src/HOL/SetInterval.thy
author hoelzl
Thu Jul 01 09:01:09 2010 +0200 (2010-07-01)
changeset 37664 2946b8f057df
parent 37388 793618618f78
child 39072 1030b1a166ef
permissions -rw-r--r--
Instantiate product type as euclidean space.
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(*  Title:      HOL/SetInterval.thy
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    Author:     Tobias Nipkow
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    Author:     Clemens Ballarin
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    Author:     Jeremy Avigad
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lessThan, greaterThan, atLeast, atMost and two-sided intervals
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*)
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header {* Set intervals *}
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theory SetInterval
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imports Int Nat_Transfer
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begin
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context ord
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begin
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definition
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  lessThan    :: "'a => 'a set" ("(1{..<_})") where
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  "{..<u} == {x. x < u}"
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definition
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  atMost      :: "'a => 'a set" ("(1{.._})") where
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  "{..u} == {x. x \<le> u}"
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definition
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  greaterThan :: "'a => 'a set" ("(1{_<..})") where
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  "{l<..} == {x. l<x}"
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definition
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  atLeast     :: "'a => 'a set" ("(1{_..})") where
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  "{l..} == {x. l\<le>x}"
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definition
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  greaterThanLessThan :: "'a => 'a => 'a set"  ("(1{_<..<_})") where
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  "{l<..<u} == {l<..} Int {..<u}"
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definition
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  atLeastLessThan :: "'a => 'a => 'a set"      ("(1{_..<_})") where
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  "{l..<u} == {l..} Int {..<u}"
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definition
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  greaterThanAtMost :: "'a => 'a => 'a set"    ("(1{_<.._})") where
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  "{l<..u} == {l<..} Int {..u}"
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definition
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  atLeastAtMost :: "'a => 'a => 'a set"        ("(1{_.._})") where
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  "{l..u} == {l..} Int {..u}"
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end
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text{* A note of warning when using @{term"{..<n}"} on type @{typ
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nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
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@{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
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syntax
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  "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3UN _<=_./ _)" [0, 0, 10] 10)
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  "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3UN _<_./ _)" [0, 0, 10] 10)
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  "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" [0, 0, 10] 10)
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  "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3INT _<_./ _)" [0, 0, 10] 10)
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syntax (xsymbols)
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  "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" [0, 0, 10] 10)
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  "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _<_./ _)" [0, 0, 10] 10)
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  "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" [0, 0, 10] 10)
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  "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" [0, 0, 10] 10)
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syntax (latex output)
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  "_UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ \<le> _)/ _)" [0, 0, 10] 10)
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  "_UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ < _)/ _)" [0, 0, 10] 10)
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  "_INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ \<le> _)/ _)" [0, 0, 10] 10)
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  "_INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ < _)/ _)" [0, 0, 10] 10)
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translations
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  "UN i<=n. A"  == "UN i:{..n}. A"
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  "UN i<n. A"   == "UN i:{..<n}. A"
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  "INT i<=n. A" == "INT i:{..n}. A"
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  "INT i<n. A"  == "INT i:{..<n}. A"
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subsection {* Various equivalences *}
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lemma (in ord) lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
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by (simp add: lessThan_def)
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lemma Compl_lessThan [simp]:
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    "!!k:: 'a::linorder. -lessThan k = atLeast k"
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apply (auto simp add: lessThan_def atLeast_def)
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done
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lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
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by auto
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lemma (in ord) greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
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by (simp add: greaterThan_def)
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lemma Compl_greaterThan [simp]:
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    "!!k:: 'a::linorder. -greaterThan k = atMost k"
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  by (auto simp add: greaterThan_def atMost_def)
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lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
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apply (subst Compl_greaterThan [symmetric])
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apply (rule double_complement)
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done
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lemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
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by (simp add: atLeast_def)
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lemma Compl_atLeast [simp]:
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    "!!k:: 'a::linorder. -atLeast k = lessThan k"
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  by (auto simp add: lessThan_def atLeast_def)
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lemma (in ord) atMost_iff [iff]: "(i: atMost k) = (i<=k)"
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by (simp add: atMost_def)
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lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
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by (blast intro: order_antisym)
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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,no_atp]:
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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,no_atp]:
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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,no_atp]:
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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,no_atp]:
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  "(i : {l..u}) = (l <= i & i <= u)"
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by (simp add: atLeastAtMost_def)
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text {* The above four lemmas could be declared as iffs. Unfortunately this
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breaks many proofs. Since it only helps blast, it is better to leave well
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alone *}
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end
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subsubsection{* Emptyness, singletons, subset *}
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context order
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begin
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lemma atLeastatMost_empty[simp]:
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  "b < a \<Longrightarrow> {a..b} = {}"
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by(auto simp: atLeastAtMost_def atLeast_def atMost_def)
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lemma atLeastatMost_empty_iff[simp]:
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  "{a..b} = {} \<longleftrightarrow> (~ a <= b)"
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by auto (blast intro: order_trans)
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lemma atLeastatMost_empty_iff2[simp]:
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  "{} = {a..b} \<longleftrightarrow> (~ a <= b)"
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by auto (blast intro: order_trans)
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lemma atLeastLessThan_empty[simp]:
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  "b <= a \<Longrightarrow> {a..<b} = {}"
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by(auto simp: atLeastLessThan_def)
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lemma atLeastLessThan_empty_iff[simp]:
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  "{a..<b} = {} \<longleftrightarrow> (~ a < b)"
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by auto (blast intro: le_less_trans)
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lemma atLeastLessThan_empty_iff2[simp]:
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  "{} = {a..<b} \<longleftrightarrow> (~ a < b)"
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by auto (blast intro: le_less_trans)
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lemma greaterThanAtMost_empty[simp]: "l \<le> k ==> {k<..l} = {}"
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by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
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lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {} \<longleftrightarrow> ~ k < l"
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by auto (blast intro: less_le_trans)
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lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} \<longleftrightarrow> ~ k < l"
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by auto (blast intro: less_le_trans)
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lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..<l} = {}"
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by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
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lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"
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by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
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lemma atLeastAtMost_singleton': "a = b \<Longrightarrow> {a .. b} = {a}" by simp
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lemma atLeastatMost_subset_iff[simp]:
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  "{a..b} <= {c..d} \<longleftrightarrow> (~ a <= b) | c <= a & b <= d"
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unfolding atLeastAtMost_def atLeast_def atMost_def
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by (blast intro: order_trans)
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lemma atLeastatMost_psubset_iff:
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  "{a..b} < {c..d} \<longleftrightarrow>
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   ((~ a <= b) | c <= a & b <= d & (c < a | b < d))  &  c <= d"
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by(simp add: psubset_eq expand_set_eq less_le_not_le)(blast intro: order_trans)
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lemma atLeastAtMost_singleton_iff[simp]:
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  "{a .. b} = {c} \<longleftrightarrow> a = b \<and> b = c"
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proof
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  assume "{a..b} = {c}"
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  hence "\<not> (\<not> a \<le> b)" unfolding atLeastatMost_empty_iff[symmetric] by simp
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  moreover with `{a..b} = {c}` have "c \<le> a \<and> b \<le> c" by auto
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  ultimately show "a = b \<and> b = c" by auto
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qed simp
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end
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lemma (in linorder) atLeastLessThan_subset_iff:
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  "{a..<b} <= {c..<d} \<Longrightarrow> b <= a | c<=a & b<=d"
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apply (auto simp:subset_eq Ball_def)
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apply(frule_tac x=a in spec)
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apply(erule_tac x=d in allE)
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apply (simp add: less_imp_le)
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done
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subsubsection {* Intersection *}
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context linorder
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begin
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lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"
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by auto
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lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"
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by auto
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lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"
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by auto
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lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"
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by auto
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lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"
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by auto
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lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"
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by auto
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lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
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by auto
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lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
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by auto
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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} *}
paulson@15047
   329
paulson@14485
   330
lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
paulson@14485
   331
by (unfold atLeast_def UNIV_def, simp)
paulson@14485
   332
paulson@14485
   333
lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
paulson@14485
   334
apply (simp add: atLeast_def)
paulson@14485
   335
apply (simp add: Suc_le_eq)
paulson@14485
   336
apply (simp add: order_le_less, blast)
paulson@14485
   337
done
paulson@14485
   338
paulson@14485
   339
lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
paulson@14485
   340
  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
paulson@14485
   341
paulson@14485
   342
lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
paulson@14485
   343
by blast
paulson@14485
   344
paulson@15047
   345
subsubsection {* The Constant @{term atMost} *}
paulson@15047
   346
paulson@14485
   347
lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
paulson@14485
   348
by (simp add: atMost_def)
paulson@14485
   349
paulson@14485
   350
lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
paulson@14485
   351
apply (simp add: atMost_def)
paulson@14485
   352
apply (simp add: less_Suc_eq order_le_less, blast)
paulson@14485
   353
done
paulson@14485
   354
paulson@14485
   355
lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
paulson@14485
   356
by blast
paulson@14485
   357
paulson@15047
   358
subsubsection {* The Constant @{term atLeastLessThan} *}
paulson@15047
   359
nipkow@28068
   360
text{*The orientation of the following 2 rules is tricky. The lhs is
nipkow@24449
   361
defined in terms of the rhs.  Hence the chosen orientation makes sense
nipkow@24449
   362
in this theory --- the reverse orientation complicates proofs (eg
nipkow@24449
   363
nontermination). But outside, when the definition of the lhs is rarely
nipkow@24449
   364
used, the opposite orientation seems preferable because it reduces a
nipkow@24449
   365
specific concept to a more general one. *}
nipkow@28068
   366
paulson@15047
   367
lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
nipkow@15042
   368
by(simp add:lessThan_def atLeastLessThan_def)
nipkow@24449
   369
nipkow@28068
   370
lemma atLeast0AtMost: "{0..n::nat} = {..n}"
nipkow@28068
   371
by(simp add:atMost_def atLeastAtMost_def)
nipkow@28068
   372
haftmann@31998
   373
declare atLeast0LessThan[symmetric, code_unfold]
haftmann@31998
   374
        atLeast0AtMost[symmetric, code_unfold]
nipkow@24449
   375
nipkow@24449
   376
lemma atLeastLessThan0: "{m..<0::nat} = {}"
paulson@15047
   377
by (simp add: atLeastLessThan_def)
nipkow@24449
   378
paulson@15047
   379
subsubsection {* Intervals of nats with @{term Suc} *}
paulson@15047
   380
paulson@15047
   381
text{*Not a simprule because the RHS is too messy.*}
paulson@15047
   382
lemma atLeastLessThanSuc:
paulson@15047
   383
    "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
paulson@15418
   384
by (auto simp add: atLeastLessThan_def)
paulson@15047
   385
paulson@15418
   386
lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
paulson@15047
   387
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   388
(*
paulson@15047
   389
lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
paulson@15047
   390
by (induct k, simp_all add: atLeastLessThanSuc)
paulson@15047
   391
paulson@15047
   392
lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
paulson@15047
   393
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   394
*)
nipkow@15045
   395
lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
paulson@14485
   396
  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   397
paulson@15418
   398
lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
paulson@15418
   399
  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
paulson@14485
   400
    greaterThanAtMost_def)
paulson@14485
   401
paulson@15418
   402
lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
paulson@15418
   403
  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
paulson@14485
   404
    greaterThanLessThan_def)
paulson@14485
   405
nipkow@15554
   406
lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
nipkow@15554
   407
by (auto simp add: atLeastAtMost_def)
nipkow@15554
   408
paulson@33044
   409
lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"
paulson@33044
   410
  apply (induct k) 
paulson@33044
   411
  apply (simp_all add: atLeastLessThanSuc)   
paulson@33044
   412
  done
paulson@33044
   413
nipkow@16733
   414
subsubsection {* Image *}
nipkow@16733
   415
nipkow@16733
   416
lemma image_add_atLeastAtMost:
nipkow@16733
   417
  "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
nipkow@16733
   418
proof
nipkow@16733
   419
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   420
next
nipkow@16733
   421
  show "?B \<subseteq> ?A"
nipkow@16733
   422
  proof
nipkow@16733
   423
    fix n assume a: "n : ?B"
webertj@20217
   424
    hence "n - k : {i..j}" by auto
nipkow@16733
   425
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   426
    ultimately show "n : ?A" by blast
nipkow@16733
   427
  qed
nipkow@16733
   428
qed
nipkow@16733
   429
nipkow@16733
   430
lemma image_add_atLeastLessThan:
nipkow@16733
   431
  "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
nipkow@16733
   432
proof
nipkow@16733
   433
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   434
next
nipkow@16733
   435
  show "?B \<subseteq> ?A"
nipkow@16733
   436
  proof
nipkow@16733
   437
    fix n assume a: "n : ?B"
webertj@20217
   438
    hence "n - k : {i..<j}" by auto
nipkow@16733
   439
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   440
    ultimately show "n : ?A" by blast
nipkow@16733
   441
  qed
nipkow@16733
   442
qed
nipkow@16733
   443
nipkow@16733
   444
corollary image_Suc_atLeastAtMost[simp]:
nipkow@16733
   445
  "Suc ` {i..j} = {Suc i..Suc j}"
huffman@30079
   446
using image_add_atLeastAtMost[where k="Suc 0"] by simp
nipkow@16733
   447
nipkow@16733
   448
corollary image_Suc_atLeastLessThan[simp]:
nipkow@16733
   449
  "Suc ` {i..<j} = {Suc i..<Suc j}"
huffman@30079
   450
using image_add_atLeastLessThan[where k="Suc 0"] by simp
nipkow@16733
   451
nipkow@16733
   452
lemma image_add_int_atLeastLessThan:
nipkow@16733
   453
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
nipkow@16733
   454
  apply (auto simp add: image_def)
nipkow@16733
   455
  apply (rule_tac x = "x - l" in bexI)
nipkow@16733
   456
  apply auto
nipkow@16733
   457
  done
nipkow@16733
   458
hoelzl@37664
   459
lemma image_minus_const_atLeastLessThan_nat:
hoelzl@37664
   460
  fixes c :: nat
hoelzl@37664
   461
  shows "(\<lambda>i. i - c) ` {x ..< y} =
hoelzl@37664
   462
      (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"
hoelzl@37664
   463
    (is "_ = ?right")
hoelzl@37664
   464
proof safe
hoelzl@37664
   465
  fix a assume a: "a \<in> ?right"
hoelzl@37664
   466
  show "a \<in> (\<lambda>i. i - c) ` {x ..< y}"
hoelzl@37664
   467
  proof cases
hoelzl@37664
   468
    assume "c < y" with a show ?thesis
hoelzl@37664
   469
      by (auto intro!: image_eqI[of _ _ "a + c"])
hoelzl@37664
   470
  next
hoelzl@37664
   471
    assume "\<not> c < y" with a show ?thesis
hoelzl@37664
   472
      by (auto intro!: image_eqI[of _ _ x] split: split_if_asm)
hoelzl@37664
   473
  qed
hoelzl@37664
   474
qed auto
hoelzl@37664
   475
hoelzl@35580
   476
context ordered_ab_group_add
hoelzl@35580
   477
begin
hoelzl@35580
   478
hoelzl@35580
   479
lemma
hoelzl@35580
   480
  fixes x :: 'a
hoelzl@35580
   481
  shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}"
hoelzl@35580
   482
  and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}"
hoelzl@35580
   483
proof safe
hoelzl@35580
   484
  fix y assume "y < -x"
hoelzl@35580
   485
  hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp
hoelzl@35580
   486
  have "- (-y) \<in> uminus ` {x<..}"
hoelzl@35580
   487
    by (rule imageI) (simp add: *)
hoelzl@35580
   488
  thus "y \<in> uminus ` {x<..}" by simp
hoelzl@35580
   489
next
hoelzl@35580
   490
  fix y assume "y \<le> -x"
hoelzl@35580
   491
  have "- (-y) \<in> uminus ` {x..}"
hoelzl@35580
   492
    by (rule imageI) (insert `y \<le> -x`[THEN le_imp_neg_le], simp)
hoelzl@35580
   493
  thus "y \<in> uminus ` {x..}" by simp
hoelzl@35580
   494
qed simp_all
hoelzl@35580
   495
hoelzl@35580
   496
lemma
hoelzl@35580
   497
  fixes x :: 'a
hoelzl@35580
   498
  shows image_uminus_lessThan[simp]: "uminus ` {..<x} = {-x<..}"
hoelzl@35580
   499
  and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"
hoelzl@35580
   500
proof -
hoelzl@35580
   501
  have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"
hoelzl@35580
   502
    and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all
hoelzl@35580
   503
  thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"
hoelzl@35580
   504
    by (simp_all add: image_image
hoelzl@35580
   505
        del: image_uminus_greaterThan image_uminus_atLeast)
hoelzl@35580
   506
qed
hoelzl@35580
   507
hoelzl@35580
   508
lemma
hoelzl@35580
   509
  fixes x :: 'a
hoelzl@35580
   510
  shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"
hoelzl@35580
   511
  and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"
hoelzl@35580
   512
  and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"
hoelzl@35580
   513
  and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"
hoelzl@35580
   514
  by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def
hoelzl@35580
   515
      greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)
hoelzl@35580
   516
end
nipkow@16733
   517
paulson@14485
   518
subsubsection {* Finiteness *}
paulson@14485
   519
nipkow@15045
   520
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
paulson@14485
   521
  by (induct k) (simp_all add: lessThan_Suc)
paulson@14485
   522
paulson@14485
   523
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
paulson@14485
   524
  by (induct k) (simp_all add: atMost_Suc)
paulson@14485
   525
paulson@14485
   526
lemma finite_greaterThanLessThan [iff]:
nipkow@15045
   527
  fixes l :: nat shows "finite {l<..<u}"
paulson@14485
   528
by (simp add: greaterThanLessThan_def)
paulson@14485
   529
paulson@14485
   530
lemma finite_atLeastLessThan [iff]:
nipkow@15045
   531
  fixes l :: nat shows "finite {l..<u}"
paulson@14485
   532
by (simp add: atLeastLessThan_def)
paulson@14485
   533
paulson@14485
   534
lemma finite_greaterThanAtMost [iff]:
nipkow@15045
   535
  fixes l :: nat shows "finite {l<..u}"
paulson@14485
   536
by (simp add: greaterThanAtMost_def)
paulson@14485
   537
paulson@14485
   538
lemma finite_atLeastAtMost [iff]:
paulson@14485
   539
  fixes l :: nat shows "finite {l..u}"
paulson@14485
   540
by (simp add: atLeastAtMost_def)
paulson@14485
   541
nipkow@28068
   542
text {* A bounded set of natural numbers is finite. *}
paulson@14485
   543
lemma bounded_nat_set_is_finite:
nipkow@24853
   544
  "(ALL i:N. i < (n::nat)) ==> finite N"
nipkow@28068
   545
apply (rule finite_subset)
nipkow@28068
   546
 apply (rule_tac [2] finite_lessThan, auto)
nipkow@28068
   547
done
nipkow@28068
   548
nipkow@31044
   549
text {* A set of natural numbers is finite iff it is bounded. *}
nipkow@31044
   550
lemma finite_nat_set_iff_bounded:
nipkow@31044
   551
  "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
nipkow@31044
   552
proof
nipkow@31044
   553
  assume f:?F  show ?B
nipkow@31044
   554
    using Max_ge[OF `?F`, simplified less_Suc_eq_le[symmetric]] by blast
nipkow@31044
   555
next
nipkow@31044
   556
  assume ?B show ?F using `?B` by(blast intro:bounded_nat_set_is_finite)
nipkow@31044
   557
qed
nipkow@31044
   558
nipkow@31044
   559
lemma finite_nat_set_iff_bounded_le:
nipkow@31044
   560
  "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
nipkow@31044
   561
apply(simp add:finite_nat_set_iff_bounded)
nipkow@31044
   562
apply(blast dest:less_imp_le_nat le_imp_less_Suc)
nipkow@31044
   563
done
nipkow@31044
   564
nipkow@28068
   565
lemma finite_less_ub:
nipkow@28068
   566
     "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
nipkow@28068
   567
by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
paulson@14485
   568
nipkow@24853
   569
text{* Any subset of an interval of natural numbers the size of the
nipkow@24853
   570
subset is exactly that interval. *}
nipkow@24853
   571
nipkow@24853
   572
lemma subset_card_intvl_is_intvl:
nipkow@24853
   573
  "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")
nipkow@24853
   574
proof cases
nipkow@24853
   575
  assume "finite A"
nipkow@24853
   576
  thus "PROP ?P"
nipkow@32006
   577
  proof(induct A rule:finite_linorder_max_induct)
nipkow@24853
   578
    case empty thus ?case by auto
nipkow@24853
   579
  next
nipkow@33434
   580
    case (insert b A)
nipkow@24853
   581
    moreover hence "b ~: A" by auto
nipkow@24853
   582
    moreover have "A <= {k..<k+card A}" and "b = k+card A"
nipkow@24853
   583
      using `b ~: A` insert by fastsimp+
nipkow@24853
   584
    ultimately show ?case by auto
nipkow@24853
   585
  qed
nipkow@24853
   586
next
nipkow@24853
   587
  assume "~finite A" thus "PROP ?P" by simp
nipkow@24853
   588
qed
nipkow@24853
   589
nipkow@24853
   590
paulson@32596
   591
subsubsection {* Proving Inclusions and Equalities between Unions *}
paulson@32596
   592
nipkow@36755
   593
lemma UN_le_eq_Un0:
nipkow@36755
   594
  "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> M 0" (is "?A = ?B")
nipkow@36755
   595
proof
nipkow@36755
   596
  show "?A <= ?B"
nipkow@36755
   597
  proof
nipkow@36755
   598
    fix x assume "x : ?A"
nipkow@36755
   599
    then obtain i where i: "i\<le>n" "x : M i" by auto
nipkow@36755
   600
    show "x : ?B"
nipkow@36755
   601
    proof(cases i)
nipkow@36755
   602
      case 0 with i show ?thesis by simp
nipkow@36755
   603
    next
nipkow@36755
   604
      case (Suc j) with i show ?thesis by auto
nipkow@36755
   605
    qed
nipkow@36755
   606
  qed
nipkow@36755
   607
next
nipkow@36755
   608
  show "?B <= ?A" by auto
nipkow@36755
   609
qed
nipkow@36755
   610
nipkow@36755
   611
lemma UN_le_add_shift:
nipkow@36755
   612
  "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")
nipkow@36755
   613
proof
nipkow@36755
   614
  show "?A <= ?B" by fastsimp
nipkow@36755
   615
next
nipkow@36755
   616
  show "?B <= ?A"
nipkow@36755
   617
  proof
nipkow@36755
   618
    fix x assume "x : ?B"
nipkow@36755
   619
    then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto
nipkow@36755
   620
    hence "i-k\<le>n & x : M((i-k)+k)" by auto
nipkow@36755
   621
    thus "x : ?A" by blast
nipkow@36755
   622
  qed
nipkow@36755
   623
qed
nipkow@36755
   624
paulson@32596
   625
lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"
paulson@32596
   626
  by (auto simp add: atLeast0LessThan) 
paulson@32596
   627
paulson@32596
   628
lemma UN_finite_subset: "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"
paulson@32596
   629
  by (subst UN_UN_finite_eq [symmetric]) blast
paulson@32596
   630
paulson@33044
   631
lemma UN_finite2_subset: 
paulson@33044
   632
     "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n+k}. B i)) \<Longrightarrow> (\<Union>n. A n) \<subseteq> (\<Union>n. B n)"
paulson@33044
   633
  apply (rule UN_finite_subset)
paulson@33044
   634
  apply (subst UN_UN_finite_eq [symmetric, of B]) 
paulson@33044
   635
  apply blast
paulson@33044
   636
  done
paulson@32596
   637
paulson@32596
   638
lemma UN_finite2_eq:
paulson@33044
   639
  "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) = (\<Union>i\<in>{0..<n+k}. B i)) \<Longrightarrow> (\<Union>n. A n) = (\<Union>n. B n)"
paulson@33044
   640
  apply (rule subset_antisym)
paulson@33044
   641
   apply (rule UN_finite2_subset, blast)
paulson@33044
   642
 apply (rule UN_finite2_subset [where k=k])
huffman@35216
   643
 apply (force simp add: atLeastLessThan_add_Un [of 0])
paulson@33044
   644
 done
paulson@32596
   645
paulson@32596
   646
paulson@14485
   647
subsubsection {* Cardinality *}
paulson@14485
   648
nipkow@15045
   649
lemma card_lessThan [simp]: "card {..<u} = u"
paulson@15251
   650
  by (induct u, simp_all add: lessThan_Suc)
paulson@14485
   651
paulson@14485
   652
lemma card_atMost [simp]: "card {..u} = Suc u"
paulson@14485
   653
  by (simp add: lessThan_Suc_atMost [THEN sym])
paulson@14485
   654
nipkow@15045
   655
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
nipkow@15045
   656
  apply (subgoal_tac "card {l..<u} = card {..<u-l}")
paulson@14485
   657
  apply (erule ssubst, rule card_lessThan)
nipkow@15045
   658
  apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
paulson@14485
   659
  apply (erule subst)
paulson@14485
   660
  apply (rule card_image)
paulson@14485
   661
  apply (simp add: inj_on_def)
paulson@14485
   662
  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
paulson@14485
   663
  apply (rule_tac x = "x - l" in exI)
paulson@14485
   664
  apply arith
paulson@14485
   665
  done
paulson@14485
   666
paulson@15418
   667
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
paulson@14485
   668
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
paulson@14485
   669
paulson@15418
   670
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
paulson@14485
   671
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
paulson@14485
   672
nipkow@15045
   673
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
paulson@14485
   674
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
paulson@14485
   675
nipkow@26105
   676
lemma ex_bij_betw_nat_finite:
nipkow@26105
   677
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
nipkow@26105
   678
apply(drule finite_imp_nat_seg_image_inj_on)
nipkow@26105
   679
apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
nipkow@26105
   680
done
nipkow@26105
   681
nipkow@26105
   682
lemma ex_bij_betw_finite_nat:
nipkow@26105
   683
  "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
nipkow@26105
   684
by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
nipkow@26105
   685
nipkow@31438
   686
lemma finite_same_card_bij:
nipkow@31438
   687
  "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"
nipkow@31438
   688
apply(drule ex_bij_betw_finite_nat)
nipkow@31438
   689
apply(drule ex_bij_betw_nat_finite)
nipkow@31438
   690
apply(auto intro!:bij_betw_trans)
nipkow@31438
   691
done
nipkow@31438
   692
nipkow@31438
   693
lemma ex_bij_betw_nat_finite_1:
nipkow@31438
   694
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
nipkow@31438
   695
by (rule finite_same_card_bij) auto
nipkow@31438
   696
nipkow@26105
   697
paulson@14485
   698
subsection {* Intervals of integers *}
paulson@14485
   699
nipkow@15045
   700
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
paulson@14485
   701
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   702
paulson@15418
   703
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
paulson@14485
   704
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
paulson@14485
   705
paulson@15418
   706
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
paulson@15418
   707
    "{l+1..<u} = {l<..<u::int}"
paulson@14485
   708
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
paulson@14485
   709
paulson@14485
   710
subsubsection {* Finiteness *}
paulson@14485
   711
paulson@15418
   712
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
nipkow@15045
   713
    {(0::int)..<u} = int ` {..<nat u}"
paulson@14485
   714
  apply (unfold image_def lessThan_def)
paulson@14485
   715
  apply auto
paulson@14485
   716
  apply (rule_tac x = "nat x" in exI)
huffman@35216
   717
  apply (auto simp add: zless_nat_eq_int_zless [THEN sym])
paulson@14485
   718
  done
paulson@14485
   719
nipkow@15045
   720
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
paulson@14485
   721
  apply (case_tac "0 \<le> u")
paulson@14485
   722
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   723
  apply (rule finite_imageI)
paulson@14485
   724
  apply auto
paulson@14485
   725
  done
paulson@14485
   726
nipkow@15045
   727
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
nipkow@15045
   728
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   729
  apply (erule subst)
paulson@14485
   730
  apply (rule finite_imageI)
paulson@14485
   731
  apply (rule finite_atLeastZeroLessThan_int)
nipkow@16733
   732
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   733
  done
paulson@14485
   734
paulson@15418
   735
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
paulson@14485
   736
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
paulson@14485
   737
paulson@15418
   738
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
paulson@14485
   739
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   740
paulson@15418
   741
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
paulson@14485
   742
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   743
nipkow@24853
   744
paulson@14485
   745
subsubsection {* Cardinality *}
paulson@14485
   746
nipkow@15045
   747
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
paulson@14485
   748
  apply (case_tac "0 \<le> u")
paulson@14485
   749
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   750
  apply (subst card_image)
paulson@14485
   751
  apply (auto simp add: inj_on_def)
paulson@14485
   752
  done
paulson@14485
   753
nipkow@15045
   754
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
nipkow@15045
   755
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
paulson@14485
   756
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
nipkow@15045
   757
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   758
  apply (erule subst)
paulson@14485
   759
  apply (rule card_image)
paulson@14485
   760
  apply (simp add: inj_on_def)
nipkow@16733
   761
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   762
  done
paulson@14485
   763
paulson@14485
   764
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
nipkow@29667
   765
apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
nipkow@29667
   766
apply (auto simp add: algebra_simps)
nipkow@29667
   767
done
paulson@14485
   768
paulson@15418
   769
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
nipkow@29667
   770
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   771
nipkow@15045
   772
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
nipkow@29667
   773
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   774
bulwahn@27656
   775
lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
bulwahn@27656
   776
proof -
bulwahn@27656
   777
  have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
bulwahn@27656
   778
  with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
bulwahn@27656
   779
qed
bulwahn@27656
   780
bulwahn@27656
   781
lemma card_less:
bulwahn@27656
   782
assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   783
shows "card {k \<in> M. k < Suc i} \<noteq> 0"
bulwahn@27656
   784
proof -
bulwahn@27656
   785
  from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
bulwahn@27656
   786
  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
bulwahn@27656
   787
qed
bulwahn@27656
   788
bulwahn@27656
   789
lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"
haftmann@37388
   790
apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])
bulwahn@27656
   791
apply simp
bulwahn@27656
   792
apply fastsimp
bulwahn@27656
   793
apply auto
bulwahn@27656
   794
apply (rule inj_on_diff_nat)
bulwahn@27656
   795
apply auto
bulwahn@27656
   796
apply (case_tac x)
bulwahn@27656
   797
apply auto
bulwahn@27656
   798
apply (case_tac xa)
bulwahn@27656
   799
apply auto
bulwahn@27656
   800
apply (case_tac xa)
bulwahn@27656
   801
apply auto
bulwahn@27656
   802
done
bulwahn@27656
   803
bulwahn@27656
   804
lemma card_less_Suc:
bulwahn@27656
   805
  assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   806
    shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
bulwahn@27656
   807
proof -
bulwahn@27656
   808
  from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
bulwahn@27656
   809
  hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
bulwahn@27656
   810
    by (auto simp only: insert_Diff)
bulwahn@27656
   811
  have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
bulwahn@27656
   812
  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
   813
    apply (subst card_insert)
bulwahn@27656
   814
    apply simp_all
bulwahn@27656
   815
    apply (subst b)
bulwahn@27656
   816
    apply (subst card_less_Suc2[symmetric])
bulwahn@27656
   817
    apply simp_all
bulwahn@27656
   818
    done
bulwahn@27656
   819
  with c show ?thesis by simp
bulwahn@27656
   820
qed
bulwahn@27656
   821
paulson@14485
   822
paulson@13850
   823
subsection {*Lemmas useful with the summation operator setsum*}
paulson@13850
   824
ballarin@16102
   825
text {* For examples, see Algebra/poly/UnivPoly2.thy *}
ballarin@13735
   826
wenzelm@14577
   827
subsubsection {* Disjoint Unions *}
ballarin@13735
   828
wenzelm@14577
   829
text {* Singletons and open intervals *}
ballarin@13735
   830
ballarin@13735
   831
lemma ivl_disj_un_singleton:
nipkow@15045
   832
  "{l::'a::linorder} Un {l<..} = {l..}"
nipkow@15045
   833
  "{..<u} Un {u::'a::linorder} = {..u}"
nipkow@15045
   834
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
nipkow@15045
   835
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
nipkow@15045
   836
  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
nipkow@15045
   837
  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
ballarin@14398
   838
by auto
ballarin@13735
   839
wenzelm@14577
   840
text {* One- and two-sided intervals *}
ballarin@13735
   841
ballarin@13735
   842
lemma ivl_disj_un_one:
nipkow@15045
   843
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
nipkow@15045
   844
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
nipkow@15045
   845
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
nipkow@15045
   846
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
nipkow@15045
   847
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
nipkow@15045
   848
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
nipkow@15045
   849
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
nipkow@15045
   850
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
ballarin@14398
   851
by auto
ballarin@13735
   852
wenzelm@14577
   853
text {* Two- and two-sided intervals *}
ballarin@13735
   854
ballarin@13735
   855
lemma ivl_disj_un_two:
nipkow@15045
   856
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
nipkow@15045
   857
  "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
nipkow@15045
   858
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
nipkow@15045
   859
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
nipkow@15045
   860
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
nipkow@15045
   861
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
nipkow@15045
   862
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
nipkow@15045
   863
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
ballarin@14398
   864
by auto
ballarin@13735
   865
ballarin@13735
   866
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
ballarin@13735
   867
wenzelm@14577
   868
subsubsection {* Disjoint Intersections *}
ballarin@13735
   869
wenzelm@14577
   870
text {* One- and two-sided intervals *}
ballarin@13735
   871
ballarin@13735
   872
lemma ivl_disj_int_one:
nipkow@15045
   873
  "{..l::'a::order} Int {l<..<u} = {}"
nipkow@15045
   874
  "{..<l} Int {l..<u} = {}"
nipkow@15045
   875
  "{..l} Int {l<..u} = {}"
nipkow@15045
   876
  "{..<l} Int {l..u} = {}"
nipkow@15045
   877
  "{l<..u} Int {u<..} = {}"
nipkow@15045
   878
  "{l<..<u} Int {u..} = {}"
nipkow@15045
   879
  "{l..u} Int {u<..} = {}"
nipkow@15045
   880
  "{l..<u} Int {u..} = {}"
ballarin@14398
   881
  by auto
ballarin@13735
   882
wenzelm@14577
   883
text {* Two- and two-sided intervals *}
ballarin@13735
   884
ballarin@13735
   885
lemma ivl_disj_int_two:
nipkow@15045
   886
  "{l::'a::order<..<m} Int {m..<u} = {}"
nipkow@15045
   887
  "{l<..m} Int {m<..<u} = {}"
nipkow@15045
   888
  "{l..<m} Int {m..<u} = {}"
nipkow@15045
   889
  "{l..m} Int {m<..<u} = {}"
nipkow@15045
   890
  "{l<..<m} Int {m..u} = {}"
nipkow@15045
   891
  "{l<..m} Int {m<..u} = {}"
nipkow@15045
   892
  "{l..<m} Int {m..u} = {}"
nipkow@15045
   893
  "{l..m} Int {m<..u} = {}"
ballarin@14398
   894
  by auto
ballarin@13735
   895
nipkow@32456
   896
lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two
ballarin@13735
   897
nipkow@15542
   898
subsubsection {* Some Differences *}
nipkow@15542
   899
nipkow@15542
   900
lemma ivl_diff[simp]:
nipkow@15542
   901
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
nipkow@15542
   902
by(auto)
nipkow@15542
   903
nipkow@15542
   904
nipkow@15542
   905
subsubsection {* Some Subset Conditions *}
nipkow@15542
   906
blanchet@35828
   907
lemma ivl_subset [simp,no_atp]:
nipkow@15542
   908
 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
nipkow@15542
   909
apply(auto simp:linorder_not_le)
nipkow@15542
   910
apply(rule ccontr)
nipkow@15542
   911
apply(insert linorder_le_less_linear[of i n])
nipkow@15542
   912
apply(clarsimp simp:linorder_not_le)
nipkow@15542
   913
apply(fastsimp)
nipkow@15542
   914
done
nipkow@15542
   915
nipkow@15041
   916
nipkow@15042
   917
subsection {* Summation indexed over intervals *}
nipkow@15042
   918
nipkow@15042
   919
syntax
nipkow@15042
   920
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   921
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   922
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
nipkow@16052
   923
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
nipkow@15042
   924
syntax (xsymbols)
nipkow@15042
   925
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   926
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   927
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
   928
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15042
   929
syntax (HTML output)
nipkow@15042
   930
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   931
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
   932
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
   933
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15056
   934
syntax (latex_sum output)
nipkow@15052
   935
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   936
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052
   937
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   938
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@16052
   939
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
   940
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15052
   941
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
   942
 ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15041
   943
nipkow@15048
   944
translations
nipkow@28853
   945
  "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
nipkow@28853
   946
  "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
nipkow@28853
   947
  "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
nipkow@28853
   948
  "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
nipkow@15041
   949
nipkow@15052
   950
text{* The above introduces some pretty alternative syntaxes for
nipkow@15056
   951
summation over intervals:
nipkow@15052
   952
\begin{center}
nipkow@15052
   953
\begin{tabular}{lll}
nipkow@15056
   954
Old & New & \LaTeX\\
nipkow@15056
   955
@{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
   956
@{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
   957
@{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
nipkow@15056
   958
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
nipkow@15052
   959
\end{tabular}
nipkow@15052
   960
\end{center}
nipkow@15056
   961
The left column shows the term before introduction of the new syntax,
nipkow@15056
   962
the middle column shows the new (default) syntax, and the right column
nipkow@15056
   963
shows a special syntax. The latter is only meaningful for latex output
nipkow@15056
   964
and has to be activated explicitly by setting the print mode to
wenzelm@21502
   965
@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
nipkow@15056
   966
antiquotations). It is not the default \LaTeX\ output because it only
nipkow@15056
   967
works well with italic-style formulae, not tt-style.
nipkow@15052
   968
nipkow@15052
   969
Note that for uniformity on @{typ nat} it is better to use
nipkow@15052
   970
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
nipkow@15052
   971
not provide all lemmas available for @{term"{m..<n}"} also in the
nipkow@15052
   972
special form for @{term"{..<n}"}. *}
nipkow@15052
   973
nipkow@15542
   974
text{* This congruence rule should be used for sums over intervals as
nipkow@15542
   975
the standard theorem @{text[source]setsum_cong} does not work well
nipkow@15542
   976
with the simplifier who adds the unsimplified premise @{term"x:B"} to
nipkow@15542
   977
the context. *}
nipkow@15542
   978
nipkow@15542
   979
lemma setsum_ivl_cong:
nipkow@15542
   980
 "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
nipkow@15542
   981
 setsum f {a..<b} = setsum g {c..<d}"
nipkow@15542
   982
by(rule setsum_cong, simp_all)
nipkow@15041
   983
nipkow@16041
   984
(* FIXME why are the following simp rules but the corresponding eqns
nipkow@16041
   985
on intervals are not? *)
nipkow@16041
   986
nipkow@16052
   987
lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
nipkow@16052
   988
by (simp add:atMost_Suc add_ac)
nipkow@16052
   989
nipkow@16041
   990
lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
nipkow@16041
   991
by (simp add:lessThan_Suc add_ac)
nipkow@15041
   992
nipkow@15911
   993
lemma setsum_cl_ivl_Suc[simp]:
nipkow@15561
   994
  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
nipkow@15561
   995
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@15561
   996
nipkow@15911
   997
lemma setsum_op_ivl_Suc[simp]:
nipkow@15561
   998
  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
nipkow@15561
   999
by (auto simp:add_ac atLeastLessThanSuc)
nipkow@16041
  1000
(*
nipkow@15561
  1001
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
nipkow@15561
  1002
    (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
nipkow@15561
  1003
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@16041
  1004
*)
nipkow@28068
  1005
nipkow@28068
  1006
lemma setsum_head:
nipkow@28068
  1007
  fixes n :: nat
nipkow@28068
  1008
  assumes mn: "m <= n" 
nipkow@28068
  1009
  shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
nipkow@28068
  1010
proof -
nipkow@28068
  1011
  from mn
nipkow@28068
  1012
  have "{m..n} = {m} \<union> {m<..n}"
nipkow@28068
  1013
    by (auto intro: ivl_disj_un_singleton)
nipkow@28068
  1014
  hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
nipkow@28068
  1015
    by (simp add: atLeast0LessThan)
nipkow@28068
  1016
  also have "\<dots> = ?rhs" by simp
nipkow@28068
  1017
  finally show ?thesis .
nipkow@28068
  1018
qed
nipkow@28068
  1019
nipkow@28068
  1020
lemma setsum_head_Suc:
nipkow@28068
  1021
  "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
nipkow@28068
  1022
by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
nipkow@28068
  1023
nipkow@28068
  1024
lemma setsum_head_upt_Suc:
nipkow@28068
  1025
  "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
huffman@30079
  1026
apply(insert setsum_head_Suc[of m "n - Suc 0" f])
nipkow@29667
  1027
apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
nipkow@28068
  1028
done
nipkow@28068
  1029
nipkow@31501
  1030
lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"
nipkow@31501
  1031
  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
nipkow@31501
  1032
proof-
nipkow@31501
  1033
  have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using `m \<le> n+1` by auto
nipkow@31501
  1034
  thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint
nipkow@31501
  1035
    atLeastSucAtMost_greaterThanAtMost)
nipkow@31501
  1036
qed
nipkow@28068
  1037
nipkow@15539
  1038
lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
  1039
  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
nipkow@15539
  1040
by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
nipkow@15539
  1041
nipkow@15539
  1042
lemma setsum_diff_nat_ivl:
nipkow@15539
  1043
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
nipkow@15539
  1044
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
  1045
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
nipkow@15539
  1046
using setsum_add_nat_ivl [of m n p f,symmetric]
nipkow@15539
  1047
apply (simp add: add_ac)
nipkow@15539
  1048
done
nipkow@15539
  1049
nipkow@31505
  1050
lemma setsum_natinterval_difff:
nipkow@31505
  1051
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
nipkow@31505
  1052
  shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
nipkow@31505
  1053
          (if m <= n then f m - f(n + 1) else 0)"
nipkow@31505
  1054
by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
nipkow@31505
  1055
nipkow@31509
  1056
lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
nipkow@31509
  1057
nipkow@31509
  1058
lemma setsum_setsum_restrict:
nipkow@31509
  1059
  "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
  1060
  by (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
nipkow@31509
  1061
     (rule setsum_commute)
nipkow@31509
  1062
nipkow@31509
  1063
lemma setsum_image_gen: assumes fS: "finite S"
nipkow@31509
  1064
  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
nipkow@31509
  1065
proof-
nipkow@31509
  1066
  { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
nipkow@31509
  1067
  hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
nipkow@31509
  1068
    by simp
nipkow@31509
  1069
  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
nipkow@31509
  1070
    by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])
nipkow@31509
  1071
  finally show ?thesis .
nipkow@31509
  1072
qed
nipkow@31509
  1073
hoelzl@35171
  1074
lemma setsum_le_included:
haftmann@36307
  1075
  fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"
hoelzl@35171
  1076
  assumes "finite s" "finite t"
hoelzl@35171
  1077
  and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)"
hoelzl@35171
  1078
  shows "setsum f s \<le> setsum g t"
hoelzl@35171
  1079
proof -
hoelzl@35171
  1080
  have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"
hoelzl@35171
  1081
  proof (rule setsum_mono)
hoelzl@35171
  1082
    fix y assume "y \<in> s"
hoelzl@35171
  1083
    with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
hoelzl@35171
  1084
    with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
hoelzl@35171
  1085
      using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]
hoelzl@35171
  1086
      by (auto intro!: setsum_mono2)
hoelzl@35171
  1087
  qed
hoelzl@35171
  1088
  also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
hoelzl@35171
  1089
    using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)
hoelzl@35171
  1090
  also have "... \<le> setsum g t"
hoelzl@35171
  1091
    using assms by (auto simp: setsum_image_gen[symmetric])
hoelzl@35171
  1092
  finally show ?thesis .
hoelzl@35171
  1093
qed
hoelzl@35171
  1094
nipkow@31509
  1095
lemma setsum_multicount_gen:
nipkow@31509
  1096
  assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
nipkow@31509
  1097
  shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
nipkow@31509
  1098
proof-
nipkow@31509
  1099
  have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
nipkow@31509
  1100
  also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]
nipkow@31509
  1101
    using assms(3) by auto
nipkow@31509
  1102
  finally show ?thesis .
nipkow@31509
  1103
qed
nipkow@31509
  1104
nipkow@31509
  1105
lemma setsum_multicount:
nipkow@31509
  1106
  assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
nipkow@31509
  1107
  shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
nipkow@31509
  1108
proof-
nipkow@31509
  1109
  have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)
huffman@35216
  1110
  also have "\<dots> = ?r" by(simp add: mult_commute)
nipkow@31509
  1111
  finally show ?thesis by auto
nipkow@31509
  1112
qed
nipkow@31509
  1113
nipkow@28068
  1114
nipkow@16733
  1115
subsection{* Shifting bounds *}
nipkow@16733
  1116
nipkow@15539
  1117
lemma setsum_shift_bounds_nat_ivl:
nipkow@15539
  1118
  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
nipkow@15539
  1119
by (induct "n", auto simp:atLeastLessThanSuc)
nipkow@15539
  1120
nipkow@16733
  1121
lemma setsum_shift_bounds_cl_nat_ivl:
nipkow@16733
  1122
  "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
nipkow@16733
  1123
apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
nipkow@16733
  1124
apply (simp add:image_add_atLeastAtMost o_def)
nipkow@16733
  1125
done
nipkow@16733
  1126
nipkow@16733
  1127
corollary setsum_shift_bounds_cl_Suc_ivl:
nipkow@16733
  1128
  "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
huffman@30079
  1129
by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
  1130
nipkow@16733
  1131
corollary setsum_shift_bounds_Suc_ivl:
nipkow@16733
  1132
  "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
huffman@30079
  1133
by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
  1134
nipkow@28068
  1135
lemma setsum_shift_lb_Suc0_0:
nipkow@28068
  1136
  "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
nipkow@28068
  1137
by(simp add:setsum_head_Suc)
kleing@19106
  1138
nipkow@28068
  1139
lemma setsum_shift_lb_Suc0_0_upt:
nipkow@28068
  1140
  "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
nipkow@28068
  1141
apply(cases k)apply simp
nipkow@28068
  1142
apply(simp add:setsum_head_upt_Suc)
nipkow@28068
  1143
done
kleing@19022
  1144
ballarin@17149
  1145
subsection {* The formula for geometric sums *}
ballarin@17149
  1146
ballarin@17149
  1147
lemma geometric_sum:
haftmann@36307
  1148
  assumes "x \<noteq> 1"
haftmann@36307
  1149
  shows "(\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"
haftmann@36307
  1150
proof -
haftmann@36307
  1151
  from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all
haftmann@36307
  1152
  moreover have "(\<Sum>i=0..<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"
haftmann@36307
  1153
  proof (induct n)
haftmann@36307
  1154
    case 0 then show ?case by simp
haftmann@36307
  1155
  next
haftmann@36307
  1156
    case (Suc n)
haftmann@36307
  1157
    moreover with `y \<noteq> 0` have "(1 + y) ^ n = (y * inverse y) * (1 + y) ^ n" by simp 
haftmann@36350
  1158
    ultimately show ?case by (simp add: field_simps divide_inverse)
haftmann@36307
  1159
  qed
haftmann@36307
  1160
  ultimately show ?thesis by simp
haftmann@36307
  1161
qed
haftmann@36307
  1162
ballarin@17149
  1163
kleing@19469
  1164
subsection {* The formula for arithmetic sums *}
kleing@19469
  1165
kleing@19469
  1166
lemma gauss_sum:
huffman@23277
  1167
  "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
kleing@19469
  1168
   of_nat n*((of_nat n)+1)"
kleing@19469
  1169
proof (induct n)
kleing@19469
  1170
  case 0
kleing@19469
  1171
  show ?case by simp
kleing@19469
  1172
next
kleing@19469
  1173
  case (Suc n)
nipkow@29667
  1174
  then show ?case by (simp add: algebra_simps)
kleing@19469
  1175
qed
kleing@19469
  1176
kleing@19469
  1177
theorem arith_series_general:
huffman@23277
  1178
  "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1179
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1180
proof cases
kleing@19469
  1181
  assume ngt1: "n > 1"
kleing@19469
  1182
  let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
kleing@19469
  1183
  have
kleing@19469
  1184
    "(\<Sum>i\<in>{..<n}. a+?I i*d) =
kleing@19469
  1185
     ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
kleing@19469
  1186
    by (rule setsum_addf)
kleing@19469
  1187
  also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
kleing@19469
  1188
  also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
huffman@30079
  1189
    unfolding One_nat_def
nipkow@28068
  1190
    by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)
kleing@19469
  1191
  also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
kleing@19469
  1192
    by (simp add: left_distrib right_distrib)
kleing@19469
  1193
  also from ngt1 have "{1..<n} = {1..n - 1}"
nipkow@28068
  1194
    by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
nipkow@28068
  1195
  also from ngt1
kleing@19469
  1196
  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
  1197
    by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)
huffman@23431
  1198
       (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
nipkow@29667
  1199
  finally show ?thesis by (simp add: algebra_simps)
kleing@19469
  1200
next
kleing@19469
  1201
  assume "\<not>(n > 1)"
kleing@19469
  1202
  hence "n = 1 \<or> n = 0" by auto
nipkow@29667
  1203
  thus ?thesis by (auto simp: algebra_simps)
kleing@19469
  1204
qed
kleing@19469
  1205
kleing@19469
  1206
lemma arith_series_nat:
kleing@19469
  1207
  "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
kleing@19469
  1208
proof -
kleing@19469
  1209
  have
kleing@19469
  1210
    "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
kleing@19469
  1211
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1212
    by (rule arith_series_general)
huffman@30079
  1213
  thus ?thesis
huffman@35216
  1214
    unfolding One_nat_def by auto
kleing@19469
  1215
qed
kleing@19469
  1216
kleing@19469
  1217
lemma arith_series_int:
kleing@19469
  1218
  "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1219
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1220
proof -
kleing@19469
  1221
  have
kleing@19469
  1222
    "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1223
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1224
    by (rule arith_series_general)
kleing@19469
  1225
  thus ?thesis by simp
kleing@19469
  1226
qed
paulson@15418
  1227
kleing@19022
  1228
lemma sum_diff_distrib:
kleing@19022
  1229
  fixes P::"nat\<Rightarrow>nat"
kleing@19022
  1230
  shows
kleing@19022
  1231
  "\<forall>x. Q x \<le> P x  \<Longrightarrow>
kleing@19022
  1232
  (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
kleing@19022
  1233
proof (induct n)
kleing@19022
  1234
  case 0 show ?case by simp
kleing@19022
  1235
next
kleing@19022
  1236
  case (Suc n)
kleing@19022
  1237
kleing@19022
  1238
  let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
kleing@19022
  1239
  let ?rhs = "\<Sum>x<n. P x - Q x"
kleing@19022
  1240
kleing@19022
  1241
  from Suc have "?lhs = ?rhs" by simp
kleing@19022
  1242
  moreover
kleing@19022
  1243
  from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
kleing@19022
  1244
  moreover
kleing@19022
  1245
  from Suc have
kleing@19022
  1246
    "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
kleing@19022
  1247
    by (subst diff_diff_left[symmetric],
kleing@19022
  1248
        subst diff_add_assoc2)
kleing@19022
  1249
       (auto simp: diff_add_assoc2 intro: setsum_mono)
kleing@19022
  1250
  ultimately
kleing@19022
  1251
  show ?case by simp
kleing@19022
  1252
qed
kleing@19022
  1253
paulson@29960
  1254
subsection {* Products indexed over intervals *}
paulson@29960
  1255
paulson@29960
  1256
syntax
paulson@29960
  1257
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1258
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1259
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)
paulson@29960
  1260
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
paulson@29960
  1261
syntax (xsymbols)
paulson@29960
  1262
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1263
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1264
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1265
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1266
syntax (HTML output)
paulson@29960
  1267
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1268
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1269
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1270
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1271
syntax (latex_prod output)
paulson@29960
  1272
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1273
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1274
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1275
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1276
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1277
 ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1278
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1279
 ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1280
paulson@29960
  1281
translations
paulson@29960
  1282
  "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"
paulson@29960
  1283
  "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"
paulson@29960
  1284
  "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"
paulson@29960
  1285
  "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"
paulson@29960
  1286
haftmann@33318
  1287
subsection {* Transfer setup *}
haftmann@33318
  1288
haftmann@33318
  1289
lemma transfer_nat_int_set_functions:
haftmann@33318
  1290
    "{..n} = nat ` {0..int n}"
haftmann@33318
  1291
    "{m..n} = nat ` {int m..int n}"  (* need all variants of these! *)
haftmann@33318
  1292
  apply (auto simp add: image_def)
haftmann@33318
  1293
  apply (rule_tac x = "int x" in bexI)
haftmann@33318
  1294
  apply auto
haftmann@33318
  1295
  apply (rule_tac x = "int x" in bexI)
haftmann@33318
  1296
  apply auto
haftmann@33318
  1297
  done
haftmann@33318
  1298
haftmann@33318
  1299
lemma transfer_nat_int_set_function_closures:
haftmann@33318
  1300
    "x >= 0 \<Longrightarrow> nat_set {x..y}"
haftmann@33318
  1301
  by (simp add: nat_set_def)
haftmann@33318
  1302
haftmann@35644
  1303
declare transfer_morphism_nat_int[transfer add
haftmann@33318
  1304
  return: transfer_nat_int_set_functions
haftmann@33318
  1305
    transfer_nat_int_set_function_closures
haftmann@33318
  1306
]
haftmann@33318
  1307
haftmann@33318
  1308
lemma transfer_int_nat_set_functions:
haftmann@33318
  1309
    "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}"
haftmann@33318
  1310
  by (simp only: is_nat_def transfer_nat_int_set_functions
haftmann@33318
  1311
    transfer_nat_int_set_function_closures
haftmann@33318
  1312
    transfer_nat_int_set_return_embed nat_0_le
haftmann@33318
  1313
    cong: transfer_nat_int_set_cong)
haftmann@33318
  1314
haftmann@33318
  1315
lemma transfer_int_nat_set_function_closures:
haftmann@33318
  1316
    "is_nat x \<Longrightarrow> nat_set {x..y}"
haftmann@33318
  1317
  by (simp only: transfer_nat_int_set_function_closures is_nat_def)
haftmann@33318
  1318
haftmann@35644
  1319
declare transfer_morphism_int_nat[transfer add
haftmann@33318
  1320
  return: transfer_int_nat_set_functions
haftmann@33318
  1321
    transfer_int_nat_set_function_closures
haftmann@33318
  1322
]
haftmann@33318
  1323
nipkow@8924
  1324
end