# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1427816928 -3600
# Node ID 8a20dd967385afb2917e0b3d9b85da119e07d4ff
# Parent  30519ff3dffb45ebe02864f8f12409c8510e904a
rationalised and generalised some theorems concerning abs and x^2.

diff -r 30519ff3dffb -r 8a20dd967385 src/HOL/Multivariate_Analysis/Bounded_Continuous_Function.thy
--- a/src/HOL/Multivariate_Analysis/Bounded_Continuous_Function.thy	Tue Mar 31 15:01:06 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Bounded_Continuous_Function.thy	Tue Mar 31 16:48:48 2015 +0100
@@ -407,7 +407,7 @@
     hence "(\<Sum>i\<in>Basis. (dist (clamp a b y \<bullet> i) (clamp a b x \<bullet> i))\<^sup>2)
         \<le> (\<Sum>i\<in>Basis. (dist (y \<bullet> i) (x \<bullet> i))\<^sup>2)"
       by (auto intro!: setsum_mono
-        simp add: clamp_def dist_real_def real_abs_le_square_iff[symmetric])
+        simp add: clamp_def dist_real_def abs_le_square_iff[symmetric])
     thus ?thesis
       by (auto intro: real_sqrt_le_mono
         simp add: euclidean_dist_l2[where y=x] euclidean_dist_l2[where y="clamp a b x"] setL2_def)
diff -r 30519ff3dffb -r 8a20dd967385 src/HOL/Multivariate_Analysis/Linear_Algebra.thy
--- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Tue Mar 31 15:01:06 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy	Tue Mar 31 16:48:48 2015 +0100
@@ -68,25 +68,14 @@
 lemma norm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x = a\<^sup>2"
   by (auto simp add: norm_eq_sqrt_inner)
 
-lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)\<^sup>2 \<le> y\<^sup>2"
-proof
-  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
-  then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
-  then show "x\<^sup>2 \<le> y\<^sup>2" by simp
-next
-  assume "x\<^sup>2 \<le> y\<^sup>2"
-  then have "sqrt (x\<^sup>2) \<le> sqrt (y\<^sup>2)" by (rule real_sqrt_le_mono)
-  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>" by simp
-qed
-
 lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>2"
-  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
+  apply (simp add: dot_square_norm abs_le_square_iff[symmetric])
   using norm_ge_zero[of x]
   apply arith
   done
 
 lemma norm_ge_square: "norm x \<ge> a \<longleftrightarrow> a \<le> 0 \<or> x \<bullet> x \<ge> a\<^sup>2"
-  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
+  apply (simp add: dot_square_norm abs_le_square_iff[symmetric])
   using norm_ge_zero[of x]
   apply arith
   done
diff -r 30519ff3dffb -r 8a20dd967385 src/HOL/Power.thy
--- a/src/HOL/Power.thy	Tue Mar 31 15:01:06 2015 +0100
+++ b/src/HOL/Power.thy	Tue Mar 31 16:48:48 2015 +0100
@@ -645,6 +645,29 @@
   "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
   unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
 
+lemma abs_le_square_iff:
+   "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
+proof
+  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
+  then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
+  then show "x\<^sup>2 \<le> y\<^sup>2" by simp
+next
+  assume "x\<^sup>2 \<le> y\<^sup>2"
+  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>"
+    by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
+qed
+
+lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> abs(x) \<le> 1"
+  using abs_le_square_iff [of x 1]
+  by simp
+
+lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> abs(x) = 1"
+  by (auto simp add: abs_if power2_eq_1_iff)
+  
+lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> abs(x) < 1"
+  using  abs_square_eq_1 [of x] abs_square_le_1 [of x]
+  by (auto simp add: le_less)
+
 end
 
 
diff -r 30519ff3dffb -r 8a20dd967385 src/HOL/Rings.thy
--- a/src/HOL/Rings.thy	Tue Mar 31 15:01:06 2015 +0100
+++ b/src/HOL/Rings.thy	Tue Mar 31 16:48:48 2015 +0100
@@ -1260,6 +1260,10 @@
   "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
   by (auto simp add: diff_less_eq ac_simps abs_less_iff)
 
+lemma abs_diff_le_iff:
+   "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
+  by (auto simp add: diff_le_eq ac_simps abs_le_iff)
+
 end
 
 hide_fact (open) comm_mult_left_mono comm_mult_strict_left_mono distrib
diff -r 30519ff3dffb -r 8a20dd967385 src/HOL/Transcendental.thy
--- a/src/HOL/Transcendental.thy	Tue Mar 31 15:01:06 2015 +0100
+++ b/src/HOL/Transcendental.thy	Tue Mar 31 16:48:48 2015 +0100
@@ -4542,16 +4542,6 @@
   (is "summable (?c x)")
   by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
 
-lemma less_one_imp_sqr_less_one:
-  fixes x :: real
-  assumes "\<bar>x\<bar> < 1"
-  shows "x\<^sup>2 < 1"
-proof -
-  have "\<bar>x\<^sup>2\<bar> < 1"
-    by (metis abs_power2 assms pos2 power2_abs power_0 power_strict_decreasing zero_eq_power2 zero_less_abs_iff)
-  thus ?thesis using zero_le_power2 by auto
-qed
-
 lemma DERIV_arctan_series:
   assumes "\<bar> x \<bar> < 1"
   shows "DERIV (\<lambda> x'. \<Sum> k. (-1)^k * (1 / real (k*2+1) * x' ^ (k*2+1))) x :> (\<Sum> k. (-1)^k * x^(k*2))"
@@ -4568,7 +4558,7 @@
   {
     fix x :: real
     assume "\<bar>x\<bar> < 1"
-    hence "x\<^sup>2 < 1" by (rule less_one_imp_sqr_less_one)
+    hence "x\<^sup>2 < 1" by (simp add: abs_square_less_1)
     have "summable (\<lambda> n. (- 1) ^ n * (x\<^sup>2) ^n)"
       by (rule summable_Leibniz(1), auto intro!: LIMSEQ_realpow_zero monoseq_realpow `x\<^sup>2 < 1` order_less_imp_le[OF `x\<^sup>2 < 1`])
     hence "summable (\<lambda> n. (- 1) ^ n * x^(2*n))" unfolding power_mult .
@@ -4676,7 +4666,7 @@
             hence "\<bar>x\<bar> < r" by auto
             hence "\<bar>x\<bar> < 1" using `r < 1` by auto
             have "\<bar> - (x\<^sup>2) \<bar> < 1"
-              using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto
+              using abs_square_less_1 `\<bar>x\<bar> < 1` by auto
             hence "(\<lambda> n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))"
               unfolding real_norm_def[symmetric] by (rule geometric_sums)
             hence "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))"