merged
authornipkow
Sun, 22 Feb 2009 09:52:49 +0100
changeset 30048 6cf1fe60ac73
parent 30046 49f603f92c47 (diff)
parent 30047 46c88406e6c0 (current diff)
child 30051 a416ed407f82
child 30052 410fefc247aa
merged
--- a/src/HOL/Library/Euclidean_Space.thy	Sun Feb 22 09:52:28 2009 +0100
+++ b/src/HOL/Library/Euclidean_Space.thy	Sun Feb 22 09:52:49 2009 +0100
@@ -8,6 +8,7 @@
 theory Euclidean_Space
   imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Complex_Main 
   Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type
+  Inner_Product
   uses ("normarith.ML")
 begin
 
@@ -547,6 +548,38 @@
 
 end
 
+subsection {* Inner products *}
+
+instantiation "^" :: (real_inner, type) real_inner
+begin
+
+definition vector_inner_def:
+  "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) {1 .. dimindex(UNIV::'b set)}"
+
+instance proof
+  fix r :: real and x y z :: "'a ^ 'b"
+  show "inner x y = inner y x"
+    unfolding vector_inner_def
+    by (simp add: inner_commute)
+  show "inner (x + y) z = inner x z + inner y z"
+    unfolding vector_inner_def
+    by (vector inner_left_distrib)
+  show "inner (scaleR r x) y = r * inner x y"
+    unfolding vector_inner_def
+    by (vector inner_scaleR_left)
+  show "0 \<le> inner x x"
+    unfolding vector_inner_def
+    by (simp add: setsum_nonneg)
+  show "inner x x = 0 \<longleftrightarrow> x = 0"
+    unfolding vector_inner_def
+    by (simp add: Cart_eq setsum_nonneg_eq_0_iff)
+  show "norm x = sqrt (inner x x)"
+    unfolding vector_inner_def vector_norm_def setL2_def
+    by (simp add: power2_norm_eq_inner)
+qed
+
+end
+
 subsection{* Properties of the dot product.  *}
 
 lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \<bullet> y = y \<bullet> x" 
--- a/src/HOL/Library/Inner_Product.thy	Sun Feb 22 09:52:28 2009 +0100
+++ b/src/HOL/Library/Inner_Product.thy	Sun Feb 22 09:52:49 2009 +0100
@@ -65,7 +65,7 @@
 lemma power2_norm_eq_inner: "(norm x)\<twosuperior> = inner x x"
   by (simp add: norm_eq_sqrt_inner)
 
-lemma Cauchy_Schwartz_ineq:
+lemma Cauchy_Schwarz_ineq:
   "(inner x y)\<twosuperior> \<le> inner x x * inner y y"
 proof (cases)
   assume "y = 0"
@@ -86,11 +86,11 @@
     by (simp add: pos_divide_le_eq y)
 qed
 
-lemma Cauchy_Schwartz_ineq2:
+lemma Cauchy_Schwarz_ineq2:
   "\<bar>inner x y\<bar> \<le> norm x * norm y"
 proof (rule power2_le_imp_le)
   have "(inner x y)\<twosuperior> \<le> inner x x * inner y y"
-    using Cauchy_Schwartz_ineq .
+    using Cauchy_Schwarz_ineq .
   thus "\<bar>inner x y\<bar>\<twosuperior> \<le> (norm x * norm y)\<twosuperior>"
     by (simp add: power_mult_distrib power2_norm_eq_inner)
   show "0 \<le> norm x * norm y"
@@ -108,7 +108,7 @@
   show "norm (x + y) \<le> norm x + norm y"
     proof (rule power2_le_imp_le)
       have "inner x y \<le> norm x * norm y"
-        by (rule order_trans [OF abs_ge_self Cauchy_Schwartz_ineq2])
+        by (rule order_trans [OF abs_ge_self Cauchy_Schwarz_ineq2])
       thus "(norm (x + y))\<twosuperior> \<le> (norm x + norm y)\<twosuperior>"
         unfolding power2_sum power2_norm_eq_inner
         by (simp add: inner_distrib inner_commute)
@@ -140,7 +140,7 @@
   show "\<exists>K. \<forall>x y::'a. norm (inner x y) \<le> norm x * norm y * K"
   proof
     show "\<forall>x y::'a. norm (inner x y) \<le> norm x * norm y * 1"
-      by (simp add: Cauchy_Schwartz_ineq2)
+      by (simp add: Cauchy_Schwarz_ineq2)
   qed
 qed