src/HOL/SetInterval.thy
author huffman
Fri Mar 30 14:25:32 2012 +0200 (2012-03-30)
changeset 47222 1b7c909a6fad
parent 47108 2a1953f0d20d
permissions -rw-r--r--
rephrase lemmas about arithmetic series using numeral '2'
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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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hoelzl@40703
   313
lemma atLeastLessThan_inj:
hoelzl@40703
   314
  fixes a b c d :: "'a::linorder"
hoelzl@40703
   315
  assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"
hoelzl@40703
   316
  shows "a = c" "b = d"
hoelzl@40703
   317
using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le linorder_antisym_conv2 subset_refl)+
hoelzl@40703
   318
hoelzl@40703
   319
lemma atLeastLessThan_eq_iff:
hoelzl@40703
   320
  fixes a b c d :: "'a::linorder"
hoelzl@40703
   321
  assumes "a < b" "c < d"
hoelzl@40703
   322
  shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"
hoelzl@40703
   323
  using atLeastLessThan_inj assms by auto
hoelzl@40703
   324
nipkow@32456
   325
subsubsection {* Intersection *}
nipkow@32456
   326
nipkow@32456
   327
context linorder
nipkow@32456
   328
begin
nipkow@32456
   329
nipkow@32456
   330
lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"
nipkow@32456
   331
by auto
nipkow@32456
   332
nipkow@32456
   333
lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"
nipkow@32456
   334
by auto
nipkow@32456
   335
nipkow@32456
   336
lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"
nipkow@32456
   337
by auto
nipkow@32456
   338
nipkow@32456
   339
lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"
nipkow@32456
   340
by auto
nipkow@32456
   341
nipkow@32456
   342
lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"
nipkow@32456
   343
by auto
nipkow@32456
   344
nipkow@32456
   345
lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"
nipkow@32456
   346
by auto
nipkow@32456
   347
nipkow@32456
   348
lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
nipkow@32456
   349
by auto
nipkow@32456
   350
nipkow@32456
   351
lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
nipkow@32456
   352
by auto
nipkow@32456
   353
nipkow@32456
   354
end
nipkow@32456
   355
nipkow@32456
   356
paulson@14485
   357
subsection {* Intervals of natural numbers *}
paulson@14485
   358
paulson@15047
   359
subsubsection {* The Constant @{term lessThan} *}
paulson@15047
   360
paulson@14485
   361
lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
paulson@14485
   362
by (simp add: lessThan_def)
paulson@14485
   363
paulson@14485
   364
lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
paulson@14485
   365
by (simp add: lessThan_def less_Suc_eq, blast)
paulson@14485
   366
kleing@43156
   367
text {* The following proof is convenient in induction proofs where
hoelzl@39072
   368
new elements get indices at the beginning. So it is used to transform
hoelzl@39072
   369
@{term "{..<Suc n}"} to @{term "0::nat"} and @{term "{..< n}"}. *}
hoelzl@39072
   370
hoelzl@39072
   371
lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"
hoelzl@39072
   372
proof safe
hoelzl@39072
   373
  fix x assume "x < Suc n" "x \<notin> Suc ` {..<n}"
hoelzl@39072
   374
  then have "x \<noteq> Suc (x - 1)" by auto
hoelzl@39072
   375
  with `x < Suc n` show "x = 0" by auto
hoelzl@39072
   376
qed
hoelzl@39072
   377
paulson@14485
   378
lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
paulson@14485
   379
by (simp add: lessThan_def atMost_def less_Suc_eq_le)
paulson@14485
   380
paulson@14485
   381
lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
paulson@14485
   382
by blast
paulson@14485
   383
paulson@15047
   384
subsubsection {* The Constant @{term greaterThan} *}
paulson@15047
   385
paulson@14485
   386
lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
paulson@14485
   387
apply (simp add: greaterThan_def)
paulson@14485
   388
apply (blast dest: gr0_conv_Suc [THEN iffD1])
paulson@14485
   389
done
paulson@14485
   390
paulson@14485
   391
lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
paulson@14485
   392
apply (simp add: greaterThan_def)
paulson@14485
   393
apply (auto elim: linorder_neqE)
paulson@14485
   394
done
paulson@14485
   395
paulson@14485
   396
lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
paulson@14485
   397
by blast
paulson@14485
   398
paulson@15047
   399
subsubsection {* The Constant @{term atLeast} *}
paulson@15047
   400
paulson@14485
   401
lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
paulson@14485
   402
by (unfold atLeast_def UNIV_def, simp)
paulson@14485
   403
paulson@14485
   404
lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
paulson@14485
   405
apply (simp add: atLeast_def)
paulson@14485
   406
apply (simp add: Suc_le_eq)
paulson@14485
   407
apply (simp add: order_le_less, blast)
paulson@14485
   408
done
paulson@14485
   409
paulson@14485
   410
lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
paulson@14485
   411
  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
paulson@14485
   412
paulson@14485
   413
lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
paulson@14485
   414
by blast
paulson@14485
   415
paulson@15047
   416
subsubsection {* The Constant @{term atMost} *}
paulson@15047
   417
paulson@14485
   418
lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
paulson@14485
   419
by (simp add: atMost_def)
paulson@14485
   420
paulson@14485
   421
lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
paulson@14485
   422
apply (simp add: atMost_def)
paulson@14485
   423
apply (simp add: less_Suc_eq order_le_less, blast)
paulson@14485
   424
done
paulson@14485
   425
paulson@14485
   426
lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
paulson@14485
   427
by blast
paulson@14485
   428
paulson@15047
   429
subsubsection {* The Constant @{term atLeastLessThan} *}
paulson@15047
   430
nipkow@28068
   431
text{*The orientation of the following 2 rules is tricky. The lhs is
nipkow@24449
   432
defined in terms of the rhs.  Hence the chosen orientation makes sense
nipkow@24449
   433
in this theory --- the reverse orientation complicates proofs (eg
nipkow@24449
   434
nontermination). But outside, when the definition of the lhs is rarely
nipkow@24449
   435
used, the opposite orientation seems preferable because it reduces a
nipkow@24449
   436
specific concept to a more general one. *}
nipkow@28068
   437
paulson@15047
   438
lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
nipkow@15042
   439
by(simp add:lessThan_def atLeastLessThan_def)
nipkow@24449
   440
nipkow@28068
   441
lemma atLeast0AtMost: "{0..n::nat} = {..n}"
nipkow@28068
   442
by(simp add:atMost_def atLeastAtMost_def)
nipkow@28068
   443
haftmann@31998
   444
declare atLeast0LessThan[symmetric, code_unfold]
haftmann@31998
   445
        atLeast0AtMost[symmetric, code_unfold]
nipkow@24449
   446
nipkow@24449
   447
lemma atLeastLessThan0: "{m..<0::nat} = {}"
paulson@15047
   448
by (simp add: atLeastLessThan_def)
nipkow@24449
   449
paulson@15047
   450
subsubsection {* Intervals of nats with @{term Suc} *}
paulson@15047
   451
paulson@15047
   452
text{*Not a simprule because the RHS is too messy.*}
paulson@15047
   453
lemma atLeastLessThanSuc:
paulson@15047
   454
    "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
paulson@15418
   455
by (auto simp add: atLeastLessThan_def)
paulson@15047
   456
paulson@15418
   457
lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
paulson@15047
   458
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   459
(*
paulson@15047
   460
lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
paulson@15047
   461
by (induct k, simp_all add: atLeastLessThanSuc)
paulson@15047
   462
paulson@15047
   463
lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
paulson@15047
   464
by (auto simp add: atLeastLessThan_def)
nipkow@16041
   465
*)
nipkow@15045
   466
lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
paulson@14485
   467
  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   468
paulson@15418
   469
lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
paulson@15418
   470
  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
paulson@14485
   471
    greaterThanAtMost_def)
paulson@14485
   472
paulson@15418
   473
lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
paulson@15418
   474
  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
paulson@14485
   475
    greaterThanLessThan_def)
paulson@14485
   476
nipkow@15554
   477
lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
nipkow@15554
   478
by (auto simp add: atLeastAtMost_def)
nipkow@15554
   479
noschinl@45932
   480
lemma atLeastAtMost_insertL: "m \<le> n \<Longrightarrow> insert m {Suc m..n} = {m ..n}"
noschinl@45932
   481
by auto
noschinl@45932
   482
kleing@43157
   483
text {* The analogous result is useful on @{typ int}: *}
kleing@43157
   484
(* here, because we don't have an own int section *)
kleing@43157
   485
lemma atLeastAtMostPlus1_int_conv:
kleing@43157
   486
  "m <= 1+n \<Longrightarrow> {m..1+n} = insert (1+n) {m..n::int}"
kleing@43157
   487
  by (auto intro: set_eqI)
kleing@43157
   488
paulson@33044
   489
lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"
paulson@33044
   490
  apply (induct k) 
paulson@33044
   491
  apply (simp_all add: atLeastLessThanSuc)   
paulson@33044
   492
  done
paulson@33044
   493
nipkow@16733
   494
subsubsection {* Image *}
nipkow@16733
   495
nipkow@16733
   496
lemma image_add_atLeastAtMost:
nipkow@16733
   497
  "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
nipkow@16733
   498
proof
nipkow@16733
   499
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   500
next
nipkow@16733
   501
  show "?B \<subseteq> ?A"
nipkow@16733
   502
  proof
nipkow@16733
   503
    fix n assume a: "n : ?B"
webertj@20217
   504
    hence "n - k : {i..j}" by auto
nipkow@16733
   505
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   506
    ultimately show "n : ?A" by blast
nipkow@16733
   507
  qed
nipkow@16733
   508
qed
nipkow@16733
   509
nipkow@16733
   510
lemma image_add_atLeastLessThan:
nipkow@16733
   511
  "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
nipkow@16733
   512
proof
nipkow@16733
   513
  show "?A \<subseteq> ?B" by auto
nipkow@16733
   514
next
nipkow@16733
   515
  show "?B \<subseteq> ?A"
nipkow@16733
   516
  proof
nipkow@16733
   517
    fix n assume a: "n : ?B"
webertj@20217
   518
    hence "n - k : {i..<j}" by auto
nipkow@16733
   519
    moreover have "n = (n - k) + k" using a by auto
nipkow@16733
   520
    ultimately show "n : ?A" by blast
nipkow@16733
   521
  qed
nipkow@16733
   522
qed
nipkow@16733
   523
nipkow@16733
   524
corollary image_Suc_atLeastAtMost[simp]:
nipkow@16733
   525
  "Suc ` {i..j} = {Suc i..Suc j}"
huffman@30079
   526
using image_add_atLeastAtMost[where k="Suc 0"] by simp
nipkow@16733
   527
nipkow@16733
   528
corollary image_Suc_atLeastLessThan[simp]:
nipkow@16733
   529
  "Suc ` {i..<j} = {Suc i..<Suc j}"
huffman@30079
   530
using image_add_atLeastLessThan[where k="Suc 0"] by simp
nipkow@16733
   531
nipkow@16733
   532
lemma image_add_int_atLeastLessThan:
nipkow@16733
   533
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
nipkow@16733
   534
  apply (auto simp add: image_def)
nipkow@16733
   535
  apply (rule_tac x = "x - l" in bexI)
nipkow@16733
   536
  apply auto
nipkow@16733
   537
  done
nipkow@16733
   538
hoelzl@37664
   539
lemma image_minus_const_atLeastLessThan_nat:
hoelzl@37664
   540
  fixes c :: nat
hoelzl@37664
   541
  shows "(\<lambda>i. i - c) ` {x ..< y} =
hoelzl@37664
   542
      (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"
hoelzl@37664
   543
    (is "_ = ?right")
hoelzl@37664
   544
proof safe
hoelzl@37664
   545
  fix a assume a: "a \<in> ?right"
hoelzl@37664
   546
  show "a \<in> (\<lambda>i. i - c) ` {x ..< y}"
hoelzl@37664
   547
  proof cases
hoelzl@37664
   548
    assume "c < y" with a show ?thesis
hoelzl@37664
   549
      by (auto intro!: image_eqI[of _ _ "a + c"])
hoelzl@37664
   550
  next
hoelzl@37664
   551
    assume "\<not> c < y" with a show ?thesis
hoelzl@37664
   552
      by (auto intro!: image_eqI[of _ _ x] split: split_if_asm)
hoelzl@37664
   553
  qed
hoelzl@37664
   554
qed auto
hoelzl@37664
   555
hoelzl@35580
   556
context ordered_ab_group_add
hoelzl@35580
   557
begin
hoelzl@35580
   558
hoelzl@35580
   559
lemma
hoelzl@35580
   560
  fixes x :: 'a
hoelzl@35580
   561
  shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}"
hoelzl@35580
   562
  and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}"
hoelzl@35580
   563
proof safe
hoelzl@35580
   564
  fix y assume "y < -x"
hoelzl@35580
   565
  hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp
hoelzl@35580
   566
  have "- (-y) \<in> uminus ` {x<..}"
hoelzl@35580
   567
    by (rule imageI) (simp add: *)
hoelzl@35580
   568
  thus "y \<in> uminus ` {x<..}" by simp
hoelzl@35580
   569
next
hoelzl@35580
   570
  fix y assume "y \<le> -x"
hoelzl@35580
   571
  have "- (-y) \<in> uminus ` {x..}"
hoelzl@35580
   572
    by (rule imageI) (insert `y \<le> -x`[THEN le_imp_neg_le], simp)
hoelzl@35580
   573
  thus "y \<in> uminus ` {x..}" by simp
hoelzl@35580
   574
qed simp_all
hoelzl@35580
   575
hoelzl@35580
   576
lemma
hoelzl@35580
   577
  fixes x :: 'a
hoelzl@35580
   578
  shows image_uminus_lessThan[simp]: "uminus ` {..<x} = {-x<..}"
hoelzl@35580
   579
  and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"
hoelzl@35580
   580
proof -
hoelzl@35580
   581
  have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"
hoelzl@35580
   582
    and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all
hoelzl@35580
   583
  thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"
hoelzl@35580
   584
    by (simp_all add: image_image
hoelzl@35580
   585
        del: image_uminus_greaterThan image_uminus_atLeast)
hoelzl@35580
   586
qed
hoelzl@35580
   587
hoelzl@35580
   588
lemma
hoelzl@35580
   589
  fixes x :: 'a
hoelzl@35580
   590
  shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"
hoelzl@35580
   591
  and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"
hoelzl@35580
   592
  and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"
hoelzl@35580
   593
  and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"
hoelzl@35580
   594
  by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def
hoelzl@35580
   595
      greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)
hoelzl@35580
   596
end
nipkow@16733
   597
paulson@14485
   598
subsubsection {* Finiteness *}
paulson@14485
   599
nipkow@15045
   600
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
paulson@14485
   601
  by (induct k) (simp_all add: lessThan_Suc)
paulson@14485
   602
paulson@14485
   603
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
paulson@14485
   604
  by (induct k) (simp_all add: atMost_Suc)
paulson@14485
   605
paulson@14485
   606
lemma finite_greaterThanLessThan [iff]:
nipkow@15045
   607
  fixes l :: nat shows "finite {l<..<u}"
paulson@14485
   608
by (simp add: greaterThanLessThan_def)
paulson@14485
   609
paulson@14485
   610
lemma finite_atLeastLessThan [iff]:
nipkow@15045
   611
  fixes l :: nat shows "finite {l..<u}"
paulson@14485
   612
by (simp add: atLeastLessThan_def)
paulson@14485
   613
paulson@14485
   614
lemma finite_greaterThanAtMost [iff]:
nipkow@15045
   615
  fixes l :: nat shows "finite {l<..u}"
paulson@14485
   616
by (simp add: greaterThanAtMost_def)
paulson@14485
   617
paulson@14485
   618
lemma finite_atLeastAtMost [iff]:
paulson@14485
   619
  fixes l :: nat shows "finite {l..u}"
paulson@14485
   620
by (simp add: atLeastAtMost_def)
paulson@14485
   621
nipkow@28068
   622
text {* A bounded set of natural numbers is finite. *}
paulson@14485
   623
lemma bounded_nat_set_is_finite:
nipkow@24853
   624
  "(ALL i:N. i < (n::nat)) ==> finite N"
nipkow@28068
   625
apply (rule finite_subset)
nipkow@28068
   626
 apply (rule_tac [2] finite_lessThan, auto)
nipkow@28068
   627
done
nipkow@28068
   628
nipkow@31044
   629
text {* A set of natural numbers is finite iff it is bounded. *}
nipkow@31044
   630
lemma finite_nat_set_iff_bounded:
nipkow@31044
   631
  "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
nipkow@31044
   632
proof
nipkow@31044
   633
  assume f:?F  show ?B
nipkow@31044
   634
    using Max_ge[OF `?F`, simplified less_Suc_eq_le[symmetric]] by blast
nipkow@31044
   635
next
nipkow@31044
   636
  assume ?B show ?F using `?B` by(blast intro:bounded_nat_set_is_finite)
nipkow@31044
   637
qed
nipkow@31044
   638
nipkow@31044
   639
lemma finite_nat_set_iff_bounded_le:
nipkow@31044
   640
  "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
nipkow@31044
   641
apply(simp add:finite_nat_set_iff_bounded)
nipkow@31044
   642
apply(blast dest:less_imp_le_nat le_imp_less_Suc)
nipkow@31044
   643
done
nipkow@31044
   644
nipkow@28068
   645
lemma finite_less_ub:
nipkow@28068
   646
     "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
nipkow@28068
   647
by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
paulson@14485
   648
nipkow@24853
   649
text{* Any subset of an interval of natural numbers the size of the
nipkow@24853
   650
subset is exactly that interval. *}
nipkow@24853
   651
nipkow@24853
   652
lemma subset_card_intvl_is_intvl:
nipkow@24853
   653
  "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")
nipkow@24853
   654
proof cases
nipkow@24853
   655
  assume "finite A"
nipkow@24853
   656
  thus "PROP ?P"
nipkow@32006
   657
  proof(induct A rule:finite_linorder_max_induct)
nipkow@24853
   658
    case empty thus ?case by auto
nipkow@24853
   659
  next
nipkow@33434
   660
    case (insert b A)
nipkow@24853
   661
    moreover hence "b ~: A" by auto
nipkow@24853
   662
    moreover have "A <= {k..<k+card A}" and "b = k+card A"
nipkow@44890
   663
      using `b ~: A` insert by fastforce+
nipkow@24853
   664
    ultimately show ?case by auto
nipkow@24853
   665
  qed
nipkow@24853
   666
next
nipkow@24853
   667
  assume "~finite A" thus "PROP ?P" by simp
nipkow@24853
   668
qed
nipkow@24853
   669
nipkow@24853
   670
paulson@32596
   671
subsubsection {* Proving Inclusions and Equalities between Unions *}
paulson@32596
   672
nipkow@36755
   673
lemma UN_le_eq_Un0:
nipkow@36755
   674
  "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> M 0" (is "?A = ?B")
nipkow@36755
   675
proof
nipkow@36755
   676
  show "?A <= ?B"
nipkow@36755
   677
  proof
nipkow@36755
   678
    fix x assume "x : ?A"
nipkow@36755
   679
    then obtain i where i: "i\<le>n" "x : M i" by auto
nipkow@36755
   680
    show "x : ?B"
nipkow@36755
   681
    proof(cases i)
nipkow@36755
   682
      case 0 with i show ?thesis by simp
nipkow@36755
   683
    next
nipkow@36755
   684
      case (Suc j) with i show ?thesis by auto
nipkow@36755
   685
    qed
nipkow@36755
   686
  qed
nipkow@36755
   687
next
nipkow@36755
   688
  show "?B <= ?A" by auto
nipkow@36755
   689
qed
nipkow@36755
   690
nipkow@36755
   691
lemma UN_le_add_shift:
nipkow@36755
   692
  "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")
nipkow@36755
   693
proof
nipkow@44890
   694
  show "?A <= ?B" by fastforce
nipkow@36755
   695
next
nipkow@36755
   696
  show "?B <= ?A"
nipkow@36755
   697
  proof
nipkow@36755
   698
    fix x assume "x : ?B"
nipkow@36755
   699
    then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto
nipkow@36755
   700
    hence "i-k\<le>n & x : M((i-k)+k)" by auto
nipkow@36755
   701
    thus "x : ?A" by blast
nipkow@36755
   702
  qed
nipkow@36755
   703
qed
nipkow@36755
   704
paulson@32596
   705
lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"
paulson@32596
   706
  by (auto simp add: atLeast0LessThan) 
paulson@32596
   707
paulson@32596
   708
lemma UN_finite_subset: "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"
paulson@32596
   709
  by (subst UN_UN_finite_eq [symmetric]) blast
paulson@32596
   710
paulson@33044
   711
lemma UN_finite2_subset: 
paulson@33044
   712
     "(!!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
   713
  apply (rule UN_finite_subset)
paulson@33044
   714
  apply (subst UN_UN_finite_eq [symmetric, of B]) 
paulson@33044
   715
  apply blast
paulson@33044
   716
  done
paulson@32596
   717
paulson@32596
   718
lemma UN_finite2_eq:
paulson@33044
   719
  "(!!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
   720
  apply (rule subset_antisym)
paulson@33044
   721
   apply (rule UN_finite2_subset, blast)
paulson@33044
   722
 apply (rule UN_finite2_subset [where k=k])
huffman@35216
   723
 apply (force simp add: atLeastLessThan_add_Un [of 0])
paulson@33044
   724
 done
paulson@32596
   725
paulson@32596
   726
paulson@14485
   727
subsubsection {* Cardinality *}
paulson@14485
   728
nipkow@15045
   729
lemma card_lessThan [simp]: "card {..<u} = u"
paulson@15251
   730
  by (induct u, simp_all add: lessThan_Suc)
paulson@14485
   731
paulson@14485
   732
lemma card_atMost [simp]: "card {..u} = Suc u"
paulson@14485
   733
  by (simp add: lessThan_Suc_atMost [THEN sym])
paulson@14485
   734
nipkow@15045
   735
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
nipkow@15045
   736
  apply (subgoal_tac "card {l..<u} = card {..<u-l}")
paulson@14485
   737
  apply (erule ssubst, rule card_lessThan)
nipkow@15045
   738
  apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
paulson@14485
   739
  apply (erule subst)
paulson@14485
   740
  apply (rule card_image)
paulson@14485
   741
  apply (simp add: inj_on_def)
paulson@14485
   742
  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
paulson@14485
   743
  apply (rule_tac x = "x - l" in exI)
paulson@14485
   744
  apply arith
paulson@14485
   745
  done
paulson@14485
   746
paulson@15418
   747
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
paulson@14485
   748
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
paulson@14485
   749
paulson@15418
   750
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
paulson@14485
   751
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
paulson@14485
   752
nipkow@15045
   753
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
paulson@14485
   754
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
paulson@14485
   755
nipkow@26105
   756
lemma ex_bij_betw_nat_finite:
nipkow@26105
   757
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
nipkow@26105
   758
apply(drule finite_imp_nat_seg_image_inj_on)
nipkow@26105
   759
apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
nipkow@26105
   760
done
nipkow@26105
   761
nipkow@26105
   762
lemma ex_bij_betw_finite_nat:
nipkow@26105
   763
  "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
nipkow@26105
   764
by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
nipkow@26105
   765
nipkow@31438
   766
lemma finite_same_card_bij:
nipkow@31438
   767
  "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"
nipkow@31438
   768
apply(drule ex_bij_betw_finite_nat)
nipkow@31438
   769
apply(drule ex_bij_betw_nat_finite)
nipkow@31438
   770
apply(auto intro!:bij_betw_trans)
nipkow@31438
   771
done
nipkow@31438
   772
nipkow@31438
   773
lemma ex_bij_betw_nat_finite_1:
nipkow@31438
   774
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
nipkow@31438
   775
by (rule finite_same_card_bij) auto
nipkow@31438
   776
hoelzl@40703
   777
lemma bij_betw_iff_card:
hoelzl@40703
   778
  assumes FIN: "finite A" and FIN': "finite B"
hoelzl@40703
   779
  shows BIJ: "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"
hoelzl@40703
   780
using assms
hoelzl@40703
   781
proof(auto simp add: bij_betw_same_card)
hoelzl@40703
   782
  assume *: "card A = card B"
hoelzl@40703
   783
  obtain f where "bij_betw f A {0 ..< card A}"
hoelzl@40703
   784
  using FIN ex_bij_betw_finite_nat by blast
hoelzl@40703
   785
  moreover obtain g where "bij_betw g {0 ..< card B} B"
hoelzl@40703
   786
  using FIN' ex_bij_betw_nat_finite by blast
hoelzl@40703
   787
  ultimately have "bij_betw (g o f) A B"
hoelzl@40703
   788
  using * by (auto simp add: bij_betw_trans)
hoelzl@40703
   789
  thus "(\<exists>f. bij_betw f A B)" by blast
hoelzl@40703
   790
qed
hoelzl@40703
   791
hoelzl@40703
   792
lemma inj_on_iff_card_le:
hoelzl@40703
   793
  assumes FIN: "finite A" and FIN': "finite B"
hoelzl@40703
   794
  shows "(\<exists>f. inj_on f A \<and> f ` A \<le> B) = (card A \<le> card B)"
hoelzl@40703
   795
proof (safe intro!: card_inj_on_le)
hoelzl@40703
   796
  assume *: "card A \<le> card B"
hoelzl@40703
   797
  obtain f where 1: "inj_on f A" and 2: "f ` A = {0 ..< card A}"
hoelzl@40703
   798
  using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force
hoelzl@40703
   799
  moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g ` {0 ..< card B} = B"
hoelzl@40703
   800
  using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force
hoelzl@40703
   801
  ultimately have "inj_on g (f ` A)" using subset_inj_on[of g _ "f ` A"] * by force
hoelzl@40703
   802
  hence "inj_on (g o f) A" using 1 comp_inj_on by blast
hoelzl@40703
   803
  moreover
hoelzl@40703
   804
  {have "{0 ..< card A} \<le> {0 ..< card B}" using * by force
hoelzl@40703
   805
   with 2 have "f ` A  \<le> {0 ..< card B}" by blast
hoelzl@40703
   806
   hence "(g o f) ` A \<le> B" unfolding comp_def using 3 by force
hoelzl@40703
   807
  }
hoelzl@40703
   808
  ultimately show "(\<exists>f. inj_on f A \<and> f ` A \<le> B)" by blast
hoelzl@40703
   809
qed (insert assms, auto)
nipkow@26105
   810
paulson@14485
   811
subsection {* Intervals of integers *}
paulson@14485
   812
nipkow@15045
   813
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
paulson@14485
   814
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   815
paulson@15418
   816
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
paulson@14485
   817
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
paulson@14485
   818
paulson@15418
   819
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
paulson@15418
   820
    "{l+1..<u} = {l<..<u::int}"
paulson@14485
   821
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
paulson@14485
   822
paulson@14485
   823
subsubsection {* Finiteness *}
paulson@14485
   824
paulson@15418
   825
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
nipkow@15045
   826
    {(0::int)..<u} = int ` {..<nat u}"
paulson@14485
   827
  apply (unfold image_def lessThan_def)
paulson@14485
   828
  apply auto
paulson@14485
   829
  apply (rule_tac x = "nat x" in exI)
huffman@35216
   830
  apply (auto simp add: zless_nat_eq_int_zless [THEN sym])
paulson@14485
   831
  done
paulson@14485
   832
nipkow@15045
   833
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
paulson@14485
   834
  apply (case_tac "0 \<le> u")
paulson@14485
   835
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   836
  apply (rule finite_imageI)
paulson@14485
   837
  apply auto
paulson@14485
   838
  done
paulson@14485
   839
nipkow@15045
   840
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
nipkow@15045
   841
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   842
  apply (erule subst)
paulson@14485
   843
  apply (rule finite_imageI)
paulson@14485
   844
  apply (rule finite_atLeastZeroLessThan_int)
nipkow@16733
   845
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   846
  done
paulson@14485
   847
paulson@15418
   848
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
paulson@14485
   849
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
paulson@14485
   850
paulson@15418
   851
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
paulson@14485
   852
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   853
paulson@15418
   854
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
paulson@14485
   855
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   856
nipkow@24853
   857
paulson@14485
   858
subsubsection {* Cardinality *}
paulson@14485
   859
nipkow@15045
   860
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
paulson@14485
   861
  apply (case_tac "0 \<le> u")
paulson@14485
   862
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   863
  apply (subst card_image)
paulson@14485
   864
  apply (auto simp add: inj_on_def)
paulson@14485
   865
  done
paulson@14485
   866
nipkow@15045
   867
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
nipkow@15045
   868
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
paulson@14485
   869
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
nipkow@15045
   870
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   871
  apply (erule subst)
paulson@14485
   872
  apply (rule card_image)
paulson@14485
   873
  apply (simp add: inj_on_def)
nipkow@16733
   874
  apply (rule image_add_int_atLeastLessThan)
paulson@14485
   875
  done
paulson@14485
   876
paulson@14485
   877
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
nipkow@29667
   878
apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
nipkow@29667
   879
apply (auto simp add: algebra_simps)
nipkow@29667
   880
done
paulson@14485
   881
paulson@15418
   882
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
nipkow@29667
   883
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   884
nipkow@15045
   885
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
nipkow@29667
   886
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   887
bulwahn@27656
   888
lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
bulwahn@27656
   889
proof -
bulwahn@27656
   890
  have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
bulwahn@27656
   891
  with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
bulwahn@27656
   892
qed
bulwahn@27656
   893
bulwahn@27656
   894
lemma card_less:
bulwahn@27656
   895
assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   896
shows "card {k \<in> M. k < Suc i} \<noteq> 0"
bulwahn@27656
   897
proof -
bulwahn@27656
   898
  from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
bulwahn@27656
   899
  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
bulwahn@27656
   900
qed
bulwahn@27656
   901
bulwahn@27656
   902
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
   903
apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])
bulwahn@27656
   904
apply simp
nipkow@44890
   905
apply fastforce
bulwahn@27656
   906
apply auto
bulwahn@27656
   907
apply (rule inj_on_diff_nat)
bulwahn@27656
   908
apply auto
bulwahn@27656
   909
apply (case_tac x)
bulwahn@27656
   910
apply auto
bulwahn@27656
   911
apply (case_tac xa)
bulwahn@27656
   912
apply auto
bulwahn@27656
   913
apply (case_tac xa)
bulwahn@27656
   914
apply auto
bulwahn@27656
   915
done
bulwahn@27656
   916
bulwahn@27656
   917
lemma card_less_Suc:
bulwahn@27656
   918
  assumes zero_in_M: "0 \<in> M"
bulwahn@27656
   919
    shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
bulwahn@27656
   920
proof -
bulwahn@27656
   921
  from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
bulwahn@27656
   922
  hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
bulwahn@27656
   923
    by (auto simp only: insert_Diff)
bulwahn@27656
   924
  have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
bulwahn@27656
   925
  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
   926
    apply (subst card_insert)
bulwahn@27656
   927
    apply simp_all
bulwahn@27656
   928
    apply (subst b)
bulwahn@27656
   929
    apply (subst card_less_Suc2[symmetric])
bulwahn@27656
   930
    apply simp_all
bulwahn@27656
   931
    done
bulwahn@27656
   932
  with c show ?thesis by simp
bulwahn@27656
   933
qed
bulwahn@27656
   934
paulson@14485
   935
paulson@13850
   936
subsection {*Lemmas useful with the summation operator setsum*}
paulson@13850
   937
ballarin@16102
   938
text {* For examples, see Algebra/poly/UnivPoly2.thy *}
ballarin@13735
   939
wenzelm@14577
   940
subsubsection {* Disjoint Unions *}
ballarin@13735
   941
wenzelm@14577
   942
text {* Singletons and open intervals *}
ballarin@13735
   943
ballarin@13735
   944
lemma ivl_disj_un_singleton:
nipkow@15045
   945
  "{l::'a::linorder} Un {l<..} = {l..}"
nipkow@15045
   946
  "{..<u} Un {u::'a::linorder} = {..u}"
nipkow@15045
   947
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
nipkow@15045
   948
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
nipkow@15045
   949
  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
nipkow@15045
   950
  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
ballarin@14398
   951
by auto
ballarin@13735
   952
wenzelm@14577
   953
text {* One- and two-sided intervals *}
ballarin@13735
   954
ballarin@13735
   955
lemma ivl_disj_un_one:
nipkow@15045
   956
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
nipkow@15045
   957
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
nipkow@15045
   958
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
nipkow@15045
   959
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
nipkow@15045
   960
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
nipkow@15045
   961
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
nipkow@15045
   962
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
nipkow@15045
   963
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
ballarin@14398
   964
by auto
ballarin@13735
   965
wenzelm@14577
   966
text {* Two- and two-sided intervals *}
ballarin@13735
   967
ballarin@13735
   968
lemma ivl_disj_un_two:
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}"
nipkow@15045
   973
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
nipkow@15045
   974
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
nipkow@15045
   975
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
nipkow@15045
   976
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
ballarin@14398
   977
by auto
ballarin@13735
   978
ballarin@13735
   979
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
ballarin@13735
   980
wenzelm@14577
   981
subsubsection {* Disjoint Intersections *}
ballarin@13735
   982
wenzelm@14577
   983
text {* One- and two-sided intervals *}
ballarin@13735
   984
ballarin@13735
   985
lemma ivl_disj_int_one:
nipkow@15045
   986
  "{..l::'a::order} Int {l<..<u} = {}"
nipkow@15045
   987
  "{..<l} Int {l..<u} = {}"
nipkow@15045
   988
  "{..l} Int {l<..u} = {}"
nipkow@15045
   989
  "{..<l} Int {l..u} = {}"
nipkow@15045
   990
  "{l<..u} Int {u<..} = {}"
nipkow@15045
   991
  "{l<..<u} Int {u..} = {}"
nipkow@15045
   992
  "{l..u} Int {u<..} = {}"
nipkow@15045
   993
  "{l..<u} Int {u..} = {}"
ballarin@14398
   994
  by auto
ballarin@13735
   995
wenzelm@14577
   996
text {* Two- and two-sided intervals *}
ballarin@13735
   997
ballarin@13735
   998
lemma ivl_disj_int_two:
nipkow@15045
   999
  "{l::'a::order<..<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} = {}"
nipkow@15045
  1003
  "{l<..<m} Int {m..u} = {}"
nipkow@15045
  1004
  "{l<..m} Int {m<..u} = {}"
nipkow@15045
  1005
  "{l..<m} Int {m..u} = {}"
nipkow@15045
  1006
  "{l..m} Int {m<..u} = {}"
ballarin@14398
  1007
  by auto
ballarin@13735
  1008
nipkow@32456
  1009
lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two
ballarin@13735
  1010
nipkow@15542
  1011
subsubsection {* Some Differences *}
nipkow@15542
  1012
nipkow@15542
  1013
lemma ivl_diff[simp]:
nipkow@15542
  1014
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
nipkow@15542
  1015
by(auto)
nipkow@15542
  1016
nipkow@15542
  1017
nipkow@15542
  1018
subsubsection {* Some Subset Conditions *}
nipkow@15542
  1019
blanchet@35828
  1020
lemma ivl_subset [simp,no_atp]:
nipkow@15542
  1021
 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
nipkow@15542
  1022
apply(auto simp:linorder_not_le)
nipkow@15542
  1023
apply(rule ccontr)
nipkow@15542
  1024
apply(insert linorder_le_less_linear[of i n])
nipkow@15542
  1025
apply(clarsimp simp:linorder_not_le)
nipkow@44890
  1026
apply(fastforce)
nipkow@15542
  1027
done
nipkow@15542
  1028
nipkow@15041
  1029
nipkow@15042
  1030
subsection {* Summation indexed over intervals *}
nipkow@15042
  1031
nipkow@15042
  1032
syntax
nipkow@15042
  1033
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
  1034
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
  1035
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
nipkow@16052
  1036
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
nipkow@15042
  1037
syntax (xsymbols)
nipkow@15042
  1038
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
  1039
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
  1040
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
  1041
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15042
  1042
syntax (HTML output)
nipkow@15042
  1043
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
  1044
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052
  1045
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052
  1046
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15056
  1047
syntax (latex_sum output)
nipkow@15052
  1048
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
  1049
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052
  1050
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
  1051
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@16052
  1052
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
  1053
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15052
  1054
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052
  1055
 ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15041
  1056
nipkow@15048
  1057
translations
nipkow@28853
  1058
  "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
nipkow@28853
  1059
  "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
nipkow@28853
  1060
  "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
nipkow@28853
  1061
  "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
nipkow@15041
  1062
nipkow@15052
  1063
text{* The above introduces some pretty alternative syntaxes for
nipkow@15056
  1064
summation over intervals:
nipkow@15052
  1065
\begin{center}
nipkow@15052
  1066
\begin{tabular}{lll}
nipkow@15056
  1067
Old & New & \LaTeX\\
nipkow@15056
  1068
@{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
  1069
@{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
  1070
@{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
nipkow@15056
  1071
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
nipkow@15052
  1072
\end{tabular}
nipkow@15052
  1073
\end{center}
nipkow@15056
  1074
The left column shows the term before introduction of the new syntax,
nipkow@15056
  1075
the middle column shows the new (default) syntax, and the right column
nipkow@15056
  1076
shows a special syntax. The latter is only meaningful for latex output
nipkow@15056
  1077
and has to be activated explicitly by setting the print mode to
wenzelm@21502
  1078
@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
nipkow@15056
  1079
antiquotations). It is not the default \LaTeX\ output because it only
nipkow@15056
  1080
works well with italic-style formulae, not tt-style.
nipkow@15052
  1081
nipkow@15052
  1082
Note that for uniformity on @{typ nat} it is better to use
nipkow@15052
  1083
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
nipkow@15052
  1084
not provide all lemmas available for @{term"{m..<n}"} also in the
nipkow@15052
  1085
special form for @{term"{..<n}"}. *}
nipkow@15052
  1086
nipkow@15542
  1087
text{* This congruence rule should be used for sums over intervals as
nipkow@15542
  1088
the standard theorem @{text[source]setsum_cong} does not work well
nipkow@15542
  1089
with the simplifier who adds the unsimplified premise @{term"x:B"} to
nipkow@15542
  1090
the context. *}
nipkow@15542
  1091
nipkow@15542
  1092
lemma setsum_ivl_cong:
nipkow@15542
  1093
 "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
nipkow@15542
  1094
 setsum f {a..<b} = setsum g {c..<d}"
nipkow@15542
  1095
by(rule setsum_cong, simp_all)
nipkow@15041
  1096
nipkow@16041
  1097
(* FIXME why are the following simp rules but the corresponding eqns
nipkow@16041
  1098
on intervals are not? *)
nipkow@16041
  1099
nipkow@16052
  1100
lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
nipkow@16052
  1101
by (simp add:atMost_Suc add_ac)
nipkow@16052
  1102
nipkow@16041
  1103
lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
nipkow@16041
  1104
by (simp add:lessThan_Suc add_ac)
nipkow@15041
  1105
nipkow@15911
  1106
lemma setsum_cl_ivl_Suc[simp]:
nipkow@15561
  1107
  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
nipkow@15561
  1108
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@15561
  1109
nipkow@15911
  1110
lemma setsum_op_ivl_Suc[simp]:
nipkow@15561
  1111
  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
nipkow@15561
  1112
by (auto simp:add_ac atLeastLessThanSuc)
nipkow@16041
  1113
(*
nipkow@15561
  1114
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
nipkow@15561
  1115
    (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
nipkow@15561
  1116
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@16041
  1117
*)
nipkow@28068
  1118
nipkow@28068
  1119
lemma setsum_head:
nipkow@28068
  1120
  fixes n :: nat
nipkow@28068
  1121
  assumes mn: "m <= n" 
nipkow@28068
  1122
  shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
nipkow@28068
  1123
proof -
nipkow@28068
  1124
  from mn
nipkow@28068
  1125
  have "{m..n} = {m} \<union> {m<..n}"
nipkow@28068
  1126
    by (auto intro: ivl_disj_un_singleton)
nipkow@28068
  1127
  hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
nipkow@28068
  1128
    by (simp add: atLeast0LessThan)
nipkow@28068
  1129
  also have "\<dots> = ?rhs" by simp
nipkow@28068
  1130
  finally show ?thesis .
nipkow@28068
  1131
qed
nipkow@28068
  1132
nipkow@28068
  1133
lemma setsum_head_Suc:
nipkow@28068
  1134
  "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
nipkow@28068
  1135
by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
nipkow@28068
  1136
nipkow@28068
  1137
lemma setsum_head_upt_Suc:
nipkow@28068
  1138
  "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
huffman@30079
  1139
apply(insert setsum_head_Suc[of m "n - Suc 0" f])
nipkow@29667
  1140
apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
nipkow@28068
  1141
done
nipkow@28068
  1142
nipkow@31501
  1143
lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"
nipkow@31501
  1144
  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
nipkow@31501
  1145
proof-
nipkow@31501
  1146
  have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using `m \<le> n+1` by auto
nipkow@31501
  1147
  thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint
nipkow@31501
  1148
    atLeastSucAtMost_greaterThanAtMost)
nipkow@31501
  1149
qed
nipkow@28068
  1150
nipkow@15539
  1151
lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
  1152
  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
nipkow@15539
  1153
by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
nipkow@15539
  1154
nipkow@15539
  1155
lemma setsum_diff_nat_ivl:
nipkow@15539
  1156
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
nipkow@15539
  1157
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
  1158
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
nipkow@15539
  1159
using setsum_add_nat_ivl [of m n p f,symmetric]
nipkow@15539
  1160
apply (simp add: add_ac)
nipkow@15539
  1161
done
nipkow@15539
  1162
nipkow@31505
  1163
lemma setsum_natinterval_difff:
nipkow@31505
  1164
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
nipkow@31505
  1165
  shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
nipkow@31505
  1166
          (if m <= n then f m - f(n + 1) else 0)"
nipkow@31505
  1167
by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
nipkow@31505
  1168
haftmann@44008
  1169
lemma setsum_restrict_set':
haftmann@44008
  1170
  "finite A \<Longrightarrow> setsum f {x \<in> A. x \<in> B} = (\<Sum>x\<in>A. if x \<in> B then f x else 0)"
haftmann@44008
  1171
  by (simp add: setsum_restrict_set [symmetric] Int_def)
haftmann@44008
  1172
haftmann@44008
  1173
lemma setsum_restrict_set'':
haftmann@44008
  1174
  "finite A \<Longrightarrow> setsum f {x \<in> A. P x} = (\<Sum>x\<in>A. if P x  then f x else 0)"
haftmann@44008
  1175
  by (simp add: setsum_restrict_set' [of A f "{x. P x}", simplified mem_Collect_eq])
nipkow@31509
  1176
nipkow@31509
  1177
lemma setsum_setsum_restrict:
haftmann@44008
  1178
  "finite S \<Longrightarrow> finite T \<Longrightarrow>
haftmann@44008
  1179
    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"
haftmann@44008
  1180
  by (simp add: setsum_restrict_set'') (rule setsum_commute)
nipkow@31509
  1181
nipkow@31509
  1182
lemma setsum_image_gen: assumes fS: "finite S"
nipkow@31509
  1183
  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
nipkow@31509
  1184
proof-
nipkow@31509
  1185
  { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
nipkow@31509
  1186
  hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
nipkow@31509
  1187
    by simp
nipkow@31509
  1188
  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
nipkow@31509
  1189
    by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])
nipkow@31509
  1190
  finally show ?thesis .
nipkow@31509
  1191
qed
nipkow@31509
  1192
hoelzl@35171
  1193
lemma setsum_le_included:
haftmann@36307
  1194
  fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"
hoelzl@35171
  1195
  assumes "finite s" "finite t"
hoelzl@35171
  1196
  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
  1197
  shows "setsum f s \<le> setsum g t"
hoelzl@35171
  1198
proof -
hoelzl@35171
  1199
  have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"
hoelzl@35171
  1200
  proof (rule setsum_mono)
hoelzl@35171
  1201
    fix y assume "y \<in> s"
hoelzl@35171
  1202
    with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
hoelzl@35171
  1203
    with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
hoelzl@35171
  1204
      using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]
hoelzl@35171
  1205
      by (auto intro!: setsum_mono2)
hoelzl@35171
  1206
  qed
hoelzl@35171
  1207
  also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
hoelzl@35171
  1208
    using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)
hoelzl@35171
  1209
  also have "... \<le> setsum g t"
hoelzl@35171
  1210
    using assms by (auto simp: setsum_image_gen[symmetric])
hoelzl@35171
  1211
  finally show ?thesis .
hoelzl@35171
  1212
qed
hoelzl@35171
  1213
nipkow@31509
  1214
lemma setsum_multicount_gen:
nipkow@31509
  1215
  assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
nipkow@31509
  1216
  shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
nipkow@31509
  1217
proof-
nipkow@31509
  1218
  have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
nipkow@31509
  1219
  also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]
nipkow@31509
  1220
    using assms(3) by auto
nipkow@31509
  1221
  finally show ?thesis .
nipkow@31509
  1222
qed
nipkow@31509
  1223
nipkow@31509
  1224
lemma setsum_multicount:
nipkow@31509
  1225
  assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
nipkow@31509
  1226
  shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
nipkow@31509
  1227
proof-
nipkow@31509
  1228
  have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)
huffman@35216
  1229
  also have "\<dots> = ?r" by(simp add: mult_commute)
nipkow@31509
  1230
  finally show ?thesis by auto
nipkow@31509
  1231
qed
nipkow@31509
  1232
nipkow@28068
  1233
nipkow@16733
  1234
subsection{* Shifting bounds *}
nipkow@16733
  1235
nipkow@15539
  1236
lemma setsum_shift_bounds_nat_ivl:
nipkow@15539
  1237
  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
nipkow@15539
  1238
by (induct "n", auto simp:atLeastLessThanSuc)
nipkow@15539
  1239
nipkow@16733
  1240
lemma setsum_shift_bounds_cl_nat_ivl:
nipkow@16733
  1241
  "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
nipkow@16733
  1242
apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
nipkow@16733
  1243
apply (simp add:image_add_atLeastAtMost o_def)
nipkow@16733
  1244
done
nipkow@16733
  1245
nipkow@16733
  1246
corollary setsum_shift_bounds_cl_Suc_ivl:
nipkow@16733
  1247
  "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
huffman@30079
  1248
by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
  1249
nipkow@16733
  1250
corollary setsum_shift_bounds_Suc_ivl:
nipkow@16733
  1251
  "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
huffman@30079
  1252
by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
nipkow@16733
  1253
nipkow@28068
  1254
lemma setsum_shift_lb_Suc0_0:
nipkow@28068
  1255
  "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
nipkow@28068
  1256
by(simp add:setsum_head_Suc)
kleing@19106
  1257
nipkow@28068
  1258
lemma setsum_shift_lb_Suc0_0_upt:
nipkow@28068
  1259
  "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
nipkow@28068
  1260
apply(cases k)apply simp
nipkow@28068
  1261
apply(simp add:setsum_head_upt_Suc)
nipkow@28068
  1262
done
kleing@19022
  1263
ballarin@17149
  1264
subsection {* The formula for geometric sums *}
ballarin@17149
  1265
ballarin@17149
  1266
lemma geometric_sum:
haftmann@36307
  1267
  assumes "x \<noteq> 1"
haftmann@36307
  1268
  shows "(\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"
haftmann@36307
  1269
proof -
haftmann@36307
  1270
  from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all
haftmann@36307
  1271
  moreover have "(\<Sum>i=0..<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"
haftmann@36307
  1272
  proof (induct n)
haftmann@36307
  1273
    case 0 then show ?case by simp
haftmann@36307
  1274
  next
haftmann@36307
  1275
    case (Suc n)
haftmann@36307
  1276
    moreover with `y \<noteq> 0` have "(1 + y) ^ n = (y * inverse y) * (1 + y) ^ n" by simp 
haftmann@36350
  1277
    ultimately show ?case by (simp add: field_simps divide_inverse)
haftmann@36307
  1278
  qed
haftmann@36307
  1279
  ultimately show ?thesis by simp
haftmann@36307
  1280
qed
haftmann@36307
  1281
ballarin@17149
  1282
kleing@19469
  1283
subsection {* The formula for arithmetic sums *}
kleing@19469
  1284
huffman@47222
  1285
lemma gauss_sum:
huffman@47222
  1286
  "(2::'a::comm_semiring_1)*(\<Sum>i\<in>{1..n}. of_nat i) =
kleing@19469
  1287
   of_nat n*((of_nat n)+1)"
kleing@19469
  1288
proof (induct n)
kleing@19469
  1289
  case 0
kleing@19469
  1290
  show ?case by simp
kleing@19469
  1291
next
kleing@19469
  1292
  case (Suc n)
huffman@47222
  1293
  then show ?case
huffman@47222
  1294
    by (simp add: algebra_simps add: one_add_one [symmetric] del: one_add_one)
huffman@47222
  1295
      (* FIXME: make numeral cancellation simprocs work for semirings *)
kleing@19469
  1296
qed
kleing@19469
  1297
kleing@19469
  1298
theorem arith_series_general:
huffman@47222
  1299
  "(2::'a::comm_semiring_1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469
  1300
  of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1301
proof cases
kleing@19469
  1302
  assume ngt1: "n > 1"
kleing@19469
  1303
  let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
kleing@19469
  1304
  have
kleing@19469
  1305
    "(\<Sum>i\<in>{..<n}. a+?I i*d) =
kleing@19469
  1306
     ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
kleing@19469
  1307
    by (rule setsum_addf)
kleing@19469
  1308
  also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
kleing@19469
  1309
  also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
huffman@30079
  1310
    unfolding One_nat_def
nipkow@28068
  1311
    by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)
huffman@47222
  1312
  also have "2*\<dots> = 2*?n*a + d*2*(\<Sum>i\<in>{1..<n}. ?I i)"
huffman@47222
  1313
    by (simp add: algebra_simps)
kleing@19469
  1314
  also from ngt1 have "{1..<n} = {1..n - 1}"
nipkow@28068
  1315
    by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
nipkow@28068
  1316
  also from ngt1
huffman@47222
  1317
  have "2*?n*a + d*2*(\<Sum>i\<in>{1..n - 1}. ?I i) = (2*?n*a + d*?I (n - 1)*?I n)"
huffman@30079
  1318
    by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)
huffman@23431
  1319
       (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
huffman@47222
  1320
  finally show ?thesis
huffman@47222
  1321
    unfolding mult_2 by (simp add: algebra_simps)
kleing@19469
  1322
next
kleing@19469
  1323
  assume "\<not>(n > 1)"
kleing@19469
  1324
  hence "n = 1 \<or> n = 0" by auto
huffman@47222
  1325
  thus ?thesis by (auto simp: mult_2)
kleing@19469
  1326
qed
kleing@19469
  1327
kleing@19469
  1328
lemma arith_series_nat:
huffman@47222
  1329
  "(2::nat) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
kleing@19469
  1330
proof -
kleing@19469
  1331
  have
huffman@47222
  1332
    "2 * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
kleing@19469
  1333
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469
  1334
    by (rule arith_series_general)
huffman@30079
  1335
  thus ?thesis
huffman@35216
  1336
    unfolding One_nat_def by auto
kleing@19469
  1337
qed
kleing@19469
  1338
kleing@19469
  1339
lemma arith_series_int:
huffman@47222
  1340
  "2 * (\<Sum>i\<in>{..<n}. a + int i * d) = int n * (a + (a + int(n - 1)*d))"
huffman@47222
  1341
  by (fact arith_series_general) (* FIXME: duplicate *)
paulson@15418
  1342
kleing@19022
  1343
lemma sum_diff_distrib:
kleing@19022
  1344
  fixes P::"nat\<Rightarrow>nat"
kleing@19022
  1345
  shows
kleing@19022
  1346
  "\<forall>x. Q x \<le> P x  \<Longrightarrow>
kleing@19022
  1347
  (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
kleing@19022
  1348
proof (induct n)
kleing@19022
  1349
  case 0 show ?case by simp
kleing@19022
  1350
next
kleing@19022
  1351
  case (Suc n)
kleing@19022
  1352
kleing@19022
  1353
  let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
kleing@19022
  1354
  let ?rhs = "\<Sum>x<n. P x - Q x"
kleing@19022
  1355
kleing@19022
  1356
  from Suc have "?lhs = ?rhs" by simp
kleing@19022
  1357
  moreover
kleing@19022
  1358
  from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
kleing@19022
  1359
  moreover
kleing@19022
  1360
  from Suc have
kleing@19022
  1361
    "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
kleing@19022
  1362
    by (subst diff_diff_left[symmetric],
kleing@19022
  1363
        subst diff_add_assoc2)
kleing@19022
  1364
       (auto simp: diff_add_assoc2 intro: setsum_mono)
kleing@19022
  1365
  ultimately
kleing@19022
  1366
  show ?case by simp
kleing@19022
  1367
qed
kleing@19022
  1368
paulson@29960
  1369
subsection {* Products indexed over intervals *}
paulson@29960
  1370
paulson@29960
  1371
syntax
paulson@29960
  1372
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1373
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1374
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)
paulson@29960
  1375
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
paulson@29960
  1376
syntax (xsymbols)
paulson@29960
  1377
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1378
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1379
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1380
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1381
syntax (HTML output)
paulson@29960
  1382
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960
  1383
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960
  1384
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960
  1385
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960
  1386
syntax (latex_prod output)
paulson@29960
  1387
  "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1388
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1389
  "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1390
 ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960
  1391
  "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1392
 ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1393
  "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960
  1394
 ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960
  1395
paulson@29960
  1396
translations
paulson@29960
  1397
  "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"
paulson@29960
  1398
  "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"
paulson@29960
  1399
  "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"
paulson@29960
  1400
  "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"
paulson@29960
  1401
haftmann@33318
  1402
subsection {* Transfer setup *}
haftmann@33318
  1403
haftmann@33318
  1404
lemma transfer_nat_int_set_functions:
haftmann@33318
  1405
    "{..n} = nat ` {0..int n}"
haftmann@33318
  1406
    "{m..n} = nat ` {int m..int n}"  (* need all variants of these! *)
haftmann@33318
  1407
  apply (auto simp add: image_def)
haftmann@33318
  1408
  apply (rule_tac x = "int x" in bexI)
haftmann@33318
  1409
  apply auto
haftmann@33318
  1410
  apply (rule_tac x = "int x" in bexI)
haftmann@33318
  1411
  apply auto
haftmann@33318
  1412
  done
haftmann@33318
  1413
haftmann@33318
  1414
lemma transfer_nat_int_set_function_closures:
haftmann@33318
  1415
    "x >= 0 \<Longrightarrow> nat_set {x..y}"
haftmann@33318
  1416
  by (simp add: nat_set_def)
haftmann@33318
  1417
haftmann@35644
  1418
declare transfer_morphism_nat_int[transfer add
haftmann@33318
  1419
  return: transfer_nat_int_set_functions
haftmann@33318
  1420
    transfer_nat_int_set_function_closures
haftmann@33318
  1421
]
haftmann@33318
  1422
haftmann@33318
  1423
lemma transfer_int_nat_set_functions:
haftmann@33318
  1424
    "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}"
haftmann@33318
  1425
  by (simp only: is_nat_def transfer_nat_int_set_functions
haftmann@33318
  1426
    transfer_nat_int_set_function_closures
haftmann@33318
  1427
    transfer_nat_int_set_return_embed nat_0_le
haftmann@33318
  1428
    cong: transfer_nat_int_set_cong)
haftmann@33318
  1429
haftmann@33318
  1430
lemma transfer_int_nat_set_function_closures:
haftmann@33318
  1431
    "is_nat x \<Longrightarrow> nat_set {x..y}"
haftmann@33318
  1432
  by (simp only: transfer_nat_int_set_function_closures is_nat_def)
haftmann@33318
  1433
haftmann@35644
  1434
declare transfer_morphism_int_nat[transfer add
haftmann@33318
  1435
  return: transfer_int_nat_set_functions
haftmann@33318
  1436
    transfer_int_nat_set_function_closures
haftmann@33318
  1437
]
haftmann@33318
  1438
nipkow@8924
  1439
end