author | huffman |
Thu, 29 Apr 2010 11:41:04 -0700 | |
changeset 36593 | fb69c8cd27bd |
parent 36581 | bbea7f52e8e1 |
child 37489 | 44e42d392c6e |
permissions | -rw-r--r-- |
36581 | 1 |
(* Title: Library/Operator_Norm.thy |
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Author: Amine Chaieb, University of Cambridge |
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*) |
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header {* Operator Norm *} |
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theory Operator_Norm |
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imports Euclidean_Space |
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begin |
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definition "onorm f = Sup {norm (f x)| x. norm x = 1}" |
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lemma norm_bound_generalize: |
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fixes f:: "real ^'n \<Rightarrow> real^'m" |
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assumes lf: "linear f" |
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shows "(\<forall>x. norm x = 1 \<longrightarrow> norm (f x) \<le> b) \<longleftrightarrow> (\<forall>x. norm (f x) \<le> b * norm x)" (is "?lhs \<longleftrightarrow> ?rhs") |
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proof- |
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{assume H: ?rhs |
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{fix x :: "real^'n" assume x: "norm x = 1" |
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from H[rule_format, of x] x have "norm (f x) \<le> b" by simp} |
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then have ?lhs by blast } |
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moreover |
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{assume H: ?lhs |
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from H[rule_format, of "basis arbitrary"] |
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have bp: "b \<ge> 0" using norm_ge_zero[of "f (basis arbitrary)"] |
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by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero]) |
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{fix x :: "real ^'n" |
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{assume "x = 0" |
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then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] bp)} |
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moreover |
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{assume x0: "x \<noteq> 0" |
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hence n0: "norm x \<noteq> 0" by (metis norm_eq_zero) |
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let ?c = "1/ norm x" |
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36593
fb69c8cd27bd
define linear algebra concepts using scaleR instead of (op *s); generalized many lemmas, though a few theorems that used to work on type int^'n are a bit less general
huffman
parents:
36581
diff
changeset
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have "norm (?c *\<^sub>R x) = 1" using x0 by (simp add: n0) |
fb69c8cd27bd
define linear algebra concepts using scaleR instead of (op *s); generalized many lemmas, though a few theorems that used to work on type int^'n are a bit less general
huffman
parents:
36581
diff
changeset
|
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with H have "norm (f (?c *\<^sub>R x)) \<le> b" by blast |
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hence "?c * norm (f x) \<le> b" |
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by (simp add: linear_cmul[OF lf]) |
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hence "norm (f x) \<le> b * norm x" |
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using n0 norm_ge_zero[of x] by (auto simp add: field_simps)} |
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ultimately have "norm (f x) \<le> b * norm x" by blast} |
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then have ?rhs by blast} |
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ultimately show ?thesis by blast |
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qed |
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lemma onorm: |
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fixes f:: "real ^'n \<Rightarrow> real ^'m" |
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assumes lf: "linear f" |
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shows "norm (f x) <= onorm f * norm x" |
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and "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b" |
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proof- |
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{ |
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let ?S = "{norm (f x) |x. norm x = 1}" |
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have Se: "?S \<noteq> {}" using norm_basis by auto |
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from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b" |
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unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def) |
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{from Sup[OF Se b, unfolded onorm_def[symmetric]] |
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show "norm (f x) <= onorm f * norm x" |
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apply - |
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apply (rule spec[where x = x]) |
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unfolding norm_bound_generalize[OF lf, symmetric] |
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by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)} |
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{ |
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show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b" |
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using Sup[OF Se b, unfolded onorm_def[symmetric]] |
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unfolding norm_bound_generalize[OF lf, symmetric] |
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by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)} |
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} |
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qed |
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lemma onorm_pos_le: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" shows "0 <= onorm f" |
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using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis arbitrary"], unfolded norm_basis] by simp |
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lemma onorm_eq_0: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" |
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shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)" |
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using onorm[OF lf] |
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apply (auto simp add: onorm_pos_le) |
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apply atomize |
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apply (erule allE[where x="0::real"]) |
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using onorm_pos_le[OF lf] |
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apply arith |
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done |
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lemma onorm_const: "onorm(\<lambda>x::real^'n. (y::real ^'m)) = norm y" |
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proof- |
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let ?f = "\<lambda>x::real^'n. (y::real ^ 'm)" |
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have th: "{norm (?f x)| x. norm x = 1} = {norm y}" |
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by(auto intro: vector_choose_size set_ext) |
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show ?thesis |
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unfolding onorm_def th |
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apply (rule Sup_unique) by (simp_all add: setle_def) |
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qed |
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lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n \<Rightarrow> real ^'m)" |
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shows "0 < onorm f \<longleftrightarrow> ~(\<forall>x. f x = 0)" |
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unfolding onorm_eq_0[OF lf, symmetric] |
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using onorm_pos_le[OF lf] by arith |
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lemma onorm_compose: |
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assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" |
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and lg: "linear (g::real^'k \<Rightarrow> real^'n)" |
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shows "onorm (f o g) <= onorm f * onorm g" |
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apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format]) |
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unfolding o_def |
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apply (subst mult_assoc) |
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apply (rule order_trans) |
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apply (rule onorm(1)[OF lf]) |
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apply (rule mult_mono1) |
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apply (rule onorm(1)[OF lg]) |
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apply (rule onorm_pos_le[OF lf]) |
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done |
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lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)" |
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shows "onorm (\<lambda>x. - f x) \<le> onorm f" |
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using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf] |
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unfolding norm_minus_cancel by metis |
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lemma onorm_neg: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)" |
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shows "onorm (\<lambda>x. - f x) = onorm f" |
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using onorm_neg_lemma[OF lf] onorm_neg_lemma[OF linear_compose_neg[OF lf]] |
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by simp |
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lemma onorm_triangle: |
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assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and lg: "linear g" |
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shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g" |
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apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format]) |
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apply (rule order_trans) |
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apply (rule norm_triangle_ineq) |
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apply (simp add: distrib) |
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apply (rule add_mono) |
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apply (rule onorm(1)[OF lf]) |
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apply (rule onorm(1)[OF lg]) |
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done |
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lemma onorm_triangle_le: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) <= e |
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\<Longrightarrow> onorm(\<lambda>x. f x + g x) <= e" |
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apply (rule order_trans) |
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apply (rule onorm_triangle) |
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apply assumption+ |
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done |
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lemma onorm_triangle_lt: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) < e |
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==> onorm(\<lambda>x. f x + g x) < e" |
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apply (rule order_le_less_trans) |
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apply (rule onorm_triangle) |
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by assumption+ |
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end |