src/HOL/Set_Interval.thy
author hoelzl
Mon Jun 30 15:45:21 2014 +0200 (2014-06-30)
changeset 57447 87429bdecad5
parent 57418 6ab1c7cb0b8d
child 57448 159e45728ceb
permissions -rw-r--r--
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
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(*  Title:      HOL/Set_Interval.thy
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    Author:     Tobias Nipkow
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    Author:     Clemens Ballarin
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    Author:     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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header {* Set intervals *}
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theory Set_Interval
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imports Lattices_Big Nat_Transfer
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begin
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context ord
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begin
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definition
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  lessThan    :: "'a => 'a set" ("(1{..<_})") where
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  "{..<u} == {x. x < u}"
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definition
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  atMost      :: "'a => 'a set" ("(1{.._})") where
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  "{..u} == {x. x \<le> u}"
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definition
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  greaterThan :: "'a => 'a set" ("(1{_<..})") where
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  "{l<..} == {x. l<x}"
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definition
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  atLeast     :: "'a => 'a set" ("(1{_..})") where
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  "{l..} == {x. l\<le>x}"
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definition
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  greaterThanLessThan :: "'a => 'a => 'a set"  ("(1{_<..<_})") where
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  "{l<..<u} == {l<..} Int {..<u}"
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definition
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  atLeastLessThan :: "'a => 'a => 'a set"      ("(1{_..<_})") where
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  "{l..<u} == {l..} Int {..<u}"
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definition
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  greaterThanAtMost :: "'a => 'a => 'a set"    ("(1{_<.._})") where
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  "{l<..u} == {l<..} Int {..u}"
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definition
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  atLeastAtMost :: "'a => 'a => 'a set"        ("(1{_.._})") where
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  "{l..u} == {l..} Int {..u}"
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end
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text{* A note of warning when using @{term"{..<n}"} on type @{typ
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nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
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@{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
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syntax
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  "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3UN _<=_./ _)" [0, 0, 10] 10)
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  "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3UN _<_./ _)" [0, 0, 10] 10)
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  "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" [0, 0, 10] 10)
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  "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3INT _<_./ _)" [0, 0, 10] 10)
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syntax (xsymbols)
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  "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" [0, 0, 10] 10)
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  "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _<_./ _)" [0, 0, 10] 10)
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  "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" [0, 0, 10] 10)
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  "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" [0, 0, 10] 10)
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syntax (latex output)
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  "_UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ \<le> _)/ _)" [0, 0, 10] 10)
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  "_UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ < _)/ _)" [0, 0, 10] 10)
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  "_INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ \<le> _)/ _)" [0, 0, 10] 10)
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  "_INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ < _)/ _)" [0, 0, 10] 10)
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translations
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  "UN i<=n. A"  == "UN i:{..n}. A"
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  "UN i<n. A"   == "UN i:{..<n}. A"
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  "INT i<=n. A" == "INT i:{..n}. A"
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  "INT i<n. A"  == "INT i:{..<n}. A"
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subsection {* Various equivalences *}
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lemma (in ord) lessThan_iff [iff]: "(i: 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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apply (auto simp add: lessThan_def atLeast_def)
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done
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lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
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by auto
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lemma (in ord) greaterThan_iff [iff]: "(i: 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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apply (subst Compl_greaterThan [symmetric])
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apply (rule double_complement)
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done
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lemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
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by (simp add: atLeast_def)
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lemma Compl_atLeast [simp]:
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    "!!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: atMost k) = (i<=k)"
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by (simp add: atMost_def)
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lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
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by (blast intro: order_antisym)
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lemma (in linorder) lessThan_Int_lessThan: "{ a <..} \<inter> { b <..} = { max a b <..}"
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  by auto
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lemma (in linorder) greaterThan_Int_greaterThan: "{..< a} \<inter> {..< b} = {..< min a b}"
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  by auto
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subsection {* Logical Equivalences for Set Inclusion and Equality *}
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lemma atLeast_subset_iff [iff]:
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     "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
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by (blast intro: order_trans)
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lemma atLeast_eq_iff [iff]:
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     "(atLeast x = atLeast y) = (x = (y::'a::linorder))"
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by (blast intro: order_antisym order_trans)
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lemma greaterThan_subset_iff [iff]:
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     "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
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apply (auto simp add: greaterThan_def)
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 apply (subst linorder_not_less [symmetric], blast)
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done
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lemma greaterThan_eq_iff [iff]:
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     "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
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apply (rule iffI)
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 apply (erule equalityE)
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 apply simp_all
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done
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lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))"
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by (blast intro: order_trans)
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lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"
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by (blast intro: order_antisym order_trans)
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lemma lessThan_subset_iff [iff]:
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     "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
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apply (auto simp add: lessThan_def)
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 apply (subst linorder_not_less [symmetric], blast)
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done
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lemma lessThan_eq_iff [iff]:
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     "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
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apply (rule iffI)
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 apply (erule equalityE)
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 apply simp_all
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done
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lemma lessThan_strict_subset_iff:
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  fixes m n :: "'a::linorder"
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  shows "{..<m} < {..<n} \<longleftrightarrow> m < n"
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  by (metis leD lessThan_subset_iff linorder_linear not_less_iff_gr_or_eq psubset_eq)
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subsection {*Two-sided intervals*}
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context ord
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begin
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lemma greaterThanLessThan_iff [simp]:
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  "(i : {l<..<u}) = (l < i & i < u)"
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by (simp add: greaterThanLessThan_def)
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lemma atLeastLessThan_iff [simp]:
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  "(i : {l..<u}) = (l <= i & i < u)"
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by (simp add: atLeastLessThan_def)
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lemma greaterThanAtMost_iff [simp]:
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  "(i : {l<..u}) = (l < i & i <= u)"
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by (simp add: greaterThanAtMost_def)
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lemma atLeastAtMost_iff [simp]:
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  "(i : {l..u}) = (l <= i & i <= u)"
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by (simp add: atLeastAtMost_def)
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text {* The above four lemmas could be declared as iffs. Unfortunately this
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breaks many proofs. Since it only helps blast, it is better to leave them
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alone. *}
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lemma greaterThanLessThan_eq: "{ a <..< b} = { a <..} \<inter> {..< b }"
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  by auto
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end
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subsubsection{* Emptyness, singletons, subset *}
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context order
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begin
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lemma atLeastatMost_empty[simp]:
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  "b < a \<Longrightarrow> {a..b} = {}"
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by(auto simp: atLeastAtMost_def atLeast_def atMost_def)
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lemma atLeastatMost_empty_iff[simp]:
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  "{a..b} = {} \<longleftrightarrow> (~ a <= b)"
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by auto (blast intro: order_trans)
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lemma atLeastatMost_empty_iff2[simp]:
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  "{} = {a..b} \<longleftrightarrow> (~ a <= b)"
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by auto (blast intro: order_trans)
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lemma atLeastLessThan_empty[simp]:
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  "b <= a \<Longrightarrow> {a..<b} = {}"
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by(auto simp: atLeastLessThan_def)
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lemma atLeastLessThan_empty_iff[simp]:
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  "{a..<b} = {} \<longleftrightarrow> (~ a < b)"
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by auto (blast intro: le_less_trans)
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lemma atLeastLessThan_empty_iff2[simp]:
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  "{} = {a..<b} \<longleftrightarrow> (~ a < b)"
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by auto (blast intro: le_less_trans)
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lemma greaterThanAtMost_empty[simp]: "l \<le> k ==> {k<..l} = {}"
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by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
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lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {} \<longleftrightarrow> ~ k < l"
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by auto (blast intro: less_le_trans)
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lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} \<longleftrightarrow> ~ k < l"
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by auto (blast intro: less_le_trans)
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lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..<l} = {}"
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by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
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lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"
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by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
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lemma atLeastAtMost_singleton': "a = b \<Longrightarrow> {a .. b} = {a}" by simp
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lemma atLeastatMost_subset_iff[simp]:
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  "{a..b} <= {c..d} \<longleftrightarrow> (~ a <= b) | c <= a & b <= d"
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unfolding atLeastAtMost_def atLeast_def atMost_def
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by (blast intro: order_trans)
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lemma atLeastatMost_psubset_iff:
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  "{a..b} < {c..d} \<longleftrightarrow>
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   ((~ a <= b) | c <= a & b <= d & (c < a | b < d))  &  c <= d"
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by(simp add: psubset_eq set_eq_iff less_le_not_le)(blast intro: order_trans)
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lemma Icc_eq_Icc[simp]:
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  "{l..h} = {l'..h'} = (l=l' \<and> h=h' \<or> \<not> l\<le>h \<and> \<not> l'\<le>h')"
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by(simp add: order_class.eq_iff)(auto intro: order_trans)
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lemma atLeastAtMost_singleton_iff[simp]:
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  "{a .. b} = {c} \<longleftrightarrow> a = b \<and> b = c"
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proof
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  assume "{a..b} = {c}"
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  hence *: "\<not> (\<not> a \<le> b)" unfolding atLeastatMost_empty_iff[symmetric] by simp
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  with `{a..b} = {c}` have "c \<le> a \<and> b \<le> c" by auto
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  with * show "a = b \<and> b = c" by auto
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qed simp
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lemma Icc_subset_Ici_iff[simp]:
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  "{l..h} \<subseteq> {l'..} = (~ l\<le>h \<or> l\<ge>l')"
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by(auto simp: subset_eq intro: order_trans)
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lemma Icc_subset_Iic_iff[simp]:
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  "{l..h} \<subseteq> {..h'} = (~ l\<le>h \<or> h\<le>h')"
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by(auto simp: subset_eq intro: order_trans)
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lemma not_Ici_eq_empty[simp]: "{l..} \<noteq> {}"
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by(auto simp: set_eq_iff)
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lemma not_Iic_eq_empty[simp]: "{..h} \<noteq> {}"
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by(auto simp: set_eq_iff)
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lemmas not_empty_eq_Ici_eq_empty[simp] = not_Ici_eq_empty[symmetric]
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lemmas not_empty_eq_Iic_eq_empty[simp] = not_Iic_eq_empty[symmetric]
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end
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context no_top
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begin
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(* also holds for no_bot but no_top should suffice *)
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lemma not_UNIV_le_Icc[simp]: "\<not> UNIV \<subseteq> {l..h}"
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using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
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lemma not_UNIV_le_Iic[simp]: "\<not> UNIV \<subseteq> {..h}"
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using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
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lemma not_Ici_le_Icc[simp]: "\<not> {l..} \<subseteq> {l'..h'}"
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using gt_ex[of h']
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by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
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lemma not_Ici_le_Iic[simp]: "\<not> {l..} \<subseteq> {..h'}"
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using gt_ex[of h']
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by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
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end
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context no_bot
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begin
nipkow@51334
   318
nipkow@51334
   319
lemma not_UNIV_le_Ici[simp]: "\<not> UNIV \<subseteq> {l..}"
nipkow@51334
   320
using lt_ex[of l] by(auto simp: subset_eq less_le_not_le)
nipkow@51334
   321
nipkow@51334
   322
lemma not_Iic_le_Icc[simp]: "\<not> {..h} \<subseteq> {l'..h'}"
nipkow@51334
   323
using lt_ex[of l']
nipkow@51334
   324
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
nipkow@51334
   325
nipkow@51334
   326
lemma not_Iic_le_Ici[simp]: "\<not> {..h} \<subseteq> {l'..}"
nipkow@51334
   327
using lt_ex[of l']
nipkow@51334
   328
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
nipkow@51334
   329
nipkow@51334
   330
end
nipkow@51334
   331
nipkow@51334
   332
nipkow@51334
   333
context no_top
nipkow@51334
   334
begin
nipkow@51334
   335
nipkow@51334
   336
(* also holds for no_bot but no_top should suffice *)
nipkow@51334
   337
lemma not_UNIV_eq_Icc[simp]: "\<not> UNIV = {l'..h'}"
nipkow@51334
   338
using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
nipkow@51334
   339
nipkow@51334
   340
lemmas not_Icc_eq_UNIV[simp] = not_UNIV_eq_Icc[symmetric]
nipkow@51334
   341
nipkow@51334
   342
lemma not_UNIV_eq_Iic[simp]: "\<not> UNIV = {..h'}"
nipkow@51334
   343
using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
nipkow@51334
   344
nipkow@51334
   345
lemmas not_Iic_eq_UNIV[simp] = not_UNIV_eq_Iic[symmetric]
nipkow@51334
   346
nipkow@51334
   347
lemma not_Icc_eq_Ici[simp]: "\<not> {l..h} = {l'..}"
nipkow@51334
   348
unfolding atLeastAtMost_def using not_Ici_le_Iic[of l'] by blast
nipkow@51334
   349
nipkow@51334
   350
lemmas not_Ici_eq_Icc[simp] = not_Icc_eq_Ici[symmetric]
nipkow@51334
   351
nipkow@51334
   352
(* also holds for no_bot but no_top should suffice *)
nipkow@51334
   353
lemma not_Iic_eq_Ici[simp]: "\<not> {..h} = {l'..}"
nipkow@51334
   354
using not_Ici_le_Iic[of l' h] by blast
nipkow@51334
   355
nipkow@51334
   356
lemmas not_Ici_eq_Iic[simp] = not_Iic_eq_Ici[symmetric]
nipkow@51334
   357
nipkow@51334
   358
end
nipkow@51334
   359
nipkow@51334
   360
context no_bot
nipkow@51334
   361
begin
nipkow@51334
   362
nipkow@51334
   363
lemma not_UNIV_eq_Ici[simp]: "\<not> UNIV = {l'..}"
nipkow@51334
   364
using lt_ex[of l'] by(auto simp: set_eq_iff  less_le_not_le)
nipkow@51334
   365
nipkow@51334
   366
lemmas not_Ici_eq_UNIV[simp] = not_UNIV_eq_Ici[symmetric]
nipkow@51334
   367
nipkow@51334
   368
lemma not_Icc_eq_Iic[simp]: "\<not> {l..h} = {..h'}"
nipkow@51334
   369
unfolding atLeastAtMost_def using not_Iic_le_Ici[of h'] by blast
nipkow@51334
   370
nipkow@51334
   371
lemmas not_Iic_eq_Icc[simp] = not_Icc_eq_Iic[symmetric]
nipkow@51334
   372
nipkow@51334
   373
end
nipkow@51334
   374
nipkow@51334
   375
hoelzl@53216
   376
context dense_linorder
hoelzl@42891
   377
begin
hoelzl@42891
   378
hoelzl@42891
   379
lemma greaterThanLessThan_empty_iff[simp]:
hoelzl@42891
   380
  "{ a <..< b } = {} \<longleftrightarrow> b \<le> a"
hoelzl@42891
   381
  using dense[of a b] by (cases "a < b") auto
hoelzl@42891
   382
hoelzl@42891
   383
lemma greaterThanLessThan_empty_iff2[simp]:
hoelzl@42891
   384
  "{} = { a <..< b } \<longleftrightarrow> b \<le> a"
hoelzl@42891
   385
  using dense[of a b] by (cases "a < b") auto
hoelzl@42891
   386
hoelzl@42901
   387
lemma atLeastLessThan_subseteq_atLeastAtMost_iff:
hoelzl@42901
   388
  "{a ..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
hoelzl@42901
   389
  using dense[of "max a d" "b"]
hoelzl@42901
   390
  by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@42901
   391
hoelzl@42901
   392
lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:
hoelzl@42901
   393
  "{a <.. b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
hoelzl@42901
   394
  using dense[of "a" "min c b"]
hoelzl@42901
   395
  by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@42901
   396
hoelzl@42901
   397
lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:
hoelzl@42901
   398
  "{a <..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
hoelzl@42901
   399
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
hoelzl@42901
   400
  by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@42901
   401
hoelzl@43657
   402
lemma atLeastAtMost_subseteq_atLeastLessThan_iff:
hoelzl@43657
   403
  "{a .. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a \<le> b \<longrightarrow> c \<le> a \<and> b < d)"
hoelzl@43657
   404
  using dense[of "max a d" "b"]
hoelzl@43657
   405
  by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@43657
   406
hoelzl@43657
   407
lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:
hoelzl@43657
   408
  "{a <.. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b < d)"
hoelzl@43657
   409
  using dense[of "a" "min c b"]
hoelzl@43657
   410
  by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@43657
   411
hoelzl@43657
   412
lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:
hoelzl@43657
   413
  "{a <..< b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
hoelzl@43657
   414
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
hoelzl@43657
   415
  by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@43657
   416
hoelzl@56328
   417
lemma greaterThanLessThan_subseteq_greaterThanAtMost_iff:
hoelzl@56328
   418
  "{a <..< b} \<subseteq> { c <.. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
hoelzl@56328
   419
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
hoelzl@56328
   420
  by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@56328
   421
hoelzl@42891
   422
end
hoelzl@42891
   423
hoelzl@51329
   424
context no_top
hoelzl@51329
   425
begin
hoelzl@51329
   426
nipkow@51334
   427
lemma greaterThan_non_empty[simp]: "{x <..} \<noteq> {}"
hoelzl@51329
   428
  using gt_ex[of x] by auto
hoelzl@51329
   429
hoelzl@51329
   430
end
hoelzl@51329
   431
hoelzl@51329
   432
context no_bot
hoelzl@51329
   433
begin
hoelzl@51329
   434
nipkow@51334
   435
lemma lessThan_non_empty[simp]: "{..< x} \<noteq> {}"
hoelzl@51329
   436
  using lt_ex[of x] by auto
hoelzl@51329
   437
hoelzl@51329
   438
end
hoelzl@51329
   439
nipkow@32408
   440
lemma (in linorder) atLeastLessThan_subset_iff:
nipkow@32408
   441
  "{a..<b} <= {c..<d} \<Longrightarrow> b <= a | c<=a & b<=d"
nipkow@32408
   442
apply (auto simp:subset_eq Ball_def)
nipkow@32408
   443
apply(frule_tac x=a in spec)
nipkow@32408
   444
apply(erule_tac x=d in allE)
nipkow@32408
   445
apply (simp add: less_imp_le)
nipkow@32408
   446
done
nipkow@32408
   447
hoelzl@40703
   448
lemma atLeastLessThan_inj:
hoelzl@40703
   449
  fixes a b c d :: "'a::linorder"
hoelzl@40703
   450
  assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"
hoelzl@40703
   451
  shows "a = c" "b = d"
hoelzl@40703
   452
using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le linorder_antisym_conv2 subset_refl)+
hoelzl@40703
   453
hoelzl@40703
   454
lemma atLeastLessThan_eq_iff:
hoelzl@40703
   455
  fixes a b c d :: "'a::linorder"
hoelzl@40703
   456
  assumes "a < b" "c < d"
hoelzl@40703
   457
  shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"
hoelzl@40703
   458
  using atLeastLessThan_inj assms by auto
hoelzl@40703
   459
hoelzl@57447
   460
lemma (in linorder) Ioc_inj: "{a <.. b} = {c <.. d} \<longleftrightarrow> (b \<le> a \<and> d \<le> c) \<or> a = c \<and> b = d"
hoelzl@57447
   461
  by (metis eq_iff greaterThanAtMost_empty_iff2 greaterThanAtMost_iff le_cases not_le)
hoelzl@57447
   462
hoelzl@57447
   463
lemma (in order) Iio_Int_singleton: "{..<k} \<inter> {x} = (if x < k then {x} else {})"
hoelzl@57447
   464
  by auto
hoelzl@57447
   465
hoelzl@57447
   466
lemma (in linorder) Ioc_subset_iff: "{a<..b} \<subseteq> {c<..d} \<longleftrightarrow> (b \<le> a \<or> c \<le> a \<and> b \<le> d)"
hoelzl@57447
   467
  by (auto simp: subset_eq Ball_def) (metis less_le not_less)
hoelzl@57447
   468
haftmann@52729
   469
lemma (in order_bot) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"
nipkow@51334
   470
by (auto simp: set_eq_iff intro: le_bot)
hoelzl@51328
   471
haftmann@52729
   472
lemma (in order_top) atMost_eq_UNIV_iff: "{..x} = UNIV \<longleftrightarrow> x = top"
nipkow@51334
   473
by (auto simp: set_eq_iff intro: top_le)
hoelzl@51328
   474
nipkow@51334
   475
lemma (in bounded_lattice) atLeastAtMost_eq_UNIV_iff:
nipkow@51334
   476
  "{x..y} = UNIV \<longleftrightarrow> (x = bot \<and> y = top)"
nipkow@51334
   477
by (auto simp: set_eq_iff intro: top_le le_bot)
hoelzl@51328
   478
hoelzl@56949
   479
lemma Iio_eq_empty_iff: "{..< n::'a::{linorder, order_bot}} = {} \<longleftrightarrow> n = bot"
hoelzl@56949
   480
  by (auto simp: set_eq_iff not_less le_bot)
hoelzl@56949
   481
hoelzl@56949
   482
lemma Iio_eq_empty_iff_nat: "{..< n::nat} = {} \<longleftrightarrow> n = 0"
hoelzl@56949
   483
  by (simp add: Iio_eq_empty_iff bot_nat_def)
hoelzl@56949
   484
hoelzl@51328
   485
hoelzl@56328
   486
subsection {* Infinite intervals *}
hoelzl@56328
   487
hoelzl@56328
   488
context dense_linorder
hoelzl@56328
   489
begin
hoelzl@56328
   490
hoelzl@56328
   491
lemma infinite_Ioo:
hoelzl@56328
   492
  assumes "a < b"
hoelzl@56328
   493
  shows "\<not> finite {a<..<b}"
hoelzl@56328
   494
proof
hoelzl@56328
   495
  assume fin: "finite {a<..<b}"
hoelzl@56328
   496
  moreover have ne: "{a<..<b} \<noteq> {}"
hoelzl@56328
   497
    using `a < b` by auto
hoelzl@56328
   498
  ultimately have "a < Max {a <..< b}" "Max {a <..< b} < b"
hoelzl@56328
   499
    using Max_in[of "{a <..< b}"] by auto
hoelzl@56328
   500
  then obtain x where "Max {a <..< b} < x" "x < b"
hoelzl@56328
   501
    using dense[of "Max {a<..<b}" b] by auto
hoelzl@56328
   502
  then have "x \<in> {a <..< b}"
hoelzl@56328
   503
    using `a < Max {a <..< b}` by auto
hoelzl@56328
   504
  then have "x \<le> Max {a <..< b}"
hoelzl@56328
   505
    using fin by auto
hoelzl@56328
   506
  with `Max {a <..< b} < x` show False by auto
hoelzl@56328
   507
qed
hoelzl@56328
   508
hoelzl@56328
   509
lemma infinite_Icc: "a < b \<Longrightarrow> \<not> finite {a .. b}"
hoelzl@56328
   510
  using greaterThanLessThan_subseteq_atLeastAtMost_iff[of a b a b] infinite_Ioo[of a b]
hoelzl@56328
   511
  by (auto dest: finite_subset)
hoelzl@56328
   512
hoelzl@56328
   513
lemma infinite_Ico: "a < b \<Longrightarrow> \<not> finite {a ..< b}"
hoelzl@56328
   514
  using greaterThanLessThan_subseteq_atLeastLessThan_iff[of a b a b] infinite_Ioo[of a b]
hoelzl@56328
   515
  by (auto dest: finite_subset)
hoelzl@56328
   516
hoelzl@56328
   517
lemma infinite_Ioc: "a < b \<Longrightarrow> \<not> finite {a <.. b}"
hoelzl@56328
   518
  using greaterThanLessThan_subseteq_greaterThanAtMost_iff[of a b a b] infinite_Ioo[of a b]
hoelzl@56328
   519
  by (auto dest: finite_subset)
hoelzl@56328
   520
hoelzl@56328
   521
end
hoelzl@56328
   522
hoelzl@56328
   523
lemma infinite_Iio: "\<not> finite {..< a :: 'a :: {no_bot, linorder}}"
hoelzl@56328
   524
proof
hoelzl@56328
   525
  assume "finite {..< a}"
hoelzl@56328
   526
  then have *: "\<And>x. x < a \<Longrightarrow> Min {..< a} \<le> x"
hoelzl@56328
   527
    by auto
hoelzl@56328
   528
  obtain x where "x < a"
hoelzl@56328
   529
    using lt_ex by auto
hoelzl@56328
   530
hoelzl@56328
   531
  obtain y where "y < Min {..< a}"
hoelzl@56328
   532
    using lt_ex by auto
hoelzl@56328
   533
  also have "Min {..< a} \<le> x"
hoelzl@56328
   534
    using `x < a` by fact
hoelzl@56328
   535
  also note `x < a`
hoelzl@56328
   536
  finally have "Min {..< a} \<le> y"
hoelzl@56328
   537
    by fact
hoelzl@56328
   538
  with `y < Min {..< a}` show False by auto
hoelzl@56328
   539
qed
hoelzl@56328
   540
hoelzl@56328
   541
lemma infinite_Iic: "\<not> finite {.. a :: 'a :: {no_bot, linorder}}"
hoelzl@56328
   542
  using infinite_Iio[of a] finite_subset[of "{..< a}" "{.. a}"]
hoelzl@56328
   543
  by (auto simp: subset_eq less_imp_le)
hoelzl@56328
   544
hoelzl@56328
   545
lemma infinite_Ioi: "\<not> finite {a :: 'a :: {no_top, linorder} <..}"
hoelzl@56328
   546
proof
hoelzl@56328
   547
  assume "finite {a <..}"
hoelzl@56328
   548
  then have *: "\<And>x. a < x \<Longrightarrow> x \<le> Max {a <..}"
hoelzl@56328
   549
    by auto
hoelzl@56328
   550
hoelzl@56328
   551
  obtain y where "Max {a <..} < y"
hoelzl@56328
   552
    using gt_ex by auto
hoelzl@56328
   553
hoelzl@56328
   554
  obtain x where "a < x"
hoelzl@56328
   555
    using gt_ex by auto
hoelzl@56328
   556
  also then have "x \<le> Max {a <..}"
hoelzl@56328
   557
    by fact
hoelzl@56328
   558
  also note `Max {a <..} < y`
hoelzl@56328
   559
  finally have "y \<le> Max { a <..}"
hoelzl@56328
   560
    by fact
hoelzl@56328
   561
  with `Max {a <..} < y` show False by auto
hoelzl@56328
   562
qed
hoelzl@56328
   563
hoelzl@56328
   564
lemma infinite_Ici: "\<not> finite {a :: 'a :: {no_top, linorder} ..}"
hoelzl@56328
   565
  using infinite_Ioi[of a] finite_subset[of "{a <..}" "{a ..}"]
hoelzl@56328
   566
  by (auto simp: subset_eq less_imp_le)
hoelzl@56328
   567
nipkow@32456
   568
subsubsection {* Intersection *}
nipkow@32456
   569
nipkow@32456
   570
context linorder
nipkow@32456
   571
begin
nipkow@32456
   572
nipkow@32456
   573
lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"
nipkow@32456
   574
by auto
nipkow@32456
   575
nipkow@32456
   576
lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"
nipkow@32456
   577
by auto
nipkow@32456
   578
nipkow@32456
   579
lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"
nipkow@32456
   580
by auto
nipkow@32456
   581
nipkow@32456
   582
lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"
nipkow@32456
   583
by auto
nipkow@32456
   584
nipkow@32456
   585
lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"
nipkow@32456
   586
by auto
nipkow@32456
   587
nipkow@32456
   588
lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"
nipkow@32456
   589
by auto
nipkow@32456
   590
nipkow@32456
   591
lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
nipkow@32456
   592
by auto
nipkow@32456
   593
nipkow@32456
   594
lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
nipkow@32456
   595
by auto
nipkow@32456
   596
hoelzl@50417
   597
lemma Int_atMost[simp]: "{..a} \<inter> {..b} = {.. min a b}"
hoelzl@50417
   598
  by (auto simp: min_def)
hoelzl@50417
   599
hoelzl@57447
   600
lemma Ioc_disjoint: "{a<..b} \<inter> {c<..d} = {} \<longleftrightarrow> b \<le> a \<or> d \<le> c \<or> b \<le> c \<or> d \<le> a"
hoelzl@57447
   601
  using assms by auto
hoelzl@57447
   602
nipkow@32456
   603
end
nipkow@32456
   604
hoelzl@51329
   605
context complete_lattice
hoelzl@51329
   606
begin
hoelzl@51329
   607
hoelzl@51329
   608
lemma
hoelzl@51329
   609
  shows Sup_atLeast[simp]: "Sup {x ..} = top"
hoelzl@51329
   610
    and Sup_greaterThanAtLeast[simp]: "x < top \<Longrightarrow> Sup {x <..} = top"
hoelzl@51329
   611
    and Sup_atMost[simp]: "Sup {.. y} = y"
hoelzl@51329
   612
    and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"
hoelzl@51329
   613
    and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"
hoelzl@51329
   614
  by (auto intro!: Sup_eqI)
hoelzl@51329
   615
hoelzl@51329
   616
lemma
hoelzl@51329
   617
  shows Inf_atMost[simp]: "Inf {.. x} = bot"
hoelzl@51329
   618
    and Inf_atMostLessThan[simp]: "top < x \<Longrightarrow> Inf {..< x} = bot"
hoelzl@51329
   619
    and Inf_atLeast[simp]: "Inf {x ..} = x"
hoelzl@51329
   620
    and Inf_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Inf { x .. y} = x"
hoelzl@51329
   621
    and Inf_atLeastLessThan[simp]: "x < y \<Longrightarrow> Inf { x ..< y} = x"
hoelzl@51329
   622
  by (auto intro!: Inf_eqI)
hoelzl@51329
   623
hoelzl@51329
   624
end
hoelzl@51329
   625
hoelzl@51329
   626
lemma
hoelzl@53216
   627
  fixes x y :: "'a :: {complete_lattice, dense_linorder}"
hoelzl@51329
   628
  shows Sup_lessThan[simp]: "Sup {..< y} = y"
hoelzl@51329
   629
    and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"
hoelzl@51329
   630
    and Sup_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Sup { x <..< y} = y"
hoelzl@51329
   631
    and Inf_greaterThan[simp]: "Inf {x <..} = x"
hoelzl@51329
   632
    and Inf_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Inf { x <.. y} = x"
hoelzl@51329
   633
    and Inf_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Inf { x <..< y} = x"
hoelzl@51329
   634
  by (auto intro!: Inf_eqI Sup_eqI intro: dense_le dense_le_bounded dense_ge dense_ge_bounded)
nipkow@32456
   635
paulson@14485
   636
subsection {* Intervals of natural numbers *}
paulson@14485
   637
paulson@15047
   638
subsubsection {* The Constant @{term lessThan} *}
paulson@15047
   639
paulson@14485
   640
lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
paulson@14485
   641
by (simp add: lessThan_def)
paulson@14485
   642
paulson@14485
   643
lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
paulson@14485
   644
by (simp add: lessThan_def less_Suc_eq, blast)
paulson@14485
   645
kleing@43156
   646
text {* The following proof is convenient in induction proofs where
hoelzl@39072
   647
new elements get indices at the beginning. So it is used to transform
hoelzl@39072
   648
@{term "{..<Suc n}"} to @{term "0::nat"} and @{term "{..< n}"}. *}
hoelzl@39072
   649
hoelzl@39072
   650
lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"
hoelzl@39072
   651
proof safe
hoelzl@39072
   652
  fix x assume "x < Suc n" "x \<notin> Suc ` {..<n}"
hoelzl@39072
   653
  then have "x \<noteq> Suc (x - 1)" by auto
hoelzl@39072
   654
  with `x < Suc n` show "x = 0" by auto
hoelzl@39072
   655
qed
hoelzl@39072
   656
paulson@14485
   657
lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
paulson@14485
   658
by (simp add: lessThan_def atMost_def less_Suc_eq_le)
paulson@14485
   659
paulson@14485
   660
lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
paulson@14485
   661
by blast
paulson@14485
   662
paulson@15047
   663
subsubsection {* The Constant @{term greaterThan} *}
paulson@15047
   664
paulson@14485
   665
lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
paulson@14485
   666
apply (simp add: greaterThan_def)
paulson@14485
   667
apply (blast dest: gr0_conv_Suc [THEN iffD1])
paulson@14485
   668
done
paulson@14485
   669
paulson@14485
   670
lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
paulson@14485
   671
apply (simp add: greaterThan_def)
paulson@14485
   672
apply (auto elim: linorder_neqE)
paulson@14485
   673
done
paulson@14485
   674
paulson@14485
   675
lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
paulson@14485
   676
by blast
paulson@14485
   677
paulson@15047
   678
subsubsection {* The Constant @{term atLeast} *}
paulson@15047
   679
paulson@14485
   680
lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
paulson@14485
   681
by (unfold atLeast_def UNIV_def, simp)
paulson@14485
   682
paulson@14485
   683
lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
paulson@14485
   684
apply (simp add: atLeast_def)
paulson@14485
   685
apply (simp add: Suc_le_eq)
paulson@14485
   686
apply (simp add: order_le_less, blast)
paulson@14485
   687
done
paulson@14485
   688
paulson@14485
   689
lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
paulson@14485
   690
  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
paulson@14485
   691
paulson@14485
   692
lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
paulson@14485
   693
by blast
paulson@14485
   694
paulson@15047
   695
subsubsection {* The Constant @{term atMost} *}
paulson@15047
   696
paulson@14485
   697
lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
paulson@14485
   698
by (simp add: atMost_def)
paulson@14485
   699
paulson@14485
   700
lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
paulson@14485
   701
apply (simp add: atMost_def)
paulson@14485
   702
apply (simp add: less_Suc_eq order_le_less, blast)
paulson@14485
   703
done
paulson@14485
   704
paulson@14485
   705
lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
paulson@14485
   706
by blast
paulson@14485
   707
paulson@15047
   708
subsubsection {* The Constant @{term atLeastLessThan} *}
paulson@15047
   709
nipkow@28068
   710
text{*The orientation of the following 2 rules is tricky. The lhs is
nipkow@24449
   711
defined in terms of the rhs.  Hence the chosen orientation makes sense
nipkow@24449
   712
in this theory --- the reverse orientation complicates proofs (eg
nipkow@24449
   713
nontermination). But outside, when the definition of the lhs is rarely
nipkow@24449
   714
used, the opposite orientation seems preferable because it reduces a
nipkow@24449
   715
specific concept to a more general one. *}
nipkow@28068
   716
paulson@15047
   717
lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
nipkow@15042
   718
by(simp add:lessThan_def atLeastLessThan_def)
nipkow@24449
   719
nipkow@28068
   720
lemma atLeast0AtMost: "{0..n::nat} = {..n}"
nipkow@28068
   721
by(simp add:atMost_def atLeastAtMost_def)
nipkow@28068
   722
haftmann@31998
   723
declare atLeast0LessThan[symmetric, code_unfold]
haftmann@31998
   724
        atLeast0AtMost[symmetric, code_unfold]
nipkow@24449
   725
nipkow@24449
   726
lemma atLeastLessThan0: "{m..<0::nat} = {}"
paulson@15047
   727
by (simp add: atLeastLessThan_def)
nipkow@24449
   728
paulson@15047
   729
subsubsection {* Intervals of nats with @{term Suc} *}
paulson@15047
   730
paulson@15047
   731
text{*Not a simprule because the RHS is too messy.*}
paulson@15047
   732
lemma atLeastLessThanSuc:
paulson@15047
   733
    "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
paulson@15418
   734
by (auto simp add: atLeastLessThan_def)
paulson@15047
   735
paulson@15418
   736
lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
paulson@15047
   737
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   738
(*
paulson@15047
   739
lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
paulson@15047
   740
by (induct k, simp_all add: atLeastLessThanSuc)
paulson@15047
   741
paulson@15047
   742
lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
paulson@15047
   743
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   744
*)
nipkow@15045
   745
lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
paulson@14485
   746
  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   747
paulson@15418
   748
lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
paulson@15418
   749
  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
paulson@14485
   750
    greaterThanAtMost_def)
paulson@14485
   751
paulson@15418
   752
lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
paulson@15418
   753
  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
paulson@14485
   754
    greaterThanLessThan_def)
paulson@14485
   755
nipkow@15554
   756
lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
nipkow@15554
   757
by (auto simp add: atLeastAtMost_def)
nipkow@15554
   758
noschinl@45932
   759
lemma atLeastAtMost_insertL: "m \<le> n \<Longrightarrow> insert m {Suc m..n} = {m ..n}"
noschinl@45932
   760
by auto
noschinl@45932
   761
kleing@43157
   762
text {* The analogous result is useful on @{typ int}: *}
kleing@43157
   763
(* here, because we don't have an own int section *)
kleing@43157
   764
lemma atLeastAtMostPlus1_int_conv:
kleing@43157
   765
  "m <= 1+n \<Longrightarrow> {m..1+n} = insert (1+n) {m..n::int}"
kleing@43157
   766
  by (auto intro: set_eqI)
kleing@43157
   767
paulson@33044
   768
lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"
paulson@33044
   769
  apply (induct k) 
paulson@33044
   770
  apply (simp_all add: atLeastLessThanSuc)   
paulson@33044
   771
  done
paulson@33044
   772
lp15@57113
   773
subsubsection {* Intervals and numerals *}
lp15@57113
   774
lp15@57113
   775
lemma lessThan_nat_numeral:  --{*Evaluation for specific numerals*}
lp15@57113
   776
  "lessThan (numeral k :: nat) = insert (pred_numeral k) (lessThan (pred_numeral k))"
lp15@57113
   777
  by (simp add: numeral_eq_Suc lessThan_Suc)
lp15@57113
   778
lp15@57113
   779
lemma atMost_nat_numeral:  --{*Evaluation for specific numerals*}
lp15@57113
   780
  "atMost (numeral k :: nat) = insert (numeral k) (atMost (pred_numeral k))"
lp15@57113
   781
  by (simp add: numeral_eq_Suc atMost_Suc)
lp15@57113
   782
lp15@57113
   783
lemma atLeastLessThan_nat_numeral:  --{*Evaluation for specific numerals*}
lp15@57113
   784
  "atLeastLessThan m (numeral k :: nat) = 
lp15@57113
   785
     (if m \<le> (pred_numeral k) then insert (pred_numeral k) (atLeastLessThan m (pred_numeral k))
lp15@57113
   786
                 else {})"
lp15@57113
   787
  by (simp add: numeral_eq_Suc atLeastLessThanSuc)
lp15@57113
   788
nipkow@16733
   789
subsubsection {* Image *}
nipkow@16733
   790
nipkow@16733
   791
lemma image_add_atLeastAtMost:
nipkow@16733
   792
  "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
nipkow@16733
   793
proof
nipkow@16733
   794
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   795
next
nipkow@16733
   796
  show "?B \<subseteq> ?A"
nipkow@16733
   797
  proof
nipkow@16733
   798
    fix n assume a: "n : ?B"
webertj@20217
   799
    hence "n - k : {i..j}" by auto
nipkow@16733
   800
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   801
    ultimately show "n : ?A" by blast
nipkow@16733
   802
  qed
nipkow@16733
   803
qed
nipkow@16733
   804
nipkow@16733
   805
lemma image_add_atLeastLessThan:
nipkow@16733
   806
  "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
nipkow@16733
   807
proof
nipkow@16733
   808
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   809
next
nipkow@16733
   810
  show "?B \<subseteq> ?A"
nipkow@16733
   811
  proof
nipkow@16733
   812
    fix n assume a: "n : ?B"
webertj@20217
   813
    hence "n - k : {i..<j}" by auto
nipkow@16733
   814
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   815
    ultimately show "n : ?A" by blast
nipkow@16733
   816
  qed
nipkow@16733
   817
qed
nipkow@16733
   818
nipkow@16733
   819
corollary image_Suc_atLeastAtMost[simp]:
nipkow@16733
   820
  "Suc ` {i..j} = {Suc i..Suc j}"
huffman@30079
   821
using image_add_atLeastAtMost[where k="Suc 0"] by simp
nipkow@16733
   822
nipkow@16733
   823
corollary image_Suc_atLeastLessThan[simp]:
nipkow@16733
   824
  "Suc ` {i..<j} = {Suc i..<Suc j}"
huffman@30079
   825
using image_add_atLeastLessThan[where k="Suc 0"] by simp
nipkow@16733
   826
nipkow@16733
   827
lemma image_add_int_atLeastLessThan:
nipkow@16733
   828
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
nipkow@16733
   829
  apply (auto simp add: image_def)
nipkow@16733
   830
  apply (rule_tac x = "x - l" in bexI)
nipkow@16733
   831
  apply auto
nipkow@16733
   832
  done
nipkow@16733
   833
hoelzl@37664
   834
lemma image_minus_const_atLeastLessThan_nat:
hoelzl@37664
   835
  fixes c :: nat
hoelzl@37664
   836
  shows "(\<lambda>i. i - c) ` {x ..< y} =
hoelzl@37664
   837
      (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"
hoelzl@37664
   838
    (is "_ = ?right")
hoelzl@37664
   839
proof safe
hoelzl@37664
   840
  fix a assume a: "a \<in> ?right"
hoelzl@37664
   841
  show "a \<in> (\<lambda>i. i - c) ` {x ..< y}"
hoelzl@37664
   842
  proof cases
hoelzl@37664
   843
    assume "c < y" with a show ?thesis
hoelzl@37664
   844
      by (auto intro!: image_eqI[of _ _ "a + c"])
hoelzl@37664
   845
  next
hoelzl@37664
   846
    assume "\<not> c < y" with a show ?thesis
hoelzl@37664
   847
      by (auto intro!: image_eqI[of _ _ x] split: split_if_asm)
hoelzl@37664
   848
  qed
hoelzl@37664
   849
qed auto
hoelzl@37664
   850
Andreas@51152
   851
lemma image_int_atLeastLessThan: "int ` {a..<b} = {int a..<int b}"
wenzelm@55143
   852
  by (auto intro!: image_eqI [where x = "nat x" for x])
Andreas@51152
   853
hoelzl@35580
   854
context ordered_ab_group_add
hoelzl@35580
   855
begin
hoelzl@35580
   856
hoelzl@35580
   857
lemma
hoelzl@35580
   858
  fixes x :: 'a
hoelzl@35580
   859
  shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}"
hoelzl@35580
   860
  and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}"
hoelzl@35580
   861
proof safe
hoelzl@35580
   862
  fix y assume "y < -x"
hoelzl@35580
   863
  hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp
hoelzl@35580
   864
  have "- (-y) \<in> uminus ` {x<..}"
hoelzl@35580
   865
    by (rule imageI) (simp add: *)
hoelzl@35580
   866
  thus "y \<in> uminus ` {x<..}" by simp
hoelzl@35580
   867
next
hoelzl@35580
   868
  fix y assume "y \<le> -x"
hoelzl@35580
   869
  have "- (-y) \<in> uminus ` {x..}"
hoelzl@35580
   870
    by (rule imageI) (insert `y \<le> -x`[THEN le_imp_neg_le], simp)
hoelzl@35580
   871
  thus "y \<in> uminus ` {x..}" by simp
hoelzl@35580
   872
qed simp_all
hoelzl@35580
   873
hoelzl@35580
   874
lemma
hoelzl@35580
   875
  fixes x :: 'a
hoelzl@35580
   876
  shows image_uminus_lessThan[simp]: "uminus ` {..<x} = {-x<..}"
hoelzl@35580
   877
  and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"
hoelzl@35580
   878
proof -
hoelzl@35580
   879
  have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"
hoelzl@35580
   880
    and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all
hoelzl@35580
   881
  thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"
hoelzl@35580
   882
    by (simp_all add: image_image
hoelzl@35580
   883
        del: image_uminus_greaterThan image_uminus_atLeast)
hoelzl@35580
   884
qed
hoelzl@35580
   885
hoelzl@35580
   886
lemma
hoelzl@35580
   887
  fixes x :: 'a
hoelzl@35580
   888
  shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"
hoelzl@35580
   889
  and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"
hoelzl@35580
   890
  and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"
hoelzl@35580
   891
  and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"
hoelzl@35580
   892
  by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def
hoelzl@35580
   893
      greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)
hoelzl@35580
   894
end
nipkow@16733
   895
paulson@14485
   896
subsubsection {* Finiteness *}
paulson@14485
   897
nipkow@15045
   898
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
paulson@14485
   899
  by (induct k) (simp_all add: lessThan_Suc)
paulson@14485
   900
paulson@14485
   901
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
paulson@14485
   902
  by (induct k) (simp_all add: atMost_Suc)
paulson@14485
   903
paulson@14485
   904
lemma finite_greaterThanLessThan [iff]:
nipkow@15045
   905
  fixes l :: nat shows "finite {l<..<u}"
paulson@14485
   906
by (simp add: greaterThanLessThan_def)
paulson@14485
   907
paulson@14485
   908
lemma finite_atLeastLessThan [iff]:
nipkow@15045
   909
  fixes l :: nat shows "finite {l..<u}"
paulson@14485
   910
by (simp add: atLeastLessThan_def)
paulson@14485
   911
paulson@14485
   912
lemma finite_greaterThanAtMost [iff]:
nipkow@15045
   913
  fixes l :: nat shows "finite {l<..u}"
paulson@14485
   914
by (simp add: greaterThanAtMost_def)
paulson@14485
   915
paulson@14485
   916
lemma finite_atLeastAtMost [iff]:
paulson@14485
   917
  fixes l :: nat shows "finite {l..u}"
paulson@14485
   918
by (simp add: atLeastAtMost_def)
paulson@14485
   919
nipkow@28068
   920
text {* A bounded set of natural numbers is finite. *}
paulson@14485
   921
lemma bounded_nat_set_is_finite:
nipkow@24853
   922
  "(ALL i:N. i < (n::nat)) ==> finite N"
nipkow@28068
   923
apply (rule finite_subset)
nipkow@28068
   924
 apply (rule_tac [2] finite_lessThan, auto)
nipkow@28068
   925
done
nipkow@28068
   926
nipkow@31044
   927
text {* A set of natural numbers is finite iff it is bounded. *}
nipkow@31044
   928
lemma finite_nat_set_iff_bounded:
nipkow@31044
   929
  "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
nipkow@31044
   930
proof
nipkow@31044
   931
  assume f:?F  show ?B
nipkow@31044
   932
    using Max_ge[OF `?F`, simplified less_Suc_eq_le[symmetric]] by blast
nipkow@31044
   933
next
nipkow@31044
   934
  assume ?B show ?F using `?B` by(blast intro:bounded_nat_set_is_finite)
nipkow@31044
   935
qed
nipkow@31044
   936
nipkow@31044
   937
lemma finite_nat_set_iff_bounded_le:
nipkow@31044
   938
  "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
nipkow@31044
   939
apply(simp add:finite_nat_set_iff_bounded)
nipkow@31044
   940
apply(blast dest:less_imp_le_nat le_imp_less_Suc)
nipkow@31044
   941
done
nipkow@31044
   942
nipkow@28068
   943
lemma finite_less_ub:
nipkow@28068
   944
     "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
nipkow@28068
   945
by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
paulson@14485
   946
hoelzl@56328
   947
nipkow@24853
   948
text{* Any subset of an interval of natural numbers the size of the
nipkow@24853
   949
subset is exactly that interval. *}
nipkow@24853
   950
nipkow@24853
   951
lemma subset_card_intvl_is_intvl:
blanchet@55085
   952
  assumes "A \<subseteq> {k..<k + card A}"
blanchet@55085
   953
  shows "A = {k..<k + card A}"
wenzelm@53374
   954
proof (cases "finite A")
wenzelm@53374
   955
  case True
wenzelm@53374
   956
  from this and assms show ?thesis
wenzelm@53374
   957
  proof (induct A rule: finite_linorder_max_induct)
nipkow@24853
   958
    case empty thus ?case by auto
nipkow@24853
   959
  next
nipkow@33434
   960
    case (insert b A)
wenzelm@53374
   961
    hence *: "b \<notin> A" by auto
blanchet@55085
   962
    with insert have "A <= {k..<k + card A}" and "b = k + card A"
wenzelm@53374
   963
      by fastforce+
wenzelm@53374
   964
    with insert * show ?case by auto
nipkow@24853
   965
  qed
nipkow@24853
   966
next
wenzelm@53374
   967
  case False
wenzelm@53374
   968
  with assms show ?thesis by simp
nipkow@24853
   969
qed
nipkow@24853
   970
nipkow@24853
   971
paulson@32596
   972
subsubsection {* Proving Inclusions and Equalities between Unions *}
paulson@32596
   973
nipkow@36755
   974
lemma UN_le_eq_Un0:
nipkow@36755
   975
  "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> M 0" (is "?A = ?B")
nipkow@36755
   976
proof
nipkow@36755
   977
  show "?A <= ?B"
nipkow@36755
   978
  proof
nipkow@36755
   979
    fix x assume "x : ?A"
nipkow@36755
   980
    then obtain i where i: "i\<le>n" "x : M i" by auto
nipkow@36755
   981
    show "x : ?B"
nipkow@36755
   982
    proof(cases i)
nipkow@36755
   983
      case 0 with i show ?thesis by simp
nipkow@36755
   984
    next
nipkow@36755
   985
      case (Suc j) with i show ?thesis by auto
nipkow@36755
   986
    qed
nipkow@36755
   987
  qed
nipkow@36755
   988
next
nipkow@36755
   989
  show "?B <= ?A" by auto
nipkow@36755
   990
qed
nipkow@36755
   991
nipkow@36755
   992
lemma UN_le_add_shift:
nipkow@36755
   993
  "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")
nipkow@36755
   994
proof
nipkow@44890
   995
  show "?A <= ?B" by fastforce
nipkow@36755
   996
next
nipkow@36755
   997
  show "?B <= ?A"
nipkow@36755
   998
  proof
nipkow@36755
   999
    fix x assume "x : ?B"
nipkow@36755
  1000
    then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto
nipkow@36755
  1001
    hence "i-k\<le>n & x : M((i-k)+k)" by auto
nipkow@36755
  1002
    thus "x : ?A" by blast
nipkow@36755
  1003
  qed
nipkow@36755
  1004
qed
nipkow@36755
  1005
paulson@32596
  1006
lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"
paulson@32596
  1007
  by (auto simp add: atLeast0LessThan) 
paulson@32596
  1008
paulson@32596
  1009
lemma UN_finite_subset: "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"
paulson@32596
  1010
  by (subst UN_UN_finite_eq [symmetric]) blast
paulson@32596
  1011
paulson@33044
  1012
lemma UN_finite2_subset: 
paulson@33044
  1013
     "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n+k}. B i)) \<Longrightarrow> (\<Union>n. A n) \<subseteq> (\<Union>n. B n)"
paulson@33044
  1014
  apply (rule UN_finite_subset)
paulson@33044
  1015
  apply (subst UN_UN_finite_eq [symmetric, of B]) 
paulson@33044
  1016
  apply blast
paulson@33044
  1017
  done
paulson@32596
  1018
paulson@32596
  1019
lemma UN_finite2_eq:
paulson@33044
  1020
  "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) = (\<Union>i\<in>{0..<n+k}. B i)) \<Longrightarrow> (\<Union>n. A n) = (\<Union>n. B n)"
paulson@33044
  1021
  apply (rule subset_antisym)
paulson@33044
  1022
   apply (rule UN_finite2_subset, blast)
paulson@33044
  1023
 apply (rule UN_finite2_subset [where k=k])
huffman@35216
  1024
 apply (force simp add: atLeastLessThan_add_Un [of 0])
paulson@33044
  1025
 done
paulson@32596
  1026
paulson@32596
  1027
paulson@14485
  1028
subsubsection {* Cardinality *}
paulson@14485
  1029
nipkow@15045
  1030
lemma card_lessThan [simp]: "card {..<u} = u"
paulson@15251
  1031
  by (induct u, simp_all add: lessThan_Suc)
paulson@14485
  1032
paulson@14485
  1033
lemma card_atMost [simp]: "card {..u} = Suc u"
paulson@14485
  1034
  by (simp add: lessThan_Suc_atMost [THEN sym])
paulson@14485
  1035
nipkow@15045
  1036
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
lp15@57113
  1037
proof -
lp15@57113
  1038
  have "{l..<u} = (%x. x + l) ` {..<u-l}"
lp15@57113
  1039
    apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
lp15@57113
  1040
    apply (rule_tac x = "x - l" in exI)
lp15@57113
  1041
    apply arith
lp15@57113
  1042
    done
lp15@57113
  1043
  then have "card {l..<u} = card {..<u-l}"
lp15@57113
  1044
    by (simp add: card_image inj_on_def)
lp15@57113
  1045
  then show ?thesis
lp15@57113
  1046
    by simp
lp15@57113
  1047
qed
paulson@14485
  1048
paulson@15418
  1049
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
paulson@14485
  1050
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
paulson@14485
  1051
paulson@15418
  1052
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
paulson@14485
  1053
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
paulson@14485
  1054
nipkow@15045
  1055
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
paulson@14485
  1056
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
paulson@14485
  1057
nipkow@26105
  1058
lemma ex_bij_betw_nat_finite:
nipkow@26105
  1059
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
nipkow@26105
  1060
apply(drule finite_imp_nat_seg_image_inj_on)
nipkow@26105
  1061
apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
nipkow@26105
  1062
done
nipkow@26105
  1063
nipkow@26105
  1064
lemma ex_bij_betw_finite_nat:
nipkow@26105
  1065
  "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
nipkow@26105
  1066
by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
nipkow@26105
  1067
nipkow@31438
  1068
lemma finite_same_card_bij:
nipkow@31438
  1069
  "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"
nipkow@31438
  1070
apply(drule ex_bij_betw_finite_nat)
nipkow@31438
  1071
apply(drule ex_bij_betw_nat_finite)
nipkow@31438
  1072
apply(auto intro!:bij_betw_trans)
nipkow@31438
  1073
done
nipkow@31438
  1074
nipkow@31438
  1075
lemma ex_bij_betw_nat_finite_1:
nipkow@31438
  1076
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
nipkow@31438
  1077
by (rule finite_same_card_bij) auto
nipkow@31438
  1078
hoelzl@40703
  1079
lemma bij_betw_iff_card:
hoelzl@40703
  1080
  assumes FIN: "finite A" and FIN': "finite B"
hoelzl@40703
  1081
  shows BIJ: "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"
hoelzl@40703
  1082
using assms
hoelzl@40703
  1083
proof(auto simp add: bij_betw_same_card)
hoelzl@40703
  1084
  assume *: "card A = card B"
hoelzl@40703
  1085
  obtain f where "bij_betw f A {0 ..< card A}"
hoelzl@40703
  1086
  using FIN ex_bij_betw_finite_nat by blast
hoelzl@40703
  1087
  moreover obtain g where "bij_betw g {0 ..< card B} B"
hoelzl@40703
  1088
  using FIN' ex_bij_betw_nat_finite by blast
hoelzl@40703
  1089
  ultimately have "bij_betw (g o f) A B"
hoelzl@40703
  1090
  using * by (auto simp add: bij_betw_trans)
hoelzl@40703
  1091
  thus "(\<exists>f. bij_betw f A B)" by blast
hoelzl@40703
  1092
qed
hoelzl@40703
  1093
hoelzl@40703
  1094
lemma inj_on_iff_card_le:
hoelzl@40703
  1095
  assumes FIN: "finite A" and FIN': "finite B"
hoelzl@40703
  1096
  shows "(\<exists>f. inj_on f A \<and> f ` A \<le> B) = (card A \<le> card B)"
hoelzl@40703
  1097
proof (safe intro!: card_inj_on_le)
hoelzl@40703
  1098
  assume *: "card A \<le> card B"
hoelzl@40703
  1099
  obtain f where 1: "inj_on f A" and 2: "f ` A = {0 ..< card A}"
hoelzl@40703
  1100
  using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force
hoelzl@40703
  1101
  moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g ` {0 ..< card B} = B"
hoelzl@40703
  1102
  using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force
hoelzl@40703
  1103
  ultimately have "inj_on g (f ` A)" using subset_inj_on[of g _ "f ` A"] * by force
hoelzl@40703
  1104
  hence "inj_on (g o f) A" using 1 comp_inj_on by blast
hoelzl@40703
  1105
  moreover
hoelzl@40703
  1106
  {have "{0 ..< card A} \<le> {0 ..< card B}" using * by force
hoelzl@40703
  1107
   with 2 have "f ` A  \<le> {0 ..< card B}" by blast
hoelzl@40703
  1108
   hence "(g o f) ` A \<le> B" unfolding comp_def using 3 by force
hoelzl@40703
  1109
  }
hoelzl@40703
  1110
  ultimately show "(\<exists>f. inj_on f A \<and> f ` A \<le> B)" by blast
hoelzl@40703
  1111
qed (insert assms, auto)
nipkow@26105
  1112
paulson@14485
  1113
subsection {* Intervals of integers *}
paulson@14485
  1114
nipkow@15045
  1115
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
paulson@14485
  1116
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
paulson@14485
  1117
paulson@15418
  1118
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
paulson@14485
  1119
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
paulson@14485
  1120
paulson@15418
  1121
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
paulson@15418
  1122
    "{l+1..<u} = {l<..<u::int}"
paulson@14485
  1123
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
paulson@14485
  1124
paulson@14485
  1125
subsubsection {* Finiteness *}
paulson@14485
  1126
paulson@15418
  1127
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
nipkow@15045
  1128
    {(0::int)..<u} = int ` {..<nat u}"
paulson@14485
  1129
  apply (unfold image_def lessThan_def)
paulson@14485
  1130
  apply auto
paulson@14485
  1131
  apply (rule_tac x = "nat x" in exI)
huffman@35216
  1132
  apply (auto simp add: zless_nat_eq_int_zless [THEN sym])
paulson@14485
  1133
  done
paulson@14485
  1134
nipkow@15045
  1135
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
wenzelm@47988
  1136
  apply (cases "0 \<le> u")
paulson@14485
  1137
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
  1138
  apply (rule finite_imageI)
paulson@14485
  1139
  apply auto
paulson@14485
  1140
  done
paulson@14485
  1141
nipkow@15045
  1142
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
nipkow@15045
  1143
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
  1144
  apply (erule subst)
paulson@14485
  1145
  apply (rule finite_imageI)
paulson@14485
  1146
  apply (rule finite_atLeastZeroLessThan_int)
nipkow@16733
  1147
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
  1148
  done
paulson@14485
  1149
paulson@15418
  1150
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
paulson@14485
  1151
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
paulson@14485
  1152
paulson@15418
  1153
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
paulson@14485
  1154
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
  1155
paulson@15418
  1156
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
paulson@14485
  1157
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
  1158
nipkow@24853
  1159
paulson@14485
  1160
subsubsection {* Cardinality *}
paulson@14485
  1161
nipkow@15045
  1162
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
wenzelm@47988
  1163
  apply (cases "0 \<le> u")
paulson@14485
  1164
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
  1165
  apply (subst card_image)
paulson@14485
  1166
  apply (auto simp add: inj_on_def)
paulson@14485
  1167
  done
paulson@14485
  1168
nipkow@15045
  1169
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
nipkow@15045
  1170
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
paulson@14485
  1171
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
nipkow@15045
  1172
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
  1173
  apply (erule subst)
paulson@14485
  1174
  apply (rule card_image)
paulson@14485
  1175
  apply (simp add: inj_on_def)
nipkow@16733
  1176
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
  1177
  done
paulson@14485
  1178
paulson@14485
  1179
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
nipkow@29667
  1180
apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
nipkow@29667
  1181
apply (auto simp add: algebra_simps)
nipkow@29667
  1182
done
paulson@14485
  1183
paulson@15418
  1184
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
nipkow@29667
  1185
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
  1186
nipkow@15045
  1187
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
nipkow@29667
  1188
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
  1189
bulwahn@27656
  1190
lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
bulwahn@27656
  1191
proof -
bulwahn@27656
  1192
  have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
bulwahn@27656
  1193
  with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
bulwahn@27656
  1194
qed
bulwahn@27656
  1195
bulwahn@27656
  1196
lemma card_less:
bulwahn@27656
  1197
assumes zero_in_M: "0 \<in> M"
bulwahn@27656
  1198
shows "card {k \<in> M. k < Suc i} \<noteq> 0"
bulwahn@27656
  1199
proof -
bulwahn@27656
  1200
  from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
bulwahn@27656
  1201
  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
bulwahn@27656
  1202
qed
bulwahn@27656
  1203
bulwahn@27656
  1204
lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"
haftmann@37388
  1205
apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])
bulwahn@27656
  1206
apply auto
bulwahn@27656
  1207
apply (rule inj_on_diff_nat)
bulwahn@27656
  1208
apply auto
bulwahn@27656
  1209
apply (case_tac x)
bulwahn@27656
  1210
apply auto
bulwahn@27656
  1211
apply (case_tac xa)
bulwahn@27656
  1212
apply auto
bulwahn@27656
  1213
apply (case_tac xa)
bulwahn@27656
  1214
apply auto
bulwahn@27656
  1215
done
bulwahn@27656
  1216
bulwahn@27656
  1217
lemma card_less_Suc:
bulwahn@27656
  1218
  assumes zero_in_M: "0 \<in> M"
bulwahn@27656
  1219
    shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
bulwahn@27656
  1220
proof -
bulwahn@27656
  1221
  from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
bulwahn@27656
  1222
  hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
bulwahn@27656
  1223
    by (auto simp only: insert_Diff)
bulwahn@27656
  1224
  have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
lp15@57113
  1225
  from finite_M_bounded_by_nat[of "\<lambda>x. x \<in> M" "Suc i"] 
lp15@57113
  1226
  have "Suc (card {k. Suc k \<in> M \<and> k < i}) = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"
bulwahn@27656
  1227
    apply (subst card_insert)
bulwahn@27656
  1228
    apply simp_all
bulwahn@27656
  1229
    apply (subst b)
bulwahn@27656
  1230
    apply (subst card_less_Suc2[symmetric])
bulwahn@27656
  1231
    apply simp_all
bulwahn@27656
  1232
    done
bulwahn@27656
  1233
  with c show ?thesis by simp
bulwahn@27656
  1234
qed
bulwahn@27656
  1235
paulson@14485
  1236
paulson@13850
  1237
subsection {*Lemmas useful with the summation operator setsum*}
paulson@13850
  1238
ballarin@16102
  1239
text {* For examples, see Algebra/poly/UnivPoly2.thy *}
ballarin@13735
  1240
wenzelm@14577
  1241
subsubsection {* Disjoint Unions *}
ballarin@13735
  1242
wenzelm@14577
  1243
text {* Singletons and open intervals *}
ballarin@13735
  1244
ballarin@13735
  1245
lemma ivl_disj_un_singleton:
nipkow@15045
  1246
  "{l::'a::linorder} Un {l<..} = {l..}"
nipkow@15045
  1247
  "{..<u} Un {u::'a::linorder} = {..u}"
nipkow@15045
  1248
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
nipkow@15045
  1249
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
nipkow@15045
  1250
  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
nipkow@15045
  1251
  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
ballarin@14398
  1252
by auto
ballarin@13735
  1253
wenzelm@14577
  1254
text {* One- and two-sided intervals *}
ballarin@13735
  1255
ballarin@13735
  1256
lemma ivl_disj_un_one:
nipkow@15045
  1257
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
nipkow@15045
  1258
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
nipkow@15045
  1259
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
nipkow@15045
  1260
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
nipkow@15045
  1261
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
nipkow@15045
  1262
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
nipkow@15045
  1263
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
nipkow@15045
  1264
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
ballarin@14398
  1265
by auto
ballarin@13735
  1266
wenzelm@14577
  1267
text {* Two- and two-sided intervals *}
ballarin@13735
  1268
ballarin@13735
  1269
lemma ivl_disj_un_two:
nipkow@15045
  1270
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
nipkow@15045
  1271
  "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
nipkow@15045
  1272
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
nipkow@15045
  1273
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
nipkow@15045
  1274
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
nipkow@15045
  1275
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
nipkow@15045
  1276
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
nipkow@15045
  1277
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
ballarin@14398
  1278
by auto
ballarin@13735
  1279
ballarin@13735
  1280
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
ballarin@13735
  1281
wenzelm@14577
  1282
subsubsection {* Disjoint Intersections *}
ballarin@13735
  1283
wenzelm@14577
  1284
text {* One- and two-sided intervals *}
ballarin@13735
  1285
ballarin@13735
  1286
lemma ivl_disj_int_one:
nipkow@15045
  1287
  "{..l::'a::order} Int {l<..<u} = {}"
nipkow@15045
  1288
  "{..<l} Int {l..<u} = {}"
nipkow@15045
  1289
  "{..l} Int {l<..u} = {}"
nipkow@15045
  1290
  "{..<l} Int {l..u} = {}"
nipkow@15045
  1291
  "{l<..u} Int {u<..} = {}"
nipkow@15045
  1292
  "{l<..<u} Int {u..} = {}"
nipkow@15045
  1293
  "{l..u} Int {u<..} = {}"
nipkow@15045
  1294
  "{l..<u} Int {u..} = {}"
ballarin@14398
  1295
  by auto
ballarin@13735
  1296
wenzelm@14577
  1297
text {* Two- and two-sided intervals *}
ballarin@13735
  1298
ballarin@13735
  1299
lemma ivl_disj_int_two:
nipkow@15045
  1300
  "{l::'a::order<..<m} Int {m..<u} = {}"
nipkow@15045
  1301
  "{l<..m} Int {m<..<u} = {}"
nipkow@15045
  1302
  "{l..<m} Int {m..<u} = {}"
nipkow@15045
  1303
  "{l..m} Int {m<..<u} = {}"
nipkow@15045
  1304
  "{l<..<m} Int {m..u} = {}"
nipkow@15045
  1305
  "{l<..m} Int {m<..u} = {}"
nipkow@15045
  1306
  "{l..<m} Int {m..u} = {}"
nipkow@15045
  1307
  "{l..m} Int {m<..u} = {}"
ballarin@14398
  1308
  by auto
ballarin@13735
  1309
nipkow@32456
  1310
lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two
ballarin@13735
  1311
nipkow@15542
  1312
subsubsection {* Some Differences *}
nipkow@15542
  1313
nipkow@15542
  1314
lemma ivl_diff[simp]:
nipkow@15542
  1315
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
nipkow@15542
  1316
by(auto)
nipkow@15542
  1317
hoelzl@56194
  1318
lemma (in linorder) lessThan_minus_lessThan [simp]:
hoelzl@56194
  1319
  "{..< n} - {..< m} = {m ..< n}"
hoelzl@56194
  1320
  by auto
hoelzl@56194
  1321
nipkow@15542
  1322
nipkow@15542
  1323
subsubsection {* Some Subset Conditions *}
nipkow@15542
  1324
blanchet@54147
  1325
lemma ivl_subset [simp]:
nipkow@15542
  1326
 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
nipkow@15542
  1327
apply(auto simp:linorder_not_le)
nipkow@15542
  1328
apply(rule ccontr)
nipkow@15542
  1329
apply(insert linorder_le_less_linear[of i n])
nipkow@15542
  1330
apply(clarsimp simp:linorder_not_le)
nipkow@44890
  1331
apply(fastforce)
nipkow@15542
  1332
done
nipkow@15542
  1333
nipkow@15041
  1334
nipkow@15042
  1335
subsection {* Summation indexed over intervals *}
nipkow@15042
  1336
nipkow@15042
  1337
syntax
nipkow@15042
  1338
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
  1339
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
  1340
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
nipkow@16052
  1341
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
nipkow@15042
  1342
syntax (xsymbols)
nipkow@15042
  1343
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
  1344
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
  1345
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
  1346
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15042
  1347
syntax (HTML output)
nipkow@15042
  1348
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
  1349
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
  1350
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
  1351
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15056
  1352
syntax (latex_sum output)
nipkow@15052
  1353
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
  1354
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052
  1355
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
  1356
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@16052
  1357
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
  1358
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15052
  1359
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
  1360
 ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15041
  1361
nipkow@15048
  1362
translations
nipkow@28853
  1363
  "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
nipkow@28853
  1364
  "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
nipkow@28853
  1365
  "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
nipkow@28853
  1366
  "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
nipkow@15041
  1367
nipkow@15052
  1368
text{* The above introduces some pretty alternative syntaxes for
nipkow@15056
  1369
summation over intervals:
nipkow@15052
  1370
\begin{center}
nipkow@15052
  1371
\begin{tabular}{lll}
nipkow@15056
  1372
Old & New & \LaTeX\\
nipkow@15056
  1373
@{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
nipkow@15056
  1374
@{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
nipkow@16052
  1375
@{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
nipkow@15056
  1376
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
nipkow@15052
  1377
\end{tabular}
nipkow@15052
  1378
\end{center}
nipkow@15056
  1379
The left column shows the term before introduction of the new syntax,
nipkow@15056
  1380
the middle column shows the new (default) syntax, and the right column
nipkow@15056
  1381
shows a special syntax. The latter is only meaningful for latex output
nipkow@15056
  1382
and has to be activated explicitly by setting the print mode to
wenzelm@21502
  1383
@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
nipkow@15056
  1384
antiquotations). It is not the default \LaTeX\ output because it only
nipkow@15056
  1385
works well with italic-style formulae, not tt-style.
nipkow@15052
  1386
nipkow@15052
  1387
Note that for uniformity on @{typ nat} it is better to use
nipkow@15052
  1388
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
nipkow@15052
  1389
not provide all lemmas available for @{term"{m..<n}"} also in the
nipkow@15052
  1390
special form for @{term"{..<n}"}. *}
nipkow@15052
  1391
nipkow@15542
  1392
text{* This congruence rule should be used for sums over intervals as
haftmann@57418
  1393
the standard theorem @{text[source]setsum.cong} does not work well
nipkow@15542
  1394
with the simplifier who adds the unsimplified premise @{term"x:B"} to
nipkow@15542
  1395
the context. *}
nipkow@15542
  1396
nipkow@15542
  1397
lemma setsum_ivl_cong:
nipkow@15542
  1398
 "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
nipkow@15542
  1399
 setsum f {a..<b} = setsum g {c..<d}"
haftmann@57418
  1400
by(rule setsum.cong, simp_all)
nipkow@15041
  1401
nipkow@16041
  1402
(* FIXME why are the following simp rules but the corresponding eqns
nipkow@16041
  1403
on intervals are not? *)
nipkow@16041
  1404
nipkow@16052
  1405
lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
nipkow@16052
  1406
by (simp add:atMost_Suc add_ac)
nipkow@16052
  1407
nipkow@16041
  1408
lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
nipkow@16041
  1409
by (simp add:lessThan_Suc add_ac)
nipkow@15041
  1410
nipkow@15911
  1411
lemma setsum_cl_ivl_Suc[simp]:
nipkow@15561
  1412
  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
nipkow@15561
  1413
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@15561
  1414
nipkow@15911
  1415
lemma setsum_op_ivl_Suc[simp]:
nipkow@15561
  1416
  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
nipkow@15561
  1417
by (auto simp:add_ac atLeastLessThanSuc)
nipkow@16041
  1418
(*
nipkow@15561
  1419
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
nipkow@15561
  1420
    (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
nipkow@15561
  1421
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@16041
  1422
*)
nipkow@28068
  1423
nipkow@28068
  1424
lemma setsum_head:
nipkow@28068
  1425
  fixes n :: nat
nipkow@28068
  1426
  assumes mn: "m <= n" 
nipkow@28068
  1427
  shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
nipkow@28068
  1428
proof -
nipkow@28068
  1429
  from mn
nipkow@28068
  1430
  have "{m..n} = {m} \<union> {m<..n}"
nipkow@28068
  1431
    by (auto intro: ivl_disj_un_singleton)
nipkow@28068
  1432
  hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
nipkow@28068
  1433
    by (simp add: atLeast0LessThan)
nipkow@28068
  1434
  also have "\<dots> = ?rhs" by simp
nipkow@28068
  1435
  finally show ?thesis .
nipkow@28068
  1436
qed
nipkow@28068
  1437
nipkow@28068
  1438
lemma setsum_head_Suc:
nipkow@28068
  1439
  "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
nipkow@28068
  1440
by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
nipkow@28068
  1441
nipkow@28068
  1442
lemma setsum_head_upt_Suc:
nipkow@28068
  1443
  "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
huffman@30079
  1444
apply(insert setsum_head_Suc[of m "n - Suc 0" f])
nipkow@29667
  1445
apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
nipkow@28068
  1446
done
nipkow@28068
  1447
nipkow@31501
  1448
lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"
nipkow@31501
  1449
  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
nipkow@31501
  1450
proof-
nipkow@31501
  1451
  have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using `m \<le> n+1` by auto
haftmann@57418
  1452
  thus ?thesis by (auto simp: ivl_disj_int setsum.union_disjoint
nipkow@31501
  1453
    atLeastSucAtMost_greaterThanAtMost)
nipkow@31501
  1454
qed
nipkow@28068
  1455
nipkow@15539
  1456
lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
  1457
  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
haftmann@57418
  1458
by (simp add:setsum.union_disjoint[symmetric] ivl_disj_int ivl_disj_un)
nipkow@15539
  1459
nipkow@15539
  1460
lemma setsum_diff_nat_ivl:
nipkow@15539
  1461
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
nipkow@15539
  1462
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
  1463
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
nipkow@15539
  1464
using setsum_add_nat_ivl [of m n p f,symmetric]
nipkow@15539
  1465
apply (simp add: add_ac)
nipkow@15539
  1466
done
nipkow@15539
  1467
nipkow@31505
  1468
lemma setsum_natinterval_difff:
nipkow@31505
  1469
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
nipkow@31505
  1470
  shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
nipkow@31505
  1471
          (if m <= n then f m - f(n + 1) else 0)"
nipkow@31505
  1472
by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
nipkow@31505
  1473
hoelzl@56194
  1474
lemma setsum_nat_group: "(\<Sum>m<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {..< n * k}"
hoelzl@56194
  1475
  apply (subgoal_tac "k = 0 | 0 < k", auto)
hoelzl@56194
  1476
  apply (induct "n")
hoelzl@56194
  1477
  apply (simp_all add: setsum_add_nat_ivl add_commute atLeast0LessThan[symmetric])
hoelzl@56194
  1478
  done
nipkow@28068
  1479
nipkow@16733
  1480
subsection{* Shifting bounds *}
nipkow@16733
  1481
nipkow@15539
  1482
lemma setsum_shift_bounds_nat_ivl:
nipkow@15539
  1483
  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
nipkow@15539
  1484
by (induct "n", auto simp:atLeastLessThanSuc)
nipkow@15539
  1485
nipkow@16733
  1486
lemma setsum_shift_bounds_cl_nat_ivl:
nipkow@16733
  1487
  "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
hoelzl@57129
  1488
  by (rule setsum.reindex_bij_witness[where i="\<lambda>i. i + k" and j="\<lambda>i. i - k"]) auto
nipkow@16733
  1489
nipkow@16733
  1490
corollary setsum_shift_bounds_cl_Suc_ivl:
nipkow@16733
  1491
  "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
huffman@30079
  1492
by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
  1493
nipkow@16733
  1494
corollary setsum_shift_bounds_Suc_ivl:
nipkow@16733
  1495
  "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
huffman@30079
  1496
by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
  1497
nipkow@28068
  1498
lemma setsum_shift_lb_Suc0_0:
nipkow@28068
  1499
  "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
nipkow@28068
  1500
by(simp add:setsum_head_Suc)
kleing@19106
  1501
nipkow@28068
  1502
lemma setsum_shift_lb_Suc0_0_upt:
nipkow@28068
  1503
  "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
nipkow@28068
  1504
apply(cases k)apply simp
nipkow@28068
  1505
apply(simp add:setsum_head_upt_Suc)
nipkow@28068
  1506
done
kleing@19022
  1507
haftmann@52380
  1508
lemma setsum_atMost_Suc_shift:
haftmann@52380
  1509
  fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
haftmann@52380
  1510
  shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
haftmann@52380
  1511
proof (induct n)
haftmann@52380
  1512
  case 0 show ?case by simp
haftmann@52380
  1513
next
haftmann@52380
  1514
  case (Suc n) note IH = this
haftmann@52380
  1515
  have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"
haftmann@52380
  1516
    by (rule setsum_atMost_Suc)
haftmann@52380
  1517
  also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
haftmann@52380
  1518
    by (rule IH)
haftmann@52380
  1519
  also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
haftmann@52380
  1520
             f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
haftmann@52380
  1521
    by (rule add_assoc)
haftmann@52380
  1522
  also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
haftmann@52380
  1523
    by (rule setsum_atMost_Suc [symmetric])
haftmann@52380
  1524
  finally show ?case .
haftmann@52380
  1525
qed
haftmann@52380
  1526
lp15@56238
  1527
lemma setsum_last_plus: fixes n::nat shows "m <= n \<Longrightarrow> (\<Sum>i = m..n. f i) = f n + (\<Sum>i = m..<n. f i)"
lp15@56238
  1528
  by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost add_commute)
lp15@56238
  1529
lp15@56238
  1530
lemma setsum_Suc_diff:
lp15@56238
  1531
  fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
lp15@56238
  1532
  assumes "m \<le> Suc n"
lp15@56238
  1533
  shows "(\<Sum>i = m..n. f(Suc i) - f i) = f (Suc n) - f m"
lp15@56238
  1534
using assms by (induct n) (auto simp: le_Suc_eq)
lp15@55718
  1535
lp15@55718
  1536
lemma nested_setsum_swap:
lp15@55718
  1537
     "(\<Sum>i = 0..n. (\<Sum>j = 0..<i. a i j)) = (\<Sum>j = 0..<n. \<Sum>i = Suc j..n. a i j)"
haftmann@57418
  1538
  by (induction n) (auto simp: setsum.distrib)
lp15@55718
  1539
lp15@56215
  1540
lemma nested_setsum_swap':
lp15@56215
  1541
     "(\<Sum>i\<le>n. (\<Sum>j<i. a i j)) = (\<Sum>j<n. \<Sum>i = Suc j..n. a i j)"
haftmann@57418
  1542
  by (induction n) (auto simp: setsum.distrib)
lp15@56215
  1543
lp15@56215
  1544
lemma setsum_zero_power [simp]:
lp15@56215
  1545
  fixes c :: "nat \<Rightarrow> 'a::division_ring"
lp15@56215
  1546
  shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
lp15@56215
  1547
apply (cases "finite A")
lp15@56215
  1548
  by (induction A rule: finite_induct) auto
lp15@56215
  1549
lp15@56238
  1550
lemma setsum_zero_power' [simp]:
lp15@56238
  1551
  fixes c :: "nat \<Rightarrow> 'a::field"
lp15@56238
  1552
  shows "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"
lp15@56238
  1553
  using setsum_zero_power [of "\<lambda>i. c i / d i" A]
lp15@56238
  1554
  by auto
lp15@56238
  1555
haftmann@52380
  1556
ballarin@17149
  1557
subsection {* The formula for geometric sums *}
ballarin@17149
  1558
ballarin@17149
  1559
lemma geometric_sum:
haftmann@36307
  1560
  assumes "x \<noteq> 1"
hoelzl@56193
  1561
  shows "(\<Sum>i<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"
haftmann@36307
  1562
proof -
haftmann@36307
  1563
  from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all
hoelzl@56193
  1564
  moreover have "(\<Sum>i<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"
hoelzl@56480
  1565
    by (induct n) (simp_all add: field_simps `y \<noteq> 0`)
haftmann@36307
  1566
  ultimately show ?thesis by simp
haftmann@36307
  1567
qed
haftmann@36307
  1568
ballarin@17149
  1569
kleing@19469
  1570
subsection {* The formula for arithmetic sums *}
kleing@19469
  1571
huffman@47222
  1572
lemma gauss_sum:
hoelzl@56193
  1573
  "(2::'a::comm_semiring_1)*(\<Sum>i\<in>{1..n}. of_nat i) = of_nat n*((of_nat n)+1)"
kleing@19469
  1574
proof (induct n)
kleing@19469
  1575
  case 0
kleing@19469
  1576
  show ?case by simp
kleing@19469
  1577
next
kleing@19469
  1578
  case (Suc n)
huffman@47222
  1579
  then show ?case
huffman@47222
  1580
    by (simp add: algebra_simps add: one_add_one [symmetric] del: one_add_one)
huffman@47222
  1581
      (* FIXME: make numeral cancellation simprocs work for semirings *)
kleing@19469
  1582
qed
kleing@19469
  1583
kleing@19469
  1584
theorem arith_series_general:
huffman@47222
  1585
  "(2::'a::comm_semiring_1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1586
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1587
proof cases
kleing@19469
  1588
  assume ngt1: "n > 1"
kleing@19469
  1589
  let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
kleing@19469
  1590
  have
kleing@19469
  1591
    "(\<Sum>i\<in>{..<n}. a+?I i*d) =
kleing@19469
  1592
     ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
haftmann@57418
  1593
    by (rule setsum.distrib)
kleing@19469
  1594
  also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
kleing@19469
  1595
  also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
huffman@30079
  1596
    unfolding One_nat_def
nipkow@28068
  1597
    by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)
huffman@47222
  1598
  also have "2*\<dots> = 2*?n*a + d*2*(\<Sum>i\<in>{1..<n}. ?I i)"
huffman@47222
  1599
    by (simp add: algebra_simps)
kleing@19469
  1600
  also from ngt1 have "{1..<n} = {1..n - 1}"
nipkow@28068
  1601
    by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
nipkow@28068
  1602
  also from ngt1
huffman@47222
  1603
  have "2*?n*a + d*2*(\<Sum>i\<in>{1..n - 1}. ?I i) = (2*?n*a + d*?I (n - 1)*?I n)"
huffman@30079
  1604
    by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)
huffman@23431
  1605
       (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
huffman@47222
  1606
  finally show ?thesis
huffman@47222
  1607
    unfolding mult_2 by (simp add: algebra_simps)
kleing@19469
  1608
next
kleing@19469
  1609
  assume "\<not>(n > 1)"
kleing@19469
  1610
  hence "n = 1 \<or> n = 0" by auto
huffman@47222
  1611
  thus ?thesis by (auto simp: mult_2)
kleing@19469
  1612
qed
kleing@19469
  1613
kleing@19469
  1614
lemma arith_series_nat:
huffman@47222
  1615
  "(2::nat) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
kleing@19469
  1616
proof -
kleing@19469
  1617
  have
huffman@47222
  1618
    "2 * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
kleing@19469
  1619
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1620
    by (rule arith_series_general)
huffman@30079
  1621
  thus ?thesis
huffman@35216
  1622
    unfolding One_nat_def by auto
kleing@19469
  1623
qed
kleing@19469
  1624
kleing@19469
  1625
lemma arith_series_int:
huffman@47222
  1626
  "2 * (\<Sum>i\<in>{..<n}. a + int i * d) = int n * (a + (a + int(n - 1)*d))"
huffman@47222
  1627
  by (fact arith_series_general) (* FIXME: duplicate *)
paulson@15418
  1628
kleing@19022
  1629
lemma sum_diff_distrib:
kleing@19022
  1630
  fixes P::"nat\<Rightarrow>nat"
kleing@19022
  1631
  shows
kleing@19022
  1632
  "\<forall>x. Q x \<le> P x  \<Longrightarrow>
kleing@19022
  1633
  (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
kleing@19022
  1634
proof (induct n)
kleing@19022
  1635
  case 0 show ?case by simp
kleing@19022
  1636
next
kleing@19022
  1637
  case (Suc n)
kleing@19022
  1638
kleing@19022
  1639
  let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
kleing@19022
  1640
  let ?rhs = "\<Sum>x<n. P x - Q x"
kleing@19022
  1641
kleing@19022
  1642
  from Suc have "?lhs = ?rhs" by simp
kleing@19022
  1643
  moreover
kleing@19022
  1644
  from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
kleing@19022
  1645
  moreover
kleing@19022
  1646
  from Suc have
kleing@19022
  1647
    "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
kleing@19022
  1648
    by (subst diff_diff_left[symmetric],
kleing@19022
  1649
        subst diff_add_assoc2)
kleing@19022
  1650
       (auto simp: diff_add_assoc2 intro: setsum_mono)
kleing@19022
  1651
  ultimately
kleing@19022
  1652
  show ?case by simp
kleing@19022
  1653
qed
kleing@19022
  1654
hoelzl@57129
  1655
lemma nat_diff_setsum_reindex: "(\<Sum>i<n. f (n - Suc i)) = (\<Sum>i<n. f i)"
hoelzl@57129
  1656
  by (rule setsum.reindex_bij_witness[where i="\<lambda>i. n - Suc i" and j="\<lambda>i. n - Suc i"]) auto
lp15@55718
  1657
paulson@29960
  1658
subsection {* Products indexed over intervals *}
paulson@29960
  1659
paulson@29960
  1660
syntax
paulson@29960
  1661
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1662
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1663
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)
paulson@29960
  1664
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
paulson@29960
  1665
syntax (xsymbols)
paulson@29960
  1666
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1667
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1668
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1669
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1670
syntax (HTML output)
paulson@29960
  1671
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1672
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1673
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1674
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1675
syntax (latex_prod output)
paulson@29960
  1676
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1677
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1678
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1679
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1680
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1681
 ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1682
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1683
 ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1684
paulson@29960
  1685
translations
paulson@29960
  1686
  "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"
paulson@29960
  1687
  "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"
paulson@29960
  1688
  "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"
paulson@29960
  1689
  "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"
paulson@29960
  1690
haftmann@33318
  1691
subsection {* Transfer setup *}
haftmann@33318
  1692
haftmann@33318
  1693
lemma transfer_nat_int_set_functions:
haftmann@33318
  1694
    "{..n} = nat ` {0..int n}"
haftmann@33318
  1695
    "{m..n} = nat ` {int m..int n}"  (* need all variants of these! *)
haftmann@33318
  1696
  apply (auto simp add: image_def)
haftmann@33318
  1697
  apply (rule_tac x = "int x" in bexI)
haftmann@33318
  1698
  apply auto
haftmann@33318
  1699
  apply (rule_tac x = "int x" in bexI)
haftmann@33318
  1700
  apply auto
haftmann@33318
  1701
  done
haftmann@33318
  1702
haftmann@33318
  1703
lemma transfer_nat_int_set_function_closures:
haftmann@33318
  1704
    "x >= 0 \<Longrightarrow> nat_set {x..y}"
haftmann@33318
  1705
  by (simp add: nat_set_def)
haftmann@33318
  1706
haftmann@35644
  1707
declare transfer_morphism_nat_int[transfer add
haftmann@33318
  1708
  return: transfer_nat_int_set_functions
haftmann@33318
  1709
    transfer_nat_int_set_function_closures
haftmann@33318
  1710
]
haftmann@33318
  1711
haftmann@33318
  1712
lemma transfer_int_nat_set_functions:
haftmann@33318
  1713
    "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}"
haftmann@33318
  1714
  by (simp only: is_nat_def transfer_nat_int_set_functions
haftmann@33318
  1715
    transfer_nat_int_set_function_closures
haftmann@33318
  1716
    transfer_nat_int_set_return_embed nat_0_le
haftmann@33318
  1717
    cong: transfer_nat_int_set_cong)
haftmann@33318
  1718
haftmann@33318
  1719
lemma transfer_int_nat_set_function_closures:
haftmann@33318
  1720
    "is_nat x \<Longrightarrow> nat_set {x..y}"
haftmann@33318
  1721
  by (simp only: transfer_nat_int_set_function_closures is_nat_def)
haftmann@33318
  1722
haftmann@35644
  1723
declare transfer_morphism_int_nat[transfer add
haftmann@33318
  1724
  return: transfer_int_nat_set_functions
haftmann@33318
  1725
    transfer_int_nat_set_function_closures
haftmann@33318
  1726
]
haftmann@33318
  1727
lp15@55242
  1728
lemma setprod_int_plus_eq: "setprod int {i..i+j} =  \<Prod>{int i..int (i+j)}"
lp15@55242
  1729
  by (induct j) (auto simp add: atLeastAtMostSuc_conv atLeastAtMostPlus1_int_conv)
lp15@55242
  1730
lp15@55242
  1731
lemma setprod_int_eq: "setprod int {i..j} =  \<Prod>{int i..int j}"
lp15@55242
  1732
proof (cases "i \<le> j")
lp15@55242
  1733
  case True
lp15@55242
  1734
  then show ?thesis
lp15@55242
  1735
    by (metis Nat.le_iff_add setprod_int_plus_eq)
lp15@55242
  1736
next
lp15@55242
  1737
  case False
lp15@55242
  1738
  then show ?thesis
lp15@55242
  1739
    by auto
lp15@55242
  1740
qed
lp15@55242
  1741
nipkow@8924
  1742
end