule has_derivative_transform_within[OF _ \<open>d > 0\<close> \<open>x \<in> X\<close>]) (auto simp: powr_def dest: pos')
qed
@@ -3158,7 +3160,7 @@
proof (rule filterlim_mono_eventually)
from reals_Archimedean2 [of "\<bar>x\<bar>"] obtain n :: nat where *: "real n > \<bar>x\<bar>" ..
then have "eventually (\<lambda>n :: nat. 0 < 1 + x / real n) at_top"
- by (intro eventually_sequentiallyI [of n]) (auto simp: divide_simps)
+ by (intro eventually_sequentiallyI [of n]) (auto simp: field_split_simps)
then show "eventually (\<lambda>n. (1 + x / n) powr n = (1 + x / n) ^ n) at_top"
by (rule eventually_mono) (erule powr_realpow)
show "(\<lambda>n. (1 + x / real n) powr real n) \<longlonglongrightarrow> exp x"
@@ -3636,7 +3638,7 @@
then have "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
by (intro mult_strict_right_mono zero_less_power \<open>0 < x\<close>)
then show "0 < ?f n"
- by (simp add: divide_simps mult_ac del: mult_Suc)
+ by (simp add: ac_simps divide_less_eq)
qed
have sums: "?f sums sin x"
by (rule sin_paired [THEN sums_group]) simp
@@ -4014,12 +4016,12 @@
lemma sin_pi_divide_n_ge_0 [simp]:
assumes "n \<noteq> 0"
shows "0 \<le> sin (pi / real n)"
- by (rule sin_ge_zero) (use assms in \<open>simp_all add: divide_simps\<close>)
+ by (rule sin_ge_zero) (use assms in \<open>simp_all add: field_split_simps\<close>)
lemma sin_pi_divide_n_gt_0:
assumes "2 \<le> n"
shows "0 < sin (pi / real n)"
- by (rule sin_gt_zero) (use assms in \<open>simp_all add: divide_simps\<close>)
+ by (rule sin_gt_zero) (use assms in \<open>simp_all add: field_split_simps\<close>)
text\<open>Proof resembles that of \<open>cos_is_zero\<close> but with \<^term>\<open>pi\<close> for the upper bound\<close>
lemma cos_total:
@@ -4482,9 +4484,9 @@
lemma sincos_principal_value: "\<exists>y. (- pi < y \<and> y \<le> pi) \<and> (sin y = sin x \<and> cos y = cos x)"
apply (rule exI [where x="pi - (2 * pi) * frac ((pi - x) / (2 * pi))"])
apply (auto simp: field_simps frac_lt_1)
- apply (simp_all add: frac_def divide_simps)
+ apply (simp_all add: frac_def field_simps)
apply (simp_all add: add_divide_distrib diff_divide_distrib)
- apply (simp_all add: sin_diff cos_diff mult.assoc [symmetric] cos_integer_2pi sin_integer_2pi)
+ apply (simp_all add: sin_add cos_add mult.assoc [symmetric])
done
@@ -4810,11 +4812,11 @@
lemma cos_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> cos x = 1 / sqrt (1 + tan x ^ 2)"
using cos_gt_zero_pi [of x]
- by (simp add: divide_simps tan_def real_sqrt_divide abs_if split: if_split_asm)
+ by (simp add: field_split_simps tan_def real_sqrt_divide abs_if split: if_split_asm)
lemma sin_tan: "\<bar>x\<bar> < pi/2 \<Longrightarrow> sin x = tan x / sqrt (1 + tan x ^ 2)"
using cos_gt_zero [of "x"] cos_gt_zero [of "-x"]
- by (force simp: divide_simps tan_def real_sqrt_divide abs_if split: if_split_asm)
+ by (force simp: field_split_simps tan_def real_sqrt_divide abs_if split: if_split_asm)
lemma tan_mono_le: "-(pi/2) < x \<Longrightarrow> x \<le> y \<Longrightarrow> y < pi/2 \<Longrightarrow> tan x \<le> tan y"
using less_eq_real_def tan_monotone by auto
@@ -6450,8 +6452,23 @@
lemma cosh_ln_real: "x > 0 \<Longrightarrow> cosh (ln x :: real) = (x + inverse x) / 2"
by (simp add: cosh_def exp_minus)
-lemma tanh_ln_real: "x > 0 \<Longrightarrow> tanh (ln x :: real) = (x ^ 2 - 1) / (x ^ 2 + 1)"
- by (simp add: tanh_def sinh_ln_real cosh_ln_real divide_simps power2_eq_square)
+lemma tanh_ln_real:
+ "tanh (ln x :: real) = (x ^ 2 - 1) / (x ^ 2 + 1)" if "x > 0"
+proof -
+ from that have "(x * 2 - inverse x * 2) * (x\<^sup>2 + 1) =
+ (x\<^sup>2 - 1) * (2 * x + 2 * inverse x)"
+ by (simp add: field_simps power2_eq_square)
+ moreover have "x\<^sup>2 + 1 > 0"
+ using that by (simp add: ac_simps add_pos_nonneg)
+ moreover have "2 * x + 2 * inverse x > 0"
+ using that by (simp add: add_pos_pos)
+ ultimately have "(x * 2 - inverse x * 2) /
+ (2 * x + 2 * inverse x) =
+ (x\<^sup>2 - 1) / (x\<^sup>2 + 1)"
+ by (simp add: frac_eq_eq)
+ with that show ?thesis
+ by (simp add: tanh_def sinh_ln_real cosh_ln_real)
+qed
lemma has_field_derivative_scaleR_right [derivative_intros]:
"(f has_field_derivative D) F \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_field_derivative (c *\<^sub>R D)) F"
@@ -6470,7 +6487,7 @@
lemma has_field_derivative_tanh [THEN DERIV_chain2, derivative_intros]:
"cosh x \<noteq> 0 \<Longrightarrow> (tanh has_field_derivative 1 - tanh x ^ 2)
(at (x :: 'a :: {banach, real_normed_field}))"
- unfolding tanh_def by (auto intro!: derivative_eq_intros simp: power2_eq_square divide_simps)
+ unfolding tanh_def by (auto intro!: derivative_eq_intros simp: power2_eq_square field_split_simps)
lemma has_derivative_sinh [derivative_intros]:
fixes g :: "'a \<Rightarrow> ('a :: {banach, real_normed_field})"
@@ -6560,8 +6577,21 @@
by (simp add: sinh_def cosh_def algebra_simps scaleR_conv_of_real exp_add [symmetric])
lemma tanh_add:
- "cosh x \<noteq> 0 \<Longrightarrow> cosh y \<noteq> 0 \<Longrightarrow> tanh (x + y) = (tanh x + tanh y) / (1 + tanh x * tanh y)"
- by (simp add: tanh_def sinh_add cosh_add divide_simps)
+ "tanh (x + y) = (tanh x + tanh y) / (1 + tanh x * tanh y)"
+ if "cosh x \<noteq> 0" "cosh y \<noteq> 0"
+proof -
+ have "(sinh x * cosh y + cosh x * sinh y) * (1 + sinh x * sinh y / (cosh x * cosh y)) =
+ (cosh x * cosh y + sinh x * sinh y) * ((sinh x * cosh y + sinh y * cosh x) / (cosh y * cosh x))"
+ using that by (simp add: field_split_simps)
+ also have "(sinh x * cosh y + sinh y * cosh x) / (cosh y * cosh x) = sinh x / cosh x + sinh y / cosh y"
+ using that by (simp add: field_split_simps)
+ finally have "(sinh x * cosh y + cosh x * sinh y) * (1 + sinh x * sinh y / (cosh x * cosh y)) =
+ (sinh x / cosh x + sinh y / cosh y) * (cosh x * cosh y + sinh x * sinh y)"
+ by simp
+ then show ?thesis
+ using that by (auto simp add: tanh_def sinh_add cosh_add eq_divide_eq)
+ (simp_all add: field_split_simps)
+qed
lemma sinh_double: "sinh (2 * x) = 2 * sinh x * cosh x"
using sinh_add[of x] by simp
@@ -6658,7 +6688,7 @@
have "arsinh (-x) = ln (sqrt (x\<^sup>2 + 1) - x)"
by (simp add: arsinh_real_def)
also have "sqrt (x^2 + 1) - x = inverse (sqrt (x^2 + 1) + x)"
- using arsinh_real_aux[of x] by (simp add: divide_simps algebra_simps power2_eq_square)
+ using arsinh_real_aux[of x] by (simp add: field_split_simps algebra_simps power2_eq_square)
also have "ln \<dots> = -arsinh x"
using arsinh_real_aux[of x] by (simp add: arsinh_real_def ln_inverse)
finally show ?thesis .
@@ -6680,7 +6710,7 @@
lemma tanh_real_gt_neg1: "tanh (x :: real) > -1"
proof -
- have "- cosh x < sinh x" by (simp add: sinh_def cosh_def divide_simps)
+ have "- cosh x < sinh x" by (simp add: sinh_def cosh_def field_split_simps)
thus ?thesis by (simp add: tanh_def field_simps)
qed
@@ -6700,7 +6730,7 @@
lemma artanh_tanh_real: "artanh (tanh x) = x"
proof -
have "artanh (tanh x) = ln (cosh x * (cosh x + sinh x) / (cosh x * (cosh x - sinh x))) / 2"
- by (simp add: artanh_def tanh_def divide_simps)
+ by (simp add: artanh_def tanh_def field_split_simps)
also have "cosh x * (cosh x + sinh x) / (cosh x * (cosh x - sinh x)) =
(cosh x + sinh x) / (cosh x - sinh x)" by simp
also have "\<dots> = (exp x)^2"
@@ -6921,7 +6951,7 @@
proof -
have pos: "1 + x ^ 2 > 0" by (intro add_pos_nonneg) auto
from pos arsinh_real_aux[of x] show ?thesis unfolding arsinh_def [abs_def]
- by (auto intro!: derivative_eq_intros simp: powr_minus powr_half_sqrt divide_simps)
+ by (auto intro!: derivative_eq_intros simp: powr_minus powr_half_sqrt field_split_simps)
qed
lemma arcosh_real_has_field_derivative [derivative_intros]:
@@ -6932,21 +6962,21 @@
from assms have "x + sqrt (x\<^sup>2 - 1) > 0" by (simp add: add_pos_pos)
thus ?thesis using assms unfolding arcosh_def [abs_def]
by (auto intro!: derivative_eq_intros
- simp: powr_minus powr_half_sqrt divide_simps power2_eq_1_iff)
+ simp: powr_minus powr_half_sqrt field_split_simps power2_eq_1_iff)
qed
lemma artanh_real_has_field_derivative [derivative_intros]:
- fixes x :: real
- assumes "abs x < 1"
- shows "(artanh has_field_derivative (1 / (1 - x ^ 2))) (at x within A)"
-proof -
- from assms have "x > -1" "x < 1" by linarith+
+ "(artanh has_field_derivative (1 / (1 - x ^ 2))) (at x within A)" if
+ "\<bar>x\<bar> < 1" for x :: real
+proof -
+ from that have "- 1 < x" "x < 1" by linarith+
hence "(artanh has_field_derivative (4 - 4 * x) / ((1 + x) * (1 - x) * (1 - x) * 4))
(at x within A)" unfolding artanh_def [abs_def]
by (auto intro!: derivative_eq_intros simp: powr_minus powr_half_sqrt)
also have "(4 - 4 * x) / ((1 + x) * (1 - x) * (1 - x) * 4) = 1 / ((1 + x) * (1 - x))"
- by (simp add: divide_simps)
- also have "(1 + x) * (1 - x) = 1 - x ^ 2" by (simp add: algebra_simps power2_eq_square)
+ using \<open>-1 < x\<close> \<open>x < 1\<close> by (simp add: frac_eq_eq)
+ also have "(1 + x) * (1 - x) = 1 - x ^ 2"
+ by (simp add: algebra_simps power2_eq_square)
finally show ?thesis .
qed