src/HOL/Set_Interval.thy
author wenzelm
Wed, 12 Mar 2025 11:39:00 +0100
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(*  Title:      HOL/Set_Interval.thy
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    Author:     Tobias Nipkow, Clemens Ballarin, Jeremy Avigad
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lessThan, greaterThan, atLeast, atMost and two-sided intervals
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Modern convention: Ixy stands for an interval where x and y
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describe the lower and upper bound and x,y : {c,o,i}
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where c = closed, o = open, i = infinite.
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Examples: Ico = {_ ..< _} and Ici = {_ ..}
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*)
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section \<open>Set intervals\<close>
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theory Set_Interval
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imports Parity
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begin
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(* Belongs in Finite_Set but 2 is not available there *)
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lemma card_2_iff: "card S = 2 \<longleftrightarrow> (\<exists>x y. S = {x,y} \<and> x \<noteq> y)"
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  by (auto simp: card_Suc_eq numeral_eq_Suc)
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lemma card_2_iff': "card S = 2 \<longleftrightarrow> (\<exists>x\<in>S. \<exists>y\<in>S. x \<noteq> y \<and> (\<forall>z\<in>S. z = x \<or> z = y))"
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  by (auto simp: card_Suc_eq numeral_eq_Suc)
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lemma card_3_iff: "card S = 3 \<longleftrightarrow> (\<exists>x y z. S = {x,y,z} \<and> x \<noteq> y \<and> y \<noteq> z \<and> x \<noteq> z)"
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  by (fastforce simp: card_Suc_eq numeral_eq_Suc)
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context ord
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begin
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definition
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  lessThan    :: "'a => 'a set" (\<open>(\<open>indent=1 notation=\<open>mixfix set interval\<close>\<close>{..<_})\<close>) where
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  "{..<u} == {x. x < u}"
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definition
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  atMost      :: "'a => 'a set" (\<open>(\<open>indent=1 notation=\<open>mixfix set interval\<close>\<close>{.._})\<close>) where
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  "{..u} == {x. x \<le> u}"
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definition
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  greaterThan :: "'a => 'a set" (\<open>(\<open>indent=1 notation=\<open>mixfix set interval\<close>\<close>{_<..})\<close>) where
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  "{l<..} == {x. l<x}"
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definition
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  atLeast     :: "'a => 'a set" (\<open>(\<open>indent=1 notation=\<open>mixfix set interval\<close>\<close>{_..})\<close>) where
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  "{l..} == {x. l\<le>x}"
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definition
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  greaterThanLessThan :: "'a => 'a => 'a set"  (\<open>(\<open>indent=1 notation=\<open>mixfix set interval\<close>\<close>{_/<..<_})\<close>) where
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  "{l<..<u} == {l<..} Int {..<u}"
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definition
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  atLeastLessThan :: "'a => 'a => 'a set"      (\<open>(\<open>indent=1 notation=\<open>mixfix set interval\<close>\<close>{_/..<_})\<close>) where
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  "{l..<u} == {l..} Int {..<u}"
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definition
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  greaterThanAtMost :: "'a => 'a => 'a set"    (\<open>(\<open>indent=1 notation=\<open>mixfix set interval\<close>\<close>{_/<.._})\<close>) where
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  "{l<..u} == {l<..} Int {..u}"
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definition
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  atLeastAtMost :: "'a => 'a => 'a set"        (\<open>(\<open>indent=1 notation=\<open>mixfix set interval\<close>\<close>{_/.._})\<close>) where
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  "{l..u} == {l..} Int {..u}"
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end
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text\<open>A note of warning when using \<^term>\<open>{..<n}\<close> on type \<^typ>\<open>nat\<close>: it is equivalent to \<^term>\<open>{0::nat..<n}\<close> but some lemmas involving
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\<^term>\<open>{m..<n}\<close> may not exist in \<^term>\<open>{..<n}\<close>-form as well.\<close>
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syntax (ASCII)
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  "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       (\<open>(\<open>indent=3 notation=\<open>binder UN\<close>\<close>UN _<=_./ _)\<close> [0, 0, 10] 10)
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  "_UNION_less" :: "'a => 'a => 'b set => 'b set"       (\<open>(\<open>indent=3 notation=\<open>binder UN\<close>\<close>UN _<_./ _)\<close> [0, 0, 10] 10)
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  "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       (\<open>(\<open>indent=3 notation=\<open>binder INT\<close>\<close>INT _<=_./ _)\<close> [0, 0, 10] 10)
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  "_INTER_less" :: "'a => 'a => 'b set => 'b set"       (\<open>(\<open>indent=3 notation=\<open>binder INT\<close>\<close>INT _<_./ _)\<close> [0, 0, 10] 10)
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syntax (latex output)
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  "_UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       (\<open>(3\<Union>(\<open>unbreakable\<close>_ \<le> _)/ _)\<close> [0, 0, 10] 10)
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  "_UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       (\<open>(3\<Union>(\<open>unbreakable\<close>_ < _)/ _)\<close> [0, 0, 10] 10)
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  "_INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       (\<open>(3\<Inter>(\<open>unbreakable\<close>_ \<le> _)/ _)\<close> [0, 0, 10] 10)
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  "_INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       (\<open>(3\<Inter>(\<open>unbreakable\<close>_ < _)/ _)\<close> [0, 0, 10] 10)
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syntax
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  "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       (\<open>(\<open>indent=3 notation=\<open>binder \<Union>\<close>\<close>\<Union>_\<le>_./ _)\<close> [0, 0, 10] 10)
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  "_UNION_less" :: "'a => 'a => 'b set => 'b set"       (\<open>(\<open>indent=3 notation=\<open>binder \<Union>\<close>\<close>\<Union>_<_./ _)\<close> [0, 0, 10] 10)
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  "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       (\<open>(\<open>indent=3 notation=\<open>binder \<Inter>\<close>\<close>\<Inter>_\<le>_./ _)\<close> [0, 0, 10] 10)
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  "_INTER_less" :: "'a => 'a => 'b set => 'b set"       (\<open>(\<open>indent=3 notation=\<open>binder \<Inter>\<close>\<close>\<Inter>_<_./ _)\<close> [0, 0, 10] 10)
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syntax_consts
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  "_UNION_le" "_UNION_less" \<rightleftharpoons> Union and
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  "_INTER_le" "_INTER_less" \<rightleftharpoons> Inter
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translations
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  "\<Union>i\<le>n. A" \<rightleftharpoons> "\<Union>i\<in>{..n}. A"
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  "\<Union>i<n. A" \<rightleftharpoons> "\<Union>i\<in>{..<n}. A"
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  "\<Inter>i\<le>n. A" \<rightleftharpoons> "\<Inter>i\<in>{..n}. A"
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  "\<Inter>i<n. A" \<rightleftharpoons> "\<Inter>i\<in>{..<n}. A"
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subsection \<open>Various equivalences\<close>
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lemma (in ord) lessThan_iff [iff]: "(i \<in> lessThan k) = (i<k)"
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by (simp add: lessThan_def)
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lemma Compl_lessThan [simp]:
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    "!!k:: 'a::linorder. -lessThan k = atLeast k"
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  by (auto simp add: lessThan_def atLeast_def)
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lemma single_Diff_lessThan [simp]: "!!k:: 'a::preorder. {k} - lessThan k = {k}"
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  by auto
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lemma (in ord) greaterThan_iff [iff]: "(i \<in> greaterThan k) = (k<i)"
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  by (simp add: greaterThan_def)
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lemma Compl_greaterThan [simp]:
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    "!!k:: 'a::linorder. -greaterThan k = atMost k"
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  by (auto simp add: greaterThan_def atMost_def)
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lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
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  by (metis Compl_greaterThan double_complement)
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lemma (in ord) atLeast_iff [iff]: "(i \<in> atLeast k) = (k<=i)"
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  by (simp add: atLeast_def)
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lemma Compl_atLeast [simp]: "!!k:: 'a::linorder. -atLeast k = lessThan k"
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  by (auto simp add: lessThan_def atLeast_def)
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lemma (in ord) atMost_iff [iff]: "(i \<in> atMost k) = (i<=k)"
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   127
by (simp add: atMost_def)
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diff changeset
   128
14485
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parents: 14478
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   129
lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
ea2707645af8 new material from Avigad
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   130
by (blast intro: order_antisym)
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   131
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   132
lemma (in linorder) lessThan_Int_lessThan: "{ a <..} \<inter> { b <..} = { max a b <..}"
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   133
  by auto
3de230ed0547 introduce order topology
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   134
3de230ed0547 introduce order topology
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   135
lemma (in linorder) greaterThan_Int_greaterThan: "{..< a} \<inter> {..< b} = {..< min a b}"
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hoelzl
parents: 50417
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   136
  by auto
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parents: 13735
diff changeset
   137
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   138
subsection \<open>Logical Equivalences for Set Inclusion and Equality\<close>
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   139
63879
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paulson <lp15@cam.ac.uk>
parents: 63721
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   140
lemma atLeast_empty_triv [simp]: "{{}..} = UNIV"
15bbf6360339 simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents: 63721
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   141
  by auto
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paulson <lp15@cam.ac.uk>
parents: 63721
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   142
15bbf6360339 simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents: 63721
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   143
lemma atMost_UNIV_triv [simp]: "{..UNIV} = UNIV"
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paulson <lp15@cam.ac.uk>
parents: 63721
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   144
  by auto
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paulson <lp15@cam.ac.uk>
parents: 63721
diff changeset
   145
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   146
lemma atLeast_subset_iff [iff]:
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paulson <lp15@cam.ac.uk>
parents: 75455
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   147
  "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::preorder))"
1910054f8c39 some additional lemmas and a little tidying up
paulson <lp15@cam.ac.uk>
parents: 75455
diff changeset
   148
  by (blast intro: order_trans)
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parents: 13735
diff changeset
   149
6d1bb3059818 new logical equivalences
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   150
lemma atLeast_eq_iff [iff]:
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parents: 75455
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   151
  "(atLeast x = atLeast y) = (x = (y::'a::order))"
1910054f8c39 some additional lemmas and a little tidying up
paulson <lp15@cam.ac.uk>
parents: 75455
diff changeset
   152
  by (blast intro: order_antisym order_trans)
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parents: 13735
diff changeset
   153
6d1bb3059818 new logical equivalences
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   154
lemma greaterThan_subset_iff [iff]:
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paulson <lp15@cam.ac.uk>
parents: 75455
diff changeset
   155
  "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
68618
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paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   156
  unfolding greaterThan_def by (auto simp: linorder_not_less [symmetric])
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parents: 13735
diff changeset
   157
6d1bb3059818 new logical equivalences
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parents: 13735
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   158
lemma greaterThan_eq_iff [iff]:
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paulson
parents: 15402
diff changeset
   159
     "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
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paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   160
  by (auto simp: elim!: equalityE)
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parents: 13735
diff changeset
   161
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parents: 70723
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   162
lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::preorder))"
68618
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paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   163
  by (blast intro: order_trans)
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paulson
parents: 13735
diff changeset
   164
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parents: 70723
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   165
lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::order))"
68618
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paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   166
  by (blast intro: order_antisym order_trans)
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paulson
parents: 13735
diff changeset
   167
6d1bb3059818 new logical equivalences
paulson
parents: 13735
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   168
lemma lessThan_subset_iff [iff]:
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paulson
parents: 15402
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   169
     "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
68618
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paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   170
  unfolding lessThan_def by (auto simp: linorder_not_less [symmetric])
13850
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paulson
parents: 13735
diff changeset
   171
6d1bb3059818 new logical equivalences
paulson
parents: 13735
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   172
lemma lessThan_eq_iff [iff]:
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paulson
parents: 15402
diff changeset
   173
     "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
68618
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paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   174
  by (auto simp: elim!: equalityE)
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ballarin
parents: 11609
diff changeset
   175
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   176
lemma lessThan_strict_subset_iff:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   177
  fixes m n :: "'a::linorder"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   178
  shows "{..<m} < {..<n} \<longleftrightarrow> m < n"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   179
  by (metis leD lessThan_subset_iff linorder_linear not_less_iff_gr_or_eq psubset_eq)
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ballarin
parents: 11609
diff changeset
   180
57448
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57447
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   181
lemma (in linorder) Ici_subset_Ioi_iff: "{a ..} \<subseteq> {b <..} \<longleftrightarrow> b < a"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57447
diff changeset
   182
  by auto
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57447
diff changeset
   183
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57447
diff changeset
   184
lemma (in linorder) Iic_subset_Iio_iff: "{.. a} \<subseteq> {..< b} \<longleftrightarrow> a < b"
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hoelzl
parents: 57447
diff changeset
   185
  by auto
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57447
diff changeset
   186
62369
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62343
diff changeset
   187
lemma (in preorder) Ioi_le_Ico: "{a <..} \<subseteq> {a ..}"
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62343
diff changeset
   188
  by (auto intro: less_imp_le)
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62343
diff changeset
   189
60758
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wenzelm
parents: 60615
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   190
subsection \<open>Two-sided intervals\<close>
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parents: 11609
diff changeset
   191
24691
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nipkow
parents: 24449
diff changeset
   192
context ord
e7f46ee04809 localized { .. } (but only a few thms)
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parents: 24449
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   193
begin
e7f46ee04809 localized { .. } (but only a few thms)
nipkow
parents: 24449
diff changeset
   194
68618
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paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   195
lemma greaterThanLessThan_iff [simp]: "(i \<in> {l<..<u}) = (l < i \<and> i < u)"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   196
  by (simp add: greaterThanLessThan_def)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   197
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   198
lemma atLeastLessThan_iff [simp]: "(i \<in> {l..<u}) = (l \<le> i \<and> i < u)"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   199
  by (simp add: atLeastLessThan_def)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   200
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   201
lemma greaterThanAtMost_iff [simp]: "(i \<in> {l<..u}) = (l < i \<and> i \<le> u)"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   202
  by (simp add: greaterThanAtMost_def)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   203
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   204
lemma atLeastAtMost_iff [simp]: "(i \<in> {l..u}) = (l \<le> i \<and> i \<le> u)"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   205
  by (simp add: atLeastAtMost_def)
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ballarin
parents: 11609
diff changeset
   206
60758
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wenzelm
parents: 60615
diff changeset
   207
text \<open>The above four lemmas could be declared as iffs. Unfortunately this
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52380
diff changeset
   208
breaks many proofs. Since it only helps blast, it is better to leave them
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   209
alone.\<close>
32436
10cd49e0c067 Turned "x <= y ==> sup x y = y" (and relatives) into simp rules
nipkow
parents: 32408
diff changeset
   210
50999
3de230ed0547 introduce order topology
hoelzl
parents: 50417
diff changeset
   211
lemma greaterThanLessThan_eq: "{ a <..< b} = { a <..} \<inter> {..< b }"
3de230ed0547 introduce order topology
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parents: 50417
diff changeset
   212
  by auto
3de230ed0547 introduce order topology
hoelzl
parents: 50417
diff changeset
   213
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
   214
lemma (in order) atLeastLessThan_eq_atLeastAtMost_diff:
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   215
  "{a..<b} = {a..b} - {b}"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   216
  by (auto simp add: atLeastLessThan_def atLeastAtMost_def)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   217
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   218
lemma (in order) greaterThanAtMost_eq_atLeastAtMost_diff:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   219
  "{a<..b} = {a..b} - {a}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   220
  by (auto simp add: greaterThanAtMost_def atLeastAtMost_def)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   221
24691
e7f46ee04809 localized { .. } (but only a few thms)
nipkow
parents: 24449
diff changeset
   222
end
13735
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ballarin
parents: 11609
diff changeset
   223
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   224
subsubsection\<open>Emptyness, singletons, subset\<close>
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15542
diff changeset
   225
70746
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   226
context preorder
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   227
begin
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   228
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   229
lemma atLeastatMost_empty_iff[simp]:
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   230
  "{a..b} = {} \<longleftrightarrow> (\<not> a \<le> b)"
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   231
  by auto (blast intro: order_trans)
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   232
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   233
lemma atLeastatMost_empty_iff2[simp]:
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   234
  "{} = {a..b} \<longleftrightarrow> (\<not> a \<le> b)"
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   235
  by auto (blast intro: order_trans)
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   236
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   237
lemma atLeastLessThan_empty_iff[simp]:
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   238
  "{a..<b} = {} \<longleftrightarrow> (\<not> a < b)"
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   239
  by auto (blast intro: le_less_trans)
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   240
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   241
lemma atLeastLessThan_empty_iff2[simp]:
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   242
  "{} = {a..<b} \<longleftrightarrow> (\<not> a < b)"
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   243
  by auto (blast intro: le_less_trans)
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   244
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   245
lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {} \<longleftrightarrow> \<not> k < l"
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   246
  by auto (blast intro: less_le_trans)
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   247
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   248
lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} \<longleftrightarrow> \<not> k < l"
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   249
  by auto (blast intro: less_le_trans)
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   250
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   251
lemma atLeastatMost_subset_iff[simp]:
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   252
  "{a..b} \<le> {c..d} \<longleftrightarrow> (\<not> a \<le> b) \<or> c \<le> a \<and> b \<le> d"
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   253
  unfolding atLeastAtMost_def atLeast_def atMost_def
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   254
  by (blast intro: order_trans)
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   255
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   256
lemma atLeastatMost_psubset_iff:
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   257
  "{a..b} < {c..d} \<longleftrightarrow>
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   258
   ((\<not> a \<le> b) \<or> c \<le> a \<and> b \<le> d \<and> (c < a \<or> b < d)) \<and> c \<le> d"
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   259
  by(simp add: psubset_eq set_eq_iff less_le_not_le)(blast intro: order_trans)
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   260
70749
5d06b7bb9d22 More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents: 70746
diff changeset
   261
lemma atLeastAtMost_subseteq_atLeastLessThan_iff:
5d06b7bb9d22 More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents: 70746
diff changeset
   262
  "{a..b} \<subseteq> {c ..< d} \<longleftrightarrow> (a \<le> b \<longrightarrow> c \<le> a \<and> b < d)" 
5d06b7bb9d22 More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents: 70746
diff changeset
   263
  by auto (blast intro: local.order_trans local.le_less_trans elim: )+
5d06b7bb9d22 More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents: 70746
diff changeset
   264
70746
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   265
lemma Icc_subset_Ici_iff[simp]:
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   266
  "{l..h} \<subseteq> {l'..} = (\<not> l\<le>h \<or> l\<ge>l')"
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   267
  by(auto simp: subset_eq intro: order_trans)
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   268
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   269
lemma Icc_subset_Iic_iff[simp]:
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   270
  "{l..h} \<subseteq> {..h'} = (\<not> l\<le>h \<or> h\<le>h')"
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   271
  by(auto simp: subset_eq intro: order_trans)
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   272
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   273
lemma not_Ici_eq_empty[simp]: "{l..} \<noteq> {}"
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   274
  by(auto simp: set_eq_iff)
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   275
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   276
lemma not_Iic_eq_empty[simp]: "{..h} \<noteq> {}"
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   277
  by(auto simp: set_eq_iff)
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   278
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   279
lemmas not_empty_eq_Ici_eq_empty[simp] = not_Ici_eq_empty[symmetric]
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   280
lemmas not_empty_eq_Iic_eq_empty[simp] = not_Iic_eq_empty[symmetric]
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   281
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   282
end
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   283
24691
e7f46ee04809 localized { .. } (but only a few thms)
nipkow
parents: 24449
diff changeset
   284
context order
e7f46ee04809 localized { .. } (but only a few thms)
nipkow
parents: 24449
diff changeset
   285
begin
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15542
diff changeset
   286
77935
7f240b0dabd9 More new theorems, and a necessary correction
paulson <lp15@cam.ac.uk>
parents: 76224
diff changeset
   287
lemma atLeastatMost_empty[simp]: "b < a \<Longrightarrow> {a..b} = {}" 
7f240b0dabd9 More new theorems, and a necessary correction
paulson <lp15@cam.ac.uk>
parents: 76224
diff changeset
   288
  and atLeastatMost_empty'[simp]: "\<not> a \<le> b \<Longrightarrow> {a..b} = {}"
70746
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   289
  by(auto simp: atLeastAtMost_def atLeast_def atMost_def)
32400
6c62363cf0d7 new lemmas
nipkow
parents: 32006
diff changeset
   290
6c62363cf0d7 new lemmas
nipkow
parents: 32006
diff changeset
   291
lemma atLeastLessThan_empty[simp]:
70746
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   292
  "b \<le> a \<Longrightarrow> {a..<b} = {}"
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   293
  by(auto simp: atLeastLessThan_def)
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15542
diff changeset
   294
32400
6c62363cf0d7 new lemmas
nipkow
parents: 32006
diff changeset
   295
lemma greaterThanAtMost_empty[simp]: "l \<le> k ==> {k<..l} = {}"
70746
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   296
  by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
32400
6c62363cf0d7 new lemmas
nipkow
parents: 32006
diff changeset
   297
29709
cf8476cc440d fixed proposition slip
haftmann
parents: 29667
diff changeset
   298
lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..<l} = {}"
70746
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   299
  by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
17719
2e75155c5ed5 Added a few lemmas
nipkow
parents: 17149
diff changeset
   300
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24853
diff changeset
   301
lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"
70746
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   302
  by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
24691
e7f46ee04809 localized { .. } (but only a few thms)
nipkow
parents: 24449
diff changeset
   303
36846
0f67561ed5a6 Added atLeastAtMost_singleton_iff, atLeastAtMost_singleton'
hoelzl
parents: 36755
diff changeset
   304
lemma atLeastAtMost_singleton': "a = b \<Longrightarrow> {a .. b} = {a}" by simp
0f67561ed5a6 Added atLeastAtMost_singleton_iff, atLeastAtMost_singleton'
hoelzl
parents: 36755
diff changeset
   305
51334
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   306
lemma Icc_eq_Icc[simp]:
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   307
  "{l..h} = {l'..h'} = (l=l' \<and> h=h' \<or> \<not> l\<le>h \<and> \<not> l'\<le>h')"
73411
1f1366966296 avoid name clash
haftmann
parents: 73139
diff changeset
   308
  by (simp add: order_class.order.eq_iff) (auto intro: order_trans)
51334
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   309
75543
1910054f8c39 some additional lemmas and a little tidying up
paulson <lp15@cam.ac.uk>
parents: 75455
diff changeset
   310
lemma (in linorder) Ico_eq_Ico:
1910054f8c39 some additional lemmas and a little tidying up
paulson <lp15@cam.ac.uk>
parents: 75455
diff changeset
   311
  "{l..<h} = {l'..<h'} = (l=l' \<and> h=h' \<or> \<not> l<h \<and> \<not> l'<h')"
1910054f8c39 some additional lemmas and a little tidying up
paulson <lp15@cam.ac.uk>
parents: 75455
diff changeset
   312
  by (metis atLeastLessThan_empty_iff2 nle_le not_less ord.atLeastLessThan_iff)
1910054f8c39 some additional lemmas and a little tidying up
paulson <lp15@cam.ac.uk>
parents: 75455
diff changeset
   313
36846
0f67561ed5a6 Added atLeastAtMost_singleton_iff, atLeastAtMost_singleton'
hoelzl
parents: 36755
diff changeset
   314
lemma atLeastAtMost_singleton_iff[simp]:
0f67561ed5a6 Added atLeastAtMost_singleton_iff, atLeastAtMost_singleton'
hoelzl
parents: 36755
diff changeset
   315
  "{a .. b} = {c} \<longleftrightarrow> a = b \<and> b = c"
0f67561ed5a6 Added atLeastAtMost_singleton_iff, atLeastAtMost_singleton'
hoelzl
parents: 36755
diff changeset
   316
proof
0f67561ed5a6 Added atLeastAtMost_singleton_iff, atLeastAtMost_singleton'
hoelzl
parents: 36755
diff changeset
   317
  assume "{a..b} = {c}"
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
   318
  hence *: "\<not> (\<not> a \<le> b)" unfolding atLeastatMost_empty_iff[symmetric] by simp
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   319
  with \<open>{a..b} = {c}\<close> have "c \<le> a \<and> b \<le> c" by auto
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
   320
  with * show "a = b \<and> b = c" by auto
36846
0f67561ed5a6 Added atLeastAtMost_singleton_iff, atLeastAtMost_singleton'
hoelzl
parents: 36755
diff changeset
   321
qed simp
0f67561ed5a6 Added atLeastAtMost_singleton_iff, atLeastAtMost_singleton'
hoelzl
parents: 36755
diff changeset
   322
80612
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   323
text \<open>The following results generalise their namesakes in @{theory HOL.Nat} to intervals\<close>
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   324
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   325
lemma lift_Suc_mono_le_ivl:
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   326
  assumes mono: "\<And>n. n\<in>N \<Longrightarrow> f n \<le> f (Suc n)"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   327
    and "n \<le> n'" and subN: "{n..<n'} \<subseteq> N"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   328
  shows "f n \<le> f n'"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   329
proof (cases "n < n'")
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   330
  case True
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   331
  then show ?thesis
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   332
    using subN
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   333
  proof (induction n n' rule: less_Suc_induct)
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   334
    case (1 i)
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   335
    then show ?case
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   336
      by (simp add: mono subsetD) 
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   337
  next
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   338
    case (2 i j k)
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   339
    have "f i \<le> f j" "f j \<le> f k"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   340
      using 2 by force+
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   341
    then show ?case by auto 
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   342
  qed
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   343
next
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   344
  case False
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   345
  with \<open>n \<le> n'\<close> show ?thesis by auto
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   346
qed
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   347
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   348
lemma lift_Suc_antimono_le_ivl:
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   349
  assumes mono: "\<And>n. n\<in>N \<Longrightarrow> f n \<ge> f (Suc n)"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   350
    and "n \<le> n'" and subN: "{n..<n'} \<subseteq> N"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   351
  shows "f n \<ge> f n'"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   352
proof (cases "n < n'")
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   353
  case True
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   354
  then show ?thesis
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   355
    using subN
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   356
  proof (induction n n' rule: less_Suc_induct)
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   357
    case (1 i)
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   358
    then show ?case
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   359
      by (simp add: mono subsetD) 
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   360
  next
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   361
    case (2 i j k)
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   362
    have "f i \<ge> f j" "f j \<ge> f k"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   363
      using 2 by force+
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   364
    then show ?case by auto 
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   365
  qed
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   366
next
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   367
  case False
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   368
  with \<open>n \<le> n'\<close> show ?thesis by auto
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   369
qed
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   370
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   371
lemma lift_Suc_mono_less_ivl:
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   372
  assumes mono: "\<And>n. n\<in>N \<Longrightarrow> f n < f (Suc n)"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   373
    and "n < n'" and subN: "{n..<n'} \<subseteq> N"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   374
  shows "f n < f n'"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   375
  using \<open>n < n'\<close>
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   376
  using subN
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   377
proof (induction n n' rule: less_Suc_induct)
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   378
  case (1 i)
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   379
  then show ?case
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   380
    by (simp add: mono subsetD) 
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   381
next
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   382
  case (2 i j k)
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   383
  have "f i < f j" "f j < f k"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   384
    using 2 by force+
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   385
  then show ?case by auto 
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   386
qed
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
   387
24691
e7f46ee04809 localized { .. } (but only a few thms)
nipkow
parents: 24449
diff changeset
   388
end
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   389
51334
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   390
context no_top
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   391
begin
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   392
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   393
(* also holds for no_bot but no_top should suffice *)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   394
lemma not_UNIV_le_Icc[simp]: "\<not> UNIV \<subseteq> {l..h}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   395
using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   396
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   397
lemma not_UNIV_le_Iic[simp]: "\<not> UNIV \<subseteq> {..h}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   398
using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   399
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   400
lemma not_Ici_le_Icc[simp]: "\<not> {l..} \<subseteq> {l'..h'}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   401
using gt_ex[of h']
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   402
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   403
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   404
lemma not_Ici_le_Iic[simp]: "\<not> {l..} \<subseteq> {..h'}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   405
using gt_ex[of h']
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   406
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   407
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   408
end
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   409
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   410
context no_bot
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   411
begin
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   412
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   413
lemma not_UNIV_le_Ici[simp]: "\<not> UNIV \<subseteq> {l..}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   414
using lt_ex[of l] by(auto simp: subset_eq less_le_not_le)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   415
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   416
lemma not_Iic_le_Icc[simp]: "\<not> {..h} \<subseteq> {l'..h'}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   417
using lt_ex[of l']
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   418
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   419
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   420
lemma not_Iic_le_Ici[simp]: "\<not> {..h} \<subseteq> {l'..}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   421
using lt_ex[of l']
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   422
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   423
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   424
end
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   425
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   426
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   427
context no_top
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   428
begin
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   429
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   430
(* also holds for no_bot but no_top should suffice *)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   431
lemma not_UNIV_eq_Icc[simp]: "\<not> UNIV = {l'..h'}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   432
using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   433
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   434
lemmas not_Icc_eq_UNIV[simp] = not_UNIV_eq_Icc[symmetric]
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   435
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   436
lemma not_UNIV_eq_Iic[simp]: "\<not> UNIV = {..h'}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   437
using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   438
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   439
lemmas not_Iic_eq_UNIV[simp] = not_UNIV_eq_Iic[symmetric]
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   440
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   441
lemma not_Icc_eq_Ici[simp]: "\<not> {l..h} = {l'..}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   442
unfolding atLeastAtMost_def using not_Ici_le_Iic[of l'] by blast
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   443
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   444
lemmas not_Ici_eq_Icc[simp] = not_Icc_eq_Ici[symmetric]
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   445
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   446
(* also holds for no_bot but no_top should suffice *)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   447
lemma not_Iic_eq_Ici[simp]: "\<not> {..h} = {l'..}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   448
using not_Ici_le_Iic[of l' h] by blast
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   449
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   450
lemmas not_Ici_eq_Iic[simp] = not_Iic_eq_Ici[symmetric]
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   451
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   452
end
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   453
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   454
context no_bot
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   455
begin
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   456
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   457
lemma not_UNIV_eq_Ici[simp]: "\<not> UNIV = {l'..}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   458
using lt_ex[of l'] by(auto simp: set_eq_iff  less_le_not_le)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   459
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   460
lemmas not_Ici_eq_UNIV[simp] = not_UNIV_eq_Ici[symmetric]
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   461
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   462
lemma not_Icc_eq_Iic[simp]: "\<not> {l..h} = {..h'}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   463
unfolding atLeastAtMost_def using not_Iic_le_Ici[of h'] by blast
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   464
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   465
lemmas not_Iic_eq_Icc[simp] = not_Icc_eq_Iic[symmetric]
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   466
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   467
end
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   468
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   469
53216
ad2e09c30aa8 renamed inner_dense_linorder to dense_linorder
hoelzl
parents: 52729
diff changeset
   470
context dense_linorder
42891
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   471
begin
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   472
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   473
lemma greaterThanLessThan_empty_iff[simp]:
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   474
  "{ a <..< b } = {} \<longleftrightarrow> b \<le> a"
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   475
  using dense[of a b] by (cases "a < b") auto
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   476
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   477
lemma greaterThanLessThan_empty_iff2[simp]:
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   478
  "{} = { a <..< b } \<longleftrightarrow> b \<le> a"
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   479
  using dense[of a b] by (cases "a < b") auto
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   480
42901
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   481
lemma atLeastLessThan_subseteq_atLeastAtMost_iff:
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   482
  "{a ..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   483
  using dense[of "max a d" "b"]
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   484
  by (force simp: subset_eq Ball_def not_less[symmetric])
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   485
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   486
lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   487
  "{a <.. b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   488
  using dense[of "a" "min c b"]
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   489
  by (force simp: subset_eq Ball_def not_less[symmetric])
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   490
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   491
lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   492
  "{a <..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   493
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   494
  by (force simp: subset_eq Ball_def not_less[symmetric])
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   495
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61378
diff changeset
   496
lemma greaterThanLessThan_subseteq_greaterThanLessThan:
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61378
diff changeset
   497
  "{a <..< b} \<subseteq> {c <..< d} \<longleftrightarrow> (a < b \<longrightarrow> a \<ge> c \<and> b \<le> d)"
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61378
diff changeset
   498
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61378
diff changeset
   499
  by (force simp: subset_eq Ball_def not_less[symmetric])
43657
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   500
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   501
lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   502
  "{a <.. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b < d)"
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   503
  using dense[of "a" "min c b"]
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   504
  by (force simp: subset_eq Ball_def not_less[symmetric])
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   505
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   506
lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   507
  "{a <..< b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   508
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   509
  by (force simp: subset_eq Ball_def not_less[symmetric])
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   510
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   511
lemma greaterThanLessThan_subseteq_greaterThanAtMost_iff:
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   512
  "{a <..< b} \<subseteq> { c <.. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   513
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   514
  by (force simp: subset_eq Ball_def not_less[symmetric])
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   515
42891
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   516
end
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   517
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   518
context no_top
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   519
begin
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   520
51334
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   521
lemma greaterThan_non_empty[simp]: "{x <..} \<noteq> {}"
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   522
  using gt_ex[of x] by auto
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   523
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   524
end
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   525
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   526
context no_bot
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   527
begin
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   528
51334
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   529
lemma lessThan_non_empty[simp]: "{..< x} \<noteq> {}"
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   530
  using lt_ex[of x] by auto
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   531
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   532
end
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   533
32408
a1a85b0a26f7 new interval lemma
nipkow
parents: 32400
diff changeset
   534
lemma (in linorder) atLeastLessThan_subset_iff:
67091
1393c2340eec more symbols;
wenzelm
parents: 66936
diff changeset
   535
  "{a..<b} \<subseteq> {c..<d} \<Longrightarrow> b \<le> a \<or> c\<le>a \<and> b\<le>d"
75669
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
   536
proof (cases "a < b")
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
   537
  case True
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
   538
  assume assm: "{a..<b} \<subseteq> {c..<d}"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
   539
  then have 1: "c \<le> a \<and> a \<le> d"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
   540
    using True by (auto simp add: subset_eq Ball_def)
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
   541
  then have 2: "b \<le> d"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
   542
    using assm by (auto simp add: subset_eq)
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
   543
  from 1 2 show ?thesis
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
   544
    by simp
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
   545
qed (auto)
32408
a1a85b0a26f7 new interval lemma
nipkow
parents: 32400
diff changeset
   546
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   547
lemma atLeastLessThan_inj:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   548
  fixes a b c d :: "'a::linorder"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   549
  assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   550
  shows "a = c" "b = d"
70749
5d06b7bb9d22 More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents: 70746
diff changeset
   551
using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le antisym_conv2 subset_refl)+
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   552
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   553
lemma atLeastLessThan_eq_iff:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   554
  fixes a b c d :: "'a::linorder"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   555
  assumes "a < b" "c < d"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   556
  shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   557
  using atLeastLessThan_inj assms by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   558
73411
1f1366966296 avoid name clash
haftmann
parents: 73139
diff changeset
   559
lemma (in linorder) Ioc_inj: 
1f1366966296 avoid name clash
haftmann
parents: 73139
diff changeset
   560
  \<open>{a <.. b} = {c <.. d} \<longleftrightarrow> (b \<le> a \<and> d \<le> c) \<or> a = c \<and> b = d\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>)
1f1366966296 avoid name clash
haftmann
parents: 73139
diff changeset
   561
proof
1f1366966296 avoid name clash
haftmann
parents: 73139
diff changeset
   562
  assume ?Q
1f1366966296 avoid name clash
haftmann
parents: 73139
diff changeset
   563
  then show ?P
1f1366966296 avoid name clash
haftmann
parents: 73139
diff changeset
   564
    by auto
1f1366966296 avoid name clash
haftmann
parents: 73139
diff changeset
   565
next
1f1366966296 avoid name clash
haftmann
parents: 73139
diff changeset
   566
  assume ?P
1f1366966296 avoid name clash
haftmann
parents: 73139
diff changeset
   567
  then have \<open>a < x \<and> x \<le> b \<longleftrightarrow> c < x \<and> x \<le> d\<close> for x
1f1366966296 avoid name clash
haftmann
parents: 73139
diff changeset
   568
    by (simp add: set_eq_iff)
1f1366966296 avoid name clash
haftmann
parents: 73139
diff changeset
   569
  from this [of a] this [of b] this [of c] this [of d] show ?Q
1f1366966296 avoid name clash
haftmann
parents: 73139
diff changeset
   570
    by auto
1f1366966296 avoid name clash
haftmann
parents: 73139
diff changeset
   571
qed
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   572
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   573
lemma (in order) Iio_Int_singleton: "{..<k} \<inter> {x} = (if x < k then {x} else {})"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   574
  by auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   575
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   576
lemma (in linorder) Ioc_subset_iff: "{a<..b} \<subseteq> {c<..d} \<longleftrightarrow> (b \<le> a \<or> c \<le> a \<and> b \<le> d)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   577
  by (auto simp: subset_eq Ball_def) (metis less_le not_less)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   578
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52380
diff changeset
   579
lemma (in order_bot) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"
51334
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   580
by (auto simp: set_eq_iff intro: le_bot)
51328
d63ec23c9125 move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents: 51152
diff changeset
   581
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52380
diff changeset
   582
lemma (in order_top) atMost_eq_UNIV_iff: "{..x} = UNIV \<longleftrightarrow> x = top"
51334
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   583
by (auto simp: set_eq_iff intro: top_le)
51328
d63ec23c9125 move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents: 51152
diff changeset
   584
51334
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   585
lemma (in bounded_lattice) atLeastAtMost_eq_UNIV_iff:
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   586
  "{x..y} = UNIV \<longleftrightarrow> (x = bot \<and> y = top)"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   587
by (auto simp: set_eq_iff intro: top_le le_bot)
51328
d63ec23c9125 move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents: 51152
diff changeset
   588
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56480
diff changeset
   589
lemma Iio_eq_empty_iff: "{..< n::'a::{linorder, order_bot}} = {} \<longleftrightarrow> n = bot"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56480
diff changeset
   590
  by (auto simp: set_eq_iff not_less le_bot)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56480
diff changeset
   591
68361
20375f232f3b infinite product material
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   592
lemma lessThan_empty_iff: "{..< n::nat} = {} \<longleftrightarrow> n = 0"
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56480
diff changeset
   593
  by (simp add: Iio_eq_empty_iff bot_nat_def)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56480
diff changeset
   594
58970
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   595
lemma mono_image_least:
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   596
  assumes f_mono: "mono f" and f_img: "f ` {m ..< n} = {m' ..< n'}" "m < n"
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   597
  shows "f m = m'"
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   598
proof -
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   599
  from f_img have "{m' ..< n'} \<noteq> {}"
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   600
    by (metis atLeastLessThan_empty_iff image_is_empty)
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   601
  with f_img have "m' \<in> f ` {m ..< n}" by auto
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   602
  then obtain k where "f k = m'" "m \<le> k" by auto
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   603
  moreover have "m' \<le> f m" using f_img by auto
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   604
  ultimately show "f m = m'"
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   605
    using f_mono by (auto elim: monoE[where x=m and y=k])
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   606
qed
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   607
51328
d63ec23c9125 move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents: 51152
diff changeset
   608
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   609
subsection \<open>Infinite intervals\<close>
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   610
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   611
context dense_linorder
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   612
begin
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   613
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   614
lemma infinite_Ioo:
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   615
  assumes "a < b"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   616
  shows "\<not> finite {a<..<b}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   617
proof
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   618
  assume fin: "finite {a<..<b}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   619
  moreover have ne: "{a<..<b} \<noteq> {}"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   620
    using \<open>a < b\<close> by auto
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   621
  ultimately have "a < Max {a <..< b}" "Max {a <..< b} < b"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   622
    using Max_in[of "{a <..< b}"] by auto
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   623
  then obtain x where "Max {a <..< b} < x" "x < b"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   624
    using dense[of "Max {a<..<b}" b] by auto
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   625
  then have "x \<in> {a <..< b}"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   626
    using \<open>a < Max {a <..< b}\<close> by auto
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   627
  then have "x \<le> Max {a <..< b}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   628
    using fin by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   629
  with \<open>Max {a <..< b} < x\<close> show False by auto
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   630
qed
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   631
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   632
lemma infinite_Icc: "a < b \<Longrightarrow> \<not> finite {a .. b}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   633
  using greaterThanLessThan_subseteq_atLeastAtMost_iff[of a b a b] infinite_Ioo[of a b]
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   634
  by (auto dest: finite_subset)
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   635
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   636
lemma infinite_Ico: "a < b \<Longrightarrow> \<not> finite {a ..< b}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   637
  using greaterThanLessThan_subseteq_atLeastLessThan_iff[of a b a b] infinite_Ioo[of a b]
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   638
  by (auto dest: finite_subset)
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   639
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   640
lemma infinite_Ioc: "a < b \<Longrightarrow> \<not> finite {a <.. b}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   641
  using greaterThanLessThan_subseteq_greaterThanAtMost_iff[of a b a b] infinite_Ioo[of a b]
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   642
  by (auto dest: finite_subset)
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   643
63967
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63935
diff changeset
   644
lemma infinite_Ioo_iff [simp]: "infinite {a<..<b} \<longleftrightarrow> a < b"
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63935
diff changeset
   645
  using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ioo)
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63935
diff changeset
   646
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63935
diff changeset
   647
lemma infinite_Icc_iff [simp]: "infinite {a .. b} \<longleftrightarrow> a < b"
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63935
diff changeset
   648
  using not_less_iff_gr_or_eq by (fastforce simp: infinite_Icc)
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63935
diff changeset
   649
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63935
diff changeset
   650
lemma infinite_Ico_iff [simp]: "infinite {a..<b} \<longleftrightarrow> a < b"
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63935
diff changeset
   651
  using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ico)
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63935
diff changeset
   652
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63935
diff changeset
   653
lemma infinite_Ioc_iff [simp]: "infinite {a<..b} \<longleftrightarrow> a < b"
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63935
diff changeset
   654
  using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ioc)
2aa42596edc3 new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents: 63935
diff changeset
   655
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   656
end
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   657
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   658
lemma infinite_Iio: "\<not> finite {..< a :: 'a :: {no_bot, linorder}}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   659
proof
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   660
  assume "finite {..< a}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   661
  then have *: "\<And>x. x < a \<Longrightarrow> Min {..< a} \<le> x"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   662
    by auto
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   663
  obtain x where "x < a"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   664
    using lt_ex by auto
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   665
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   666
  obtain y where "y < Min {..< a}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   667
    using lt_ex by auto
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   668
  also have "Min {..< a} \<le> x"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   669
    using \<open>x < a\<close> by fact
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   670
  also note \<open>x < a\<close>
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   671
  finally have "Min {..< a} \<le> y"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   672
    by fact
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   673
  with \<open>y < Min {..< a}\<close> show False by auto
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   674
qed
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   675
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   676
lemma infinite_Iic: "\<not> finite {.. a :: 'a :: {no_bot, linorder}}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   677
  using infinite_Iio[of a] finite_subset[of "{..< a}" "{.. a}"]
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   678
  by (auto simp: subset_eq less_imp_le)
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   679
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   680
lemma infinite_Ioi: "\<not> finite {a :: 'a :: {no_top, linorder} <..}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   681
proof
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   682
  assume "finite {a <..}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   683
  then have *: "\<And>x. a < x \<Longrightarrow> x \<le> Max {a <..}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   684
    by auto
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   685
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   686
  obtain y where "Max {a <..} < y"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   687
    using gt_ex by auto
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   688
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63417
diff changeset
   689
  obtain x where x: "a < x"
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   690
    using gt_ex by auto
63540
f8652d0534fa tuned proofs -- avoid unstructured calculation;
wenzelm
parents: 63417
diff changeset
   691
  also from x have "x \<le> Max {a <..}"
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   692
    by fact
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   693
  also note \<open>Max {a <..} < y\<close>
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   694
  finally have "y \<le> Max { a <..}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   695
    by fact
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   696
  with \<open>Max {a <..} < y\<close> show False by auto
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   697
qed
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   698
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   699
lemma infinite_Ici: "\<not> finite {a :: 'a :: {no_top, linorder} ..}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   700
  using infinite_Ioi[of a] finite_subset[of "{a <..}" "{a ..}"]
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   701
  by (auto simp: subset_eq less_imp_le)
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   702
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   703
subsubsection \<open>Intersection\<close>
32456
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   704
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   705
context linorder
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   706
begin
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   707
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   708
lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   709
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   710
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   711
lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   712
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   713
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   714
lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   715
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   716
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   717
lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   718
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   719
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   720
lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   721
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   722
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   723
lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   724
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   725
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   726
lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   727
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   728
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   729
lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   730
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   731
50417
f18b92f91818 add Int_atMost
hoelzl
parents: 47988
diff changeset
   732
lemma Int_atMost[simp]: "{..a} \<inter> {..b} = {.. min a b}"
f18b92f91818 add Int_atMost
hoelzl
parents: 47988
diff changeset
   733
  by (auto simp: min_def)
f18b92f91818 add Int_atMost
hoelzl
parents: 47988
diff changeset
   734
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   735
lemma Ioc_disjoint: "{a<..b} \<inter> {c<..d} = {} \<longleftrightarrow> b \<le> a \<or> d \<le> c \<or> b \<le> c \<or> d \<le> a"
63092
a949b2a5f51d eliminated use of empty "assms";
wenzelm
parents: 62789
diff changeset
   736
  by auto
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   737
32456
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   738
end
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   739
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   740
context complete_lattice
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   741
begin
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   742
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   743
lemma
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   744
  shows Sup_atLeast[simp]: "Sup {x ..} = top"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   745
    and Sup_greaterThanAtLeast[simp]: "x < top \<Longrightarrow> Sup {x <..} = top"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   746
    and Sup_atMost[simp]: "Sup {.. y} = y"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   747
    and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   748
    and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   749
  by (auto intro!: Sup_eqI)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   750
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   751
lemma
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   752
  shows Inf_atMost[simp]: "Inf {.. x} = bot"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   753
    and Inf_atMostLessThan[simp]: "top < x \<Longrightarrow> Inf {..< x} = bot"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   754
    and Inf_atLeast[simp]: "Inf {x ..} = x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   755
    and Inf_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Inf { x .. y} = x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   756
    and Inf_atLeastLessThan[simp]: "x < y \<Longrightarrow> Inf { x ..< y} = x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   757
  by (auto intro!: Inf_eqI)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   758
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   759
end
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   760
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   761
lemma
53216
ad2e09c30aa8 renamed inner_dense_linorder to dense_linorder
hoelzl
parents: 52729
diff changeset
   762
  fixes x y :: "'a :: {complete_lattice, dense_linorder}"
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   763
  shows Sup_lessThan[simp]: "Sup {..< y} = y"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   764
    and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   765
    and Sup_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Sup { x <..< y} = y"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   766
    and Inf_greaterThan[simp]: "Inf {x <..} = x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   767
    and Inf_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Inf { x <.. y} = x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   768
    and Inf_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Inf { x <..< y} = x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   769
  by (auto intro!: Inf_eqI Sup_eqI intro: dense_le dense_le_bounded dense_ge dense_ge_bounded)
32456
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   770
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   771
subsection \<open>Intervals of natural numbers\<close>
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   772
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
   773
subsubsection \<open>The Constant \<^term>\<open>lessThan\<close>\<close>
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   774
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   775
lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   776
by (simp add: lessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   777
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   778
lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   779
by (simp add: lessThan_def less_Suc_eq, blast)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   780
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   781
text \<open>The following proof is convenient in induction proofs where
39072
1030b1a166ef Add lessThan_Suc_eq_insert_0
hoelzl
parents: 37664
diff changeset
   782
new elements get indices at the beginning. So it is used to transform
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
   783
\<^term>\<open>{..<Suc n}\<close> to \<^term>\<open>0::nat\<close> and \<^term>\<open>{..< n}\<close>.\<close>
39072
1030b1a166ef Add lessThan_Suc_eq_insert_0
hoelzl
parents: 37664
diff changeset
   784
69700
7a92cbec7030 new material about summations and powers, along with some tweaks
paulson <lp15@cam.ac.uk>
parents: 69593
diff changeset
   785
lemma zero_notin_Suc_image [simp]: "0 \<notin> Suc ` A"
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58970
diff changeset
   786
  by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58970
diff changeset
   787
39072
1030b1a166ef Add lessThan_Suc_eq_insert_0
hoelzl
parents: 37664
diff changeset
   788
lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58970
diff changeset
   789
  by (auto simp: image_iff less_Suc_eq_0_disj)
39072
1030b1a166ef Add lessThan_Suc_eq_insert_0
hoelzl
parents: 37664
diff changeset
   790
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   791
lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   792
by (simp add: lessThan_def atMost_def less_Suc_eq_le)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   793
68361
20375f232f3b infinite product material
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   794
lemma atMost_Suc_eq_insert_0: "{.. Suc n} = insert 0 (Suc ` {.. n})"
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58970
diff changeset
   795
  unfolding lessThan_Suc_atMost[symmetric] lessThan_Suc_eq_insert_0[of "Suc n"] ..
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58970
diff changeset
   796
69276
3d954183b707 replaced some ancient ASCII syntax
haftmann
parents: 69235
diff changeset
   797
lemma UN_lessThan_UNIV: "(\<Union>m::nat. lessThan m) = UNIV"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   798
by blast
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   799
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
   800
subsubsection \<open>The Constant \<^term>\<open>greaterThan\<close>\<close>
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   801
65273
917ae0ba03a2 Removal of [simp] status for greaterThan_0. Moved two theorems into main HOL.
paulson <lp15@cam.ac.uk>
parents: 64773
diff changeset
   802
lemma greaterThan_0: "greaterThan 0 = range Suc"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   803
  unfolding greaterThan_def
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   804
  by (blast dest: gr0_conv_Suc [THEN iffD1])
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   805
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   806
lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   807
  unfolding greaterThan_def
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   808
  by (auto elim: linorder_neqE)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   809
69276
3d954183b707 replaced some ancient ASCII syntax
haftmann
parents: 69235
diff changeset
   810
lemma INT_greaterThan_UNIV: "(\<Inter>m::nat. greaterThan m) = {}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   811
  by blast
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   812
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
   813
subsubsection \<open>The Constant \<^term>\<open>atLeast\<close>\<close>
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   814
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   815
lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   816
by (unfold atLeast_def UNIV_def, simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   817
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   818
lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   819
  unfolding atLeast_def by (auto simp: order_le_less Suc_le_eq)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   820
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   821
lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   822
  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   823
69276
3d954183b707 replaced some ancient ASCII syntax
haftmann
parents: 69235
diff changeset
   824
lemma UN_atLeast_UNIV: "(\<Union>m::nat. atLeast m) = UNIV"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   825
  by blast
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   826
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
   827
subsubsection \<open>The Constant \<^term>\<open>atMost\<close>\<close>
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   828
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   829
lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   830
  by (simp add: atMost_def)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   831
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   832
lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   833
  unfolding atMost_def by (auto simp add: less_Suc_eq order_le_less)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   834
69276
3d954183b707 replaced some ancient ASCII syntax
haftmann
parents: 69235
diff changeset
   835
lemma UN_atMost_UNIV: "(\<Union>m::nat. atMost m) = UNIV"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   836
  by blast
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   837
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
   838
subsubsection \<open>The Constant \<^term>\<open>atLeastLessThan\<close>\<close>
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   839
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   840
text\<open>The orientation of the following 2 rules is tricky. The lhs is
24449
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24286
diff changeset
   841
defined in terms of the rhs.  Hence the chosen orientation makes sense
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24286
diff changeset
   842
in this theory --- the reverse orientation complicates proofs (eg
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24286
diff changeset
   843
nontermination). But outside, when the definition of the lhs is rarely
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24286
diff changeset
   844
used, the opposite orientation seems preferable because it reduces a
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   845
specific concept to a more general one.\<close>
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   846
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   847
lemma atLeast0LessThan [code_abbrev]: "{0::nat..<n} = {..<n}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   848
  by(simp add:lessThan_def atLeastLessThan_def)
24449
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24286
diff changeset
   849
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   850
lemma atLeast0AtMost [code_abbrev]: "{0..n::nat} = {..n}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   851
  by(simp add:atMost_def atLeastAtMost_def)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   852
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   853
lemma lessThan_atLeast0: "{..<n} = {0::nat..<n}"
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   854
  by (simp add: atLeast0LessThan)
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   855
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   856
lemma atMost_atLeast0: "{..n} = {0::nat..n}"
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   857
  by (simp add: atLeast0AtMost)
24449
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24286
diff changeset
   858
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24286
diff changeset
   859
lemma atLeastLessThan0: "{m..<0::nat} = {}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   860
  by (simp add: atLeastLessThan_def)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   861
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   862
lemma atLeast0_lessThan_Suc: "{0..<Suc n} = insert n {0..<n}"
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   863
  by (simp add: atLeast0LessThan lessThan_Suc)
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   864
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   865
lemma atLeast0_lessThan_Suc_eq_insert_0: "{0..<Suc n} = insert 0 (Suc ` {0..<n})"
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   866
  by (simp add: atLeast0LessThan lessThan_Suc_eq_insert_0)
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   867
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   868
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
   869
subsubsection \<open>The Constant \<^term>\<open>atLeastAtMost\<close>\<close>
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   870
69198
9218b7652839 moved lemmas
nipkow
parents: 69182
diff changeset
   871
lemma Icc_eq_insert_lb_nat: "m \<le> n \<Longrightarrow> {m..n} = insert m {Suc m..n}"
9218b7652839 moved lemmas
nipkow
parents: 69182
diff changeset
   872
by auto
9218b7652839 moved lemmas
nipkow
parents: 69182
diff changeset
   873
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   874
lemma atLeast0_atMost_Suc:
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   875
  "{0..Suc n} = insert (Suc n) {0..n}"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   876
  by (simp add: atLeast0AtMost atMost_Suc)
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   877
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   878
lemma atLeast0_atMost_Suc_eq_insert_0:
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   879
  "{0..Suc n} = insert 0 (Suc ` {0..n})"
68361
20375f232f3b infinite product material
paulson <lp15@cam.ac.uk>
parents: 68064
diff changeset
   880
  by (simp add: atLeast0AtMost atMost_Suc_eq_insert_0)
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   881
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
   882
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
   883
subsubsection \<open>Intervals of nats with \<^term>\<open>Suc\<close>\<close>
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   884
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   885
text\<open>Not a simprule because the RHS is too messy.\<close>
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   886
lemma atLeastLessThanSuc:
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   887
    "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   888
by (auto simp add: atLeastLessThan_def)
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   889
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   890
lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   891
by (auto simp add: atLeastLessThan_def)
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   892
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   893
lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   894
  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   895
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   896
lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   897
  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   898
      greaterThanAtMost_def)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   899
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   900
lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   901
  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   902
    greaterThanLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   903
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15542
diff changeset
   904
lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
71699
8e5c20e4e11a a few more applys
paulson <lp15@cam.ac.uk>
parents: 71535
diff changeset
   905
  by auto
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15542
diff changeset
   906
45932
6f08f8fe9752 add lemmas
noschinl
parents: 44890
diff changeset
   907
lemma atLeastAtMost_insertL: "m \<le> n \<Longrightarrow> insert m {Suc m..n} = {m ..n}"
71699
8e5c20e4e11a a few more applys
paulson <lp15@cam.ac.uk>
parents: 71535
diff changeset
   908
  by auto
45932
6f08f8fe9752 add lemmas
noschinl
parents: 44890
diff changeset
   909
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
   910
text \<open>The analogous result is useful on \<^typ>\<open>int\<close>:\<close>
43157
b505be6f029a atLeastAtMostSuc_conv on int
kleing
parents: 43156
diff changeset
   911
(* here, because we don't have an own int section *)
b505be6f029a atLeastAtMostSuc_conv on int
kleing
parents: 43156
diff changeset
   912
lemma atLeastAtMostPlus1_int_conv:
70746
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
   913
  "m \<le> 1+n \<Longrightarrow> {m..1+n} = insert (1+n) {m..n::int}"
43157
b505be6f029a atLeastAtMostSuc_conv on int
kleing
parents: 43156
diff changeset
   914
  by (auto intro: set_eqI)
b505be6f029a atLeastAtMostSuc_conv on int
kleing
parents: 43156
diff changeset
   915
33044
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   916
lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
   917
  by (induct k) (simp_all add: atLeastLessThanSuc)
33044
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   918
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   919
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   920
subsubsection \<open>Intervals and numerals\<close>
57113
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   921
67443
3abf6a722518 standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents: 67411
diff changeset
   922
lemma lessThan_nat_numeral:  \<comment> \<open>Evaluation for specific numerals\<close>
57113
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   923
  "lessThan (numeral k :: nat) = insert (pred_numeral k) (lessThan (pred_numeral k))"
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   924
  by (simp add: numeral_eq_Suc lessThan_Suc)
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   925
67443
3abf6a722518 standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents: 67411
diff changeset
   926
lemma atMost_nat_numeral:  \<comment> \<open>Evaluation for specific numerals\<close>
57113
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   927
  "atMost (numeral k :: nat) = insert (numeral k) (atMost (pred_numeral k))"
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   928
  by (simp add: numeral_eq_Suc atMost_Suc)
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   929
67443
3abf6a722518 standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents: 67411
diff changeset
   930
lemma atLeastLessThan_nat_numeral:  \<comment> \<open>Evaluation for specific numerals\<close>
62369
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62343
diff changeset
   931
  "atLeastLessThan m (numeral k :: nat) =
57113
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   932
     (if m \<le> (pred_numeral k) then insert (pred_numeral k) (atLeastLessThan m (pred_numeral k))
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   933
                 else {})"
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   934
  by (simp add: numeral_eq_Suc atLeastLessThanSuc)
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   935
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   936
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   937
subsubsection \<open>Image\<close>
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   938
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   939
context linordered_semidom
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   940
begin
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   941
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   942
lemma image_add_atLeast[simp]: "plus k ` {i..} = {k + i..}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   943
proof -
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   944
  have "n = k + (n - k)" if "i + k \<le> n" for n
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   945
  proof -
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   946
    have "n = (n - (k + i)) + (k + i)" using that
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   947
      by (metis add_commute le_add_diff_inverse)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   948
    then show "n = k + (n - k)"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   949
      by (metis local.add_diff_cancel_left' add_assoc add_commute)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   950
  qed
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   951
  then show ?thesis
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   952
    by (fastforce simp: add_le_imp_le_diff add.commute)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   953
qed
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   954
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
   955
lemma image_add_atLeastAtMost [simp]:
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   956
  "plus k ` {i..j} = {i + k..j + k}" (is "?A = ?B")
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   957
proof
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   958
  show "?A \<subseteq> ?B"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   959
    by (auto simp add: ac_simps)
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   960
next
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   961
  show "?B \<subseteq> ?A"
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   962
  proof
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   963
    fix n
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   964
    assume "n \<in> ?B"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   965
    then have "i \<le> n - k"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   966
      by (simp add: add_le_imp_le_diff)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   967
    have "n = n - k + k"
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60586
diff changeset
   968
    proof -
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   969
      from \<open>n \<in> ?B\<close> have "n = n - (i + k) + (i + k)"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   970
        by simp
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   971
      also have "\<dots> = n - k - i + i + k"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   972
        by (simp add: algebra_simps)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   973
      also have "\<dots> = n - k + k"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   974
        using \<open>i \<le> n - k\<close> by simp
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   975
      finally show ?thesis .
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60586
diff changeset
   976
    qed
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   977
    moreover have "n - k \<in> {i..j}"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   978
      using \<open>n \<in> ?B\<close>
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   979
      by (auto simp: add_le_imp_le_diff add_le_add_imp_diff_le)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   980
    ultimately show "n \<in> ?A"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   981
      by (simp add: ac_simps) 
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   982
  qed
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   983
qed
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   984
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
   985
lemma image_add_atLeastAtMost' [simp]:
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   986
  "(\<lambda>n. n + k) ` {i..j} = {i + k..j + k}"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   987
  by (simp add: add.commute [of _ k])
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   988
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
   989
lemma image_add_atLeastLessThan [simp]:
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   990
  "plus k ` {i..<j} = {i + k..<j + k}"
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
   991
  by (simp add: image_set_diff atLeastLessThan_eq_atLeastAtMost_diff ac_simps)
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
   992
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
   993
lemma image_add_atLeastLessThan' [simp]:
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   994
  "(\<lambda>n. n + k) ` {i..<j} = {i + k..<j + k}"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   995
  by (simp add: add.commute [of _ k])
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
   996
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   997
lemma image_add_greaterThanAtMost[simp]: "(+) c ` {a<..b} = {c + a<..c + b}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   998
  by (simp add: image_set_diff greaterThanAtMost_eq_atLeastAtMost_diff ac_simps)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
   999
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1000
end
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1001
35580
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1002
context ordered_ab_group_add
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1003
begin
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1004
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1005
lemma
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1006
  fixes x :: 'a
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1007
  shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1008
  and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1009
proof safe
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1010
  fix y assume "y < -x"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1011
  hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1012
  have "- (-y) \<in> uminus ` {x<..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1013
    by (rule imageI) (simp add: *)
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1014
  thus "y \<in> uminus ` {x<..}" by simp
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1015
next
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1016
  fix y assume "y \<le> -x"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1017
  have "- (-y) \<in> uminus ` {x..}"
75669
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1018
    by (rule imageI) (use \<open>y \<le> -x\<close>[THEN le_imp_neg_le] in \<open>simp\<close>)
35580
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1019
  thus "y \<in> uminus ` {x..}" by simp
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1020
qed simp_all
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1021
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1022
lemma
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1023
  fixes x :: 'a
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1024
  shows image_uminus_lessThan[simp]: "uminus ` {..<x} = {-x<..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1025
  and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1026
proof -
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1027
  have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1028
    and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1029
  thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1030
    by (simp_all add: image_image
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1031
        del: image_uminus_greaterThan image_uminus_atLeast)
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1032
qed
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1033
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1034
lemma
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1035
  fixes x :: 'a
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1036
  shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1037
  and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1038
  and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1039
  and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1040
  by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1041
      greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1042
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1043
lemma image_add_atMost[simp]: "(+) c ` {..a} = {..c + a}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1044
  by (auto intro!: image_eqI[where x="x - c" for x] simp: algebra_simps)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1045
35580
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
  1046
end
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1047
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1048
lemma image_Suc_atLeastAtMost [simp]:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1049
  "Suc ` {i..j} = {Suc i..Suc j}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1050
  using image_add_atLeastAtMost [of 1 i j]
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1051
    by (simp only: plus_1_eq_Suc) simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1052
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1053
lemma image_Suc_atLeastLessThan [simp]:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1054
  "Suc ` {i..<j} = {Suc i..<Suc j}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1055
  using image_add_atLeastLessThan [of 1 i j]
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1056
    by (simp only: plus_1_eq_Suc) simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1057
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1058
corollary image_Suc_atMost:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1059
  "Suc ` {..n} = {1..Suc n}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1060
  by (simp add: atMost_atLeast0 atLeastLessThanSuc_atLeastAtMost)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1061
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1062
corollary image_Suc_lessThan:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1063
  "Suc ` {..<n} = {1..n}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1064
  by (simp add: lessThan_atLeast0 atLeastLessThanSuc_atLeastAtMost)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1065
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1066
lemma image_diff_atLeastAtMost [simp]:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1067
  fixes d::"'a::linordered_idom" shows "((-) d ` {a..b}) = {d-b..d-a}"
75669
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1068
proof
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1069
  show "{d - b..d - a} \<subseteq> (-) d ` {a..b}"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1070
  proof
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1071
    fix x
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1072
    assume "x \<in> {d - b..d - a}"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1073
    then have "d - x \<in> {a..b}" and "x = d - (d - x)"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1074
      by auto
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1075
    then show "x \<in> (-) d ` {a..b}"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1076
      by (rule rev_image_eqI)
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1077
  qed
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1078
qed(auto)
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1079
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1080
lemma image_diff_atLeastLessThan [simp]:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1081
  fixes a b c::"'a::linordered_idom"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1082
  shows "(-) c ` {a..<b} = {c - b<..c - a}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1083
proof -
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1084
  have "(-) c ` {a..<b} = (+) c ` uminus ` {a ..<b}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1085
    unfolding image_image by simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1086
  also have "\<dots> = {c - b<..c - a}" by simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1087
  finally show ?thesis by simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1088
qed
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1089
67727
ce3e87a51488 moved Lipschitz continuity from AFP/Ordinary_Differential_Equations and AFP/Gromov_Hyperbolicity; moved lemmas from AFP/Gromov_Hyperbolicity/Library_Complements
immler
parents: 67685
diff changeset
  1090
lemma image_minus_const_greaterThanAtMost[simp]:
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1091
  fixes a b c::"'a::linordered_idom"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1092
  shows "(-) c ` {a<..b} = {c - b..<c - a}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1093
proof -
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1094
  have "(-) c ` {a<..b} = (+) c ` uminus ` {a<..b}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1095
    unfolding image_image by simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1096
  also have "\<dots> = {c - b..<c - a}" by simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1097
  finally show ?thesis by simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1098
qed
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1099
67727
ce3e87a51488 moved Lipschitz continuity from AFP/Ordinary_Differential_Equations and AFP/Gromov_Hyperbolicity; moved lemmas from AFP/Gromov_Hyperbolicity/Library_Complements
immler
parents: 67685
diff changeset
  1100
lemma image_minus_const_atLeast[simp]:
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1101
  fixes a c::"'a::linordered_idom"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1102
  shows "(-) c ` {a..} = {..c - a}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1103
proof -
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1104
  have "(-) c ` {a..} = (+) c ` uminus ` {a ..}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1105
    unfolding image_image by simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1106
  also have "\<dots> = {..c - a}" by simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1107
  finally show ?thesis by simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1108
qed
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1109
67727
ce3e87a51488 moved Lipschitz continuity from AFP/Ordinary_Differential_Equations and AFP/Gromov_Hyperbolicity; moved lemmas from AFP/Gromov_Hyperbolicity/Library_Complements
immler
parents: 67685
diff changeset
  1110
lemma image_minus_const_AtMost[simp]:
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1111
  fixes b c::"'a::linordered_idom"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1112
  shows "(-) c ` {..b} = {c - b..}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1113
proof -
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1114
  have "(-) c ` {..b} = (+) c ` uminus ` {..b}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1115
    unfolding image_image by simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1116
  also have "\<dots> = {c - b..}" by simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1117
  finally show ?thesis by simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1118
qed
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1119
67727
ce3e87a51488 moved Lipschitz continuity from AFP/Ordinary_Differential_Equations and AFP/Gromov_Hyperbolicity; moved lemmas from AFP/Gromov_Hyperbolicity/Library_Complements
immler
parents: 67685
diff changeset
  1120
lemma image_minus_const_atLeastAtMost' [simp]:
ce3e87a51488 moved Lipschitz continuity from AFP/Ordinary_Differential_Equations and AFP/Gromov_Hyperbolicity; moved lemmas from AFP/Gromov_Hyperbolicity/Library_Complements
immler
parents: 67685
diff changeset
  1121
  "(\<lambda>t. t-d)`{a..b} = {a-d..b-d}" for d::"'a::linordered_idom"
ce3e87a51488 moved Lipschitz continuity from AFP/Ordinary_Differential_Equations and AFP/Gromov_Hyperbolicity; moved lemmas from AFP/Gromov_Hyperbolicity/Library_Complements
immler
parents: 67685
diff changeset
  1122
  by (metis (no_types, lifting) diff_conv_add_uminus image_add_atLeastAtMost' image_cong)
ce3e87a51488 moved Lipschitz continuity from AFP/Ordinary_Differential_Equations and AFP/Gromov_Hyperbolicity; moved lemmas from AFP/Gromov_Hyperbolicity/Library_Complements
immler
parents: 67685
diff changeset
  1123
69502
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1124
context linordered_field
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1125
begin
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1126
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1127
lemma image_mult_atLeastAtMost [simp]:
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68618
diff changeset
  1128
  "((*) d ` {a..b}) = {d*a..d*b}" if "d>0"
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1129
  using that
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1130
  by (auto simp: field_simps mult_le_cancel_right intro: rev_image_eqI [where x="x/d" for x])
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1131
69502
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1132
lemma image_divide_atLeastAtMost [simp]:
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1133
  "((\<lambda>c. c / d) ` {a..b}) = {a/d..b/d}" if "d>0"
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1134
proof -
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1135
  from that have "inverse d > 0"
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1136
    by simp
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1137
  with image_mult_atLeastAtMost [of "inverse d" a b]
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1138
  have "(*) (inverse d) ` {a..b} = {inverse d * a..inverse d * b}"
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1139
    by blast
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1140
  moreover have "(*) (inverse d) = (\<lambda>c. c / d)"
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1141
    by (simp add: fun_eq_iff field_simps)
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1142
  ultimately show ?thesis
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1143
    by simp
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1144
qed
0cf906072e20 more rules
haftmann
parents: 69276
diff changeset
  1145
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1146
lemma image_mult_atLeastAtMost_if:
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68618
diff changeset
  1147
  "(*) c ` {x .. y} =
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1148
    (if c > 0 then {c * x .. c * y} else if x \<le> y then {c * y .. c * x} else {})"
69768
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1149
proof (cases "c = 0 \<or> x > y")
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1150
  case True
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1151
  then show ?thesis
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1152
    by auto
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1153
next
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1154
  case False
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1155
  then have "x \<le> y"
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1156
    by auto
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1157
  from False consider "c < 0"| "c > 0"
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1158
    by (auto simp add: neq_iff)
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1159
  then show ?thesis
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1160
  proof cases
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1161
    case 1
69768
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1162
    have "(*) c ` {x..y} = {c * y..c * x}"
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1163
    proof (rule set_eqI)
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1164
      fix d
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1165
      from 1 have "inj (\<lambda>z. z / c)"
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1166
        by (auto intro: injI)
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1167
      then have "d \<in> (*) c ` {x..y} \<longleftrightarrow> d / c \<in> (\<lambda>z. z div c) ` (*) c ` {x..y}"
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1168
        by (subst inj_image_mem_iff) simp_all
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1169
      also have "\<dots> \<longleftrightarrow> d / c \<in> {x..y}"
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1170
        using 1 by (simp add: image_image)
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1171
      also have "\<dots> \<longleftrightarrow> d \<in> {c * y..c * x}"
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1172
        by (auto simp add: field_simps 1)
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1173
      finally show "d \<in> (*) c ` {x..y} \<longleftrightarrow> d \<in> {c * y..c * x}" .
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1174
    qed
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1175
    with \<open>x \<le> y\<close> show ?thesis
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1176
      by auto
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1177
  qed (simp add: mult_left_mono_neg)
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1178
qed
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1179
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1180
lemma image_mult_atLeastAtMost_if':
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1181
  "(\<lambda>x. x * c) ` {x..y} =
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1182
    (if x \<le> y then if c > 0 then {x * c .. y * c} else {y * c .. x * c} else {})"
69768
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1183
  using image_mult_atLeastAtMost_if [of c x y] by (auto simp add: ac_simps)
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1184
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1185
lemma image_affinity_atLeastAtMost:
69768
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1186
  "((\<lambda>x. m * x + c) ` {a..b}) = (if {a..b} = {} then {}
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1187
            else if 0 \<le> m then {m * a + c .. m * b + c}
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1188
            else {m * b + c .. m * a + c})"
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1189
proof -
69768
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1190
  have *: "(\<lambda>x. m * x + c) = ((\<lambda>x. x + c) \<circ> (*) m)"
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1191
    by (simp add: fun_eq_iff)
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1192
  show ?thesis by (simp only: * image_comp [symmetric] image_mult_atLeastAtMost_if)
7e4966eaf781 proper congruence rule for image operator
haftmann
parents: 69700
diff changeset
  1193
    (auto simp add: mult_le_cancel_left)
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1194
qed
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1195
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1196
lemma image_affinity_atLeastAtMost_diff:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1197
  "((\<lambda>x. m*x - c) ` {a..b}) = (if {a..b}={} then {}
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1198
            else if 0 \<le> m then {m*a - c .. m*b - c}
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1199
            else {m*b - c .. m*a - c})"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1200
  using image_affinity_atLeastAtMost [of m "-c" a b]
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1201
  by simp
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1202
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1203
lemma image_affinity_atLeastAtMost_div:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1204
  "((\<lambda>x. x/m + c) ` {a..b}) = (if {a..b}={} then {}
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1205
            else if 0 \<le> m then {a/m + c .. b/m + c}
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1206
            else {b/m + c .. a/m + c})"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1207
  using image_affinity_atLeastAtMost [of "inverse m" c a b]
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1208
  by (simp add: field_class.field_divide_inverse algebra_simps inverse_eq_divide)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1209
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1210
lemma image_affinity_atLeastAtMost_div_diff:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1211
  "((\<lambda>x. x/m - c) ` {a..b}) = (if {a..b}={} then {}
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1212
            else if 0 \<le> m then {a/m - c .. b/m - c}
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1213
            else {b/m - c .. a/m - c})"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1214
  using image_affinity_atLeastAtMost_diff [of "inverse m" c a b]
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1215
  by (simp add: field_class.field_divide_inverse algebra_simps inverse_eq_divide)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1216
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1217
end
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1218
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1219
lemma atLeast1_lessThan_eq_remove0:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1220
  "{Suc 0..<n} = {..<n} - {0}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1221
  by auto
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1222
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1223
lemma atLeast1_atMost_eq_remove0:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1224
  "{Suc 0..n} = {..n} - {0}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1225
  by auto
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1226
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1227
lemma image_add_int_atLeastLessThan:
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1228
    "(\<lambda>x. x + (l::int)) ` {0..<u-l} = {l..<u}"
75669
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1229
  by safe auto
67685
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1230
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1231
lemma image_minus_const_atLeastLessThan_nat:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1232
  fixes c :: nat
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1233
  shows "(\<lambda>i. i - c) ` {x ..< y} =
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1234
      (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1235
    (is "_ = ?right")
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1236
proof safe
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1237
  fix a assume a: "a \<in> ?right"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1238
  show "a \<in> (\<lambda>i. i - c) ` {x ..< y}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1239
  proof cases
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1240
    assume "c < y" with a show ?thesis
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1241
      by (auto intro!: image_eqI[of _ _ "a + c"])
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1242
  next
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1243
    assume "\<not> c < y" with a show ?thesis
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1244
      by (auto intro!: image_eqI[of _ _ x] split: if_split_asm)
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1245
  qed
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1246
qed auto
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1247
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1248
lemma image_int_atLeastLessThan:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1249
  "int ` {a..<b} = {int a..<int b}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1250
  by (auto intro!: image_eqI [where x = "nat x" for x])
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1251
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1252
lemma image_int_atLeastAtMost:
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1253
  "int ` {a..b} = {int a..int b}"
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1254
  by (auto intro!: image_eqI [where x = "nat x" for x])
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1255
bdff8bf0a75b moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents: 67613
diff changeset
  1256
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1257
subsubsection \<open>Finiteness\<close>
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1258
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1259
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1260
  by (induct k) (simp_all add: lessThan_Suc)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1261
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1262
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1263
  by (induct k) (simp_all add: atMost_Suc)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1264
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1265
lemma finite_greaterThanLessThan [iff]:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1266
  fixes l :: nat shows "finite {l<..<u}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1267
  by (simp add: greaterThanLessThan_def)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1268
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1269
lemma finite_atLeastLessThan [iff]:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1270
  fixes l :: nat shows "finite {l..<u}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1271
  by (simp add: atLeastLessThan_def)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1272
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1273
lemma finite_greaterThanAtMost [iff]:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1274
  fixes l :: nat shows "finite {l<..u}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1275
  by (simp add: greaterThanAtMost_def)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1276
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1277
lemma finite_atLeastAtMost [iff]:
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1278
  fixes l :: nat shows "finite {l..u}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1279
  by (simp add: atLeastAtMost_def)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1280
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1281
text \<open>A bounded set of natural numbers is finite.\<close>
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1282
lemma bounded_nat_set_is_finite: "(\<forall>i\<in>N. i < (n::nat)) \<Longrightarrow> finite N"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1283
  by (rule finite_subset [OF _ finite_lessThan]) auto
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1284
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1285
text \<open>A set of natural numbers is finite iff it is bounded.\<close>
31044
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
  1286
lemma finite_nat_set_iff_bounded:
67091
1393c2340eec more symbols;
wenzelm
parents: 66936
diff changeset
  1287
  "finite(N::nat set) = (\<exists>m. \<forall>n\<in>N. n<m)" (is "?F = ?B")
31044
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
  1288
proof
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
  1289
  assume f:?F  show ?B
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1290
    using Max_ge[OF \<open>?F\<close>, simplified less_Suc_eq_le[symmetric]] by blast
31044
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
  1291
next
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1292
  assume ?B show ?F using \<open>?B\<close> by(blast intro:bounded_nat_set_is_finite)
31044
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
  1293
qed
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
  1294
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1295
lemma finite_nat_set_iff_bounded_le: "finite(N::nat set) = (\<exists>m. \<forall>n\<in>N. n\<le>m)"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1296
  unfolding finite_nat_set_iff_bounded
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1297
  by (blast dest:less_imp_le_nat le_imp_less_Suc)
31044
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
  1298
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1299
lemma finite_less_ub:
75669
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1300
     "\<And>f::nat\<Rightarrow>nat. (!!n. n \<le> f n) \<Longrightarrow> finite {n. f n \<le> u}"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1301
  by (rule finite_subset[of _ "{..u}"])
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1302
    (auto intro: order_trans)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1303
64773
223b2ebdda79 Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  1304
lemma bounded_Max_nat:
223b2ebdda79 Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  1305
  fixes P :: "nat \<Rightarrow> bool"
223b2ebdda79 Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  1306
  assumes x: "P x" and M: "\<And>x. P x \<Longrightarrow> x \<le> M"
223b2ebdda79 Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  1307
  obtains m where "P m" "\<And>x. P x \<Longrightarrow> x \<le> m"
223b2ebdda79 Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  1308
proof -
223b2ebdda79 Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  1309
  have "finite {x. P x}"
223b2ebdda79 Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  1310
    using M finite_nat_set_iff_bounded_le by auto
223b2ebdda79 Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  1311
  then have "Max {x. P x} \<in> {x. P x}"
223b2ebdda79 Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  1312
    using Max_in x by auto
223b2ebdda79 Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  1313
  then show ?thesis
223b2ebdda79 Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  1314
    by (simp add: \<open>finite {x. P x}\<close> that)
223b2ebdda79 Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  1315
qed
223b2ebdda79 Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents: 64272
diff changeset
  1316
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
  1317
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1318
text\<open>Any subset of an interval of natural numbers the size of the
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1319
subset is exactly that interval.\<close>
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
  1320
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
  1321
lemma subset_card_intvl_is_intvl:
55085
0e8e4dc55866 moved 'fundef_cong' attribute (and other basic 'fun' stuff) up the dependency chain
blanchet
parents: 54606
diff changeset
  1322
  assumes "A \<subseteq> {k..<k + card A}"
0e8e4dc55866 moved 'fundef_cong' attribute (and other basic 'fun' stuff) up the dependency chain
blanchet
parents: 54606
diff changeset
  1323
  shows "A = {k..<k + card A}"
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
  1324
proof (cases "finite A")
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
  1325
  case True
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
  1326
  from this and assms show ?thesis
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
  1327
  proof (induct A rule: finite_linorder_max_induct)
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
  1328
    case empty thus ?case by auto
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
  1329
  next
33434
e9de8d69c1b9 fixed order of parameters in induction rules
nipkow
parents: 33318
diff changeset
  1330
    case (insert b A)
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
  1331
    hence *: "b \<notin> A" by auto
70746
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
  1332
    with insert have "A \<le> {k..<k + card A}" and "b = k + card A"
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
  1333
      by fastforce+
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
  1334
    with insert * show ?case by auto
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
  1335
  qed
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
  1336
next
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
  1337
  case False
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
  1338
  with assms show ?thesis by simp
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
  1339
qed
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
  1340
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
  1341
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1342
subsubsection \<open>Proving Inclusions and Equalities between Unions\<close>
32596
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1343
36755
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1344
lemma UN_le_eq_Un0:
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1345
  "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> M 0" (is "?A = ?B")
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1346
proof
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1347
  show "?A \<subseteq> ?B"
36755
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1348
  proof
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1349
    fix x assume "x \<in> ?A"
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1350
    then obtain i where i: "i\<le>n" "x \<in> M i" by auto
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1351
    show "x \<in> ?B"
36755
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1352
    proof(cases i)
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1353
      case 0 with i show ?thesis by simp
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1354
    next
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1355
      case (Suc j) with i show ?thesis by auto
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1356
    qed
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1357
  qed
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1358
next
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1359
  show "?B \<subseteq> ?A" by fastforce
36755
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1360
qed
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1361
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1362
lemma UN_le_add_shift:
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1363
  "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1364
proof
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1365
  show "?A \<subseteq> ?B" by fastforce
36755
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1366
next
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1367
  show "?B \<subseteq> ?A"
36755
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1368
  proof
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1369
    fix x assume "x \<in> ?B"
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1370
    then obtain i where i: "i \<in> {k..n+k}" "x \<in> M(i)" by auto
67091
1393c2340eec more symbols;
wenzelm
parents: 66936
diff changeset
  1371
    hence "i-k\<le>n \<and> x \<in> M((i-k)+k)" by auto
1393c2340eec more symbols;
wenzelm
parents: 66936
diff changeset
  1372
    thus "x \<in> ?A" by blast
36755
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1373
  qed
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1374
qed
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1375
70723
4e39d87c9737 imported new material mostly due to Sébastien Gouëzel
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  1376
lemma UN_le_add_shift_strict:
4e39d87c9737 imported new material mostly due to Sébastien Gouëzel
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  1377
  "(\<Union>i<n::nat. M(i+k)) = (\<Union>i\<in>{k..<n+k}. M i)" (is "?A = ?B")
4e39d87c9737 imported new material mostly due to Sébastien Gouëzel
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  1378
proof
4e39d87c9737 imported new material mostly due to Sébastien Gouëzel
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  1379
  show "?B \<subseteq> ?A"
4e39d87c9737 imported new material mostly due to Sébastien Gouëzel
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  1380
  proof
4e39d87c9737 imported new material mostly due to Sébastien Gouëzel
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  1381
    fix x assume "x \<in> ?B"
4e39d87c9737 imported new material mostly due to Sébastien Gouëzel
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  1382
    then obtain i where i: "i \<in> {k..<n+k}" "x \<in> M(i)" by auto
4e39d87c9737 imported new material mostly due to Sébastien Gouëzel
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  1383
    then have "i - k < n \<and> x \<in> M((i-k) + k)" by auto
4e39d87c9737 imported new material mostly due to Sébastien Gouëzel
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  1384
    then show "x \<in> ?A" using UN_le_add_shift by blast
4e39d87c9737 imported new material mostly due to Sébastien Gouëzel
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  1385
  qed
4e39d87c9737 imported new material mostly due to Sébastien Gouëzel
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  1386
qed (fastforce)
4e39d87c9737 imported new material mostly due to Sébastien Gouëzel
paulson <lp15@cam.ac.uk>
parents: 70365
diff changeset
  1387
62369
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62343
diff changeset
  1388
lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62343
diff changeset
  1389
  by (auto simp add: atLeast0LessThan)
32596
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1390
62343
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62128
diff changeset
  1391
lemma UN_finite_subset:
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62128
diff changeset
  1392
  "(\<And>n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"
32596
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1393
  by (subst UN_UN_finite_eq [symmetric]) blast
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1394
62369
acfc4ad7b76a instantiate topologies for nat, int and enat
hoelzl
parents: 62343
diff changeset
  1395
lemma UN_finite2_subset:
62343
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62128
diff changeset
  1396
  assumes "\<And>n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n + k}. B i)"
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62128
diff changeset
  1397
  shows "(\<Union>n. A n) \<subseteq> (\<Union>n. B n)"
75669
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1398
proof (rule UN_finite_subset, rule subsetI)
62343
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62128
diff changeset
  1399
  fix n and a
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62128
diff changeset
  1400
  from assms have "(\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n + k}. B i)" .
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62128
diff changeset
  1401
  moreover assume "a \<in> (\<Union>i\<in>{0..<n}. A i)"
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62128
diff changeset
  1402
  ultimately have "a \<in> (\<Union>i\<in>{0..<n + k}. B i)" by blast
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62128
diff changeset
  1403
  then show "a \<in> (\<Union>i. B i)" by (auto simp add: UN_UN_finite_eq)
24106dc44def prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents: 62128
diff changeset
  1404
qed
32596
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1405
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1406
lemma UN_finite2_eq:
75669
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1407
  assumes "(\<And>n::nat. (\<Union>i\<in>{0..<n}. A i) = (\<Union>i\<in>{0..<n + k}. B i))"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1408
  shows "(\<Union>n. A n) = (\<Union>n. B n)"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1409
proof (rule subset_antisym [OF UN_finite_subset UN_finite2_subset])
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1410
  fix n
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1411
  show "\<Union> (A ` {0..<n}) \<subseteq> (\<Union>n. B n)"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1412
    using assms by auto
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1413
next
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1414
  fix n
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1415
  show "\<Union> (B ` {0..<n}) \<subseteq> \<Union> (A ` {0..<n + k})"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1416
    using assms by (force simp add: atLeastLessThan_add_Un [of 0])+
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1417
qed
32596
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1418
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1419
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1420
subsubsection \<open>Cardinality\<close>
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1421
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1422
lemma card_lessThan [simp]: "card {..<u} = u"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
  1423
  by (induct u, simp_all add: lessThan_Suc)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1424
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1425
lemma card_atMost [simp]: "card {..u} = Suc u"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1426
  by (simp add: lessThan_Suc_atMost [THEN sym])
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1427
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1428
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
57113
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1429
proof -
75669
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1430
  have "(\<lambda>x. x + l) ` {..<u - l} \<subseteq> {l..<u}"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1431
    by auto
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1432
  moreover have "{l..<u} \<subseteq> (\<lambda>x. x + l) ` {..<u-l}"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1433
  proof
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1434
    fix x
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1435
    assume *: "x \<in> {l..<u}"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1436
    then have "x - l \<in> {..< u -l}"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1437
      by auto
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1438
    then have "(x - l) + l \<in> (\<lambda>x. x + l) ` {..< u -l}"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1439
      by auto
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1440
    then show "x \<in> (\<lambda>x. x + l) ` {..<u - l}"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1441
      using * by auto
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1442
  qed
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1443
  ultimately have "{l..<u} = (\<lambda>x. x + l) ` {..<u-l}"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1444
    by auto
57113
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1445
  then have "card {l..<u} = card {..<u-l}"
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1446
    by (simp add: card_image inj_on_def)
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1447
  then show ?thesis
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1448
    by simp
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1449
qed
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1450
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1451
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1452
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1453
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1454
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1455
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1456
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1457
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1458
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1459
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1460
lemma subset_eq_atLeast0_lessThan_finite:
63365
5340fb6633d0 more theorems
haftmann
parents: 63317
diff changeset
  1461
  fixes n :: nat
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1462
  assumes "N \<subseteq> {0..<n}"
63915
bab633745c7f tuned proofs;
wenzelm
parents: 63879
diff changeset
  1463
  shows "finite N"
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1464
  using assms finite_atLeastLessThan by (rule finite_subset)
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1465
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1466
lemma subset_eq_atLeast0_atMost_finite:
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1467
  fixes n :: nat
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1468
  assumes "N \<subseteq> {0..n}"
63915
bab633745c7f tuned proofs;
wenzelm
parents: 63879
diff changeset
  1469
  shows "finite N"
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1470
  using assms finite_atLeastAtMost by (rule finite_subset)
63365
5340fb6633d0 more theorems
haftmann
parents: 63317
diff changeset
  1471
26105
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
  1472
lemma ex_bij_betw_nat_finite:
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
  1473
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1474
  apply(drule finite_imp_nat_seg_image_inj_on)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1475
  apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1476
  done
26105
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
  1477
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
  1478
lemma ex_bij_betw_finite_nat:
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
  1479
  "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1480
  by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
26105
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
  1481
31438
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
  1482
lemma finite_same_card_bij:
67091
1393c2340eec more symbols;
wenzelm
parents: 66936
diff changeset
  1483
  "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> \<exists>h. bij_betw h A B"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1484
  apply(drule ex_bij_betw_finite_nat)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1485
  apply(drule ex_bij_betw_nat_finite)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1486
  apply(auto intro!:bij_betw_trans)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1487
  done
31438
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
  1488
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
  1489
lemma ex_bij_betw_nat_finite_1:
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
  1490
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1491
  by (rule finite_same_card_bij) auto
31438
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
  1492
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
  1493
lemma bij_betw_iff_card:
63114
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63099
diff changeset
  1494
  assumes "finite A" "finite B"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63099
diff changeset
  1495
  shows "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63099
diff changeset
  1496
proof
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63099
diff changeset
  1497
  assume "card A = card B"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63099
diff changeset
  1498
  moreover obtain f where "bij_betw f A {0 ..< card A}"
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63099
diff changeset
  1499
    using assms ex_bij_betw_finite_nat by blast
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
  1500
  moreover obtain g where "bij_betw g {0 ..< card B} B"
63114
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63099
diff changeset
  1501
    using assms ex_bij_betw_nat_finite by blast
67091
1393c2340eec more symbols;
wenzelm
parents: 66936
diff changeset
  1502
  ultimately have "bij_betw (g \<circ> f) A B"
63114
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63099
diff changeset
  1503
    by (auto simp: bij_betw_trans)
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
  1504
  thus "(\<exists>f. bij_betw f A B)" by blast
63114
27afe7af7379 Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents: 63099
diff changeset
  1505
qed (auto simp: bij_betw_same_card)
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
  1506
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1507
lemma subset_eq_atLeast0_lessThan_card:
63365
5340fb6633d0 more theorems
haftmann
parents: 63317
diff changeset
  1508
  fixes n :: nat
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1509
  assumes "N \<subseteq> {0..<n}"
63365
5340fb6633d0 more theorems
haftmann
parents: 63317
diff changeset
  1510
  shows "card N \<le> n"
5340fb6633d0 more theorems
haftmann
parents: 63317
diff changeset
  1511
proof -
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1512
  from assms finite_lessThan have "card N \<le> card {0..<n}"
63365
5340fb6633d0 more theorems
haftmann
parents: 63317
diff changeset
  1513
    using card_mono by blast
5340fb6633d0 more theorems
haftmann
parents: 63317
diff changeset
  1514
  then show ?thesis by simp
5340fb6633d0 more theorems
haftmann
parents: 63317
diff changeset
  1515
qed
5340fb6633d0 more theorems
haftmann
parents: 63317
diff changeset
  1516
69235
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1517
text \<open>Relational version of @{thm [source] card_inj_on_le}:\<close>
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1518
lemma card_le_if_inj_on_rel:
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1519
assumes "finite B"
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1520
  "\<And>a. a \<in> A \<Longrightarrow> \<exists>b. b\<in>B \<and> r a b"
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1521
  "\<And>a1 a2 b. \<lbrakk> a1 \<in> A;  a2 \<in> A;  b \<in> B;  r a1 b;  r a2 b \<rbrakk> \<Longrightarrow> a1 = a2"
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1522
shows "card A \<le> card B"
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1523
proof -
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1524
  let ?P = "\<lambda>a b. b \<in> B \<and> r a b"
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1525
  let ?f = "\<lambda>a. SOME b. ?P a b"
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1526
  have 1: "?f ` A \<subseteq> B"  by (auto intro: someI2_ex[OF assms(2)])
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1527
  have "inj_on ?f A"
75669
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1528
    unfolding inj_on_def
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1529
  proof safe
69235
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1530
    fix a1 a2 assume asms: "a1 \<in> A" "a2 \<in> A" "?f a1 = ?f a2"
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1531
    have 0: "?f a1 \<in> B" using "1" \<open>a1 \<in> A\<close> by blast
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1532
    have 1: "r a1 (?f a1)" using someI_ex[OF assms(2)[OF \<open>a1 \<in> A\<close>]] by blast
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1533
    have 2: "r a2 (?f a1)" using someI_ex[OF assms(2)[OF \<open>a2 \<in> A\<close>]] asms(3) by auto
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1534
    show "a1 = a2" using assms(3)[OF asms(1,2) 0 1 2] .
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1535
  qed
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1536
  with 1 show ?thesis using card_inj_on_le[of ?f A B] assms(1) by simp
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1537
qed
0e156963b636 simplified proof, moved lemma, added lemma
nipkow
parents: 69198
diff changeset
  1538
73555
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1539
lemma inj_on_funpow_least: \<^marker>\<open>contributor \<open>Lars Noschinski\<close>\<close>
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1540
  \<open>inj_on (\<lambda>k. (f ^^ k) s) {0..<n}\<close>
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1541
  if \<open>(f ^^ n) s = s\<close> \<open>\<And>m. 0 < m \<Longrightarrow> m < n \<Longrightarrow> (f ^^ m) s \<noteq> s\<close>
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1542
proof -
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1543
  { fix k l assume A: "k < n" "l < n" "k \<noteq> l" "(f ^^ k) s = (f ^^ l) s"
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1544
    define k' l' where "k' = min k l" and "l' = max k l"
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1545
    with A have A': "k' < l'" "(f ^^ k') s = (f ^^ l') s" "l' < n"
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1546
      by (auto simp: min_def max_def)
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1547
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1548
    have "s = (f ^^ ((n - l') + l')) s" using that \<open>l' < n\<close> by simp
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1549
    also have "\<dots> = (f ^^ (n - l')) ((f ^^ l') s)" by (simp add: funpow_add)
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1550
    also have "(f ^^ l') s = (f ^^ k') s" by (simp add: A')
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1551
    also have "(f ^^ (n - l')) \<dots> = (f ^^ (n - l' + k')) s" by (simp add: funpow_add)
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1552
    finally have "(f ^^ (n - l' + k')) s = s" by simp
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1553
    moreover have "n - l' + k' < n" "0 < n - l' + k'"using A' by linarith+
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1554
    ultimately have False using that(2) by auto
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1555
  }
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1556
  then show ?thesis by (intro inj_onI) auto
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1557
qed
92783562ab78 collected combinatorial material
haftmann
parents: 73411
diff changeset
  1558
63365
5340fb6633d0 more theorems
haftmann
parents: 63317
diff changeset
  1559
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1560
subsection \<open>Intervals of integers\<close>
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1561
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1562
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1563
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1564
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1565
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1566
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1567
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1568
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1569
    "{l+1..<u} = {l<..<u::int}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1570
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1571
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1572
subsubsection \<open>Finiteness\<close>
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1573
75669
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1574
lemma image_atLeastZeroLessThan_int:
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1575
  assumes "0 \<le> u"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1576
  shows "{(0::int)..<u} = int ` {..<nat u}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1577
  unfolding image_def lessThan_def
75669
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1578
proof
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1579
  show "{0..<u} \<subseteq> {y. \<exists>x\<in>{x. x < nat u}. y = int x}"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1580
  proof
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1581
    fix x
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1582
    assume "x \<in> {0..<u}"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1583
    then have "x = int (nat x)" and  "nat x < nat u"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1584
      by (auto simp add: zless_nat_eq_int_zless [THEN sym])
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1585
    then have "\<exists>xa<nat u. x = int xa"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1586
      using exI[of _ "(nat x)"] by simp
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1587
    then show "x \<in> {y. \<exists>x\<in>{x. x < nat u}. y = int x}"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1588
      by simp
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1589
  qed
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1590
qed (auto)
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1591
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1592
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1593
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1594
proof (cases "0 \<le> u")
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1595
  case True
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1596
  then show ?thesis
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1597
    by (auto simp: image_atLeastZeroLessThan_int)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1598
qed auto
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1599
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1600
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1601
  by (simp only: image_add_int_atLeastLessThan [symmetric, of l] finite_imageI finite_atLeastZeroLessThan_int)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1602
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1603
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1604
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1605
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1606
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1607
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1608
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1609
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1610
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1611
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
  1612
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1613
subsubsection \<open>Cardinality\<close>
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1614
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1615
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1616
proof (cases "0 \<le> u")
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1617
  case True
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1618
  then show ?thesis
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1619
    by (auto simp: image_atLeastZeroLessThan_int card_image inj_on_def)    
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1620
qed auto
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1621
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1622
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1623
proof -
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1624
  have "card {l..<u} = card {0..<u-l}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1625
    apply (subst image_add_int_atLeastLessThan [symmetric])
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1626
    apply (rule card_image)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1627
    apply (simp add: inj_on_def)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1628
    done
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1629
  then show ?thesis
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1630
    by (simp add: card_atLeastZeroLessThan_int)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1631
qed
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1632
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1633
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1634
  apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1635
  apply (auto simp add: algebra_simps)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1636
  done
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1637
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1638
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1639
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1640
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1641
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1642
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1643
27656
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1644
lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1645
proof -
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1646
  have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1647
  with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1648
qed
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1649
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1650
lemma card_less:
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1651
  assumes zero_in_M: "0 \<in> M"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1652
  shows "card {k \<in> M. k < Suc i} \<noteq> 0"
27656
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1653
proof -
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1654
  from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1655
  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1656
qed
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1657
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1658
lemma card_less_Suc2: 
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1659
  assumes "0 \<notin> M" shows "card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1660
proof -
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1661
  have *: "\<lbrakk>j \<in> M; j < Suc i\<rbrakk> \<Longrightarrow> j - Suc 0 < i \<and> Suc (j - Suc 0) \<in> M \<and> Suc 0 \<le> j" for j
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1662
    by (cases j) (use assms in auto)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1663
  show ?thesis
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1664
  proof (rule card_bij_eq)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1665
    show "inj_on Suc {k. Suc k \<in> M \<and> k < i}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1666
      by force
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1667
    show "inj_on (\<lambda>x. x - Suc 0) {k \<in> M. k < Suc i}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1668
      by (rule inj_on_diff_nat) (use * in blast)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1669
  qed (use * in auto)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1670
qed
27656
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1671
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1672
lemma card_less_Suc:
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1673
  assumes "0 \<in> M"
27656
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1674
    shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1675
proof -
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1676
  have "Suc (card {k. Suc k \<in> M \<and> k < i}) = Suc (card {k. Suc k \<in> M - {0} \<and> k < i})"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1677
    by simp
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1678
  also have "\<dots> = Suc (card {k \<in> M - {0}. k < Suc i})"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1679
    apply (subst card_less_Suc2)
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1680
    using assms by auto
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1681
  also have "\<dots> = Suc (card ({k \<in> M. k < Suc i} - {0}))"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1682
    by (force intro: arg_cong [where f=card])
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1683
  also have "\<dots> = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"
72302
d7d90ed4c74e fixed some remarkably ugly proofs
paulson <lp15@cam.ac.uk>
parents: 72268
diff changeset
  1684
    by (simp add: card.insert_remove)
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1685
  also have "... = card {k \<in> M. k < Suc i}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1686
    using assms
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1687
    by (force simp add: intro: arg_cong [where f=card])
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1688
  finally show ?thesis.
27656
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1689
qed
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1690
73139
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  1691
lemma card_le_Suc_Max: "finite S \<Longrightarrow> card S \<le> Suc (Max S)"
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  1692
proof (rule classical)
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  1693
  assume "finite S" and "\<not> Suc (Max S) \<ge> card S"
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  1694
  then have "Suc (Max S) < card S"
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  1695
    by simp
74885
2df334453c4c isabelle update_cartouches;
wenzelm
parents: 74101
diff changeset
  1696
  with \<open>finite S\<close> have "S \<subseteq> {0..Max S}"
73139
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  1697
    by auto
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  1698
  hence "card S \<le> card {0..Max S}"
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  1699
    by (intro card_mono; auto)
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  1700
  thus "card S \<le> Suc (Max S)"
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  1701
    by simp
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  1702
qed
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1703
80671
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1704
lemma finite_countable_subset:
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1705
  assumes "finite A" and A: "A \<subseteq> (\<Union>i::nat. B i)"
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1706
  obtains n where "A \<subseteq> (\<Union>i<n. B i)"
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1707
proof -
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1708
  obtain f where f: "\<And>x. x \<in> A \<Longrightarrow> x \<in> B(f x)"
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1709
    by (metis in_mono UN_iff A)
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1710
  define n where "n = Suc (Max (f`A))"
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1711
  have "finite (f ` A)"
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1712
    by (simp add: \<open>finite A\<close>)
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1713
  then have "A \<subseteq> (\<Union>i<n. B i)"
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1714
    unfolding UN_iff f n_def subset_iff
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1715
    by (meson Max_ge f imageI le_imp_less_Suc lessThan_iff)
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1716
  then show ?thesis ..
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1717
qed
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1718
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1719
lemma finite_countable_equals:
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1720
  assumes "finite A" "A = (\<Union>i::nat. B i)"
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1721
  obtains n where "A = (\<Union>i<n. B i)"
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1722
proof -
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1723
  obtain n where "A \<subseteq> (\<Union>i<n. B i)"
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1724
  proof (rule finite_countable_subset)
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1725
    show "A \<subseteq> \<Union> (range B)"
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1726
      by (force simp: assms)
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1727
  qed (use assms in auto)
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1728
  with that show ?thesis
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1729
    by (force simp: assms)
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1730
qed
daa604a00491 Two little lemmas
paulson <lp15@cam.ac.uk>
parents: 80612
diff changeset
  1731
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  1732
subsection \<open>Lemmas useful with the summation operator sum\<close>
13850
6d1bb3059818 new logical equivalences
paulson
parents: 13735
diff changeset
  1733
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1734
text \<open>For examples, see Algebra/poly/UnivPoly2.thy\<close>
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1735
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1736
subsubsection \<open>Disjoint Unions\<close>
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1737
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1738
text \<open>Singletons and open intervals\<close>
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1739
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1740
lemma ivl_disj_un_singleton:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1741
  "{l::'a::linorder} Un {l<..} = {l..}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1742
  "{..<u} Un {u::'a::linorder} = {..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1743
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1744
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1745
  "(l::'a::linorder) \<le> u ==> {l} Un {l<..u} = {l..u}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1746
  "(l::'a::linorder) \<le> u ==> {l..<u} Un {u} = {l..u}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
  1747
by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1748
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1749
text \<open>One- and two-sided intervals\<close>
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1750
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1751
lemma ivl_disj_un_one:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1752
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1753
  "(l::'a::linorder) \<le> u ==> {..<l} Un {l..<u} = {..<u}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1754
  "(l::'a::linorder) \<le> u ==> {..l} Un {l<..u} = {..u}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1755
  "(l::'a::linorder) \<le> u ==> {..<l} Un {l..u} = {..u}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1756
  "(l::'a::linorder) \<le> u ==> {l<..u} Un {u<..} = {l<..}"
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1757
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1758
  "(l::'a::linorder) \<le> u ==> {l..u} Un {u<..} = {l..}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1759
  "(l::'a::linorder) \<le> u ==> {l..<u} Un {u..} = {l..}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
  1760
by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1761
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1762
text \<open>Two- and two-sided intervals\<close>
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1763
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1764
lemma ivl_disj_un_two:
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1765
  "[| (l::'a::linorder) < m; m \<le> u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1766
  "[| (l::'a::linorder) \<le> m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1767
  "[| (l::'a::linorder) \<le> m; m \<le> u |] ==> {l..<m} Un {m..<u} = {l..<u}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1768
  "[| (l::'a::linorder) \<le> m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1769
  "[| (l::'a::linorder) < m; m \<le> u |] ==> {l<..<m} Un {m..u} = {l<..u}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1770
  "[| (l::'a::linorder) \<le> m; m \<le> u |] ==> {l<..m} Un {m<..u} = {l<..u}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1771
  "[| (l::'a::linorder) \<le> m; m \<le> u |] ==> {l..<m} Un {m..u} = {l..u}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1772
  "[| (l::'a::linorder) \<le> m; m \<le> u |] ==> {l..m} Un {m<..u} = {l..u}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
  1773
by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1774
60150
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  1775
lemma ivl_disj_un_two_touch:
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  1776
  "[| (l::'a::linorder) < m; m < u |] ==> {l<..m} Un {m..<u} = {l<..<u}"
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1777
  "[| (l::'a::linorder) \<le> m; m < u |] ==> {l..m} Un {m..<u} = {l..<u}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1778
  "[| (l::'a::linorder) < m; m \<le> u |] ==> {l<..m} Un {m..u} = {l<..u}"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1779
  "[| (l::'a::linorder) \<le> m; m \<le> u |] ==> {l..m} Un {m..u} = {l..u}"
60150
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  1780
by auto
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  1781
bd773c47ad0b New material about complex transcendental functions (especially Ln, Arg) and polynomials
paulson <lp15@cam.ac.uk>
parents: 60017
diff changeset
  1782
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two ivl_disj_un_two_touch
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1783
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1784
subsubsection \<open>Disjoint Intersections\<close>
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1785
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1786
text \<open>One- and two-sided intervals\<close>
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1787
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1788
lemma ivl_disj_int_one:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1789
  "{..l::'a::order} Int {l<..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1790
  "{..<l} Int {l..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1791
  "{..l} Int {l<..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1792
  "{..<l} Int {l..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1793
  "{l<..u} Int {u<..} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1794
  "{l<..<u} Int {u..} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1795
  "{l..u} Int {u<..} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1796
  "{l..<u} Int {u..} = {}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
  1797
  by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1798
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1799
text \<open>Two- and two-sided intervals\<close>
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1800
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1801
lemma ivl_disj_int_two:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1802
  "{l::'a::order<..<m} Int {m..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1803
  "{l<..m} Int {m<..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1804
  "{l..<m} Int {m..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1805
  "{l..m} Int {m<..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1806
  "{l<..<m} Int {m..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1807
  "{l<..m} Int {m<..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1808
  "{l..<m} Int {m..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1809
  "{l..m} Int {m<..u} = {}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
  1810
  by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1811
32456
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
  1812
lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1813
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1814
subsubsection \<open>Some Differences\<close>
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1815
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1816
lemma ivl_diff[simp]:
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1817
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1818
by(auto)
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1819
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56193
diff changeset
  1820
lemma (in linorder) lessThan_minus_lessThan [simp]:
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56193
diff changeset
  1821
  "{..< n} - {..< m} = {m ..< n}"
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56193
diff changeset
  1822
  by auto
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56193
diff changeset
  1823
60762
bf0c76ccee8d new material for multivariate analysis, etc.
paulson
parents: 60758
diff changeset
  1824
lemma (in linorder) atLeastAtMost_diff_ends:
bf0c76ccee8d new material for multivariate analysis, etc.
paulson
parents: 60758
diff changeset
  1825
  "{a..b} - {a, b} = {a<..<b}"
bf0c76ccee8d new material for multivariate analysis, etc.
paulson
parents: 60758
diff changeset
  1826
  by auto
bf0c76ccee8d new material for multivariate analysis, etc.
paulson
parents: 60758
diff changeset
  1827
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1828
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1829
subsubsection \<open>Some Subset Conditions\<close>
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1830
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1831
lemma ivl_subset [simp]: "({i..<j} \<subseteq> {m..<n}) = (j \<le> i \<or> m \<le> i \<and> j \<le> (n::'a::linorder))"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  1832
  using linorder_class.le_less_linear[of i n]
75669
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1833
  by safe (force intro: leI)+
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1834
15041
a6b1f0cef7b3 Got rid of Summation and made it a translation into setsum instead.
nipkow
parents: 14846
diff changeset
  1835
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1836
subsection \<open>Generic big monoid operation over intervals\<close>
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1837
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1838
context semiring_char_0
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1839
begin
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1840
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1841
lemma inj_on_of_nat [simp]:
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1842
  "inj_on of_nat N"
75669
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  1843
  by (rule inj_onI) simp
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1844
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1845
lemma bij_betw_of_nat [simp]:
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1846
  "bij_betw of_nat N A \<longleftrightarrow> of_nat ` N = A"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1847
  by (simp add: bij_betw_def)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1848
75101
f0e2023f361a one new lemma
paulson <lp15@cam.ac.uk>
parents: 74969
diff changeset
  1849
lemma Nats_infinite: "infinite (\<nat> :: 'a set)"
f0e2023f361a one new lemma
paulson <lp15@cam.ac.uk>
parents: 74969
diff changeset
  1850
  by (metis Nats_def finite_imageD infinite_UNIV_char_0 inj_on_of_nat)
f0e2023f361a one new lemma
paulson <lp15@cam.ac.uk>
parents: 74969
diff changeset
  1851
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1852
end
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1853
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1854
context comm_monoid_set
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1855
begin
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1856
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1857
lemma atLeastLessThan_reindex:
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1858
  "F g {h m..<h n} = F (g \<circ> h) {m..<n}"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1859
  if "bij_betw h {m..<n} {h m..<h n}" for m n ::nat
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1860
proof -
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1861
  from that have "inj_on h {m..<n}" and "h ` {m..<n} = {h m..<h n}"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1862
    by (simp_all add: bij_betw_def)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1863
  then show ?thesis
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1864
    using reindex [of h "{m..<n}" g] by simp
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1865
qed
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1866
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1867
lemma atLeastAtMost_reindex:
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1868
  "F g {h m..h n} = F (g \<circ> h) {m..n}"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1869
  if "bij_betw h {m..n} {h m..h n}" for m n ::nat
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1870
proof -
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1871
  from that have "inj_on h {m..n}" and "h ` {m..n} = {h m..h n}"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1872
    by (simp_all add: bij_betw_def)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1873
  then show ?thesis
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1874
    using reindex [of h "{m..n}" g] by simp
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1875
qed
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1876
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1877
lemma atLeastLessThan_shift_bounds:
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1878
  "F g {m + k..<n + k} = F (g \<circ> plus k) {m..<n}"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1879
  for m n k :: nat
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1880
  using atLeastLessThan_reindex [of "plus k" m n g]
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1881
  by (simp add: ac_simps)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1882
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1883
lemma atLeastAtMost_shift_bounds:
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1884
  "F g {m + k..n + k} = F (g \<circ> plus k) {m..n}"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1885
  for m n k :: nat
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1886
  using atLeastAtMost_reindex [of "plus k" m n g]
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1887
  by (simp add: ac_simps)
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1888
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1889
lemma atLeast_Suc_lessThan_Suc_shift:
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1890
  "F g {Suc m..<Suc n} = F (g \<circ> Suc) {m..<n}"
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1891
  using atLeastLessThan_shift_bounds [of _ _ 1]
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1892
  by (simp add: plus_1_eq_Suc)
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1893
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1894
lemma atLeast_Suc_atMost_Suc_shift:
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1895
  "F g {Suc m..Suc n} = F (g \<circ> Suc) {m..n}"
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1896
  using atLeastAtMost_shift_bounds [of _ _ 1]
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1897
  by (simp add: plus_1_eq_Suc)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1898
74969
fa0020b47868 New and simplified theorems
paulson <lp15@cam.ac.uk>
parents: 74965
diff changeset
  1899
lemma atLeast_atMost_pred_shift:
fa0020b47868 New and simplified theorems
paulson <lp15@cam.ac.uk>
parents: 74965
diff changeset
  1900
  "F (g \<circ> (\<lambda>n. n - Suc 0)) {Suc m..Suc n} = F g {m..n}"
fa0020b47868 New and simplified theorems
paulson <lp15@cam.ac.uk>
parents: 74965
diff changeset
  1901
  unfolding atLeast_Suc_atMost_Suc_shift by simp
fa0020b47868 New and simplified theorems
paulson <lp15@cam.ac.uk>
parents: 74965
diff changeset
  1902
fa0020b47868 New and simplified theorems
paulson <lp15@cam.ac.uk>
parents: 74965
diff changeset
  1903
lemma atLeast_lessThan_pred_shift:
fa0020b47868 New and simplified theorems
paulson <lp15@cam.ac.uk>
parents: 74965
diff changeset
  1904
  "F (g \<circ> (\<lambda>n. n - Suc 0)) {Suc m..<Suc n} = F g {m..<n}"
fa0020b47868 New and simplified theorems
paulson <lp15@cam.ac.uk>
parents: 74965
diff changeset
  1905
  unfolding atLeast_Suc_lessThan_Suc_shift by simp
fa0020b47868 New and simplified theorems
paulson <lp15@cam.ac.uk>
parents: 74965
diff changeset
  1906
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1907
lemma atLeast_int_lessThan_int_shift:
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1908
  "F g {int m..<int n} = F (g \<circ> int) {m..<n}"
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1909
  by (rule atLeastLessThan_reindex)
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1910
    (simp add: image_int_atLeastLessThan)
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1911
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1912
lemma atLeast_int_atMost_int_shift:
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  1913
  "F g {int m..int n} = F (g \<circ> int) {m..n}"
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1914
  by (rule atLeastAtMost_reindex)
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1915
    (simp add: image_int_atLeastAtMost)
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1916
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1917
lemma atLeast0_lessThan_Suc:
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1918
  "F g {0..<Suc n} = F g {0..<n} \<^bold>* g n"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1919
  by (simp add: atLeast0_lessThan_Suc ac_simps)
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1920
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1921
lemma atLeast0_atMost_Suc:
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1922
  "F g {0..Suc n} = F g {0..n} \<^bold>* g (Suc n)"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1923
  by (simp add: atLeast0_atMost_Suc ac_simps)
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1924
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1925
lemma atLeast0_lessThan_Suc_shift:
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1926
  "F g {0..<Suc n} = g 0 \<^bold>* F (g \<circ> Suc) {0..<n}"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1927
  by (simp add: atLeast0_lessThan_Suc_eq_insert_0 atLeast_Suc_lessThan_Suc_shift)
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1928
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1929
lemma atLeast0_atMost_Suc_shift:
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1930
  "F g {0..Suc n} = g 0 \<^bold>* F (g \<circ> Suc) {0..n}"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1931
  by (simp add: atLeast0_atMost_Suc_eq_insert_0 atLeast_Suc_atMost_Suc_shift)
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1932
67987
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1933
lemma atLeast_Suc_lessThan:
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1934
  "F g {m..<n} = g m \<^bold>* F g {Suc m..<n}" if "m < n"
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1935
proof -
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1936
  from that have "{m..<n} = insert m {Suc m..<n}"
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1937
    by auto
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1938
  then show ?thesis by simp
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1939
qed
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1940
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1941
lemma atLeast_Suc_atMost:
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1942
  "F g {m..n} = g m \<^bold>* F g {Suc m..n}" if "m \<le> n"
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1943
proof -
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1944
  from that have "{m..n} = insert m {Suc m..n}"
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1945
    by auto
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1946
  then show ?thesis by simp
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1947
qed
9044e1f1d324 more and generalized lemmas
haftmann
parents: 67907
diff changeset
  1948
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1949
lemma ivl_cong:
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1950
  "a = c \<Longrightarrow> b = d \<Longrightarrow> (\<And>x. c \<le> x \<Longrightarrow> x < d \<Longrightarrow> g x = h x)
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1951
    \<Longrightarrow> F g {a..<b} = F h {c..<d}"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1952
  by (rule cong) simp_all
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1953
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1954
lemma atLeastLessThan_shift_0:
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1955
  fixes m n p :: nat
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1956
  shows "F g {m..<n} = F (g \<circ> plus m) {0..<n - m}"
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1957
  using atLeastLessThan_shift_bounds [of g 0 m "n - m"]
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1958
  by (cases "m \<le> n") simp_all
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1959
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1960
lemma atLeastAtMost_shift_0:
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1961
  fixes m n p :: nat
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1962
  assumes "m \<le> n"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1963
  shows "F g {m..n} = F (g \<circ> plus m) {0..n - m}"
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1964
  using assms atLeastAtMost_shift_bounds [of g 0 m "n - m"] by simp
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1965
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1966
lemma atLeastLessThan_concat:
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1967
  fixes m n p :: nat
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1968
  shows "m \<le> n \<Longrightarrow> n \<le> p \<Longrightarrow> F g {m..<n} \<^bold>* F g {n..<p} = F g {m..<p}"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1969
  by (simp add: union_disjoint [symmetric] ivl_disj_un)
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1970
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1971
lemma atLeastLessThan_rev:
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1972
  "F g {n..<m} = F (\<lambda>i. g (m + n - Suc i)) {n..<m}"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1973
  by (rule reindex_bij_witness [where i="\<lambda>i. m + n - Suc i" and j="\<lambda>i. m + n - Suc i"], auto)
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1974
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1975
lemma atLeastAtMost_rev:
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1976
  fixes n m :: nat
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1977
  shows "F g {n..m} = F (\<lambda>i. g (m + n - i)) {n..m}"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1978
  by (rule reindex_bij_witness [where i="\<lambda>i. m + n - i" and j="\<lambda>i. m + n - i"]) auto
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1979
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1980
lemma atLeastLessThan_rev_at_least_Suc_atMost:
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1981
  "F g {n..<m} = F (\<lambda>i. g (m + n - i)) {Suc n..m}"
67411
3f4b0c84630f restored naming of lemmas after corresponding constants
haftmann
parents: 67399
diff changeset
  1982
  unfolding atLeastLessThan_rev [of g n m]
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1983
  by (cases m) (simp_all add: atLeast_Suc_atMost_Suc_shift atLeastLessThanSuc_atLeastAtMost)
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1984
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1985
end
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1986
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  1987
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1988
subsection \<open>Summation indexed over intervals\<close>
15042
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
  1989
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61799
diff changeset
  1990
syntax (ASCII)
80934
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  1991
  "_from_to_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>notation=\<open>binder SUM\<close>\<close>SUM _ = _.._./ _)\<close> [0,0,0,10] 10)
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  1992
  "_from_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>notation=\<open>binder SUM\<close>\<close>SUM _ = _..<_./ _)\<close> [0,0,0,10] 10)
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  1993
  "_upt_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>notation=\<open>binder SUM\<close>\<close>SUM _<_./ _)\<close> [0,0,10] 10)
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  1994
  "_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>notation=\<open>binder SUM\<close>\<close>SUM _<=_./ _)\<close> [0,0,10] 10)
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61799
diff changeset
  1995
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  1996
syntax (latex_sum output)
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  1997
  "_from_to_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
80932
261cd8722677 standardize mixfix annotations via "isabelle update -u mixfix_cartouches -l Pure HOL" --- to simplify systematic editing;
wenzelm
parents: 80760
diff changeset
  1998
 (\<open>(3\<^latex>\<open>$\sum_{\<close>_ = _\<^latex>\<open>}^{\<close>_\<^latex>\<open>}$\<close> _)\<close> [0,0,0,10] 10)
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  1999
  "_from_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
80932
261cd8722677 standardize mixfix annotations via "isabelle update -u mixfix_cartouches -l Pure HOL" --- to simplify systematic editing;
wenzelm
parents: 80760
diff changeset
  2000
 (\<open>(3\<^latex>\<open>$\sum_{\<close>_ = _\<^latex>\<open>}^{<\<close>_\<^latex>\<open>}$\<close> _)\<close> [0,0,0,10] 10)
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2001
  "_upt_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
80932
261cd8722677 standardize mixfix annotations via "isabelle update -u mixfix_cartouches -l Pure HOL" --- to simplify systematic editing;
wenzelm
parents: 80760
diff changeset
  2002
 (\<open>(3\<^latex>\<open>$\sum_{\<close>_ < _\<^latex>\<open>}$\<close> _)\<close> [0,0,10] 10)
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2003
  "_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
80932
261cd8722677 standardize mixfix annotations via "isabelle update -u mixfix_cartouches -l Pure HOL" --- to simplify systematic editing;
wenzelm
parents: 80760
diff changeset
  2004
 (\<open>(3\<^latex>\<open>$\sum_{\<close>_ \<le> _\<^latex>\<open>}$\<close> _)\<close> [0,0,10] 10)
15041
a6b1f0cef7b3 Got rid of Summation and made it a translation into setsum instead.
nipkow
parents: 14846
diff changeset
  2005
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61799
diff changeset
  2006
syntax
80934
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  2007
  "_from_to_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>indent=3 notation=\<open>binder \<Sum>\<close>\<close>\<Sum>_ = _.._./ _)\<close> [0,0,0,10] 10)
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  2008
  "_from_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>indent=3 notation=\<open>binder \<Sum>\<close>\<close>\<Sum>_ = _..<_./ _)\<close> [0,0,0,10] 10)
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  2009
  "_upt_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>indent=3 notation=\<open>binder \<Sum>\<close>\<close>\<Sum>_<_./ _)\<close> [0,0,10] 10)
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  2010
  "_upto_sum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>indent=3 notation=\<open>binder \<Sum>\<close>\<close>\<Sum>_\<le>_./ _)\<close> [0,0,10] 10)
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61799
diff changeset
  2011
80760
be8c0e039a5e more markup for syntax consts;
wenzelm
parents: 80671
diff changeset
  2012
syntax_consts
be8c0e039a5e more markup for syntax consts;
wenzelm
parents: 80671
diff changeset
  2013
  "_from_to_sum" "_from_upto_sum" "_upt_sum" "_upto_sum" == sum
be8c0e039a5e more markup for syntax consts;
wenzelm
parents: 80671
diff changeset
  2014
15048
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
  2015
translations
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2016
  "\<Sum>x=a..b. t" == "CONST sum (\<lambda>x. t) {a..b}"
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2017
  "\<Sum>x=a..<b. t" == "CONST sum (\<lambda>x. t) {a..<b}"
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2018
  "\<Sum>i\<le>n. t" == "CONST sum (\<lambda>i. t) {..n}"
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2019
  "\<Sum>i<n. t" == "CONST sum (\<lambda>i. t) {..<n}"
15041
a6b1f0cef7b3 Got rid of Summation and made it a translation into setsum instead.
nipkow
parents: 14846
diff changeset
  2020
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  2021
text\<open>The above introduces some pretty alternative syntaxes for
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  2022
summation over intervals:
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  2023
\begin{center}
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  2024
\begin{tabular}{lll}
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  2025
Old & New & \LaTeX\\
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
  2026
@{term[source]"\<Sum>x\<in>{a..b}. e"} & \<^term>\<open>\<Sum>x=a..b. e\<close> & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
  2027
@{term[source]"\<Sum>x\<in>{a..<b}. e"} & \<^term>\<open>\<Sum>x=a..<b. e\<close> & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
  2028
@{term[source]"\<Sum>x\<in>{..b}. e"} & \<^term>\<open>\<Sum>x\<le>b. e\<close> & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
  2029
@{term[source]"\<Sum>x\<in>{..<b}. e"} & \<^term>\<open>\<Sum>x<b. e\<close> & @{term[mode=latex_sum]"\<Sum>x<b. e"}
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  2030
\end{tabular}
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  2031
\end{center}
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  2032
The left column shows the term before introduction of the new syntax,
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  2033
the middle column shows the new (default) syntax, and the right column
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  2034
shows a special syntax. The latter is only meaningful for latex output
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  2035
and has to be activated explicitly by setting the print mode to
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61524
diff changeset
  2036
\<open>latex_sum\<close> (e.g.\ via \<open>mode = latex_sum\<close> in
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  2037
antiquotations). It is not the default \LaTeX\ output because it only
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  2038
works well with italic-style formulae, not tt-style.
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  2039
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
  2040
Note that for uniformity on \<^typ>\<open>nat\<close> it is better to use
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
  2041
\<^term>\<open>\<Sum>x::nat=0..<n. e\<close> rather than \<open>\<Sum>x<n. e\<close>: \<open>sum\<close> may
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
  2042
not provide all lemmas available for \<^term>\<open>{m..<n}\<close> also in the
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
  2043
special form for \<^term>\<open>{..<n}\<close>.\<close>
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  2044
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  2045
text\<open>This congruence rule should be used for sums over intervals as
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2046
the standard theorem @{text[source]sum.cong} does not work well
69593
3dda49e08b9d isabelle update -u control_cartouches;
wenzelm
parents: 69502
diff changeset
  2047
with the simplifier who adds the unsimplified premise \<^term>\<open>x\<in>B\<close> to
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  2048
the context.\<close>
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  2049
70097
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2050
context comm_monoid_set
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2051
begin
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2052
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2053
lemma zero_middle:
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2054
  assumes "1 \<le> p" "k \<le> p"
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2055
  shows "F (\<lambda>j. if j < k then g j else if j = k then \<^bold>1 else h (j - Suc 0)) {..p}
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2056
       = F (\<lambda>j. if j < k then g j else h j) {..p - Suc 0}"  (is "?lhs = ?rhs")
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2057
proof -
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2058
  have [simp]: "{..p - Suc 0} \<inter> {j. j < k} = {..<k}" "{..p - Suc 0} \<inter> - {j. j < k} = {k..p - Suc 0}"
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2059
    using assms by auto
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2060
  have "?lhs = F g {..<k} \<^bold>* F (\<lambda>j. if j = k then \<^bold>1 else h (j - Suc 0)) {k..p}"
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2061
    using union_disjoint [of "{..<k}" "{k..p}"] assms
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2062
    by (simp add: ivl_disj_int_one ivl_disj_un_one)
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2063
  also have "\<dots> = F g {..<k} \<^bold>* F (\<lambda>j.  h (j - Suc 0)) {Suc k..p}"
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2064
    by (simp add: atLeast_Suc_atMost [of k p] assms)
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2065
  also have "\<dots> = F g {..<k} \<^bold>* F h {k .. p - Suc 0}"
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2066
    using reindex [of Suc "{k..p - Suc 0}"] assms by simp
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2067
  also have "\<dots> = ?rhs"
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2068
    by (simp add: If_cases)
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2069
  finally show ?thesis .
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2070
qed
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2071
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2072
lemma atMost_Suc [simp]:
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2073
  "F g {..Suc n} = F g {..n} \<^bold>* g (Suc n)"
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2074
  by (simp add: atMost_Suc ac_simps)
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2075
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2076
lemma lessThan_Suc [simp]:
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2077
  "F g {..<Suc n} = F g {..<n} \<^bold>* g n"
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2078
  by (simp add: lessThan_Suc ac_simps)
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2079
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2080
lemma cl_ivl_Suc [simp]:
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2081
  "F g {m..Suc n} = (if Suc n < m then \<^bold>1 else F g {m..n} \<^bold>* g(Suc n))"
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2082
  by (auto simp: ac_simps atLeastAtMostSuc_conv)
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
  2083
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2084
lemma op_ivl_Suc [simp]:
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2085
  "F g {m..<Suc n} = (if n < m then \<^bold>1 else F g {m..<n} \<^bold>* g(n))"
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2086
  by (auto simp: ac_simps atLeastLessThanSuc)
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  2087
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2088
lemma head:
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  2089
  fixes n :: nat
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  2090
  assumes mn: "m \<le> n"
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2091
  shows "F g {m..n} = g m \<^bold>* F g {m<..n}" (is "?lhs = ?rhs")
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  2092
proof -
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  2093
  from mn
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  2094
  have "{m..n} = {m} \<union> {m<..n}"
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  2095
    by (auto intro: ivl_disj_un_singleton)
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2096
  hence "?lhs = F g ({m} \<union> {m<..n})"
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  2097
    by (simp add: atLeast0LessThan)
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  2098
  also have "\<dots> = ?rhs" by simp
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  2099
  finally show ?thesis .
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  2100
qed
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  2101
72686
703b601d71b5 cleanup of old proofs
paulson <lp15@cam.ac.uk>
parents: 72302
diff changeset
  2102
lemma last_plus: 
703b601d71b5 cleanup of old proofs
paulson <lp15@cam.ac.uk>
parents: 72302
diff changeset
  2103
  fixes n::nat  shows "m \<le> n \<Longrightarrow> F g {m..n} = g n \<^bold>* F g {m..<n}"
703b601d71b5 cleanup of old proofs
paulson <lp15@cam.ac.uk>
parents: 72302
diff changeset
  2104
  by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost commute)
703b601d71b5 cleanup of old proofs
paulson <lp15@cam.ac.uk>
parents: 72302
diff changeset
  2105
703b601d71b5 cleanup of old proofs
paulson <lp15@cam.ac.uk>
parents: 72302
diff changeset
  2106
lemma head_if:
703b601d71b5 cleanup of old proofs
paulson <lp15@cam.ac.uk>
parents: 72302
diff changeset
  2107
  fixes n :: nat
703b601d71b5 cleanup of old proofs
paulson <lp15@cam.ac.uk>
parents: 72302
diff changeset
  2108
  shows "F g {m..n} = (if n < m then \<^bold>1 else  F g {m..<n} \<^bold>* g(n))"
703b601d71b5 cleanup of old proofs
paulson <lp15@cam.ac.uk>
parents: 72302
diff changeset
  2109
  by (simp add: commute last_plus)
703b601d71b5 cleanup of old proofs
paulson <lp15@cam.ac.uk>
parents: 72302
diff changeset
  2110
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2111
lemma ub_add_nat: 
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2112
  assumes "(m::nat) \<le> n + 1"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2113
  shows "F g {m..n + p} = F g {m..n} \<^bold>* F g {n + 1..n + p}"
31501
2a60c9b951e0 New lemma
nipkow
parents: 31438
diff changeset
  2114
proof-
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  2115
  have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using \<open>m \<le> n+1\<close> by auto
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2116
  thus ?thesis by (auto simp: ivl_disj_int union_disjoint atLeastSucAtMost_greaterThanAtMost)
31501
2a60c9b951e0 New lemma
nipkow
parents: 31438
diff changeset
  2117
qed
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  2118
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2119
lemma nat_group: 
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2120
  fixes k::nat shows "F (\<lambda>m. F g {m * k ..< m*k + k}) {..<n} = F g {..< n * k}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2121
proof (cases k)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2122
  case (Suc l)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2123
  then have "k > 0"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2124
    by auto
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2125
  then show ?thesis
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2126
    by (induct n) (simp_all add: atLeastLessThan_concat add.commute atLeast0LessThan[symmetric])
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2127
qed auto   
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2128
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2129
lemma triangle_reindex:
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2130
  fixes n :: nat
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2131
  shows "F (\<lambda>(i,j). g i j) {(i,j). i+j < n} = F (\<lambda>k. F (\<lambda>i. g i (k - i)) {..k}) {..<n}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2132
  apply (simp add: Sigma)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2133
  apply (rule reindex_bij_witness[where j="\<lambda>(i, j). (i+j, i)" and i="\<lambda>(k, i). (i, k - i)"])
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2134
  apply auto
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2135
  done
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2136
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2137
lemma triangle_reindex_eq:
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2138
  fixes n :: nat
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2139
  shows "F (\<lambda>(i,j). g i j) {(i,j). i+j \<le> n} = F (\<lambda>k. F (\<lambda>i. g i (k - i)) {..k}) {..n}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2140
using triangle_reindex [of g "Suc n"]
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2141
by (simp only: Nat.less_Suc_eq_le lessThan_Suc_atMost)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2142
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2143
lemma nat_diff_reindex: "F (\<lambda>i. g (n - Suc i)) {..<n} = F g {..<n}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2144
  by (rule reindex_bij_witness[where i="\<lambda>i. n - Suc i" and j="\<lambda>i. n - Suc i"]) auto
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2145
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2146
lemma shift_bounds_nat_ivl:
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2147
  "F g {m+k..<n+k} = F (\<lambda>i. g(i + k)){m..<n::nat}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2148
by (induct "n", auto simp: atLeastLessThanSuc)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2149
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2150
lemma shift_bounds_cl_nat_ivl:
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2151
  "F g {m+k..n+k} = F (\<lambda>i. g(i + k)){m..n::nat}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2152
  by (rule reindex_bij_witness[where i="\<lambda>i. i + k" and j="\<lambda>i. i - k"]) auto
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2153
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2154
corollary shift_bounds_cl_Suc_ivl:
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2155
  "F g {Suc m..Suc n} = F (\<lambda>i. g(Suc i)){m..n}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2156
by (simp add: shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2157
71167
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 71094
diff changeset
  2158
corollary Suc_reindex_ivl: "m \<le> n \<Longrightarrow> F g {m..n} \<^bold>* g (Suc n) = g m \<^bold>* F (\<lambda>i. g (Suc i)) {m..n}"
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 71094
diff changeset
  2159
  by (simp add: assoc atLeast_Suc_atMost flip: shift_bounds_cl_Suc_ivl)
b4d409c65a76 Rearrangement of material in Complex_Analysis_Basics, which contained much that had nothing to do with complex analysis.
paulson <lp15@cam.ac.uk>
parents: 71094
diff changeset
  2160
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2161
corollary shift_bounds_Suc_ivl:
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2162
  "F g {Suc m..<Suc n} = F (\<lambda>i. g(Suc i)){m..<n}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2163
by (simp add: shift_bounds_nat_ivl[where k="Suc 0", simplified])
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2164
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2165
lemma atMost_Suc_shift:
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2166
  shows "F g {..Suc n} = g 0 \<^bold>* F (\<lambda>i. g (Suc i)) {..n}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2167
proof (induct n)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2168
  case 0 show ?case by simp
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2169
next
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2170
  case (Suc n) note IH = this
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2171
  have "F g {..Suc (Suc n)} = F g {..Suc n} \<^bold>* g (Suc (Suc n))"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2172
    by (rule atMost_Suc)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2173
  also have "F g {..Suc n}  = g 0 \<^bold>* F (\<lambda>i. g (Suc i)) {..n}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2174
    by (rule IH)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2175
  also have "g 0 \<^bold>* F (\<lambda>i. g (Suc i)) {..n} \<^bold>* g (Suc (Suc n)) =
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2176
             g 0 \<^bold>* (F (\<lambda>i. g (Suc i)) {..n} \<^bold>* g (Suc (Suc n)))"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2177
    by (rule assoc)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2178
  also have "F (\<lambda>i. g (Suc i)) {..n} \<^bold>* g (Suc (Suc n)) = F (\<lambda>i. g (Suc i)) {..Suc n}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2179
    by (rule atMost_Suc [symmetric])
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2180
  finally show ?case .
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2181
qed
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2182
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2183
lemma lessThan_Suc_shift:
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2184
  "F g {..<Suc n} = g 0 \<^bold>* F (\<lambda>i. g (Suc i)) {..<n}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2185
  by (induction n) (simp_all add: ac_simps)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2186
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2187
lemma atMost_shift:
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2188
  "F g {..n} = g 0 \<^bold>* F (\<lambda>i. g (Suc i)) {..<n}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2189
  by (metis atLeast0AtMost atLeast0LessThan atLeastLessThanSuc_atLeastAtMost 
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2190
       atLeastSucAtMost_greaterThanAtMost le0 head shift_bounds_Suc_ivl)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2191
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2192
lemma nested_swap:
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2193
     "F (\<lambda>i. F (\<lambda>j. a i j) {0..<i}) {0..n} = F (\<lambda>j. F (\<lambda>i. a i j) {Suc j..n}) {0..<n}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2194
  by (induction n) (auto simp: distrib)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2195
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2196
lemma nested_swap':
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2197
     "F (\<lambda>i. F (\<lambda>j. a i j) {..<i}) {..n} = F (\<lambda>j. F (\<lambda>i. a i j) {Suc j..n}) {..<n}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2198
  by (induction n) (auto simp: distrib)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2199
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2200
lemma atLeast1_atMost_eq:
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2201
  "F g {Suc 0..n} = F (\<lambda>k. g (Suc k)) {..<n}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2202
proof -
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2203
  have "F g {Suc 0..n} = F g (Suc ` {..<n})"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2204
    by (simp add: image_Suc_lessThan)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2205
  also have "\<dots> = F (\<lambda>k. g (Suc k)) {..<n}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2206
    by (simp add: reindex)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2207
  finally show ?thesis .
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2208
qed
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2209
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2210
lemma atLeastLessThan_Suc: "a \<le> b \<Longrightarrow> F g {a..<Suc b} = F g {a..<b} \<^bold>* g b"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2211
  by (simp add: atLeastLessThanSuc commute)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2212
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2213
lemma nat_ivl_Suc':
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2214
  assumes "m \<le> Suc n"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2215
  shows   "F g {m..Suc n} = g (Suc n) \<^bold>* F g {m..n}"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2216
proof -
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2217
  from assms have "{m..Suc n} = insert (Suc n) {m..n}" by auto
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2218
  also have "F g \<dots> = g (Suc n) \<^bold>* F g {m..n}" by simp
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2219
  finally show ?thesis .
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2220
qed
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2221
70365
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70340
diff changeset
  2222
lemma in_pairs: "F g {2*m..Suc(2*n)} = F (\<lambda>i. g(2*i) \<^bold>* g(Suc(2*i))) {m..n}"
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70340
diff changeset
  2223
proof (induction n)
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70340
diff changeset
  2224
  case 0
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70340
diff changeset
  2225
  show ?case
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70340
diff changeset
  2226
    by (cases "m=0") auto
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70340
diff changeset
  2227
next
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70340
diff changeset
  2228
  case (Suc n)
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70340
diff changeset
  2229
  then show ?case
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70340
diff changeset
  2230
    by (auto simp: assoc split: if_split_asm)
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70340
diff changeset
  2231
qed
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70340
diff changeset
  2232
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70340
diff changeset
  2233
lemma in_pairs_0: "F g {..Suc(2*n)} = F (\<lambda>i. g(2*i) \<^bold>* g(Suc(2*i))) {..n}"
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70340
diff changeset
  2234
  using in_pairs [of _ 0 n] by (simp add: atLeast0AtMost)
4df0628e8545 a few new lemmas and a bit of tidying
paulson <lp15@cam.ac.uk>
parents: 70340
diff changeset
  2235
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2236
end
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2237
73139
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2238
lemma card_sum_le_nat_sum: "\<Sum> {0..<card S} \<le> \<Sum> S"
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2239
proof (cases "finite S")
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2240
  case True
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2241
  then show ?thesis
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2242
  proof (induction "card S" arbitrary: S)
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2243
    case (Suc x)
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2244
    then have "Max S \<ge> x" using card_le_Suc_Max by fastforce
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2245
    let ?S' = "S - {Max S}"
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2246
    from Suc have "Max S \<in> S" by (auto intro: Max_in)
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2247
    hence cards: "card S = Suc (card ?S')"
74885
2df334453c4c isabelle update_cartouches;
wenzelm
parents: 74101
diff changeset
  2248
      using \<open>finite S\<close> by (intro card.remove; auto)
73139
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2249
    hence "\<Sum> {0..<card ?S'} \<le> \<Sum> ?S'"
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2250
      using Suc by (intro Suc; auto)
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2251
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2252
    hence "\<Sum> {0..<card ?S'} + x \<le> \<Sum> ?S' + Max S"
74885
2df334453c4c isabelle update_cartouches;
wenzelm
parents: 74101
diff changeset
  2253
      using \<open>Max S \<ge> x\<close> by simp
73139
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2254
    also have "... = \<Sum> S"
74885
2df334453c4c isabelle update_cartouches;
wenzelm
parents: 74101
diff changeset
  2255
      using sum.remove[OF \<open>finite S\<close> \<open>Max S \<in> S\<close>, where g="\<lambda>x. x"]
73139
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2256
      by simp
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2257
    finally show ?case
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2258
      using cards Suc by auto
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2259
  qed simp
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2260
qed simp
be9b73dfd3e0 added lemmas
nipkow
parents: 72686
diff changeset
  2261
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2262
lemma sum_natinterval_diff:
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2263
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2264
  shows  "sum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2265
          (if m \<le> n then f m - f(n + 1) else 0)"
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2266
by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2267
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2268
lemma sum_diff_nat_ivl:
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  2269
  fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  2270
  shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow> sum f {m..<p} - sum f {m..<n} = sum f {n..<p}"
70097
4005298550a6 The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents: 69768
diff changeset
  2271
  using sum.atLeastLessThan_concat [of m n p f,symmetric]
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  2272
  by (simp add: ac_simps)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  2273
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2274
lemma sum_diff_distrib: "\<forall>x. Q x \<le> P x  \<Longrightarrow> (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x :: nat)"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2275
  by (subst sum_subtractf_nat) auto
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2276
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2277
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2278
subsubsection \<open>Shifting bounds\<close>
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  2279
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2280
context comm_monoid_add
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2281
begin
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2282
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2283
context
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2284
  fixes f :: "nat \<Rightarrow> 'a"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2285
  assumes "f 0 = 0"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2286
begin
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2287
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2288
lemma sum_shift_lb_Suc0_0_upt:
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2289
  "sum f {Suc 0..<k} = sum f {0..<k}"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2290
proof (cases k)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2291
  case 0
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2292
  then show ?thesis
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2293
    by simp
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2294
next
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2295
  case (Suc k)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2296
  moreover have "{0..<Suc k} = insert 0 {Suc 0..<Suc k}"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2297
    by auto
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2298
  ultimately show ?thesis
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2299
    using \<open>f 0 = 0\<close> by simp
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2300
qed
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2301
68618
3db8520941a4 de-applying (mostly Set_Interval)
paulson <lp15@cam.ac.uk>
parents: 68361
diff changeset
  2302
lemma sum_shift_lb_Suc0_0: "sum f {Suc 0..k} = sum f {0..k}"
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2303
proof (cases k)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2304
  case 0
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2305
  with \<open>f 0 = 0\<close> show ?thesis
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2306
    by simp
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2307
next
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2308
  case (Suc k)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2309
  moreover have "{0..Suc k} = insert 0 {Suc 0..Suc k}"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2310
    by auto
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2311
  ultimately show ?thesis
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2312
    using \<open>f 0 = 0\<close> by simp
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2313
qed
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2314
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2315
end
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2316
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2317
end
19022
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
  2318
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2319
lemma sum_Suc_diff:
56238
5d147e1e18d1 a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents: 56215
diff changeset
  2320
  fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
5d147e1e18d1 a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents: 56215
diff changeset
  2321
  assumes "m \<le> Suc n"
5d147e1e18d1 a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents: 56215
diff changeset
  2322
  shows "(\<Sum>i = m..n. f(Suc i) - f i) = f (Suc n) - f m"
5d147e1e18d1 a few new lemmas and generalisations of old ones
paulson <lp15@cam.ac.uk>
parents: 56215
diff changeset
  2323
using assms by (induct n) (auto simp: le_Suc_eq)
55718
34618f031ba9 A few lemmas about summations, etc.
paulson <lp15@cam.ac.uk>
parents: 55242
diff changeset
  2324
65273
917ae0ba03a2 Removal of [simp] status for greaterThan_0. Moved two theorems into main HOL.
paulson <lp15@cam.ac.uk>
parents: 64773
diff changeset
  2325
lemma sum_Suc_diff':
917ae0ba03a2 Removal of [simp] status for greaterThan_0. Moved two theorems into main HOL.
paulson <lp15@cam.ac.uk>
parents: 64773
diff changeset
  2326
  fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
917ae0ba03a2 Removal of [simp] status for greaterThan_0. Moved two theorems into main HOL.
paulson <lp15@cam.ac.uk>
parents: 64773
diff changeset
  2327
  assumes "m \<le> n"
917ae0ba03a2 Removal of [simp] status for greaterThan_0. Moved two theorems into main HOL.
paulson <lp15@cam.ac.uk>
parents: 64773
diff changeset
  2328
  shows "(\<Sum>i = m..<n. f (Suc i) - f i) = f n - f m"
917ae0ba03a2 Removal of [simp] status for greaterThan_0. Moved two theorems into main HOL.
paulson <lp15@cam.ac.uk>
parents: 64773
diff changeset
  2329
using assms by (induct n) (auto simp: le_Suc_eq)
917ae0ba03a2 Removal of [simp] status for greaterThan_0. Moved two theorems into main HOL.
paulson <lp15@cam.ac.uk>
parents: 64773
diff changeset
  2330
78663
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2331
lemma sum_diff_split:
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2332
  fixes f:: "nat \<Rightarrow> 'a::ab_group_add"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2333
  assumes "m \<le> n"
79566
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2334
  shows "(\<Sum>i\<le>n. f i) - (\<Sum>i<m. f i) = (\<Sum>i\<le>n - m. f(n - i))"
78663
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2335
proof -
79566
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2336
  have "\<And>i. i \<le> n-m \<Longrightarrow> \<exists>k\<ge>m. k \<le> n \<and> i = n-k"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2337
    by (metis Nat.le_diff_conv2 add.commute \<open>m\<le>n\<close> diff_diff_cancel diff_le_self order.trans)
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2338
  then have eq: "{..n-m} = (-)n ` {m..n}"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2339
    by force
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2340
  have inj: "inj_on ((-)n) {m..n}"
78663
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2341
    by (auto simp: inj_on_def)
79566
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2342
  have "(\<Sum>i\<le>n - m. f(n - i)) = (\<Sum>i=m..n. f i)"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2343
    by (simp add: eq sum.reindex_cong [OF inj])
78663
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2344
  also have "\<dots> = (\<Sum>i\<le>n. f i) - (\<Sum>i<m. f i)"
79566
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2345
    using sum_diff_nat_ivl[of 0 "m" "Suc n" f] assms
78663
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2346
    by (simp only: atLeast0AtMost atLeast0LessThan atLeastLessThanSuc_atLeastAtMost)
79566
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2347
  finally show ?thesis by metis
78663
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2348
qed
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2349
80612
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2350
lemma prod_divide_nat_ivl:
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2351
  fixes f :: "nat \<Rightarrow> 'a::idom_divide"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2352
  shows "\<lbrakk> m \<le> n; n \<le> p; prod f {m..<n} \<noteq> 0\<rbrakk> \<Longrightarrow> prod f {m..<p} div prod f {m..<n} = prod f {n..<p}"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2353
  using prod.atLeastLessThan_concat [of m n p f,symmetric]
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2354
  by (simp add: ac_simps)
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2355
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2356
lemma prod_divide_split: (*FIXME: why is \<Prod> syntax not available?*)
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2357
  fixes f:: "nat \<Rightarrow> 'a::idom_divide"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2358
  assumes "m \<le> n" "prod f {..<m} \<noteq> 0"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2359
  shows "(prod f {..n}) div (prod f {..<m}) = prod (\<lambda>i. f(n - i)) {..n - m}"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2360
proof -
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2361
  have "\<And>i. i \<le> n-m \<Longrightarrow> \<exists>k\<ge>m. k \<le> n \<and> i = n-k"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2362
    by (metis Nat.le_diff_conv2 add.commute \<open>m\<le>n\<close> diff_diff_cancel diff_le_self order.trans)
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2363
  then have eq: "{..n-m} = (-)n ` {m..n}"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2364
    by force
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2365
  have inj: "inj_on ((-)n) {m..n}"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2366
    by (auto simp: inj_on_def)
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2367
  have "prod (\<lambda>i. f(n - i)) {..n - m} = prod f {m..n}"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2368
    by (simp add: eq prod.reindex_cong [OF inj])
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2369
  also have "\<dots> = prod f {..n} div prod f {..<m}"
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2370
    using prod_divide_nat_ivl[of 0 "m" "Suc n" f] assms
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2371
    by (force simp: atLeast0AtMost atLeast0LessThan atLeastLessThanSuc_atLeastAtMost)
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2372
  finally show ?thesis by metis
e65eed943bee A lot of new material from the Ramsey development, including a couple of new simprules.
paulson <lp15@cam.ac.uk>
parents: 79566
diff changeset
  2373
qed
52380
3cc46b8cca5e lifting for primitive definitions;
haftmann
parents: 51334
diff changeset
  2374
79566
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2375
subsubsection \<open>Telescoping sums\<close>
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61378
diff changeset
  2376
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2377
lemma sum_telescope:
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61378
diff changeset
  2378
  fixes f::"nat \<Rightarrow> 'a::ab_group_add"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2379
  shows "sum (\<lambda>i. f i - f (Suc i)) {.. i} = f 0 - f (Suc i)"
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61378
diff changeset
  2380
  by (induct i) simp_all
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61378
diff changeset
  2381
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2382
lemma sum_telescope'':
61524
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61378
diff changeset
  2383
  assumes "m \<le> n"
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61378
diff changeset
  2384
  shows   "(\<Sum>k\<in>{Suc m..n}. f k - f (k - 1)) = f n - (f m :: 'a :: ab_group_add)"
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61378
diff changeset
  2385
  by (rule dec_induct[OF assms]) (simp_all add: algebra_simps)
f2e51e704a96 added many small lemmas about setsum/setprod/powr/...
eberlm
parents: 61378
diff changeset
  2386
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2387
lemma sum_lessThan_telescope:
63721
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63540
diff changeset
  2388
  "(\<Sum>n<m. f (Suc n) - f n :: 'a :: ab_group_add) = f m - f 0"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63540
diff changeset
  2389
  by (induction m) (simp_all add: algebra_simps)
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63540
diff changeset
  2390
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2391
lemma sum_lessThan_telescope':
63721
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63540
diff changeset
  2392
  "(\<Sum>n<m. f n - f (Suc n) :: 'a :: ab_group_add) = f 0 - f m"
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63540
diff changeset
  2393
  by (induction m) (simp_all add: algebra_simps)
492bb53c3420 More analysis lemmas
Manuel Eberl <eberlm@in.tum.de>
parents: 63540
diff changeset
  2394
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2395
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2396
subsubsection \<open>The formula for geometric sums\<close>
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
  2397
66490
cc66ab2373ce added lemma
nipkow
parents: 65578
diff changeset
  2398
lemma sum_power2: "(\<Sum>i=0..<k. (2::nat)^i) = 2^k-1"
72268
71a8935eb5da removal of needless premises
paulson <lp15@cam.ac.uk>
parents: 71822
diff changeset
  2399
  by (induction k) (auto simp: mult_2)
66490
cc66ab2373ce added lemma
nipkow
parents: 65578
diff changeset
  2400
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
  2401
lemma geometric_sum:
36307
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  2402
  assumes "x \<noteq> 1"
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 55719
diff changeset
  2403
  shows "(\<Sum>i<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"
36307
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  2404
proof -
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  2405
  from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all
56193
c726ecfb22b6 cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents: 55719
diff changeset
  2406
  moreover have "(\<Sum>i<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2407
    by (induct n) (simp_all add: field_simps \<open>y \<noteq> 0\<close>)
36307
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  2408
  ultimately show ?thesis by simp
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  2409
qed
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  2410
78256
71e1aa0d9421 A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents: 77935
diff changeset
  2411
lemma geometric_sum_less:
71e1aa0d9421 A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents: 77935
diff changeset
  2412
  assumes "0 < x" "x < 1" "finite S"
71e1aa0d9421 A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents: 77935
diff changeset
  2413
  shows "(\<Sum>i\<in>S. x ^ i) < 1 / (1 - x::'a::linordered_field)"
71e1aa0d9421 A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents: 77935
diff changeset
  2414
proof -
71e1aa0d9421 A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents: 77935
diff changeset
  2415
  define n where "n \<equiv> Suc (Max S)" 
71e1aa0d9421 A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents: 77935
diff changeset
  2416
  have "(\<Sum>i\<in>S. x ^ i) \<le> (\<Sum>i<n. x ^ i)"
71e1aa0d9421 A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents: 77935
diff changeset
  2417
    unfolding n_def using assms  by (fastforce intro!: sum_mono2 le_imp_less_Suc)
71e1aa0d9421 A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents: 77935
diff changeset
  2418
  also have "\<dots> = (1 - x ^ n) / (1 - x)"
71e1aa0d9421 A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents: 77935
diff changeset
  2419
    using assms by (simp add: geometric_sum field_simps)
71e1aa0d9421 A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents: 77935
diff changeset
  2420
  also have "\<dots> < 1 / (1-x)"
71e1aa0d9421 A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents: 77935
diff changeset
  2421
    using assms by (simp add: field_simps power_Suc_less)
71e1aa0d9421 A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents: 77935
diff changeset
  2422
  finally show ?thesis .
71e1aa0d9421 A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents: 77935
diff changeset
  2423
qed
71e1aa0d9421 A couple of new lemmas involving cardinality
paulson <lp15@cam.ac.uk>
parents: 77935
diff changeset
  2424
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2425
lemma diff_power_eq_sum:
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2426
  fixes y :: "'a::{comm_ring,monoid_mult}"
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2427
  shows
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2428
    "x ^ (Suc n) - y ^ (Suc n) =
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2429
      (x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))"
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2430
proof (induct n)
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2431
  case (Suc n)
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2432
  have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x^n) - y * (y * y ^ n)"
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2433
    by simp
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2434
  also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x^n)"
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2435
    by (simp add: algebra_simps)
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2436
  also have "... = y * ((x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2437
    by (simp only: Suc)
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2438
  also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2439
    by (simp only: mult.left_commute)
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2440
  also have "... = (x - y) * (\<Sum>p<Suc (Suc n). x ^ p * y ^ (Suc n - p))"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2441
    by (simp add: field_simps Suc_diff_le sum_distrib_right sum_distrib_left)
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2442
  finally show ?case .
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2443
qed simp
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2444
67443
3abf6a722518 standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents: 67411
diff changeset
  2445
corollary power_diff_sumr2: \<comment> \<open>\<open>COMPLEX_POLYFUN\<close> in HOL Light\<close>
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2446
  fixes x :: "'a::{comm_ring,monoid_mult}"
72268
71a8935eb5da removal of needless premises
paulson <lp15@cam.ac.uk>
parents: 71822
diff changeset
  2447
  shows "x^n - y^n = (x - y) * (\<Sum>i<n. y^(n - Suc i) * x^i)"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2448
using diff_power_eq_sum[of x "n - 1" y]
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2449
by (cases "n = 0") (simp_all add: field_simps)
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2450
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2451
lemma power_diff_1_eq:
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2452
  fixes x :: "'a::{comm_ring,monoid_mult}"
72268
71a8935eb5da removal of needless premises
paulson <lp15@cam.ac.uk>
parents: 71822
diff changeset
  2453
  shows "x^n - 1 = (x - 1) * (\<Sum>i<n. (x^i))"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2454
using diff_power_eq_sum [of x _ 1]
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2455
  by (cases n) auto
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2456
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2457
lemma one_diff_power_eq':
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2458
  fixes x :: "'a::{comm_ring,monoid_mult}"
72268
71a8935eb5da removal of needless premises
paulson <lp15@cam.ac.uk>
parents: 71822
diff changeset
  2459
  shows "1 - x^n = (1 - x) * (\<Sum>i<n. x^(n - Suc i))"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2460
using diff_power_eq_sum [of 1 _ x]
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2461
  by (cases n) auto
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2462
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2463
lemma one_diff_power_eq:
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2464
  fixes x :: "'a::{comm_ring,monoid_mult}"
72268
71a8935eb5da removal of needless premises
paulson <lp15@cam.ac.uk>
parents: 71822
diff changeset
  2465
  shows "1 - x^n = (1 - x) * (\<Sum>i<n. x^i)"
71a8935eb5da removal of needless premises
paulson <lp15@cam.ac.uk>
parents: 71822
diff changeset
  2466
by (metis one_diff_power_eq' sum.nat_diff_reindex)
60162
645058aa9d6f tidying some messy proofs
paulson <lp15@cam.ac.uk>
parents: 60150
diff changeset
  2467
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2468
lemma sum_gp_basic:
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2469
  fixes x :: "'a::{comm_ring,monoid_mult}"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2470
  shows "(1 - x) * (\<Sum>i\<le>n. x^i) = 1 - x^Suc n"
72268
71a8935eb5da removal of needless premises
paulson <lp15@cam.ac.uk>
parents: 71822
diff changeset
  2471
  by (simp only: one_diff_power_eq lessThan_Suc_atMost)
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2472
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2473
lemma sum_power_shift:
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2474
  fixes x :: "'a::{comm_ring,monoid_mult}"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2475
  assumes "m \<le> n"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2476
  shows "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i\<le>n-m. x^i)"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2477
proof -
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2478
  have "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i=m..n. x^(i-m))"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2479
    by (simp add: sum_distrib_left power_add [symmetric])
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2480
  also have "(\<Sum>i=m..n. x^(i-m)) = (\<Sum>i\<le>n-m. x^i)"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2481
    using \<open>m \<le> n\<close> by (intro sum.reindex_bij_witness[where j="\<lambda>i. i - m" and i="\<lambda>i. i + m"]) auto
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2482
  finally show ?thesis .
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2483
qed
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2484
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2485
lemma sum_gp_multiplied:
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2486
  fixes x :: "'a::{comm_ring,monoid_mult}"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2487
  assumes "m \<le> n"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2488
  shows "(1 - x) * (\<Sum>i=m..n. x^i) = x^m - x^Suc n"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2489
proof -
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2490
  have  "(1 - x) * (\<Sum>i=m..n. x^i) = x^m * (1 - x) * (\<Sum>i\<le>n-m. x^i)"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2491
    by (metis mult.assoc mult.commute assms sum_power_shift)
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2492
  also have "... =x^m * (1 - x^Suc(n-m))"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2493
    by (metis mult.assoc sum_gp_basic)
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2494
  also have "... = x^m - x^Suc n"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2495
    using assms
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2496
    by (simp add: algebra_simps) (metis le_add_diff_inverse power_add)
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2497
  finally show ?thesis .
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2498
qed
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2499
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2500
lemma sum_gp:
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2501
  fixes x :: "'a::{comm_ring,division_ring}"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2502
  shows   "(\<Sum>i=m..n. x^i) =
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2503
               (if n < m then 0
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2504
                else if x = 1 then of_nat((n + 1) - m)
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2505
                else (x^m - x^Suc n) / (1 - x))"
75669
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2506
proof (cases "n < m")
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2507
  case False
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2508
  assume *: "\<not> n < m"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2509
  then show ?thesis
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2510
  proof (cases "x = 1")
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2511
    case False
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2512
    assume "x \<noteq> 1"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2513
    then have not_zero: "1 - x \<noteq> 0"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2514
      by auto
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2515
    have "(1 - x) * (\<Sum>i=m..n. x^i) = x ^ m - x * x ^ n"
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2516
      using sum_gp_multiplied [of m n x] * by auto
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2517
    then have "(\<Sum>i=m..n. x^i) = (x ^ m - x * x ^ n) / (1 - x) "
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2518
      using nonzero_divide_eq_eq mult.commute not_zero
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2519
      by metis
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2520
    then show ?thesis
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2521
      by auto
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2522
  qed (auto)
43f5dfb7fa35 tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents: 75668
diff changeset
  2523
qed (auto)
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2524
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2525
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2526
subsubsection\<open>Geometric progressions\<close>
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2527
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2528
lemma sum_gp0:
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2529
  fixes x :: "'a::{comm_ring,division_ring}"
70113
c8deb8ba6d05 Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents: 70097
diff changeset
  2530
  shows "(\<Sum>i\<le>n. x^i) = (if x = 1 then of_nat(n + 1) else (1 - x^Suc n) / (1 - x))"
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2531
  using sum_gp_basic[of x n]
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70749
diff changeset
  2532
  by (simp add: mult.commute field_split_simps)
65578
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2533
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2534
lemma sum_power_add:
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2535
  fixes x :: "'a::{comm_ring,monoid_mult}"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2536
  shows "(\<Sum>i\<in>I. x^(m+i)) = x^m * (\<Sum>i\<in>I. x^i)"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2537
  by (simp add: sum_distrib_left power_add)
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2538
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2539
lemma sum_gp_offset:
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2540
  fixes x :: "'a::{comm_ring,division_ring}"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2541
  shows   "(\<Sum>i=m..m+n. x^i) =
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2542
       (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2543
  using sum_gp [of x m "m+n"]
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2544
  by (auto simp: power_add algebra_simps)
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2545
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2546
lemma sum_gp_strict:
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2547
  fixes x :: "'a::{comm_ring,division_ring}"
e4997c181cce New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents: 65273
diff changeset
  2548
  shows "(\<Sum>i<n. x^i) = (if x = 1 then of_nat n else (1 - x^n) / (1 - x))"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70749
diff changeset
  2549
  by (induct n) (auto simp: algebra_simps field_split_simps)
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
  2550
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2551
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2552
subsubsection \<open>The formulae for arithmetic sums\<close>
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2553
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2554
context comm_semiring_1
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2555
begin
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2556
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2557
lemma double_gauss_sum:
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2558
  "2 * (\<Sum>i = 0..n. of_nat i) = of_nat n * (of_nat n + 1)"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2559
  by (induct n) (simp_all add: sum.atLeast0_atMost_Suc algebra_simps left_add_twice)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2560
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2561
lemma double_gauss_sum_from_Suc_0:
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2562
  "2 * (\<Sum>i = Suc 0..n. of_nat i) = of_nat n * (of_nat n + 1)"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2563
proof -
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2564
  have "sum of_nat {Suc 0..n} = sum of_nat (insert 0 {Suc 0..n})"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2565
    by simp
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2566
  also have "\<dots> = sum of_nat {0..n}"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2567
    by (cases n) (simp_all add: atLeast0_atMost_Suc_eq_insert_0)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2568
  finally show ?thesis
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2569
    by (simp add: double_gauss_sum)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2570
qed
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2571
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2572
lemma double_arith_series:
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2573
  "2 * (\<Sum>i = 0..n. a + of_nat i * d) = (of_nat n + 1) * (2 * a + of_nat n * d)"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2574
proof -
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2575
  have "(\<Sum>i = 0..n. a + of_nat i * d) = ((\<Sum>i = 0..n. a) + (\<Sum>i = 0..n. of_nat i * d))"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2576
    by (rule sum.distrib)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2577
  also have "\<dots> = (of_nat (Suc n) * a + d * (\<Sum>i = 0..n. of_nat i))"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2578
    by (simp add: sum_distrib_left algebra_simps)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2579
  finally show ?thesis
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2580
    by (simp add: algebra_simps double_gauss_sum left_add_twice)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2581
qed
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2582
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2583
end
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2584
78937
5e6b195eee83 slightly less technical formulation of very specific type class
haftmann
parents: 78663
diff changeset
  2585
context linordered_euclidean_semiring
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2586
begin
19469
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
  2587
47222
1b7c909a6fad rephrase lemmas about arithmetic series using numeral '2'
huffman
parents: 47108
diff changeset
  2588
lemma gauss_sum:
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2589
  "(\<Sum>i = 0..n. of_nat i) = of_nat n * (of_nat n + 1) div 2"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2590
  using double_gauss_sum [of n, symmetric] by simp
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2591
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2592
lemma gauss_sum_from_Suc_0:
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2593
  "(\<Sum>i = Suc 0..n. of_nat i) = of_nat n * (of_nat n + 1) div 2"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2594
  using double_gauss_sum_from_Suc_0 [of n, symmetric] by simp
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2595
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2596
lemma arith_series:
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2597
  "(\<Sum>i = 0..n. a + of_nat i * d) = (of_nat n + 1) * (2 * a + of_nat n * d) div 2"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2598
  using double_arith_series [of a d n, symmetric] by simp
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2599
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2600
end
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2601
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2602
lemma gauss_sum_nat:
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2603
  "\<Sum>{0..n} = (n * Suc n) div 2"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2604
  using gauss_sum [of n, where ?'a = nat] by simp
19469
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
  2605
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
  2606
lemma arith_series_nat:
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2607
  "(\<Sum>i = 0..n. a + i * d) = Suc n * (2 * a + n * d) div 2"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2608
  using arith_series [of a d n] by simp
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2609
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2610
lemma Sum_Icc_int:
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2611
  "\<Sum>{m..n} = (n * (n + 1) - m * (m - 1)) div 2"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2612
  if "m \<le> n" for m n :: int
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2613
using that proof (induct i \<equiv> "nat (n - m)" arbitrary: m n)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2614
  case 0
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2615
  then have "m = n"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2616
    by arith
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2617
  then show ?case
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2618
    by (simp add: algebra_simps mult_2 [symmetric])
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2619
next
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2620
  case (Suc i)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2621
  have 0: "i = nat((n-1) - m)" "m \<le> n-1" using Suc(2,3) by arith+
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2622
  have "\<Sum> {m..n} = \<Sum> {m..1+(n-1)}" by simp
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2623
  also have "\<dots> = \<Sum> {m..n-1} + n" using \<open>m \<le> n\<close>
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2624
    by(subst atLeastAtMostPlus1_int_conv) simp_all
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2625
  also have "\<dots> = ((n-1)*(n-1+1) - m*(m-1)) div 2 + n"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2626
    by(simp add: Suc(1)[OF 0])
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2627
  also have "\<dots> = ((n-1)*(n-1+1) - m*(m-1) + 2*n) div 2" by simp
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2628
  also have "\<dots> = (n*(n+1) - m*(m-1)) div 2"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2629
    by (simp add: algebra_simps mult_2_right)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2630
  finally show ?case .
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2631
qed
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2632
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2633
lemma Sum_Icc_nat:
69182
2424301cc73d more and generalized lemmas
haftmann
parents: 69064
diff changeset
  2634
  "\<Sum>{m..n} = (n * (n + 1) - m * (m - 1)) div 2" for m n :: nat
2424301cc73d more and generalized lemmas
haftmann
parents: 69064
diff changeset
  2635
proof (cases "m \<le> n")
2424301cc73d more and generalized lemmas
haftmann
parents: 69064
diff changeset
  2636
  case True
2424301cc73d more and generalized lemmas
haftmann
parents: 69064
diff changeset
  2637
  then have *: "m * (m - 1) \<le> n * (n + 1)"
2424301cc73d more and generalized lemmas
haftmann
parents: 69064
diff changeset
  2638
    by (meson diff_le_self order_trans le_add1 mult_le_mono)
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2639
  have "int (\<Sum>{m..n}) = (\<Sum>{int m..int n})"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2640
    by (simp add: sum.atLeast_int_atMost_int_shift)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2641
  also have "\<dots> = (int n * (int n + 1) - int m * (int m - 1)) div 2"
69182
2424301cc73d more and generalized lemmas
haftmann
parents: 69064
diff changeset
  2642
    using \<open>m \<le> n\<close> by (simp add: Sum_Icc_int)
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2643
  also have "\<dots> = int ((n * (n + 1) - m * (m - 1)) div 2)"
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2644
    using le_square * by (simp add: algebra_simps of_nat_div of_nat_diff)
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2645
  finally show ?thesis
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2646
    by (simp only: of_nat_eq_iff)
69182
2424301cc73d more and generalized lemmas
haftmann
parents: 69064
diff changeset
  2647
next
2424301cc73d more and generalized lemmas
haftmann
parents: 69064
diff changeset
  2648
  case False
2424301cc73d more and generalized lemmas
haftmann
parents: 69064
diff changeset
  2649
  then show ?thesis
2424301cc73d more and generalized lemmas
haftmann
parents: 69064
diff changeset
  2650
    by (auto dest: less_imp_Suc_add simp add: not_le algebra_simps)
19469
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
  2651
qed
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
  2652
66936
cf8d8fc23891 tuned some proofs and added some lemmas
haftmann
parents: 66836
diff changeset
  2653
lemma Sum_Ico_nat: 
69182
2424301cc73d more and generalized lemmas
haftmann
parents: 69064
diff changeset
  2654
  "\<Sum>{m..<n} = (n * (n - 1) - m * (m - 1)) div 2" for m n :: nat
2424301cc73d more and generalized lemmas
haftmann
parents: 69064
diff changeset
  2655
  by (cases n) (simp_all add: atLeastLessThanSuc_atLeastAtMost Sum_Icc_nat)
19022
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
  2656
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61799
diff changeset
  2657
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2658
subsubsection \<open>Division remainder\<close>
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2659
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2660
lemma range_mod:
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2661
  fixes n :: nat
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2662
  assumes "n > 0"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2663
  shows "range (\<lambda>m. m mod n) = {0..<n}" (is "?A = ?B")
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2664
proof (rule set_eqI)
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2665
  fix m
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2666
  show "m \<in> ?A \<longleftrightarrow> m \<in> ?B"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2667
  proof
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2668
    assume "m \<in> ?A"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2669
    with assms show "m \<in> ?B"
63915
bab633745c7f tuned proofs;
wenzelm
parents: 63879
diff changeset
  2670
      by auto
63417
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2671
  next
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2672
    assume "m \<in> ?B"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2673
    moreover have "m mod n \<in> ?A"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2674
      by (rule rangeI)
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2675
    ultimately show "m \<in> ?A"
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2676
      by simp
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2677
  qed
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2678
qed
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2679
c184ec919c70 more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents: 63365
diff changeset
  2680
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  2681
subsection \<open>Products indexed over intervals\<close>
29960
9d5c6f376768 Syntactic support for products over set intervals
paulson
parents: 29920
diff changeset
  2682
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61799
diff changeset
  2683
syntax (ASCII)
80934
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  2684
  "_from_to_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>notation=\<open>binder PROD\<close>\<close>PROD _ = _.._./ _)\<close> [0,0,0,10] 10)
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  2685
  "_from_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>notation=\<open>binder PROD\<close>\<close>PROD _ = _..<_./ _)\<close> [0,0,0,10] 10)
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  2686
  "_upt_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>notation=\<open>binder PROD\<close>\<close>PROD _<_./ _)\<close> [0,0,10] 10)
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  2687
  "_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>notation=\<open>binder PROD\<close>\<close>PROD _<=_./ _)\<close> [0,0,10] 10)
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61799
diff changeset
  2688
29960
9d5c6f376768 Syntactic support for products over set intervals
paulson
parents: 29920
diff changeset
  2689
syntax (latex_prod output)
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  2690
  "_from_to_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
80932
261cd8722677 standardize mixfix annotations via "isabelle update -u mixfix_cartouches -l Pure HOL" --- to simplify systematic editing;
wenzelm
parents: 80760
diff changeset
  2691
 (\<open>(3\<^latex>\<open>$\prod_{\<close>_ = _\<^latex>\<open>}^{\<close>_\<^latex>\<open>}$\<close> _)\<close> [0,0,0,10] 10)
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  2692
  "_from_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
80932
261cd8722677 standardize mixfix annotations via "isabelle update -u mixfix_cartouches -l Pure HOL" --- to simplify systematic editing;
wenzelm
parents: 80760
diff changeset
  2693
 (\<open>(3\<^latex>\<open>$\prod_{\<close>_ = _\<^latex>\<open>}^{<\<close>_\<^latex>\<open>}$\<close> _)\<close> [0,0,0,10] 10)
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  2694
  "_upt_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
80932
261cd8722677 standardize mixfix annotations via "isabelle update -u mixfix_cartouches -l Pure HOL" --- to simplify systematic editing;
wenzelm
parents: 80760
diff changeset
  2695
 (\<open>(3\<^latex>\<open>$\prod_{\<close>_ < _\<^latex>\<open>}$\<close> _)\<close> [0,0,10] 10)
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  2696
  "_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
80932
261cd8722677 standardize mixfix annotations via "isabelle update -u mixfix_cartouches -l Pure HOL" --- to simplify systematic editing;
wenzelm
parents: 80760
diff changeset
  2697
 (\<open>(3\<^latex>\<open>$\prod_{\<close>_ \<le> _\<^latex>\<open>}$\<close> _)\<close> [0,0,10] 10)
29960
9d5c6f376768 Syntactic support for products over set intervals
paulson
parents: 29920
diff changeset
  2698
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61799
diff changeset
  2699
syntax
80934
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  2700
  "_from_to_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>indent=3 notation=\<open>binder \<Prod>\<close>\<close>\<Prod>_ = _.._./ _)\<close> [0,0,0,10] 10)
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  2701
  "_from_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>indent=3 notation=\<open>binder \<Prod>\<close>\<close>\<Prod>_ = _..<_./ _)\<close> [0,0,0,10] 10)
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  2702
  "_upt_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>indent=3 notation=\<open>binder \<Prod>\<close>\<close>\<Prod>_<_./ _)\<close> [0,0,10] 10)
8e72f55295fd more inner syntax markup: HOL;
wenzelm
parents: 80932
diff changeset
  2703
  "_upto_prod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"  (\<open>(\<open>indent=3 notation=\<open>binder \<Prod>\<close>\<close>\<Prod>_\<le>_./ _)\<close> [0,0,10] 10)
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61799
diff changeset
  2704
80760
be8c0e039a5e more markup for syntax consts;
wenzelm
parents: 80671
diff changeset
  2705
syntax_consts
be8c0e039a5e more markup for syntax consts;
wenzelm
parents: 80671
diff changeset
  2706
  "_from_to_prod" "_from_upto_prod" "_upt_prod" "_upto_prod" \<rightleftharpoons> prod
be8c0e039a5e more markup for syntax consts;
wenzelm
parents: 80671
diff changeset
  2707
29960
9d5c6f376768 Syntactic support for products over set intervals
paulson
parents: 29920
diff changeset
  2708
translations
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  2709
  "\<Prod>x=a..b. t" \<rightleftharpoons> "CONST prod (\<lambda>x. t) {a..b}"
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  2710
  "\<Prod>x=a..<b. t" \<rightleftharpoons> "CONST prod (\<lambda>x. t) {a..<b}"
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  2711
  "\<Prod>i\<le>n. t" \<rightleftharpoons> "CONST prod (\<lambda>i. t) {..n}"
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  2712
  "\<Prod>i<n. t" \<rightleftharpoons> "CONST prod (\<lambda>i. t) {..<n}"
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  2713
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  2714
lemma prod_int_plus_eq: "prod int {i..i+j} =  \<Prod>{int i..int (i+j)}"
55242
413ec965f95d Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
  2715
  by (induct j) (auto simp add: atLeastAtMostSuc_conv atLeastAtMostPlus1_int_conv)
413ec965f95d Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
  2716
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  2717
lemma prod_int_eq: "prod int {i..j} =  \<Prod>{int i..int j}"
55242
413ec965f95d Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
  2718
proof (cases "i \<le> j")
413ec965f95d Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
  2719
  case True
413ec965f95d Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
  2720
  then show ?thesis
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  2721
    by (metis le_iff_add prod_int_plus_eq)
55242
413ec965f95d Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
  2722
next
413ec965f95d Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
  2723
  case False
413ec965f95d Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
  2724
  then show ?thesis
413ec965f95d Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
  2725
    by auto
413ec965f95d Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
  2726
qed
413ec965f95d Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents: 55143
diff changeset
  2727
79566
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2728
subsubsection \<open>Telescoping products\<close>
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2729
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2730
lemma prod_telescope:
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2731
  fixes f::"nat \<Rightarrow> 'a::field"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2732
  assumes "\<And>i. i\<le>n \<Longrightarrow> f (Suc i) \<noteq> 0"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2733
  shows "(\<Prod>i\<le>n. f i / f (Suc i)) = f 0 / f (Suc n)"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2734
  using assms by (induction n) auto
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2735
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2736
lemma prod_telescope'':
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2737
  fixes f::"nat \<Rightarrow> 'a::field"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2738
  assumes "m \<le> n"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2739
  assumes "\<And>i. i \<in> {m..n} \<Longrightarrow> f i \<noteq> 0"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2740
  shows   "(\<Prod>i = Suc m..n. f i / f (i - 1)) = f n / f m"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2741
  by (rule dec_induct[OF \<open>m \<le> n\<close>]) (auto simp add: assms)
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2742
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2743
lemma prod_lessThan_telescope:
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2744
  fixes f::"nat \<Rightarrow> 'a::field"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2745
  assumes "\<And>i. i\<le>n \<Longrightarrow> f i \<noteq> 0"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2746
  shows "(\<Prod>i<n. f (Suc i) / f i) = f n / f 0"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2747
  using assms by (induction n) auto
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2748
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2749
lemma prod_lessThan_telescope':
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2750
  fixes f::"nat \<Rightarrow> 'a::field"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2751
  assumes "\<And>i. i\<le>n \<Longrightarrow> f i \<noteq> 0"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2752
  shows "(\<Prod>i<n. f i / f (Suc i)) = f 0 / f n"
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2753
  using assms by (induction n) auto
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2754
f783490c6c99 A small number of new lemmas
paulson <lp15@cam.ac.uk>
parents: 78937
diff changeset
  2755
62128
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2756
subsection \<open>Efficient folding over intervals\<close>
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2757
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2758
function fold_atLeastAtMost_nat where
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2759
  [simp del]: "fold_atLeastAtMost_nat f a (b::nat) acc =
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2760
                 (if a > b then acc else fold_atLeastAtMost_nat f (a+1) b (f a acc))"
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2761
by pat_completeness auto
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2762
termination by (relation "measure (\<lambda>(_,a,b,_). Suc b - a)") auto
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2763
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2764
lemma fold_atLeastAtMost_nat:
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2765
  assumes "comp_fun_commute f"
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2766
  shows   "fold_atLeastAtMost_nat f a b acc = Finite_Set.fold f acc {a..b}"
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2767
using assms
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2768
proof (induction f a b acc rule: fold_atLeastAtMost_nat.induct, goal_cases)
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2769
  case (1 f a b acc)
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2770
  interpret comp_fun_commute f by fact
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2771
  show ?case
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2772
  proof (cases "a > b")
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2773
    case True
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2774
    thus ?thesis by (subst fold_atLeastAtMost_nat.simps) auto
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2775
  next
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2776
    case False
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2777
    with 1 show ?thesis
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2778
      by (subst fold_atLeastAtMost_nat.simps)
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2779
         (auto simp: atLeastAtMost_insertL[symmetric] fold_fun_left_comm)
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2780
  qed
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2781
qed
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2782
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2783
lemma sum_atLeastAtMost_code:
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2784
  "sum f {a..b} = fold_atLeastAtMost_nat (\<lambda>a acc. f a + acc) a b 0"
62128
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2785
proof -
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67091
diff changeset
  2786
  have "comp_fun_commute (\<lambda>a. (+) (f a))"
62128
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2787
    by unfold_locales (auto simp: o_def add_ac)
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2788
  thus ?thesis
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63967
diff changeset
  2789
    by (simp add: sum.eq_fold fold_atLeastAtMost_nat o_def)
62128
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2790
qed
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2791
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  2792
lemma prod_atLeastAtMost_code:
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  2793
  "prod f {a..b} = fold_atLeastAtMost_nat (\<lambda>a acc. f a * acc) a b 1"
62128
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2794
proof -
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68618
diff changeset
  2795
  have "comp_fun_commute (\<lambda>a. (*) (f a))"
62128
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2796
    by unfold_locales (auto simp: o_def mult_ac)
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2797
  thus ?thesis
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 64267
diff changeset
  2798
    by (simp add: prod.eq_fold fold_atLeastAtMost_nat o_def)
62128
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2799
qed
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2800
70746
cf7b5020c207 Generalisation of many theorems to a more abstract type class (suggested by Mr Anonymous)
paulson <lp15@cam.ac.uk>
parents: 70723
diff changeset
  2801
(* TODO: Add support for folding over more kinds of intervals here *)
62128
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61955
diff changeset
  2802
78663
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2803
lemma pairs_le_eq_Sigma: "{(i, j). i + j \<le> m} = Sigma (atMost m) (\<lambda>r. atMost (m - r))"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2804
  for m :: nat
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2805
  by auto
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2806
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2807
lemma sum_up_index_split: "(\<Sum>k\<le>m + n. f k) = (\<Sum>k\<le>m. f k) + (\<Sum>k = Suc m..m + n. f k)"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2808
  by (metis atLeast0AtMost Suc_eq_plus1 le0 sum.ub_add_nat)
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2809
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2810
lemma Sigma_interval_disjoint: "(SIGMA i:A. {..v i}) \<inter> (SIGMA i:A.{v i<..w}) = {}"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2811
  for w :: "'a::order"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2812
  by auto
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2813
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2814
lemma product_atMost_eq_Un: "A \<times> {..m} = (SIGMA i:A.{..m - i}) \<union> (SIGMA i:A.{m - i<..m})"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2815
  for m :: nat
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2816
  by auto
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2817
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2818
lemma polynomial_product: (*with thanks to Chaitanya Mangla*)
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2819
  fixes x :: "'a::idom"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2820
  assumes m: "\<And>i. i > m \<Longrightarrow> a i = 0"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2821
    and n: "\<And>j. j > n \<Longrightarrow> b j = 0"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2822
  shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) =
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2823
         (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2824
proof -
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2825
  have "\<And>i j. \<lbrakk>m + n - i < j; a i \<noteq> 0\<rbrakk> \<Longrightarrow> b j = 0"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2826
    by (meson le_add_diff leI le_less_trans m n)
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2827
  then have \<section>: "(\<Sum>(i,j)\<in>(SIGMA i:{..m+n}. {m+n - i<..m+n}). a i * x ^ i * (b j * x ^ j)) = 0"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2828
    by (clarsimp simp add: sum_Un Sigma_interval_disjoint intro!: sum.neutral)
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2829
  have "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) = (\<Sum>i\<le>m. \<Sum>j\<le>n. (a i * x ^ i) * (b j * x ^ j))"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2830
    by (rule sum_product)
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2831
  also have "\<dots> = (\<Sum>i\<le>m + n. \<Sum>j\<le>n + m. a i * x ^ i * (b j * x ^ j))"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2832
    using assms by (auto simp: sum_up_index_split)
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2833
  also have "\<dots> = (\<Sum>r\<le>m + n. \<Sum>j\<le>m + n - r. a r * x ^ r * (b j * x ^ j))"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2834
    by (simp add: add_ac sum.Sigma product_atMost_eq_Un sum_Un Sigma_interval_disjoint \<section>)
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2835
  also have "\<dots> = (\<Sum>(i,j)\<in>{(i,j). i+j \<le> m+n}. (a i * x ^ i) * (b j * x ^ j))"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2836
    by (auto simp: pairs_le_eq_Sigma sum.Sigma)
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2837
  also have "... = (\<Sum>k\<le>m + n. \<Sum>i\<le>k. a i * x ^ i * (b (k - i) * x ^ (k - i)))"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2838
    by (rule sum.triangle_reindex_eq)
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2839
  also have "\<dots> = (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2840
    by (auto simp: algebra_simps sum_distrib_left simp flip: power_add intro!: sum.cong)
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2841
  finally show ?thesis .
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2842
qed
3032bc7d613d A little reorganisation
paulson <lp15@cam.ac.uk>
parents: 78256
diff changeset
  2843
8924
c434283b4cfa Added SetInterval
nipkow
parents:
diff changeset
  2844
end