src/HOL/Set_Interval.thy
author wenzelm
Fri Mar 07 22:30:58 2014 +0100 (2014-03-07)
changeset 55990 41c6b99c5fb7
parent 55719 cdddd073bff8
child 56193 c726ecfb22b6
permissions -rw-r--r--
more antiquotations;
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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
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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@42891
   417
end
hoelzl@42891
   418
hoelzl@51329
   419
context no_top
hoelzl@51329
   420
begin
hoelzl@51329
   421
nipkow@51334
   422
lemma greaterThan_non_empty[simp]: "{x <..} \<noteq> {}"
hoelzl@51329
   423
  using gt_ex[of x] by auto
hoelzl@51329
   424
hoelzl@51329
   425
end
hoelzl@51329
   426
hoelzl@51329
   427
context no_bot
hoelzl@51329
   428
begin
hoelzl@51329
   429
nipkow@51334
   430
lemma lessThan_non_empty[simp]: "{..< x} \<noteq> {}"
hoelzl@51329
   431
  using lt_ex[of x] by auto
hoelzl@51329
   432
hoelzl@51329
   433
end
hoelzl@51329
   434
nipkow@32408
   435
lemma (in linorder) atLeastLessThan_subset_iff:
nipkow@32408
   436
  "{a..<b} <= {c..<d} \<Longrightarrow> b <= a | c<=a & b<=d"
nipkow@32408
   437
apply (auto simp:subset_eq Ball_def)
nipkow@32408
   438
apply(frule_tac x=a in spec)
nipkow@32408
   439
apply(erule_tac x=d in allE)
nipkow@32408
   440
apply (simp add: less_imp_le)
nipkow@32408
   441
done
nipkow@32408
   442
hoelzl@40703
   443
lemma atLeastLessThan_inj:
hoelzl@40703
   444
  fixes a b c d :: "'a::linorder"
hoelzl@40703
   445
  assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"
hoelzl@40703
   446
  shows "a = c" "b = d"
hoelzl@40703
   447
using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le linorder_antisym_conv2 subset_refl)+
hoelzl@40703
   448
hoelzl@40703
   449
lemma atLeastLessThan_eq_iff:
hoelzl@40703
   450
  fixes a b c d :: "'a::linorder"
hoelzl@40703
   451
  assumes "a < b" "c < d"
hoelzl@40703
   452
  shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"
hoelzl@40703
   453
  using atLeastLessThan_inj assms by auto
hoelzl@40703
   454
haftmann@52729
   455
lemma (in order_bot) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"
nipkow@51334
   456
by (auto simp: set_eq_iff intro: le_bot)
hoelzl@51328
   457
haftmann@52729
   458
lemma (in order_top) atMost_eq_UNIV_iff: "{..x} = UNIV \<longleftrightarrow> x = top"
nipkow@51334
   459
by (auto simp: set_eq_iff intro: top_le)
hoelzl@51328
   460
nipkow@51334
   461
lemma (in bounded_lattice) atLeastAtMost_eq_UNIV_iff:
nipkow@51334
   462
  "{x..y} = UNIV \<longleftrightarrow> (x = bot \<and> y = top)"
nipkow@51334
   463
by (auto simp: set_eq_iff intro: top_le le_bot)
hoelzl@51328
   464
hoelzl@51328
   465
nipkow@32456
   466
subsubsection {* Intersection *}
nipkow@32456
   467
nipkow@32456
   468
context linorder
nipkow@32456
   469
begin
nipkow@32456
   470
nipkow@32456
   471
lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"
nipkow@32456
   472
by auto
nipkow@32456
   473
nipkow@32456
   474
lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"
nipkow@32456
   475
by auto
nipkow@32456
   476
nipkow@32456
   477
lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"
nipkow@32456
   478
by auto
nipkow@32456
   479
nipkow@32456
   480
lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"
nipkow@32456
   481
by auto
nipkow@32456
   482
nipkow@32456
   483
lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"
nipkow@32456
   484
by auto
nipkow@32456
   485
nipkow@32456
   486
lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"
nipkow@32456
   487
by auto
nipkow@32456
   488
nipkow@32456
   489
lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
nipkow@32456
   490
by auto
nipkow@32456
   491
nipkow@32456
   492
lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
nipkow@32456
   493
by auto
nipkow@32456
   494
hoelzl@50417
   495
lemma Int_atMost[simp]: "{..a} \<inter> {..b} = {.. min a b}"
hoelzl@50417
   496
  by (auto simp: min_def)
hoelzl@50417
   497
nipkow@32456
   498
end
nipkow@32456
   499
hoelzl@51329
   500
context complete_lattice
hoelzl@51329
   501
begin
hoelzl@51329
   502
hoelzl@51329
   503
lemma
hoelzl@51329
   504
  shows Sup_atLeast[simp]: "Sup {x ..} = top"
hoelzl@51329
   505
    and Sup_greaterThanAtLeast[simp]: "x < top \<Longrightarrow> Sup {x <..} = top"
hoelzl@51329
   506
    and Sup_atMost[simp]: "Sup {.. y} = y"
hoelzl@51329
   507
    and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"
hoelzl@51329
   508
    and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"
hoelzl@51329
   509
  by (auto intro!: Sup_eqI)
hoelzl@51329
   510
hoelzl@51329
   511
lemma
hoelzl@51329
   512
  shows Inf_atMost[simp]: "Inf {.. x} = bot"
hoelzl@51329
   513
    and Inf_atMostLessThan[simp]: "top < x \<Longrightarrow> Inf {..< x} = bot"
hoelzl@51329
   514
    and Inf_atLeast[simp]: "Inf {x ..} = x"
hoelzl@51329
   515
    and Inf_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Inf { x .. y} = x"
hoelzl@51329
   516
    and Inf_atLeastLessThan[simp]: "x < y \<Longrightarrow> Inf { x ..< y} = x"
hoelzl@51329
   517
  by (auto intro!: Inf_eqI)
hoelzl@51329
   518
hoelzl@51329
   519
end
hoelzl@51329
   520
hoelzl@51329
   521
lemma
hoelzl@53216
   522
  fixes x y :: "'a :: {complete_lattice, dense_linorder}"
hoelzl@51329
   523
  shows Sup_lessThan[simp]: "Sup {..< y} = y"
hoelzl@51329
   524
    and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"
hoelzl@51329
   525
    and Sup_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Sup { x <..< y} = y"
hoelzl@51329
   526
    and Inf_greaterThan[simp]: "Inf {x <..} = x"
hoelzl@51329
   527
    and Inf_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Inf { x <.. y} = x"
hoelzl@51329
   528
    and Inf_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Inf { x <..< y} = x"
hoelzl@51329
   529
  by (auto intro!: Inf_eqI Sup_eqI intro: dense_le dense_le_bounded dense_ge dense_ge_bounded)
nipkow@32456
   530
paulson@14485
   531
subsection {* Intervals of natural numbers *}
paulson@14485
   532
paulson@15047
   533
subsubsection {* The Constant @{term lessThan} *}
paulson@15047
   534
paulson@14485
   535
lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
paulson@14485
   536
by (simp add: lessThan_def)
paulson@14485
   537
paulson@14485
   538
lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
paulson@14485
   539
by (simp add: lessThan_def less_Suc_eq, blast)
paulson@14485
   540
kleing@43156
   541
text {* The following proof is convenient in induction proofs where
hoelzl@39072
   542
new elements get indices at the beginning. So it is used to transform
hoelzl@39072
   543
@{term "{..<Suc n}"} to @{term "0::nat"} and @{term "{..< n}"}. *}
hoelzl@39072
   544
hoelzl@39072
   545
lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"
hoelzl@39072
   546
proof safe
hoelzl@39072
   547
  fix x assume "x < Suc n" "x \<notin> Suc ` {..<n}"
hoelzl@39072
   548
  then have "x \<noteq> Suc (x - 1)" by auto
hoelzl@39072
   549
  with `x < Suc n` show "x = 0" by auto
hoelzl@39072
   550
qed
hoelzl@39072
   551
paulson@14485
   552
lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
paulson@14485
   553
by (simp add: lessThan_def atMost_def less_Suc_eq_le)
paulson@14485
   554
paulson@14485
   555
lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
paulson@14485
   556
by blast
paulson@14485
   557
paulson@15047
   558
subsubsection {* The Constant @{term greaterThan} *}
paulson@15047
   559
paulson@14485
   560
lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
paulson@14485
   561
apply (simp add: greaterThan_def)
paulson@14485
   562
apply (blast dest: gr0_conv_Suc [THEN iffD1])
paulson@14485
   563
done
paulson@14485
   564
paulson@14485
   565
lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
paulson@14485
   566
apply (simp add: greaterThan_def)
paulson@14485
   567
apply (auto elim: linorder_neqE)
paulson@14485
   568
done
paulson@14485
   569
paulson@14485
   570
lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
paulson@14485
   571
by blast
paulson@14485
   572
paulson@15047
   573
subsubsection {* The Constant @{term atLeast} *}
paulson@15047
   574
paulson@14485
   575
lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
paulson@14485
   576
by (unfold atLeast_def UNIV_def, simp)
paulson@14485
   577
paulson@14485
   578
lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
paulson@14485
   579
apply (simp add: atLeast_def)
paulson@14485
   580
apply (simp add: Suc_le_eq)
paulson@14485
   581
apply (simp add: order_le_less, blast)
paulson@14485
   582
done
paulson@14485
   583
paulson@14485
   584
lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
paulson@14485
   585
  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
paulson@14485
   586
paulson@14485
   587
lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
paulson@14485
   588
by blast
paulson@14485
   589
paulson@15047
   590
subsubsection {* The Constant @{term atMost} *}
paulson@15047
   591
paulson@14485
   592
lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
paulson@14485
   593
by (simp add: atMost_def)
paulson@14485
   594
paulson@14485
   595
lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
paulson@14485
   596
apply (simp add: atMost_def)
paulson@14485
   597
apply (simp add: less_Suc_eq order_le_less, blast)
paulson@14485
   598
done
paulson@14485
   599
paulson@14485
   600
lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
paulson@14485
   601
by blast
paulson@14485
   602
paulson@15047
   603
subsubsection {* The Constant @{term atLeastLessThan} *}
paulson@15047
   604
nipkow@28068
   605
text{*The orientation of the following 2 rules is tricky. The lhs is
nipkow@24449
   606
defined in terms of the rhs.  Hence the chosen orientation makes sense
nipkow@24449
   607
in this theory --- the reverse orientation complicates proofs (eg
nipkow@24449
   608
nontermination). But outside, when the definition of the lhs is rarely
nipkow@24449
   609
used, the opposite orientation seems preferable because it reduces a
nipkow@24449
   610
specific concept to a more general one. *}
nipkow@28068
   611
paulson@15047
   612
lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
nipkow@15042
   613
by(simp add:lessThan_def atLeastLessThan_def)
nipkow@24449
   614
nipkow@28068
   615
lemma atLeast0AtMost: "{0..n::nat} = {..n}"
nipkow@28068
   616
by(simp add:atMost_def atLeastAtMost_def)
nipkow@28068
   617
haftmann@31998
   618
declare atLeast0LessThan[symmetric, code_unfold]
haftmann@31998
   619
        atLeast0AtMost[symmetric, code_unfold]
nipkow@24449
   620
nipkow@24449
   621
lemma atLeastLessThan0: "{m..<0::nat} = {}"
paulson@15047
   622
by (simp add: atLeastLessThan_def)
nipkow@24449
   623
paulson@15047
   624
subsubsection {* Intervals of nats with @{term Suc} *}
paulson@15047
   625
paulson@15047
   626
text{*Not a simprule because the RHS is too messy.*}
paulson@15047
   627
lemma atLeastLessThanSuc:
paulson@15047
   628
    "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
paulson@15418
   629
by (auto simp add: atLeastLessThan_def)
paulson@15047
   630
paulson@15418
   631
lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
paulson@15047
   632
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   633
(*
paulson@15047
   634
lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
paulson@15047
   635
by (induct k, simp_all add: atLeastLessThanSuc)
paulson@15047
   636
paulson@15047
   637
lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
paulson@15047
   638
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   639
*)
nipkow@15045
   640
lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
paulson@14485
   641
  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   642
paulson@15418
   643
lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
paulson@15418
   644
  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
paulson@14485
   645
    greaterThanAtMost_def)
paulson@14485
   646
paulson@15418
   647
lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
paulson@15418
   648
  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
paulson@14485
   649
    greaterThanLessThan_def)
paulson@14485
   650
nipkow@15554
   651
lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
nipkow@15554
   652
by (auto simp add: atLeastAtMost_def)
nipkow@15554
   653
noschinl@45932
   654
lemma atLeastAtMost_insertL: "m \<le> n \<Longrightarrow> insert m {Suc m..n} = {m ..n}"
noschinl@45932
   655
by auto
noschinl@45932
   656
kleing@43157
   657
text {* The analogous result is useful on @{typ int}: *}
kleing@43157
   658
(* here, because we don't have an own int section *)
kleing@43157
   659
lemma atLeastAtMostPlus1_int_conv:
kleing@43157
   660
  "m <= 1+n \<Longrightarrow> {m..1+n} = insert (1+n) {m..n::int}"
kleing@43157
   661
  by (auto intro: set_eqI)
kleing@43157
   662
paulson@33044
   663
lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"
paulson@33044
   664
  apply (induct k) 
paulson@33044
   665
  apply (simp_all add: atLeastLessThanSuc)   
paulson@33044
   666
  done
paulson@33044
   667
nipkow@16733
   668
subsubsection {* Image *}
nipkow@16733
   669
nipkow@16733
   670
lemma image_add_atLeastAtMost:
nipkow@16733
   671
  "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
nipkow@16733
   672
proof
nipkow@16733
   673
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   674
next
nipkow@16733
   675
  show "?B \<subseteq> ?A"
nipkow@16733
   676
  proof
nipkow@16733
   677
    fix n assume a: "n : ?B"
webertj@20217
   678
    hence "n - k : {i..j}" by auto
nipkow@16733
   679
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   680
    ultimately show "n : ?A" by blast
nipkow@16733
   681
  qed
nipkow@16733
   682
qed
nipkow@16733
   683
nipkow@16733
   684
lemma image_add_atLeastLessThan:
nipkow@16733
   685
  "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
nipkow@16733
   686
proof
nipkow@16733
   687
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   688
next
nipkow@16733
   689
  show "?B \<subseteq> ?A"
nipkow@16733
   690
  proof
nipkow@16733
   691
    fix n assume a: "n : ?B"
webertj@20217
   692
    hence "n - k : {i..<j}" by auto
nipkow@16733
   693
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   694
    ultimately show "n : ?A" by blast
nipkow@16733
   695
  qed
nipkow@16733
   696
qed
nipkow@16733
   697
nipkow@16733
   698
corollary image_Suc_atLeastAtMost[simp]:
nipkow@16733
   699
  "Suc ` {i..j} = {Suc i..Suc j}"
huffman@30079
   700
using image_add_atLeastAtMost[where k="Suc 0"] by simp
nipkow@16733
   701
nipkow@16733
   702
corollary image_Suc_atLeastLessThan[simp]:
nipkow@16733
   703
  "Suc ` {i..<j} = {Suc i..<Suc j}"
huffman@30079
   704
using image_add_atLeastLessThan[where k="Suc 0"] by simp
nipkow@16733
   705
nipkow@16733
   706
lemma image_add_int_atLeastLessThan:
nipkow@16733
   707
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
nipkow@16733
   708
  apply (auto simp add: image_def)
nipkow@16733
   709
  apply (rule_tac x = "x - l" in bexI)
nipkow@16733
   710
  apply auto
nipkow@16733
   711
  done
nipkow@16733
   712
hoelzl@37664
   713
lemma image_minus_const_atLeastLessThan_nat:
hoelzl@37664
   714
  fixes c :: nat
hoelzl@37664
   715
  shows "(\<lambda>i. i - c) ` {x ..< y} =
hoelzl@37664
   716
      (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"
hoelzl@37664
   717
    (is "_ = ?right")
hoelzl@37664
   718
proof safe
hoelzl@37664
   719
  fix a assume a: "a \<in> ?right"
hoelzl@37664
   720
  show "a \<in> (\<lambda>i. i - c) ` {x ..< y}"
hoelzl@37664
   721
  proof cases
hoelzl@37664
   722
    assume "c < y" with a show ?thesis
hoelzl@37664
   723
      by (auto intro!: image_eqI[of _ _ "a + c"])
hoelzl@37664
   724
  next
hoelzl@37664
   725
    assume "\<not> c < y" with a show ?thesis
hoelzl@37664
   726
      by (auto intro!: image_eqI[of _ _ x] split: split_if_asm)
hoelzl@37664
   727
  qed
hoelzl@37664
   728
qed auto
hoelzl@37664
   729
Andreas@51152
   730
lemma image_int_atLeastLessThan: "int ` {a..<b} = {int a..<int b}"
wenzelm@55143
   731
  by (auto intro!: image_eqI [where x = "nat x" for x])
Andreas@51152
   732
hoelzl@35580
   733
context ordered_ab_group_add
hoelzl@35580
   734
begin
hoelzl@35580
   735
hoelzl@35580
   736
lemma
hoelzl@35580
   737
  fixes x :: 'a
hoelzl@35580
   738
  shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}"
hoelzl@35580
   739
  and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}"
hoelzl@35580
   740
proof safe
hoelzl@35580
   741
  fix y assume "y < -x"
hoelzl@35580
   742
  hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp
hoelzl@35580
   743
  have "- (-y) \<in> uminus ` {x<..}"
hoelzl@35580
   744
    by (rule imageI) (simp add: *)
hoelzl@35580
   745
  thus "y \<in> uminus ` {x<..}" by simp
hoelzl@35580
   746
next
hoelzl@35580
   747
  fix y assume "y \<le> -x"
hoelzl@35580
   748
  have "- (-y) \<in> uminus ` {x..}"
hoelzl@35580
   749
    by (rule imageI) (insert `y \<le> -x`[THEN le_imp_neg_le], simp)
hoelzl@35580
   750
  thus "y \<in> uminus ` {x..}" by simp
hoelzl@35580
   751
qed simp_all
hoelzl@35580
   752
hoelzl@35580
   753
lemma
hoelzl@35580
   754
  fixes x :: 'a
hoelzl@35580
   755
  shows image_uminus_lessThan[simp]: "uminus ` {..<x} = {-x<..}"
hoelzl@35580
   756
  and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"
hoelzl@35580
   757
proof -
hoelzl@35580
   758
  have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"
hoelzl@35580
   759
    and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all
hoelzl@35580
   760
  thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"
hoelzl@35580
   761
    by (simp_all add: image_image
hoelzl@35580
   762
        del: image_uminus_greaterThan image_uminus_atLeast)
hoelzl@35580
   763
qed
hoelzl@35580
   764
hoelzl@35580
   765
lemma
hoelzl@35580
   766
  fixes x :: 'a
hoelzl@35580
   767
  shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"
hoelzl@35580
   768
  and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"
hoelzl@35580
   769
  and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"
hoelzl@35580
   770
  and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"
hoelzl@35580
   771
  by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def
hoelzl@35580
   772
      greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)
hoelzl@35580
   773
end
nipkow@16733
   774
paulson@14485
   775
subsubsection {* Finiteness *}
paulson@14485
   776
nipkow@15045
   777
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
paulson@14485
   778
  by (induct k) (simp_all add: lessThan_Suc)
paulson@14485
   779
paulson@14485
   780
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
paulson@14485
   781
  by (induct k) (simp_all add: atMost_Suc)
paulson@14485
   782
paulson@14485
   783
lemma finite_greaterThanLessThan [iff]:
nipkow@15045
   784
  fixes l :: nat shows "finite {l<..<u}"
paulson@14485
   785
by (simp add: greaterThanLessThan_def)
paulson@14485
   786
paulson@14485
   787
lemma finite_atLeastLessThan [iff]:
nipkow@15045
   788
  fixes l :: nat shows "finite {l..<u}"
paulson@14485
   789
by (simp add: atLeastLessThan_def)
paulson@14485
   790
paulson@14485
   791
lemma finite_greaterThanAtMost [iff]:
nipkow@15045
   792
  fixes l :: nat shows "finite {l<..u}"
paulson@14485
   793
by (simp add: greaterThanAtMost_def)
paulson@14485
   794
paulson@14485
   795
lemma finite_atLeastAtMost [iff]:
paulson@14485
   796
  fixes l :: nat shows "finite {l..u}"
paulson@14485
   797
by (simp add: atLeastAtMost_def)
paulson@14485
   798
nipkow@28068
   799
text {* A bounded set of natural numbers is finite. *}
paulson@14485
   800
lemma bounded_nat_set_is_finite:
nipkow@24853
   801
  "(ALL i:N. i < (n::nat)) ==> finite N"
nipkow@28068
   802
apply (rule finite_subset)
nipkow@28068
   803
 apply (rule_tac [2] finite_lessThan, auto)
nipkow@28068
   804
done
nipkow@28068
   805
nipkow@31044
   806
text {* A set of natural numbers is finite iff it is bounded. *}
nipkow@31044
   807
lemma finite_nat_set_iff_bounded:
nipkow@31044
   808
  "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
nipkow@31044
   809
proof
nipkow@31044
   810
  assume f:?F  show ?B
nipkow@31044
   811
    using Max_ge[OF `?F`, simplified less_Suc_eq_le[symmetric]] by blast
nipkow@31044
   812
next
nipkow@31044
   813
  assume ?B show ?F using `?B` by(blast intro:bounded_nat_set_is_finite)
nipkow@31044
   814
qed
nipkow@31044
   815
nipkow@31044
   816
lemma finite_nat_set_iff_bounded_le:
nipkow@31044
   817
  "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
nipkow@31044
   818
apply(simp add:finite_nat_set_iff_bounded)
nipkow@31044
   819
apply(blast dest:less_imp_le_nat le_imp_less_Suc)
nipkow@31044
   820
done
nipkow@31044
   821
nipkow@28068
   822
lemma finite_less_ub:
nipkow@28068
   823
     "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
nipkow@28068
   824
by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
paulson@14485
   825
nipkow@24853
   826
text{* Any subset of an interval of natural numbers the size of the
nipkow@24853
   827
subset is exactly that interval. *}
nipkow@24853
   828
nipkow@24853
   829
lemma subset_card_intvl_is_intvl:
blanchet@55085
   830
  assumes "A \<subseteq> {k..<k + card A}"
blanchet@55085
   831
  shows "A = {k..<k + card A}"
wenzelm@53374
   832
proof (cases "finite A")
wenzelm@53374
   833
  case True
wenzelm@53374
   834
  from this and assms show ?thesis
wenzelm@53374
   835
  proof (induct A rule: finite_linorder_max_induct)
nipkow@24853
   836
    case empty thus ?case by auto
nipkow@24853
   837
  next
nipkow@33434
   838
    case (insert b A)
wenzelm@53374
   839
    hence *: "b \<notin> A" by auto
blanchet@55085
   840
    with insert have "A <= {k..<k + card A}" and "b = k + card A"
wenzelm@53374
   841
      by fastforce+
wenzelm@53374
   842
    with insert * show ?case by auto
nipkow@24853
   843
  qed
nipkow@24853
   844
next
wenzelm@53374
   845
  case False
wenzelm@53374
   846
  with assms show ?thesis by simp
nipkow@24853
   847
qed
nipkow@24853
   848
nipkow@24853
   849
paulson@32596
   850
subsubsection {* Proving Inclusions and Equalities between Unions *}
paulson@32596
   851
nipkow@36755
   852
lemma UN_le_eq_Un0:
nipkow@36755
   853
  "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> M 0" (is "?A = ?B")
nipkow@36755
   854
proof
nipkow@36755
   855
  show "?A <= ?B"
nipkow@36755
   856
  proof
nipkow@36755
   857
    fix x assume "x : ?A"
nipkow@36755
   858
    then obtain i where i: "i\<le>n" "x : M i" by auto
nipkow@36755
   859
    show "x : ?B"
nipkow@36755
   860
    proof(cases i)
nipkow@36755
   861
      case 0 with i show ?thesis by simp
nipkow@36755
   862
    next
nipkow@36755
   863
      case (Suc j) with i show ?thesis by auto
nipkow@36755
   864
    qed
nipkow@36755
   865
  qed
nipkow@36755
   866
next
nipkow@36755
   867
  show "?B <= ?A" by auto
nipkow@36755
   868
qed
nipkow@36755
   869
nipkow@36755
   870
lemma UN_le_add_shift:
nipkow@36755
   871
  "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")
nipkow@36755
   872
proof
nipkow@44890
   873
  show "?A <= ?B" by fastforce
nipkow@36755
   874
next
nipkow@36755
   875
  show "?B <= ?A"
nipkow@36755
   876
  proof
nipkow@36755
   877
    fix x assume "x : ?B"
nipkow@36755
   878
    then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto
nipkow@36755
   879
    hence "i-k\<le>n & x : M((i-k)+k)" by auto
nipkow@36755
   880
    thus "x : ?A" by blast
nipkow@36755
   881
  qed
nipkow@36755
   882
qed
nipkow@36755
   883
paulson@32596
   884
lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"
paulson@32596
   885
  by (auto simp add: atLeast0LessThan) 
paulson@32596
   886
paulson@32596
   887
lemma UN_finite_subset: "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"
paulson@32596
   888
  by (subst UN_UN_finite_eq [symmetric]) blast
paulson@32596
   889
paulson@33044
   890
lemma UN_finite2_subset: 
paulson@33044
   891
     "(!!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
   892
  apply (rule UN_finite_subset)
paulson@33044
   893
  apply (subst UN_UN_finite_eq [symmetric, of B]) 
paulson@33044
   894
  apply blast
paulson@33044
   895
  done
paulson@32596
   896
paulson@32596
   897
lemma UN_finite2_eq:
paulson@33044
   898
  "(!!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
   899
  apply (rule subset_antisym)
paulson@33044
   900
   apply (rule UN_finite2_subset, blast)
paulson@33044
   901
 apply (rule UN_finite2_subset [where k=k])
huffman@35216
   902
 apply (force simp add: atLeastLessThan_add_Un [of 0])
paulson@33044
   903
 done
paulson@32596
   904
paulson@32596
   905
paulson@14485
   906
subsubsection {* Cardinality *}
paulson@14485
   907
nipkow@15045
   908
lemma card_lessThan [simp]: "card {..<u} = u"
paulson@15251
   909
  by (induct u, simp_all add: lessThan_Suc)
paulson@14485
   910
paulson@14485
   911
lemma card_atMost [simp]: "card {..u} = Suc u"
paulson@14485
   912
  by (simp add: lessThan_Suc_atMost [THEN sym])
paulson@14485
   913
nipkow@15045
   914
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
nipkow@15045
   915
  apply (subgoal_tac "card {l..<u} = card {..<u-l}")
paulson@14485
   916
  apply (erule ssubst, rule card_lessThan)
nipkow@15045
   917
  apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
paulson@14485
   918
  apply (erule subst)
paulson@14485
   919
  apply (rule card_image)
paulson@14485
   920
  apply (simp add: inj_on_def)
paulson@14485
   921
  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
paulson@14485
   922
  apply (rule_tac x = "x - l" in exI)
paulson@14485
   923
  apply arith
paulson@14485
   924
  done
paulson@14485
   925
paulson@15418
   926
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
paulson@14485
   927
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
paulson@14485
   928
paulson@15418
   929
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
paulson@14485
   930
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
paulson@14485
   931
nipkow@15045
   932
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
paulson@14485
   933
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
paulson@14485
   934
nipkow@26105
   935
lemma ex_bij_betw_nat_finite:
nipkow@26105
   936
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
nipkow@26105
   937
apply(drule finite_imp_nat_seg_image_inj_on)
nipkow@26105
   938
apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
nipkow@26105
   939
done
nipkow@26105
   940
nipkow@26105
   941
lemma ex_bij_betw_finite_nat:
nipkow@26105
   942
  "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
nipkow@26105
   943
by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
nipkow@26105
   944
nipkow@31438
   945
lemma finite_same_card_bij:
nipkow@31438
   946
  "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"
nipkow@31438
   947
apply(drule ex_bij_betw_finite_nat)
nipkow@31438
   948
apply(drule ex_bij_betw_nat_finite)
nipkow@31438
   949
apply(auto intro!:bij_betw_trans)
nipkow@31438
   950
done
nipkow@31438
   951
nipkow@31438
   952
lemma ex_bij_betw_nat_finite_1:
nipkow@31438
   953
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
nipkow@31438
   954
by (rule finite_same_card_bij) auto
nipkow@31438
   955
hoelzl@40703
   956
lemma bij_betw_iff_card:
hoelzl@40703
   957
  assumes FIN: "finite A" and FIN': "finite B"
hoelzl@40703
   958
  shows BIJ: "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"
hoelzl@40703
   959
using assms
hoelzl@40703
   960
proof(auto simp add: bij_betw_same_card)
hoelzl@40703
   961
  assume *: "card A = card B"
hoelzl@40703
   962
  obtain f where "bij_betw f A {0 ..< card A}"
hoelzl@40703
   963
  using FIN ex_bij_betw_finite_nat by blast
hoelzl@40703
   964
  moreover obtain g where "bij_betw g {0 ..< card B} B"
hoelzl@40703
   965
  using FIN' ex_bij_betw_nat_finite by blast
hoelzl@40703
   966
  ultimately have "bij_betw (g o f) A B"
hoelzl@40703
   967
  using * by (auto simp add: bij_betw_trans)
hoelzl@40703
   968
  thus "(\<exists>f. bij_betw f A B)" by blast
hoelzl@40703
   969
qed
hoelzl@40703
   970
hoelzl@40703
   971
lemma inj_on_iff_card_le:
hoelzl@40703
   972
  assumes FIN: "finite A" and FIN': "finite B"
hoelzl@40703
   973
  shows "(\<exists>f. inj_on f A \<and> f ` A \<le> B) = (card A \<le> card B)"
hoelzl@40703
   974
proof (safe intro!: card_inj_on_le)
hoelzl@40703
   975
  assume *: "card A \<le> card B"
hoelzl@40703
   976
  obtain f where 1: "inj_on f A" and 2: "f ` A = {0 ..< card A}"
hoelzl@40703
   977
  using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force
hoelzl@40703
   978
  moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g ` {0 ..< card B} = B"
hoelzl@40703
   979
  using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force
hoelzl@40703
   980
  ultimately have "inj_on g (f ` A)" using subset_inj_on[of g _ "f ` A"] * by force
hoelzl@40703
   981
  hence "inj_on (g o f) A" using 1 comp_inj_on by blast
hoelzl@40703
   982
  moreover
hoelzl@40703
   983
  {have "{0 ..< card A} \<le> {0 ..< card B}" using * by force
hoelzl@40703
   984
   with 2 have "f ` A  \<le> {0 ..< card B}" by blast
hoelzl@40703
   985
   hence "(g o f) ` A \<le> B" unfolding comp_def using 3 by force
hoelzl@40703
   986
  }
hoelzl@40703
   987
  ultimately show "(\<exists>f. inj_on f A \<and> f ` A \<le> B)" by blast
hoelzl@40703
   988
qed (insert assms, auto)
nipkow@26105
   989
paulson@14485
   990
subsection {* Intervals of integers *}
paulson@14485
   991
nipkow@15045
   992
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
paulson@14485
   993
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   994
paulson@15418
   995
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
paulson@14485
   996
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
paulson@14485
   997
paulson@15418
   998
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
paulson@15418
   999
    "{l+1..<u} = {l<..<u::int}"
paulson@14485
  1000
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
paulson@14485
  1001
paulson@14485
  1002
subsubsection {* Finiteness *}
paulson@14485
  1003
paulson@15418
  1004
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
nipkow@15045
  1005
    {(0::int)..<u} = int ` {..<nat u}"
paulson@14485
  1006
  apply (unfold image_def lessThan_def)
paulson@14485
  1007
  apply auto
paulson@14485
  1008
  apply (rule_tac x = "nat x" in exI)
huffman@35216
  1009
  apply (auto simp add: zless_nat_eq_int_zless [THEN sym])
paulson@14485
  1010
  done
paulson@14485
  1011
nipkow@15045
  1012
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
wenzelm@47988
  1013
  apply (cases "0 \<le> u")
paulson@14485
  1014
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
  1015
  apply (rule finite_imageI)
paulson@14485
  1016
  apply auto
paulson@14485
  1017
  done
paulson@14485
  1018
nipkow@15045
  1019
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
nipkow@15045
  1020
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
  1021
  apply (erule subst)
paulson@14485
  1022
  apply (rule finite_imageI)
paulson@14485
  1023
  apply (rule finite_atLeastZeroLessThan_int)
nipkow@16733
  1024
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
  1025
  done
paulson@14485
  1026
paulson@15418
  1027
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
paulson@14485
  1028
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
paulson@14485
  1029
paulson@15418
  1030
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
paulson@14485
  1031
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
  1032
paulson@15418
  1033
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
paulson@14485
  1034
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
  1035
nipkow@24853
  1036
paulson@14485
  1037
subsubsection {* Cardinality *}
paulson@14485
  1038
nipkow@15045
  1039
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
wenzelm@47988
  1040
  apply (cases "0 \<le> u")
paulson@14485
  1041
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
  1042
  apply (subst card_image)
paulson@14485
  1043
  apply (auto simp add: inj_on_def)
paulson@14485
  1044
  done
paulson@14485
  1045
nipkow@15045
  1046
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
nipkow@15045
  1047
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
paulson@14485
  1048
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
nipkow@15045
  1049
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
  1050
  apply (erule subst)
paulson@14485
  1051
  apply (rule card_image)
paulson@14485
  1052
  apply (simp add: inj_on_def)
nipkow@16733
  1053
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
  1054
  done
paulson@14485
  1055
paulson@14485
  1056
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
nipkow@29667
  1057
apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
nipkow@29667
  1058
apply (auto simp add: algebra_simps)
nipkow@29667
  1059
done
paulson@14485
  1060
paulson@15418
  1061
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
nipkow@29667
  1062
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
  1063
nipkow@15045
  1064
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
nipkow@29667
  1065
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
  1066
bulwahn@27656
  1067
lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
bulwahn@27656
  1068
proof -
bulwahn@27656
  1069
  have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
bulwahn@27656
  1070
  with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
bulwahn@27656
  1071
qed
bulwahn@27656
  1072
bulwahn@27656
  1073
lemma card_less:
bulwahn@27656
  1074
assumes zero_in_M: "0 \<in> M"
bulwahn@27656
  1075
shows "card {k \<in> M. k < Suc i} \<noteq> 0"
bulwahn@27656
  1076
proof -
bulwahn@27656
  1077
  from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
bulwahn@27656
  1078
  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
bulwahn@27656
  1079
qed
bulwahn@27656
  1080
bulwahn@27656
  1081
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
  1082
apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])
bulwahn@27656
  1083
apply simp
nipkow@44890
  1084
apply fastforce
bulwahn@27656
  1085
apply auto
bulwahn@27656
  1086
apply (rule inj_on_diff_nat)
bulwahn@27656
  1087
apply auto
bulwahn@27656
  1088
apply (case_tac x)
bulwahn@27656
  1089
apply auto
bulwahn@27656
  1090
apply (case_tac xa)
bulwahn@27656
  1091
apply auto
bulwahn@27656
  1092
apply (case_tac xa)
bulwahn@27656
  1093
apply auto
bulwahn@27656
  1094
done
bulwahn@27656
  1095
bulwahn@27656
  1096
lemma card_less_Suc:
bulwahn@27656
  1097
  assumes zero_in_M: "0 \<in> M"
bulwahn@27656
  1098
    shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
bulwahn@27656
  1099
proof -
bulwahn@27656
  1100
  from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
bulwahn@27656
  1101
  hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
bulwahn@27656
  1102
    by (auto simp only: insert_Diff)
bulwahn@27656
  1103
  have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
bulwahn@27656
  1104
  from finite_M_bounded_by_nat[of "\<lambda>x. x \<in> M" "Suc i"] have "Suc (card {k. Suc k \<in> M \<and> k < i}) = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"
bulwahn@27656
  1105
    apply (subst card_insert)
bulwahn@27656
  1106
    apply simp_all
bulwahn@27656
  1107
    apply (subst b)
bulwahn@27656
  1108
    apply (subst card_less_Suc2[symmetric])
bulwahn@27656
  1109
    apply simp_all
bulwahn@27656
  1110
    done
bulwahn@27656
  1111
  with c show ?thesis by simp
bulwahn@27656
  1112
qed
bulwahn@27656
  1113
paulson@14485
  1114
paulson@13850
  1115
subsection {*Lemmas useful with the summation operator setsum*}
paulson@13850
  1116
ballarin@16102
  1117
text {* For examples, see Algebra/poly/UnivPoly2.thy *}
ballarin@13735
  1118
wenzelm@14577
  1119
subsubsection {* Disjoint Unions *}
ballarin@13735
  1120
wenzelm@14577
  1121
text {* Singletons and open intervals *}
ballarin@13735
  1122
ballarin@13735
  1123
lemma ivl_disj_un_singleton:
nipkow@15045
  1124
  "{l::'a::linorder} Un {l<..} = {l..}"
nipkow@15045
  1125
  "{..<u} Un {u::'a::linorder} = {..u}"
nipkow@15045
  1126
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
nipkow@15045
  1127
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
nipkow@15045
  1128
  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
nipkow@15045
  1129
  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
ballarin@14398
  1130
by auto
ballarin@13735
  1131
wenzelm@14577
  1132
text {* One- and two-sided intervals *}
ballarin@13735
  1133
ballarin@13735
  1134
lemma ivl_disj_un_one:
nipkow@15045
  1135
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
nipkow@15045
  1136
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
nipkow@15045
  1137
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
nipkow@15045
  1138
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
nipkow@15045
  1139
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
nipkow@15045
  1140
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
nipkow@15045
  1141
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
nipkow@15045
  1142
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
ballarin@14398
  1143
by auto
ballarin@13735
  1144
wenzelm@14577
  1145
text {* Two- and two-sided intervals *}
ballarin@13735
  1146
ballarin@13735
  1147
lemma ivl_disj_un_two:
nipkow@15045
  1148
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
nipkow@15045
  1149
  "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
nipkow@15045
  1150
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
nipkow@15045
  1151
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
nipkow@15045
  1152
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
nipkow@15045
  1153
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
nipkow@15045
  1154
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
nipkow@15045
  1155
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
ballarin@14398
  1156
by auto
ballarin@13735
  1157
ballarin@13735
  1158
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
ballarin@13735
  1159
wenzelm@14577
  1160
subsubsection {* Disjoint Intersections *}
ballarin@13735
  1161
wenzelm@14577
  1162
text {* One- and two-sided intervals *}
ballarin@13735
  1163
ballarin@13735
  1164
lemma ivl_disj_int_one:
nipkow@15045
  1165
  "{..l::'a::order} Int {l<..<u} = {}"
nipkow@15045
  1166
  "{..<l} Int {l..<u} = {}"
nipkow@15045
  1167
  "{..l} Int {l<..u} = {}"
nipkow@15045
  1168
  "{..<l} Int {l..u} = {}"
nipkow@15045
  1169
  "{l<..u} Int {u<..} = {}"
nipkow@15045
  1170
  "{l<..<u} Int {u..} = {}"
nipkow@15045
  1171
  "{l..u} Int {u<..} = {}"
nipkow@15045
  1172
  "{l..<u} Int {u..} = {}"
ballarin@14398
  1173
  by auto
ballarin@13735
  1174
wenzelm@14577
  1175
text {* Two- and two-sided intervals *}
ballarin@13735
  1176
ballarin@13735
  1177
lemma ivl_disj_int_two:
nipkow@15045
  1178
  "{l::'a::order<..<m} Int {m..<u} = {}"
nipkow@15045
  1179
  "{l<..m} Int {m<..<u} = {}"
nipkow@15045
  1180
  "{l..<m} Int {m..<u} = {}"
nipkow@15045
  1181
  "{l..m} Int {m<..<u} = {}"
nipkow@15045
  1182
  "{l<..<m} Int {m..u} = {}"
nipkow@15045
  1183
  "{l<..m} Int {m<..u} = {}"
nipkow@15045
  1184
  "{l..<m} Int {m..u} = {}"
nipkow@15045
  1185
  "{l..m} Int {m<..u} = {}"
ballarin@14398
  1186
  by auto
ballarin@13735
  1187
nipkow@32456
  1188
lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two
ballarin@13735
  1189
nipkow@15542
  1190
subsubsection {* Some Differences *}
nipkow@15542
  1191
nipkow@15542
  1192
lemma ivl_diff[simp]:
nipkow@15542
  1193
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
nipkow@15542
  1194
by(auto)
nipkow@15542
  1195
nipkow@15542
  1196
nipkow@15542
  1197
subsubsection {* Some Subset Conditions *}
nipkow@15542
  1198
blanchet@54147
  1199
lemma ivl_subset [simp]:
nipkow@15542
  1200
 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
nipkow@15542
  1201
apply(auto simp:linorder_not_le)
nipkow@15542
  1202
apply(rule ccontr)
nipkow@15542
  1203
apply(insert linorder_le_less_linear[of i n])
nipkow@15542
  1204
apply(clarsimp simp:linorder_not_le)
nipkow@44890
  1205
apply(fastforce)
nipkow@15542
  1206
done
nipkow@15542
  1207
nipkow@15041
  1208
nipkow@15042
  1209
subsection {* Summation indexed over intervals *}
nipkow@15042
  1210
nipkow@15042
  1211
syntax
nipkow@15042
  1212
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
  1213
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
  1214
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
nipkow@16052
  1215
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
nipkow@15042
  1216
syntax (xsymbols)
nipkow@15042
  1217
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
  1218
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
  1219
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
  1220
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15042
  1221
syntax (HTML output)
nipkow@15042
  1222
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
  1223
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
  1224
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
  1225
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15056
  1226
syntax (latex_sum output)
nipkow@15052
  1227
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
  1228
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052
  1229
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
  1230
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@16052
  1231
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
  1232
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15052
  1233
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
  1234
 ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15041
  1235
nipkow@15048
  1236
translations
nipkow@28853
  1237
  "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
nipkow@28853
  1238
  "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
nipkow@28853
  1239
  "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
nipkow@28853
  1240
  "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
nipkow@15041
  1241
nipkow@15052
  1242
text{* The above introduces some pretty alternative syntaxes for
nipkow@15056
  1243
summation over intervals:
nipkow@15052
  1244
\begin{center}
nipkow@15052
  1245
\begin{tabular}{lll}
nipkow@15056
  1246
Old & New & \LaTeX\\
nipkow@15056
  1247
@{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
  1248
@{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
  1249
@{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
nipkow@15056
  1250
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
nipkow@15052
  1251
\end{tabular}
nipkow@15052
  1252
\end{center}
nipkow@15056
  1253
The left column shows the term before introduction of the new syntax,
nipkow@15056
  1254
the middle column shows the new (default) syntax, and the right column
nipkow@15056
  1255
shows a special syntax. The latter is only meaningful for latex output
nipkow@15056
  1256
and has to be activated explicitly by setting the print mode to
wenzelm@21502
  1257
@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
nipkow@15056
  1258
antiquotations). It is not the default \LaTeX\ output because it only
nipkow@15056
  1259
works well with italic-style formulae, not tt-style.
nipkow@15052
  1260
nipkow@15052
  1261
Note that for uniformity on @{typ nat} it is better to use
nipkow@15052
  1262
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
nipkow@15052
  1263
not provide all lemmas available for @{term"{m..<n}"} also in the
nipkow@15052
  1264
special form for @{term"{..<n}"}. *}
nipkow@15052
  1265
nipkow@15542
  1266
text{* This congruence rule should be used for sums over intervals as
nipkow@15542
  1267
the standard theorem @{text[source]setsum_cong} does not work well
nipkow@15542
  1268
with the simplifier who adds the unsimplified premise @{term"x:B"} to
nipkow@15542
  1269
the context. *}
nipkow@15542
  1270
nipkow@15542
  1271
lemma setsum_ivl_cong:
nipkow@15542
  1272
 "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
nipkow@15542
  1273
 setsum f {a..<b} = setsum g {c..<d}"
nipkow@15542
  1274
by(rule setsum_cong, simp_all)
nipkow@15041
  1275
nipkow@16041
  1276
(* FIXME why are the following simp rules but the corresponding eqns
nipkow@16041
  1277
on intervals are not? *)
nipkow@16041
  1278
nipkow@16052
  1279
lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
nipkow@16052
  1280
by (simp add:atMost_Suc add_ac)
nipkow@16052
  1281
nipkow@16041
  1282
lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
nipkow@16041
  1283
by (simp add:lessThan_Suc add_ac)
nipkow@15041
  1284
nipkow@15911
  1285
lemma setsum_cl_ivl_Suc[simp]:
nipkow@15561
  1286
  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
nipkow@15561
  1287
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@15561
  1288
nipkow@15911
  1289
lemma setsum_op_ivl_Suc[simp]:
nipkow@15561
  1290
  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
nipkow@15561
  1291
by (auto simp:add_ac atLeastLessThanSuc)
nipkow@16041
  1292
(*
nipkow@15561
  1293
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
nipkow@15561
  1294
    (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
nipkow@15561
  1295
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@16041
  1296
*)
nipkow@28068
  1297
nipkow@28068
  1298
lemma setsum_head:
nipkow@28068
  1299
  fixes n :: nat
nipkow@28068
  1300
  assumes mn: "m <= n" 
nipkow@28068
  1301
  shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
nipkow@28068
  1302
proof -
nipkow@28068
  1303
  from mn
nipkow@28068
  1304
  have "{m..n} = {m} \<union> {m<..n}"
nipkow@28068
  1305
    by (auto intro: ivl_disj_un_singleton)
nipkow@28068
  1306
  hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
nipkow@28068
  1307
    by (simp add: atLeast0LessThan)
nipkow@28068
  1308
  also have "\<dots> = ?rhs" by simp
nipkow@28068
  1309
  finally show ?thesis .
nipkow@28068
  1310
qed
nipkow@28068
  1311
nipkow@28068
  1312
lemma setsum_head_Suc:
nipkow@28068
  1313
  "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
nipkow@28068
  1314
by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
nipkow@28068
  1315
nipkow@28068
  1316
lemma setsum_head_upt_Suc:
nipkow@28068
  1317
  "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
huffman@30079
  1318
apply(insert setsum_head_Suc[of m "n - Suc 0" f])
nipkow@29667
  1319
apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
nipkow@28068
  1320
done
nipkow@28068
  1321
nipkow@31501
  1322
lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"
nipkow@31501
  1323
  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
nipkow@31501
  1324
proof-
nipkow@31501
  1325
  have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using `m \<le> n+1` by auto
nipkow@31501
  1326
  thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint
nipkow@31501
  1327
    atLeastSucAtMost_greaterThanAtMost)
nipkow@31501
  1328
qed
nipkow@28068
  1329
nipkow@15539
  1330
lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
  1331
  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
nipkow@15539
  1332
by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
nipkow@15539
  1333
nipkow@15539
  1334
lemma setsum_diff_nat_ivl:
nipkow@15539
  1335
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
nipkow@15539
  1336
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
  1337
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
nipkow@15539
  1338
using setsum_add_nat_ivl [of m n p f,symmetric]
nipkow@15539
  1339
apply (simp add: add_ac)
nipkow@15539
  1340
done
nipkow@15539
  1341
nipkow@31505
  1342
lemma setsum_natinterval_difff:
nipkow@31505
  1343
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
nipkow@31505
  1344
  shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
nipkow@31505
  1345
          (if m <= n then f m - f(n + 1) else 0)"
nipkow@31505
  1346
by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
nipkow@31505
  1347
haftmann@44008
  1348
lemma setsum_restrict_set':
haftmann@44008
  1349
  "finite A \<Longrightarrow> setsum f {x \<in> A. x \<in> B} = (\<Sum>x\<in>A. if x \<in> B then f x else 0)"
haftmann@44008
  1350
  by (simp add: setsum_restrict_set [symmetric] Int_def)
haftmann@44008
  1351
haftmann@44008
  1352
lemma setsum_restrict_set'':
haftmann@44008
  1353
  "finite A \<Longrightarrow> setsum f {x \<in> A. P x} = (\<Sum>x\<in>A. if P x  then f x else 0)"
haftmann@44008
  1354
  by (simp add: setsum_restrict_set' [of A f "{x. P x}", simplified mem_Collect_eq])
nipkow@31509
  1355
nipkow@31509
  1356
lemma setsum_setsum_restrict:
haftmann@44008
  1357
  "finite S \<Longrightarrow> finite T \<Longrightarrow>
haftmann@44008
  1358
    setsum (\<lambda>x. setsum (\<lambda>y. f x y) {y. y \<in> T \<and> R x y}) S = setsum (\<lambda>y. setsum (\<lambda>x. f x y) {x. x \<in> S \<and> R x y}) T"
haftmann@44008
  1359
  by (simp add: setsum_restrict_set'') (rule setsum_commute)
nipkow@31509
  1360
nipkow@31509
  1361
lemma setsum_image_gen: assumes fS: "finite S"
nipkow@31509
  1362
  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
nipkow@31509
  1363
proof-
nipkow@31509
  1364
  { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
nipkow@31509
  1365
  hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
nipkow@31509
  1366
    by simp
nipkow@31509
  1367
  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
nipkow@31509
  1368
    by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])
nipkow@31509
  1369
  finally show ?thesis .
nipkow@31509
  1370
qed
nipkow@31509
  1371
hoelzl@35171
  1372
lemma setsum_le_included:
haftmann@36307
  1373
  fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"
hoelzl@35171
  1374
  assumes "finite s" "finite t"
hoelzl@35171
  1375
  and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)"
hoelzl@35171
  1376
  shows "setsum f s \<le> setsum g t"
hoelzl@35171
  1377
proof -
hoelzl@35171
  1378
  have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"
hoelzl@35171
  1379
  proof (rule setsum_mono)
hoelzl@35171
  1380
    fix y assume "y \<in> s"
hoelzl@35171
  1381
    with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
hoelzl@35171
  1382
    with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
hoelzl@35171
  1383
      using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]
hoelzl@35171
  1384
      by (auto intro!: setsum_mono2)
hoelzl@35171
  1385
  qed
hoelzl@35171
  1386
  also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
hoelzl@35171
  1387
    using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)
hoelzl@35171
  1388
  also have "... \<le> setsum g t"
hoelzl@35171
  1389
    using assms by (auto simp: setsum_image_gen[symmetric])
hoelzl@35171
  1390
  finally show ?thesis .
hoelzl@35171
  1391
qed
hoelzl@35171
  1392
nipkow@31509
  1393
lemma setsum_multicount_gen:
nipkow@31509
  1394
  assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
nipkow@31509
  1395
  shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
nipkow@31509
  1396
proof-
nipkow@31509
  1397
  have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
nipkow@31509
  1398
  also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]
nipkow@31509
  1399
    using assms(3) by auto
nipkow@31509
  1400
  finally show ?thesis .
nipkow@31509
  1401
qed
nipkow@31509
  1402
nipkow@31509
  1403
lemma setsum_multicount:
nipkow@31509
  1404
  assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
nipkow@31509
  1405
  shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
nipkow@31509
  1406
proof-
nipkow@31509
  1407
  have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)
huffman@35216
  1408
  also have "\<dots> = ?r" by(simp add: mult_commute)
nipkow@31509
  1409
  finally show ?thesis by auto
nipkow@31509
  1410
qed
nipkow@31509
  1411
nipkow@28068
  1412
nipkow@16733
  1413
subsection{* Shifting bounds *}
nipkow@16733
  1414
nipkow@15539
  1415
lemma setsum_shift_bounds_nat_ivl:
nipkow@15539
  1416
  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
nipkow@15539
  1417
by (induct "n", auto simp:atLeastLessThanSuc)
nipkow@15539
  1418
nipkow@16733
  1419
lemma setsum_shift_bounds_cl_nat_ivl:
nipkow@16733
  1420
  "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
nipkow@16733
  1421
apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
nipkow@16733
  1422
apply (simp add:image_add_atLeastAtMost o_def)
nipkow@16733
  1423
done
nipkow@16733
  1424
nipkow@16733
  1425
corollary setsum_shift_bounds_cl_Suc_ivl:
nipkow@16733
  1426
  "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
huffman@30079
  1427
by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
  1428
nipkow@16733
  1429
corollary setsum_shift_bounds_Suc_ivl:
nipkow@16733
  1430
  "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
huffman@30079
  1431
by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
  1432
nipkow@28068
  1433
lemma setsum_shift_lb_Suc0_0:
nipkow@28068
  1434
  "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
nipkow@28068
  1435
by(simp add:setsum_head_Suc)
kleing@19106
  1436
nipkow@28068
  1437
lemma setsum_shift_lb_Suc0_0_upt:
nipkow@28068
  1438
  "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
nipkow@28068
  1439
apply(cases k)apply simp
nipkow@28068
  1440
apply(simp add:setsum_head_upt_Suc)
nipkow@28068
  1441
done
kleing@19022
  1442
haftmann@52380
  1443
lemma setsum_atMost_Suc_shift:
haftmann@52380
  1444
  fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
haftmann@52380
  1445
  shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
haftmann@52380
  1446
proof (induct n)
haftmann@52380
  1447
  case 0 show ?case by simp
haftmann@52380
  1448
next
haftmann@52380
  1449
  case (Suc n) note IH = this
haftmann@52380
  1450
  have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"
haftmann@52380
  1451
    by (rule setsum_atMost_Suc)
haftmann@52380
  1452
  also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
haftmann@52380
  1453
    by (rule IH)
haftmann@52380
  1454
  also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
haftmann@52380
  1455
             f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
haftmann@52380
  1456
    by (rule add_assoc)
haftmann@52380
  1457
  also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
haftmann@52380
  1458
    by (rule setsum_atMost_Suc [symmetric])
haftmann@52380
  1459
  finally show ?case .
haftmann@52380
  1460
qed
haftmann@52380
  1461
lp15@55718
  1462
lemma setsum_last_plus: "n \<noteq> 0 \<Longrightarrow> (\<Sum>i = 0..n. f i) = f n + (\<Sum>i = 0..n - Suc 0. f i)"
lp15@55718
  1463
  using atLeastAtMostSuc_conv [of 0 "n - 1"]
lp15@55718
  1464
  by auto
lp15@55718
  1465
lp15@55718
  1466
lemma nested_setsum_swap:
lp15@55718
  1467
     "(\<Sum>i = 0..n. (\<Sum>j = 0..<i. a i j)) = (\<Sum>j = 0..<n. \<Sum>i = Suc j..n. a i j)"
lp15@55718
  1468
  by (induction n) (auto simp: setsum_addf)
lp15@55718
  1469
haftmann@52380
  1470
ballarin@17149
  1471
subsection {* The formula for geometric sums *}
ballarin@17149
  1472
ballarin@17149
  1473
lemma geometric_sum:
haftmann@36307
  1474
  assumes "x \<noteq> 1"
haftmann@36307
  1475
  shows "(\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"
haftmann@36307
  1476
proof -
haftmann@36307
  1477
  from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all
haftmann@36307
  1478
  moreover have "(\<Sum>i=0..<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"
haftmann@36307
  1479
  proof (induct n)
haftmann@36307
  1480
    case 0 then show ?case by simp
haftmann@36307
  1481
  next
haftmann@36307
  1482
    case (Suc n)
wenzelm@53374
  1483
    moreover from Suc `y \<noteq> 0` have "(1 + y) ^ n = (y * inverse y) * (1 + y) ^ n" by simp 
haftmann@36350
  1484
    ultimately show ?case by (simp add: field_simps divide_inverse)
haftmann@36307
  1485
  qed
haftmann@36307
  1486
  ultimately show ?thesis by simp
haftmann@36307
  1487
qed
haftmann@36307
  1488
ballarin@17149
  1489
kleing@19469
  1490
subsection {* The formula for arithmetic sums *}
kleing@19469
  1491
huffman@47222
  1492
lemma gauss_sum:
huffman@47222
  1493
  "(2::'a::comm_semiring_1)*(\<Sum>i\<in>{1..n}. of_nat i) =
kleing@19469
  1494
   of_nat n*((of_nat n)+1)"
kleing@19469
  1495
proof (induct n)
kleing@19469
  1496
  case 0
kleing@19469
  1497
  show ?case by simp
kleing@19469
  1498
next
kleing@19469
  1499
  case (Suc n)
huffman@47222
  1500
  then show ?case
huffman@47222
  1501
    by (simp add: algebra_simps add: one_add_one [symmetric] del: one_add_one)
huffman@47222
  1502
      (* FIXME: make numeral cancellation simprocs work for semirings *)
kleing@19469
  1503
qed
kleing@19469
  1504
kleing@19469
  1505
theorem arith_series_general:
huffman@47222
  1506
  "(2::'a::comm_semiring_1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1507
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1508
proof cases
kleing@19469
  1509
  assume ngt1: "n > 1"
kleing@19469
  1510
  let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
kleing@19469
  1511
  have
kleing@19469
  1512
    "(\<Sum>i\<in>{..<n}. a+?I i*d) =
kleing@19469
  1513
     ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
kleing@19469
  1514
    by (rule setsum_addf)
kleing@19469
  1515
  also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
kleing@19469
  1516
  also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
huffman@30079
  1517
    unfolding One_nat_def
nipkow@28068
  1518
    by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)
huffman@47222
  1519
  also have "2*\<dots> = 2*?n*a + d*2*(\<Sum>i\<in>{1..<n}. ?I i)"
huffman@47222
  1520
    by (simp add: algebra_simps)
kleing@19469
  1521
  also from ngt1 have "{1..<n} = {1..n - 1}"
nipkow@28068
  1522
    by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
nipkow@28068
  1523
  also from ngt1
huffman@47222
  1524
  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
  1525
    by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)
huffman@23431
  1526
       (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
huffman@47222
  1527
  finally show ?thesis
huffman@47222
  1528
    unfolding mult_2 by (simp add: algebra_simps)
kleing@19469
  1529
next
kleing@19469
  1530
  assume "\<not>(n > 1)"
kleing@19469
  1531
  hence "n = 1 \<or> n = 0" by auto
huffman@47222
  1532
  thus ?thesis by (auto simp: mult_2)
kleing@19469
  1533
qed
kleing@19469
  1534
kleing@19469
  1535
lemma arith_series_nat:
huffman@47222
  1536
  "(2::nat) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
kleing@19469
  1537
proof -
kleing@19469
  1538
  have
huffman@47222
  1539
    "2 * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
kleing@19469
  1540
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1541
    by (rule arith_series_general)
huffman@30079
  1542
  thus ?thesis
huffman@35216
  1543
    unfolding One_nat_def by auto
kleing@19469
  1544
qed
kleing@19469
  1545
kleing@19469
  1546
lemma arith_series_int:
huffman@47222
  1547
  "2 * (\<Sum>i\<in>{..<n}. a + int i * d) = int n * (a + (a + int(n - 1)*d))"
huffman@47222
  1548
  by (fact arith_series_general) (* FIXME: duplicate *)
paulson@15418
  1549
kleing@19022
  1550
lemma sum_diff_distrib:
kleing@19022
  1551
  fixes P::"nat\<Rightarrow>nat"
kleing@19022
  1552
  shows
kleing@19022
  1553
  "\<forall>x. Q x \<le> P x  \<Longrightarrow>
kleing@19022
  1554
  (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
kleing@19022
  1555
proof (induct n)
kleing@19022
  1556
  case 0 show ?case by simp
kleing@19022
  1557
next
kleing@19022
  1558
  case (Suc n)
kleing@19022
  1559
kleing@19022
  1560
  let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
kleing@19022
  1561
  let ?rhs = "\<Sum>x<n. P x - Q x"
kleing@19022
  1562
kleing@19022
  1563
  from Suc have "?lhs = ?rhs" by simp
kleing@19022
  1564
  moreover
kleing@19022
  1565
  from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
kleing@19022
  1566
  moreover
kleing@19022
  1567
  from Suc have
kleing@19022
  1568
    "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
kleing@19022
  1569
    by (subst diff_diff_left[symmetric],
kleing@19022
  1570
        subst diff_add_assoc2)
kleing@19022
  1571
       (auto simp: diff_add_assoc2 intro: setsum_mono)
kleing@19022
  1572
  ultimately
kleing@19022
  1573
  show ?case by simp
kleing@19022
  1574
qed
kleing@19022
  1575
lp15@55718
  1576
lemma nat_diff_setsum_reindex:
lp15@55718
  1577
  fixes x :: "'a::{comm_ring,monoid_mult}"
lp15@55718
  1578
  shows "(\<Sum>i=0..<n. f (n - Suc i)) = (\<Sum>i=0..<n. f i)"
lp15@55718
  1579
apply (subst setsum_reindex_cong [of "%i. n - Suc i" "{0..< n}"])
lp15@55718
  1580
apply (auto simp: inj_on_def)
lp15@55718
  1581
apply (rule_tac x="n - Suc x" in image_eqI, auto)
lp15@55718
  1582
done
lp15@55718
  1583
paulson@29960
  1584
subsection {* Products indexed over intervals *}
paulson@29960
  1585
paulson@29960
  1586
syntax
paulson@29960
  1587
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1588
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1589
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)
paulson@29960
  1590
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
paulson@29960
  1591
syntax (xsymbols)
paulson@29960
  1592
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1593
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1594
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1595
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1596
syntax (HTML output)
paulson@29960
  1597
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1598
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1599
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1600
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1601
syntax (latex_prod output)
paulson@29960
  1602
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1603
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1604
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1605
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1606
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1607
 ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1608
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1609
 ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1610
paulson@29960
  1611
translations
paulson@29960
  1612
  "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"
paulson@29960
  1613
  "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"
paulson@29960
  1614
  "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"
paulson@29960
  1615
  "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"
paulson@29960
  1616
haftmann@33318
  1617
subsection {* Transfer setup *}
haftmann@33318
  1618
haftmann@33318
  1619
lemma transfer_nat_int_set_functions:
haftmann@33318
  1620
    "{..n} = nat ` {0..int n}"
haftmann@33318
  1621
    "{m..n} = nat ` {int m..int n}"  (* need all variants of these! *)
haftmann@33318
  1622
  apply (auto simp add: image_def)
haftmann@33318
  1623
  apply (rule_tac x = "int x" in bexI)
haftmann@33318
  1624
  apply auto
haftmann@33318
  1625
  apply (rule_tac x = "int x" in bexI)
haftmann@33318
  1626
  apply auto
haftmann@33318
  1627
  done
haftmann@33318
  1628
haftmann@33318
  1629
lemma transfer_nat_int_set_function_closures:
haftmann@33318
  1630
    "x >= 0 \<Longrightarrow> nat_set {x..y}"
haftmann@33318
  1631
  by (simp add: nat_set_def)
haftmann@33318
  1632
haftmann@35644
  1633
declare transfer_morphism_nat_int[transfer add
haftmann@33318
  1634
  return: transfer_nat_int_set_functions
haftmann@33318
  1635
    transfer_nat_int_set_function_closures
haftmann@33318
  1636
]
haftmann@33318
  1637
haftmann@33318
  1638
lemma transfer_int_nat_set_functions:
haftmann@33318
  1639
    "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}"
haftmann@33318
  1640
  by (simp only: is_nat_def transfer_nat_int_set_functions
haftmann@33318
  1641
    transfer_nat_int_set_function_closures
haftmann@33318
  1642
    transfer_nat_int_set_return_embed nat_0_le
haftmann@33318
  1643
    cong: transfer_nat_int_set_cong)
haftmann@33318
  1644
haftmann@33318
  1645
lemma transfer_int_nat_set_function_closures:
haftmann@33318
  1646
    "is_nat x \<Longrightarrow> nat_set {x..y}"
haftmann@33318
  1647
  by (simp only: transfer_nat_int_set_function_closures is_nat_def)
haftmann@33318
  1648
haftmann@35644
  1649
declare transfer_morphism_int_nat[transfer add
haftmann@33318
  1650
  return: transfer_int_nat_set_functions
haftmann@33318
  1651
    transfer_int_nat_set_function_closures
haftmann@33318
  1652
]
haftmann@33318
  1653
lp15@55242
  1654
lemma setprod_int_plus_eq: "setprod int {i..i+j} =  \<Prod>{int i..int (i+j)}"
lp15@55242
  1655
  by (induct j) (auto simp add: atLeastAtMostSuc_conv atLeastAtMostPlus1_int_conv)
lp15@55242
  1656
lp15@55242
  1657
lemma setprod_int_eq: "setprod int {i..j} =  \<Prod>{int i..int j}"
lp15@55242
  1658
proof (cases "i \<le> j")
lp15@55242
  1659
  case True
lp15@55242
  1660
  then show ?thesis
lp15@55242
  1661
    by (metis Nat.le_iff_add setprod_int_plus_eq)
lp15@55242
  1662
next
lp15@55242
  1663
  case False
lp15@55242
  1664
  then show ?thesis
lp15@55242
  1665
    by auto
lp15@55242
  1666
qed
lp15@55242
  1667
lp15@55718
  1668
lemma setprod_power_distrib:
lp15@55718
  1669
  fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
lp15@55719
  1670
  shows "setprod f A ^ n = setprod (\<lambda>x. (f x)^n) A"
lp15@55719
  1671
proof (cases "finite A") 
lp15@55719
  1672
  case True then show ?thesis 
lp15@55719
  1673
    by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
lp15@55719
  1674
next
lp15@55719
  1675
  case False then show ?thesis 
lp15@55719
  1676
    by (metis setprod_infinite power_one)
lp15@55719
  1677
qed
lp15@55718
  1678
nipkow@8924
  1679
end