src/HOL/Library/Product_Vector.thy
changeset 44066 d74182c93f04
parent 37678 0040bafffdef
child 44126 ce44e70d0c47
     1.1 --- a/src/HOL/Library/Product_Vector.thy	Mon Aug 08 10:26:26 2011 -0700
     1.2 +++ b/src/HOL/Library/Product_Vector.thy	Mon Aug 08 10:32:55 2011 -0700
     1.3 @@ -28,13 +28,13 @@
     1.4  instance proof
     1.5    fix a b :: real and x y :: "'a \<times> 'b"
     1.6    show "scaleR a (x + y) = scaleR a x + scaleR a y"
     1.7 -    by (simp add: expand_prod_eq scaleR_right_distrib)
     1.8 +    by (simp add: prod_eq_iff scaleR_right_distrib)
     1.9    show "scaleR (a + b) x = scaleR a x + scaleR b x"
    1.10 -    by (simp add: expand_prod_eq scaleR_left_distrib)
    1.11 +    by (simp add: prod_eq_iff scaleR_left_distrib)
    1.12    show "scaleR a (scaleR b x) = scaleR (a * b) x"
    1.13 -    by (simp add: expand_prod_eq)
    1.14 +    by (simp add: prod_eq_iff)
    1.15    show "scaleR 1 x = x"
    1.16 -    by (simp add: expand_prod_eq)
    1.17 +    by (simp add: prod_eq_iff)
    1.18  qed
    1.19  
    1.20  end
    1.21 @@ -174,7 +174,7 @@
    1.22  instance proof
    1.23    fix x y :: "'a \<times> 'b"
    1.24    show "dist x y = 0 \<longleftrightarrow> x = y"
    1.25 -    unfolding dist_prod_def expand_prod_eq by simp
    1.26 +    unfolding dist_prod_def prod_eq_iff by simp
    1.27  next
    1.28    fix x y z :: "'a \<times> 'b"
    1.29    show "dist x y \<le> dist x z + dist y z"
    1.30 @@ -373,7 +373,7 @@
    1.31      unfolding norm_prod_def by simp
    1.32    show "norm x = 0 \<longleftrightarrow> x = 0"
    1.33      unfolding norm_prod_def
    1.34 -    by (simp add: expand_prod_eq)
    1.35 +    by (simp add: prod_eq_iff)
    1.36    show "norm (x + y) \<le> norm x + norm y"
    1.37      unfolding norm_prod_def
    1.38      apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
    1.39 @@ -423,7 +423,7 @@
    1.40      unfolding inner_prod_def
    1.41      by (intro add_nonneg_nonneg inner_ge_zero)
    1.42    show "inner x x = 0 \<longleftrightarrow> x = 0"
    1.43 -    unfolding inner_prod_def expand_prod_eq
    1.44 +    unfolding inner_prod_def prod_eq_iff
    1.45      by (simp add: add_nonneg_eq_0_iff)
    1.46    show "norm x = sqrt (inner x x)"
    1.47      unfolding norm_prod_def inner_prod_def