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author | huffman |

Sat, 13 Jun 2009 08:29:34 -0700 | |

changeset 31656 | abadaf4922f8 |

parent 31655 | bcb1eb2197f8 |

child 31657 | 1dfa55a46d7d |

new continuous/vimage lemmas; cleaned up proofs

--- a/src/HOL/Library/Topology_Euclidean_Space.thy Sat Jun 13 08:18:14 2009 -0700 +++ b/src/HOL/Library/Topology_Euclidean_Space.thy Sat Jun 13 08:29:34 2009 -0700 @@ -3730,6 +3730,16 @@ shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}" using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto +lemma continuous_open_vimage: + fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *) + shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f -` s)" + unfolding vimage_def by (rule continuous_open_preimage_univ) + +lemma continuous_closed_vimage: + fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" (* FIXME: generalize *) + shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f -` s)" + unfolding vimage_def by (rule continuous_closed_preimage_univ) + text{* Equality of continuous functions on closure and related results. *} lemma continuous_closed_in_preimage_constant: @@ -5164,16 +5174,12 @@ lemma closed_halfspace_le: "closed {x. inner a x \<le> b}" proof- - have *:"{x \<in> UNIV. inner a x \<in> {r. \<exists>x. inner a x = r \<and> r \<le> b}} = {x. inner a x \<le> b}" by auto - let ?T = "{..b}" - have "closed ?T" by (rule closed_real_atMost) - moreover have "{r. \<exists>x. inner a x = r \<and> r \<le> b} = range (inner a) \<inter> ?T" - unfolding image_def by auto - ultimately have "\<exists>T. closed T \<and> {r. \<exists>x. inner a x = r \<and> r \<le> b} = range (inner a) \<inter> T" by fast - hence "closedin euclidean {x \<in> UNIV. inner a x \<in> {r. \<exists>x. inner a x = r \<and> r \<le> b}}" - using continuous_on_inner[of UNIV a, unfolded continuous_on_closed subtopology_UNIV] unfolding closedin_closed - by (fast elim!: allE[where x="{r. (\<exists>x. inner a x = r \<and> r \<le> b)}"]) - thus ?thesis unfolding closed_closedin[THEN sym] and * by auto + have "\<forall>x. continuous (at x) (inner a)" + unfolding continuous_at by (rule allI) (intro tendsto_intros) + hence "closed (inner a -` {..b})" + using closed_real_atMost by (rule continuous_closed_vimage) + moreover have "{x. inner a x \<le> b} = inner a -` {..b}" by auto + ultimately show ?thesis by simp qed lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"