src/HOL/SetInterval.thy
author nipkow
Wed Aug 26 10:48:12 2009 +0200 (2009-08-26)
changeset 32400 6c62363cf0d7
parent 32006 0e209ff7f236
child 32408 a1a85b0a26f7
permissions -rw-r--r--
new lemmas
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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.
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  If we do so, a call to blast in Hyperreal/Star.ML, lemma @{text STAR_Int}
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  seems to take forever (more than one hour). *}
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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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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]
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        atLeast0AtMost[symmetric, code_unfold]
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lemma atLeastLessThan0: "{m..<0::nat} = {}"
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by (simp add: atLeastLessThan_def)
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subsubsection {* Intervals of nats with @{term Suc} *}
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text{*Not a simprule because the RHS is too messy.*}
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lemma atLeastLessThanSuc:
paulson@15047
   331
    "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
paulson@15418
   332
by (auto simp add: atLeastLessThan_def)
paulson@15047
   333
paulson@15418
   334
lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
paulson@15047
   335
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   336
(*
paulson@15047
   337
lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
paulson@15047
   338
by (induct k, simp_all add: atLeastLessThanSuc)
paulson@15047
   339
paulson@15047
   340
lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
paulson@15047
   341
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   342
*)
nipkow@15045
   343
lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
paulson@14485
   344
  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   345
paulson@15418
   346
lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
paulson@15418
   347
  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
paulson@14485
   348
    greaterThanAtMost_def)
paulson@14485
   349
paulson@15418
   350
lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
paulson@15418
   351
  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
paulson@14485
   352
    greaterThanLessThan_def)
paulson@14485
   353
nipkow@15554
   354
lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
nipkow@15554
   355
by (auto simp add: atLeastAtMost_def)
nipkow@15554
   356
nipkow@16733
   357
subsubsection {* Image *}
nipkow@16733
   358
nipkow@16733
   359
lemma image_add_atLeastAtMost:
nipkow@16733
   360
  "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
nipkow@16733
   361
proof
nipkow@16733
   362
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   363
next
nipkow@16733
   364
  show "?B \<subseteq> ?A"
nipkow@16733
   365
  proof
nipkow@16733
   366
    fix n assume a: "n : ?B"
webertj@20217
   367
    hence "n - k : {i..j}" by auto
nipkow@16733
   368
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   369
    ultimately show "n : ?A" by blast
nipkow@16733
   370
  qed
nipkow@16733
   371
qed
nipkow@16733
   372
nipkow@16733
   373
lemma image_add_atLeastLessThan:
nipkow@16733
   374
  "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
nipkow@16733
   375
proof
nipkow@16733
   376
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   377
next
nipkow@16733
   378
  show "?B \<subseteq> ?A"
nipkow@16733
   379
  proof
nipkow@16733
   380
    fix n assume a: "n : ?B"
webertj@20217
   381
    hence "n - k : {i..<j}" by auto
nipkow@16733
   382
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   383
    ultimately show "n : ?A" by blast
nipkow@16733
   384
  qed
nipkow@16733
   385
qed
nipkow@16733
   386
nipkow@16733
   387
corollary image_Suc_atLeastAtMost[simp]:
nipkow@16733
   388
  "Suc ` {i..j} = {Suc i..Suc j}"
huffman@30079
   389
using image_add_atLeastAtMost[where k="Suc 0"] by simp
nipkow@16733
   390
nipkow@16733
   391
corollary image_Suc_atLeastLessThan[simp]:
nipkow@16733
   392
  "Suc ` {i..<j} = {Suc i..<Suc j}"
huffman@30079
   393
using image_add_atLeastLessThan[where k="Suc 0"] by simp
nipkow@16733
   394
nipkow@16733
   395
lemma image_add_int_atLeastLessThan:
nipkow@16733
   396
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
nipkow@16733
   397
  apply (auto simp add: image_def)
nipkow@16733
   398
  apply (rule_tac x = "x - l" in bexI)
nipkow@16733
   399
  apply auto
nipkow@16733
   400
  done
nipkow@16733
   401
nipkow@16733
   402
paulson@14485
   403
subsubsection {* Finiteness *}
paulson@14485
   404
nipkow@15045
   405
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
paulson@14485
   406
  by (induct k) (simp_all add: lessThan_Suc)
paulson@14485
   407
paulson@14485
   408
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
paulson@14485
   409
  by (induct k) (simp_all add: atMost_Suc)
paulson@14485
   410
paulson@14485
   411
lemma finite_greaterThanLessThan [iff]:
nipkow@15045
   412
  fixes l :: nat shows "finite {l<..<u}"
paulson@14485
   413
by (simp add: greaterThanLessThan_def)
paulson@14485
   414
paulson@14485
   415
lemma finite_atLeastLessThan [iff]:
nipkow@15045
   416
  fixes l :: nat shows "finite {l..<u}"
paulson@14485
   417
by (simp add: atLeastLessThan_def)
paulson@14485
   418
paulson@14485
   419
lemma finite_greaterThanAtMost [iff]:
nipkow@15045
   420
  fixes l :: nat shows "finite {l<..u}"
paulson@14485
   421
by (simp add: greaterThanAtMost_def)
paulson@14485
   422
paulson@14485
   423
lemma finite_atLeastAtMost [iff]:
paulson@14485
   424
  fixes l :: nat shows "finite {l..u}"
paulson@14485
   425
by (simp add: atLeastAtMost_def)
paulson@14485
   426
nipkow@28068
   427
text {* A bounded set of natural numbers is finite. *}
paulson@14485
   428
lemma bounded_nat_set_is_finite:
nipkow@24853
   429
  "(ALL i:N. i < (n::nat)) ==> finite N"
nipkow@28068
   430
apply (rule finite_subset)
nipkow@28068
   431
 apply (rule_tac [2] finite_lessThan, auto)
nipkow@28068
   432
done
nipkow@28068
   433
nipkow@31044
   434
text {* A set of natural numbers is finite iff it is bounded. *}
nipkow@31044
   435
lemma finite_nat_set_iff_bounded:
nipkow@31044
   436
  "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
nipkow@31044
   437
proof
nipkow@31044
   438
  assume f:?F  show ?B
nipkow@31044
   439
    using Max_ge[OF `?F`, simplified less_Suc_eq_le[symmetric]] by blast
nipkow@31044
   440
next
nipkow@31044
   441
  assume ?B show ?F using `?B` by(blast intro:bounded_nat_set_is_finite)
nipkow@31044
   442
qed
nipkow@31044
   443
nipkow@31044
   444
lemma finite_nat_set_iff_bounded_le:
nipkow@31044
   445
  "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
nipkow@31044
   446
apply(simp add:finite_nat_set_iff_bounded)
nipkow@31044
   447
apply(blast dest:less_imp_le_nat le_imp_less_Suc)
nipkow@31044
   448
done
nipkow@31044
   449
nipkow@28068
   450
lemma finite_less_ub:
nipkow@28068
   451
     "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
nipkow@28068
   452
by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
paulson@14485
   453
nipkow@24853
   454
text{* Any subset of an interval of natural numbers the size of the
nipkow@24853
   455
subset is exactly that interval. *}
nipkow@24853
   456
nipkow@24853
   457
lemma subset_card_intvl_is_intvl:
nipkow@24853
   458
  "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")
nipkow@24853
   459
proof cases
nipkow@24853
   460
  assume "finite A"
nipkow@24853
   461
  thus "PROP ?P"
nipkow@32006
   462
  proof(induct A rule:finite_linorder_max_induct)
nipkow@24853
   463
    case empty thus ?case by auto
nipkow@24853
   464
  next
nipkow@24853
   465
    case (insert A b)
nipkow@24853
   466
    moreover hence "b ~: A" by auto
nipkow@24853
   467
    moreover have "A <= {k..<k+card A}" and "b = k+card A"
nipkow@24853
   468
      using `b ~: A` insert by fastsimp+
nipkow@24853
   469
    ultimately show ?case by auto
nipkow@24853
   470
  qed
nipkow@24853
   471
next
nipkow@24853
   472
  assume "~finite A" thus "PROP ?P" by simp
nipkow@24853
   473
qed
nipkow@24853
   474
nipkow@24853
   475
paulson@14485
   476
subsubsection {* Cardinality *}
paulson@14485
   477
nipkow@15045
   478
lemma card_lessThan [simp]: "card {..<u} = u"
paulson@15251
   479
  by (induct u, simp_all add: lessThan_Suc)
paulson@14485
   480
paulson@14485
   481
lemma card_atMost [simp]: "card {..u} = Suc u"
paulson@14485
   482
  by (simp add: lessThan_Suc_atMost [THEN sym])
paulson@14485
   483
nipkow@15045
   484
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
nipkow@15045
   485
  apply (subgoal_tac "card {l..<u} = card {..<u-l}")
paulson@14485
   486
  apply (erule ssubst, rule card_lessThan)
nipkow@15045
   487
  apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
paulson@14485
   488
  apply (erule subst)
paulson@14485
   489
  apply (rule card_image)
paulson@14485
   490
  apply (simp add: inj_on_def)
paulson@14485
   491
  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
paulson@14485
   492
  apply (rule_tac x = "x - l" in exI)
paulson@14485
   493
  apply arith
paulson@14485
   494
  done
paulson@14485
   495
paulson@15418
   496
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
paulson@14485
   497
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
paulson@14485
   498
paulson@15418
   499
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
paulson@14485
   500
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
paulson@14485
   501
nipkow@15045
   502
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
paulson@14485
   503
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
paulson@14485
   504
nipkow@26105
   505
lemma ex_bij_betw_nat_finite:
nipkow@26105
   506
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
nipkow@26105
   507
apply(drule finite_imp_nat_seg_image_inj_on)
nipkow@26105
   508
apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
nipkow@26105
   509
done
nipkow@26105
   510
nipkow@26105
   511
lemma ex_bij_betw_finite_nat:
nipkow@26105
   512
  "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
nipkow@26105
   513
by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
nipkow@26105
   514
nipkow@31438
   515
lemma finite_same_card_bij:
nipkow@31438
   516
  "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"
nipkow@31438
   517
apply(drule ex_bij_betw_finite_nat)
nipkow@31438
   518
apply(drule ex_bij_betw_nat_finite)
nipkow@31438
   519
apply(auto intro!:bij_betw_trans)
nipkow@31438
   520
done
nipkow@31438
   521
nipkow@31438
   522
lemma ex_bij_betw_nat_finite_1:
nipkow@31438
   523
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
nipkow@31438
   524
by (rule finite_same_card_bij) auto
nipkow@31438
   525
nipkow@26105
   526
paulson@14485
   527
subsection {* Intervals of integers *}
paulson@14485
   528
nipkow@15045
   529
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
paulson@14485
   530
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   531
paulson@15418
   532
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
paulson@14485
   533
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
paulson@14485
   534
paulson@15418
   535
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
paulson@15418
   536
    "{l+1..<u} = {l<..<u::int}"
paulson@14485
   537
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
paulson@14485
   538
paulson@14485
   539
subsubsection {* Finiteness *}
paulson@14485
   540
paulson@15418
   541
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
nipkow@15045
   542
    {(0::int)..<u} = int ` {..<nat u}"
paulson@14485
   543
  apply (unfold image_def lessThan_def)
paulson@14485
   544
  apply auto
paulson@14485
   545
  apply (rule_tac x = "nat x" in exI)
paulson@14485
   546
  apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
paulson@14485
   547
  done
paulson@14485
   548
nipkow@15045
   549
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
paulson@14485
   550
  apply (case_tac "0 \<le> u")
paulson@14485
   551
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   552
  apply (rule finite_imageI)
paulson@14485
   553
  apply auto
paulson@14485
   554
  done
paulson@14485
   555
nipkow@15045
   556
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
nipkow@15045
   557
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   558
  apply (erule subst)
paulson@14485
   559
  apply (rule finite_imageI)
paulson@14485
   560
  apply (rule finite_atLeastZeroLessThan_int)
nipkow@16733
   561
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   562
  done
paulson@14485
   563
paulson@15418
   564
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
paulson@14485
   565
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
paulson@14485
   566
paulson@15418
   567
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
paulson@14485
   568
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   569
paulson@15418
   570
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
paulson@14485
   571
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   572
nipkow@24853
   573
paulson@14485
   574
subsubsection {* Cardinality *}
paulson@14485
   575
nipkow@15045
   576
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
paulson@14485
   577
  apply (case_tac "0 \<le> u")
paulson@14485
   578
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   579
  apply (subst card_image)
paulson@14485
   580
  apply (auto simp add: inj_on_def)
paulson@14485
   581
  done
paulson@14485
   582
nipkow@15045
   583
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
nipkow@15045
   584
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
paulson@14485
   585
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
nipkow@15045
   586
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   587
  apply (erule subst)
paulson@14485
   588
  apply (rule card_image)
paulson@14485
   589
  apply (simp add: inj_on_def)
nipkow@16733
   590
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   591
  done
paulson@14485
   592
paulson@14485
   593
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
nipkow@29667
   594
apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
nipkow@29667
   595
apply (auto simp add: algebra_simps)
nipkow@29667
   596
done
paulson@14485
   597
paulson@15418
   598
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
nipkow@29667
   599
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   600
nipkow@15045
   601
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
nipkow@29667
   602
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   603
bulwahn@27656
   604
lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
bulwahn@27656
   605
proof -
bulwahn@27656
   606
  have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
bulwahn@27656
   607
  with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
bulwahn@27656
   608
qed
bulwahn@27656
   609
bulwahn@27656
   610
lemma card_less:
bulwahn@27656
   611
assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   612
shows "card {k \<in> M. k < Suc i} \<noteq> 0"
bulwahn@27656
   613
proof -
bulwahn@27656
   614
  from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
bulwahn@27656
   615
  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
bulwahn@27656
   616
qed
bulwahn@27656
   617
bulwahn@27656
   618
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
   619
apply (rule card_bij_eq [of "Suc" _ _ "\<lambda>x. x - Suc 0"])
bulwahn@27656
   620
apply simp
bulwahn@27656
   621
apply fastsimp
bulwahn@27656
   622
apply auto
bulwahn@27656
   623
apply (rule inj_on_diff_nat)
bulwahn@27656
   624
apply auto
bulwahn@27656
   625
apply (case_tac x)
bulwahn@27656
   626
apply auto
bulwahn@27656
   627
apply (case_tac xa)
bulwahn@27656
   628
apply auto
bulwahn@27656
   629
apply (case_tac xa)
bulwahn@27656
   630
apply auto
bulwahn@27656
   631
done
bulwahn@27656
   632
bulwahn@27656
   633
lemma card_less_Suc:
bulwahn@27656
   634
  assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   635
    shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
bulwahn@27656
   636
proof -
bulwahn@27656
   637
  from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
bulwahn@27656
   638
  hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
bulwahn@27656
   639
    by (auto simp only: insert_Diff)
bulwahn@27656
   640
  have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
bulwahn@27656
   641
  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
   642
    apply (subst card_insert)
bulwahn@27656
   643
    apply simp_all
bulwahn@27656
   644
    apply (subst b)
bulwahn@27656
   645
    apply (subst card_less_Suc2[symmetric])
bulwahn@27656
   646
    apply simp_all
bulwahn@27656
   647
    done
bulwahn@27656
   648
  with c show ?thesis by simp
bulwahn@27656
   649
qed
bulwahn@27656
   650
paulson@14485
   651
paulson@13850
   652
subsection {*Lemmas useful with the summation operator setsum*}
paulson@13850
   653
ballarin@16102
   654
text {* For examples, see Algebra/poly/UnivPoly2.thy *}
ballarin@13735
   655
wenzelm@14577
   656
subsubsection {* Disjoint Unions *}
ballarin@13735
   657
wenzelm@14577
   658
text {* Singletons and open intervals *}
ballarin@13735
   659
ballarin@13735
   660
lemma ivl_disj_un_singleton:
nipkow@15045
   661
  "{l::'a::linorder} Un {l<..} = {l..}"
nipkow@15045
   662
  "{..<u} Un {u::'a::linorder} = {..u}"
nipkow@15045
   663
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
nipkow@15045
   664
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
nipkow@15045
   665
  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
nipkow@15045
   666
  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
ballarin@14398
   667
by auto
ballarin@13735
   668
wenzelm@14577
   669
text {* One- and two-sided intervals *}
ballarin@13735
   670
ballarin@13735
   671
lemma ivl_disj_un_one:
nipkow@15045
   672
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
nipkow@15045
   673
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
nipkow@15045
   674
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
nipkow@15045
   675
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
nipkow@15045
   676
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
nipkow@15045
   677
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
nipkow@15045
   678
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
nipkow@15045
   679
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
ballarin@14398
   680
by auto
ballarin@13735
   681
wenzelm@14577
   682
text {* Two- and two-sided intervals *}
ballarin@13735
   683
ballarin@13735
   684
lemma ivl_disj_un_two:
nipkow@15045
   685
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
nipkow@15045
   686
  "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
nipkow@15045
   687
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
nipkow@15045
   688
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
nipkow@15045
   689
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
nipkow@15045
   690
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
nipkow@15045
   691
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
nipkow@15045
   692
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
ballarin@14398
   693
by auto
ballarin@13735
   694
ballarin@13735
   695
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
ballarin@13735
   696
wenzelm@14577
   697
subsubsection {* Disjoint Intersections *}
ballarin@13735
   698
wenzelm@14577
   699
text {* Singletons and open intervals *}
ballarin@13735
   700
ballarin@13735
   701
lemma ivl_disj_int_singleton:
nipkow@15045
   702
  "{l::'a::order} Int {l<..} = {}"
nipkow@15045
   703
  "{..<u} Int {u} = {}"
nipkow@15045
   704
  "{l} Int {l<..<u} = {}"
nipkow@15045
   705
  "{l<..<u} Int {u} = {}"
nipkow@15045
   706
  "{l} Int {l<..u} = {}"
nipkow@15045
   707
  "{l..<u} Int {u} = {}"
ballarin@13735
   708
  by simp+
ballarin@13735
   709
wenzelm@14577
   710
text {* One- and two-sided intervals *}
ballarin@13735
   711
ballarin@13735
   712
lemma ivl_disj_int_one:
nipkow@15045
   713
  "{..l::'a::order} Int {l<..<u} = {}"
nipkow@15045
   714
  "{..<l} Int {l..<u} = {}"
nipkow@15045
   715
  "{..l} Int {l<..u} = {}"
nipkow@15045
   716
  "{..<l} Int {l..u} = {}"
nipkow@15045
   717
  "{l<..u} Int {u<..} = {}"
nipkow@15045
   718
  "{l<..<u} Int {u..} = {}"
nipkow@15045
   719
  "{l..u} Int {u<..} = {}"
nipkow@15045
   720
  "{l..<u} Int {u..} = {}"
ballarin@14398
   721
  by auto
ballarin@13735
   722
wenzelm@14577
   723
text {* Two- and two-sided intervals *}
ballarin@13735
   724
ballarin@13735
   725
lemma ivl_disj_int_two:
nipkow@15045
   726
  "{l::'a::order<..<m} Int {m..<u} = {}"
nipkow@15045
   727
  "{l<..m} Int {m<..<u} = {}"
nipkow@15045
   728
  "{l..<m} Int {m..<u} = {}"
nipkow@15045
   729
  "{l..m} Int {m<..<u} = {}"
nipkow@15045
   730
  "{l<..<m} Int {m..u} = {}"
nipkow@15045
   731
  "{l<..m} Int {m<..u} = {}"
nipkow@15045
   732
  "{l..<m} Int {m..u} = {}"
nipkow@15045
   733
  "{l..m} Int {m<..u} = {}"
ballarin@14398
   734
  by auto
ballarin@13735
   735
ballarin@13735
   736
lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
ballarin@13735
   737
nipkow@15542
   738
subsubsection {* Some Differences *}
nipkow@15542
   739
nipkow@15542
   740
lemma ivl_diff[simp]:
nipkow@15542
   741
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
nipkow@15542
   742
by(auto)
nipkow@15542
   743
nipkow@15542
   744
nipkow@15542
   745
subsubsection {* Some Subset Conditions *}
nipkow@15542
   746
paulson@24286
   747
lemma ivl_subset [simp,noatp]:
nipkow@15542
   748
 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
nipkow@15542
   749
apply(auto simp:linorder_not_le)
nipkow@15542
   750
apply(rule ccontr)
nipkow@15542
   751
apply(insert linorder_le_less_linear[of i n])
nipkow@15542
   752
apply(clarsimp simp:linorder_not_le)
nipkow@15542
   753
apply(fastsimp)
nipkow@15542
   754
done
nipkow@15542
   755
nipkow@15041
   756
nipkow@15042
   757
subsection {* Summation indexed over intervals *}
nipkow@15042
   758
nipkow@15042
   759
syntax
nipkow@15042
   760
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   761
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   762
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
nipkow@16052
   763
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
nipkow@15042
   764
syntax (xsymbols)
nipkow@15042
   765
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   766
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   767
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
   768
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15042
   769
syntax (HTML output)
nipkow@15042
   770
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   771
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   772
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
   773
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15056
   774
syntax (latex_sum output)
nipkow@15052
   775
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   776
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052
   777
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   778
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@16052
   779
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
   780
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15052
   781
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
   782
 ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15041
   783
nipkow@15048
   784
translations
nipkow@28853
   785
  "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
nipkow@28853
   786
  "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
nipkow@28853
   787
  "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
nipkow@28853
   788
  "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
nipkow@15041
   789
nipkow@15052
   790
text{* The above introduces some pretty alternative syntaxes for
nipkow@15056
   791
summation over intervals:
nipkow@15052
   792
\begin{center}
nipkow@15052
   793
\begin{tabular}{lll}
nipkow@15056
   794
Old & New & \LaTeX\\
nipkow@15056
   795
@{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
   796
@{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
   797
@{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
nipkow@15056
   798
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
nipkow@15052
   799
\end{tabular}
nipkow@15052
   800
\end{center}
nipkow@15056
   801
The left column shows the term before introduction of the new syntax,
nipkow@15056
   802
the middle column shows the new (default) syntax, and the right column
nipkow@15056
   803
shows a special syntax. The latter is only meaningful for latex output
nipkow@15056
   804
and has to be activated explicitly by setting the print mode to
wenzelm@21502
   805
@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
nipkow@15056
   806
antiquotations). It is not the default \LaTeX\ output because it only
nipkow@15056
   807
works well with italic-style formulae, not tt-style.
nipkow@15052
   808
nipkow@15052
   809
Note that for uniformity on @{typ nat} it is better to use
nipkow@15052
   810
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
nipkow@15052
   811
not provide all lemmas available for @{term"{m..<n}"} also in the
nipkow@15052
   812
special form for @{term"{..<n}"}. *}
nipkow@15052
   813
nipkow@15542
   814
text{* This congruence rule should be used for sums over intervals as
nipkow@15542
   815
the standard theorem @{text[source]setsum_cong} does not work well
nipkow@15542
   816
with the simplifier who adds the unsimplified premise @{term"x:B"} to
nipkow@15542
   817
the context. *}
nipkow@15542
   818
nipkow@15542
   819
lemma setsum_ivl_cong:
nipkow@15542
   820
 "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
nipkow@15542
   821
 setsum f {a..<b} = setsum g {c..<d}"
nipkow@15542
   822
by(rule setsum_cong, simp_all)
nipkow@15041
   823
nipkow@16041
   824
(* FIXME why are the following simp rules but the corresponding eqns
nipkow@16041
   825
on intervals are not? *)
nipkow@16041
   826
nipkow@16052
   827
lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
nipkow@16052
   828
by (simp add:atMost_Suc add_ac)
nipkow@16052
   829
nipkow@16041
   830
lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
nipkow@16041
   831
by (simp add:lessThan_Suc add_ac)
nipkow@15041
   832
nipkow@15911
   833
lemma setsum_cl_ivl_Suc[simp]:
nipkow@15561
   834
  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
nipkow@15561
   835
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@15561
   836
nipkow@15911
   837
lemma setsum_op_ivl_Suc[simp]:
nipkow@15561
   838
  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
nipkow@15561
   839
by (auto simp:add_ac atLeastLessThanSuc)
nipkow@16041
   840
(*
nipkow@15561
   841
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
nipkow@15561
   842
    (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
nipkow@15561
   843
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@16041
   844
*)
nipkow@28068
   845
nipkow@28068
   846
lemma setsum_head:
nipkow@28068
   847
  fixes n :: nat
nipkow@28068
   848
  assumes mn: "m <= n" 
nipkow@28068
   849
  shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
nipkow@28068
   850
proof -
nipkow@28068
   851
  from mn
nipkow@28068
   852
  have "{m..n} = {m} \<union> {m<..n}"
nipkow@28068
   853
    by (auto intro: ivl_disj_un_singleton)
nipkow@28068
   854
  hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
nipkow@28068
   855
    by (simp add: atLeast0LessThan)
nipkow@28068
   856
  also have "\<dots> = ?rhs" by simp
nipkow@28068
   857
  finally show ?thesis .
nipkow@28068
   858
qed
nipkow@28068
   859
nipkow@28068
   860
lemma setsum_head_Suc:
nipkow@28068
   861
  "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
nipkow@28068
   862
by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
nipkow@28068
   863
nipkow@28068
   864
lemma setsum_head_upt_Suc:
nipkow@28068
   865
  "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
huffman@30079
   866
apply(insert setsum_head_Suc[of m "n - Suc 0" f])
nipkow@29667
   867
apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
nipkow@28068
   868
done
nipkow@28068
   869
nipkow@31501
   870
lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"
nipkow@31501
   871
  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
nipkow@31501
   872
proof-
nipkow@31501
   873
  have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using `m \<le> n+1` by auto
nipkow@31501
   874
  thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint
nipkow@31501
   875
    atLeastSucAtMost_greaterThanAtMost)
nipkow@31501
   876
qed
nipkow@28068
   877
nipkow@15539
   878
lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
   879
  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
nipkow@15539
   880
by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
nipkow@15539
   881
nipkow@15539
   882
lemma setsum_diff_nat_ivl:
nipkow@15539
   883
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
nipkow@15539
   884
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
   885
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
nipkow@15539
   886
using setsum_add_nat_ivl [of m n p f,symmetric]
nipkow@15539
   887
apply (simp add: add_ac)
nipkow@15539
   888
done
nipkow@15539
   889
nipkow@31505
   890
lemma setsum_natinterval_difff:
nipkow@31505
   891
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
nipkow@31505
   892
  shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
nipkow@31505
   893
          (if m <= n then f m - f(n + 1) else 0)"
nipkow@31505
   894
by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
nipkow@31505
   895
nipkow@31509
   896
lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
nipkow@31509
   897
nipkow@31509
   898
lemma setsum_setsum_restrict:
nipkow@31509
   899
  "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
   900
  by (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
nipkow@31509
   901
     (rule setsum_commute)
nipkow@31509
   902
nipkow@31509
   903
lemma setsum_image_gen: assumes fS: "finite S"
nipkow@31509
   904
  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
nipkow@31509
   905
proof-
nipkow@31509
   906
  { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
nipkow@31509
   907
  hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
nipkow@31509
   908
    by simp
nipkow@31509
   909
  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
nipkow@31509
   910
    by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])
nipkow@31509
   911
  finally show ?thesis .
nipkow@31509
   912
qed
nipkow@31509
   913
nipkow@31509
   914
lemma setsum_multicount_gen:
nipkow@31509
   915
  assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
nipkow@31509
   916
  shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
nipkow@31509
   917
proof-
nipkow@31509
   918
  have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
nipkow@31509
   919
  also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]
nipkow@31509
   920
    using assms(3) by auto
nipkow@31509
   921
  finally show ?thesis .
nipkow@31509
   922
qed
nipkow@31509
   923
nipkow@31509
   924
lemma setsum_multicount:
nipkow@31509
   925
  assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
nipkow@31509
   926
  shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
nipkow@31509
   927
proof-
nipkow@31509
   928
  have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)
nipkow@31509
   929
  also have "\<dots> = ?r" by(simp add: setsum_constant mult_commute)
nipkow@31509
   930
  finally show ?thesis by auto
nipkow@31509
   931
qed
nipkow@31509
   932
nipkow@28068
   933
nipkow@16733
   934
subsection{* Shifting bounds *}
nipkow@16733
   935
nipkow@15539
   936
lemma setsum_shift_bounds_nat_ivl:
nipkow@15539
   937
  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
nipkow@15539
   938
by (induct "n", auto simp:atLeastLessThanSuc)
nipkow@15539
   939
nipkow@16733
   940
lemma setsum_shift_bounds_cl_nat_ivl:
nipkow@16733
   941
  "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
nipkow@16733
   942
apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
nipkow@16733
   943
apply (simp add:image_add_atLeastAtMost o_def)
nipkow@16733
   944
done
nipkow@16733
   945
nipkow@16733
   946
corollary setsum_shift_bounds_cl_Suc_ivl:
nipkow@16733
   947
  "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
huffman@30079
   948
by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
   949
nipkow@16733
   950
corollary setsum_shift_bounds_Suc_ivl:
nipkow@16733
   951
  "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
huffman@30079
   952
by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
   953
nipkow@28068
   954
lemma setsum_shift_lb_Suc0_0:
nipkow@28068
   955
  "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
nipkow@28068
   956
by(simp add:setsum_head_Suc)
kleing@19106
   957
nipkow@28068
   958
lemma setsum_shift_lb_Suc0_0_upt:
nipkow@28068
   959
  "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
nipkow@28068
   960
apply(cases k)apply simp
nipkow@28068
   961
apply(simp add:setsum_head_upt_Suc)
nipkow@28068
   962
done
kleing@19022
   963
ballarin@17149
   964
subsection {* The formula for geometric sums *}
ballarin@17149
   965
ballarin@17149
   966
lemma geometric_sum:
ballarin@17149
   967
  "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =
haftmann@31017
   968
  (x ^ n - 1) / (x - 1::'a::{field})"
nipkow@23496
   969
by (induct "n") (simp_all add:field_simps power_Suc)
ballarin@17149
   970
kleing@19469
   971
subsection {* The formula for arithmetic sums *}
kleing@19469
   972
kleing@19469
   973
lemma gauss_sum:
huffman@23277
   974
  "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
kleing@19469
   975
   of_nat n*((of_nat n)+1)"
kleing@19469
   976
proof (induct n)
kleing@19469
   977
  case 0
kleing@19469
   978
  show ?case by simp
kleing@19469
   979
next
kleing@19469
   980
  case (Suc n)
nipkow@29667
   981
  then show ?case by (simp add: algebra_simps)
kleing@19469
   982
qed
kleing@19469
   983
kleing@19469
   984
theorem arith_series_general:
huffman@23277
   985
  "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
   986
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
   987
proof cases
kleing@19469
   988
  assume ngt1: "n > 1"
kleing@19469
   989
  let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
kleing@19469
   990
  have
kleing@19469
   991
    "(\<Sum>i\<in>{..<n}. a+?I i*d) =
kleing@19469
   992
     ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
kleing@19469
   993
    by (rule setsum_addf)
kleing@19469
   994
  also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
kleing@19469
   995
  also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
huffman@30079
   996
    unfolding One_nat_def
nipkow@28068
   997
    by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)
kleing@19469
   998
  also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
kleing@19469
   999
    by (simp add: left_distrib right_distrib)
kleing@19469
  1000
  also from ngt1 have "{1..<n} = {1..n - 1}"
nipkow@28068
  1001
    by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
nipkow@28068
  1002
  also from ngt1
kleing@19469
  1003
  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
  1004
    by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)
huffman@23431
  1005
       (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
nipkow@29667
  1006
  finally show ?thesis by (simp add: algebra_simps)
kleing@19469
  1007
next
kleing@19469
  1008
  assume "\<not>(n > 1)"
kleing@19469
  1009
  hence "n = 1 \<or> n = 0" by auto
nipkow@29667
  1010
  thus ?thesis by (auto simp: algebra_simps)
kleing@19469
  1011
qed
kleing@19469
  1012
kleing@19469
  1013
lemma arith_series_nat:
kleing@19469
  1014
  "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
kleing@19469
  1015
proof -
kleing@19469
  1016
  have
kleing@19469
  1017
    "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
kleing@19469
  1018
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1019
    by (rule arith_series_general)
huffman@30079
  1020
  thus ?thesis
huffman@30079
  1021
    unfolding One_nat_def by (auto simp add: of_nat_id)
kleing@19469
  1022
qed
kleing@19469
  1023
kleing@19469
  1024
lemma arith_series_int:
kleing@19469
  1025
  "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1026
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1027
proof -
kleing@19469
  1028
  have
kleing@19469
  1029
    "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1030
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1031
    by (rule arith_series_general)
kleing@19469
  1032
  thus ?thesis by simp
kleing@19469
  1033
qed
paulson@15418
  1034
kleing@19022
  1035
lemma sum_diff_distrib:
kleing@19022
  1036
  fixes P::"nat\<Rightarrow>nat"
kleing@19022
  1037
  shows
kleing@19022
  1038
  "\<forall>x. Q x \<le> P x  \<Longrightarrow>
kleing@19022
  1039
  (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
kleing@19022
  1040
proof (induct n)
kleing@19022
  1041
  case 0 show ?case by simp
kleing@19022
  1042
next
kleing@19022
  1043
  case (Suc n)
kleing@19022
  1044
kleing@19022
  1045
  let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
kleing@19022
  1046
  let ?rhs = "\<Sum>x<n. P x - Q x"
kleing@19022
  1047
kleing@19022
  1048
  from Suc have "?lhs = ?rhs" by simp
kleing@19022
  1049
  moreover
kleing@19022
  1050
  from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
kleing@19022
  1051
  moreover
kleing@19022
  1052
  from Suc have
kleing@19022
  1053
    "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
kleing@19022
  1054
    by (subst diff_diff_left[symmetric],
kleing@19022
  1055
        subst diff_add_assoc2)
kleing@19022
  1056
       (auto simp: diff_add_assoc2 intro: setsum_mono)
kleing@19022
  1057
  ultimately
kleing@19022
  1058
  show ?case by simp
kleing@19022
  1059
qed
kleing@19022
  1060
paulson@29960
  1061
subsection {* Products indexed over intervals *}
paulson@29960
  1062
paulson@29960
  1063
syntax
paulson@29960
  1064
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1065
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1066
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)
paulson@29960
  1067
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
paulson@29960
  1068
syntax (xsymbols)
paulson@29960
  1069
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1070
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1071
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1072
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1073
syntax (HTML output)
paulson@29960
  1074
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1075
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1076
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1077
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1078
syntax (latex_prod output)
paulson@29960
  1079
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1080
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1081
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1082
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1083
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1084
 ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1085
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1086
 ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1087
paulson@29960
  1088
translations
paulson@29960
  1089
  "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"
paulson@29960
  1090
  "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"
paulson@29960
  1091
  "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"
paulson@29960
  1092
  "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"
paulson@29960
  1093
nipkow@8924
  1094
end