src/HOL/Set_Interval.thy
author paulson <lp15@cam.ac.uk>
Tue Jul 28 16:16:13 2015 +0100 (2015-07-28)
changeset 60809 457abb82fb9e
parent 60762 bf0c76ccee8d
child 61204 3e491e34a62e
permissions -rw-r--r--
the Cauchy integral theorem and related material
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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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section \<open>Set intervals\<close>
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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\<open>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.\<close>
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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 \<open>Various equivalences\<close>
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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 \<open>Logical Equivalences for Set Inclusion and Equality\<close>
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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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lemma (in linorder) Ici_subset_Ioi_iff: "{a ..} \<subseteq> {b <..} \<longleftrightarrow> b < a"
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  by auto
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lemma (in linorder) Iic_subset_Iio_iff: "{.. a} \<subseteq> {..< b} \<longleftrightarrow> a < b"
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  by auto
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subsection \<open>Two-sided intervals\<close>
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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 \<open>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.\<close>
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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\<open>Emptyness, singletons, subset\<close>
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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 \<open>{a..b} = {c}\<close> 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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   315
nipkow@51334
   316
lemma not_Ici_le_Iic[simp]: "\<not> {l..} \<subseteq> {..h'}"
nipkow@51334
   317
using gt_ex[of h']
nipkow@51334
   318
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
nipkow@51334
   319
nipkow@51334
   320
end
nipkow@51334
   321
nipkow@51334
   322
context no_bot
nipkow@51334
   323
begin
nipkow@51334
   324
nipkow@51334
   325
lemma not_UNIV_le_Ici[simp]: "\<not> UNIV \<subseteq> {l..}"
nipkow@51334
   326
using lt_ex[of l] by(auto simp: subset_eq less_le_not_le)
nipkow@51334
   327
nipkow@51334
   328
lemma not_Iic_le_Icc[simp]: "\<not> {..h} \<subseteq> {l'..h'}"
nipkow@51334
   329
using lt_ex[of l']
nipkow@51334
   330
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
nipkow@51334
   331
nipkow@51334
   332
lemma not_Iic_le_Ici[simp]: "\<not> {..h} \<subseteq> {l'..}"
nipkow@51334
   333
using lt_ex[of l']
nipkow@51334
   334
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
nipkow@51334
   335
nipkow@51334
   336
end
nipkow@51334
   337
nipkow@51334
   338
nipkow@51334
   339
context no_top
nipkow@51334
   340
begin
nipkow@51334
   341
nipkow@51334
   342
(* also holds for no_bot but no_top should suffice *)
nipkow@51334
   343
lemma not_UNIV_eq_Icc[simp]: "\<not> UNIV = {l'..h'}"
nipkow@51334
   344
using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
nipkow@51334
   345
nipkow@51334
   346
lemmas not_Icc_eq_UNIV[simp] = not_UNIV_eq_Icc[symmetric]
nipkow@51334
   347
nipkow@51334
   348
lemma not_UNIV_eq_Iic[simp]: "\<not> UNIV = {..h'}"
nipkow@51334
   349
using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
nipkow@51334
   350
nipkow@51334
   351
lemmas not_Iic_eq_UNIV[simp] = not_UNIV_eq_Iic[symmetric]
nipkow@51334
   352
nipkow@51334
   353
lemma not_Icc_eq_Ici[simp]: "\<not> {l..h} = {l'..}"
nipkow@51334
   354
unfolding atLeastAtMost_def using not_Ici_le_Iic[of l'] by blast
nipkow@51334
   355
nipkow@51334
   356
lemmas not_Ici_eq_Icc[simp] = not_Icc_eq_Ici[symmetric]
nipkow@51334
   357
nipkow@51334
   358
(* also holds for no_bot but no_top should suffice *)
nipkow@51334
   359
lemma not_Iic_eq_Ici[simp]: "\<not> {..h} = {l'..}"
nipkow@51334
   360
using not_Ici_le_Iic[of l' h] by blast
nipkow@51334
   361
nipkow@51334
   362
lemmas not_Ici_eq_Iic[simp] = not_Iic_eq_Ici[symmetric]
nipkow@51334
   363
nipkow@51334
   364
end
nipkow@51334
   365
nipkow@51334
   366
context no_bot
nipkow@51334
   367
begin
nipkow@51334
   368
nipkow@51334
   369
lemma not_UNIV_eq_Ici[simp]: "\<not> UNIV = {l'..}"
nipkow@51334
   370
using lt_ex[of l'] by(auto simp: set_eq_iff  less_le_not_le)
nipkow@51334
   371
nipkow@51334
   372
lemmas not_Ici_eq_UNIV[simp] = not_UNIV_eq_Ici[symmetric]
nipkow@51334
   373
nipkow@51334
   374
lemma not_Icc_eq_Iic[simp]: "\<not> {l..h} = {..h'}"
nipkow@51334
   375
unfolding atLeastAtMost_def using not_Iic_le_Ici[of h'] by blast
nipkow@51334
   376
nipkow@51334
   377
lemmas not_Iic_eq_Icc[simp] = not_Icc_eq_Iic[symmetric]
nipkow@51334
   378
nipkow@51334
   379
end
nipkow@51334
   380
nipkow@51334
   381
hoelzl@53216
   382
context dense_linorder
hoelzl@42891
   383
begin
hoelzl@42891
   384
hoelzl@42891
   385
lemma greaterThanLessThan_empty_iff[simp]:
hoelzl@42891
   386
  "{ a <..< b } = {} \<longleftrightarrow> b \<le> a"
hoelzl@42891
   387
  using dense[of a b] by (cases "a < b") auto
hoelzl@42891
   388
hoelzl@42891
   389
lemma greaterThanLessThan_empty_iff2[simp]:
hoelzl@42891
   390
  "{} = { a <..< b } \<longleftrightarrow> b \<le> a"
hoelzl@42891
   391
  using dense[of a b] by (cases "a < b") auto
hoelzl@42891
   392
hoelzl@42901
   393
lemma atLeastLessThan_subseteq_atLeastAtMost_iff:
hoelzl@42901
   394
  "{a ..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
hoelzl@42901
   395
  using dense[of "max a d" "b"]
hoelzl@42901
   396
  by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@42901
   397
hoelzl@42901
   398
lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:
hoelzl@42901
   399
  "{a <.. b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
hoelzl@42901
   400
  using dense[of "a" "min c b"]
hoelzl@42901
   401
  by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@42901
   402
hoelzl@42901
   403
lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:
hoelzl@42901
   404
  "{a <..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
hoelzl@42901
   405
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
hoelzl@42901
   406
  by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@42901
   407
hoelzl@43657
   408
lemma atLeastAtMost_subseteq_atLeastLessThan_iff:
hoelzl@43657
   409
  "{a .. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a \<le> b \<longrightarrow> c \<le> a \<and> b < d)"
hoelzl@43657
   410
  using dense[of "max a d" "b"]
hoelzl@43657
   411
  by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@43657
   412
hoelzl@43657
   413
lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:
hoelzl@43657
   414
  "{a <.. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b < d)"
hoelzl@43657
   415
  using dense[of "a" "min c b"]
hoelzl@43657
   416
  by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@43657
   417
hoelzl@43657
   418
lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:
hoelzl@43657
   419
  "{a <..< b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
hoelzl@43657
   420
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
hoelzl@43657
   421
  by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@43657
   422
hoelzl@56328
   423
lemma greaterThanLessThan_subseteq_greaterThanAtMost_iff:
hoelzl@56328
   424
  "{a <..< b} \<subseteq> { c <.. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
hoelzl@56328
   425
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
hoelzl@56328
   426
  by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@56328
   427
hoelzl@42891
   428
end
hoelzl@42891
   429
hoelzl@51329
   430
context no_top
hoelzl@51329
   431
begin
hoelzl@51329
   432
nipkow@51334
   433
lemma greaterThan_non_empty[simp]: "{x <..} \<noteq> {}"
hoelzl@51329
   434
  using gt_ex[of x] by auto
hoelzl@51329
   435
hoelzl@51329
   436
end
hoelzl@51329
   437
hoelzl@51329
   438
context no_bot
hoelzl@51329
   439
begin
hoelzl@51329
   440
nipkow@51334
   441
lemma lessThan_non_empty[simp]: "{..< x} \<noteq> {}"
hoelzl@51329
   442
  using lt_ex[of x] by auto
hoelzl@51329
   443
hoelzl@51329
   444
end
hoelzl@51329
   445
nipkow@32408
   446
lemma (in linorder) atLeastLessThan_subset_iff:
nipkow@32408
   447
  "{a..<b} <= {c..<d} \<Longrightarrow> b <= a | c<=a & b<=d"
nipkow@32408
   448
apply (auto simp:subset_eq Ball_def)
nipkow@32408
   449
apply(frule_tac x=a in spec)
nipkow@32408
   450
apply(erule_tac x=d in allE)
nipkow@32408
   451
apply (simp add: less_imp_le)
nipkow@32408
   452
done
nipkow@32408
   453
hoelzl@40703
   454
lemma atLeastLessThan_inj:
hoelzl@40703
   455
  fixes a b c d :: "'a::linorder"
hoelzl@40703
   456
  assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"
hoelzl@40703
   457
  shows "a = c" "b = d"
hoelzl@40703
   458
using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le linorder_antisym_conv2 subset_refl)+
hoelzl@40703
   459
hoelzl@40703
   460
lemma atLeastLessThan_eq_iff:
hoelzl@40703
   461
  fixes a b c d :: "'a::linorder"
hoelzl@40703
   462
  assumes "a < b" "c < d"
hoelzl@40703
   463
  shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"
hoelzl@40703
   464
  using atLeastLessThan_inj assms by auto
hoelzl@40703
   465
hoelzl@57447
   466
lemma (in linorder) Ioc_inj: "{a <.. b} = {c <.. d} \<longleftrightarrow> (b \<le> a \<and> d \<le> c) \<or> a = c \<and> b = d"
hoelzl@57447
   467
  by (metis eq_iff greaterThanAtMost_empty_iff2 greaterThanAtMost_iff le_cases not_le)
hoelzl@57447
   468
hoelzl@57447
   469
lemma (in order) Iio_Int_singleton: "{..<k} \<inter> {x} = (if x < k then {x} else {})"
hoelzl@57447
   470
  by auto
hoelzl@57447
   471
hoelzl@57447
   472
lemma (in linorder) Ioc_subset_iff: "{a<..b} \<subseteq> {c<..d} \<longleftrightarrow> (b \<le> a \<or> c \<le> a \<and> b \<le> d)"
hoelzl@57447
   473
  by (auto simp: subset_eq Ball_def) (metis less_le not_less)
hoelzl@57447
   474
haftmann@52729
   475
lemma (in order_bot) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"
nipkow@51334
   476
by (auto simp: set_eq_iff intro: le_bot)
hoelzl@51328
   477
haftmann@52729
   478
lemma (in order_top) atMost_eq_UNIV_iff: "{..x} = UNIV \<longleftrightarrow> x = top"
nipkow@51334
   479
by (auto simp: set_eq_iff intro: top_le)
hoelzl@51328
   480
nipkow@51334
   481
lemma (in bounded_lattice) atLeastAtMost_eq_UNIV_iff:
nipkow@51334
   482
  "{x..y} = UNIV \<longleftrightarrow> (x = bot \<and> y = top)"
nipkow@51334
   483
by (auto simp: set_eq_iff intro: top_le le_bot)
hoelzl@51328
   484
hoelzl@56949
   485
lemma Iio_eq_empty_iff: "{..< n::'a::{linorder, order_bot}} = {} \<longleftrightarrow> n = bot"
hoelzl@56949
   486
  by (auto simp: set_eq_iff not_less le_bot)
hoelzl@56949
   487
hoelzl@56949
   488
lemma Iio_eq_empty_iff_nat: "{..< n::nat} = {} \<longleftrightarrow> n = 0"
hoelzl@56949
   489
  by (simp add: Iio_eq_empty_iff bot_nat_def)
hoelzl@56949
   490
noschinl@58970
   491
lemma mono_image_least:
noschinl@58970
   492
  assumes f_mono: "mono f" and f_img: "f ` {m ..< n} = {m' ..< n'}" "m < n"
noschinl@58970
   493
  shows "f m = m'"
noschinl@58970
   494
proof -
noschinl@58970
   495
  from f_img have "{m' ..< n'} \<noteq> {}"
noschinl@58970
   496
    by (metis atLeastLessThan_empty_iff image_is_empty)
noschinl@58970
   497
  with f_img have "m' \<in> f ` {m ..< n}" by auto
noschinl@58970
   498
  then obtain k where "f k = m'" "m \<le> k" by auto
noschinl@58970
   499
  moreover have "m' \<le> f m" using f_img by auto
noschinl@58970
   500
  ultimately show "f m = m'"
noschinl@58970
   501
    using f_mono by (auto elim: monoE[where x=m and y=k])
noschinl@58970
   502
qed
noschinl@58970
   503
hoelzl@51328
   504
wenzelm@60758
   505
subsection \<open>Infinite intervals\<close>
hoelzl@56328
   506
hoelzl@56328
   507
context dense_linorder
hoelzl@56328
   508
begin
hoelzl@56328
   509
hoelzl@56328
   510
lemma infinite_Ioo:
hoelzl@56328
   511
  assumes "a < b"
hoelzl@56328
   512
  shows "\<not> finite {a<..<b}"
hoelzl@56328
   513
proof
hoelzl@56328
   514
  assume fin: "finite {a<..<b}"
hoelzl@56328
   515
  moreover have ne: "{a<..<b} \<noteq> {}"
wenzelm@60758
   516
    using \<open>a < b\<close> by auto
hoelzl@56328
   517
  ultimately have "a < Max {a <..< b}" "Max {a <..< b} < b"
hoelzl@56328
   518
    using Max_in[of "{a <..< b}"] by auto
hoelzl@56328
   519
  then obtain x where "Max {a <..< b} < x" "x < b"
hoelzl@56328
   520
    using dense[of "Max {a<..<b}" b] by auto
hoelzl@56328
   521
  then have "x \<in> {a <..< b}"
wenzelm@60758
   522
    using \<open>a < Max {a <..< b}\<close> by auto
hoelzl@56328
   523
  then have "x \<le> Max {a <..< b}"
hoelzl@56328
   524
    using fin by auto
wenzelm@60758
   525
  with \<open>Max {a <..< b} < x\<close> show False by auto
hoelzl@56328
   526
qed
hoelzl@56328
   527
hoelzl@56328
   528
lemma infinite_Icc: "a < b \<Longrightarrow> \<not> finite {a .. b}"
hoelzl@56328
   529
  using greaterThanLessThan_subseteq_atLeastAtMost_iff[of a b a b] infinite_Ioo[of a b]
hoelzl@56328
   530
  by (auto dest: finite_subset)
hoelzl@56328
   531
hoelzl@56328
   532
lemma infinite_Ico: "a < b \<Longrightarrow> \<not> finite {a ..< b}"
hoelzl@56328
   533
  using greaterThanLessThan_subseteq_atLeastLessThan_iff[of a b a b] infinite_Ioo[of a b]
hoelzl@56328
   534
  by (auto dest: finite_subset)
hoelzl@56328
   535
hoelzl@56328
   536
lemma infinite_Ioc: "a < b \<Longrightarrow> \<not> finite {a <.. b}"
hoelzl@56328
   537
  using greaterThanLessThan_subseteq_greaterThanAtMost_iff[of a b a b] infinite_Ioo[of a b]
hoelzl@56328
   538
  by (auto dest: finite_subset)
hoelzl@56328
   539
hoelzl@56328
   540
end
hoelzl@56328
   541
hoelzl@56328
   542
lemma infinite_Iio: "\<not> finite {..< a :: 'a :: {no_bot, linorder}}"
hoelzl@56328
   543
proof
hoelzl@56328
   544
  assume "finite {..< a}"
hoelzl@56328
   545
  then have *: "\<And>x. x < a \<Longrightarrow> Min {..< a} \<le> x"
hoelzl@56328
   546
    by auto
hoelzl@56328
   547
  obtain x where "x < a"
hoelzl@56328
   548
    using lt_ex by auto
hoelzl@56328
   549
hoelzl@56328
   550
  obtain y where "y < Min {..< a}"
hoelzl@56328
   551
    using lt_ex by auto
hoelzl@56328
   552
  also have "Min {..< a} \<le> x"
wenzelm@60758
   553
    using \<open>x < a\<close> by fact
wenzelm@60758
   554
  also note \<open>x < a\<close>
hoelzl@56328
   555
  finally have "Min {..< a} \<le> y"
hoelzl@56328
   556
    by fact
wenzelm@60758
   557
  with \<open>y < Min {..< a}\<close> show False by auto
hoelzl@56328
   558
qed
hoelzl@56328
   559
hoelzl@56328
   560
lemma infinite_Iic: "\<not> finite {.. a :: 'a :: {no_bot, linorder}}"
hoelzl@56328
   561
  using infinite_Iio[of a] finite_subset[of "{..< a}" "{.. a}"]
hoelzl@56328
   562
  by (auto simp: subset_eq less_imp_le)
hoelzl@56328
   563
hoelzl@56328
   564
lemma infinite_Ioi: "\<not> finite {a :: 'a :: {no_top, linorder} <..}"
hoelzl@56328
   565
proof
hoelzl@56328
   566
  assume "finite {a <..}"
hoelzl@56328
   567
  then have *: "\<And>x. a < x \<Longrightarrow> x \<le> Max {a <..}"
hoelzl@56328
   568
    by auto
hoelzl@56328
   569
hoelzl@56328
   570
  obtain y where "Max {a <..} < y"
hoelzl@56328
   571
    using gt_ex by auto
hoelzl@56328
   572
hoelzl@56328
   573
  obtain x where "a < x"
hoelzl@56328
   574
    using gt_ex by auto
hoelzl@56328
   575
  also then have "x \<le> Max {a <..}"
hoelzl@56328
   576
    by fact
wenzelm@60758
   577
  also note \<open>Max {a <..} < y\<close>
hoelzl@56328
   578
  finally have "y \<le> Max { a <..}"
hoelzl@56328
   579
    by fact
wenzelm@60758
   580
  with \<open>Max {a <..} < y\<close> show False by auto
hoelzl@56328
   581
qed
hoelzl@56328
   582
hoelzl@56328
   583
lemma infinite_Ici: "\<not> finite {a :: 'a :: {no_top, linorder} ..}"
hoelzl@56328
   584
  using infinite_Ioi[of a] finite_subset[of "{a <..}" "{a ..}"]
hoelzl@56328
   585
  by (auto simp: subset_eq less_imp_le)
hoelzl@56328
   586
wenzelm@60758
   587
subsubsection \<open>Intersection\<close>
nipkow@32456
   588
nipkow@32456
   589
context linorder
nipkow@32456
   590
begin
nipkow@32456
   591
nipkow@32456
   592
lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"
nipkow@32456
   593
by auto
nipkow@32456
   594
nipkow@32456
   595
lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"
nipkow@32456
   596
by auto
nipkow@32456
   597
nipkow@32456
   598
lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"
nipkow@32456
   599
by auto
nipkow@32456
   600
nipkow@32456
   601
lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"
nipkow@32456
   602
by auto
nipkow@32456
   603
nipkow@32456
   604
lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"
nipkow@32456
   605
by auto
nipkow@32456
   606
nipkow@32456
   607
lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"
nipkow@32456
   608
by auto
nipkow@32456
   609
nipkow@32456
   610
lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
nipkow@32456
   611
by auto
nipkow@32456
   612
nipkow@32456
   613
lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
nipkow@32456
   614
by auto
nipkow@32456
   615
hoelzl@50417
   616
lemma Int_atMost[simp]: "{..a} \<inter> {..b} = {.. min a b}"
hoelzl@50417
   617
  by (auto simp: min_def)
hoelzl@50417
   618
hoelzl@57447
   619
lemma Ioc_disjoint: "{a<..b} \<inter> {c<..d} = {} \<longleftrightarrow> b \<le> a \<or> d \<le> c \<or> b \<le> c \<or> d \<le> a"
hoelzl@57447
   620
  using assms by auto
hoelzl@57447
   621
nipkow@32456
   622
end
nipkow@32456
   623
hoelzl@51329
   624
context complete_lattice
hoelzl@51329
   625
begin
hoelzl@51329
   626
hoelzl@51329
   627
lemma
hoelzl@51329
   628
  shows Sup_atLeast[simp]: "Sup {x ..} = top"
hoelzl@51329
   629
    and Sup_greaterThanAtLeast[simp]: "x < top \<Longrightarrow> Sup {x <..} = top"
hoelzl@51329
   630
    and Sup_atMost[simp]: "Sup {.. y} = y"
hoelzl@51329
   631
    and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"
hoelzl@51329
   632
    and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"
hoelzl@51329
   633
  by (auto intro!: Sup_eqI)
hoelzl@51329
   634
hoelzl@51329
   635
lemma
hoelzl@51329
   636
  shows Inf_atMost[simp]: "Inf {.. x} = bot"
hoelzl@51329
   637
    and Inf_atMostLessThan[simp]: "top < x \<Longrightarrow> Inf {..< x} = bot"
hoelzl@51329
   638
    and Inf_atLeast[simp]: "Inf {x ..} = x"
hoelzl@51329
   639
    and Inf_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Inf { x .. y} = x"
hoelzl@51329
   640
    and Inf_atLeastLessThan[simp]: "x < y \<Longrightarrow> Inf { x ..< y} = x"
hoelzl@51329
   641
  by (auto intro!: Inf_eqI)
hoelzl@51329
   642
hoelzl@51329
   643
end
hoelzl@51329
   644
hoelzl@51329
   645
lemma
hoelzl@53216
   646
  fixes x y :: "'a :: {complete_lattice, dense_linorder}"
hoelzl@51329
   647
  shows Sup_lessThan[simp]: "Sup {..< y} = y"
hoelzl@51329
   648
    and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"
hoelzl@51329
   649
    and Sup_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Sup { x <..< y} = y"
hoelzl@51329
   650
    and Inf_greaterThan[simp]: "Inf {x <..} = x"
hoelzl@51329
   651
    and Inf_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Inf { x <.. y} = x"
hoelzl@51329
   652
    and Inf_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Inf { x <..< y} = x"
hoelzl@51329
   653
  by (auto intro!: Inf_eqI Sup_eqI intro: dense_le dense_le_bounded dense_ge dense_ge_bounded)
nipkow@32456
   654
wenzelm@60758
   655
subsection \<open>Intervals of natural numbers\<close>
paulson@14485
   656
wenzelm@60758
   657
subsubsection \<open>The Constant @{term lessThan}\<close>
paulson@15047
   658
paulson@14485
   659
lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
paulson@14485
   660
by (simp add: lessThan_def)
paulson@14485
   661
paulson@14485
   662
lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
paulson@14485
   663
by (simp add: lessThan_def less_Suc_eq, blast)
paulson@14485
   664
wenzelm@60758
   665
text \<open>The following proof is convenient in induction proofs where
hoelzl@39072
   666
new elements get indices at the beginning. So it is used to transform
wenzelm@60758
   667
@{term "{..<Suc n}"} to @{term "0::nat"} and @{term "{..< n}"}.\<close>
hoelzl@39072
   668
hoelzl@59000
   669
lemma zero_notin_Suc_image: "0 \<notin> Suc ` A"
hoelzl@59000
   670
  by auto
hoelzl@59000
   671
hoelzl@39072
   672
lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"
hoelzl@59000
   673
  by (auto simp: image_iff less_Suc_eq_0_disj)
hoelzl@39072
   674
paulson@14485
   675
lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
paulson@14485
   676
by (simp add: lessThan_def atMost_def less_Suc_eq_le)
paulson@14485
   677
hoelzl@59000
   678
lemma Iic_Suc_eq_insert_0: "{.. Suc n} = insert 0 (Suc ` {.. n})"
hoelzl@59000
   679
  unfolding lessThan_Suc_atMost[symmetric] lessThan_Suc_eq_insert_0[of "Suc n"] ..
hoelzl@59000
   680
paulson@14485
   681
lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
paulson@14485
   682
by blast
paulson@14485
   683
wenzelm@60758
   684
subsubsection \<open>The Constant @{term greaterThan}\<close>
paulson@15047
   685
paulson@14485
   686
lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
paulson@14485
   687
apply (simp add: greaterThan_def)
paulson@14485
   688
apply (blast dest: gr0_conv_Suc [THEN iffD1])
paulson@14485
   689
done
paulson@14485
   690
paulson@14485
   691
lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
paulson@14485
   692
apply (simp add: greaterThan_def)
paulson@14485
   693
apply (auto elim: linorder_neqE)
paulson@14485
   694
done
paulson@14485
   695
paulson@14485
   696
lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
paulson@14485
   697
by blast
paulson@14485
   698
wenzelm@60758
   699
subsubsection \<open>The Constant @{term atLeast}\<close>
paulson@15047
   700
paulson@14485
   701
lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
paulson@14485
   702
by (unfold atLeast_def UNIV_def, simp)
paulson@14485
   703
paulson@14485
   704
lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
paulson@14485
   705
apply (simp add: atLeast_def)
paulson@14485
   706
apply (simp add: Suc_le_eq)
paulson@14485
   707
apply (simp add: order_le_less, blast)
paulson@14485
   708
done
paulson@14485
   709
paulson@14485
   710
lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
paulson@14485
   711
  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
paulson@14485
   712
paulson@14485
   713
lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
paulson@14485
   714
by blast
paulson@14485
   715
wenzelm@60758
   716
subsubsection \<open>The Constant @{term atMost}\<close>
paulson@15047
   717
paulson@14485
   718
lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
paulson@14485
   719
by (simp add: atMost_def)
paulson@14485
   720
paulson@14485
   721
lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
paulson@14485
   722
apply (simp add: atMost_def)
paulson@14485
   723
apply (simp add: less_Suc_eq order_le_less, blast)
paulson@14485
   724
done
paulson@14485
   725
paulson@14485
   726
lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
paulson@14485
   727
by blast
paulson@14485
   728
wenzelm@60758
   729
subsubsection \<open>The Constant @{term atLeastLessThan}\<close>
paulson@15047
   730
wenzelm@60758
   731
text\<open>The orientation of the following 2 rules is tricky. The lhs is
nipkow@24449
   732
defined in terms of the rhs.  Hence the chosen orientation makes sense
nipkow@24449
   733
in this theory --- the reverse orientation complicates proofs (eg
nipkow@24449
   734
nontermination). But outside, when the definition of the lhs is rarely
nipkow@24449
   735
used, the opposite orientation seems preferable because it reduces a
wenzelm@60758
   736
specific concept to a more general one.\<close>
nipkow@28068
   737
paulson@15047
   738
lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
nipkow@15042
   739
by(simp add:lessThan_def atLeastLessThan_def)
nipkow@24449
   740
nipkow@28068
   741
lemma atLeast0AtMost: "{0..n::nat} = {..n}"
nipkow@28068
   742
by(simp add:atMost_def atLeastAtMost_def)
nipkow@28068
   743
haftmann@31998
   744
declare atLeast0LessThan[symmetric, code_unfold]
haftmann@31998
   745
        atLeast0AtMost[symmetric, code_unfold]
nipkow@24449
   746
nipkow@24449
   747
lemma atLeastLessThan0: "{m..<0::nat} = {}"
paulson@15047
   748
by (simp add: atLeastLessThan_def)
nipkow@24449
   749
wenzelm@60758
   750
subsubsection \<open>Intervals of nats with @{term Suc}\<close>
paulson@15047
   751
wenzelm@60758
   752
text\<open>Not a simprule because the RHS is too messy.\<close>
paulson@15047
   753
lemma atLeastLessThanSuc:
paulson@15047
   754
    "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
paulson@15418
   755
by (auto simp add: atLeastLessThan_def)
paulson@15047
   756
paulson@15418
   757
lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
paulson@15047
   758
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   759
(*
paulson@15047
   760
lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
paulson@15047
   761
by (induct k, simp_all add: atLeastLessThanSuc)
paulson@15047
   762
paulson@15047
   763
lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
paulson@15047
   764
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   765
*)
nipkow@15045
   766
lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
paulson@14485
   767
  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   768
paulson@15418
   769
lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
paulson@15418
   770
  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
paulson@14485
   771
    greaterThanAtMost_def)
paulson@14485
   772
paulson@15418
   773
lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
paulson@15418
   774
  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
paulson@14485
   775
    greaterThanLessThan_def)
paulson@14485
   776
nipkow@15554
   777
lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
nipkow@15554
   778
by (auto simp add: atLeastAtMost_def)
nipkow@15554
   779
noschinl@45932
   780
lemma atLeastAtMost_insertL: "m \<le> n \<Longrightarrow> insert m {Suc m..n} = {m ..n}"
noschinl@45932
   781
by auto
noschinl@45932
   782
wenzelm@60758
   783
text \<open>The analogous result is useful on @{typ int}:\<close>
kleing@43157
   784
(* here, because we don't have an own int section *)
kleing@43157
   785
lemma atLeastAtMostPlus1_int_conv:
kleing@43157
   786
  "m <= 1+n \<Longrightarrow> {m..1+n} = insert (1+n) {m..n::int}"
kleing@43157
   787
  by (auto intro: set_eqI)
kleing@43157
   788
paulson@33044
   789
lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"
paulson@33044
   790
  apply (induct k) 
paulson@33044
   791
  apply (simp_all add: atLeastLessThanSuc)   
paulson@33044
   792
  done
paulson@33044
   793
wenzelm@60758
   794
subsubsection \<open>Intervals and numerals\<close>
lp15@57113
   795
wenzelm@60758
   796
lemma lessThan_nat_numeral:  --\<open>Evaluation for specific numerals\<close>
lp15@57113
   797
  "lessThan (numeral k :: nat) = insert (pred_numeral k) (lessThan (pred_numeral k))"
lp15@57113
   798
  by (simp add: numeral_eq_Suc lessThan_Suc)
lp15@57113
   799
wenzelm@60758
   800
lemma atMost_nat_numeral:  --\<open>Evaluation for specific numerals\<close>
lp15@57113
   801
  "atMost (numeral k :: nat) = insert (numeral k) (atMost (pred_numeral k))"
lp15@57113
   802
  by (simp add: numeral_eq_Suc atMost_Suc)
lp15@57113
   803
wenzelm@60758
   804
lemma atLeastLessThan_nat_numeral:  --\<open>Evaluation for specific numerals\<close>
lp15@57113
   805
  "atLeastLessThan m (numeral k :: nat) = 
lp15@57113
   806
     (if m \<le> (pred_numeral k) then insert (pred_numeral k) (atLeastLessThan m (pred_numeral k))
lp15@57113
   807
                 else {})"
lp15@57113
   808
  by (simp add: numeral_eq_Suc atLeastLessThanSuc)
lp15@57113
   809
wenzelm@60758
   810
subsubsection \<open>Image\<close>
nipkow@16733
   811
lp15@60809
   812
lemma image_add_atLeastAtMost [simp]:
lp15@60615
   813
  fixes k ::"'a::linordered_semidom"
lp15@60615
   814
  shows "(\<lambda>n. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
nipkow@16733
   815
proof
nipkow@16733
   816
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   817
next
nipkow@16733
   818
  show "?B \<subseteq> ?A"
nipkow@16733
   819
  proof
nipkow@16733
   820
    fix n assume a: "n : ?B"
lp15@60615
   821
    hence "n - k : {i..j}"
lp15@60615
   822
      by (auto simp: add_le_imp_le_diff add_le_add_imp_diff_le)
lp15@60615
   823
    moreover have "n = (n - k) + k" using a
lp15@60615
   824
    proof -
lp15@60615
   825
      have "k + i \<le> n"
lp15@60615
   826
        by (metis a add.commute atLeastAtMost_iff)
lp15@60615
   827
      hence "k + (n - k) = n"
lp15@60615
   828
        by (metis (no_types) ab_semigroup_add_class.add_ac(1) add_diff_cancel_left' le_add_diff_inverse)
lp15@60615
   829
      thus ?thesis
lp15@60615
   830
        by (simp add: add.commute)
lp15@60615
   831
    qed
nipkow@16733
   832
    ultimately show "n : ?A" by blast
nipkow@16733
   833
  qed
nipkow@16733
   834
qed
nipkow@16733
   835
lp15@60809
   836
lemma image_diff_atLeastAtMost [simp]:
lp15@60809
   837
  fixes d::"'a::linordered_idom" shows "(op - d ` {a..b}) = {d-b..d-a}"
lp15@60809
   838
  apply auto
lp15@60809
   839
  apply (rule_tac x="d-x" in rev_image_eqI, auto)
lp15@60809
   840
  done
lp15@60809
   841
lp15@60809
   842
lemma image_mult_atLeastAtMost [simp]:
lp15@60809
   843
  fixes d::"'a::linordered_field"
lp15@60809
   844
  assumes "d>0" shows "(op * d ` {a..b}) = {d*a..d*b}"
lp15@60809
   845
  using assms
lp15@60809
   846
  by (auto simp: field_simps mult_le_cancel_right intro: rev_image_eqI [where x="x/d" for x])
lp15@60809
   847
lp15@60809
   848
lemma image_affinity_atLeastAtMost:
lp15@60809
   849
  fixes c :: "'a::linordered_field"
lp15@60809
   850
  shows "((\<lambda>x. m*x + c) ` {a..b}) = (if {a..b}={} then {}
lp15@60809
   851
            else if 0 \<le> m then {m*a + c .. m *b + c}
lp15@60809
   852
            else {m*b + c .. m*a + c})"
lp15@60809
   853
  apply (case_tac "m=0", auto simp: mult_le_cancel_left)
lp15@60809
   854
  apply (rule_tac x="inverse m*(x-c)" in rev_image_eqI, auto simp: field_simps)
lp15@60809
   855
  apply (rule_tac x="inverse m*(x-c)" in rev_image_eqI, auto simp: field_simps)
lp15@60809
   856
  done
lp15@60809
   857
lp15@60809
   858
lemma image_affinity_atLeastAtMost_diff:
lp15@60809
   859
  fixes c :: "'a::linordered_field"
lp15@60809
   860
  shows "((\<lambda>x. m*x - c) ` {a..b}) = (if {a..b}={} then {}
lp15@60809
   861
            else if 0 \<le> m then {m*a - c .. m*b - c}
lp15@60809
   862
            else {m*b - c .. m*a - c})"
lp15@60809
   863
  using image_affinity_atLeastAtMost [of m "-c" a b]
lp15@60809
   864
  by simp
lp15@60809
   865
lp15@60809
   866
lemma image_affinity_atLeastAtMost_div_diff:
lp15@60809
   867
  fixes c :: "'a::linordered_field"
lp15@60809
   868
  shows "((\<lambda>x. x/m - c) ` {a..b}) = (if {a..b}={} then {}
lp15@60809
   869
            else if 0 \<le> m then {a/m - c .. b/m - c}
lp15@60809
   870
            else {b/m - c .. a/m - c})"
lp15@60809
   871
  using image_affinity_atLeastAtMost_diff [of "inverse m" c a b]
lp15@60809
   872
  by (simp add: field_class.field_divide_inverse algebra_simps)
lp15@60809
   873
nipkow@16733
   874
lemma image_add_atLeastLessThan:
nipkow@16733
   875
  "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
nipkow@16733
   876
proof
nipkow@16733
   877
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   878
next
nipkow@16733
   879
  show "?B \<subseteq> ?A"
nipkow@16733
   880
  proof
nipkow@16733
   881
    fix n assume a: "n : ?B"
webertj@20217
   882
    hence "n - k : {i..<j}" by auto
nipkow@16733
   883
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   884
    ultimately show "n : ?A" by blast
nipkow@16733
   885
  qed
nipkow@16733
   886
qed
nipkow@16733
   887
nipkow@16733
   888
corollary image_Suc_atLeastAtMost[simp]:
nipkow@16733
   889
  "Suc ` {i..j} = {Suc i..Suc j}"
huffman@30079
   890
using image_add_atLeastAtMost[where k="Suc 0"] by simp
nipkow@16733
   891
nipkow@16733
   892
corollary image_Suc_atLeastLessThan[simp]:
nipkow@16733
   893
  "Suc ` {i..<j} = {Suc i..<Suc j}"
huffman@30079
   894
using image_add_atLeastLessThan[where k="Suc 0"] by simp
nipkow@16733
   895
nipkow@16733
   896
lemma image_add_int_atLeastLessThan:
nipkow@16733
   897
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
nipkow@16733
   898
  apply (auto simp add: image_def)
nipkow@16733
   899
  apply (rule_tac x = "x - l" in bexI)
nipkow@16733
   900
  apply auto
nipkow@16733
   901
  done
nipkow@16733
   902
hoelzl@37664
   903
lemma image_minus_const_atLeastLessThan_nat:
hoelzl@37664
   904
  fixes c :: nat
hoelzl@37664
   905
  shows "(\<lambda>i. i - c) ` {x ..< y} =
hoelzl@37664
   906
      (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"
hoelzl@37664
   907
    (is "_ = ?right")
hoelzl@37664
   908
proof safe
hoelzl@37664
   909
  fix a assume a: "a \<in> ?right"
hoelzl@37664
   910
  show "a \<in> (\<lambda>i. i - c) ` {x ..< y}"
hoelzl@37664
   911
  proof cases
hoelzl@37664
   912
    assume "c < y" with a show ?thesis
hoelzl@37664
   913
      by (auto intro!: image_eqI[of _ _ "a + c"])
hoelzl@37664
   914
  next
hoelzl@37664
   915
    assume "\<not> c < y" with a show ?thesis
hoelzl@37664
   916
      by (auto intro!: image_eqI[of _ _ x] split: split_if_asm)
hoelzl@37664
   917
  qed
hoelzl@37664
   918
qed auto
hoelzl@37664
   919
Andreas@51152
   920
lemma image_int_atLeastLessThan: "int ` {a..<b} = {int a..<int b}"
wenzelm@55143
   921
  by (auto intro!: image_eqI [where x = "nat x" for x])
Andreas@51152
   922
hoelzl@35580
   923
context ordered_ab_group_add
hoelzl@35580
   924
begin
hoelzl@35580
   925
hoelzl@35580
   926
lemma
hoelzl@35580
   927
  fixes x :: 'a
hoelzl@35580
   928
  shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}"
hoelzl@35580
   929
  and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}"
hoelzl@35580
   930
proof safe
hoelzl@35580
   931
  fix y assume "y < -x"
hoelzl@35580
   932
  hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp
hoelzl@35580
   933
  have "- (-y) \<in> uminus ` {x<..}"
hoelzl@35580
   934
    by (rule imageI) (simp add: *)
hoelzl@35580
   935
  thus "y \<in> uminus ` {x<..}" by simp
hoelzl@35580
   936
next
hoelzl@35580
   937
  fix y assume "y \<le> -x"
hoelzl@35580
   938
  have "- (-y) \<in> uminus ` {x..}"
wenzelm@60758
   939
    by (rule imageI) (insert \<open>y \<le> -x\<close>[THEN le_imp_neg_le], simp)
hoelzl@35580
   940
  thus "y \<in> uminus ` {x..}" by simp
hoelzl@35580
   941
qed simp_all
hoelzl@35580
   942
hoelzl@35580
   943
lemma
hoelzl@35580
   944
  fixes x :: 'a
hoelzl@35580
   945
  shows image_uminus_lessThan[simp]: "uminus ` {..<x} = {-x<..}"
hoelzl@35580
   946
  and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"
hoelzl@35580
   947
proof -
hoelzl@35580
   948
  have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"
hoelzl@35580
   949
    and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all
hoelzl@35580
   950
  thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"
hoelzl@35580
   951
    by (simp_all add: image_image
hoelzl@35580
   952
        del: image_uminus_greaterThan image_uminus_atLeast)
hoelzl@35580
   953
qed
hoelzl@35580
   954
hoelzl@35580
   955
lemma
hoelzl@35580
   956
  fixes x :: 'a
hoelzl@35580
   957
  shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"
hoelzl@35580
   958
  and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"
hoelzl@35580
   959
  and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"
hoelzl@35580
   960
  and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"
hoelzl@35580
   961
  by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def
hoelzl@35580
   962
      greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)
hoelzl@35580
   963
end
nipkow@16733
   964
wenzelm@60758
   965
subsubsection \<open>Finiteness\<close>
paulson@14485
   966
nipkow@15045
   967
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
paulson@14485
   968
  by (induct k) (simp_all add: lessThan_Suc)
paulson@14485
   969
paulson@14485
   970
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
paulson@14485
   971
  by (induct k) (simp_all add: atMost_Suc)
paulson@14485
   972
paulson@14485
   973
lemma finite_greaterThanLessThan [iff]:
nipkow@15045
   974
  fixes l :: nat shows "finite {l<..<u}"
paulson@14485
   975
by (simp add: greaterThanLessThan_def)
paulson@14485
   976
paulson@14485
   977
lemma finite_atLeastLessThan [iff]:
nipkow@15045
   978
  fixes l :: nat shows "finite {l..<u}"
paulson@14485
   979
by (simp add: atLeastLessThan_def)
paulson@14485
   980
paulson@14485
   981
lemma finite_greaterThanAtMost [iff]:
nipkow@15045
   982
  fixes l :: nat shows "finite {l<..u}"
paulson@14485
   983
by (simp add: greaterThanAtMost_def)
paulson@14485
   984
paulson@14485
   985
lemma finite_atLeastAtMost [iff]:
paulson@14485
   986
  fixes l :: nat shows "finite {l..u}"
paulson@14485
   987
by (simp add: atLeastAtMost_def)
paulson@14485
   988
wenzelm@60758
   989
text \<open>A bounded set of natural numbers is finite.\<close>
paulson@14485
   990
lemma bounded_nat_set_is_finite:
nipkow@24853
   991
  "(ALL i:N. i < (n::nat)) ==> finite N"
nipkow@28068
   992
apply (rule finite_subset)
nipkow@28068
   993
 apply (rule_tac [2] finite_lessThan, auto)
nipkow@28068
   994
done
nipkow@28068
   995
wenzelm@60758
   996
text \<open>A set of natural numbers is finite iff it is bounded.\<close>
nipkow@31044
   997
lemma finite_nat_set_iff_bounded:
nipkow@31044
   998
  "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
nipkow@31044
   999
proof
nipkow@31044
  1000
  assume f:?F  show ?B
wenzelm@60758
  1001
    using Max_ge[OF \<open>?F\<close>, simplified less_Suc_eq_le[symmetric]] by blast
nipkow@31044
  1002
next
wenzelm@60758
  1003
  assume ?B show ?F using \<open>?B\<close> by(blast intro:bounded_nat_set_is_finite)
nipkow@31044
  1004
qed
nipkow@31044
  1005
nipkow@31044
  1006
lemma finite_nat_set_iff_bounded_le:
nipkow@31044
  1007
  "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
nipkow@31044
  1008
apply(simp add:finite_nat_set_iff_bounded)
nipkow@31044
  1009
apply(blast dest:less_imp_le_nat le_imp_less_Suc)
nipkow@31044
  1010
done
nipkow@31044
  1011
nipkow@28068
  1012
lemma finite_less_ub:
nipkow@28068
  1013
     "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
nipkow@28068
  1014
by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
paulson@14485
  1015
hoelzl@56328
  1016
wenzelm@60758
  1017
text\<open>Any subset of an interval of natural numbers the size of the
wenzelm@60758
  1018
subset is exactly that interval.\<close>
nipkow@24853
  1019
nipkow@24853
  1020
lemma subset_card_intvl_is_intvl:
blanchet@55085
  1021
  assumes "A \<subseteq> {k..<k + card A}"
blanchet@55085
  1022
  shows "A = {k..<k + card A}"
wenzelm@53374
  1023
proof (cases "finite A")
wenzelm@53374
  1024
  case True
wenzelm@53374
  1025
  from this and assms show ?thesis
wenzelm@53374
  1026
  proof (induct A rule: finite_linorder_max_induct)
nipkow@24853
  1027
    case empty thus ?case by auto
nipkow@24853
  1028
  next
nipkow@33434
  1029
    case (insert b A)
wenzelm@53374
  1030
    hence *: "b \<notin> A" by auto
blanchet@55085
  1031
    with insert have "A <= {k..<k + card A}" and "b = k + card A"
wenzelm@53374
  1032
      by fastforce+
wenzelm@53374
  1033
    with insert * show ?case by auto
nipkow@24853
  1034
  qed
nipkow@24853
  1035
next
wenzelm@53374
  1036
  case False
wenzelm@53374
  1037
  with assms show ?thesis by simp
nipkow@24853
  1038
qed
nipkow@24853
  1039
nipkow@24853
  1040
wenzelm@60758
  1041
subsubsection \<open>Proving Inclusions and Equalities between Unions\<close>
paulson@32596
  1042
nipkow@36755
  1043
lemma UN_le_eq_Un0:
nipkow@36755
  1044
  "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> M 0" (is "?A = ?B")
nipkow@36755
  1045
proof
nipkow@36755
  1046
  show "?A <= ?B"
nipkow@36755
  1047
  proof
nipkow@36755
  1048
    fix x assume "x : ?A"
nipkow@36755
  1049
    then obtain i where i: "i\<le>n" "x : M i" by auto
nipkow@36755
  1050
    show "x : ?B"
nipkow@36755
  1051
    proof(cases i)
nipkow@36755
  1052
      case 0 with i show ?thesis by simp
nipkow@36755
  1053
    next
nipkow@36755
  1054
      case (Suc j) with i show ?thesis by auto
nipkow@36755
  1055
    qed
nipkow@36755
  1056
  qed
nipkow@36755
  1057
next
nipkow@36755
  1058
  show "?B <= ?A" by auto
nipkow@36755
  1059
qed
nipkow@36755
  1060
nipkow@36755
  1061
lemma UN_le_add_shift:
nipkow@36755
  1062
  "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")
nipkow@36755
  1063
proof
nipkow@44890
  1064
  show "?A <= ?B" by fastforce
nipkow@36755
  1065
next
nipkow@36755
  1066
  show "?B <= ?A"
nipkow@36755
  1067
  proof
nipkow@36755
  1068
    fix x assume "x : ?B"
nipkow@36755
  1069
    then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto
nipkow@36755
  1070
    hence "i-k\<le>n & x : M((i-k)+k)" by auto
nipkow@36755
  1071
    thus "x : ?A" by blast
nipkow@36755
  1072
  qed
nipkow@36755
  1073
qed
nipkow@36755
  1074
paulson@32596
  1075
lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"
paulson@32596
  1076
  by (auto simp add: atLeast0LessThan) 
paulson@32596
  1077
paulson@32596
  1078
lemma UN_finite_subset: "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"
paulson@32596
  1079
  by (subst UN_UN_finite_eq [symmetric]) blast
paulson@32596
  1080
paulson@33044
  1081
lemma UN_finite2_subset: 
paulson@33044
  1082
     "(!!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
  1083
  apply (rule UN_finite_subset)
paulson@33044
  1084
  apply (subst UN_UN_finite_eq [symmetric, of B]) 
paulson@33044
  1085
  apply blast
paulson@33044
  1086
  done
paulson@32596
  1087
paulson@32596
  1088
lemma UN_finite2_eq:
paulson@33044
  1089
  "(!!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
  1090
  apply (rule subset_antisym)
paulson@33044
  1091
   apply (rule UN_finite2_subset, blast)
paulson@33044
  1092
 apply (rule UN_finite2_subset [where k=k])
huffman@35216
  1093
 apply (force simp add: atLeastLessThan_add_Un [of 0])
paulson@33044
  1094
 done
paulson@32596
  1095
paulson@32596
  1096
wenzelm@60758
  1097
subsubsection \<open>Cardinality\<close>
paulson@14485
  1098
nipkow@15045
  1099
lemma card_lessThan [simp]: "card {..<u} = u"
paulson@15251
  1100
  by (induct u, simp_all add: lessThan_Suc)
paulson@14485
  1101
paulson@14485
  1102
lemma card_atMost [simp]: "card {..u} = Suc u"
paulson@14485
  1103
  by (simp add: lessThan_Suc_atMost [THEN sym])
paulson@14485
  1104
nipkow@15045
  1105
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
lp15@57113
  1106
proof -
lp15@57113
  1107
  have "{l..<u} = (%x. x + l) ` {..<u-l}"
lp15@57113
  1108
    apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
lp15@57113
  1109
    apply (rule_tac x = "x - l" in exI)
lp15@57113
  1110
    apply arith
lp15@57113
  1111
    done
lp15@57113
  1112
  then have "card {l..<u} = card {..<u-l}"
lp15@57113
  1113
    by (simp add: card_image inj_on_def)
lp15@57113
  1114
  then show ?thesis
lp15@57113
  1115
    by simp
lp15@57113
  1116
qed
paulson@14485
  1117
paulson@15418
  1118
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
paulson@14485
  1119
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
paulson@14485
  1120
paulson@15418
  1121
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
paulson@14485
  1122
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
paulson@14485
  1123
nipkow@15045
  1124
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
paulson@14485
  1125
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
paulson@14485
  1126
nipkow@26105
  1127
lemma ex_bij_betw_nat_finite:
nipkow@26105
  1128
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
nipkow@26105
  1129
apply(drule finite_imp_nat_seg_image_inj_on)
nipkow@26105
  1130
apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
nipkow@26105
  1131
done
nipkow@26105
  1132
nipkow@26105
  1133
lemma ex_bij_betw_finite_nat:
nipkow@26105
  1134
  "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
nipkow@26105
  1135
by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
nipkow@26105
  1136
nipkow@31438
  1137
lemma finite_same_card_bij:
nipkow@31438
  1138
  "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"
nipkow@31438
  1139
apply(drule ex_bij_betw_finite_nat)
nipkow@31438
  1140
apply(drule ex_bij_betw_nat_finite)
nipkow@31438
  1141
apply(auto intro!:bij_betw_trans)
nipkow@31438
  1142
done
nipkow@31438
  1143
nipkow@31438
  1144
lemma ex_bij_betw_nat_finite_1:
nipkow@31438
  1145
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
nipkow@31438
  1146
by (rule finite_same_card_bij) auto
nipkow@31438
  1147
hoelzl@40703
  1148
lemma bij_betw_iff_card:
hoelzl@40703
  1149
  assumes FIN: "finite A" and FIN': "finite B"
hoelzl@40703
  1150
  shows BIJ: "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"
hoelzl@40703
  1151
using assms
hoelzl@40703
  1152
proof(auto simp add: bij_betw_same_card)
hoelzl@40703
  1153
  assume *: "card A = card B"
hoelzl@40703
  1154
  obtain f where "bij_betw f A {0 ..< card A}"
hoelzl@40703
  1155
  using FIN ex_bij_betw_finite_nat by blast
hoelzl@40703
  1156
  moreover obtain g where "bij_betw g {0 ..< card B} B"
hoelzl@40703
  1157
  using FIN' ex_bij_betw_nat_finite by blast
hoelzl@40703
  1158
  ultimately have "bij_betw (g o f) A B"
hoelzl@40703
  1159
  using * by (auto simp add: bij_betw_trans)
hoelzl@40703
  1160
  thus "(\<exists>f. bij_betw f A B)" by blast
hoelzl@40703
  1161
qed
hoelzl@40703
  1162
hoelzl@40703
  1163
lemma inj_on_iff_card_le:
hoelzl@40703
  1164
  assumes FIN: "finite A" and FIN': "finite B"
hoelzl@40703
  1165
  shows "(\<exists>f. inj_on f A \<and> f ` A \<le> B) = (card A \<le> card B)"
hoelzl@40703
  1166
proof (safe intro!: card_inj_on_le)
hoelzl@40703
  1167
  assume *: "card A \<le> card B"
hoelzl@40703
  1168
  obtain f where 1: "inj_on f A" and 2: "f ` A = {0 ..< card A}"
hoelzl@40703
  1169
  using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force
hoelzl@40703
  1170
  moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g ` {0 ..< card B} = B"
hoelzl@40703
  1171
  using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force
hoelzl@40703
  1172
  ultimately have "inj_on g (f ` A)" using subset_inj_on[of g _ "f ` A"] * by force
hoelzl@40703
  1173
  hence "inj_on (g o f) A" using 1 comp_inj_on by blast
hoelzl@40703
  1174
  moreover
hoelzl@40703
  1175
  {have "{0 ..< card A} \<le> {0 ..< card B}" using * by force
hoelzl@40703
  1176
   with 2 have "f ` A  \<le> {0 ..< card B}" by blast
hoelzl@40703
  1177
   hence "(g o f) ` A \<le> B" unfolding comp_def using 3 by force
hoelzl@40703
  1178
  }
hoelzl@40703
  1179
  ultimately show "(\<exists>f. inj_on f A \<and> f ` A \<le> B)" by blast
hoelzl@40703
  1180
qed (insert assms, auto)
nipkow@26105
  1181
wenzelm@60758
  1182
subsection \<open>Intervals of integers\<close>
paulson@14485
  1183
nipkow@15045
  1184
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
paulson@14485
  1185
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
paulson@14485
  1186
paulson@15418
  1187
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
paulson@14485
  1188
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
paulson@14485
  1189
paulson@15418
  1190
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
paulson@15418
  1191
    "{l+1..<u} = {l<..<u::int}"
paulson@14485
  1192
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
paulson@14485
  1193
wenzelm@60758
  1194
subsubsection \<open>Finiteness\<close>
paulson@14485
  1195
paulson@15418
  1196
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
nipkow@15045
  1197
    {(0::int)..<u} = int ` {..<nat u}"
paulson@14485
  1198
  apply (unfold image_def lessThan_def)
paulson@14485
  1199
  apply auto
paulson@14485
  1200
  apply (rule_tac x = "nat x" in exI)
huffman@35216
  1201
  apply (auto simp add: zless_nat_eq_int_zless [THEN sym])
paulson@14485
  1202
  done
paulson@14485
  1203
nipkow@15045
  1204
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
wenzelm@47988
  1205
  apply (cases "0 \<le> u")
paulson@14485
  1206
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
  1207
  apply (rule finite_imageI)
paulson@14485
  1208
  apply auto
paulson@14485
  1209
  done
paulson@14485
  1210
nipkow@15045
  1211
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
nipkow@15045
  1212
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
  1213
  apply (erule subst)
paulson@14485
  1214
  apply (rule finite_imageI)
paulson@14485
  1215
  apply (rule finite_atLeastZeroLessThan_int)
nipkow@16733
  1216
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
  1217
  done
paulson@14485
  1218
paulson@15418
  1219
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
paulson@14485
  1220
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
paulson@14485
  1221
paulson@15418
  1222
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
paulson@14485
  1223
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
  1224
paulson@15418
  1225
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
paulson@14485
  1226
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
  1227
nipkow@24853
  1228
wenzelm@60758
  1229
subsubsection \<open>Cardinality\<close>
paulson@14485
  1230
nipkow@15045
  1231
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
wenzelm@47988
  1232
  apply (cases "0 \<le> u")
paulson@14485
  1233
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
  1234
  apply (subst card_image)
paulson@14485
  1235
  apply (auto simp add: inj_on_def)
paulson@14485
  1236
  done
paulson@14485
  1237
nipkow@15045
  1238
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
nipkow@15045
  1239
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
paulson@14485
  1240
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
nipkow@15045
  1241
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
  1242
  apply (erule subst)
paulson@14485
  1243
  apply (rule card_image)
paulson@14485
  1244
  apply (simp add: inj_on_def)
nipkow@16733
  1245
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
  1246
  done
paulson@14485
  1247
paulson@14485
  1248
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
nipkow@29667
  1249
apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
nipkow@29667
  1250
apply (auto simp add: algebra_simps)
nipkow@29667
  1251
done
paulson@14485
  1252
paulson@15418
  1253
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
nipkow@29667
  1254
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
  1255
nipkow@15045
  1256
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
nipkow@29667
  1257
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
  1258
bulwahn@27656
  1259
lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
bulwahn@27656
  1260
proof -
bulwahn@27656
  1261
  have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
bulwahn@27656
  1262
  with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
bulwahn@27656
  1263
qed
bulwahn@27656
  1264
bulwahn@27656
  1265
lemma card_less:
bulwahn@27656
  1266
assumes zero_in_M: "0 \<in> M"
bulwahn@27656
  1267
shows "card {k \<in> M. k < Suc i} \<noteq> 0"
bulwahn@27656
  1268
proof -
bulwahn@27656
  1269
  from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
bulwahn@27656
  1270
  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
bulwahn@27656
  1271
qed
bulwahn@27656
  1272
bulwahn@27656
  1273
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
  1274
apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])
bulwahn@27656
  1275
apply auto
bulwahn@27656
  1276
apply (rule inj_on_diff_nat)
bulwahn@27656
  1277
apply auto
bulwahn@27656
  1278
apply (case_tac x)
bulwahn@27656
  1279
apply auto
bulwahn@27656
  1280
apply (case_tac xa)
bulwahn@27656
  1281
apply auto
bulwahn@27656
  1282
apply (case_tac xa)
bulwahn@27656
  1283
apply auto
bulwahn@27656
  1284
done
bulwahn@27656
  1285
bulwahn@27656
  1286
lemma card_less_Suc:
bulwahn@27656
  1287
  assumes zero_in_M: "0 \<in> M"
bulwahn@27656
  1288
    shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
bulwahn@27656
  1289
proof -
bulwahn@27656
  1290
  from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
bulwahn@27656
  1291
  hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
bulwahn@27656
  1292
    by (auto simp only: insert_Diff)
bulwahn@27656
  1293
  have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
lp15@57113
  1294
  from finite_M_bounded_by_nat[of "\<lambda>x. x \<in> M" "Suc i"] 
lp15@57113
  1295
  have "Suc (card {k. Suc k \<in> M \<and> k < i}) = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"
bulwahn@27656
  1296
    apply (subst card_insert)
bulwahn@27656
  1297
    apply simp_all
bulwahn@27656
  1298
    apply (subst b)
bulwahn@27656
  1299
    apply (subst card_less_Suc2[symmetric])
bulwahn@27656
  1300
    apply simp_all
bulwahn@27656
  1301
    done
bulwahn@27656
  1302
  with c show ?thesis by simp
bulwahn@27656
  1303
qed
bulwahn@27656
  1304
paulson@14485
  1305
wenzelm@60758
  1306
subsection \<open>Lemmas useful with the summation operator setsum\<close>
paulson@13850
  1307
wenzelm@60758
  1308
text \<open>For examples, see Algebra/poly/UnivPoly2.thy\<close>
ballarin@13735
  1309
wenzelm@60758
  1310
subsubsection \<open>Disjoint Unions\<close>
ballarin@13735
  1311
wenzelm@60758
  1312
text \<open>Singletons and open intervals\<close>
ballarin@13735
  1313
ballarin@13735
  1314
lemma ivl_disj_un_singleton:
nipkow@15045
  1315
  "{l::'a::linorder} Un {l<..} = {l..}"
nipkow@15045
  1316
  "{..<u} Un {u::'a::linorder} = {..u}"
nipkow@15045
  1317
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
nipkow@15045
  1318
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
nipkow@15045
  1319
  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
nipkow@15045
  1320
  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
ballarin@14398
  1321
by auto
ballarin@13735
  1322
wenzelm@60758
  1323
text \<open>One- and two-sided intervals\<close>
ballarin@13735
  1324
ballarin@13735
  1325
lemma ivl_disj_un_one:
nipkow@15045
  1326
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
nipkow@15045
  1327
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
nipkow@15045
  1328
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
nipkow@15045
  1329
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
nipkow@15045
  1330
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
nipkow@15045
  1331
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
nipkow@15045
  1332
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
nipkow@15045
  1333
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
ballarin@14398
  1334
by auto
ballarin@13735
  1335
wenzelm@60758
  1336
text \<open>Two- and two-sided intervals\<close>
ballarin@13735
  1337
ballarin@13735
  1338
lemma ivl_disj_un_two:
nipkow@15045
  1339
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
nipkow@15045
  1340
  "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
nipkow@15045
  1341
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
nipkow@15045
  1342
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
nipkow@15045
  1343
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
nipkow@15045
  1344
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
nipkow@15045
  1345
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
nipkow@15045
  1346
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
ballarin@14398
  1347
by auto
ballarin@13735
  1348
lp15@60150
  1349
lemma ivl_disj_un_two_touch:
lp15@60150
  1350
  "[| (l::'a::linorder) < m; m < u |] ==> {l<..m} Un {m..<u} = {l<..<u}"
lp15@60150
  1351
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m..<u} = {l..<u}"
lp15@60150
  1352
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..m} Un {m..u} = {l<..u}"
lp15@60150
  1353
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m..u} = {l..u}"
lp15@60150
  1354
by auto
lp15@60150
  1355
lp15@60150
  1356
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two ivl_disj_un_two_touch
ballarin@13735
  1357
wenzelm@60758
  1358
subsubsection \<open>Disjoint Intersections\<close>
ballarin@13735
  1359
wenzelm@60758
  1360
text \<open>One- and two-sided intervals\<close>
ballarin@13735
  1361
ballarin@13735
  1362
lemma ivl_disj_int_one:
nipkow@15045
  1363
  "{..l::'a::order} Int {l<..<u} = {}"
nipkow@15045
  1364
  "{..<l} Int {l..<u} = {}"
nipkow@15045
  1365
  "{..l} Int {l<..u} = {}"
nipkow@15045
  1366
  "{..<l} Int {l..u} = {}"
nipkow@15045
  1367
  "{l<..u} Int {u<..} = {}"
nipkow@15045
  1368
  "{l<..<u} Int {u..} = {}"
nipkow@15045
  1369
  "{l..u} Int {u<..} = {}"
nipkow@15045
  1370
  "{l..<u} Int {u..} = {}"
ballarin@14398
  1371
  by auto
ballarin@13735
  1372
wenzelm@60758
  1373
text \<open>Two- and two-sided intervals\<close>
ballarin@13735
  1374
ballarin@13735
  1375
lemma ivl_disj_int_two:
nipkow@15045
  1376
  "{l::'a::order<..<m} Int {m..<u} = {}"
nipkow@15045
  1377
  "{l<..m} Int {m<..<u} = {}"
nipkow@15045
  1378
  "{l..<m} Int {m..<u} = {}"
nipkow@15045
  1379
  "{l..m} Int {m<..<u} = {}"
nipkow@15045
  1380
  "{l<..<m} Int {m..u} = {}"
nipkow@15045
  1381
  "{l<..m} Int {m<..u} = {}"
nipkow@15045
  1382
  "{l..<m} Int {m..u} = {}"
nipkow@15045
  1383
  "{l..m} Int {m<..u} = {}"
ballarin@14398
  1384
  by auto
ballarin@13735
  1385
nipkow@32456
  1386
lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two
ballarin@13735
  1387
wenzelm@60758
  1388
subsubsection \<open>Some Differences\<close>
nipkow@15542
  1389
nipkow@15542
  1390
lemma ivl_diff[simp]:
nipkow@15542
  1391
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
nipkow@15542
  1392
by(auto)
nipkow@15542
  1393
hoelzl@56194
  1394
lemma (in linorder) lessThan_minus_lessThan [simp]:
hoelzl@56194
  1395
  "{..< n} - {..< m} = {m ..< n}"
hoelzl@56194
  1396
  by auto
hoelzl@56194
  1397
paulson@60762
  1398
lemma (in linorder) atLeastAtMost_diff_ends:
paulson@60762
  1399
  "{a..b} - {a, b} = {a<..<b}"
paulson@60762
  1400
  by auto
paulson@60762
  1401
nipkow@15542
  1402
wenzelm@60758
  1403
subsubsection \<open>Some Subset Conditions\<close>
nipkow@15542
  1404
blanchet@54147
  1405
lemma ivl_subset [simp]:
nipkow@15542
  1406
 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
nipkow@15542
  1407
apply(auto simp:linorder_not_le)
nipkow@15542
  1408
apply(rule ccontr)
nipkow@15542
  1409
apply(insert linorder_le_less_linear[of i n])
nipkow@15542
  1410
apply(clarsimp simp:linorder_not_le)
nipkow@44890
  1411
apply(fastforce)
nipkow@15542
  1412
done
nipkow@15542
  1413
nipkow@15041
  1414
wenzelm@60758
  1415
subsection \<open>Summation indexed over intervals\<close>
nipkow@15042
  1416
nipkow@15042
  1417
syntax
nipkow@15042
  1418
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
  1419
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
  1420
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
nipkow@16052
  1421
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
nipkow@15042
  1422
syntax (xsymbols)
nipkow@15042
  1423
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
  1424
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
  1425
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
  1426
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15042
  1427
syntax (HTML output)
nipkow@15042
  1428
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
  1429
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
  1430
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
  1431
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15056
  1432
syntax (latex_sum output)
nipkow@15052
  1433
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
  1434
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052
  1435
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
  1436
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@16052
  1437
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
  1438
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15052
  1439
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
  1440
 ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15041
  1441
nipkow@15048
  1442
translations
nipkow@28853
  1443
  "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
nipkow@28853
  1444
  "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
nipkow@28853
  1445
  "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
nipkow@28853
  1446
  "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
nipkow@15041
  1447
wenzelm@60758
  1448
text\<open>The above introduces some pretty alternative syntaxes for
nipkow@15056
  1449
summation over intervals:
nipkow@15052
  1450
\begin{center}
nipkow@15052
  1451
\begin{tabular}{lll}
nipkow@15056
  1452
Old & New & \LaTeX\\
nipkow@15056
  1453
@{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
  1454
@{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
  1455
@{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
nipkow@15056
  1456
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
nipkow@15052
  1457
\end{tabular}
nipkow@15052
  1458
\end{center}
nipkow@15056
  1459
The left column shows the term before introduction of the new syntax,
nipkow@15056
  1460
the middle column shows the new (default) syntax, and the right column
nipkow@15056
  1461
shows a special syntax. The latter is only meaningful for latex output
nipkow@15056
  1462
and has to be activated explicitly by setting the print mode to
wenzelm@21502
  1463
@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
nipkow@15056
  1464
antiquotations). It is not the default \LaTeX\ output because it only
nipkow@15056
  1465
works well with italic-style formulae, not tt-style.
nipkow@15052
  1466
nipkow@15052
  1467
Note that for uniformity on @{typ nat} it is better to use
nipkow@15052
  1468
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
nipkow@15052
  1469
not provide all lemmas available for @{term"{m..<n}"} also in the
wenzelm@60758
  1470
special form for @{term"{..<n}"}.\<close>
nipkow@15052
  1471
wenzelm@60758
  1472
text\<open>This congruence rule should be used for sums over intervals as
haftmann@57418
  1473
the standard theorem @{text[source]setsum.cong} does not work well
nipkow@15542
  1474
with the simplifier who adds the unsimplified premise @{term"x:B"} to
wenzelm@60758
  1475
the context.\<close>
nipkow@15542
  1476
nipkow@15542
  1477
lemma setsum_ivl_cong:
nipkow@15542
  1478
 "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
nipkow@15542
  1479
 setsum f {a..<b} = setsum g {c..<d}"
haftmann@57418
  1480
by(rule setsum.cong, simp_all)
nipkow@15041
  1481
nipkow@16041
  1482
(* FIXME why are the following simp rules but the corresponding eqns
nipkow@16041
  1483
on intervals are not? *)
nipkow@16041
  1484
nipkow@16052
  1485
lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
haftmann@57514
  1486
by (simp add:atMost_Suc ac_simps)
nipkow@16052
  1487
nipkow@16041
  1488
lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
haftmann@57514
  1489
by (simp add:lessThan_Suc ac_simps)
nipkow@15041
  1490
nipkow@15911
  1491
lemma setsum_cl_ivl_Suc[simp]:
nipkow@15561
  1492
  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
haftmann@57514
  1493
by (auto simp:ac_simps atLeastAtMostSuc_conv)
nipkow@15561
  1494
nipkow@15911
  1495
lemma setsum_op_ivl_Suc[simp]:
nipkow@15561
  1496
  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
haftmann@57514
  1497
by (auto simp:ac_simps atLeastLessThanSuc)
nipkow@16041
  1498
(*
nipkow@15561
  1499
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
nipkow@15561
  1500
    (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
haftmann@57514
  1501
by (auto simp:ac_simps atLeastAtMostSuc_conv)
nipkow@16041
  1502
*)
nipkow@28068
  1503
nipkow@28068
  1504
lemma setsum_head:
nipkow@28068
  1505
  fixes n :: nat
nipkow@28068
  1506
  assumes mn: "m <= n" 
nipkow@28068
  1507
  shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
nipkow@28068
  1508
proof -
nipkow@28068
  1509
  from mn
nipkow@28068
  1510
  have "{m..n} = {m} \<union> {m<..n}"
nipkow@28068
  1511
    by (auto intro: ivl_disj_un_singleton)
nipkow@28068
  1512
  hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
nipkow@28068
  1513
    by (simp add: atLeast0LessThan)
nipkow@28068
  1514
  also have "\<dots> = ?rhs" by simp
nipkow@28068
  1515
  finally show ?thesis .
nipkow@28068
  1516
qed
nipkow@28068
  1517
nipkow@28068
  1518
lemma setsum_head_Suc:
nipkow@28068
  1519
  "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
nipkow@28068
  1520
by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
nipkow@28068
  1521
nipkow@28068
  1522
lemma setsum_head_upt_Suc:
nipkow@28068
  1523
  "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
huffman@30079
  1524
apply(insert setsum_head_Suc[of m "n - Suc 0" f])
nipkow@29667
  1525
apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
nipkow@28068
  1526
done
nipkow@28068
  1527
nipkow@31501
  1528
lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"
nipkow@31501
  1529
  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
nipkow@31501
  1530
proof-
wenzelm@60758
  1531
  have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using \<open>m \<le> n+1\<close> by auto
haftmann@57418
  1532
  thus ?thesis by (auto simp: ivl_disj_int setsum.union_disjoint
nipkow@31501
  1533
    atLeastSucAtMost_greaterThanAtMost)
nipkow@31501
  1534
qed
nipkow@28068
  1535
nipkow@15539
  1536
lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
  1537
  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
haftmann@57418
  1538
by (simp add:setsum.union_disjoint[symmetric] ivl_disj_int ivl_disj_un)
nipkow@15539
  1539
nipkow@15539
  1540
lemma setsum_diff_nat_ivl:
nipkow@15539
  1541
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
nipkow@15539
  1542
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
  1543
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
nipkow@15539
  1544
using setsum_add_nat_ivl [of m n p f,symmetric]
haftmann@57514
  1545
apply (simp add: ac_simps)
nipkow@15539
  1546
done
nipkow@15539
  1547
nipkow@31505
  1548
lemma setsum_natinterval_difff:
nipkow@31505
  1549
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
nipkow@31505
  1550
  shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
nipkow@31505
  1551
          (if m <= n then f m - f(n + 1) else 0)"
nipkow@31505
  1552
by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
nipkow@31505
  1553
hoelzl@56194
  1554
lemma setsum_nat_group: "(\<Sum>m<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {..< n * k}"
hoelzl@56194
  1555
  apply (subgoal_tac "k = 0 | 0 < k", auto)
hoelzl@56194
  1556
  apply (induct "n")
haftmann@57512
  1557
  apply (simp_all add: setsum_add_nat_ivl add.commute atLeast0LessThan[symmetric])
hoelzl@56194
  1558
  done
nipkow@28068
  1559
lp15@60150
  1560
lemma setsum_triangle_reindex:
lp15@60150
  1561
  fixes n :: nat
lp15@60150
  1562
  shows "(\<Sum>(i,j)\<in>{(i,j). i+j < n}. f i j) = (\<Sum>k<n. \<Sum>i\<le>k. f i (k - i))"
lp15@60150
  1563
  apply (simp add: setsum.Sigma)
lp15@60150
  1564
  apply (rule setsum.reindex_bij_witness[where j="\<lambda>(i, j). (i+j, i)" and i="\<lambda>(k, i). (i, k - i)"])
lp15@60150
  1565
  apply auto
lp15@60150
  1566
  done
lp15@60150
  1567
lp15@60150
  1568
lemma setsum_triangle_reindex_eq:
lp15@60150
  1569
  fixes n :: nat
lp15@60150
  1570
  shows "(\<Sum>(i,j)\<in>{(i,j). i+j \<le> n}. f i j) = (\<Sum>k\<le>n. \<Sum>i\<le>k. f i (k - i))"
lp15@60150
  1571
using setsum_triangle_reindex [of f "Suc n"]
lp15@60150
  1572
by (simp only: Nat.less_Suc_eq_le lessThan_Suc_atMost)
lp15@60150
  1573
lp15@60162
  1574
lemma nat_diff_setsum_reindex: "(\<Sum>i<n. f (n - Suc i)) = (\<Sum>i<n. f i)"
lp15@60162
  1575
  by (rule setsum.reindex_bij_witness[where i="\<lambda>i. n - Suc i" and j="\<lambda>i. n - Suc i"]) auto
lp15@60162
  1576
wenzelm@60758
  1577
subsection\<open>Shifting bounds\<close>
nipkow@16733
  1578
nipkow@15539
  1579
lemma setsum_shift_bounds_nat_ivl:
nipkow@15539
  1580
  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
nipkow@15539
  1581
by (induct "n", auto simp:atLeastLessThanSuc)
nipkow@15539
  1582
nipkow@16733
  1583
lemma setsum_shift_bounds_cl_nat_ivl:
nipkow@16733
  1584
  "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
hoelzl@57129
  1585
  by (rule setsum.reindex_bij_witness[where i="\<lambda>i. i + k" and j="\<lambda>i. i - k"]) auto
nipkow@16733
  1586
nipkow@16733
  1587
corollary setsum_shift_bounds_cl_Suc_ivl:
nipkow@16733
  1588
  "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
huffman@30079
  1589
by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
  1590
nipkow@16733
  1591
corollary setsum_shift_bounds_Suc_ivl:
nipkow@16733
  1592
  "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
huffman@30079
  1593
by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
  1594
nipkow@28068
  1595
lemma setsum_shift_lb_Suc0_0:
nipkow@28068
  1596
  "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
nipkow@28068
  1597
by(simp add:setsum_head_Suc)
kleing@19106
  1598
nipkow@28068
  1599
lemma setsum_shift_lb_Suc0_0_upt:
nipkow@28068
  1600
  "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
nipkow@28068
  1601
apply(cases k)apply simp
nipkow@28068
  1602
apply(simp add:setsum_head_upt_Suc)
nipkow@28068
  1603
done
kleing@19022
  1604
haftmann@52380
  1605
lemma setsum_atMost_Suc_shift:
haftmann@52380
  1606
  fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
haftmann@52380
  1607
  shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
haftmann@52380
  1608
proof (induct n)
haftmann@52380
  1609
  case 0 show ?case by simp
haftmann@52380
  1610
next
haftmann@52380
  1611
  case (Suc n) note IH = this
haftmann@52380
  1612
  have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"
haftmann@52380
  1613
    by (rule setsum_atMost_Suc)
haftmann@52380
  1614
  also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
haftmann@52380
  1615
    by (rule IH)
haftmann@52380
  1616
  also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
haftmann@52380
  1617
             f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
haftmann@57512
  1618
    by (rule add.assoc)
haftmann@52380
  1619
  also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
haftmann@52380
  1620
    by (rule setsum_atMost_Suc [symmetric])
haftmann@52380
  1621
  finally show ?case .
haftmann@52380
  1622
qed
haftmann@52380
  1623
lp15@56238
  1624
lemma setsum_last_plus: fixes n::nat shows "m <= n \<Longrightarrow> (\<Sum>i = m..n. f i) = f n + (\<Sum>i = m..<n. f i)"
haftmann@57512
  1625
  by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost add.commute)
lp15@56238
  1626
lp15@56238
  1627
lemma setsum_Suc_diff:
lp15@56238
  1628
  fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
lp15@56238
  1629
  assumes "m \<le> Suc n"
lp15@56238
  1630
  shows "(\<Sum>i = m..n. f(Suc i) - f i) = f (Suc n) - f m"
lp15@56238
  1631
using assms by (induct n) (auto simp: le_Suc_eq)
lp15@55718
  1632
lp15@55718
  1633
lemma nested_setsum_swap:
lp15@55718
  1634
     "(\<Sum>i = 0..n. (\<Sum>j = 0..<i. a i j)) = (\<Sum>j = 0..<n. \<Sum>i = Suc j..n. a i j)"
haftmann@57418
  1635
  by (induction n) (auto simp: setsum.distrib)
lp15@55718
  1636
lp15@56215
  1637
lemma nested_setsum_swap':
lp15@56215
  1638
     "(\<Sum>i\<le>n. (\<Sum>j<i. a i j)) = (\<Sum>j<n. \<Sum>i = Suc j..n. a i j)"
haftmann@57418
  1639
  by (induction n) (auto simp: setsum.distrib)
lp15@56215
  1640
lp15@56238
  1641
lemma setsum_zero_power' [simp]:
lp15@56238
  1642
  fixes c :: "nat \<Rightarrow> 'a::field"
lp15@56238
  1643
  shows "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"
lp15@56238
  1644
  using setsum_zero_power [of "\<lambda>i. c i / d i" A]
lp15@56238
  1645
  by auto
lp15@56238
  1646
haftmann@52380
  1647
wenzelm@60758
  1648
subsection \<open>The formula for geometric sums\<close>
ballarin@17149
  1649
ballarin@17149
  1650
lemma geometric_sum:
haftmann@36307
  1651
  assumes "x \<noteq> 1"
hoelzl@56193
  1652
  shows "(\<Sum>i<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"
haftmann@36307
  1653
proof -
haftmann@36307
  1654
  from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all
hoelzl@56193
  1655
  moreover have "(\<Sum>i<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"
wenzelm@60758
  1656
    by (induct n) (simp_all add: power_Suc field_simps \<open>y \<noteq> 0\<close>)
haftmann@36307
  1657
  ultimately show ?thesis by simp
haftmann@36307
  1658
qed
haftmann@36307
  1659
lp15@60162
  1660
lemma diff_power_eq_setsum:
lp15@60162
  1661
  fixes y :: "'a::{comm_ring,monoid_mult}"
lp15@60162
  1662
  shows
lp15@60162
  1663
    "x ^ (Suc n) - y ^ (Suc n) =
lp15@60162
  1664
      (x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))"
lp15@60162
  1665
proof (induct n)
lp15@60162
  1666
  case (Suc n)
lp15@60162
  1667
  have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x^n) - y * (y * y ^ n)"
lp15@60162
  1668
    by (simp add: power_Suc)
lp15@60162
  1669
  also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x^n)"
lp15@60162
  1670
    by (simp add: power_Suc algebra_simps)
lp15@60162
  1671
  also have "... = y * ((x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
lp15@60162
  1672
    by (simp only: Suc)
lp15@60162
  1673
  also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
lp15@60162
  1674
    by (simp only: mult.left_commute)
lp15@60162
  1675
  also have "... = (x - y) * (\<Sum>p<Suc (Suc n). x ^ p * y ^ (Suc n - p))"
lp15@60162
  1676
    by (simp add: power_Suc field_simps Suc_diff_le setsum_left_distrib setsum_right_distrib)
lp15@60162
  1677
  finally show ?case .
lp15@60162
  1678
qed simp
lp15@60162
  1679
wenzelm@60758
  1680
corollary power_diff_sumr2: --\<open>@{text COMPLEX_POLYFUN} in HOL Light\<close>
lp15@60162
  1681
  fixes x :: "'a::{comm_ring,monoid_mult}"
lp15@60162
  1682
  shows   "x^n - y^n = (x - y) * (\<Sum>i<n. y^(n - Suc i) * x^i)"
lp15@60162
  1683
using diff_power_eq_setsum[of x "n - 1" y]
lp15@60162
  1684
by (cases "n = 0") (simp_all add: field_simps)
lp15@60162
  1685
lp15@60162
  1686
lemma power_diff_1_eq:
lp15@60162
  1687
  fixes x :: "'a::{comm_ring,monoid_mult}"
lp15@60162
  1688
  shows "n \<noteq> 0 \<Longrightarrow> x^n - 1 = (x - 1) * (\<Sum>i<n. (x^i))"
lp15@60162
  1689
using diff_power_eq_setsum [of x _ 1]
lp15@60162
  1690
  by (cases n) auto
lp15@60162
  1691
lp15@60162
  1692
lemma one_diff_power_eq':
lp15@60162
  1693
  fixes x :: "'a::{comm_ring,monoid_mult}"
lp15@60162
  1694
  shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^(n - Suc i))"
lp15@60162
  1695
using diff_power_eq_setsum [of 1 _ x]
lp15@60162
  1696
  by (cases n) auto
lp15@60162
  1697
lp15@60162
  1698
lemma one_diff_power_eq:
lp15@60162
  1699
  fixes x :: "'a::{comm_ring,monoid_mult}"
lp15@60162
  1700
  shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^i)"
lp15@60162
  1701
by (metis one_diff_power_eq' [of n x] nat_diff_setsum_reindex)
lp15@60162
  1702
ballarin@17149
  1703
wenzelm@60758
  1704
subsection \<open>The formula for arithmetic sums\<close>
kleing@19469
  1705
huffman@47222
  1706
lemma gauss_sum:
hoelzl@56193
  1707
  "(2::'a::comm_semiring_1)*(\<Sum>i\<in>{1..n}. of_nat i) = of_nat n*((of_nat n)+1)"
kleing@19469
  1708
proof (induct n)
kleing@19469
  1709
  case 0
kleing@19469
  1710
  show ?case by simp
kleing@19469
  1711
next
kleing@19469
  1712
  case (Suc n)
huffman@47222
  1713
  then show ?case
huffman@47222
  1714
    by (simp add: algebra_simps add: one_add_one [symmetric] del: one_add_one)
huffman@47222
  1715
      (* FIXME: make numeral cancellation simprocs work for semirings *)
kleing@19469
  1716
qed
kleing@19469
  1717
kleing@19469
  1718
theorem arith_series_general:
huffman@47222
  1719
  "(2::'a::comm_semiring_1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1720
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1721
proof cases
kleing@19469
  1722
  assume ngt1: "n > 1"
kleing@19469
  1723
  let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
kleing@19469
  1724
  have
kleing@19469
  1725
    "(\<Sum>i\<in>{..<n}. a+?I i*d) =
kleing@19469
  1726
     ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
haftmann@57418
  1727
    by (rule setsum.distrib)
kleing@19469
  1728
  also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
kleing@19469
  1729
  also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
huffman@30079
  1730
    unfolding One_nat_def
haftmann@57514
  1731
    by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt ac_simps)
huffman@47222
  1732
  also have "2*\<dots> = 2*?n*a + d*2*(\<Sum>i\<in>{1..<n}. ?I i)"
huffman@47222
  1733
    by (simp add: algebra_simps)
kleing@19469
  1734
  also from ngt1 have "{1..<n} = {1..n - 1}"
nipkow@28068
  1735
    by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
nipkow@28068
  1736
  also from ngt1
huffman@47222
  1737
  have "2*?n*a + d*2*(\<Sum>i\<in>{1..n - 1}. ?I i) = (2*?n*a + d*?I (n - 1)*?I n)"
haftmann@57514
  1738
    by (simp only: mult.assoc gauss_sum [of "n - 1"], unfold One_nat_def)
haftmann@57514
  1739
      (simp add:  mult.commute trans [OF add.commute of_nat_Suc [symmetric]])
huffman@47222
  1740
  finally show ?thesis
huffman@47222
  1741
    unfolding mult_2 by (simp add: algebra_simps)
kleing@19469
  1742
next
kleing@19469
  1743
  assume "\<not>(n > 1)"
kleing@19469
  1744
  hence "n = 1 \<or> n = 0" by auto
huffman@47222
  1745
  thus ?thesis by (auto simp: mult_2)
kleing@19469
  1746
qed
kleing@19469
  1747
kleing@19469
  1748
lemma arith_series_nat:
huffman@47222
  1749
  "(2::nat) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
kleing@19469
  1750
proof -
kleing@19469
  1751
  have
huffman@47222
  1752
    "2 * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
kleing@19469
  1753
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1754
    by (rule arith_series_general)
huffman@30079
  1755
  thus ?thesis
huffman@35216
  1756
    unfolding One_nat_def by auto
kleing@19469
  1757
qed
kleing@19469
  1758
kleing@19469
  1759
lemma arith_series_int:
huffman@47222
  1760
  "2 * (\<Sum>i\<in>{..<n}. a + int i * d) = int n * (a + (a + int(n - 1)*d))"
huffman@47222
  1761
  by (fact arith_series_general) (* FIXME: duplicate *)
paulson@15418
  1762
hoelzl@59416
  1763
lemma sum_diff_distrib: "\<forall>x. Q x \<le> P x  \<Longrightarrow> (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x :: nat)"
hoelzl@59416
  1764
  by (subst setsum_subtractf_nat) auto
kleing@19022
  1765
wenzelm@60758
  1766
subsection \<open>Products indexed over intervals\<close>
paulson@29960
  1767
paulson@29960
  1768
syntax
paulson@29960
  1769
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1770
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1771
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)
paulson@29960
  1772
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
paulson@29960
  1773
syntax (xsymbols)
paulson@29960
  1774
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1775
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1776
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1777
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1778
syntax (HTML output)
paulson@29960
  1779
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1780
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1781
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1782
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1783
syntax (latex_prod output)
paulson@29960
  1784
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1785
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1786
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1787
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1788
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1789
 ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1790
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1791
 ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1792
paulson@29960
  1793
translations
paulson@29960
  1794
  "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"
paulson@29960
  1795
  "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"
paulson@29960
  1796
  "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"
paulson@29960
  1797
  "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"
paulson@29960
  1798
wenzelm@60758
  1799
subsection \<open>Transfer setup\<close>
haftmann@33318
  1800
haftmann@33318
  1801
lemma transfer_nat_int_set_functions:
haftmann@33318
  1802
    "{..n} = nat ` {0..int n}"
haftmann@33318
  1803
    "{m..n} = nat ` {int m..int n}"  (* need all variants of these! *)
haftmann@33318
  1804
  apply (auto simp add: image_def)
haftmann@33318
  1805
  apply (rule_tac x = "int x" in bexI)
haftmann@33318
  1806
  apply auto
haftmann@33318
  1807
  apply (rule_tac x = "int x" in bexI)
haftmann@33318
  1808
  apply auto
haftmann@33318
  1809
  done
haftmann@33318
  1810
haftmann@33318
  1811
lemma transfer_nat_int_set_function_closures:
haftmann@33318
  1812
    "x >= 0 \<Longrightarrow> nat_set {x..y}"
haftmann@33318
  1813
  by (simp add: nat_set_def)
haftmann@33318
  1814
haftmann@35644
  1815
declare transfer_morphism_nat_int[transfer add
haftmann@33318
  1816
  return: transfer_nat_int_set_functions
haftmann@33318
  1817
    transfer_nat_int_set_function_closures
haftmann@33318
  1818
]
haftmann@33318
  1819
haftmann@33318
  1820
lemma transfer_int_nat_set_functions:
haftmann@33318
  1821
    "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}"
haftmann@33318
  1822
  by (simp only: is_nat_def transfer_nat_int_set_functions
haftmann@33318
  1823
    transfer_nat_int_set_function_closures
haftmann@33318
  1824
    transfer_nat_int_set_return_embed nat_0_le
haftmann@33318
  1825
    cong: transfer_nat_int_set_cong)
haftmann@33318
  1826
haftmann@33318
  1827
lemma transfer_int_nat_set_function_closures:
haftmann@33318
  1828
    "is_nat x \<Longrightarrow> nat_set {x..y}"
haftmann@33318
  1829
  by (simp only: transfer_nat_int_set_function_closures is_nat_def)
haftmann@33318
  1830
haftmann@35644
  1831
declare transfer_morphism_int_nat[transfer add
haftmann@33318
  1832
  return: transfer_int_nat_set_functions
haftmann@33318
  1833
    transfer_int_nat_set_function_closures
haftmann@33318
  1834
]
haftmann@33318
  1835
lp15@55242
  1836
lemma setprod_int_plus_eq: "setprod int {i..i+j} =  \<Prod>{int i..int (i+j)}"
lp15@55242
  1837
  by (induct j) (auto simp add: atLeastAtMostSuc_conv atLeastAtMostPlus1_int_conv)
lp15@55242
  1838
lp15@55242
  1839
lemma setprod_int_eq: "setprod int {i..j} =  \<Prod>{int i..int j}"
lp15@55242
  1840
proof (cases "i \<le> j")
lp15@55242
  1841
  case True
lp15@55242
  1842
  then show ?thesis
lp15@55242
  1843
    by (metis Nat.le_iff_add setprod_int_plus_eq)
lp15@55242
  1844
next
lp15@55242
  1845
  case False
lp15@55242
  1846
  then show ?thesis
lp15@55242
  1847
    by auto
lp15@55242
  1848
qed
lp15@55242
  1849
nipkow@8924
  1850
end