src/HOL/SetInterval.thy
author hoelzl
Mon Jul 04 10:15:49 2011 +0200 (2011-07-04)
changeset 43657 537ea3846f64
parent 43157 b505be6f029a
child 44008 2e09299ce807
permissions -rw-r--r--
equalities of subsets of atLeastLessThan
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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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lemma lessThan_strict_subset_iff:
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  fixes m n :: "'a::linorder"
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  shows "{..<m} < {..<n} \<longleftrightarrow> m < n"
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  by (metis leD lessThan_subset_iff linorder_linear not_less_iff_gr_or_eq psubset_eq)
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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 set_eq_iff 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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context dense_linorder
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begin
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lemma greaterThanLessThan_empty_iff[simp]:
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  "{ a <..< b } = {} \<longleftrightarrow> b \<le> a"
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  using dense[of a b] by (cases "a < b") auto
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lemma greaterThanLessThan_empty_iff2[simp]:
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  "{} = { a <..< b } \<longleftrightarrow> b \<le> a"
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  using dense[of a b] by (cases "a < b") auto
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lemma atLeastLessThan_subseteq_atLeastAtMost_iff:
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  "{a ..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
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  using dense[of "max a d" "b"]
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  by (force simp: subset_eq Ball_def not_less[symmetric])
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lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:
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  "{a <.. b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
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  using dense[of "a" "min c b"]
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  by (force simp: subset_eq Ball_def not_less[symmetric])
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lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:
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  "{a <..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
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  using dense[of "a" "min c b"] dense[of "max a d" "b"]
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  by (force simp: subset_eq Ball_def not_less[symmetric])
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lemma atLeastAtMost_subseteq_atLeastLessThan_iff:
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  "{a .. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a \<le> b \<longrightarrow> c \<le> a \<and> b < d)"
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  using dense[of "max a d" "b"]
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  by (force simp: subset_eq Ball_def not_less[symmetric])
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lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:
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  "{a <.. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b < d)"
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  using dense[of "a" "min c b"]
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  by (force simp: subset_eq Ball_def not_less[symmetric])
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lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:
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  "{a <..< b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
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  using dense[of "a" "min c b"] dense[of "max a d" "b"]
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  by (force simp: subset_eq Ball_def not_less[symmetric])
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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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lemma atLeastLessThan_inj:
hoelzl@40703
   313
  fixes a b c d :: "'a::linorder"
hoelzl@40703
   314
  assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"
hoelzl@40703
   315
  shows "a = c" "b = d"
hoelzl@40703
   316
using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le linorder_antisym_conv2 subset_refl)+
hoelzl@40703
   317
hoelzl@40703
   318
lemma atLeastLessThan_eq_iff:
hoelzl@40703
   319
  fixes a b c d :: "'a::linorder"
hoelzl@40703
   320
  assumes "a < b" "c < d"
hoelzl@40703
   321
  shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"
hoelzl@40703
   322
  using atLeastLessThan_inj assms by auto
hoelzl@40703
   323
nipkow@32456
   324
subsubsection {* Intersection *}
nipkow@32456
   325
nipkow@32456
   326
context linorder
nipkow@32456
   327
begin
nipkow@32456
   328
nipkow@32456
   329
lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"
nipkow@32456
   330
by auto
nipkow@32456
   331
nipkow@32456
   332
lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"
nipkow@32456
   333
by auto
nipkow@32456
   334
nipkow@32456
   335
lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"
nipkow@32456
   336
by auto
nipkow@32456
   337
nipkow@32456
   338
lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"
nipkow@32456
   339
by auto
nipkow@32456
   340
nipkow@32456
   341
lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"
nipkow@32456
   342
by auto
nipkow@32456
   343
nipkow@32456
   344
lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"
nipkow@32456
   345
by auto
nipkow@32456
   346
nipkow@32456
   347
lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
nipkow@32456
   348
by auto
nipkow@32456
   349
nipkow@32456
   350
lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
nipkow@32456
   351
by auto
nipkow@32456
   352
nipkow@32456
   353
end
nipkow@32456
   354
nipkow@32456
   355
paulson@14485
   356
subsection {* Intervals of natural numbers *}
paulson@14485
   357
paulson@15047
   358
subsubsection {* The Constant @{term lessThan} *}
paulson@15047
   359
paulson@14485
   360
lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
paulson@14485
   361
by (simp add: lessThan_def)
paulson@14485
   362
paulson@14485
   363
lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
paulson@14485
   364
by (simp add: lessThan_def less_Suc_eq, blast)
paulson@14485
   365
kleing@43156
   366
text {* The following proof is convenient in induction proofs where
hoelzl@39072
   367
new elements get indices at the beginning. So it is used to transform
hoelzl@39072
   368
@{term "{..<Suc n}"} to @{term "0::nat"} and @{term "{..< n}"}. *}
hoelzl@39072
   369
hoelzl@39072
   370
lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"
hoelzl@39072
   371
proof safe
hoelzl@39072
   372
  fix x assume "x < Suc n" "x \<notin> Suc ` {..<n}"
hoelzl@39072
   373
  then have "x \<noteq> Suc (x - 1)" by auto
hoelzl@39072
   374
  with `x < Suc n` show "x = 0" by auto
hoelzl@39072
   375
qed
hoelzl@39072
   376
paulson@14485
   377
lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
paulson@14485
   378
by (simp add: lessThan_def atMost_def less_Suc_eq_le)
paulson@14485
   379
paulson@14485
   380
lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
paulson@14485
   381
by blast
paulson@14485
   382
paulson@15047
   383
subsubsection {* The Constant @{term greaterThan} *}
paulson@15047
   384
paulson@14485
   385
lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
paulson@14485
   386
apply (simp add: greaterThan_def)
paulson@14485
   387
apply (blast dest: gr0_conv_Suc [THEN iffD1])
paulson@14485
   388
done
paulson@14485
   389
paulson@14485
   390
lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
paulson@14485
   391
apply (simp add: greaterThan_def)
paulson@14485
   392
apply (auto elim: linorder_neqE)
paulson@14485
   393
done
paulson@14485
   394
paulson@14485
   395
lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
paulson@14485
   396
by blast
paulson@14485
   397
paulson@15047
   398
subsubsection {* The Constant @{term atLeast} *}
paulson@15047
   399
paulson@14485
   400
lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
paulson@14485
   401
by (unfold atLeast_def UNIV_def, simp)
paulson@14485
   402
paulson@14485
   403
lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
paulson@14485
   404
apply (simp add: atLeast_def)
paulson@14485
   405
apply (simp add: Suc_le_eq)
paulson@14485
   406
apply (simp add: order_le_less, blast)
paulson@14485
   407
done
paulson@14485
   408
paulson@14485
   409
lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
paulson@14485
   410
  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
paulson@14485
   411
paulson@14485
   412
lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
paulson@14485
   413
by blast
paulson@14485
   414
paulson@15047
   415
subsubsection {* The Constant @{term atMost} *}
paulson@15047
   416
paulson@14485
   417
lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
paulson@14485
   418
by (simp add: atMost_def)
paulson@14485
   419
paulson@14485
   420
lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
paulson@14485
   421
apply (simp add: atMost_def)
paulson@14485
   422
apply (simp add: less_Suc_eq order_le_less, blast)
paulson@14485
   423
done
paulson@14485
   424
paulson@14485
   425
lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
paulson@14485
   426
by blast
paulson@14485
   427
paulson@15047
   428
subsubsection {* The Constant @{term atLeastLessThan} *}
paulson@15047
   429
nipkow@28068
   430
text{*The orientation of the following 2 rules is tricky. The lhs is
nipkow@24449
   431
defined in terms of the rhs.  Hence the chosen orientation makes sense
nipkow@24449
   432
in this theory --- the reverse orientation complicates proofs (eg
nipkow@24449
   433
nontermination). But outside, when the definition of the lhs is rarely
nipkow@24449
   434
used, the opposite orientation seems preferable because it reduces a
nipkow@24449
   435
specific concept to a more general one. *}
nipkow@28068
   436
paulson@15047
   437
lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
nipkow@15042
   438
by(simp add:lessThan_def atLeastLessThan_def)
nipkow@24449
   439
nipkow@28068
   440
lemma atLeast0AtMost: "{0..n::nat} = {..n}"
nipkow@28068
   441
by(simp add:atMost_def atLeastAtMost_def)
nipkow@28068
   442
haftmann@31998
   443
declare atLeast0LessThan[symmetric, code_unfold]
haftmann@31998
   444
        atLeast0AtMost[symmetric, code_unfold]
nipkow@24449
   445
nipkow@24449
   446
lemma atLeastLessThan0: "{m..<0::nat} = {}"
paulson@15047
   447
by (simp add: atLeastLessThan_def)
nipkow@24449
   448
paulson@15047
   449
subsubsection {* Intervals of nats with @{term Suc} *}
paulson@15047
   450
paulson@15047
   451
text{*Not a simprule because the RHS is too messy.*}
paulson@15047
   452
lemma atLeastLessThanSuc:
paulson@15047
   453
    "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
paulson@15418
   454
by (auto simp add: atLeastLessThan_def)
paulson@15047
   455
paulson@15418
   456
lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
paulson@15047
   457
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   458
(*
paulson@15047
   459
lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
paulson@15047
   460
by (induct k, simp_all add: atLeastLessThanSuc)
paulson@15047
   461
paulson@15047
   462
lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
paulson@15047
   463
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   464
*)
nipkow@15045
   465
lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
paulson@14485
   466
  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   467
paulson@15418
   468
lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
paulson@15418
   469
  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
paulson@14485
   470
    greaterThanAtMost_def)
paulson@14485
   471
paulson@15418
   472
lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
paulson@15418
   473
  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
paulson@14485
   474
    greaterThanLessThan_def)
paulson@14485
   475
nipkow@15554
   476
lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
nipkow@15554
   477
by (auto simp add: atLeastAtMost_def)
nipkow@15554
   478
kleing@43157
   479
text {* The analogous result is useful on @{typ int}: *}
kleing@43157
   480
(* here, because we don't have an own int section *)
kleing@43157
   481
lemma atLeastAtMostPlus1_int_conv:
kleing@43157
   482
  "m <= 1+n \<Longrightarrow> {m..1+n} = insert (1+n) {m..n::int}"
kleing@43157
   483
  by (auto intro: set_eqI)
kleing@43157
   484
paulson@33044
   485
lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"
paulson@33044
   486
  apply (induct k) 
paulson@33044
   487
  apply (simp_all add: atLeastLessThanSuc)   
paulson@33044
   488
  done
paulson@33044
   489
nipkow@16733
   490
subsubsection {* Image *}
nipkow@16733
   491
nipkow@16733
   492
lemma image_add_atLeastAtMost:
nipkow@16733
   493
  "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
nipkow@16733
   494
proof
nipkow@16733
   495
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   496
next
nipkow@16733
   497
  show "?B \<subseteq> ?A"
nipkow@16733
   498
  proof
nipkow@16733
   499
    fix n assume a: "n : ?B"
webertj@20217
   500
    hence "n - k : {i..j}" by auto
nipkow@16733
   501
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   502
    ultimately show "n : ?A" by blast
nipkow@16733
   503
  qed
nipkow@16733
   504
qed
nipkow@16733
   505
nipkow@16733
   506
lemma image_add_atLeastLessThan:
nipkow@16733
   507
  "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
nipkow@16733
   508
proof
nipkow@16733
   509
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   510
next
nipkow@16733
   511
  show "?B \<subseteq> ?A"
nipkow@16733
   512
  proof
nipkow@16733
   513
    fix n assume a: "n : ?B"
webertj@20217
   514
    hence "n - k : {i..<j}" by auto
nipkow@16733
   515
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   516
    ultimately show "n : ?A" by blast
nipkow@16733
   517
  qed
nipkow@16733
   518
qed
nipkow@16733
   519
nipkow@16733
   520
corollary image_Suc_atLeastAtMost[simp]:
nipkow@16733
   521
  "Suc ` {i..j} = {Suc i..Suc j}"
huffman@30079
   522
using image_add_atLeastAtMost[where k="Suc 0"] by simp
nipkow@16733
   523
nipkow@16733
   524
corollary image_Suc_atLeastLessThan[simp]:
nipkow@16733
   525
  "Suc ` {i..<j} = {Suc i..<Suc j}"
huffman@30079
   526
using image_add_atLeastLessThan[where k="Suc 0"] by simp
nipkow@16733
   527
nipkow@16733
   528
lemma image_add_int_atLeastLessThan:
nipkow@16733
   529
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
nipkow@16733
   530
  apply (auto simp add: image_def)
nipkow@16733
   531
  apply (rule_tac x = "x - l" in bexI)
nipkow@16733
   532
  apply auto
nipkow@16733
   533
  done
nipkow@16733
   534
hoelzl@37664
   535
lemma image_minus_const_atLeastLessThan_nat:
hoelzl@37664
   536
  fixes c :: nat
hoelzl@37664
   537
  shows "(\<lambda>i. i - c) ` {x ..< y} =
hoelzl@37664
   538
      (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"
hoelzl@37664
   539
    (is "_ = ?right")
hoelzl@37664
   540
proof safe
hoelzl@37664
   541
  fix a assume a: "a \<in> ?right"
hoelzl@37664
   542
  show "a \<in> (\<lambda>i. i - c) ` {x ..< y}"
hoelzl@37664
   543
  proof cases
hoelzl@37664
   544
    assume "c < y" with a show ?thesis
hoelzl@37664
   545
      by (auto intro!: image_eqI[of _ _ "a + c"])
hoelzl@37664
   546
  next
hoelzl@37664
   547
    assume "\<not> c < y" with a show ?thesis
hoelzl@37664
   548
      by (auto intro!: image_eqI[of _ _ x] split: split_if_asm)
hoelzl@37664
   549
  qed
hoelzl@37664
   550
qed auto
hoelzl@37664
   551
hoelzl@35580
   552
context ordered_ab_group_add
hoelzl@35580
   553
begin
hoelzl@35580
   554
hoelzl@35580
   555
lemma
hoelzl@35580
   556
  fixes x :: 'a
hoelzl@35580
   557
  shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}"
hoelzl@35580
   558
  and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}"
hoelzl@35580
   559
proof safe
hoelzl@35580
   560
  fix y assume "y < -x"
hoelzl@35580
   561
  hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp
hoelzl@35580
   562
  have "- (-y) \<in> uminus ` {x<..}"
hoelzl@35580
   563
    by (rule imageI) (simp add: *)
hoelzl@35580
   564
  thus "y \<in> uminus ` {x<..}" by simp
hoelzl@35580
   565
next
hoelzl@35580
   566
  fix y assume "y \<le> -x"
hoelzl@35580
   567
  have "- (-y) \<in> uminus ` {x..}"
hoelzl@35580
   568
    by (rule imageI) (insert `y \<le> -x`[THEN le_imp_neg_le], simp)
hoelzl@35580
   569
  thus "y \<in> uminus ` {x..}" by simp
hoelzl@35580
   570
qed simp_all
hoelzl@35580
   571
hoelzl@35580
   572
lemma
hoelzl@35580
   573
  fixes x :: 'a
hoelzl@35580
   574
  shows image_uminus_lessThan[simp]: "uminus ` {..<x} = {-x<..}"
hoelzl@35580
   575
  and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"
hoelzl@35580
   576
proof -
hoelzl@35580
   577
  have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"
hoelzl@35580
   578
    and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all
hoelzl@35580
   579
  thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"
hoelzl@35580
   580
    by (simp_all add: image_image
hoelzl@35580
   581
        del: image_uminus_greaterThan image_uminus_atLeast)
hoelzl@35580
   582
qed
hoelzl@35580
   583
hoelzl@35580
   584
lemma
hoelzl@35580
   585
  fixes x :: 'a
hoelzl@35580
   586
  shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"
hoelzl@35580
   587
  and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"
hoelzl@35580
   588
  and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"
hoelzl@35580
   589
  and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"
hoelzl@35580
   590
  by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def
hoelzl@35580
   591
      greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)
hoelzl@35580
   592
end
nipkow@16733
   593
paulson@14485
   594
subsubsection {* Finiteness *}
paulson@14485
   595
nipkow@15045
   596
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
paulson@14485
   597
  by (induct k) (simp_all add: lessThan_Suc)
paulson@14485
   598
paulson@14485
   599
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
paulson@14485
   600
  by (induct k) (simp_all add: atMost_Suc)
paulson@14485
   601
paulson@14485
   602
lemma finite_greaterThanLessThan [iff]:
nipkow@15045
   603
  fixes l :: nat shows "finite {l<..<u}"
paulson@14485
   604
by (simp add: greaterThanLessThan_def)
paulson@14485
   605
paulson@14485
   606
lemma finite_atLeastLessThan [iff]:
nipkow@15045
   607
  fixes l :: nat shows "finite {l..<u}"
paulson@14485
   608
by (simp add: atLeastLessThan_def)
paulson@14485
   609
paulson@14485
   610
lemma finite_greaterThanAtMost [iff]:
nipkow@15045
   611
  fixes l :: nat shows "finite {l<..u}"
paulson@14485
   612
by (simp add: greaterThanAtMost_def)
paulson@14485
   613
paulson@14485
   614
lemma finite_atLeastAtMost [iff]:
paulson@14485
   615
  fixes l :: nat shows "finite {l..u}"
paulson@14485
   616
by (simp add: atLeastAtMost_def)
paulson@14485
   617
nipkow@28068
   618
text {* A bounded set of natural numbers is finite. *}
paulson@14485
   619
lemma bounded_nat_set_is_finite:
nipkow@24853
   620
  "(ALL i:N. i < (n::nat)) ==> finite N"
nipkow@28068
   621
apply (rule finite_subset)
nipkow@28068
   622
 apply (rule_tac [2] finite_lessThan, auto)
nipkow@28068
   623
done
nipkow@28068
   624
nipkow@31044
   625
text {* A set of natural numbers is finite iff it is bounded. *}
nipkow@31044
   626
lemma finite_nat_set_iff_bounded:
nipkow@31044
   627
  "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
nipkow@31044
   628
proof
nipkow@31044
   629
  assume f:?F  show ?B
nipkow@31044
   630
    using Max_ge[OF `?F`, simplified less_Suc_eq_le[symmetric]] by blast
nipkow@31044
   631
next
nipkow@31044
   632
  assume ?B show ?F using `?B` by(blast intro:bounded_nat_set_is_finite)
nipkow@31044
   633
qed
nipkow@31044
   634
nipkow@31044
   635
lemma finite_nat_set_iff_bounded_le:
nipkow@31044
   636
  "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
nipkow@31044
   637
apply(simp add:finite_nat_set_iff_bounded)
nipkow@31044
   638
apply(blast dest:less_imp_le_nat le_imp_less_Suc)
nipkow@31044
   639
done
nipkow@31044
   640
nipkow@28068
   641
lemma finite_less_ub:
nipkow@28068
   642
     "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
nipkow@28068
   643
by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
paulson@14485
   644
nipkow@24853
   645
text{* Any subset of an interval of natural numbers the size of the
nipkow@24853
   646
subset is exactly that interval. *}
nipkow@24853
   647
nipkow@24853
   648
lemma subset_card_intvl_is_intvl:
nipkow@24853
   649
  "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")
nipkow@24853
   650
proof cases
nipkow@24853
   651
  assume "finite A"
nipkow@24853
   652
  thus "PROP ?P"
nipkow@32006
   653
  proof(induct A rule:finite_linorder_max_induct)
nipkow@24853
   654
    case empty thus ?case by auto
nipkow@24853
   655
  next
nipkow@33434
   656
    case (insert b A)
nipkow@24853
   657
    moreover hence "b ~: A" by auto
nipkow@24853
   658
    moreover have "A <= {k..<k+card A}" and "b = k+card A"
nipkow@24853
   659
      using `b ~: A` insert by fastsimp+
nipkow@24853
   660
    ultimately show ?case by auto
nipkow@24853
   661
  qed
nipkow@24853
   662
next
nipkow@24853
   663
  assume "~finite A" thus "PROP ?P" by simp
nipkow@24853
   664
qed
nipkow@24853
   665
nipkow@24853
   666
paulson@32596
   667
subsubsection {* Proving Inclusions and Equalities between Unions *}
paulson@32596
   668
nipkow@36755
   669
lemma UN_le_eq_Un0:
nipkow@36755
   670
  "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> M 0" (is "?A = ?B")
nipkow@36755
   671
proof
nipkow@36755
   672
  show "?A <= ?B"
nipkow@36755
   673
  proof
nipkow@36755
   674
    fix x assume "x : ?A"
nipkow@36755
   675
    then obtain i where i: "i\<le>n" "x : M i" by auto
nipkow@36755
   676
    show "x : ?B"
nipkow@36755
   677
    proof(cases i)
nipkow@36755
   678
      case 0 with i show ?thesis by simp
nipkow@36755
   679
    next
nipkow@36755
   680
      case (Suc j) with i show ?thesis by auto
nipkow@36755
   681
    qed
nipkow@36755
   682
  qed
nipkow@36755
   683
next
nipkow@36755
   684
  show "?B <= ?A" by auto
nipkow@36755
   685
qed
nipkow@36755
   686
nipkow@36755
   687
lemma UN_le_add_shift:
nipkow@36755
   688
  "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")
nipkow@36755
   689
proof
nipkow@36755
   690
  show "?A <= ?B" by fastsimp
nipkow@36755
   691
next
nipkow@36755
   692
  show "?B <= ?A"
nipkow@36755
   693
  proof
nipkow@36755
   694
    fix x assume "x : ?B"
nipkow@36755
   695
    then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto
nipkow@36755
   696
    hence "i-k\<le>n & x : M((i-k)+k)" by auto
nipkow@36755
   697
    thus "x : ?A" by blast
nipkow@36755
   698
  qed
nipkow@36755
   699
qed
nipkow@36755
   700
paulson@32596
   701
lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"
paulson@32596
   702
  by (auto simp add: atLeast0LessThan) 
paulson@32596
   703
paulson@32596
   704
lemma UN_finite_subset: "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"
paulson@32596
   705
  by (subst UN_UN_finite_eq [symmetric]) blast
paulson@32596
   706
paulson@33044
   707
lemma UN_finite2_subset: 
paulson@33044
   708
     "(!!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
   709
  apply (rule UN_finite_subset)
paulson@33044
   710
  apply (subst UN_UN_finite_eq [symmetric, of B]) 
paulson@33044
   711
  apply blast
paulson@33044
   712
  done
paulson@32596
   713
paulson@32596
   714
lemma UN_finite2_eq:
paulson@33044
   715
  "(!!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
   716
  apply (rule subset_antisym)
paulson@33044
   717
   apply (rule UN_finite2_subset, blast)
paulson@33044
   718
 apply (rule UN_finite2_subset [where k=k])
huffman@35216
   719
 apply (force simp add: atLeastLessThan_add_Un [of 0])
paulson@33044
   720
 done
paulson@32596
   721
paulson@32596
   722
paulson@14485
   723
subsubsection {* Cardinality *}
paulson@14485
   724
nipkow@15045
   725
lemma card_lessThan [simp]: "card {..<u} = u"
paulson@15251
   726
  by (induct u, simp_all add: lessThan_Suc)
paulson@14485
   727
paulson@14485
   728
lemma card_atMost [simp]: "card {..u} = Suc u"
paulson@14485
   729
  by (simp add: lessThan_Suc_atMost [THEN sym])
paulson@14485
   730
nipkow@15045
   731
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
nipkow@15045
   732
  apply (subgoal_tac "card {l..<u} = card {..<u-l}")
paulson@14485
   733
  apply (erule ssubst, rule card_lessThan)
nipkow@15045
   734
  apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
paulson@14485
   735
  apply (erule subst)
paulson@14485
   736
  apply (rule card_image)
paulson@14485
   737
  apply (simp add: inj_on_def)
paulson@14485
   738
  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
paulson@14485
   739
  apply (rule_tac x = "x - l" in exI)
paulson@14485
   740
  apply arith
paulson@14485
   741
  done
paulson@14485
   742
paulson@15418
   743
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
paulson@14485
   744
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
paulson@14485
   745
paulson@15418
   746
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
paulson@14485
   747
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
paulson@14485
   748
nipkow@15045
   749
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
paulson@14485
   750
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
paulson@14485
   751
nipkow@26105
   752
lemma ex_bij_betw_nat_finite:
nipkow@26105
   753
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
nipkow@26105
   754
apply(drule finite_imp_nat_seg_image_inj_on)
nipkow@26105
   755
apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
nipkow@26105
   756
done
nipkow@26105
   757
nipkow@26105
   758
lemma ex_bij_betw_finite_nat:
nipkow@26105
   759
  "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
nipkow@26105
   760
by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
nipkow@26105
   761
nipkow@31438
   762
lemma finite_same_card_bij:
nipkow@31438
   763
  "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"
nipkow@31438
   764
apply(drule ex_bij_betw_finite_nat)
nipkow@31438
   765
apply(drule ex_bij_betw_nat_finite)
nipkow@31438
   766
apply(auto intro!:bij_betw_trans)
nipkow@31438
   767
done
nipkow@31438
   768
nipkow@31438
   769
lemma ex_bij_betw_nat_finite_1:
nipkow@31438
   770
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
nipkow@31438
   771
by (rule finite_same_card_bij) auto
nipkow@31438
   772
hoelzl@40703
   773
lemma bij_betw_iff_card:
hoelzl@40703
   774
  assumes FIN: "finite A" and FIN': "finite B"
hoelzl@40703
   775
  shows BIJ: "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"
hoelzl@40703
   776
using assms
hoelzl@40703
   777
proof(auto simp add: bij_betw_same_card)
hoelzl@40703
   778
  assume *: "card A = card B"
hoelzl@40703
   779
  obtain f where "bij_betw f A {0 ..< card A}"
hoelzl@40703
   780
  using FIN ex_bij_betw_finite_nat by blast
hoelzl@40703
   781
  moreover obtain g where "bij_betw g {0 ..< card B} B"
hoelzl@40703
   782
  using FIN' ex_bij_betw_nat_finite by blast
hoelzl@40703
   783
  ultimately have "bij_betw (g o f) A B"
hoelzl@40703
   784
  using * by (auto simp add: bij_betw_trans)
hoelzl@40703
   785
  thus "(\<exists>f. bij_betw f A B)" by blast
hoelzl@40703
   786
qed
hoelzl@40703
   787
hoelzl@40703
   788
lemma inj_on_iff_card_le:
hoelzl@40703
   789
  assumes FIN: "finite A" and FIN': "finite B"
hoelzl@40703
   790
  shows "(\<exists>f. inj_on f A \<and> f ` A \<le> B) = (card A \<le> card B)"
hoelzl@40703
   791
proof (safe intro!: card_inj_on_le)
hoelzl@40703
   792
  assume *: "card A \<le> card B"
hoelzl@40703
   793
  obtain f where 1: "inj_on f A" and 2: "f ` A = {0 ..< card A}"
hoelzl@40703
   794
  using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force
hoelzl@40703
   795
  moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g ` {0 ..< card B} = B"
hoelzl@40703
   796
  using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force
hoelzl@40703
   797
  ultimately have "inj_on g (f ` A)" using subset_inj_on[of g _ "f ` A"] * by force
hoelzl@40703
   798
  hence "inj_on (g o f) A" using 1 comp_inj_on by blast
hoelzl@40703
   799
  moreover
hoelzl@40703
   800
  {have "{0 ..< card A} \<le> {0 ..< card B}" using * by force
hoelzl@40703
   801
   with 2 have "f ` A  \<le> {0 ..< card B}" by blast
hoelzl@40703
   802
   hence "(g o f) ` A \<le> B" unfolding comp_def using 3 by force
hoelzl@40703
   803
  }
hoelzl@40703
   804
  ultimately show "(\<exists>f. inj_on f A \<and> f ` A \<le> B)" by blast
hoelzl@40703
   805
qed (insert assms, auto)
nipkow@26105
   806
paulson@14485
   807
subsection {* Intervals of integers *}
paulson@14485
   808
nipkow@15045
   809
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
paulson@14485
   810
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   811
paulson@15418
   812
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
paulson@14485
   813
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
paulson@14485
   814
paulson@15418
   815
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
paulson@15418
   816
    "{l+1..<u} = {l<..<u::int}"
paulson@14485
   817
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
paulson@14485
   818
paulson@14485
   819
subsubsection {* Finiteness *}
paulson@14485
   820
paulson@15418
   821
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
nipkow@15045
   822
    {(0::int)..<u} = int ` {..<nat u}"
paulson@14485
   823
  apply (unfold image_def lessThan_def)
paulson@14485
   824
  apply auto
paulson@14485
   825
  apply (rule_tac x = "nat x" in exI)
huffman@35216
   826
  apply (auto simp add: zless_nat_eq_int_zless [THEN sym])
paulson@14485
   827
  done
paulson@14485
   828
nipkow@15045
   829
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
paulson@14485
   830
  apply (case_tac "0 \<le> u")
paulson@14485
   831
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   832
  apply (rule finite_imageI)
paulson@14485
   833
  apply auto
paulson@14485
   834
  done
paulson@14485
   835
nipkow@15045
   836
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
nipkow@15045
   837
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   838
  apply (erule subst)
paulson@14485
   839
  apply (rule finite_imageI)
paulson@14485
   840
  apply (rule finite_atLeastZeroLessThan_int)
nipkow@16733
   841
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   842
  done
paulson@14485
   843
paulson@15418
   844
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
paulson@14485
   845
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
paulson@14485
   846
paulson@15418
   847
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
paulson@14485
   848
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   849
paulson@15418
   850
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
paulson@14485
   851
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   852
nipkow@24853
   853
paulson@14485
   854
subsubsection {* Cardinality *}
paulson@14485
   855
nipkow@15045
   856
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
paulson@14485
   857
  apply (case_tac "0 \<le> u")
paulson@14485
   858
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   859
  apply (subst card_image)
paulson@14485
   860
  apply (auto simp add: inj_on_def)
paulson@14485
   861
  done
paulson@14485
   862
nipkow@15045
   863
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
nipkow@15045
   864
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
paulson@14485
   865
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
nipkow@15045
   866
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   867
  apply (erule subst)
paulson@14485
   868
  apply (rule card_image)
paulson@14485
   869
  apply (simp add: inj_on_def)
nipkow@16733
   870
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   871
  done
paulson@14485
   872
paulson@14485
   873
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
nipkow@29667
   874
apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
nipkow@29667
   875
apply (auto simp add: algebra_simps)
nipkow@29667
   876
done
paulson@14485
   877
paulson@15418
   878
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
nipkow@29667
   879
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   880
nipkow@15045
   881
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
nipkow@29667
   882
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   883
bulwahn@27656
   884
lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
bulwahn@27656
   885
proof -
bulwahn@27656
   886
  have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
bulwahn@27656
   887
  with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
bulwahn@27656
   888
qed
bulwahn@27656
   889
bulwahn@27656
   890
lemma card_less:
bulwahn@27656
   891
assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   892
shows "card {k \<in> M. k < Suc i} \<noteq> 0"
bulwahn@27656
   893
proof -
bulwahn@27656
   894
  from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
bulwahn@27656
   895
  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
bulwahn@27656
   896
qed
bulwahn@27656
   897
bulwahn@27656
   898
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
   899
apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])
bulwahn@27656
   900
apply simp
bulwahn@27656
   901
apply fastsimp
bulwahn@27656
   902
apply auto
bulwahn@27656
   903
apply (rule inj_on_diff_nat)
bulwahn@27656
   904
apply auto
bulwahn@27656
   905
apply (case_tac x)
bulwahn@27656
   906
apply auto
bulwahn@27656
   907
apply (case_tac xa)
bulwahn@27656
   908
apply auto
bulwahn@27656
   909
apply (case_tac xa)
bulwahn@27656
   910
apply auto
bulwahn@27656
   911
done
bulwahn@27656
   912
bulwahn@27656
   913
lemma card_less_Suc:
bulwahn@27656
   914
  assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   915
    shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
bulwahn@27656
   916
proof -
bulwahn@27656
   917
  from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
bulwahn@27656
   918
  hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
bulwahn@27656
   919
    by (auto simp only: insert_Diff)
bulwahn@27656
   920
  have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
bulwahn@27656
   921
  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
   922
    apply (subst card_insert)
bulwahn@27656
   923
    apply simp_all
bulwahn@27656
   924
    apply (subst b)
bulwahn@27656
   925
    apply (subst card_less_Suc2[symmetric])
bulwahn@27656
   926
    apply simp_all
bulwahn@27656
   927
    done
bulwahn@27656
   928
  with c show ?thesis by simp
bulwahn@27656
   929
qed
bulwahn@27656
   930
paulson@14485
   931
paulson@13850
   932
subsection {*Lemmas useful with the summation operator setsum*}
paulson@13850
   933
ballarin@16102
   934
text {* For examples, see Algebra/poly/UnivPoly2.thy *}
ballarin@13735
   935
wenzelm@14577
   936
subsubsection {* Disjoint Unions *}
ballarin@13735
   937
wenzelm@14577
   938
text {* Singletons and open intervals *}
ballarin@13735
   939
ballarin@13735
   940
lemma ivl_disj_un_singleton:
nipkow@15045
   941
  "{l::'a::linorder} Un {l<..} = {l..}"
nipkow@15045
   942
  "{..<u} Un {u::'a::linorder} = {..u}"
nipkow@15045
   943
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
nipkow@15045
   944
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
nipkow@15045
   945
  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
nipkow@15045
   946
  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
ballarin@14398
   947
by auto
ballarin@13735
   948
wenzelm@14577
   949
text {* One- and two-sided intervals *}
ballarin@13735
   950
ballarin@13735
   951
lemma ivl_disj_un_one:
nipkow@15045
   952
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
nipkow@15045
   953
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
nipkow@15045
   954
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
nipkow@15045
   955
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
nipkow@15045
   956
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
nipkow@15045
   957
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
nipkow@15045
   958
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
nipkow@15045
   959
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
ballarin@14398
   960
by auto
ballarin@13735
   961
wenzelm@14577
   962
text {* Two- and two-sided intervals *}
ballarin@13735
   963
ballarin@13735
   964
lemma ivl_disj_un_two:
nipkow@15045
   965
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
nipkow@15045
   966
  "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
nipkow@15045
   967
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
nipkow@15045
   968
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
nipkow@15045
   969
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
nipkow@15045
   970
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
nipkow@15045
   971
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
nipkow@15045
   972
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
ballarin@14398
   973
by auto
ballarin@13735
   974
ballarin@13735
   975
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
ballarin@13735
   976
wenzelm@14577
   977
subsubsection {* Disjoint Intersections *}
ballarin@13735
   978
wenzelm@14577
   979
text {* One- and two-sided intervals *}
ballarin@13735
   980
ballarin@13735
   981
lemma ivl_disj_int_one:
nipkow@15045
   982
  "{..l::'a::order} Int {l<..<u} = {}"
nipkow@15045
   983
  "{..<l} Int {l..<u} = {}"
nipkow@15045
   984
  "{..l} Int {l<..u} = {}"
nipkow@15045
   985
  "{..<l} Int {l..u} = {}"
nipkow@15045
   986
  "{l<..u} Int {u<..} = {}"
nipkow@15045
   987
  "{l<..<u} Int {u..} = {}"
nipkow@15045
   988
  "{l..u} Int {u<..} = {}"
nipkow@15045
   989
  "{l..<u} Int {u..} = {}"
ballarin@14398
   990
  by auto
ballarin@13735
   991
wenzelm@14577
   992
text {* Two- and two-sided intervals *}
ballarin@13735
   993
ballarin@13735
   994
lemma ivl_disj_int_two:
nipkow@15045
   995
  "{l::'a::order<..<m} Int {m..<u} = {}"
nipkow@15045
   996
  "{l<..m} Int {m<..<u} = {}"
nipkow@15045
   997
  "{l..<m} Int {m..<u} = {}"
nipkow@15045
   998
  "{l..m} Int {m<..<u} = {}"
nipkow@15045
   999
  "{l<..<m} Int {m..u} = {}"
nipkow@15045
  1000
  "{l<..m} Int {m<..u} = {}"
nipkow@15045
  1001
  "{l..<m} Int {m..u} = {}"
nipkow@15045
  1002
  "{l..m} Int {m<..u} = {}"
ballarin@14398
  1003
  by auto
ballarin@13735
  1004
nipkow@32456
  1005
lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two
ballarin@13735
  1006
nipkow@15542
  1007
subsubsection {* Some Differences *}
nipkow@15542
  1008
nipkow@15542
  1009
lemma ivl_diff[simp]:
nipkow@15542
  1010
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
nipkow@15542
  1011
by(auto)
nipkow@15542
  1012
nipkow@15542
  1013
nipkow@15542
  1014
subsubsection {* Some Subset Conditions *}
nipkow@15542
  1015
blanchet@35828
  1016
lemma ivl_subset [simp,no_atp]:
nipkow@15542
  1017
 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
nipkow@15542
  1018
apply(auto simp:linorder_not_le)
nipkow@15542
  1019
apply(rule ccontr)
nipkow@15542
  1020
apply(insert linorder_le_less_linear[of i n])
nipkow@15542
  1021
apply(clarsimp simp:linorder_not_le)
nipkow@15542
  1022
apply(fastsimp)
nipkow@15542
  1023
done
nipkow@15542
  1024
nipkow@15041
  1025
nipkow@15042
  1026
subsection {* Summation indexed over intervals *}
nipkow@15042
  1027
nipkow@15042
  1028
syntax
nipkow@15042
  1029
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
  1030
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
  1031
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
nipkow@16052
  1032
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
nipkow@15042
  1033
syntax (xsymbols)
nipkow@15042
  1034
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
  1035
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
  1036
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
  1037
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15042
  1038
syntax (HTML output)
nipkow@15042
  1039
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
  1040
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
  1041
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
  1042
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15056
  1043
syntax (latex_sum output)
nipkow@15052
  1044
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
  1045
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052
  1046
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
  1047
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@16052
  1048
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
  1049
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15052
  1050
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
  1051
 ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15041
  1052
nipkow@15048
  1053
translations
nipkow@28853
  1054
  "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
nipkow@28853
  1055
  "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
nipkow@28853
  1056
  "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
nipkow@28853
  1057
  "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
nipkow@15041
  1058
nipkow@15052
  1059
text{* The above introduces some pretty alternative syntaxes for
nipkow@15056
  1060
summation over intervals:
nipkow@15052
  1061
\begin{center}
nipkow@15052
  1062
\begin{tabular}{lll}
nipkow@15056
  1063
Old & New & \LaTeX\\
nipkow@15056
  1064
@{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
  1065
@{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
  1066
@{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
nipkow@15056
  1067
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
nipkow@15052
  1068
\end{tabular}
nipkow@15052
  1069
\end{center}
nipkow@15056
  1070
The left column shows the term before introduction of the new syntax,
nipkow@15056
  1071
the middle column shows the new (default) syntax, and the right column
nipkow@15056
  1072
shows a special syntax. The latter is only meaningful for latex output
nipkow@15056
  1073
and has to be activated explicitly by setting the print mode to
wenzelm@21502
  1074
@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
nipkow@15056
  1075
antiquotations). It is not the default \LaTeX\ output because it only
nipkow@15056
  1076
works well with italic-style formulae, not tt-style.
nipkow@15052
  1077
nipkow@15052
  1078
Note that for uniformity on @{typ nat} it is better to use
nipkow@15052
  1079
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
nipkow@15052
  1080
not provide all lemmas available for @{term"{m..<n}"} also in the
nipkow@15052
  1081
special form for @{term"{..<n}"}. *}
nipkow@15052
  1082
nipkow@15542
  1083
text{* This congruence rule should be used for sums over intervals as
nipkow@15542
  1084
the standard theorem @{text[source]setsum_cong} does not work well
nipkow@15542
  1085
with the simplifier who adds the unsimplified premise @{term"x:B"} to
nipkow@15542
  1086
the context. *}
nipkow@15542
  1087
nipkow@15542
  1088
lemma setsum_ivl_cong:
nipkow@15542
  1089
 "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
nipkow@15542
  1090
 setsum f {a..<b} = setsum g {c..<d}"
nipkow@15542
  1091
by(rule setsum_cong, simp_all)
nipkow@15041
  1092
nipkow@16041
  1093
(* FIXME why are the following simp rules but the corresponding eqns
nipkow@16041
  1094
on intervals are not? *)
nipkow@16041
  1095
nipkow@16052
  1096
lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
nipkow@16052
  1097
by (simp add:atMost_Suc add_ac)
nipkow@16052
  1098
nipkow@16041
  1099
lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
nipkow@16041
  1100
by (simp add:lessThan_Suc add_ac)
nipkow@15041
  1101
nipkow@15911
  1102
lemma setsum_cl_ivl_Suc[simp]:
nipkow@15561
  1103
  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
nipkow@15561
  1104
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@15561
  1105
nipkow@15911
  1106
lemma setsum_op_ivl_Suc[simp]:
nipkow@15561
  1107
  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
nipkow@15561
  1108
by (auto simp:add_ac atLeastLessThanSuc)
nipkow@16041
  1109
(*
nipkow@15561
  1110
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
nipkow@15561
  1111
    (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
nipkow@15561
  1112
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@16041
  1113
*)
nipkow@28068
  1114
nipkow@28068
  1115
lemma setsum_head:
nipkow@28068
  1116
  fixes n :: nat
nipkow@28068
  1117
  assumes mn: "m <= n" 
nipkow@28068
  1118
  shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
nipkow@28068
  1119
proof -
nipkow@28068
  1120
  from mn
nipkow@28068
  1121
  have "{m..n} = {m} \<union> {m<..n}"
nipkow@28068
  1122
    by (auto intro: ivl_disj_un_singleton)
nipkow@28068
  1123
  hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
nipkow@28068
  1124
    by (simp add: atLeast0LessThan)
nipkow@28068
  1125
  also have "\<dots> = ?rhs" by simp
nipkow@28068
  1126
  finally show ?thesis .
nipkow@28068
  1127
qed
nipkow@28068
  1128
nipkow@28068
  1129
lemma setsum_head_Suc:
nipkow@28068
  1130
  "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
nipkow@28068
  1131
by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
nipkow@28068
  1132
nipkow@28068
  1133
lemma setsum_head_upt_Suc:
nipkow@28068
  1134
  "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
huffman@30079
  1135
apply(insert setsum_head_Suc[of m "n - Suc 0" f])
nipkow@29667
  1136
apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
nipkow@28068
  1137
done
nipkow@28068
  1138
nipkow@31501
  1139
lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"
nipkow@31501
  1140
  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
nipkow@31501
  1141
proof-
nipkow@31501
  1142
  have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using `m \<le> n+1` by auto
nipkow@31501
  1143
  thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint
nipkow@31501
  1144
    atLeastSucAtMost_greaterThanAtMost)
nipkow@31501
  1145
qed
nipkow@28068
  1146
nipkow@15539
  1147
lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
  1148
  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
nipkow@15539
  1149
by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
nipkow@15539
  1150
nipkow@15539
  1151
lemma setsum_diff_nat_ivl:
nipkow@15539
  1152
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
nipkow@15539
  1153
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
  1154
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
nipkow@15539
  1155
using setsum_add_nat_ivl [of m n p f,symmetric]
nipkow@15539
  1156
apply (simp add: add_ac)
nipkow@15539
  1157
done
nipkow@15539
  1158
nipkow@31505
  1159
lemma setsum_natinterval_difff:
nipkow@31505
  1160
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
nipkow@31505
  1161
  shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
nipkow@31505
  1162
          (if m <= n then f m - f(n + 1) else 0)"
nipkow@31505
  1163
by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
nipkow@31505
  1164
nipkow@31509
  1165
lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
nipkow@31509
  1166
nipkow@31509
  1167
lemma setsum_setsum_restrict:
nipkow@31509
  1168
  "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
  1169
  by (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
nipkow@31509
  1170
     (rule setsum_commute)
nipkow@31509
  1171
nipkow@31509
  1172
lemma setsum_image_gen: assumes fS: "finite S"
nipkow@31509
  1173
  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
nipkow@31509
  1174
proof-
nipkow@31509
  1175
  { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
nipkow@31509
  1176
  hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
nipkow@31509
  1177
    by simp
nipkow@31509
  1178
  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
nipkow@31509
  1179
    by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])
nipkow@31509
  1180
  finally show ?thesis .
nipkow@31509
  1181
qed
nipkow@31509
  1182
hoelzl@35171
  1183
lemma setsum_le_included:
haftmann@36307
  1184
  fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"
hoelzl@35171
  1185
  assumes "finite s" "finite t"
hoelzl@35171
  1186
  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
  1187
  shows "setsum f s \<le> setsum g t"
hoelzl@35171
  1188
proof -
hoelzl@35171
  1189
  have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"
hoelzl@35171
  1190
  proof (rule setsum_mono)
hoelzl@35171
  1191
    fix y assume "y \<in> s"
hoelzl@35171
  1192
    with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
hoelzl@35171
  1193
    with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
hoelzl@35171
  1194
      using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]
hoelzl@35171
  1195
      by (auto intro!: setsum_mono2)
hoelzl@35171
  1196
  qed
hoelzl@35171
  1197
  also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
hoelzl@35171
  1198
    using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)
hoelzl@35171
  1199
  also have "... \<le> setsum g t"
hoelzl@35171
  1200
    using assms by (auto simp: setsum_image_gen[symmetric])
hoelzl@35171
  1201
  finally show ?thesis .
hoelzl@35171
  1202
qed
hoelzl@35171
  1203
nipkow@31509
  1204
lemma setsum_multicount_gen:
nipkow@31509
  1205
  assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
nipkow@31509
  1206
  shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
nipkow@31509
  1207
proof-
nipkow@31509
  1208
  have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
nipkow@31509
  1209
  also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]
nipkow@31509
  1210
    using assms(3) by auto
nipkow@31509
  1211
  finally show ?thesis .
nipkow@31509
  1212
qed
nipkow@31509
  1213
nipkow@31509
  1214
lemma setsum_multicount:
nipkow@31509
  1215
  assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
nipkow@31509
  1216
  shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
nipkow@31509
  1217
proof-
nipkow@31509
  1218
  have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)
huffman@35216
  1219
  also have "\<dots> = ?r" by(simp add: mult_commute)
nipkow@31509
  1220
  finally show ?thesis by auto
nipkow@31509
  1221
qed
nipkow@31509
  1222
nipkow@28068
  1223
nipkow@16733
  1224
subsection{* Shifting bounds *}
nipkow@16733
  1225
nipkow@15539
  1226
lemma setsum_shift_bounds_nat_ivl:
nipkow@15539
  1227
  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
nipkow@15539
  1228
by (induct "n", auto simp:atLeastLessThanSuc)
nipkow@15539
  1229
nipkow@16733
  1230
lemma setsum_shift_bounds_cl_nat_ivl:
nipkow@16733
  1231
  "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
nipkow@16733
  1232
apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
nipkow@16733
  1233
apply (simp add:image_add_atLeastAtMost o_def)
nipkow@16733
  1234
done
nipkow@16733
  1235
nipkow@16733
  1236
corollary setsum_shift_bounds_cl_Suc_ivl:
nipkow@16733
  1237
  "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
huffman@30079
  1238
by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
  1239
nipkow@16733
  1240
corollary setsum_shift_bounds_Suc_ivl:
nipkow@16733
  1241
  "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
huffman@30079
  1242
by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
  1243
nipkow@28068
  1244
lemma setsum_shift_lb_Suc0_0:
nipkow@28068
  1245
  "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
nipkow@28068
  1246
by(simp add:setsum_head_Suc)
kleing@19106
  1247
nipkow@28068
  1248
lemma setsum_shift_lb_Suc0_0_upt:
nipkow@28068
  1249
  "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
nipkow@28068
  1250
apply(cases k)apply simp
nipkow@28068
  1251
apply(simp add:setsum_head_upt_Suc)
nipkow@28068
  1252
done
kleing@19022
  1253
ballarin@17149
  1254
subsection {* The formula for geometric sums *}
ballarin@17149
  1255
ballarin@17149
  1256
lemma geometric_sum:
haftmann@36307
  1257
  assumes "x \<noteq> 1"
haftmann@36307
  1258
  shows "(\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"
haftmann@36307
  1259
proof -
haftmann@36307
  1260
  from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all
haftmann@36307
  1261
  moreover have "(\<Sum>i=0..<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"
haftmann@36307
  1262
  proof (induct n)
haftmann@36307
  1263
    case 0 then show ?case by simp
haftmann@36307
  1264
  next
haftmann@36307
  1265
    case (Suc n)
haftmann@36307
  1266
    moreover with `y \<noteq> 0` have "(1 + y) ^ n = (y * inverse y) * (1 + y) ^ n" by simp 
haftmann@36350
  1267
    ultimately show ?case by (simp add: field_simps divide_inverse)
haftmann@36307
  1268
  qed
haftmann@36307
  1269
  ultimately show ?thesis by simp
haftmann@36307
  1270
qed
haftmann@36307
  1271
ballarin@17149
  1272
kleing@19469
  1273
subsection {* The formula for arithmetic sums *}
kleing@19469
  1274
kleing@19469
  1275
lemma gauss_sum:
huffman@23277
  1276
  "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
kleing@19469
  1277
   of_nat n*((of_nat n)+1)"
kleing@19469
  1278
proof (induct n)
kleing@19469
  1279
  case 0
kleing@19469
  1280
  show ?case by simp
kleing@19469
  1281
next
kleing@19469
  1282
  case (Suc n)
nipkow@29667
  1283
  then show ?case by (simp add: algebra_simps)
kleing@19469
  1284
qed
kleing@19469
  1285
kleing@19469
  1286
theorem arith_series_general:
huffman@23277
  1287
  "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1288
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1289
proof cases
kleing@19469
  1290
  assume ngt1: "n > 1"
kleing@19469
  1291
  let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
kleing@19469
  1292
  have
kleing@19469
  1293
    "(\<Sum>i\<in>{..<n}. a+?I i*d) =
kleing@19469
  1294
     ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
kleing@19469
  1295
    by (rule setsum_addf)
kleing@19469
  1296
  also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
kleing@19469
  1297
  also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
huffman@30079
  1298
    unfolding One_nat_def
nipkow@28068
  1299
    by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)
kleing@19469
  1300
  also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
kleing@19469
  1301
    by (simp add: left_distrib right_distrib)
kleing@19469
  1302
  also from ngt1 have "{1..<n} = {1..n - 1}"
nipkow@28068
  1303
    by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
nipkow@28068
  1304
  also from ngt1
kleing@19469
  1305
  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
  1306
    by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)
huffman@23431
  1307
       (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
nipkow@29667
  1308
  finally show ?thesis by (simp add: algebra_simps)
kleing@19469
  1309
next
kleing@19469
  1310
  assume "\<not>(n > 1)"
kleing@19469
  1311
  hence "n = 1 \<or> n = 0" by auto
nipkow@29667
  1312
  thus ?thesis by (auto simp: algebra_simps)
kleing@19469
  1313
qed
kleing@19469
  1314
kleing@19469
  1315
lemma arith_series_nat:
kleing@19469
  1316
  "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
kleing@19469
  1317
proof -
kleing@19469
  1318
  have
kleing@19469
  1319
    "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
kleing@19469
  1320
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1321
    by (rule arith_series_general)
huffman@30079
  1322
  thus ?thesis
huffman@35216
  1323
    unfolding One_nat_def by auto
kleing@19469
  1324
qed
kleing@19469
  1325
kleing@19469
  1326
lemma arith_series_int:
kleing@19469
  1327
  "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1328
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1329
proof -
kleing@19469
  1330
  have
kleing@19469
  1331
    "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1332
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1333
    by (rule arith_series_general)
kleing@19469
  1334
  thus ?thesis by simp
kleing@19469
  1335
qed
paulson@15418
  1336
kleing@19022
  1337
lemma sum_diff_distrib:
kleing@19022
  1338
  fixes P::"nat\<Rightarrow>nat"
kleing@19022
  1339
  shows
kleing@19022
  1340
  "\<forall>x. Q x \<le> P x  \<Longrightarrow>
kleing@19022
  1341
  (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
kleing@19022
  1342
proof (induct n)
kleing@19022
  1343
  case 0 show ?case by simp
kleing@19022
  1344
next
kleing@19022
  1345
  case (Suc n)
kleing@19022
  1346
kleing@19022
  1347
  let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
kleing@19022
  1348
  let ?rhs = "\<Sum>x<n. P x - Q x"
kleing@19022
  1349
kleing@19022
  1350
  from Suc have "?lhs = ?rhs" by simp
kleing@19022
  1351
  moreover
kleing@19022
  1352
  from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
kleing@19022
  1353
  moreover
kleing@19022
  1354
  from Suc have
kleing@19022
  1355
    "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
kleing@19022
  1356
    by (subst diff_diff_left[symmetric],
kleing@19022
  1357
        subst diff_add_assoc2)
kleing@19022
  1358
       (auto simp: diff_add_assoc2 intro: setsum_mono)
kleing@19022
  1359
  ultimately
kleing@19022
  1360
  show ?case by simp
kleing@19022
  1361
qed
kleing@19022
  1362
paulson@29960
  1363
subsection {* Products indexed over intervals *}
paulson@29960
  1364
paulson@29960
  1365
syntax
paulson@29960
  1366
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1367
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1368
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)
paulson@29960
  1369
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
paulson@29960
  1370
syntax (xsymbols)
paulson@29960
  1371
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1372
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1373
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1374
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1375
syntax (HTML output)
paulson@29960
  1376
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1377
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1378
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1379
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1380
syntax (latex_prod output)
paulson@29960
  1381
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1382
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1383
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1384
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1385
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1386
 ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1387
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1388
 ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1389
paulson@29960
  1390
translations
paulson@29960
  1391
  "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"
paulson@29960
  1392
  "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"
paulson@29960
  1393
  "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"
paulson@29960
  1394
  "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"
paulson@29960
  1395
haftmann@33318
  1396
subsection {* Transfer setup *}
haftmann@33318
  1397
haftmann@33318
  1398
lemma transfer_nat_int_set_functions:
haftmann@33318
  1399
    "{..n} = nat ` {0..int n}"
haftmann@33318
  1400
    "{m..n} = nat ` {int m..int n}"  (* need all variants of these! *)
haftmann@33318
  1401
  apply (auto simp add: image_def)
haftmann@33318
  1402
  apply (rule_tac x = "int x" in bexI)
haftmann@33318
  1403
  apply auto
haftmann@33318
  1404
  apply (rule_tac x = "int x" in bexI)
haftmann@33318
  1405
  apply auto
haftmann@33318
  1406
  done
haftmann@33318
  1407
haftmann@33318
  1408
lemma transfer_nat_int_set_function_closures:
haftmann@33318
  1409
    "x >= 0 \<Longrightarrow> nat_set {x..y}"
haftmann@33318
  1410
  by (simp add: nat_set_def)
haftmann@33318
  1411
haftmann@35644
  1412
declare transfer_morphism_nat_int[transfer add
haftmann@33318
  1413
  return: transfer_nat_int_set_functions
haftmann@33318
  1414
    transfer_nat_int_set_function_closures
haftmann@33318
  1415
]
haftmann@33318
  1416
haftmann@33318
  1417
lemma transfer_int_nat_set_functions:
haftmann@33318
  1418
    "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}"
haftmann@33318
  1419
  by (simp only: is_nat_def transfer_nat_int_set_functions
haftmann@33318
  1420
    transfer_nat_int_set_function_closures
haftmann@33318
  1421
    transfer_nat_int_set_return_embed nat_0_le
haftmann@33318
  1422
    cong: transfer_nat_int_set_cong)
haftmann@33318
  1423
haftmann@33318
  1424
lemma transfer_int_nat_set_function_closures:
haftmann@33318
  1425
    "is_nat x \<Longrightarrow> nat_set {x..y}"
haftmann@33318
  1426
  by (simp only: transfer_nat_int_set_function_closures is_nat_def)
haftmann@33318
  1427
haftmann@35644
  1428
declare transfer_morphism_int_nat[transfer add
haftmann@33318
  1429
  return: transfer_int_nat_set_functions
haftmann@33318
  1430
    transfer_int_nat_set_function_closures
haftmann@33318
  1431
]
haftmann@33318
  1432
nipkow@8924
  1433
end