src/HOL/SetInterval.thy
author nipkow
Fri Aug 28 18:52:41 2009 +0200 (2009-08-28)
changeset 32436 10cd49e0c067
parent 32408 a1a85b0a26f7
child 32456 341c83339aeb
permissions -rw-r--r--
Turned "x <= y ==> sup x y = y" (and relatives) into simp rules
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(*  Title:      HOL/SetInterval.thy
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    Author:     Tobias Nipkow and Clemens Ballarin
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                Additions by Jeremy Avigad in March 2004
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    Copyright   2000  TU Muenchen
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lessThan, greaterThan, atLeast, atMost and two-sided intervals
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*)
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header {* Set intervals *}
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theory SetInterval
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imports Int
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begin
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context ord
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begin
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definition
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  lessThan    :: "'a => 'a set"	("(1{..<_})") where
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  "{..<u} == {x. x < u}"
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definition
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  atMost      :: "'a => 'a set"	("(1{.._})") where
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  "{..u} == {x. x \<le> u}"
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definition
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  greaterThan :: "'a => 'a set"	("(1{_<..})") where
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  "{l<..} == {x. l<x}"
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definition
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  atLeast     :: "'a => 'a set"	("(1{_..})") where
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  "{l..} == {x. l\<le>x}"
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definition
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  greaterThanLessThan :: "'a => 'a => 'a set"  ("(1{_<..<_})") where
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  "{l<..<u} == {l<..} Int {..<u}"
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definition
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  atLeastLessThan :: "'a => 'a => 'a set"      ("(1{_..<_})") where
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  "{l..<u} == {l..} Int {..<u}"
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definition
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  greaterThanAtMost :: "'a => 'a => 'a set"    ("(1{_<.._})") where
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  "{l<..u} == {l<..} Int {..u}"
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definition
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  atLeastAtMost :: "'a => 'a => 'a set"        ("(1{_.._})") where
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  "{l..u} == {l..} Int {..u}"
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end
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text{* A note of warning when using @{term"{..<n}"} on type @{typ
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nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
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@{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
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syntax
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  "@UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3UN _<=_./ _)" 10)
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  "@UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3UN _<_./ _)" 10)
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  "@INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" 10)
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  "@INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3INT _<_./ _)" 10)
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syntax (xsymbols)
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  "@UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" 10)
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  "@UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _<_./ _)" 10)
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  "@INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" 10)
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  "@INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" 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> _)/ _)" 10)
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  "@UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ < _)/ _)" 10)
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  "@INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ \<le> _)/ _)" 10)
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  "@INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ < _)/ _)" 10)
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translations
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  "UN i<=n. A"  == "UN i:{..n}. A"
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  "UN i<n. A"   == "UN i:{..<n}. A"
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  "INT i<=n. A" == "INT i:{..n}. A"
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  "INT i<n. A"  == "INT i:{..<n}. A"
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subsection {* Various equivalences *}
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lemma (in ord) lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
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by (simp add: lessThan_def)
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lemma Compl_lessThan [simp]:
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    "!!k:: 'a::linorder. -lessThan k = atLeast k"
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apply (auto simp add: lessThan_def atLeast_def)
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done
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lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
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by auto
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lemma (in ord) greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
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by (simp add: greaterThan_def)
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lemma Compl_greaterThan [simp]:
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    "!!k:: 'a::linorder. -greaterThan k = atMost k"
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  by (auto simp add: greaterThan_def atMost_def)
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lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
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apply (subst Compl_greaterThan [symmetric])
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apply (rule double_complement)
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done
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lemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
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by (simp add: atLeast_def)
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lemma Compl_atLeast [simp]:
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    "!!k:: 'a::linorder. -atLeast k = lessThan k"
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  by (auto simp add: lessThan_def atLeast_def)
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lemma (in ord) atMost_iff [iff]: "(i: atMost k) = (i<=k)"
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by (simp add: atMost_def)
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lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
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by (blast intro: order_antisym)
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subsection {* Logical Equivalences for Set Inclusion and Equality *}
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lemma atLeast_subset_iff [iff]:
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     "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
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by (blast intro: order_trans)
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lemma atLeast_eq_iff [iff]:
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     "(atLeast x = atLeast y) = (x = (y::'a::linorder))"
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by (blast intro: order_antisym order_trans)
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lemma greaterThan_subset_iff [iff]:
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     "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
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apply (auto simp add: greaterThan_def)
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 apply (subst linorder_not_less [symmetric], blast)
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done
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lemma greaterThan_eq_iff [iff]:
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     "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
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apply (rule iffI)
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 apply (erule equalityE)
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 apply simp_all
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done
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lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))"
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by (blast intro: order_trans)
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lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"
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by (blast intro: order_antisym order_trans)
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lemma lessThan_subset_iff [iff]:
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     "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
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apply (auto simp add: lessThan_def)
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 apply (subst linorder_not_less [symmetric], blast)
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done
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lemma lessThan_eq_iff [iff]:
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     "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
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apply (rule iffI)
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 apply (erule equalityE)
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 apply simp_all
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done
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subsection {*Two-sided intervals*}
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context ord
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begin
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lemma greaterThanLessThan_iff [simp,noatp]:
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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,noatp]:
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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,noatp]:
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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,noatp]:
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  "(i : {l..u}) = (l <= i & i <= u)"
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by (simp add: atLeastAtMost_def)
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text {* The above four lemmas could be declared as iffs. Unfortunately this
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breaks many proofs. Since it only helps blast, it is better to leave well
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alone *}
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end
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subsubsection{* Emptyness, singletons, subset *}
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context order
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begin
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lemma atLeastatMost_empty[simp]:
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  "b < a \<Longrightarrow> {a..b} = {}"
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by(auto simp: atLeastAtMost_def atLeast_def atMost_def)
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lemma atLeastatMost_empty_iff[simp]:
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  "{a..b} = {} \<longleftrightarrow> (~ a <= b)"
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by auto (blast intro: order_trans)
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lemma atLeastatMost_empty_iff2[simp]:
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  "{} = {a..b} \<longleftrightarrow> (~ a <= b)"
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by auto (blast intro: order_trans)
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lemma atLeastLessThan_empty[simp]:
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  "b <= a \<Longrightarrow> {a..<b} = {}"
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by(auto simp: atLeastLessThan_def)
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lemma atLeastLessThan_empty_iff[simp]:
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  "{a..<b} = {} \<longleftrightarrow> (~ a < b)"
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by auto (blast intro: le_less_trans)
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lemma atLeastLessThan_empty_iff2[simp]:
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  "{} = {a..<b} \<longleftrightarrow> (~ a < b)"
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by auto (blast intro: le_less_trans)
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lemma greaterThanAtMost_empty[simp]: "l \<le> k ==> {k<..l} = {}"
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by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
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lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {} \<longleftrightarrow> ~ k < l"
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by auto (blast intro: less_le_trans)
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lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} \<longleftrightarrow> ~ k < l"
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by auto (blast intro: less_le_trans)
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lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..<l} = {}"
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by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
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lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"
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by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
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lemma atLeastatMost_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 expand_set_eq less_le_not_le)(blast intro: order_trans)
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end
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lemma (in linorder) atLeastLessThan_subset_iff:
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  "{a..<b} <= {c..<d} \<Longrightarrow> b <= a | c<=a & b<=d"
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apply (auto simp:subset_eq Ball_def)
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apply(frule_tac x=a in spec)
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apply(erule_tac x=d in allE)
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apply (simp add: less_imp_le)
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done
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subsection {* Intervals of natural numbers *}
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subsubsection {* The Constant @{term lessThan} *}
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lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
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by (simp add: lessThan_def)
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lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
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by (simp add: lessThan_def less_Suc_eq, blast)
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lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
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by (simp add: lessThan_def atMost_def less_Suc_eq_le)
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lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
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by blast
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subsubsection {* The Constant @{term greaterThan} *}
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lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
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apply (simp add: greaterThan_def)
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apply (blast dest: gr0_conv_Suc [THEN iffD1])
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done
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lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
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apply (simp add: greaterThan_def)
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apply (auto elim: linorder_neqE)
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done
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lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
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by blast
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subsubsection {* The Constant @{term atLeast} *}
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lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
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by (unfold atLeast_def UNIV_def, simp)
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lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
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apply (simp add: atLeast_def)
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apply (simp add: Suc_le_eq)
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apply (simp add: order_le_less, blast)
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done
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lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
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  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
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lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
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by blast
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subsubsection {* The Constant @{term atMost} *}
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lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
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by (simp add: atMost_def)
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lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
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apply (simp add: atMost_def)
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apply (simp add: less_Suc_eq order_le_less, blast)
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done
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lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
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by blast
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subsubsection {* The Constant @{term atLeastLessThan} *}
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text{*The orientation of the following 2 rules is tricky. The lhs is
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defined in terms of the rhs.  Hence the chosen orientation makes sense
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in this theory --- the reverse orientation complicates proofs (eg
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nontermination). But outside, when the definition of the lhs is rarely
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used, the opposite orientation seems preferable because it reduces a
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specific concept to a more general one. *}
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lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
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by(simp add:lessThan_def atLeastLessThan_def)
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lemma atLeast0AtMost: "{0..n::nat} = {..n}"
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by(simp add:atMost_def atLeastAtMost_def)
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declare atLeast0LessThan[symmetric, code_unfold]
haftmann@31998
   331
        atLeast0AtMost[symmetric, code_unfold]
nipkow@24449
   332
nipkow@24449
   333
lemma atLeastLessThan0: "{m..<0::nat} = {}"
paulson@15047
   334
by (simp add: atLeastLessThan_def)
nipkow@24449
   335
paulson@15047
   336
subsubsection {* Intervals of nats with @{term Suc} *}
paulson@15047
   337
paulson@15047
   338
text{*Not a simprule because the RHS is too messy.*}
paulson@15047
   339
lemma atLeastLessThanSuc:
paulson@15047
   340
    "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
paulson@15418
   341
by (auto simp add: atLeastLessThan_def)
paulson@15047
   342
paulson@15418
   343
lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
paulson@15047
   344
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   345
(*
paulson@15047
   346
lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
paulson@15047
   347
by (induct k, simp_all add: atLeastLessThanSuc)
paulson@15047
   348
paulson@15047
   349
lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
paulson@15047
   350
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   351
*)
nipkow@15045
   352
lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
paulson@14485
   353
  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   354
paulson@15418
   355
lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
paulson@15418
   356
  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
paulson@14485
   357
    greaterThanAtMost_def)
paulson@14485
   358
paulson@15418
   359
lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
paulson@15418
   360
  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
paulson@14485
   361
    greaterThanLessThan_def)
paulson@14485
   362
nipkow@15554
   363
lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
nipkow@15554
   364
by (auto simp add: atLeastAtMost_def)
nipkow@15554
   365
nipkow@16733
   366
subsubsection {* Image *}
nipkow@16733
   367
nipkow@16733
   368
lemma image_add_atLeastAtMost:
nipkow@16733
   369
  "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
nipkow@16733
   370
proof
nipkow@16733
   371
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   372
next
nipkow@16733
   373
  show "?B \<subseteq> ?A"
nipkow@16733
   374
  proof
nipkow@16733
   375
    fix n assume a: "n : ?B"
webertj@20217
   376
    hence "n - k : {i..j}" by auto
nipkow@16733
   377
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   378
    ultimately show "n : ?A" by blast
nipkow@16733
   379
  qed
nipkow@16733
   380
qed
nipkow@16733
   381
nipkow@16733
   382
lemma image_add_atLeastLessThan:
nipkow@16733
   383
  "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
nipkow@16733
   384
proof
nipkow@16733
   385
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   386
next
nipkow@16733
   387
  show "?B \<subseteq> ?A"
nipkow@16733
   388
  proof
nipkow@16733
   389
    fix n assume a: "n : ?B"
webertj@20217
   390
    hence "n - k : {i..<j}" by auto
nipkow@16733
   391
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   392
    ultimately show "n : ?A" by blast
nipkow@16733
   393
  qed
nipkow@16733
   394
qed
nipkow@16733
   395
nipkow@16733
   396
corollary image_Suc_atLeastAtMost[simp]:
nipkow@16733
   397
  "Suc ` {i..j} = {Suc i..Suc j}"
huffman@30079
   398
using image_add_atLeastAtMost[where k="Suc 0"] by simp
nipkow@16733
   399
nipkow@16733
   400
corollary image_Suc_atLeastLessThan[simp]:
nipkow@16733
   401
  "Suc ` {i..<j} = {Suc i..<Suc j}"
huffman@30079
   402
using image_add_atLeastLessThan[where k="Suc 0"] by simp
nipkow@16733
   403
nipkow@16733
   404
lemma image_add_int_atLeastLessThan:
nipkow@16733
   405
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
nipkow@16733
   406
  apply (auto simp add: image_def)
nipkow@16733
   407
  apply (rule_tac x = "x - l" in bexI)
nipkow@16733
   408
  apply auto
nipkow@16733
   409
  done
nipkow@16733
   410
nipkow@16733
   411
paulson@14485
   412
subsubsection {* Finiteness *}
paulson@14485
   413
nipkow@15045
   414
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
paulson@14485
   415
  by (induct k) (simp_all add: lessThan_Suc)
paulson@14485
   416
paulson@14485
   417
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
paulson@14485
   418
  by (induct k) (simp_all add: atMost_Suc)
paulson@14485
   419
paulson@14485
   420
lemma finite_greaterThanLessThan [iff]:
nipkow@15045
   421
  fixes l :: nat shows "finite {l<..<u}"
paulson@14485
   422
by (simp add: greaterThanLessThan_def)
paulson@14485
   423
paulson@14485
   424
lemma finite_atLeastLessThan [iff]:
nipkow@15045
   425
  fixes l :: nat shows "finite {l..<u}"
paulson@14485
   426
by (simp add: atLeastLessThan_def)
paulson@14485
   427
paulson@14485
   428
lemma finite_greaterThanAtMost [iff]:
nipkow@15045
   429
  fixes l :: nat shows "finite {l<..u}"
paulson@14485
   430
by (simp add: greaterThanAtMost_def)
paulson@14485
   431
paulson@14485
   432
lemma finite_atLeastAtMost [iff]:
paulson@14485
   433
  fixes l :: nat shows "finite {l..u}"
paulson@14485
   434
by (simp add: atLeastAtMost_def)
paulson@14485
   435
nipkow@28068
   436
text {* A bounded set of natural numbers is finite. *}
paulson@14485
   437
lemma bounded_nat_set_is_finite:
nipkow@24853
   438
  "(ALL i:N. i < (n::nat)) ==> finite N"
nipkow@28068
   439
apply (rule finite_subset)
nipkow@28068
   440
 apply (rule_tac [2] finite_lessThan, auto)
nipkow@28068
   441
done
nipkow@28068
   442
nipkow@31044
   443
text {* A set of natural numbers is finite iff it is bounded. *}
nipkow@31044
   444
lemma finite_nat_set_iff_bounded:
nipkow@31044
   445
  "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
nipkow@31044
   446
proof
nipkow@31044
   447
  assume f:?F  show ?B
nipkow@31044
   448
    using Max_ge[OF `?F`, simplified less_Suc_eq_le[symmetric]] by blast
nipkow@31044
   449
next
nipkow@31044
   450
  assume ?B show ?F using `?B` by(blast intro:bounded_nat_set_is_finite)
nipkow@31044
   451
qed
nipkow@31044
   452
nipkow@31044
   453
lemma finite_nat_set_iff_bounded_le:
nipkow@31044
   454
  "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
nipkow@31044
   455
apply(simp add:finite_nat_set_iff_bounded)
nipkow@31044
   456
apply(blast dest:less_imp_le_nat le_imp_less_Suc)
nipkow@31044
   457
done
nipkow@31044
   458
nipkow@28068
   459
lemma finite_less_ub:
nipkow@28068
   460
     "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
nipkow@28068
   461
by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
paulson@14485
   462
nipkow@24853
   463
text{* Any subset of an interval of natural numbers the size of the
nipkow@24853
   464
subset is exactly that interval. *}
nipkow@24853
   465
nipkow@24853
   466
lemma subset_card_intvl_is_intvl:
nipkow@24853
   467
  "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")
nipkow@24853
   468
proof cases
nipkow@24853
   469
  assume "finite A"
nipkow@24853
   470
  thus "PROP ?P"
nipkow@32006
   471
  proof(induct A rule:finite_linorder_max_induct)
nipkow@24853
   472
    case empty thus ?case by auto
nipkow@24853
   473
  next
nipkow@24853
   474
    case (insert A b)
nipkow@24853
   475
    moreover hence "b ~: A" by auto
nipkow@24853
   476
    moreover have "A <= {k..<k+card A}" and "b = k+card A"
nipkow@24853
   477
      using `b ~: A` insert by fastsimp+
nipkow@24853
   478
    ultimately show ?case by auto
nipkow@24853
   479
  qed
nipkow@24853
   480
next
nipkow@24853
   481
  assume "~finite A" thus "PROP ?P" by simp
nipkow@24853
   482
qed
nipkow@24853
   483
nipkow@24853
   484
paulson@14485
   485
subsubsection {* Cardinality *}
paulson@14485
   486
nipkow@15045
   487
lemma card_lessThan [simp]: "card {..<u} = u"
paulson@15251
   488
  by (induct u, simp_all add: lessThan_Suc)
paulson@14485
   489
paulson@14485
   490
lemma card_atMost [simp]: "card {..u} = Suc u"
paulson@14485
   491
  by (simp add: lessThan_Suc_atMost [THEN sym])
paulson@14485
   492
nipkow@15045
   493
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
nipkow@15045
   494
  apply (subgoal_tac "card {l..<u} = card {..<u-l}")
paulson@14485
   495
  apply (erule ssubst, rule card_lessThan)
nipkow@15045
   496
  apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
paulson@14485
   497
  apply (erule subst)
paulson@14485
   498
  apply (rule card_image)
paulson@14485
   499
  apply (simp add: inj_on_def)
paulson@14485
   500
  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
paulson@14485
   501
  apply (rule_tac x = "x - l" in exI)
paulson@14485
   502
  apply arith
paulson@14485
   503
  done
paulson@14485
   504
paulson@15418
   505
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
paulson@14485
   506
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
paulson@14485
   507
paulson@15418
   508
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
paulson@14485
   509
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
paulson@14485
   510
nipkow@15045
   511
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
paulson@14485
   512
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
paulson@14485
   513
nipkow@26105
   514
lemma ex_bij_betw_nat_finite:
nipkow@26105
   515
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
nipkow@26105
   516
apply(drule finite_imp_nat_seg_image_inj_on)
nipkow@26105
   517
apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
nipkow@26105
   518
done
nipkow@26105
   519
nipkow@26105
   520
lemma ex_bij_betw_finite_nat:
nipkow@26105
   521
  "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
nipkow@26105
   522
by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
nipkow@26105
   523
nipkow@31438
   524
lemma finite_same_card_bij:
nipkow@31438
   525
  "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"
nipkow@31438
   526
apply(drule ex_bij_betw_finite_nat)
nipkow@31438
   527
apply(drule ex_bij_betw_nat_finite)
nipkow@31438
   528
apply(auto intro!:bij_betw_trans)
nipkow@31438
   529
done
nipkow@31438
   530
nipkow@31438
   531
lemma ex_bij_betw_nat_finite_1:
nipkow@31438
   532
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
nipkow@31438
   533
by (rule finite_same_card_bij) auto
nipkow@31438
   534
nipkow@26105
   535
paulson@14485
   536
subsection {* Intervals of integers *}
paulson@14485
   537
nipkow@15045
   538
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
paulson@14485
   539
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   540
paulson@15418
   541
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
paulson@14485
   542
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
paulson@14485
   543
paulson@15418
   544
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
paulson@15418
   545
    "{l+1..<u} = {l<..<u::int}"
paulson@14485
   546
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
paulson@14485
   547
paulson@14485
   548
subsubsection {* Finiteness *}
paulson@14485
   549
paulson@15418
   550
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
nipkow@15045
   551
    {(0::int)..<u} = int ` {..<nat u}"
paulson@14485
   552
  apply (unfold image_def lessThan_def)
paulson@14485
   553
  apply auto
paulson@14485
   554
  apply (rule_tac x = "nat x" in exI)
paulson@14485
   555
  apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
paulson@14485
   556
  done
paulson@14485
   557
nipkow@15045
   558
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
paulson@14485
   559
  apply (case_tac "0 \<le> u")
paulson@14485
   560
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   561
  apply (rule finite_imageI)
paulson@14485
   562
  apply auto
paulson@14485
   563
  done
paulson@14485
   564
nipkow@15045
   565
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
nipkow@15045
   566
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   567
  apply (erule subst)
paulson@14485
   568
  apply (rule finite_imageI)
paulson@14485
   569
  apply (rule finite_atLeastZeroLessThan_int)
nipkow@16733
   570
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   571
  done
paulson@14485
   572
paulson@15418
   573
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
paulson@14485
   574
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
paulson@14485
   575
paulson@15418
   576
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
paulson@14485
   577
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   578
paulson@15418
   579
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
paulson@14485
   580
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   581
nipkow@24853
   582
paulson@14485
   583
subsubsection {* Cardinality *}
paulson@14485
   584
nipkow@15045
   585
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
paulson@14485
   586
  apply (case_tac "0 \<le> u")
paulson@14485
   587
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   588
  apply (subst card_image)
paulson@14485
   589
  apply (auto simp add: inj_on_def)
paulson@14485
   590
  done
paulson@14485
   591
nipkow@15045
   592
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
nipkow@15045
   593
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
paulson@14485
   594
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
nipkow@15045
   595
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   596
  apply (erule subst)
paulson@14485
   597
  apply (rule card_image)
paulson@14485
   598
  apply (simp add: inj_on_def)
nipkow@16733
   599
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   600
  done
paulson@14485
   601
paulson@14485
   602
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
nipkow@29667
   603
apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
nipkow@29667
   604
apply (auto simp add: algebra_simps)
nipkow@29667
   605
done
paulson@14485
   606
paulson@15418
   607
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
nipkow@29667
   608
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   609
nipkow@15045
   610
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
nipkow@29667
   611
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   612
bulwahn@27656
   613
lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
bulwahn@27656
   614
proof -
bulwahn@27656
   615
  have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
bulwahn@27656
   616
  with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
bulwahn@27656
   617
qed
bulwahn@27656
   618
bulwahn@27656
   619
lemma card_less:
bulwahn@27656
   620
assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   621
shows "card {k \<in> M. k < Suc i} \<noteq> 0"
bulwahn@27656
   622
proof -
bulwahn@27656
   623
  from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
bulwahn@27656
   624
  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
bulwahn@27656
   625
qed
bulwahn@27656
   626
bulwahn@27656
   627
lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"
huffman@30079
   628
apply (rule card_bij_eq [of "Suc" _ _ "\<lambda>x. x - Suc 0"])
bulwahn@27656
   629
apply simp
bulwahn@27656
   630
apply fastsimp
bulwahn@27656
   631
apply auto
bulwahn@27656
   632
apply (rule inj_on_diff_nat)
bulwahn@27656
   633
apply auto
bulwahn@27656
   634
apply (case_tac x)
bulwahn@27656
   635
apply auto
bulwahn@27656
   636
apply (case_tac xa)
bulwahn@27656
   637
apply auto
bulwahn@27656
   638
apply (case_tac xa)
bulwahn@27656
   639
apply auto
bulwahn@27656
   640
done
bulwahn@27656
   641
bulwahn@27656
   642
lemma card_less_Suc:
bulwahn@27656
   643
  assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   644
    shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
bulwahn@27656
   645
proof -
bulwahn@27656
   646
  from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
bulwahn@27656
   647
  hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
bulwahn@27656
   648
    by (auto simp only: insert_Diff)
bulwahn@27656
   649
  have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
bulwahn@27656
   650
  from finite_M_bounded_by_nat[of "\<lambda>x. x \<in> M" "Suc i"] have "Suc (card {k. Suc k \<in> M \<and> k < i}) = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"
bulwahn@27656
   651
    apply (subst card_insert)
bulwahn@27656
   652
    apply simp_all
bulwahn@27656
   653
    apply (subst b)
bulwahn@27656
   654
    apply (subst card_less_Suc2[symmetric])
bulwahn@27656
   655
    apply simp_all
bulwahn@27656
   656
    done
bulwahn@27656
   657
  with c show ?thesis by simp
bulwahn@27656
   658
qed
bulwahn@27656
   659
paulson@14485
   660
paulson@13850
   661
subsection {*Lemmas useful with the summation operator setsum*}
paulson@13850
   662
ballarin@16102
   663
text {* For examples, see Algebra/poly/UnivPoly2.thy *}
ballarin@13735
   664
wenzelm@14577
   665
subsubsection {* Disjoint Unions *}
ballarin@13735
   666
wenzelm@14577
   667
text {* Singletons and open intervals *}
ballarin@13735
   668
ballarin@13735
   669
lemma ivl_disj_un_singleton:
nipkow@15045
   670
  "{l::'a::linorder} Un {l<..} = {l..}"
nipkow@15045
   671
  "{..<u} Un {u::'a::linorder} = {..u}"
nipkow@15045
   672
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
nipkow@15045
   673
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
nipkow@15045
   674
  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
nipkow@15045
   675
  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
ballarin@14398
   676
by auto
ballarin@13735
   677
wenzelm@14577
   678
text {* One- and two-sided intervals *}
ballarin@13735
   679
ballarin@13735
   680
lemma ivl_disj_un_one:
nipkow@15045
   681
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
nipkow@15045
   682
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
nipkow@15045
   683
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
nipkow@15045
   684
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
nipkow@15045
   685
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
nipkow@15045
   686
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
nipkow@15045
   687
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
nipkow@15045
   688
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
ballarin@14398
   689
by auto
ballarin@13735
   690
wenzelm@14577
   691
text {* Two- and two-sided intervals *}
ballarin@13735
   692
ballarin@13735
   693
lemma ivl_disj_un_two:
nipkow@15045
   694
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
nipkow@15045
   695
  "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
nipkow@15045
   696
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
nipkow@15045
   697
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
nipkow@15045
   698
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
nipkow@15045
   699
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
nipkow@15045
   700
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
nipkow@15045
   701
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
ballarin@14398
   702
by auto
ballarin@13735
   703
ballarin@13735
   704
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
ballarin@13735
   705
wenzelm@14577
   706
subsubsection {* Disjoint Intersections *}
ballarin@13735
   707
wenzelm@14577
   708
text {* Singletons and open intervals *}
ballarin@13735
   709
ballarin@13735
   710
lemma ivl_disj_int_singleton:
nipkow@15045
   711
  "{l::'a::order} Int {l<..} = {}"
nipkow@15045
   712
  "{..<u} Int {u} = {}"
nipkow@15045
   713
  "{l} Int {l<..<u} = {}"
nipkow@15045
   714
  "{l<..<u} Int {u} = {}"
nipkow@15045
   715
  "{l} Int {l<..u} = {}"
nipkow@15045
   716
  "{l..<u} Int {u} = {}"
ballarin@13735
   717
  by simp+
ballarin@13735
   718
wenzelm@14577
   719
text {* One- and two-sided intervals *}
ballarin@13735
   720
ballarin@13735
   721
lemma ivl_disj_int_one:
nipkow@15045
   722
  "{..l::'a::order} Int {l<..<u} = {}"
nipkow@15045
   723
  "{..<l} Int {l..<u} = {}"
nipkow@15045
   724
  "{..l} Int {l<..u} = {}"
nipkow@15045
   725
  "{..<l} Int {l..u} = {}"
nipkow@15045
   726
  "{l<..u} Int {u<..} = {}"
nipkow@15045
   727
  "{l<..<u} Int {u..} = {}"
nipkow@15045
   728
  "{l..u} Int {u<..} = {}"
nipkow@15045
   729
  "{l..<u} Int {u..} = {}"
ballarin@14398
   730
  by auto
ballarin@13735
   731
wenzelm@14577
   732
text {* Two- and two-sided intervals *}
ballarin@13735
   733
ballarin@13735
   734
lemma ivl_disj_int_two:
nipkow@15045
   735
  "{l::'a::order<..<m} Int {m..<u} = {}"
nipkow@15045
   736
  "{l<..m} Int {m<..<u} = {}"
nipkow@15045
   737
  "{l..<m} Int {m..<u} = {}"
nipkow@15045
   738
  "{l..m} Int {m<..<u} = {}"
nipkow@15045
   739
  "{l<..<m} Int {m..u} = {}"
nipkow@15045
   740
  "{l<..m} Int {m<..u} = {}"
nipkow@15045
   741
  "{l..<m} Int {m..u} = {}"
nipkow@15045
   742
  "{l..m} Int {m<..u} = {}"
ballarin@14398
   743
  by auto
ballarin@13735
   744
ballarin@13735
   745
lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
ballarin@13735
   746
nipkow@15542
   747
subsubsection {* Some Differences *}
nipkow@15542
   748
nipkow@15542
   749
lemma ivl_diff[simp]:
nipkow@15542
   750
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
nipkow@15542
   751
by(auto)
nipkow@15542
   752
nipkow@15542
   753
nipkow@15542
   754
subsubsection {* Some Subset Conditions *}
nipkow@15542
   755
paulson@24286
   756
lemma ivl_subset [simp,noatp]:
nipkow@15542
   757
 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
nipkow@15542
   758
apply(auto simp:linorder_not_le)
nipkow@15542
   759
apply(rule ccontr)
nipkow@15542
   760
apply(insert linorder_le_less_linear[of i n])
nipkow@15542
   761
apply(clarsimp simp:linorder_not_le)
nipkow@15542
   762
apply(fastsimp)
nipkow@15542
   763
done
nipkow@15542
   764
nipkow@15041
   765
nipkow@15042
   766
subsection {* Summation indexed over intervals *}
nipkow@15042
   767
nipkow@15042
   768
syntax
nipkow@15042
   769
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   770
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   771
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
nipkow@16052
   772
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
nipkow@15042
   773
syntax (xsymbols)
nipkow@15042
   774
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   775
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   776
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
   777
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15042
   778
syntax (HTML output)
nipkow@15042
   779
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   780
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   781
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
   782
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15056
   783
syntax (latex_sum output)
nipkow@15052
   784
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   785
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052
   786
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   787
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@16052
   788
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
   789
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15052
   790
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
   791
 ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15041
   792
nipkow@15048
   793
translations
nipkow@28853
   794
  "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
nipkow@28853
   795
  "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
nipkow@28853
   796
  "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
nipkow@28853
   797
  "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
nipkow@15041
   798
nipkow@15052
   799
text{* The above introduces some pretty alternative syntaxes for
nipkow@15056
   800
summation over intervals:
nipkow@15052
   801
\begin{center}
nipkow@15052
   802
\begin{tabular}{lll}
nipkow@15056
   803
Old & New & \LaTeX\\
nipkow@15056
   804
@{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
   805
@{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
   806
@{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
nipkow@15056
   807
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
nipkow@15052
   808
\end{tabular}
nipkow@15052
   809
\end{center}
nipkow@15056
   810
The left column shows the term before introduction of the new syntax,
nipkow@15056
   811
the middle column shows the new (default) syntax, and the right column
nipkow@15056
   812
shows a special syntax. The latter is only meaningful for latex output
nipkow@15056
   813
and has to be activated explicitly by setting the print mode to
wenzelm@21502
   814
@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
nipkow@15056
   815
antiquotations). It is not the default \LaTeX\ output because it only
nipkow@15056
   816
works well with italic-style formulae, not tt-style.
nipkow@15052
   817
nipkow@15052
   818
Note that for uniformity on @{typ nat} it is better to use
nipkow@15052
   819
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
nipkow@15052
   820
not provide all lemmas available for @{term"{m..<n}"} also in the
nipkow@15052
   821
special form for @{term"{..<n}"}. *}
nipkow@15052
   822
nipkow@15542
   823
text{* This congruence rule should be used for sums over intervals as
nipkow@15542
   824
the standard theorem @{text[source]setsum_cong} does not work well
nipkow@15542
   825
with the simplifier who adds the unsimplified premise @{term"x:B"} to
nipkow@15542
   826
the context. *}
nipkow@15542
   827
nipkow@15542
   828
lemma setsum_ivl_cong:
nipkow@15542
   829
 "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
nipkow@15542
   830
 setsum f {a..<b} = setsum g {c..<d}"
nipkow@15542
   831
by(rule setsum_cong, simp_all)
nipkow@15041
   832
nipkow@16041
   833
(* FIXME why are the following simp rules but the corresponding eqns
nipkow@16041
   834
on intervals are not? *)
nipkow@16041
   835
nipkow@16052
   836
lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
nipkow@16052
   837
by (simp add:atMost_Suc add_ac)
nipkow@16052
   838
nipkow@16041
   839
lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
nipkow@16041
   840
by (simp add:lessThan_Suc add_ac)
nipkow@15041
   841
nipkow@15911
   842
lemma setsum_cl_ivl_Suc[simp]:
nipkow@15561
   843
  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
nipkow@15561
   844
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@15561
   845
nipkow@15911
   846
lemma setsum_op_ivl_Suc[simp]:
nipkow@15561
   847
  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
nipkow@15561
   848
by (auto simp:add_ac atLeastLessThanSuc)
nipkow@16041
   849
(*
nipkow@15561
   850
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
nipkow@15561
   851
    (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
nipkow@15561
   852
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@16041
   853
*)
nipkow@28068
   854
nipkow@28068
   855
lemma setsum_head:
nipkow@28068
   856
  fixes n :: nat
nipkow@28068
   857
  assumes mn: "m <= n" 
nipkow@28068
   858
  shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
nipkow@28068
   859
proof -
nipkow@28068
   860
  from mn
nipkow@28068
   861
  have "{m..n} = {m} \<union> {m<..n}"
nipkow@28068
   862
    by (auto intro: ivl_disj_un_singleton)
nipkow@28068
   863
  hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
nipkow@28068
   864
    by (simp add: atLeast0LessThan)
nipkow@28068
   865
  also have "\<dots> = ?rhs" by simp
nipkow@28068
   866
  finally show ?thesis .
nipkow@28068
   867
qed
nipkow@28068
   868
nipkow@28068
   869
lemma setsum_head_Suc:
nipkow@28068
   870
  "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
nipkow@28068
   871
by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
nipkow@28068
   872
nipkow@28068
   873
lemma setsum_head_upt_Suc:
nipkow@28068
   874
  "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
huffman@30079
   875
apply(insert setsum_head_Suc[of m "n - Suc 0" f])
nipkow@29667
   876
apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
nipkow@28068
   877
done
nipkow@28068
   878
nipkow@31501
   879
lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"
nipkow@31501
   880
  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
nipkow@31501
   881
proof-
nipkow@31501
   882
  have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using `m \<le> n+1` by auto
nipkow@31501
   883
  thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint
nipkow@31501
   884
    atLeastSucAtMost_greaterThanAtMost)
nipkow@31501
   885
qed
nipkow@28068
   886
nipkow@15539
   887
lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
   888
  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
nipkow@15539
   889
by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
nipkow@15539
   890
nipkow@15539
   891
lemma setsum_diff_nat_ivl:
nipkow@15539
   892
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
nipkow@15539
   893
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
   894
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
nipkow@15539
   895
using setsum_add_nat_ivl [of m n p f,symmetric]
nipkow@15539
   896
apply (simp add: add_ac)
nipkow@15539
   897
done
nipkow@15539
   898
nipkow@31505
   899
lemma setsum_natinterval_difff:
nipkow@31505
   900
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
nipkow@31505
   901
  shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
nipkow@31505
   902
          (if m <= n then f m - f(n + 1) else 0)"
nipkow@31505
   903
by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
nipkow@31505
   904
nipkow@31509
   905
lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
nipkow@31509
   906
nipkow@31509
   907
lemma setsum_setsum_restrict:
nipkow@31509
   908
  "finite S \<Longrightarrow> finite T \<Longrightarrow> setsum (\<lambda>x. setsum (\<lambda>y. f x y) {y. y\<in> T \<and> R x y}) S = setsum (\<lambda>y. setsum (\<lambda>x. f x y) {x. x \<in> S \<and> R x y}) T"
nipkow@31509
   909
  by (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
nipkow@31509
   910
     (rule setsum_commute)
nipkow@31509
   911
nipkow@31509
   912
lemma setsum_image_gen: assumes fS: "finite S"
nipkow@31509
   913
  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
nipkow@31509
   914
proof-
nipkow@31509
   915
  { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
nipkow@31509
   916
  hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
nipkow@31509
   917
    by simp
nipkow@31509
   918
  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
nipkow@31509
   919
    by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])
nipkow@31509
   920
  finally show ?thesis .
nipkow@31509
   921
qed
nipkow@31509
   922
nipkow@31509
   923
lemma setsum_multicount_gen:
nipkow@31509
   924
  assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
nipkow@31509
   925
  shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
nipkow@31509
   926
proof-
nipkow@31509
   927
  have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
nipkow@31509
   928
  also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]
nipkow@31509
   929
    using assms(3) by auto
nipkow@31509
   930
  finally show ?thesis .
nipkow@31509
   931
qed
nipkow@31509
   932
nipkow@31509
   933
lemma setsum_multicount:
nipkow@31509
   934
  assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
nipkow@31509
   935
  shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
nipkow@31509
   936
proof-
nipkow@31509
   937
  have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)
nipkow@31509
   938
  also have "\<dots> = ?r" by(simp add: setsum_constant mult_commute)
nipkow@31509
   939
  finally show ?thesis by auto
nipkow@31509
   940
qed
nipkow@31509
   941
nipkow@28068
   942
nipkow@16733
   943
subsection{* Shifting bounds *}
nipkow@16733
   944
nipkow@15539
   945
lemma setsum_shift_bounds_nat_ivl:
nipkow@15539
   946
  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
nipkow@15539
   947
by (induct "n", auto simp:atLeastLessThanSuc)
nipkow@15539
   948
nipkow@16733
   949
lemma setsum_shift_bounds_cl_nat_ivl:
nipkow@16733
   950
  "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
nipkow@16733
   951
apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
nipkow@16733
   952
apply (simp add:image_add_atLeastAtMost o_def)
nipkow@16733
   953
done
nipkow@16733
   954
nipkow@16733
   955
corollary setsum_shift_bounds_cl_Suc_ivl:
nipkow@16733
   956
  "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
huffman@30079
   957
by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
   958
nipkow@16733
   959
corollary setsum_shift_bounds_Suc_ivl:
nipkow@16733
   960
  "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
huffman@30079
   961
by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
   962
nipkow@28068
   963
lemma setsum_shift_lb_Suc0_0:
nipkow@28068
   964
  "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
nipkow@28068
   965
by(simp add:setsum_head_Suc)
kleing@19106
   966
nipkow@28068
   967
lemma setsum_shift_lb_Suc0_0_upt:
nipkow@28068
   968
  "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
nipkow@28068
   969
apply(cases k)apply simp
nipkow@28068
   970
apply(simp add:setsum_head_upt_Suc)
nipkow@28068
   971
done
kleing@19022
   972
ballarin@17149
   973
subsection {* The formula for geometric sums *}
ballarin@17149
   974
ballarin@17149
   975
lemma geometric_sum:
ballarin@17149
   976
  "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =
haftmann@31017
   977
  (x ^ n - 1) / (x - 1::'a::{field})"
nipkow@23496
   978
by (induct "n") (simp_all add:field_simps power_Suc)
ballarin@17149
   979
kleing@19469
   980
subsection {* The formula for arithmetic sums *}
kleing@19469
   981
kleing@19469
   982
lemma gauss_sum:
huffman@23277
   983
  "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
kleing@19469
   984
   of_nat n*((of_nat n)+1)"
kleing@19469
   985
proof (induct n)
kleing@19469
   986
  case 0
kleing@19469
   987
  show ?case by simp
kleing@19469
   988
next
kleing@19469
   989
  case (Suc n)
nipkow@29667
   990
  then show ?case by (simp add: algebra_simps)
kleing@19469
   991
qed
kleing@19469
   992
kleing@19469
   993
theorem arith_series_general:
huffman@23277
   994
  "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
   995
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
   996
proof cases
kleing@19469
   997
  assume ngt1: "n > 1"
kleing@19469
   998
  let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
kleing@19469
   999
  have
kleing@19469
  1000
    "(\<Sum>i\<in>{..<n}. a+?I i*d) =
kleing@19469
  1001
     ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
kleing@19469
  1002
    by (rule setsum_addf)
kleing@19469
  1003
  also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
kleing@19469
  1004
  also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
huffman@30079
  1005
    unfolding One_nat_def
nipkow@28068
  1006
    by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)
kleing@19469
  1007
  also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
kleing@19469
  1008
    by (simp add: left_distrib right_distrib)
kleing@19469
  1009
  also from ngt1 have "{1..<n} = {1..n - 1}"
nipkow@28068
  1010
    by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
nipkow@28068
  1011
  also from ngt1
kleing@19469
  1012
  have "(1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..n - 1}. ?I i) = ((1+1)*?n*a + d*?I (n - 1)*?I n)"
huffman@30079
  1013
    by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)
huffman@23431
  1014
       (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
nipkow@29667
  1015
  finally show ?thesis by (simp add: algebra_simps)
kleing@19469
  1016
next
kleing@19469
  1017
  assume "\<not>(n > 1)"
kleing@19469
  1018
  hence "n = 1 \<or> n = 0" by auto
nipkow@29667
  1019
  thus ?thesis by (auto simp: algebra_simps)
kleing@19469
  1020
qed
kleing@19469
  1021
kleing@19469
  1022
lemma arith_series_nat:
kleing@19469
  1023
  "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
kleing@19469
  1024
proof -
kleing@19469
  1025
  have
kleing@19469
  1026
    "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
kleing@19469
  1027
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1028
    by (rule arith_series_general)
huffman@30079
  1029
  thus ?thesis
huffman@30079
  1030
    unfolding One_nat_def by (auto simp add: of_nat_id)
kleing@19469
  1031
qed
kleing@19469
  1032
kleing@19469
  1033
lemma arith_series_int:
kleing@19469
  1034
  "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1035
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1036
proof -
kleing@19469
  1037
  have
kleing@19469
  1038
    "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1039
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1040
    by (rule arith_series_general)
kleing@19469
  1041
  thus ?thesis by simp
kleing@19469
  1042
qed
paulson@15418
  1043
kleing@19022
  1044
lemma sum_diff_distrib:
kleing@19022
  1045
  fixes P::"nat\<Rightarrow>nat"
kleing@19022
  1046
  shows
kleing@19022
  1047
  "\<forall>x. Q x \<le> P x  \<Longrightarrow>
kleing@19022
  1048
  (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
kleing@19022
  1049
proof (induct n)
kleing@19022
  1050
  case 0 show ?case by simp
kleing@19022
  1051
next
kleing@19022
  1052
  case (Suc n)
kleing@19022
  1053
kleing@19022
  1054
  let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
kleing@19022
  1055
  let ?rhs = "\<Sum>x<n. P x - Q x"
kleing@19022
  1056
kleing@19022
  1057
  from Suc have "?lhs = ?rhs" by simp
kleing@19022
  1058
  moreover
kleing@19022
  1059
  from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
kleing@19022
  1060
  moreover
kleing@19022
  1061
  from Suc have
kleing@19022
  1062
    "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
kleing@19022
  1063
    by (subst diff_diff_left[symmetric],
kleing@19022
  1064
        subst diff_add_assoc2)
kleing@19022
  1065
       (auto simp: diff_add_assoc2 intro: setsum_mono)
kleing@19022
  1066
  ultimately
kleing@19022
  1067
  show ?case by simp
kleing@19022
  1068
qed
kleing@19022
  1069
paulson@29960
  1070
subsection {* Products indexed over intervals *}
paulson@29960
  1071
paulson@29960
  1072
syntax
paulson@29960
  1073
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1074
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1075
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)
paulson@29960
  1076
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
paulson@29960
  1077
syntax (xsymbols)
paulson@29960
  1078
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1079
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1080
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1081
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1082
syntax (HTML output)
paulson@29960
  1083
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1084
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1085
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1086
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1087
syntax (latex_prod output)
paulson@29960
  1088
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1089
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1090
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1091
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1092
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1093
 ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1094
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1095
 ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1096
paulson@29960
  1097
translations
paulson@29960
  1098
  "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"
paulson@29960
  1099
  "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"
paulson@29960
  1100
  "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"
paulson@29960
  1101
  "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"
paulson@29960
  1102
nipkow@8924
  1103
end