src/HOL/SetInterval.thy
author nipkow
Thu Jun 04 13:26:32 2009 +0200 (2009-06-04)
changeset 31438 a1c4c1500abe
parent 31044 6896c2498ac0
child 31501 2a60c9b951e0
permissions -rw-r--r--
A few finite 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 and singletons *}
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context order
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begin
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lemma atLeastAtMost_empty [simp]: "n < m ==> {m..n} = {}";
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by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
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lemma atLeastLessThan_empty[simp]: "n \<le> m ==> {m..<n} = {}"
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by (auto simp add: atLeastLessThan_def)
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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 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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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:
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    "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
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by (auto simp add: atLeastLessThan_def)
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lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
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by (auto simp add: atLeastLessThan_def)
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(*
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lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
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by (induct k, simp_all add: atLeastLessThanSuc)
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lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
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by (auto simp add: atLeastLessThan_def)
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*)
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lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
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  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
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lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
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  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
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    greaterThanAtMost_def)
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lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
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  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
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    greaterThanLessThan_def)
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lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
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by (auto simp add: atLeastAtMost_def)
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subsubsection {* Image *}
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lemma image_add_atLeastAtMost:
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  "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
nipkow@16733
   327
proof
nipkow@16733
   328
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   329
next
nipkow@16733
   330
  show "?B \<subseteq> ?A"
nipkow@16733
   331
  proof
nipkow@16733
   332
    fix n assume a: "n : ?B"
webertj@20217
   333
    hence "n - k : {i..j}" by auto
nipkow@16733
   334
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   335
    ultimately show "n : ?A" by blast
nipkow@16733
   336
  qed
nipkow@16733
   337
qed
nipkow@16733
   338
nipkow@16733
   339
lemma image_add_atLeastLessThan:
nipkow@16733
   340
  "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
nipkow@16733
   341
proof
nipkow@16733
   342
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   343
next
nipkow@16733
   344
  show "?B \<subseteq> ?A"
nipkow@16733
   345
  proof
nipkow@16733
   346
    fix n assume a: "n : ?B"
webertj@20217
   347
    hence "n - k : {i..<j}" by auto
nipkow@16733
   348
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   349
    ultimately show "n : ?A" by blast
nipkow@16733
   350
  qed
nipkow@16733
   351
qed
nipkow@16733
   352
nipkow@16733
   353
corollary image_Suc_atLeastAtMost[simp]:
nipkow@16733
   354
  "Suc ` {i..j} = {Suc i..Suc j}"
huffman@30079
   355
using image_add_atLeastAtMost[where k="Suc 0"] by simp
nipkow@16733
   356
nipkow@16733
   357
corollary image_Suc_atLeastLessThan[simp]:
nipkow@16733
   358
  "Suc ` {i..<j} = {Suc i..<Suc j}"
huffman@30079
   359
using image_add_atLeastLessThan[where k="Suc 0"] by simp
nipkow@16733
   360
nipkow@16733
   361
lemma image_add_int_atLeastLessThan:
nipkow@16733
   362
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
nipkow@16733
   363
  apply (auto simp add: image_def)
nipkow@16733
   364
  apply (rule_tac x = "x - l" in bexI)
nipkow@16733
   365
  apply auto
nipkow@16733
   366
  done
nipkow@16733
   367
nipkow@16733
   368
paulson@14485
   369
subsubsection {* Finiteness *}
paulson@14485
   370
nipkow@15045
   371
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
paulson@14485
   372
  by (induct k) (simp_all add: lessThan_Suc)
paulson@14485
   373
paulson@14485
   374
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
paulson@14485
   375
  by (induct k) (simp_all add: atMost_Suc)
paulson@14485
   376
paulson@14485
   377
lemma finite_greaterThanLessThan [iff]:
nipkow@15045
   378
  fixes l :: nat shows "finite {l<..<u}"
paulson@14485
   379
by (simp add: greaterThanLessThan_def)
paulson@14485
   380
paulson@14485
   381
lemma finite_atLeastLessThan [iff]:
nipkow@15045
   382
  fixes l :: nat shows "finite {l..<u}"
paulson@14485
   383
by (simp add: atLeastLessThan_def)
paulson@14485
   384
paulson@14485
   385
lemma finite_greaterThanAtMost [iff]:
nipkow@15045
   386
  fixes l :: nat shows "finite {l<..u}"
paulson@14485
   387
by (simp add: greaterThanAtMost_def)
paulson@14485
   388
paulson@14485
   389
lemma finite_atLeastAtMost [iff]:
paulson@14485
   390
  fixes l :: nat shows "finite {l..u}"
paulson@14485
   391
by (simp add: atLeastAtMost_def)
paulson@14485
   392
nipkow@28068
   393
text {* A bounded set of natural numbers is finite. *}
paulson@14485
   394
lemma bounded_nat_set_is_finite:
nipkow@24853
   395
  "(ALL i:N. i < (n::nat)) ==> finite N"
nipkow@28068
   396
apply (rule finite_subset)
nipkow@28068
   397
 apply (rule_tac [2] finite_lessThan, auto)
nipkow@28068
   398
done
nipkow@28068
   399
nipkow@31044
   400
text {* A set of natural numbers is finite iff it is bounded. *}
nipkow@31044
   401
lemma finite_nat_set_iff_bounded:
nipkow@31044
   402
  "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
nipkow@31044
   403
proof
nipkow@31044
   404
  assume f:?F  show ?B
nipkow@31044
   405
    using Max_ge[OF `?F`, simplified less_Suc_eq_le[symmetric]] by blast
nipkow@31044
   406
next
nipkow@31044
   407
  assume ?B show ?F using `?B` by(blast intro:bounded_nat_set_is_finite)
nipkow@31044
   408
qed
nipkow@31044
   409
nipkow@31044
   410
lemma finite_nat_set_iff_bounded_le:
nipkow@31044
   411
  "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
nipkow@31044
   412
apply(simp add:finite_nat_set_iff_bounded)
nipkow@31044
   413
apply(blast dest:less_imp_le_nat le_imp_less_Suc)
nipkow@31044
   414
done
nipkow@31044
   415
nipkow@28068
   416
lemma finite_less_ub:
nipkow@28068
   417
     "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
nipkow@28068
   418
by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
paulson@14485
   419
nipkow@24853
   420
text{* Any subset of an interval of natural numbers the size of the
nipkow@24853
   421
subset is exactly that interval. *}
nipkow@24853
   422
nipkow@24853
   423
lemma subset_card_intvl_is_intvl:
nipkow@24853
   424
  "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")
nipkow@24853
   425
proof cases
nipkow@24853
   426
  assume "finite A"
nipkow@24853
   427
  thus "PROP ?P"
nipkow@24853
   428
  proof(induct A rule:finite_linorder_induct)
nipkow@24853
   429
    case empty thus ?case by auto
nipkow@24853
   430
  next
nipkow@24853
   431
    case (insert A b)
nipkow@24853
   432
    moreover hence "b ~: A" by auto
nipkow@24853
   433
    moreover have "A <= {k..<k+card A}" and "b = k+card A"
nipkow@24853
   434
      using `b ~: A` insert by fastsimp+
nipkow@24853
   435
    ultimately show ?case by auto
nipkow@24853
   436
  qed
nipkow@24853
   437
next
nipkow@24853
   438
  assume "~finite A" thus "PROP ?P" by simp
nipkow@24853
   439
qed
nipkow@24853
   440
nipkow@24853
   441
paulson@14485
   442
subsubsection {* Cardinality *}
paulson@14485
   443
nipkow@15045
   444
lemma card_lessThan [simp]: "card {..<u} = u"
paulson@15251
   445
  by (induct u, simp_all add: lessThan_Suc)
paulson@14485
   446
paulson@14485
   447
lemma card_atMost [simp]: "card {..u} = Suc u"
paulson@14485
   448
  by (simp add: lessThan_Suc_atMost [THEN sym])
paulson@14485
   449
nipkow@15045
   450
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
nipkow@15045
   451
  apply (subgoal_tac "card {l..<u} = card {..<u-l}")
paulson@14485
   452
  apply (erule ssubst, rule card_lessThan)
nipkow@15045
   453
  apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
paulson@14485
   454
  apply (erule subst)
paulson@14485
   455
  apply (rule card_image)
paulson@14485
   456
  apply (simp add: inj_on_def)
paulson@14485
   457
  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
paulson@14485
   458
  apply (rule_tac x = "x - l" in exI)
paulson@14485
   459
  apply arith
paulson@14485
   460
  done
paulson@14485
   461
paulson@15418
   462
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
paulson@14485
   463
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
paulson@14485
   464
paulson@15418
   465
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
paulson@14485
   466
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
paulson@14485
   467
nipkow@15045
   468
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
paulson@14485
   469
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
paulson@14485
   470
nipkow@26105
   471
lemma ex_bij_betw_nat_finite:
nipkow@26105
   472
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
nipkow@26105
   473
apply(drule finite_imp_nat_seg_image_inj_on)
nipkow@26105
   474
apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
nipkow@26105
   475
done
nipkow@26105
   476
nipkow@26105
   477
lemma ex_bij_betw_finite_nat:
nipkow@26105
   478
  "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
nipkow@26105
   479
by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
nipkow@26105
   480
nipkow@31438
   481
lemma finite_same_card_bij:
nipkow@31438
   482
  "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"
nipkow@31438
   483
apply(drule ex_bij_betw_finite_nat)
nipkow@31438
   484
apply(drule ex_bij_betw_nat_finite)
nipkow@31438
   485
apply(auto intro!:bij_betw_trans)
nipkow@31438
   486
done
nipkow@31438
   487
nipkow@31438
   488
lemma ex_bij_betw_nat_finite_1:
nipkow@31438
   489
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
nipkow@31438
   490
by (rule finite_same_card_bij) auto
nipkow@31438
   491
nipkow@26105
   492
paulson@14485
   493
subsection {* Intervals of integers *}
paulson@14485
   494
nipkow@15045
   495
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
paulson@14485
   496
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   497
paulson@15418
   498
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
paulson@14485
   499
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
paulson@14485
   500
paulson@15418
   501
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
paulson@15418
   502
    "{l+1..<u} = {l<..<u::int}"
paulson@14485
   503
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
paulson@14485
   504
paulson@14485
   505
subsubsection {* Finiteness *}
paulson@14485
   506
paulson@15418
   507
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
nipkow@15045
   508
    {(0::int)..<u} = int ` {..<nat u}"
paulson@14485
   509
  apply (unfold image_def lessThan_def)
paulson@14485
   510
  apply auto
paulson@14485
   511
  apply (rule_tac x = "nat x" in exI)
paulson@14485
   512
  apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
paulson@14485
   513
  done
paulson@14485
   514
nipkow@15045
   515
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
paulson@14485
   516
  apply (case_tac "0 \<le> u")
paulson@14485
   517
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   518
  apply (rule finite_imageI)
paulson@14485
   519
  apply auto
paulson@14485
   520
  done
paulson@14485
   521
nipkow@15045
   522
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
nipkow@15045
   523
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   524
  apply (erule subst)
paulson@14485
   525
  apply (rule finite_imageI)
paulson@14485
   526
  apply (rule finite_atLeastZeroLessThan_int)
nipkow@16733
   527
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   528
  done
paulson@14485
   529
paulson@15418
   530
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
paulson@14485
   531
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
paulson@14485
   532
paulson@15418
   533
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
paulson@14485
   534
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   535
paulson@15418
   536
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
paulson@14485
   537
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   538
nipkow@24853
   539
paulson@14485
   540
subsubsection {* Cardinality *}
paulson@14485
   541
nipkow@15045
   542
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
paulson@14485
   543
  apply (case_tac "0 \<le> u")
paulson@14485
   544
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   545
  apply (subst card_image)
paulson@14485
   546
  apply (auto simp add: inj_on_def)
paulson@14485
   547
  done
paulson@14485
   548
nipkow@15045
   549
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
nipkow@15045
   550
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
paulson@14485
   551
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
nipkow@15045
   552
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   553
  apply (erule subst)
paulson@14485
   554
  apply (rule card_image)
paulson@14485
   555
  apply (simp add: inj_on_def)
nipkow@16733
   556
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   557
  done
paulson@14485
   558
paulson@14485
   559
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
nipkow@29667
   560
apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
nipkow@29667
   561
apply (auto simp add: algebra_simps)
nipkow@29667
   562
done
paulson@14485
   563
paulson@15418
   564
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
nipkow@29667
   565
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   566
nipkow@15045
   567
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
nipkow@29667
   568
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   569
bulwahn@27656
   570
lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
bulwahn@27656
   571
proof -
bulwahn@27656
   572
  have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
bulwahn@27656
   573
  with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
bulwahn@27656
   574
qed
bulwahn@27656
   575
bulwahn@27656
   576
lemma card_less:
bulwahn@27656
   577
assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   578
shows "card {k \<in> M. k < Suc i} \<noteq> 0"
bulwahn@27656
   579
proof -
bulwahn@27656
   580
  from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
bulwahn@27656
   581
  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
bulwahn@27656
   582
qed
bulwahn@27656
   583
bulwahn@27656
   584
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
   585
apply (rule card_bij_eq [of "Suc" _ _ "\<lambda>x. x - Suc 0"])
bulwahn@27656
   586
apply simp
bulwahn@27656
   587
apply fastsimp
bulwahn@27656
   588
apply auto
bulwahn@27656
   589
apply (rule inj_on_diff_nat)
bulwahn@27656
   590
apply auto
bulwahn@27656
   591
apply (case_tac x)
bulwahn@27656
   592
apply auto
bulwahn@27656
   593
apply (case_tac xa)
bulwahn@27656
   594
apply auto
bulwahn@27656
   595
apply (case_tac xa)
bulwahn@27656
   596
apply auto
bulwahn@27656
   597
done
bulwahn@27656
   598
bulwahn@27656
   599
lemma card_less_Suc:
bulwahn@27656
   600
  assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   601
    shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
bulwahn@27656
   602
proof -
bulwahn@27656
   603
  from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
bulwahn@27656
   604
  hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
bulwahn@27656
   605
    by (auto simp only: insert_Diff)
bulwahn@27656
   606
  have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
bulwahn@27656
   607
  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
   608
    apply (subst card_insert)
bulwahn@27656
   609
    apply simp_all
bulwahn@27656
   610
    apply (subst b)
bulwahn@27656
   611
    apply (subst card_less_Suc2[symmetric])
bulwahn@27656
   612
    apply simp_all
bulwahn@27656
   613
    done
bulwahn@27656
   614
  with c show ?thesis by simp
bulwahn@27656
   615
qed
bulwahn@27656
   616
paulson@14485
   617
paulson@13850
   618
subsection {*Lemmas useful with the summation operator setsum*}
paulson@13850
   619
ballarin@16102
   620
text {* For examples, see Algebra/poly/UnivPoly2.thy *}
ballarin@13735
   621
wenzelm@14577
   622
subsubsection {* Disjoint Unions *}
ballarin@13735
   623
wenzelm@14577
   624
text {* Singletons and open intervals *}
ballarin@13735
   625
ballarin@13735
   626
lemma ivl_disj_un_singleton:
nipkow@15045
   627
  "{l::'a::linorder} Un {l<..} = {l..}"
nipkow@15045
   628
  "{..<u} Un {u::'a::linorder} = {..u}"
nipkow@15045
   629
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
nipkow@15045
   630
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
nipkow@15045
   631
  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
nipkow@15045
   632
  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
ballarin@14398
   633
by auto
ballarin@13735
   634
wenzelm@14577
   635
text {* One- and two-sided intervals *}
ballarin@13735
   636
ballarin@13735
   637
lemma ivl_disj_un_one:
nipkow@15045
   638
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
nipkow@15045
   639
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
nipkow@15045
   640
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
nipkow@15045
   641
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
nipkow@15045
   642
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
nipkow@15045
   643
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
nipkow@15045
   644
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
nipkow@15045
   645
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
ballarin@14398
   646
by auto
ballarin@13735
   647
wenzelm@14577
   648
text {* Two- and two-sided intervals *}
ballarin@13735
   649
ballarin@13735
   650
lemma ivl_disj_un_two:
nipkow@15045
   651
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
nipkow@15045
   652
  "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
nipkow@15045
   653
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
nipkow@15045
   654
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
nipkow@15045
   655
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
nipkow@15045
   656
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
nipkow@15045
   657
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
nipkow@15045
   658
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
ballarin@14398
   659
by auto
ballarin@13735
   660
ballarin@13735
   661
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
ballarin@13735
   662
wenzelm@14577
   663
subsubsection {* Disjoint Intersections *}
ballarin@13735
   664
wenzelm@14577
   665
text {* Singletons and open intervals *}
ballarin@13735
   666
ballarin@13735
   667
lemma ivl_disj_int_singleton:
nipkow@15045
   668
  "{l::'a::order} Int {l<..} = {}"
nipkow@15045
   669
  "{..<u} Int {u} = {}"
nipkow@15045
   670
  "{l} Int {l<..<u} = {}"
nipkow@15045
   671
  "{l<..<u} Int {u} = {}"
nipkow@15045
   672
  "{l} Int {l<..u} = {}"
nipkow@15045
   673
  "{l..<u} Int {u} = {}"
ballarin@13735
   674
  by simp+
ballarin@13735
   675
wenzelm@14577
   676
text {* One- and two-sided intervals *}
ballarin@13735
   677
ballarin@13735
   678
lemma ivl_disj_int_one:
nipkow@15045
   679
  "{..l::'a::order} Int {l<..<u} = {}"
nipkow@15045
   680
  "{..<l} Int {l..<u} = {}"
nipkow@15045
   681
  "{..l} Int {l<..u} = {}"
nipkow@15045
   682
  "{..<l} Int {l..u} = {}"
nipkow@15045
   683
  "{l<..u} Int {u<..} = {}"
nipkow@15045
   684
  "{l<..<u} Int {u..} = {}"
nipkow@15045
   685
  "{l..u} Int {u<..} = {}"
nipkow@15045
   686
  "{l..<u} Int {u..} = {}"
ballarin@14398
   687
  by auto
ballarin@13735
   688
wenzelm@14577
   689
text {* Two- and two-sided intervals *}
ballarin@13735
   690
ballarin@13735
   691
lemma ivl_disj_int_two:
nipkow@15045
   692
  "{l::'a::order<..<m} Int {m..<u} = {}"
nipkow@15045
   693
  "{l<..m} Int {m<..<u} = {}"
nipkow@15045
   694
  "{l..<m} Int {m..<u} = {}"
nipkow@15045
   695
  "{l..m} Int {m<..<u} = {}"
nipkow@15045
   696
  "{l<..<m} Int {m..u} = {}"
nipkow@15045
   697
  "{l<..m} Int {m<..u} = {}"
nipkow@15045
   698
  "{l..<m} Int {m..u} = {}"
nipkow@15045
   699
  "{l..m} Int {m<..u} = {}"
ballarin@14398
   700
  by auto
ballarin@13735
   701
ballarin@13735
   702
lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
ballarin@13735
   703
nipkow@15542
   704
subsubsection {* Some Differences *}
nipkow@15542
   705
nipkow@15542
   706
lemma ivl_diff[simp]:
nipkow@15542
   707
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
nipkow@15542
   708
by(auto)
nipkow@15542
   709
nipkow@15542
   710
nipkow@15542
   711
subsubsection {* Some Subset Conditions *}
nipkow@15542
   712
paulson@24286
   713
lemma ivl_subset [simp,noatp]:
nipkow@15542
   714
 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
nipkow@15542
   715
apply(auto simp:linorder_not_le)
nipkow@15542
   716
apply(rule ccontr)
nipkow@15542
   717
apply(insert linorder_le_less_linear[of i n])
nipkow@15542
   718
apply(clarsimp simp:linorder_not_le)
nipkow@15542
   719
apply(fastsimp)
nipkow@15542
   720
done
nipkow@15542
   721
nipkow@15041
   722
nipkow@15042
   723
subsection {* Summation indexed over intervals *}
nipkow@15042
   724
nipkow@15042
   725
syntax
nipkow@15042
   726
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   727
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   728
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
nipkow@16052
   729
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
nipkow@15042
   730
syntax (xsymbols)
nipkow@15042
   731
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   732
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   733
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
   734
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15042
   735
syntax (HTML output)
nipkow@15042
   736
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   737
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   738
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
   739
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15056
   740
syntax (latex_sum output)
nipkow@15052
   741
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   742
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052
   743
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   744
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@16052
   745
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
   746
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15052
   747
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
   748
 ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15041
   749
nipkow@15048
   750
translations
nipkow@28853
   751
  "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
nipkow@28853
   752
  "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
nipkow@28853
   753
  "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
nipkow@28853
   754
  "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
nipkow@15041
   755
nipkow@15052
   756
text{* The above introduces some pretty alternative syntaxes for
nipkow@15056
   757
summation over intervals:
nipkow@15052
   758
\begin{center}
nipkow@15052
   759
\begin{tabular}{lll}
nipkow@15056
   760
Old & New & \LaTeX\\
nipkow@15056
   761
@{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
   762
@{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
   763
@{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
nipkow@15056
   764
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
nipkow@15052
   765
\end{tabular}
nipkow@15052
   766
\end{center}
nipkow@15056
   767
The left column shows the term before introduction of the new syntax,
nipkow@15056
   768
the middle column shows the new (default) syntax, and the right column
nipkow@15056
   769
shows a special syntax. The latter is only meaningful for latex output
nipkow@15056
   770
and has to be activated explicitly by setting the print mode to
wenzelm@21502
   771
@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
nipkow@15056
   772
antiquotations). It is not the default \LaTeX\ output because it only
nipkow@15056
   773
works well with italic-style formulae, not tt-style.
nipkow@15052
   774
nipkow@15052
   775
Note that for uniformity on @{typ nat} it is better to use
nipkow@15052
   776
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
nipkow@15052
   777
not provide all lemmas available for @{term"{m..<n}"} also in the
nipkow@15052
   778
special form for @{term"{..<n}"}. *}
nipkow@15052
   779
nipkow@15542
   780
text{* This congruence rule should be used for sums over intervals as
nipkow@15542
   781
the standard theorem @{text[source]setsum_cong} does not work well
nipkow@15542
   782
with the simplifier who adds the unsimplified premise @{term"x:B"} to
nipkow@15542
   783
the context. *}
nipkow@15542
   784
nipkow@15542
   785
lemma setsum_ivl_cong:
nipkow@15542
   786
 "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
nipkow@15542
   787
 setsum f {a..<b} = setsum g {c..<d}"
nipkow@15542
   788
by(rule setsum_cong, simp_all)
nipkow@15041
   789
nipkow@16041
   790
(* FIXME why are the following simp rules but the corresponding eqns
nipkow@16041
   791
on intervals are not? *)
nipkow@16041
   792
nipkow@16052
   793
lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
nipkow@16052
   794
by (simp add:atMost_Suc add_ac)
nipkow@16052
   795
nipkow@16041
   796
lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
nipkow@16041
   797
by (simp add:lessThan_Suc add_ac)
nipkow@15041
   798
nipkow@15911
   799
lemma setsum_cl_ivl_Suc[simp]:
nipkow@15561
   800
  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
nipkow@15561
   801
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@15561
   802
nipkow@15911
   803
lemma setsum_op_ivl_Suc[simp]:
nipkow@15561
   804
  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
nipkow@15561
   805
by (auto simp:add_ac atLeastLessThanSuc)
nipkow@16041
   806
(*
nipkow@15561
   807
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
nipkow@15561
   808
    (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
nipkow@15561
   809
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@16041
   810
*)
nipkow@28068
   811
nipkow@28068
   812
lemma setsum_head:
nipkow@28068
   813
  fixes n :: nat
nipkow@28068
   814
  assumes mn: "m <= n" 
nipkow@28068
   815
  shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
nipkow@28068
   816
proof -
nipkow@28068
   817
  from mn
nipkow@28068
   818
  have "{m..n} = {m} \<union> {m<..n}"
nipkow@28068
   819
    by (auto intro: ivl_disj_un_singleton)
nipkow@28068
   820
  hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
nipkow@28068
   821
    by (simp add: atLeast0LessThan)
nipkow@28068
   822
  also have "\<dots> = ?rhs" by simp
nipkow@28068
   823
  finally show ?thesis .
nipkow@28068
   824
qed
nipkow@28068
   825
nipkow@28068
   826
lemma setsum_head_Suc:
nipkow@28068
   827
  "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
nipkow@28068
   828
by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
nipkow@28068
   829
nipkow@28068
   830
lemma setsum_head_upt_Suc:
nipkow@28068
   831
  "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
huffman@30079
   832
apply(insert setsum_head_Suc[of m "n - Suc 0" f])
nipkow@29667
   833
apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
nipkow@28068
   834
done
nipkow@28068
   835
nipkow@28068
   836
nipkow@15539
   837
lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
   838
  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
nipkow@15539
   839
by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
nipkow@15539
   840
nipkow@15539
   841
lemma setsum_diff_nat_ivl:
nipkow@15539
   842
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
nipkow@15539
   843
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
   844
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
nipkow@15539
   845
using setsum_add_nat_ivl [of m n p f,symmetric]
nipkow@15539
   846
apply (simp add: add_ac)
nipkow@15539
   847
done
nipkow@15539
   848
nipkow@28068
   849
nipkow@16733
   850
subsection{* Shifting bounds *}
nipkow@16733
   851
nipkow@15539
   852
lemma setsum_shift_bounds_nat_ivl:
nipkow@15539
   853
  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
nipkow@15539
   854
by (induct "n", auto simp:atLeastLessThanSuc)
nipkow@15539
   855
nipkow@16733
   856
lemma setsum_shift_bounds_cl_nat_ivl:
nipkow@16733
   857
  "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
nipkow@16733
   858
apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
nipkow@16733
   859
apply (simp add:image_add_atLeastAtMost o_def)
nipkow@16733
   860
done
nipkow@16733
   861
nipkow@16733
   862
corollary setsum_shift_bounds_cl_Suc_ivl:
nipkow@16733
   863
  "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
huffman@30079
   864
by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
   865
nipkow@16733
   866
corollary setsum_shift_bounds_Suc_ivl:
nipkow@16733
   867
  "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
huffman@30079
   868
by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
   869
nipkow@28068
   870
lemma setsum_shift_lb_Suc0_0:
nipkow@28068
   871
  "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
nipkow@28068
   872
by(simp add:setsum_head_Suc)
kleing@19106
   873
nipkow@28068
   874
lemma setsum_shift_lb_Suc0_0_upt:
nipkow@28068
   875
  "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
nipkow@28068
   876
apply(cases k)apply simp
nipkow@28068
   877
apply(simp add:setsum_head_upt_Suc)
nipkow@28068
   878
done
kleing@19022
   879
ballarin@17149
   880
subsection {* The formula for geometric sums *}
ballarin@17149
   881
ballarin@17149
   882
lemma geometric_sum:
ballarin@17149
   883
  "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =
haftmann@31017
   884
  (x ^ n - 1) / (x - 1::'a::{field})"
nipkow@23496
   885
by (induct "n") (simp_all add:field_simps power_Suc)
ballarin@17149
   886
kleing@19469
   887
subsection {* The formula for arithmetic sums *}
kleing@19469
   888
kleing@19469
   889
lemma gauss_sum:
huffman@23277
   890
  "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
kleing@19469
   891
   of_nat n*((of_nat n)+1)"
kleing@19469
   892
proof (induct n)
kleing@19469
   893
  case 0
kleing@19469
   894
  show ?case by simp
kleing@19469
   895
next
kleing@19469
   896
  case (Suc n)
nipkow@29667
   897
  then show ?case by (simp add: algebra_simps)
kleing@19469
   898
qed
kleing@19469
   899
kleing@19469
   900
theorem arith_series_general:
huffman@23277
   901
  "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
   902
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
   903
proof cases
kleing@19469
   904
  assume ngt1: "n > 1"
kleing@19469
   905
  let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
kleing@19469
   906
  have
kleing@19469
   907
    "(\<Sum>i\<in>{..<n}. a+?I i*d) =
kleing@19469
   908
     ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
kleing@19469
   909
    by (rule setsum_addf)
kleing@19469
   910
  also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
kleing@19469
   911
  also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
huffman@30079
   912
    unfolding One_nat_def
nipkow@28068
   913
    by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)
kleing@19469
   914
  also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
kleing@19469
   915
    by (simp add: left_distrib right_distrib)
kleing@19469
   916
  also from ngt1 have "{1..<n} = {1..n - 1}"
nipkow@28068
   917
    by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
nipkow@28068
   918
  also from ngt1
kleing@19469
   919
  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
   920
    by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)
huffman@23431
   921
       (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
nipkow@29667
   922
  finally show ?thesis by (simp add: algebra_simps)
kleing@19469
   923
next
kleing@19469
   924
  assume "\<not>(n > 1)"
kleing@19469
   925
  hence "n = 1 \<or> n = 0" by auto
nipkow@29667
   926
  thus ?thesis by (auto simp: algebra_simps)
kleing@19469
   927
qed
kleing@19469
   928
kleing@19469
   929
lemma arith_series_nat:
kleing@19469
   930
  "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
kleing@19469
   931
proof -
kleing@19469
   932
  have
kleing@19469
   933
    "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
kleing@19469
   934
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
   935
    by (rule arith_series_general)
huffman@30079
   936
  thus ?thesis
huffman@30079
   937
    unfolding One_nat_def by (auto simp add: of_nat_id)
kleing@19469
   938
qed
kleing@19469
   939
kleing@19469
   940
lemma arith_series_int:
kleing@19469
   941
  "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
   942
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
   943
proof -
kleing@19469
   944
  have
kleing@19469
   945
    "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
   946
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
   947
    by (rule arith_series_general)
kleing@19469
   948
  thus ?thesis by simp
kleing@19469
   949
qed
paulson@15418
   950
kleing@19022
   951
lemma sum_diff_distrib:
kleing@19022
   952
  fixes P::"nat\<Rightarrow>nat"
kleing@19022
   953
  shows
kleing@19022
   954
  "\<forall>x. Q x \<le> P x  \<Longrightarrow>
kleing@19022
   955
  (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
kleing@19022
   956
proof (induct n)
kleing@19022
   957
  case 0 show ?case by simp
kleing@19022
   958
next
kleing@19022
   959
  case (Suc n)
kleing@19022
   960
kleing@19022
   961
  let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
kleing@19022
   962
  let ?rhs = "\<Sum>x<n. P x - Q x"
kleing@19022
   963
kleing@19022
   964
  from Suc have "?lhs = ?rhs" by simp
kleing@19022
   965
  moreover
kleing@19022
   966
  from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
kleing@19022
   967
  moreover
kleing@19022
   968
  from Suc have
kleing@19022
   969
    "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
kleing@19022
   970
    by (subst diff_diff_left[symmetric],
kleing@19022
   971
        subst diff_add_assoc2)
kleing@19022
   972
       (auto simp: diff_add_assoc2 intro: setsum_mono)
kleing@19022
   973
  ultimately
kleing@19022
   974
  show ?case by simp
kleing@19022
   975
qed
kleing@19022
   976
paulson@29960
   977
subsection {* Products indexed over intervals *}
paulson@29960
   978
paulson@29960
   979
syntax
paulson@29960
   980
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
   981
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
   982
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)
paulson@29960
   983
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
paulson@29960
   984
syntax (xsymbols)
paulson@29960
   985
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
   986
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
   987
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
   988
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
   989
syntax (HTML output)
paulson@29960
   990
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
   991
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
   992
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
   993
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
   994
syntax (latex_prod output)
paulson@29960
   995
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
   996
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
   997
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
   998
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
   999
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1000
 ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1001
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1002
 ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1003
paulson@29960
  1004
translations
paulson@29960
  1005
  "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"
paulson@29960
  1006
  "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"
paulson@29960
  1007
  "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"
paulson@29960
  1008
  "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"
paulson@29960
  1009
nipkow@8924
  1010
end