src/HOL/Set_Interval.thy
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(*  Title:      HOL/Set_Interval.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 Set_Interval
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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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43657
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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:
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  fixes a b c d :: "'a::linorder"
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  assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"
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  shows "a = c" "b = d"
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using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le linorder_antisym_conv2 subset_refl)+
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lemma atLeastLessThan_eq_iff:
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  fixes a b c d :: "'a::linorder"
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  assumes "a < b" "c < d"
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  shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"
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  using atLeastLessThan_inj assms by auto
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subsubsection {* Intersection *}
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context linorder
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begin
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lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"
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by auto
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lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"
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by auto
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lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"
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by auto
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lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"
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by auto
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lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"
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by auto
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lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"
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by auto
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lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
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by auto
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lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
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by auto
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end
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subsection {* Intervals of natural numbers *}
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subsubsection {* The Constant @{term lessThan} *}
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lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
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by (simp add: lessThan_def)
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lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
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by (simp add: lessThan_def less_Suc_eq, blast)
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text {* The following proof is convenient in induction proofs where
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new elements get indices at the beginning. So it is used to transform
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@{term "{..<Suc n}"} to @{term "0::nat"} and @{term "{..< n}"}. *}
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lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"
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proof safe
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  fix x assume "x < Suc n" "x \<notin> Suc ` {..<n}"
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  then have "x \<noteq> Suc (x - 1)" by auto
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  with `x < Suc n` show "x = 0" by auto
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qed
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lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
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   379
by (simp add: lessThan_def atMost_def less_Suc_eq_le)
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lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
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by blast
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subsubsection {* The Constant @{term greaterThan} *}
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lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
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apply (simp add: greaterThan_def)
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apply (blast dest: gr0_conv_Suc [THEN iffD1])
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   389
done
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   390
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lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
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   392
apply (simp add: greaterThan_def)
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   393
apply (auto elim: linorder_neqE)
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   394
done
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   395
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   396
lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
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   397
by blast
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diff changeset
   398
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subsubsection {* The Constant @{term atLeast} *}
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   400
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   401
lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
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parents: 14478
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   402
by (unfold atLeast_def UNIV_def, simp)
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parents: 14478
diff changeset
   403
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   404
lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
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parents: 14478
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   405
apply (simp add: atLeast_def)
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parents: 14478
diff changeset
   406
apply (simp add: Suc_le_eq)
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parents: 14478
diff changeset
   407
apply (simp add: order_le_less, blast)
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diff changeset
   408
done
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parents: 14478
diff changeset
   409
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diff changeset
   410
lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
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parents: 14478
diff changeset
   411
  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
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parents: 14478
diff changeset
   412
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diff changeset
   413
lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
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parents: 14478
diff changeset
   414
by blast
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parents: 14478
diff changeset
   415
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   416
subsubsection {* The Constant @{term atMost} *}
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diff changeset
   417
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   418
lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
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   419
by (simp add: atMost_def)
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parents: 14478
diff changeset
   420
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diff changeset
   421
lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
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parents: 14478
diff changeset
   422
apply (simp add: atMost_def)
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parents: 14478
diff changeset
   423
apply (simp add: less_Suc_eq order_le_less, blast)
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diff changeset
   424
done
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parents: 14478
diff changeset
   425
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diff changeset
   426
lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
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paulson
parents: 14478
diff changeset
   427
by blast
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paulson
parents: 14478
diff changeset
   428
15047
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parents: 15045
diff changeset
   429
subsubsection {* The Constant @{term atLeastLessThan} *}
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diff changeset
   430
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   431
text{*The orientation of the following 2 rules is tricky. The lhs is
24449
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diff changeset
   432
defined in terms of the rhs.  Hence the chosen orientation makes sense
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parents: 24286
diff changeset
   433
in this theory --- the reverse orientation complicates proofs (eg
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diff changeset
   434
nontermination). But outside, when the definition of the lhs is rarely
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parents: 24286
diff changeset
   435
used, the opposite orientation seems preferable because it reduces a
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diff changeset
   436
specific concept to a more general one. *}
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parents: 27656
diff changeset
   437
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diff changeset
   438
lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
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fa7d27ef7e59 added {0::nat..n(} = {..n(}
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parents: 15041
diff changeset
   439
by(simp add:lessThan_def atLeastLessThan_def)
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parents: 24286
diff changeset
   440
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diff changeset
   441
lemma atLeast0AtMost: "{0..n::nat} = {..n}"
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diff changeset
   442
by(simp add:atMost_def atLeastAtMost_def)
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parents: 27656
diff changeset
   443
31998
2c7a24f74db9 code attributes use common underscore convention
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parents: 31509
diff changeset
   444
declare atLeast0LessThan[symmetric, code_unfold]
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parents: 31509
diff changeset
   445
        atLeast0AtMost[symmetric, code_unfold]
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diff changeset
   446
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diff changeset
   447
lemma atLeastLessThan0: "{m..<0::nat} = {}"
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parents: 15045
diff changeset
   448
by (simp add: atLeastLessThan_def)
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diff changeset
   449
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   450
subsubsection {* Intervals of nats with @{term Suc} *}
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diff changeset
   451
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   452
text{*Not a simprule because the RHS is too messy.*}
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   453
lemma atLeastLessThanSuc:
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   454
    "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   455
by (auto simp add: atLeastLessThan_def)
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   456
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   457
lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   458
by (auto simp add: atLeastLessThan_def)
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   459
(*
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   460
lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   461
by (induct k, simp_all add: atLeastLessThanSuc)
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   462
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   463
lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   464
by (auto simp add: atLeastLessThan_def)
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   465
*)
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   466
lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   467
  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   468
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   469
lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   470
  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   471
    greaterThanAtMost_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   472
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   473
lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   474
  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   475
    greaterThanLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   476
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15542
diff changeset
   477
lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15542
diff changeset
   478
by (auto simp add: atLeastAtMost_def)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15542
diff changeset
   479
45932
6f08f8fe9752 add lemmas
noschinl
parents: 44890
diff changeset
   480
lemma atLeastAtMost_insertL: "m \<le> n \<Longrightarrow> insert m {Suc m..n} = {m ..n}"
6f08f8fe9752 add lemmas
noschinl
parents: 44890
diff changeset
   481
by auto
6f08f8fe9752 add lemmas
noschinl
parents: 44890
diff changeset
   482
43157
b505be6f029a atLeastAtMostSuc_conv on int
kleing
parents: 43156
diff changeset
   483
text {* The analogous result is useful on @{typ int}: *}
b505be6f029a atLeastAtMostSuc_conv on int
kleing
parents: 43156
diff changeset
   484
(* here, because we don't have an own int section *)
b505be6f029a atLeastAtMostSuc_conv on int
kleing
parents: 43156
diff changeset
   485
lemma atLeastAtMostPlus1_int_conv:
b505be6f029a atLeastAtMostSuc_conv on int
kleing
parents: 43156
diff changeset
   486
  "m <= 1+n \<Longrightarrow> {m..1+n} = insert (1+n) {m..n::int}"
b505be6f029a atLeastAtMostSuc_conv on int
kleing
parents: 43156
diff changeset
   487
  by (auto intro: set_eqI)
b505be6f029a atLeastAtMostSuc_conv on int
kleing
parents: 43156
diff changeset
   488
33044
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   489
lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   490
  apply (induct k) 
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   491
  apply (simp_all add: atLeastLessThanSuc)   
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   492
  done
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   493
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   494
subsubsection {* Image *}
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   495
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   496
lemma image_add_atLeastAtMost:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   497
  "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   498
proof
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   499
  show "?A \<subseteq> ?B" by auto
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   500
next
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   501
  show "?B \<subseteq> ?A"
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   502
  proof
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   503
    fix n assume a: "n : ?B"
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19538
diff changeset
   504
    hence "n - k : {i..j}" by auto
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   505
    moreover have "n = (n - k) + k" using a by auto
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   506
    ultimately show "n : ?A" by blast
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   507
  qed
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   508
qed
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   509
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   510
lemma image_add_atLeastLessThan:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   511
  "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   512
proof
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   513
  show "?A \<subseteq> ?B" by auto
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   514
next
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   515
  show "?B \<subseteq> ?A"
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   516
  proof
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   517
    fix n assume a: "n : ?B"
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19538
diff changeset
   518
    hence "n - k : {i..<j}" by auto
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   519
    moreover have "n = (n - k) + k" using a by auto
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   520
    ultimately show "n : ?A" by blast
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   521
  qed
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   522
qed
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   523
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   524
corollary image_Suc_atLeastAtMost[simp]:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   525
  "Suc ` {i..j} = {Suc i..Suc j}"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 29960
diff changeset
   526
using image_add_atLeastAtMost[where k="Suc 0"] by simp
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   527
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   528
corollary image_Suc_atLeastLessThan[simp]:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   529
  "Suc ` {i..<j} = {Suc i..<Suc j}"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 29960
diff changeset
   530
using image_add_atLeastLessThan[where k="Suc 0"] by simp
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   531
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   532
lemma image_add_int_atLeastLessThan:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   533
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   534
  apply (auto simp add: image_def)
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   535
  apply (rule_tac x = "x - l" in bexI)
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   536
  apply auto
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   537
  done
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   538
37664
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   539
lemma image_minus_const_atLeastLessThan_nat:
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   540
  fixes c :: nat
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   541
  shows "(\<lambda>i. i - c) ` {x ..< y} =
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   542
      (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   543
    (is "_ = ?right")
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   544
proof safe
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   545
  fix a assume a: "a \<in> ?right"
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   546
  show "a \<in> (\<lambda>i. i - c) ` {x ..< y}"
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   547
  proof cases
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   548
    assume "c < y" with a show ?thesis
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   549
      by (auto intro!: image_eqI[of _ _ "a + c"])
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   550
  next
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   551
    assume "\<not> c < y" with a show ?thesis
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   552
      by (auto intro!: image_eqI[of _ _ x] split: split_if_asm)
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   553
  qed
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   554
qed auto
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   555
35580
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   556
context ordered_ab_group_add
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   557
begin
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   558
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   559
lemma
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   560
  fixes x :: 'a
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   561
  shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   562
  and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   563
proof safe
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   564
  fix y assume "y < -x"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   565
  hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   566
  have "- (-y) \<in> uminus ` {x<..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   567
    by (rule imageI) (simp add: *)
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   568
  thus "y \<in> uminus ` {x<..}" by simp
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   569
next
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   570
  fix y assume "y \<le> -x"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   571
  have "- (-y) \<in> uminus ` {x..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   572
    by (rule imageI) (insert `y \<le> -x`[THEN le_imp_neg_le], simp)
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   573
  thus "y \<in> uminus ` {x..}" by simp
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   574
qed simp_all
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   575
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   576
lemma
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   577
  fixes x :: 'a
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   578
  shows image_uminus_lessThan[simp]: "uminus ` {..<x} = {-x<..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   579
  and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   580
proof -
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   581
  have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   582
    and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   583
  thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   584
    by (simp_all add: image_image
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   585
        del: image_uminus_greaterThan image_uminus_atLeast)
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   586
qed
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   587
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   588
lemma
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   589
  fixes x :: 'a
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   590
  shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   591
  and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   592
  and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   593
  and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   594
  by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   595
      greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   596
end
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   597
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   598
subsubsection {* Finiteness *}
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   599
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   600
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   601
  by (induct k) (simp_all add: lessThan_Suc)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   602
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   603
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   604
  by (induct k) (simp_all add: atMost_Suc)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   605
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   606
lemma finite_greaterThanLessThan [iff]:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   607
  fixes l :: nat shows "finite {l<..<u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   608
by (simp add: greaterThanLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   609
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   610
lemma finite_atLeastLessThan [iff]:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   611
  fixes l :: nat shows "finite {l..<u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   612
by (simp add: atLeastLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   613
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   614
lemma finite_greaterThanAtMost [iff]:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   615
  fixes l :: nat shows "finite {l<..u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   616
by (simp add: greaterThanAtMost_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   617
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   618
lemma finite_atLeastAtMost [iff]:
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   619
  fixes l :: nat shows "finite {l..u}"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   620
by (simp add: atLeastAtMost_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   621
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   622
text {* A bounded set of natural numbers is finite. *}
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   623
lemma bounded_nat_set_is_finite:
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   624
  "(ALL i:N. i < (n::nat)) ==> finite N"
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   625
apply (rule finite_subset)
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   626
 apply (rule_tac [2] finite_lessThan, auto)
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   627
done
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   628
31044
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   629
text {* A set of natural numbers is finite iff it is bounded. *}
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   630
lemma finite_nat_set_iff_bounded:
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   631
  "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   632
proof
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   633
  assume f:?F  show ?B
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   634
    using Max_ge[OF `?F`, simplified less_Suc_eq_le[symmetric]] by blast
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   635
next
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   636
  assume ?B show ?F using `?B` by(blast intro:bounded_nat_set_is_finite)
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   637
qed
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   638
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   639
lemma finite_nat_set_iff_bounded_le:
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   640
  "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   641
apply(simp add:finite_nat_set_iff_bounded)
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   642
apply(blast dest:less_imp_le_nat le_imp_less_Suc)
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   643
done
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   644
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   645
lemma finite_less_ub:
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   646
     "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   647
by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   648
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   649
text{* Any subset of an interval of natural numbers the size of the
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   650
subset is exactly that interval. *}
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   651
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   652
lemma subset_card_intvl_is_intvl:
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   653
  "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   654
proof cases
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   655
  assume "finite A"
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   656
  thus "PROP ?P"
32006
0e209ff7f236 More finite set induction rules
nipkow
parents: 31998
diff changeset
   657
  proof(induct A rule:finite_linorder_max_induct)
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   658
    case empty thus ?case by auto
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   659
  next
33434
e9de8d69c1b9 fixed order of parameters in induction rules
nipkow
parents: 33318
diff changeset
   660
    case (insert b A)
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   661
    moreover hence "b ~: A" by auto
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   662
    moreover have "A <= {k..<k+card A}" and "b = k+card A"
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44008
diff changeset
   663
      using `b ~: A` insert by fastforce+
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   664
    ultimately show ?case by auto
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   665
  qed
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   666
next
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   667
  assume "~finite A" thus "PROP ?P" by simp
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   668
qed
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   669
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   670
32596
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
   671
subsubsection {* Proving Inclusions and Equalities between Unions *}
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
   672
36755
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   673
lemma UN_le_eq_Un0:
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   674
  "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> M 0" (is "?A = ?B")
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   675
proof
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   676
  show "?A <= ?B"
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   677
  proof
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   678
    fix x assume "x : ?A"
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   679
    then obtain i where i: "i\<le>n" "x : M i" by auto
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   680
    show "x : ?B"
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   681
    proof(cases i)
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   682
      case 0 with i show ?thesis by simp
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   683
    next
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   684
      case (Suc j) with i show ?thesis by auto
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   685
    qed
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   686
  qed
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   687
next
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   688
  show "?B <= ?A" by auto
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   689
qed
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   690
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   691
lemma UN_le_add_shift:
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   692
  "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   693
proof
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44008
diff changeset
   694
  show "?A <= ?B" by fastforce
36755
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   695
next
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   696
  show "?B <= ?A"
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   697
  proof
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   698
    fix x assume "x : ?B"
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   699
    then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   700
    hence "i-k\<le>n & x : M((i-k)+k)" by auto
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   701
    thus "x : ?A" by blast
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   702
  qed
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   703
qed
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
   704
32596
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
   705
lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
   706
  by (auto simp add: atLeast0LessThan) 
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
   707
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
   708
lemma UN_finite_subset: "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
   709
  by (subst UN_UN_finite_eq [symmetric]) blast
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
   710
33044
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   711
lemma UN_finite2_subset: 
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   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)"
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   713
  apply (rule UN_finite_subset)
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   714
  apply (subst UN_UN_finite_eq [symmetric, of B]) 
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   715
  apply blast
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   716
  done
32596
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
   717
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
   718
lemma UN_finite2_eq:
33044
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   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)"
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   720
  apply (rule subset_antisym)
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   721
   apply (rule UN_finite2_subset, blast)
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   722
 apply (rule UN_finite2_subset [where k=k])
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
   723
 apply (force simp add: atLeastLessThan_add_Un [of 0])
33044
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   724
 done
32596
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
   725
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
   726
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   727
subsubsection {* Cardinality *}
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   728
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   729
lemma card_lessThan [simp]: "card {..<u} = u"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   730
  by (induct u, simp_all add: lessThan_Suc)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   731
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   732
lemma card_atMost [simp]: "card {..u} = Suc u"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   733
  by (simp add: lessThan_Suc_atMost [THEN sym])
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   734
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   735
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   736
  apply (subgoal_tac "card {l..<u} = card {..<u-l}")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   737
  apply (erule ssubst, rule card_lessThan)
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   738
  apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   739
  apply (erule subst)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   740
  apply (rule card_image)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   741
  apply (simp add: inj_on_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   742
  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   743
  apply (rule_tac x = "x - l" in exI)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   744
  apply arith
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   745
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   746
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   747
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   748
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   749
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   750
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   751
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   752
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   753
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   754
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   755
26105
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   756
lemma ex_bij_betw_nat_finite:
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   757
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   758
apply(drule finite_imp_nat_seg_image_inj_on)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   759
apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   760
done
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   761
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   762
lemma ex_bij_betw_finite_nat:
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   763
  "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   764
by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   765
31438
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
   766
lemma finite_same_card_bij:
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
   767
  "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> EX h. bij_betw h A B"
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
   768
apply(drule ex_bij_betw_finite_nat)
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
   769
apply(drule ex_bij_betw_nat_finite)
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
   770
apply(auto intro!:bij_betw_trans)
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
   771
done
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
   772
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
   773
lemma ex_bij_betw_nat_finite_1:
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
   774
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
   775
by (rule finite_same_card_bij) auto
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
   776
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   777
lemma bij_betw_iff_card:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   778
  assumes FIN: "finite A" and FIN': "finite B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   779
  shows BIJ: "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   780
using assms
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   781
proof(auto simp add: bij_betw_same_card)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   782
  assume *: "card A = card B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   783
  obtain f where "bij_betw f A {0 ..< card A}"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   784
  using FIN ex_bij_betw_finite_nat by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   785
  moreover obtain g where "bij_betw g {0 ..< card B} B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   786
  using FIN' ex_bij_betw_nat_finite by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   787
  ultimately have "bij_betw (g o f) A B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   788
  using * by (auto simp add: bij_betw_trans)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   789
  thus "(\<exists>f. bij_betw f A B)" by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   790
qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   791
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   792
lemma inj_on_iff_card_le:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   793
  assumes FIN: "finite A" and FIN': "finite B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   794
  shows "(\<exists>f. inj_on f A \<and> f ` A \<le> B) = (card A \<le> card B)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   795
proof (safe intro!: card_inj_on_le)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   796
  assume *: "card A \<le> card B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   797
  obtain f where 1: "inj_on f A" and 2: "f ` A = {0 ..< card A}"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   798
  using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   799
  moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g ` {0 ..< card B} = B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   800
  using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   801
  ultimately have "inj_on g (f ` A)" using subset_inj_on[of g _ "f ` A"] * by force
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   802
  hence "inj_on (g o f) A" using 1 comp_inj_on by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   803
  moreover
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   804
  {have "{0 ..< card A} \<le> {0 ..< card B}" using * by force
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   805
   with 2 have "f ` A  \<le> {0 ..< card B}" by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   806
   hence "(g o f) ` A \<le> B" unfolding comp_def using 3 by force
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   807
  }
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   808
  ultimately show "(\<exists>f. inj_on f A \<and> f ` A \<le> B)" by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   809
qed (insert assms, auto)
26105
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   810
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   811
subsection {* Intervals of integers *}
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   812
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   813
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   814
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   815
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   816
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   817
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   818
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   819
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   820
    "{l+1..<u} = {l<..<u::int}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   821
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   822
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   823
subsubsection {* Finiteness *}
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   824
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   825
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   826
    {(0::int)..<u} = int ` {..<nat u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   827
  apply (unfold image_def lessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   828
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   829
  apply (rule_tac x = "nat x" in exI)
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
   830
  apply (auto simp add: zless_nat_eq_int_zless [THEN sym])
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   831
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   832
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   833
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   834
  apply (case_tac "0 \<le> u")
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   835
  apply (subst image_atLeastZeroLessThan_int, assumption)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   836
  apply (rule finite_imageI)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   837
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   838
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   839
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   840
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   841
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   842
  apply (erule subst)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   843
  apply (rule finite_imageI)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   844
  apply (rule finite_atLeastZeroLessThan_int)
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   845
  apply (rule image_add_int_atLeastLessThan)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   846
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   847
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   848
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   849
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   850
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   851
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   852
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   853
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   854
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   855
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   856
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   857
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   858
subsubsection {* Cardinality *}
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   859
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   860
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   861
  apply (case_tac "0 \<le> u")
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   862
  apply (subst image_atLeastZeroLessThan_int, assumption)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   863
  apply (subst card_image)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   864
  apply (auto simp add: inj_on_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   865
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   866
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   867
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   868
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   869
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   870
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   871
  apply (erule subst)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   872
  apply (rule card_image)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   873
  apply (simp add: inj_on_def)
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   874
  apply (rule image_add_int_atLeastLessThan)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   875
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   876
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   877
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28853
diff changeset
   878
apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28853
diff changeset
   879
apply (auto simp add: algebra_simps)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28853
diff changeset
   880
done
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   881
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   882
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28853
diff changeset
   883
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   884
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   885
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28853
diff changeset
   886
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   887
27656
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   888
lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   889
proof -
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   890
  have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   891
  with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   892
qed
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   893
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   894
lemma card_less:
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   895
assumes zero_in_M: "0 \<in> M"
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   896
shows "card {k \<in> M. k < Suc i} \<noteq> 0"
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   897
proof -
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   898
  from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" by auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   899
  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   900
qed
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   901
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   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}"
37388
793618618f78 tuned quotes, antiquotations and whitespace
haftmann
parents: 36846
diff changeset
   903
apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])
27656
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   904
apply simp
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44008
diff changeset
   905
apply fastforce
27656
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   906
apply auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   907
apply (rule inj_on_diff_nat)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   908
apply auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   909
apply (case_tac x)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   910
apply auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   911
apply (case_tac xa)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   912
apply auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   913
apply (case_tac xa)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   914
apply auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   915
done
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   916
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   917
lemma card_less_Suc:
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   918
  assumes zero_in_M: "0 \<in> M"
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   919
    shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   920
proof -
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   921
  from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   922
  hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   923
    by (auto simp only: insert_Diff)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   924
  have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   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}))"
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   926
    apply (subst card_insert)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   927
    apply simp_all
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   928
    apply (subst b)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   929
    apply (subst card_less_Suc2[symmetric])
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   930
    apply simp_all
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   931
    done
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   932
  with c show ?thesis by simp
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   933
qed
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   934
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   935
13850
6d1bb3059818 new logical equivalences
paulson
parents: 13735
diff changeset
   936
subsection {*Lemmas useful with the summation operator setsum*}
6d1bb3059818 new logical equivalences
paulson
parents: 13735
diff changeset
   937
16102
c5f6726d9bb1 Locale expressions: rename with optional mixfix syntax.
ballarin
parents: 16052
diff changeset
   938
text {* For examples, see Algebra/poly/UnivPoly2.thy *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   939
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   940
subsubsection {* Disjoint Unions *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   941
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   942
text {* Singletons and open intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   943
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   944
lemma ivl_disj_un_singleton:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   945
  "{l::'a::linorder} Un {l<..} = {l..}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   946
  "{..<u} Un {u::'a::linorder} = {..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   947
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   948
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   949
  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   950
  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
   951
by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   952
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   953
text {* One- and two-sided intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   954
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   955
lemma ivl_disj_un_one:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   956
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   957
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   958
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   959
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   960
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   961
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   962
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   963
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
   964
by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   965
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   966
text {* Two- and two-sided intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   967
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   968
lemma ivl_disj_un_two:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   969
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   970
  "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   971
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   972
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   973
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   974
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   975
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   976
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
   977
by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   978
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   979
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   980
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   981
subsubsection {* Disjoint Intersections *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   982
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   983
text {* One- and two-sided intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   984
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   985
lemma ivl_disj_int_one:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   986
  "{..l::'a::order} Int {l<..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   987
  "{..<l} Int {l..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   988
  "{..l} Int {l<..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   989
  "{..<l} Int {l..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   990
  "{l<..u} Int {u<..} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   991
  "{l<..<u} Int {u..} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   992
  "{l..u} Int {u<..} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   993
  "{l..<u} Int {u..} = {}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
   994
  by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   995
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   996
text {* Two- and two-sided intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   997
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   998
lemma ivl_disj_int_two:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   999
  "{l::'a::order<..<m} Int {m..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1000
  "{l<..m} Int {m<..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1001
  "{l..<m} Int {m..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1002
  "{l..m} Int {m<..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1003
  "{l<..<m} Int {m..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1004
  "{l<..m} Int {m<..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1005
  "{l..<m} Int {m..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1006
  "{l..m} Int {m<..u} = {}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
  1007
  by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1008
32456
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
  1009
lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1010
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1011
subsubsection {* Some Differences *}
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1012
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1013
lemma ivl_diff[simp]:
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1014
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1015
by(auto)
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1016
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1017
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1018
subsubsection {* Some Subset Conditions *}
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1019
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35644
diff changeset
  1020
lemma ivl_subset [simp,no_atp]:
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1021
 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1022
apply(auto simp:linorder_not_le)
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1023
apply(rule ccontr)
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1024
apply(insert linorder_le_less_linear[of i n])
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1025
apply(clarsimp simp:linorder_not_le)
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44008
diff changeset
  1026
apply(fastforce)
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1027
done
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1028
15041
a6b1f0cef7b3 Got rid of Summation and made it a translation into setsum instead.
nipkow
parents: 14846
diff changeset
  1029
15042
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
  1030
subsection {* Summation indexed over intervals *}
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
  1031
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
  1032
syntax
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
  1033
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
15048
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
  1034
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
  1035
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
  1036
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
15042
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
  1037
syntax (xsymbols)
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
  1038
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
15048
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
  1039
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
  1040
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
  1041
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
15042
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
  1042
syntax (HTML output)
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
  1043
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
15048
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
  1044
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
  1045
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
  1046
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  1047
syntax (latex_sum output)
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1048
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1049
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1050
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1051
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
  1052
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
  1053
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1054
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
  1055
 ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
15041
a6b1f0cef7b3 Got rid of Summation and made it a translation into setsum instead.
nipkow
parents: 14846
diff changeset
  1056
15048
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
  1057
translations
28853
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28068
diff changeset
  1058
  "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28068
diff changeset
  1059
  "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28068
diff changeset
  1060
  "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
69eb69659bf3 Added new fold operator and renamed the old oe to fold_image.
nipkow
parents: 28068
diff changeset
  1061
  "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
15041
a6b1f0cef7b3 Got rid of Summation and made it a translation into setsum instead.
nipkow
parents: 14846
diff changeset
  1062
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1063
text{* The above introduces some pretty alternative syntaxes for
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  1064
summation over intervals:
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1065
\begin{center}
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1066
\begin{tabular}{lll}
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  1067
Old & New & \LaTeX\\
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  1068
@{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  1069
@{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
  1070
@{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  1071
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1072
\end{tabular}
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1073
\end{center}
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  1074
The left column shows the term before introduction of the new syntax,
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  1075
the middle column shows the new (default) syntax, and the right column
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  1076
shows a special syntax. The latter is only meaningful for latex output
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  1077
and has to be activated explicitly by setting the print mode to
21502
7f3ea2b3bab6 prefer antiquotations over LaTeX macros;
wenzelm
parents: 20217
diff changeset
  1078
@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  1079
antiquotations). It is not the default \LaTeX\ output because it only
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
  1080
works well with italic-style formulae, not tt-style.
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1081
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1082
Note that for uniformity on @{typ nat} it is better to use
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1083
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1084
not provide all lemmas available for @{term"{m..<n}"} also in the
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1085
special form for @{term"{..<n}"}. *}
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
  1086
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1087
text{* This congruence rule should be used for sums over intervals as
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1088
the standard theorem @{text[source]setsum_cong} does not work well
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1089
with the simplifier who adds the unsimplified premise @{term"x:B"} to
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1090
the context. *}
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1091
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1092
lemma setsum_ivl_cong:
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1093
 "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1094
 setsum f {a..<b} = setsum g {c..<d}"
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
  1095
by(rule setsum_cong, simp_all)
15041
a6b1f0cef7b3 Got rid of Summation and made it a translation into setsum instead.
nipkow
parents: 14846
diff changeset
  1096
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
  1097
(* FIXME why are the following simp rules but the corresponding eqns
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
  1098
on intervals are not? *)
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
  1099
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
  1100
lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
  1101
by (simp add:atMost_Suc add_ac)
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
  1102
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
  1103
lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
  1104
by (simp add:lessThan_Suc add_ac)
15041
a6b1f0cef7b3 Got rid of Summation and made it a translation into setsum instead.
nipkow
parents: 14846
diff changeset
  1105
15911
b730b0edc085 turned 2 lemmas into simp rules
nipkow
parents: 15561
diff changeset
  1106
lemma setsum_cl_ivl_Suc[simp]:
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
  1107
  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
  1108
by (auto simp:add_ac atLeastAtMostSuc_conv)
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
  1109
15911
b730b0edc085 turned 2 lemmas into simp rules
nipkow
parents: 15561
diff changeset
  1110
lemma setsum_op_ivl_Suc[simp]:
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
  1111
  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
  1112
by (auto simp:add_ac atLeastLessThanSuc)
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
  1113
(*
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
  1114
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
  1115
    (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
  1116
by (auto simp:add_ac atLeastAtMostSuc_conv)
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
  1117
*)
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1118
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1119
lemma setsum_head:
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1120
  fixes n :: nat
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1121
  assumes mn: "m <= n" 
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1122
  shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1123
proof -
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1124
  from mn
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1125
  have "{m..n} = {m} \<union> {m<..n}"
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1126
    by (auto intro: ivl_disj_un_singleton)
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1127
  hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1128
    by (simp add: atLeast0LessThan)
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1129
  also have "\<dots> = ?rhs" by simp
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1130
  finally show ?thesis .
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1131
qed
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1132
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1133
lemma setsum_head_Suc:
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1134
  "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1135
by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1136
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1137
lemma setsum_head_upt_Suc:
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1138
  "m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 29960
diff changeset
  1139
apply(insert setsum_head_Suc[of m "n - Suc 0" f])
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28853
diff changeset
  1140
apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1141
done
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1142
31501
2a60c9b951e0 New lemma
nipkow
parents: 31438
diff changeset
  1143
lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"
2a60c9b951e0 New lemma
nipkow
parents: 31438
diff changeset
  1144
  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
2a60c9b951e0 New lemma
nipkow
parents: 31438
diff changeset
  1145
proof-
2a60c9b951e0 New lemma
nipkow
parents: 31438
diff changeset
  1146
  have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using `m \<le> n+1` by auto
2a60c9b951e0 New lemma
nipkow
parents: 31438
diff changeset
  1147
  thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint
2a60c9b951e0 New lemma
nipkow
parents: 31438
diff changeset
  1148
    atLeastSucAtMost_greaterThanAtMost)
2a60c9b951e0 New lemma
nipkow
parents: 31438
diff changeset
  1149
qed
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1150
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1151
lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1152
  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1153
by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1154
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1155
lemma setsum_diff_nat_ivl:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1156
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1157
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1158
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1159
using setsum_add_nat_ivl [of m n p f,symmetric]
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1160
apply (simp add: add_ac)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1161
done
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1162
31505
6f589131ba94 new lemma
nipkow
parents: 31501
diff changeset
  1163
lemma setsum_natinterval_difff:
6f589131ba94 new lemma
nipkow
parents: 31501
diff changeset
  1164
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
6f589131ba94 new lemma
nipkow
parents: 31501
diff changeset
  1165
  shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
6f589131ba94 new lemma
nipkow
parents: 31501
diff changeset
  1166
          (if m <= n then f m - f(n + 1) else 0)"
6f589131ba94 new lemma
nipkow
parents: 31501
diff changeset
  1167
by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)
6f589131ba94 new lemma
nipkow
parents: 31501
diff changeset
  1168
44008
2e09299ce807 tuned proofs
haftmann
parents: 43657
diff changeset
  1169
lemma setsum_restrict_set':
2e09299ce807 tuned proofs
haftmann
parents: 43657
diff changeset
  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)"
2e09299ce807 tuned proofs
haftmann
parents: 43657
diff changeset
  1171
  by (simp add: setsum_restrict_set [symmetric] Int_def)
2e09299ce807 tuned proofs
haftmann
parents: 43657
diff changeset
  1172
2e09299ce807 tuned proofs
haftmann
parents: 43657
diff changeset
  1173
lemma setsum_restrict_set'':
2e09299ce807 tuned proofs
haftmann
parents: 43657
diff changeset
  1174
  "finite A \<Longrightarrow> setsum f {x \<in> A. P x} = (\<Sum>x\<in>A. if P x  then f x else 0)"
2e09299ce807 tuned proofs
haftmann
parents: 43657
diff changeset
  1175
  by (simp add: setsum_restrict_set' [of A f "{x. P x}", simplified mem_Collect_eq])
31509
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1176
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1177
lemma setsum_setsum_restrict:
44008
2e09299ce807 tuned proofs
haftmann
parents: 43657
diff changeset
  1178
  "finite S \<Longrightarrow> finite T \<Longrightarrow>
2e09299ce807 tuned proofs
haftmann
parents: 43657
diff changeset
  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"
2e09299ce807 tuned proofs
haftmann
parents: 43657
diff changeset
  1180
  by (simp add: setsum_restrict_set'') (rule setsum_commute)
31509
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1181
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1182
lemma setsum_image_gen: assumes fS: "finite S"
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1183
  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1184
proof-
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1185
  { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1186
  hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1187
    by simp
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1188
  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1189
    by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1190
  finally show ?thesis .
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1191
qed
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1192
35171
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1193
lemma setsum_le_included:
36307
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  1194
  fixes f :: "'a \<Rightarrow> 'b::ordered_comm_monoid_add"
35171
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1195
  assumes "finite s" "finite t"
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  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)"
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1197
  shows "setsum f s \<le> setsum g t"
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1198
proof -
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1199
  have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1200
  proof (rule setsum_mono)
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1201
    fix y assume "y \<in> s"
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1202
    with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1203
    with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1204
      using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1205
      by (auto intro!: setsum_mono2)
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1206
  qed
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1207
  also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1208
    using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1209
  also have "... \<le> setsum g t"
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1210
    using assms by (auto simp: setsum_image_gen[symmetric])
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1211
  finally show ?thesis .
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1212
qed
28f824c7addc Moved setprod_mono, abs_setprod and setsum_le_included to the Main image. Is used in Multivariate_Analysis.
hoelzl
parents: 35115
diff changeset
  1213
31509
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1214
lemma setsum_multicount_gen:
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1215
  assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1216
  shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1217
proof-
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1218
  have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1219
  also have "\<dots> = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1220
    using assms(3) by auto
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1221
  finally show ?thesis .
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1222
qed
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1223
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1224
lemma setsum_multicount:
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1225
  assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1226
  shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1227
proof-
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1228
  have "?l = setsum (\<lambda>i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
  1229
  also have "\<dots> = ?r" by(simp add: mult_commute)
31509
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1230
  finally show ?thesis by auto
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1231
qed
00ede188c5d6 more lemmas
nipkow
parents: 31505
diff changeset
  1232
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1233
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1234
subsection{* Shifting bounds *}
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1235
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1236
lemma setsum_shift_bounds_nat_ivl:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1237
  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1238
by (induct "n", auto simp:atLeastLessThanSuc)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
  1239
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1240
lemma setsum_shift_bounds_cl_nat_ivl:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1241
  "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1242
apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1243
apply (simp add:image_add_atLeastAtMost o_def)
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1244
done
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1245
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1246
corollary setsum_shift_bounds_cl_Suc_ivl:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1247
  "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 29960
diff changeset
  1248
by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1249
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1250
corollary setsum_shift_bounds_Suc_ivl:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1251
  "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 29960
diff changeset
  1252
by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1253
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1254
lemma setsum_shift_lb_Suc0_0:
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1255
  "f(0::nat) = (0::nat) \<Longrightarrow> setsum f {Suc 0..k} = setsum f {0..k}"
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1256
by(simp add:setsum_head_Suc)
19106
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
  1257
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1258
lemma setsum_shift_lb_Suc0_0_upt:
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1259
  "f(0::nat) = 0 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1260
apply(cases k)apply simp
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1261
apply(simp add:setsum_head_upt_Suc)
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
  1262
done
19022
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
  1263
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
  1264
subsection {* The formula for geometric sums *}
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
  1265
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
  1266
lemma geometric_sum:
36307
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  1267
  assumes "x \<noteq> 1"
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  1268
  shows "(\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  1269
proof -
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  1270
  from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  1271
  moreover have "(\<Sum>i=0..<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  1272
  proof (induct n)
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  1273
    case 0 then show ?case by simp
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  1274
  next
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  1275
    case (Suc n)
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  1276
    moreover with `y \<noteq> 0` have "(1 + y) ^ n = (y * inverse y) * (1 + y) ^ n" by simp 
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36307
diff changeset
  1277
    ultimately show ?case by (simp add: field_simps divide_inverse)
36307
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  1278
  qed
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  1279
  ultimately show ?thesis by simp
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  1280
qed
1732232f9b27 sharpened constraint (c.f. 4e7f5b22dd7d); explicit is better than implicit
haftmann
parents: 35828
diff changeset
  1281
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
  1282
19469
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
  1283
subsection {* The formula for arithmetic sums *}
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
  1284
47222
1b7c909a6fad rephrase lemmas about arithmetic series using numeral '2'
huffman
parents: 47108
diff changeset
  1285
lemma gauss_sum:
1b7c909a6fad rephrase lemmas about arithmetic series using numeral '2'
huffman
parents: 47108
diff changeset
  1286
  "(2::'a::comm_semiring_1)*(\<Sum>i\<in>{1..n}. of_nat i) =
19469
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
  1287
   of_nat n*((of_nat n)+1)"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
  1288
proof (induct n)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
  1289
  case 0
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
  1290
  show ?case by simp
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
  1291
next
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
  1292
  case (Suc n)
47222
1b7c909a6fad rephrase lemmas about arithmetic series using numeral '2'
huffman
parents: 47108
diff changeset
  1293
  then show ?case
1b7c909a6fad rephrase lemmas about arithmetic series usi