author paulson Wed, 11 Jul 2018 09:14:21 +0100 changeset 68612 ffc3fd6c7498 parent 68610 4fdc9f681479 (current diff) parent 68611 4bc4b5c0ccfc (diff) child 68613 2fae3e01a2ec
merged
```--- a/src/HOL/Analysis/Inner_Product.thy	Wed Jul 11 01:04:23 2018 +0200
+++ b/src/HOL/Analysis/Inner_Product.thy	Wed Jul 11 09:14:21 2018 +0100
@@ -153,7 +153,7 @@
unfolding norm_eq_sqrt_inner by simp
qed
have "sqrt (a\<^sup>2 * inner x x) = \<bar>a\<bar> * sqrt (inner x x)"
then show "norm (a *\<^sub>R x) = \<bar>a\<bar> * norm x"
unfolding norm_eq_sqrt_inner
```--- a/src/HOL/Analysis/Product_Vector.thy	Wed Jul 11 01:04:23 2018 +0200
+++ b/src/HOL/Analysis/Product_Vector.thy	Wed Jul 11 09:14:21 2018 +0100
@@ -297,7 +297,7 @@
unfolding norm_prod_def
done
show "sgn x = scaleR (inverse (norm x)) x"
by (rule sgn_prod_def)```
```--- a/src/HOL/Deriv.thy	Wed Jul 11 01:04:23 2018 +0200
+++ b/src/HOL/Deriv.thy	Wed Jul 11 09:14:21 2018 +0100
@@ -1695,7 +1695,7 @@
assumes der: "DERIV f (g x) :> D"
and neq: "D \<noteq> 0"
and x: "a < x" "x < b"
-    and inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
+    and inj: "\<And>y. \<lbrakk>a < y; y < b\<rbrakk> \<Longrightarrow> f (g y) = y"
and cont: "isCont g x"
shows "DERIV g x :> inverse D"
unfolding DERIV_iff2```
```--- a/src/HOL/Limits.thy	Wed Jul 11 01:04:23 2018 +0200
+++ b/src/HOL/Limits.thy	Wed Jul 11 09:14:21 2018 +0100
@@ -1030,13 +1030,6 @@
shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
using assms unfolding continuous_within by (rule tendsto_inverse)

-lemma continuous_at_inverse[continuous_intros, simp]:
-  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
-  assumes "isCont f a"
-    and "f a \<noteq> 0"
-  shows "isCont (\<lambda>x. inverse (f x)) a"
-  using assms unfolding continuous_at by (rule tendsto_inverse)
-
lemma continuous_on_inverse[continuous_intros]:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
assumes "continuous_on s f"
@@ -1135,14 +1128,14 @@
shows   "filterlim (\<lambda>x. norm (f x)) at_top F"
proof -
{
-    fix r :: real
-    have "\<forall>\<^sub>F x in F. r \<le> norm (f x)" using filterlim_at_infinity[of 0 f F] assms
-      by (cases "r > 0")
+    fix r :: real
+    have "\<forall>\<^sub>F x in F. r \<le> norm (f x)" using filterlim_at_infinity[of 0 f F] assms
+      by (cases "r > 0")
(auto simp: not_less intro: always_eventually order.trans[OF _ norm_ge_zero])
}
thus ?thesis by (auto simp: filterlim_at_top)
qed
-
+
lemma filterlim_norm_at_top_imp_at_infinity:
fixes F
assumes "filterlim (\<lambda>x. norm (f x)) at_top F"
@@ -1203,7 +1196,7 @@
lemma filterlim_of_real_at_infinity [tendsto_intros]:
"filterlim (of_real :: real \<Rightarrow> 'a :: real_normed_algebra_1) at_infinity at_top"
by (intro filterlim_norm_at_top_imp_at_infinity) (auto simp: filterlim_abs_real)
-
+
lemma not_tendsto_and_filterlim_at_infinity:
fixes c :: "'a::real_normed_vector"
assumes "F \<noteq> bot"
@@ -1347,15 +1340,15 @@
unfolding filterlim_at_bot eventually_at_top_dense
by (metis leI less_minus_iff order_less_asym)

-lemma at_bot_mirror :
-  shows "(at_bot::('a::{ordered_ab_group_add,linorder} filter)) = filtermap uminus at_top"
+lemma at_bot_mirror :
+  shows "(at_bot::('a::{ordered_ab_group_add,linorder} filter)) = filtermap uminus at_top"
apply (rule filtermap_fun_inverse[of uminus, symmetric])
subgoal unfolding filterlim_at_top eventually_at_bot_linorder using le_minus_iff by auto
subgoal unfolding filterlim_at_bot eventually_at_top_linorder using minus_le_iff by auto
by auto

-lemma at_top_mirror :
-  shows "(at_top::('a::{ordered_ab_group_add,linorder} filter)) = filtermap uminus at_bot"
+lemma at_top_mirror :
+  shows "(at_top::('a::{ordered_ab_group_add,linorder} filter)) = filtermap uminus at_bot"
apply (subst at_bot_mirror)

@@ -1553,7 +1546,7 @@
by (auto intro: filterlim_at_infinity_imp_eventually_ne tendsto_imp_eventually_ne eventually_conj)
then show ?thesis
by (subst filterlim_inverse_at_iff[symmetric]) simp_all
-qed
+qed

lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) \<longlongrightarrow> 0) F"
by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
@@ -1734,7 +1727,7 @@
assumes "filterlim f at_infinity F" "n > 0"
shows   "filterlim (\<lambda>x. f x ^ n) at_infinity F"
by (rule filterlim_norm_at_top_imp_at_infinity)
-     (auto simp: norm_power intro!: filterlim_pow_at_top assms
+     (auto simp: norm_power intro!: filterlim_pow_at_top assms
intro: filterlim_at_infinity_imp_norm_at_top)

@@ -2560,7 +2553,7 @@

lemma isUCont_Cauchy: "isUCont f \<Longrightarrow> Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
by (rule uniformly_continuous_on_Cauchy[where S=UNIV and f=f]) simp_all
-
+
lemma uniformly_continuous_imp_Cauchy_continuous:
fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
shows "\<lbrakk>uniformly_continuous_on S f; Cauchy \<sigma>; \<And>n. (\<sigma> n) \<in> S\<rbrakk> \<Longrightarrow> Cauchy(f \<circ> \<sigma>)"
@@ -2824,8 +2817,8 @@
lemma isCont_inverse_function:
fixes f g :: "real \<Rightarrow> real"
assumes d: "0 < d"
-    and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
-    and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
+    and inj: "\<And>z. \<bar>z-x\<bar> \<le> d \<Longrightarrow> g (f z) = z"
+    and cont: "\<And>z. \<bar>z-x\<bar> \<le> d \<Longrightarrow> isCont f z"
shows "isCont g (f x)"
proof -
let ?A = "f (x - d)"
@@ -2854,20 +2847,13 @@
lemma isCont_inverse_function2:
fixes f g :: "real \<Rightarrow> real"
shows
-    "a < x \<Longrightarrow> x < b \<Longrightarrow>
-      \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z \<Longrightarrow>
-      \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z \<Longrightarrow> isCont g (f x)"
+    "\<lbrakk>a < x; x < b;
+      \<And>z. \<lbrakk>a \<le> z; z \<le> b\<rbrakk> \<Longrightarrow> g (f z) = z;
+      \<And>z. \<lbrakk>a \<le> z; z \<le> b\<rbrakk> \<Longrightarrow> isCont f z\<rbrakk> \<Longrightarrow> isCont g (f x)"
apply (rule isCont_inverse_function [where f=f and d="min (x - a) (b - x)"])
done

-(* need to rename second continuous_at_inverse *)
-lemma isCont_inv_fun:
-  fixes f g :: "real \<Rightarrow> real"
-  shows "0 < d \<Longrightarrow> (\<forall>z. \<bar>z - x\<bar> \<le> d \<longrightarrow> g (f z) = z) \<Longrightarrow>
-    \<forall>z. \<bar>z - x\<bar> \<le> d \<longrightarrow> isCont f z \<Longrightarrow> isCont g (f x)"
-  by (rule isCont_inverse_function)
-
text \<open>Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110.\<close>
lemma LIM_fun_gt_zero: "f \<midarrow>c\<rightarrow> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> 0 < f x)"
for f :: "real \<Rightarrow> real"```
```--- a/src/HOL/Nonstandard_Analysis/HLim.thy	Wed Jul 11 01:04:23 2018 +0200
+++ b/src/HOL/Nonstandard_Analysis/HLim.thy	Wed Jul 11 09:14:21 2018 +0100
@@ -234,7 +234,7 @@

lemma isNSCont_inverse: "isNSCont f x \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> isNSCont (\<lambda>x. inverse (f x)) x"
for f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
-  by (auto intro: continuous_at_inverse simp add: isNSCont_isCont_iff)
+  using NSLIM_inverse NSLIM_isNSCont isNSCont_NSLIM by blast

lemma isNSCont_const [simp]: "isNSCont (\<lambda>x. k) a"
```--- a/src/HOL/Nonstandard_Analysis/HTranscendental.thy	Wed Jul 11 01:04:23 2018 +0200
+++ b/src/HOL/Nonstandard_Analysis/HTranscendental.thy	Wed Jul 11 09:14:21 2018 +0100
@@ -60,7 +60,7 @@
"!!x y. [|0 < x; 0 <y |] ==>
( *f* sqrt)(x*y) = ( *f* sqrt)(x) * ( *f* sqrt)(y)"
apply transfer
-apply (auto intro: real_sqrt_mult_distrib)
+apply (auto intro: real_sqrt_mult)
done

lemma hypreal_sqrt_mult_distrib2:```
```--- a/src/HOL/NthRoot.thy	Wed Jul 11 01:04:23 2018 +0200
+++ b/src/HOL/NthRoot.thy	Wed Jul 11 09:14:21 2018 +0100
@@ -314,34 +314,28 @@
using x .
show "x < x + 1"
by simp
-  show "\<forall>y. 0 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y"
-    using n by simp
show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
by (rule DERIV_pow)
show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
using n x by simp
show "isCont (root n) x"
by (rule isCont_real_root)
-qed
+qed (use n in auto)

lemma DERIV_odd_real_root:
assumes n: "odd n"
and x: "x \<noteq> 0"
shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
proof (rule DERIV_inverse_function)
-  show "x - 1 < x"
-    by simp
-  show "x < x + 1"
-    by simp
-  show "\<forall>y. x - 1 < y \<and> y < x + 1 \<longrightarrow> root n y ^ n = y"
-    using n by (simp add: odd_real_root_pow)
+  show "x - 1 < x" "x < x + 1"
+    by auto
show "DERIV (\<lambda>x. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
by (rule DERIV_pow)
show "real n * root n x ^ (n - Suc 0) \<noteq> 0"
using odd_pos [OF n] x by simp
show "isCont (root n) x"
by (rule isCont_real_root)
-qed
+qed (use n odd_real_root_pow in auto)

lemma DERIV_even_real_root:
assumes n: "0 < n"
@@ -353,10 +347,10 @@
by simp
show "x < 0"
using x .
-  show "\<forall>y. x - 1 < y \<and> y < 0 \<longrightarrow> - (root n y ^ n) = y"
-  proof (rule allI, rule impI, erule conjE)
-    fix y assume "x - 1 < y" and "y < 0"
-    then have "root n (-y) ^ n = -y" using \<open>0 < n\<close> by simp
+  show "- (root n y ^ n) = y" if "x - 1 < y" and "y < 0" for y
+  proof -
+    have "root n (-y) ^ n = -y"
+      using that \<open>0 < n\<close> by simp
with real_root_minus and \<open>even n\<close>
show "- (root n y ^ n) = y" by simp
qed
@@ -816,8 +810,6 @@
lemmas real_root_pos2 = real_root_power_cancel
lemmas real_root_pos_pos = real_root_gt_zero [THEN order_less_imp_le]
lemmas real_root_pos_pos_le = real_root_ge_zero
-lemmas real_sqrt_mult_distrib = real_sqrt_mult
-lemmas real_sqrt_mult_distrib2 = real_sqrt_mult
lemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iff

end```
```--- a/src/HOL/Power.thy	Wed Jul 11 01:04:23 2018 +0200
+++ b/src/HOL/Power.thy	Wed Jul 11 09:14:21 2018 +0100
@@ -745,6 +745,12 @@
lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> \<bar>x\<bar> < 1"
using  abs_square_eq_1 [of x] abs_square_le_1 [of x] by (auto simp add: le_less)

+lemma square_le_1:
+  assumes "- 1 \<le> x" "x \<le> 1"
+  shows "x\<^sup>2 \<le> 1"
+    using assms
+    by (metis add.inverse_inverse linear mult_le_one neg_equal_0_iff_equal neg_le_iff_le power2_eq_square power_minus_Bit0)
+
end

```
```--- a/src/HOL/Transcendental.thy	Wed Jul 11 01:04:23 2018 +0200
+++ b/src/HOL/Transcendental.thy	Wed Jul 11 09:14:21 2018 +0100
@@ -10,7 +10,7 @@
imports Series Deriv NthRoot
begin

-text \<open>A fact theorem on reals.\<close>
+text \<open>A theorem about the factcorial function on the reals.\<close>

lemma square_fact_le_2_fact: "fact n * fact n \<le> (fact (2 * n) :: real)"
proof (induct n)
@@ -1822,7 +1822,7 @@
proof (cases "0 < x")
case True
then have "isCont ln (exp (ln x))"
-    by (intro isCont_inv_fun[where d = "\<bar>x\<bar>" and f = exp]) auto
+    by (intro isCont_inverse_function[where d = "\<bar>x\<bar>" and f = exp]) auto
with True show ?thesis
by simp
next
@@ -4540,6 +4540,9 @@
unfolding tan_def
by (auto intro!: derivative_eq_intros, simp add: divide_inverse power2_eq_square)

+declare DERIV_tan[THEN DERIV_chain2, derivative_intros]
+  and DERIV_tan[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
+
lemmas has_derivative_tan[derivative_intros] = DERIV_tan[THEN DERIV_compose_FDERIV]

lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
@@ -4633,20 +4636,42 @@
by (rule_tac x="-x" in exI) auto
qed

-lemma tan_total: "\<exists>! x. -(pi/2) < x \<and> x < (pi/2) \<and> tan x = y"
-  apply (insert lemma_tan_total1 [where y = y], auto)
-  apply hypsubst_thin
-  apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
-   apply (subgoal_tac [2] "\<exists>z. y < z \<and> z < xa \<and> DERIV tan z :> 0")
-    apply (subgoal_tac "\<exists>z. xa < z \<and> z < y \<and> DERIV tan z :> 0")
-     apply (rule_tac [4] Rolle)
-        apply (rule_tac [2] Rolle)
-           apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
-       apply (rule_tac [!] DERIV_tan asm_rl)
-       apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
-        simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
-  done
+proposition tan_total: "\<exists>! x. -(pi/2) < x \<and> x < (pi/2) \<and> tan x = y"
+proof -
+  have "u = v" if u: "- (pi / 2) < u" "u < pi / 2" and v: "- (pi / 2) < v" "v < pi / 2"
+    and eq: "tan u = tan v" for u v
+  proof (cases u v rule: linorder_cases)
+    case less
+    have "\<And>x. u \<le> x \<and> x \<le> v \<longrightarrow> isCont tan x"
+      by (metis cos_gt_zero_pi isCont_tan less_numeral_extra(3) less_trans order.not_eq_order_implies_strict u v)
+    moreover have "\<And>x. u < x \<and> x < v \<Longrightarrow> tan differentiable (at x)"
+      by (metis DERIV_tan cos_gt_zero_pi differentiableI less_numeral_extra(3) order.strict_trans u(1) v(2))
+    ultimately obtain z where "u < z" "z < v" "DERIV tan z :> 0"
+      by (metis less Rolle eq)
+    moreover have "cos z \<noteq> 0"
+      by (metis (no_types) \<open>u < z\<close> \<open>z < v\<close> cos_gt_zero_pi less_le_trans linorder_not_less not_less_iff_gr_or_eq u(1) v(2))
+    ultimately show ?thesis
+      using DERIV_unique [OF _ DERIV_tan] by fastforce
+  next
+    case greater
+    have "\<And>x. v \<le> x \<and> x \<le> u \<Longrightarrow> isCont tan x"
+      by (metis cos_gt_zero_pi isCont_tan less_numeral_extra(3) less_trans order.not_eq_order_implies_strict u v)
+    moreover have "\<And>x. v < x \<and> x < u \<Longrightarrow> tan differentiable (at x)"
+      by (metis DERIV_tan cos_gt_zero_pi differentiableI less_numeral_extra(3) order.strict_trans u(2) v(1))
+    ultimately obtain z where "v < z" "z < u" "DERIV tan z :> 0"
+      by (metis greater Rolle eq)
+    moreover have "cos z \<noteq> 0"
+      by (metis  \<open>v < z\<close> \<open>z < u\<close> cos_gt_zero_pi less_le_trans linorder_not_less not_less_iff_gr_or_eq u(2) v(1))
+    ultimately show ?thesis
+      using DERIV_unique [OF _ DERIV_tan] by fastforce
+  qed auto
+  then have "\<exists>!x. - (pi / 2) < x \<and> x < pi / 2 \<and> tan x = y"
+    if x: "- (pi / 2) < x" "x < pi / 2" "tan x = y" for x
+    using that by auto
+  then show ?thesis
+    using lemma_tan_total1 [where y = y]
+    by auto
+qed

lemma tan_monotone:
assumes "- (pi/2) < y" and "y < x" and "x < pi/2"
@@ -4898,21 +4923,21 @@
lemma arcsin_ubound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2"
by (blast dest: arcsin)

-lemma arcsin_lt_bounded: "- 1 < y \<Longrightarrow> y < 1 \<Longrightarrow> - (pi/2) < arcsin y \<and> arcsin y < pi/2"
-  apply (frule order_less_imp_le)
-  apply (frule_tac y = y in order_less_imp_le)
-  apply (frule arcsin_bounded, safe)
-    apply simp
-   apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
-   apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
-   apply (drule_tac [!] f = sin in arg_cong, auto)
-  done
+lemma arcsin_lt_bounded:
+  assumes "- 1 < y" "y < 1"
+  shows  "- (pi/2) < arcsin y \<and> arcsin y < pi/2"
+proof -
+  have "arcsin y \<noteq> pi/2"
+    by (metis arcsin assms not_less not_less_iff_gr_or_eq sin_pi_half)
+  moreover have "arcsin y \<noteq> - pi/2"
+    by (metis arcsin assms minus_divide_left not_less not_less_iff_gr_or_eq sin_minus sin_pi_half)
+  ultimately show ?thesis
+    using arcsin_bounded [of y] assms by auto
+qed

lemma arcsin_sin: "- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> arcsin (sin x) = x"
-  apply (unfold arcsin_def)
-  apply (rule the1_equality)
-   apply (rule sin_total, auto)
-  done
+  unfolding arcsin_def
+  using the1_equality [OF sin_total]  by simp

lemma arcsin_0 [simp]: "arcsin 0 = 0"
using arcsin_sin [of 0] by simp
@@ -4947,14 +4972,18 @@
lemma arccos_ubound: "- 1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> pi"
by (blast dest: arccos)

-lemma arccos_lt_bounded: "- 1 < y \<Longrightarrow> y < 1 \<Longrightarrow> 0 < arccos y \<and> arccos y < pi"
-  apply (frule order_less_imp_le)
-  apply (frule_tac y = y in order_less_imp_le)
-  apply (frule arccos_bounded, auto)
-   apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
-   apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
-   apply (drule_tac [!] f = cos in arg_cong, auto)
-  done
+lemma arccos_lt_bounded:
+  assumes "- 1 < y" "y < 1"
+  shows  "0 < arccos y \<and> arccos y < pi"
+proof -
+  have "arccos y \<noteq> 0"
+    by (metis (no_types) arccos assms(1) assms(2) cos_zero less_eq_real_def less_irrefl)
+  moreover have "arccos y \<noteq> -pi"
+    by (metis arccos assms(1) assms(2) cos_minus cos_pi not_less not_less_iff_gr_or_eq)
+  ultimately show ?thesis
+    using arccos_bounded [of y] assms
+    by (metis arccos cos_pi not_less not_less_iff_gr_or_eq)
+qed

lemma arccos_cos: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> arccos (cos x) = x"
by (auto simp: arccos_def intro!: the1_equality cos_total)
@@ -4962,31 +4991,29 @@
lemma arccos_cos2: "x \<le> 0 \<Longrightarrow> - pi \<le> x \<Longrightarrow> arccos (cos x) = -x"
by (auto simp: arccos_def intro!: the1_equality cos_total)

-lemma cos_arcsin: "- 1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<^sup>2)"
-  apply (subgoal_tac "x\<^sup>2 \<le> 1")
-   apply (rule power2_eq_imp_eq)
-    apply (rule cos_ge_zero)
-     apply (erule (1) arcsin_lbound)
-    apply (erule (1) arcsin_ubound)
-   apply simp
-  apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
-  apply (rule power_mono, simp)
-  apply simp
-  done
-
-lemma sin_arccos: "- 1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<^sup>2)"
-  apply (subgoal_tac "x\<^sup>2 \<le> 1")
-   apply (rule power2_eq_imp_eq)
-    apply (rule sin_ge_zero)
-     apply (erule (1) arccos_lbound)
-    apply (erule (1) arccos_ubound)
-   apply simp
-  apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
-  apply (rule power_mono, simp)
-  apply simp
-  done
+lemma cos_arcsin:
+  assumes "- 1 \<le> x" "x \<le> 1"
+  shows "cos (arcsin x) = sqrt (1 - x\<^sup>2)"
+proof (rule power2_eq_imp_eq)
+  show "(cos (arcsin x))\<^sup>2 = (sqrt (1 - x\<^sup>2))\<^sup>2"
+    by (simp add: square_le_1 assms cos_squared_eq)
+  show "0 \<le> cos (arcsin x)"
+    using arcsin assms cos_ge_zero by blast
+  show "0 \<le> sqrt (1 - x\<^sup>2)"
+    by (simp add: square_le_1 assms)
+qed
+
+lemma sin_arccos:
+  assumes "- 1 \<le> x" "x \<le> 1"
+  shows "sin (arccos x) = sqrt (1 - x\<^sup>2)"
+proof (rule power2_eq_imp_eq)
+  show "(sin (arccos x))\<^sup>2 = (sqrt (1 - x\<^sup>2))\<^sup>2"
+    by (simp add: square_le_1 assms sin_squared_eq)
+  show "0 \<le> sin (arccos x)"
+    by (simp add: arccos_bounded assms sin_ge_zero)
+  show "0 \<le> sqrt (1 - x\<^sup>2)"
+    by (simp add: square_le_1 assms)
+qed

lemma arccos_0 [simp]: "arccos 0 = pi/2"
by (metis arccos_cos cos_gt_zero cos_pi cos_pi_half pi_gt_zero
@@ -5063,10 +5090,7 @@

lemma tan_sec: "cos x \<noteq> 0 \<Longrightarrow> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
for x :: "'a::{real_normed_field,banach,field}"
-  apply (rule power_inverse [THEN subst])
-  apply (rule_tac c1 = "(cos x)\<^sup>2" in mult_right_cancel [THEN iffD1])
-   apply (auto simp: tan_def field_simps)
-  done

lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)
@@ -5141,14 +5165,21 @@
by (auto simp: continuous_on_eq_continuous_at subset_eq)

lemma isCont_arctan: "isCont arctan x"
-  apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
-  apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
-  apply (subgoal_tac "isCont arctan (tan (arctan x))")
-  apply (erule (1) isCont_inverse_function2 [where f=tan])
-   apply (metis arctan_tan order_le_less_trans order_less_le_trans)
-  apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
-  done
+proof -
+  obtain u where u: "- (pi / 2) < u" "u < arctan x"
+    by (meson arctan arctan_less_iff linordered_field_no_lb)
+  obtain v where v: "arctan x < v" "v < pi / 2"
+    by (meson arctan_less_iff arctan_ubound linordered_field_no_ub)
+  have "isCont arctan (tan (arctan x))"
+  proof (rule isCont_inverse_function2 [of u "arctan x" v])
+    show "\<And>z. \<lbrakk>u \<le> z; z \<le> v\<rbrakk> \<Longrightarrow> arctan (tan z) = z"
+      using arctan_unique u(1) v(2) by auto
+    then show "\<And>z. \<lbrakk>u \<le> z; z \<le> v\<rbrakk> \<Longrightarrow> isCont tan z"
+      by (metis arctan cos_gt_zero_pi isCont_tan less_irrefl)
+  qed (use u v in auto)
+  then show ?thesis
+qed

lemma tendsto_arctan [tendsto_intros]: "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) \<longlongrightarrow> arctan x) F"
by (rule isCont_tendsto_compose [OF isCont_arctan])
@@ -5160,43 +5191,36 @@
"continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))"
unfolding continuous_on_def by (auto intro: tendsto_arctan)

-lemma DERIV_arcsin: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))"
-  apply (rule DERIV_inverse_function [where f=sin and a="-1" and b=1])
-       apply (rule DERIV_cong [OF DERIV_sin])
-      apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
-      apply (rule power_strict_mono, simp)
-       apply simp
-      apply simp
-     apply assumption
-    apply assumption
-   apply simp
-  apply (erule (1) isCont_arcsin)
-  done
-
-lemma DERIV_arccos: "- 1 < x \<Longrightarrow> x < 1 \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))"
-  apply (rule DERIV_inverse_function [where f=cos and a="-1" and b=1])
-       apply (rule DERIV_cong [OF DERIV_cos])
-      apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
-      apply (rule power_strict_mono, simp)
-       apply simp
-      apply simp
-     apply assumption
-    apply assumption
-   apply simp
-  apply (erule (1) isCont_arccos)
-  done
+lemma DERIV_arcsin:
+  assumes "- 1 < x" "x < 1"
+  shows "DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))"
+proof (rule DERIV_inverse_function)
+  show "(sin has_real_derivative sqrt (1 - x\<^sup>2)) (at (arcsin x))"
+    by (rule derivative_eq_intros | use assms cos_arcsin in force)+
+  show "sqrt (1 - x\<^sup>2) \<noteq> 0"
+    using abs_square_eq_1 assms by force
+qed (use assms isCont_arcsin in auto)
+
+lemma DERIV_arccos:
+  assumes "- 1 < x" "x < 1"
+  shows "DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))"
+proof (rule DERIV_inverse_function)
+  show "(cos has_real_derivative - sqrt (1 - x\<^sup>2)) (at (arccos x))"
+    by (rule derivative_eq_intros | use assms sin_arccos in force)+
+  show "- sqrt (1 - x\<^sup>2) \<noteq> 0"
+    using abs_square_eq_1 assms by force
+qed (use assms isCont_arccos in auto)

lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)"
-  apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
-       apply (rule DERIV_cong [OF DERIV_tan])
-        apply (rule cos_arctan_not_zero)
-   apply (simp add: arctan power_inverse [symmetric] tan_sec [symmetric])
-  apply (subgoal_tac "0 < 1 + x\<^sup>2", simp)
-  done
+proof (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
+  show "(tan has_real_derivative 1 + x\<^sup>2) (at (arctan x))"
+    apply (rule derivative_eq_intros | simp)+
+    by (metis arctan cos_arctan_not_zero power_inverse tan_sec)
+  show "\<And>y. \<lbrakk>x - 1 < y; y < x + 1\<rbrakk> \<Longrightarrow> tan (arctan y) = y"
+    using tan_arctan by blast
+  show "1 + x\<^sup>2 \<noteq> 0"
+    by (metis power_one sum_power2_eq_zero_iff zero_neq_one)
+qed (use isCont_arctan in auto)

declare
DERIV_arcsin[THEN DERIV_chain2, derivative_intros]
@@ -5830,7 +5854,7 @@
moreover have "isCont (\<lambda> x. ?a x n - ?diff x n) x" for x