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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Modern convention: Ixy stands for an interval where x and y
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describe the lower and upper bound and x,y : {c,o,i}
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where c = closed, o = open, i = infinite.
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Examples: Ico = {_ ..< _} and Ici = {_ ..}
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*)
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section \<open>Set intervals\<close>
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theory Set_Interval
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imports Lattices_Big 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\<open>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.\<close>
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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 \<open>Various equivalences\<close>
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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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lemma (in linorder) lessThan_Int_lessThan: "{ a <..} \<inter> { b <..} = { max a b <..}"
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  by auto
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lemma (in linorder) greaterThan_Int_greaterThan: "{..< a} \<inter> {..< b} = {..< min a b}"
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  by auto
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subsection \<open>Logical Equivalences for Set Inclusion and Equality\<close>
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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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lemma (in linorder) Ici_subset_Ioi_iff: "{a ..} \<subseteq> {b <..} \<longleftrightarrow> b < a"
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  by auto
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lemma (in linorder) Iic_subset_Iio_iff: "{.. a} \<subseteq> {..< b} \<longleftrightarrow> a < b"
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  by auto
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subsection \<open>Two-sided intervals\<close>
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context ord
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begin
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lemma greaterThanLessThan_iff [simp]:
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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]:
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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]:
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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]:
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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 \<open>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 them
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alone.\<close>
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lemma greaterThanLessThan_eq: "{ a <..< b} = { a <..} \<inter> {..< b }"
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  by auto
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end
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subsubsection\<open>Emptyness, singletons, subset\<close>
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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 Icc_eq_Icc[simp]:
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  "{l..h} = {l'..h'} = (l=l' \<and> h=h' \<or> \<not> l\<le>h \<and> \<not> l'\<le>h')"
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by(simp add: order_class.eq_iff)(auto 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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  with \<open>{a..b} = {c}\<close> have "c \<le> a \<and> b \<le> c" by auto
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  with * show "a = b \<and> b = c" by auto
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qed simp
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lemma Icc_subset_Ici_iff[simp]:
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  "{l..h} \<subseteq> {l'..} = (~ l\<le>h \<or> l\<ge>l')"
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by(auto simp: subset_eq intro: order_trans)
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lemma Icc_subset_Iic_iff[simp]:
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  "{l..h} \<subseteq> {..h'} = (~ l\<le>h \<or> h\<le>h')"
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by(auto simp: subset_eq intro: order_trans)
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lemma not_Ici_eq_empty[simp]: "{l..} \<noteq> {}"
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by(auto simp: set_eq_iff)
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lemma not_Iic_eq_empty[simp]: "{..h} \<noteq> {}"
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by(auto simp: set_eq_iff)
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   296
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lemmas not_empty_eq_Ici_eq_empty[simp] = not_Ici_eq_empty[symmetric]
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lemmas not_empty_eq_Iic_eq_empty[simp] = not_Iic_eq_empty[symmetric]
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   299
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diff changeset
   300
end
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   301
51334
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   302
context no_top
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parents: 51329
diff changeset
   303
begin
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diff changeset
   304
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   305
(* also holds for no_bot but no_top should suffice *)
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   306
lemma not_UNIV_le_Icc[simp]: "\<not> UNIV \<subseteq> {l..h}"
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   307
using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
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parents: 51329
diff changeset
   308
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   309
lemma not_UNIV_le_Iic[simp]: "\<not> UNIV \<subseteq> {..h}"
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   310
using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
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parents: 51329
diff changeset
   311
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   312
lemma not_Ici_le_Icc[simp]: "\<not> {l..} \<subseteq> {l'..h'}"
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diff changeset
   313
using gt_ex[of h']
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diff changeset
   314
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
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parents: 51329
diff changeset
   315
fd531bd984d8 more lemmas about intervals
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diff changeset
   316
lemma not_Ici_le_Iic[simp]: "\<not> {l..} \<subseteq> {..h'}"
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   317
using gt_ex[of h']
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parents: 51329
diff changeset
   318
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
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parents: 51329
diff changeset
   319
fd531bd984d8 more lemmas about intervals
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diff changeset
   320
end
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parents: 51329
diff changeset
   321
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   322
context no_bot
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parents: 51329
diff changeset
   323
begin
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parents: 51329
diff changeset
   324
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diff changeset
   325
lemma not_UNIV_le_Ici[simp]: "\<not> UNIV \<subseteq> {l..}"
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parents: 51329
diff changeset
   326
using lt_ex[of l] by(auto simp: subset_eq less_le_not_le)
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   327
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diff changeset
   328
lemma not_Iic_le_Icc[simp]: "\<not> {..h} \<subseteq> {l'..h'}"
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parents: 51329
diff changeset
   329
using lt_ex[of l']
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parents: 51329
diff changeset
   330
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   331
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parents: 51329
diff changeset
   332
lemma not_Iic_le_Ici[simp]: "\<not> {..h} \<subseteq> {l'..}"
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parents: 51329
diff changeset
   333
using lt_ex[of l']
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   334
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   335
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   336
end
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   337
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   338
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   339
context no_top
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   340
begin
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   341
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parents: 51329
diff changeset
   342
(* also holds for no_bot but no_top should suffice *)
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parents: 51329
diff changeset
   343
lemma not_UNIV_eq_Icc[simp]: "\<not> UNIV = {l'..h'}"
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parents: 51329
diff changeset
   344
using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   345
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   346
lemmas not_Icc_eq_UNIV[simp] = not_UNIV_eq_Icc[symmetric]
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parents: 51329
diff changeset
   347
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   348
lemma not_UNIV_eq_Iic[simp]: "\<not> UNIV = {..h'}"
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parents: 51329
diff changeset
   349
using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   350
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   351
lemmas not_Iic_eq_UNIV[simp] = not_UNIV_eq_Iic[symmetric]
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   352
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   353
lemma not_Icc_eq_Ici[simp]: "\<not> {l..h} = {l'..}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   354
unfolding atLeastAtMost_def using not_Ici_le_Iic[of l'] by blast
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   355
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   356
lemmas not_Ici_eq_Icc[simp] = not_Icc_eq_Ici[symmetric]
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   357
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   358
(* also holds for no_bot but no_top should suffice *)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   359
lemma not_Iic_eq_Ici[simp]: "\<not> {..h} = {l'..}"
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nipkow
parents: 51329
diff changeset
   360
using not_Ici_le_Iic[of l' h] by blast
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   361
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   362
lemmas not_Ici_eq_Iic[simp] = not_Iic_eq_Ici[symmetric]
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   363
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   364
end
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   365
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   366
context no_bot
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   367
begin
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   368
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   369
lemma not_UNIV_eq_Ici[simp]: "\<not> UNIV = {l'..}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   370
using lt_ex[of l'] by(auto simp: set_eq_iff  less_le_not_le)
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   371
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   372
lemmas not_Ici_eq_UNIV[simp] = not_UNIV_eq_Ici[symmetric]
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   373
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   374
lemma not_Icc_eq_Iic[simp]: "\<not> {l..h} = {..h'}"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   375
unfolding atLeastAtMost_def using not_Iic_le_Ici[of h'] by blast
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   376
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   377
lemmas not_Iic_eq_Icc[simp] = not_Icc_eq_Iic[symmetric]
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   378
fd531bd984d8 more lemmas about intervals
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parents: 51329
diff changeset
   379
end
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   380
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   381
53216
ad2e09c30aa8 renamed inner_dense_linorder to dense_linorder
hoelzl
parents: 52729
diff changeset
   382
context dense_linorder
42891
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   383
begin
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   384
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   385
lemma greaterThanLessThan_empty_iff[simp]:
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   386
  "{ a <..< b } = {} \<longleftrightarrow> b \<le> a"
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   387
  using dense[of a b] by (cases "a < b") auto
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   388
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   389
lemma greaterThanLessThan_empty_iff2[simp]:
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   390
  "{} = { a <..< b } \<longleftrightarrow> b \<le> a"
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   391
  using dense[of a b] by (cases "a < b") auto
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   392
42901
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   393
lemma atLeastLessThan_subseteq_atLeastAtMost_iff:
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   394
  "{a ..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   395
  using dense[of "max a d" "b"]
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   396
  by (force simp: subset_eq Ball_def not_less[symmetric])
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   397
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   398
lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   399
  "{a <.. b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   400
  using dense[of "a" "min c b"]
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   401
  by (force simp: subset_eq Ball_def not_less[symmetric])
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   402
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   403
lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   404
  "{a <..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   405
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   406
  by (force simp: subset_eq Ball_def not_less[symmetric])
e35cf2b25f48 equations for subsets of atLeastAtMost
hoelzl
parents: 42891
diff changeset
   407
43657
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   408
lemma atLeastAtMost_subseteq_atLeastLessThan_iff:
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   409
  "{a .. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a \<le> b \<longrightarrow> c \<le> a \<and> b < d)"
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   410
  using dense[of "max a d" "b"]
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   411
  by (force simp: subset_eq Ball_def not_less[symmetric])
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   412
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   413
lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   414
  "{a <.. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b < d)"
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   415
  using dense[of "a" "min c b"]
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   416
  by (force simp: subset_eq Ball_def not_less[symmetric])
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   417
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   418
lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   419
  "{a <..< b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   420
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   421
  by (force simp: subset_eq Ball_def not_less[symmetric])
537ea3846f64 equalities of subsets of atLeastLessThan
hoelzl
parents: 43157
diff changeset
   422
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   423
lemma greaterThanLessThan_subseteq_greaterThanAtMost_iff:
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   424
  "{a <..< b} \<subseteq> { c <.. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   425
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   426
  by (force simp: subset_eq Ball_def not_less[symmetric])
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   427
42891
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   428
end
e2f473671937 simp rules for empty intervals on dense linear order
hoelzl
parents: 40703
diff changeset
   429
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   430
context no_top
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   431
begin
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   432
51334
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   433
lemma greaterThan_non_empty[simp]: "{x <..} \<noteq> {}"
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   434
  using gt_ex[of x] by auto
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   435
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   436
end
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   437
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   438
context no_bot
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   439
begin
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   440
51334
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   441
lemma lessThan_non_empty[simp]: "{..< x} \<noteq> {}"
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   442
  using lt_ex[of x] by auto
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   443
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   444
end
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   445
32408
a1a85b0a26f7 new interval lemma
nipkow
parents: 32400
diff changeset
   446
lemma (in linorder) atLeastLessThan_subset_iff:
a1a85b0a26f7 new interval lemma
nipkow
parents: 32400
diff changeset
   447
  "{a..<b} <= {c..<d} \<Longrightarrow> b <= a | c<=a & b<=d"
a1a85b0a26f7 new interval lemma
nipkow
parents: 32400
diff changeset
   448
apply (auto simp:subset_eq Ball_def)
a1a85b0a26f7 new interval lemma
nipkow
parents: 32400
diff changeset
   449
apply(frule_tac x=a in spec)
a1a85b0a26f7 new interval lemma
nipkow
parents: 32400
diff changeset
   450
apply(erule_tac x=d in allE)
a1a85b0a26f7 new interval lemma
nipkow
parents: 32400
diff changeset
   451
apply (simp add: less_imp_le)
a1a85b0a26f7 new interval lemma
nipkow
parents: 32400
diff changeset
   452
done
a1a85b0a26f7 new interval lemma
nipkow
parents: 32400
diff changeset
   453
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   454
lemma atLeastLessThan_inj:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   455
  fixes a b c d :: "'a::linorder"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   456
  assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   457
  shows "a = c" "b = d"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   458
using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le linorder_antisym_conv2 subset_refl)+
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   459
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   460
lemma atLeastLessThan_eq_iff:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   461
  fixes a b c d :: "'a::linorder"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   462
  assumes "a < b" "c < d"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   463
  shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   464
  using atLeastLessThan_inj assms by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
   465
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   466
lemma (in linorder) Ioc_inj: "{a <.. b} = {c <.. d} \<longleftrightarrow> (b \<le> a \<and> d \<le> c) \<or> a = c \<and> b = d"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   467
  by (metis eq_iff greaterThanAtMost_empty_iff2 greaterThanAtMost_iff le_cases not_le)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   468
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   469
lemma (in order) Iio_Int_singleton: "{..<k} \<inter> {x} = (if x < k then {x} else {})"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   470
  by auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   471
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   472
lemma (in linorder) Ioc_subset_iff: "{a<..b} \<subseteq> {c<..d} \<longleftrightarrow> (b \<le> a \<or> c \<le> a \<and> b \<le> d)"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   473
  by (auto simp: subset_eq Ball_def) (metis less_le not_less)
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   474
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52380
diff changeset
   475
lemma (in order_bot) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"
51334
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   476
by (auto simp: set_eq_iff intro: le_bot)
51328
d63ec23c9125 move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents: 51152
diff changeset
   477
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52380
diff changeset
   478
lemma (in order_top) atMost_eq_UNIV_iff: "{..x} = UNIV \<longleftrightarrow> x = top"
51334
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   479
by (auto simp: set_eq_iff intro: top_le)
51328
d63ec23c9125 move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents: 51152
diff changeset
   480
51334
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   481
lemma (in bounded_lattice) atLeastAtMost_eq_UNIV_iff:
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   482
  "{x..y} = UNIV \<longleftrightarrow> (x = bot \<and> y = top)"
fd531bd984d8 more lemmas about intervals
nipkow
parents: 51329
diff changeset
   483
by (auto simp: set_eq_iff intro: top_le le_bot)
51328
d63ec23c9125 move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents: 51152
diff changeset
   484
56949
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56480
diff changeset
   485
lemma Iio_eq_empty_iff: "{..< n::'a::{linorder, order_bot}} = {} \<longleftrightarrow> n = bot"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56480
diff changeset
   486
  by (auto simp: set_eq_iff not_less le_bot)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56480
diff changeset
   487
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56480
diff changeset
   488
lemma Iio_eq_empty_iff_nat: "{..< n::nat} = {} \<longleftrightarrow> n = 0"
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56480
diff changeset
   489
  by (simp add: Iio_eq_empty_iff bot_nat_def)
d1a937cbf858 clean up Lebesgue integration
hoelzl
parents: 56480
diff changeset
   490
58970
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   491
lemma mono_image_least:
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   492
  assumes f_mono: "mono f" and f_img: "f ` {m ..< n} = {m' ..< n'}" "m < n"
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   493
  shows "f m = m'"
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   494
proof -
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   495
  from f_img have "{m' ..< n'} \<noteq> {}"
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   496
    by (metis atLeastLessThan_empty_iff image_is_empty)
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   497
  with f_img have "m' \<in> f ` {m ..< n}" by auto
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   498
  then obtain k where "f k = m'" "m \<le> k" by auto
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   499
  moreover have "m' \<le> f m" using f_img by auto
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   500
  ultimately show "f m = m'"
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   501
    using f_mono by (auto elim: monoE[where x=m and y=k])
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   502
qed
2f65dcd32a59 add forgotten lemma
noschinl
parents: 58889
diff changeset
   503
51328
d63ec23c9125 move auxiliary lemmas from Library/Extended_Reals to HOL image
hoelzl
parents: 51152
diff changeset
   504
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   505
subsection \<open>Infinite intervals\<close>
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   506
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   507
context dense_linorder
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   508
begin
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   509
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   510
lemma infinite_Ioo:
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   511
  assumes "a < b"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   512
  shows "\<not> finite {a<..<b}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   513
proof
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   514
  assume fin: "finite {a<..<b}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   515
  moreover have ne: "{a<..<b} \<noteq> {}"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   516
    using \<open>a < b\<close> by auto
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   517
  ultimately have "a < Max {a <..< b}" "Max {a <..< b} < b"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   518
    using Max_in[of "{a <..< b}"] by auto
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   519
  then obtain x where "Max {a <..< b} < x" "x < b"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   520
    using dense[of "Max {a<..<b}" b] by auto
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   521
  then have "x \<in> {a <..< b}"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   522
    using \<open>a < Max {a <..< b}\<close> by auto
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   523
  then have "x \<le> Max {a <..< b}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   524
    using fin by auto
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   525
  with \<open>Max {a <..< b} < x\<close> show False by auto
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   526
qed
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   527
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   528
lemma infinite_Icc: "a < b \<Longrightarrow> \<not> finite {a .. b}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   529
  using greaterThanLessThan_subseteq_atLeastAtMost_iff[of a b a b] infinite_Ioo[of a b]
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   530
  by (auto dest: finite_subset)
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   531
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   532
lemma infinite_Ico: "a < b \<Longrightarrow> \<not> finite {a ..< b}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   533
  using greaterThanLessThan_subseteq_atLeastLessThan_iff[of a b a b] infinite_Ioo[of a b]
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   534
  by (auto dest: finite_subset)
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   535
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   536
lemma infinite_Ioc: "a < b \<Longrightarrow> \<not> finite {a <.. b}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   537
  using greaterThanLessThan_subseteq_greaterThanAtMost_iff[of a b a b] infinite_Ioo[of a b]
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   538
  by (auto dest: finite_subset)
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   539
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   540
end
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   541
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   542
lemma infinite_Iio: "\<not> finite {..< a :: 'a :: {no_bot, linorder}}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   543
proof
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   544
  assume "finite {..< a}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   545
  then have *: "\<And>x. x < a \<Longrightarrow> Min {..< a} \<le> x"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   546
    by auto
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   547
  obtain x where "x < a"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   548
    using lt_ex by auto
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   549
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   550
  obtain y where "y < Min {..< a}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   551
    using lt_ex by auto
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   552
  also have "Min {..< a} \<le> x"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   553
    using \<open>x < a\<close> by fact
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   554
  also note \<open>x < a\<close>
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   555
  finally have "Min {..< a} \<le> y"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   556
    by fact
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   557
  with \<open>y < Min {..< a}\<close> show False by auto
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   558
qed
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   559
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   560
lemma infinite_Iic: "\<not> finite {.. a :: 'a :: {no_bot, linorder}}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   561
  using infinite_Iio[of a] finite_subset[of "{..< a}" "{.. a}"]
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   562
  by (auto simp: subset_eq less_imp_le)
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   563
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   564
lemma infinite_Ioi: "\<not> finite {a :: 'a :: {no_top, linorder} <..}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   565
proof
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   566
  assume "finite {a <..}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   567
  then have *: "\<And>x. a < x \<Longrightarrow> x \<le> Max {a <..}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   568
    by auto
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   569
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   570
  obtain y where "Max {a <..} < y"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   571
    using gt_ex by auto
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   572
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   573
  obtain x where "a < x"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   574
    using gt_ex by auto
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   575
  also then have "x \<le> Max {a <..}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   576
    by fact
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   577
  also note \<open>Max {a <..} < y\<close>
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   578
  finally have "y \<le> Max { a <..}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   579
    by fact
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   580
  with \<open>Max {a <..} < y\<close> show False by auto
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   581
qed
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   582
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   583
lemma infinite_Ici: "\<not> finite {a :: 'a :: {no_top, linorder} ..}"
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   584
  using infinite_Ioi[of a] finite_subset[of "{a <..}" "{a ..}"]
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   585
  by (auto simp: subset_eq less_imp_le)
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   586
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   587
subsubsection \<open>Intersection\<close>
32456
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   588
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   589
context linorder
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   590
begin
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   591
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   592
lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   593
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   594
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   595
lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   596
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   597
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   598
lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   599
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   600
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   601
lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   602
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   603
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   604
lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   605
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   606
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   607
lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   608
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   609
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   610
lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   611
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   612
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   613
lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   614
by auto
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   615
50417
f18b92f91818 add Int_atMost
hoelzl
parents: 47988
diff changeset
   616
lemma Int_atMost[simp]: "{..a} \<inter> {..b} = {.. min a b}"
f18b92f91818 add Int_atMost
hoelzl
parents: 47988
diff changeset
   617
  by (auto simp: min_def)
f18b92f91818 add Int_atMost
hoelzl
parents: 47988
diff changeset
   618
57447
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   619
lemma Ioc_disjoint: "{a<..b} \<inter> {c<..d} = {} \<longleftrightarrow> b \<le> a \<or> d \<le> c \<or> b \<le> c \<or> d \<le> a"
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   620
  using assms by auto
87429bdecad5 import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents: 57418
diff changeset
   621
32456
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   622
end
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   623
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   624
context complete_lattice
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   625
begin
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   626
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   627
lemma
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   628
  shows Sup_atLeast[simp]: "Sup {x ..} = top"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   629
    and Sup_greaterThanAtLeast[simp]: "x < top \<Longrightarrow> Sup {x <..} = top"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   630
    and Sup_atMost[simp]: "Sup {.. y} = y"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   631
    and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   632
    and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   633
  by (auto intro!: Sup_eqI)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   634
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   635
lemma
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   636
  shows Inf_atMost[simp]: "Inf {.. x} = bot"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   637
    and Inf_atMostLessThan[simp]: "top < x \<Longrightarrow> Inf {..< x} = bot"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   638
    and Inf_atLeast[simp]: "Inf {x ..} = x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   639
    and Inf_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Inf { x .. y} = x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   640
    and Inf_atLeastLessThan[simp]: "x < y \<Longrightarrow> Inf { x ..< y} = x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   641
  by (auto intro!: Inf_eqI)
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   642
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   643
end
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   644
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   645
lemma
53216
ad2e09c30aa8 renamed inner_dense_linorder to dense_linorder
hoelzl
parents: 52729
diff changeset
   646
  fixes x y :: "'a :: {complete_lattice, dense_linorder}"
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   647
  shows Sup_lessThan[simp]: "Sup {..< y} = y"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   648
    and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   649
    and Sup_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Sup { x <..< y} = y"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   650
    and Inf_greaterThan[simp]: "Inf {x <..} = x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   651
    and Inf_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Inf { x <.. y} = x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   652
    and Inf_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Inf { x <..< y} = x"
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51328
diff changeset
   653
  by (auto intro!: Inf_eqI Sup_eqI intro: dense_le dense_le_bounded dense_ge dense_ge_bounded)
32456
341c83339aeb tuned the simp rules for Int involving insert and intervals.
nipkow
parents: 32436
diff changeset
   654
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   655
subsection \<open>Intervals of natural numbers\<close>
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   656
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   657
subsubsection \<open>The Constant @{term lessThan}\<close>
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   658
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   659
lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   660
by (simp add: lessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   661
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   662
lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   663
by (simp add: lessThan_def less_Suc_eq, blast)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   664
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   665
text \<open>The following proof is convenient in induction proofs where
39072
1030b1a166ef Add lessThan_Suc_eq_insert_0
hoelzl
parents: 37664
diff changeset
   666
new elements get indices at the beginning. So it is used to transform
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   667
@{term "{..<Suc n}"} to @{term "0::nat"} and @{term "{..< n}"}.\<close>
39072
1030b1a166ef Add lessThan_Suc_eq_insert_0
hoelzl
parents: 37664
diff changeset
   668
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58970
diff changeset
   669
lemma zero_notin_Suc_image: "0 \<notin> Suc ` A"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58970
diff changeset
   670
  by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58970
diff changeset
   671
39072
1030b1a166ef Add lessThan_Suc_eq_insert_0
hoelzl
parents: 37664
diff changeset
   672
lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58970
diff changeset
   673
  by (auto simp: image_iff less_Suc_eq_0_disj)
39072
1030b1a166ef Add lessThan_Suc_eq_insert_0
hoelzl
parents: 37664
diff changeset
   674
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   675
lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   676
by (simp add: lessThan_def atMost_def less_Suc_eq_le)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   677
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58970
diff changeset
   678
lemma Iic_Suc_eq_insert_0: "{.. Suc n} = insert 0 (Suc ` {.. n})"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58970
diff changeset
   679
  unfolding lessThan_Suc_atMost[symmetric] lessThan_Suc_eq_insert_0[of "Suc n"] ..
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58970
diff changeset
   680
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   681
lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   682
by blast
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   683
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   684
subsubsection \<open>The Constant @{term greaterThan}\<close>
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   685
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   686
lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   687
apply (simp add: greaterThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   688
apply (blast dest: gr0_conv_Suc [THEN iffD1])
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   689
done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   690
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   691
lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   692
apply (simp add: greaterThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   693
apply (auto elim: linorder_neqE)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   694
done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   695
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   696
lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   697
by blast
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   698
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   699
subsubsection \<open>The Constant @{term atLeast}\<close>
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   700
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   701
lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   702
by (unfold atLeast_def UNIV_def, simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   703
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   704
lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   705
apply (simp add: atLeast_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   706
apply (simp add: Suc_le_eq)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   707
apply (simp add: order_le_less, blast)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   708
done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   709
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   710
lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   711
  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   712
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   713
lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   714
by blast
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   715
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   716
subsubsection \<open>The Constant @{term atMost}\<close>
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   717
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   718
lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   719
by (simp add: atMost_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   720
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   721
lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   722
apply (simp add: atMost_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   723
apply (simp add: less_Suc_eq order_le_less, blast)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   724
done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   725
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   726
lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   727
by blast
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   728
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   729
subsubsection \<open>The Constant @{term atLeastLessThan}\<close>
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   730
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   731
text\<open>The orientation of the following 2 rules is tricky. The lhs is
24449
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24286
diff changeset
   732
defined in terms of the rhs.  Hence the chosen orientation makes sense
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24286
diff changeset
   733
in this theory --- the reverse orientation complicates proofs (eg
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24286
diff changeset
   734
nontermination). But outside, when the definition of the lhs is rarely
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24286
diff changeset
   735
used, the opposite orientation seems preferable because it reduces a
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   736
specific concept to a more general one.\<close>
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   737
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   738
lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
15042
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
   739
by(simp add:lessThan_def atLeastLessThan_def)
24449
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24286
diff changeset
   740
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   741
lemma atLeast0AtMost: "{0..n::nat} = {..n}"
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   742
by(simp add:atMost_def atLeastAtMost_def)
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   743
31998
2c7a24f74db9 code attributes use common underscore convention
haftmann
parents: 31509
diff changeset
   744
declare atLeast0LessThan[symmetric, code_unfold]
2c7a24f74db9 code attributes use common underscore convention
haftmann
parents: 31509
diff changeset
   745
        atLeast0AtMost[symmetric, code_unfold]
24449
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24286
diff changeset
   746
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24286
diff changeset
   747
lemma atLeastLessThan0: "{m..<0::nat} = {}"
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   748
by (simp add: atLeastLessThan_def)
24449
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24286
diff changeset
   749
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   750
subsubsection \<open>Intervals of nats with @{term Suc}\<close>
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   751
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   752
text\<open>Not a simprule because the RHS is too messy.\<close>
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   753
lemma atLeastLessThanSuc:
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   754
    "{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
   755
by (auto simp add: atLeastLessThan_def)
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   756
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   757
lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   758
by (auto simp add: atLeastLessThan_def)
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   759
(*
15047
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   760
lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   761
by (induct k, simp_all add: atLeastLessThanSuc)
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   762
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   763
lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
fa62de5862b9 redefining sumr to be a translation to setsum
paulson
parents: 15045
diff changeset
   764
by (auto simp add: atLeastLessThan_def)
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   765
*)
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   766
lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   767
  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   768
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   769
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
   770
  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   771
    greaterThanAtMost_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   772
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   773
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
   774
  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   775
    greaterThanLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   776
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15542
diff changeset
   777
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
   778
by (auto simp add: atLeastAtMost_def)
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15542
diff changeset
   779
45932
6f08f8fe9752 add lemmas
noschinl
parents: 44890
diff changeset
   780
lemma atLeastAtMost_insertL: "m \<le> n \<Longrightarrow> insert m {Suc m..n} = {m ..n}"
6f08f8fe9752 add lemmas
noschinl
parents: 44890
diff changeset
   781
by auto
6f08f8fe9752 add lemmas
noschinl
parents: 44890
diff changeset
   782
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   783
text \<open>The analogous result is useful on @{typ int}:\<close>
43157
b505be6f029a atLeastAtMostSuc_conv on int
kleing
parents: 43156
diff changeset
   784
(* here, because we don't have an own int section *)
b505be6f029a atLeastAtMostSuc_conv on int
kleing
parents: 43156
diff changeset
   785
lemma atLeastAtMostPlus1_int_conv:
b505be6f029a atLeastAtMostSuc_conv on int
kleing
parents: 43156
diff changeset
   786
  "m <= 1+n \<Longrightarrow> {m..1+n} = insert (1+n) {m..n::int}"
b505be6f029a atLeastAtMostSuc_conv on int
kleing
parents: 43156
diff changeset
   787
  by (auto intro: set_eqI)
b505be6f029a atLeastAtMostSuc_conv on int
kleing
parents: 43156
diff changeset
   788
33044
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   789
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
   790
  apply (induct k) 
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   791
  apply (simp_all add: atLeastLessThanSuc)   
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   792
  done
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
   793
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   794
subsubsection \<open>Intervals and numerals\<close>
57113
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   795
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   796
lemma lessThan_nat_numeral:  --\<open>Evaluation for specific numerals\<close>
57113
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   797
  "lessThan (numeral k :: nat) = insert (pred_numeral k) (lessThan (pred_numeral k))"
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   798
  by (simp add: numeral_eq_Suc lessThan_Suc)
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   799
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   800
lemma atMost_nat_numeral:  --\<open>Evaluation for specific numerals\<close>
57113
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   801
  "atMost (numeral k :: nat) = insert (numeral k) (atMost (pred_numeral k))"
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   802
  by (simp add: numeral_eq_Suc atMost_Suc)
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   803
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   804
lemma atLeastLessThan_nat_numeral:  --\<open>Evaluation for specific numerals\<close>
57113
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   805
  "atLeastLessThan m (numeral k :: nat) = 
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   806
     (if m \<le> (pred_numeral k) then insert (pred_numeral k) (atLeastLessThan m (pred_numeral k))
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   807
                 else {})"
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   808
  by (simp add: numeral_eq_Suc atLeastLessThanSuc)
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
   809
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   810
subsubsection \<open>Image\<close>
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   811
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   812
lemma image_add_atLeastAtMost:
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60586
diff changeset
   813
  fixes k ::"'a::linordered_semidom"
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60586
diff changeset
   814
  shows "(\<lambda>n. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   815
proof
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   816
  show "?A \<subseteq> ?B" by auto
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   817
next
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   818
  show "?B \<subseteq> ?A"
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   819
  proof
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   820
    fix n assume a: "n : ?B"
60615
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60586
diff changeset
   821
    hence "n - k : {i..j}"
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60586
diff changeset
   822
      by (auto simp: add_le_imp_le_diff add_le_add_imp_diff_le)
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60586
diff changeset
   823
    moreover have "n = (n - k) + k" using a
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60586
diff changeset
   824
    proof -
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60586
diff changeset
   825
      have "k + i \<le> n"
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60586
diff changeset
   826
        by (metis a add.commute atLeastAtMost_iff)
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60586
diff changeset
   827
      hence "k + (n - k) = n"
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60586
diff changeset
   828
        by (metis (no_types) ab_semigroup_add_class.add_ac(1) add_diff_cancel_left' le_add_diff_inverse)
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60586
diff changeset
   829
      thus ?thesis
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60586
diff changeset
   830
        by (simp add: add.commute)
e5fa1d5d3952 Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents: 60586
diff changeset
   831
    qed
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   832
    ultimately show "n : ?A" by blast
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   833
  qed
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   834
qed
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   835
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   836
lemma image_add_atLeastLessThan:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   837
  "(%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
   838
proof
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   839
  show "?A \<subseteq> ?B" by auto
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   840
next
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   841
  show "?B \<subseteq> ?A"
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   842
  proof
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   843
    fix n assume a: "n : ?B"
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19538
diff changeset
   844
    hence "n - k : {i..<j}" by auto
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   845
    moreover have "n = (n - k) + k" using a by auto
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   846
    ultimately show "n : ?A" by blast
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   847
  qed
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   848
qed
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   849
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   850
corollary image_Suc_atLeastAtMost[simp]:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   851
  "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
   852
using image_add_atLeastAtMost[where k="Suc 0"] by simp
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   853
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   854
corollary image_Suc_atLeastLessThan[simp]:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   855
  "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
   856
using image_add_atLeastLessThan[where k="Suc 0"] by simp
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   857
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   858
lemma image_add_int_atLeastLessThan:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   859
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   860
  apply (auto simp add: image_def)
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   861
  apply (rule_tac x = "x - l" in bexI)
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   862
  apply auto
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   863
  done
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   864
37664
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   865
lemma image_minus_const_atLeastLessThan_nat:
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   866
  fixes c :: nat
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   867
  shows "(\<lambda>i. i - c) ` {x ..< y} =
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   868
      (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
   869
    (is "_ = ?right")
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   870
proof safe
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   871
  fix a assume a: "a \<in> ?right"
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   872
  show "a \<in> (\<lambda>i. i - c) ` {x ..< y}"
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   873
  proof cases
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   874
    assume "c < y" with a show ?thesis
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   875
      by (auto intro!: image_eqI[of _ _ "a + c"])
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   876
  next
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   877
    assume "\<not> c < y" with a show ?thesis
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   878
      by (auto intro!: image_eqI[of _ _ x] split: split_if_asm)
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   879
  qed
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   880
qed auto
2946b8f057df Instantiate product type as euclidean space.
hoelzl
parents: 37388
diff changeset
   881
51152
b52cc015429a added lemma
Andreas Lochbihler
parents: 50999
diff changeset
   882
lemma image_int_atLeastLessThan: "int ` {a..<b} = {int a..<int b}"
55143
04448228381d explicit eigen-context for attributes "where", "of", and corresponding read_instantiate, instantiate_tac;
wenzelm
parents: 55088
diff changeset
   883
  by (auto intro!: image_eqI [where x = "nat x" for x])
51152
b52cc015429a added lemma
Andreas Lochbihler
parents: 50999
diff changeset
   884
35580
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   885
context ordered_ab_group_add
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   886
begin
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   887
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   888
lemma
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   889
  fixes x :: 'a
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   890
  shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   891
  and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   892
proof safe
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   893
  fix y assume "y < -x"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   894
  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
   895
  have "- (-y) \<in> uminus ` {x<..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   896
    by (rule imageI) (simp add: *)
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   897
  thus "y \<in> uminus ` {x<..}" by simp
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   898
next
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   899
  fix y assume "y \<le> -x"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   900
  have "- (-y) \<in> uminus ` {x..}"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   901
    by (rule imageI) (insert \<open>y \<le> -x\<close>[THEN le_imp_neg_le], simp)
35580
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   902
  thus "y \<in> uminus ` {x..}" by simp
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   903
qed simp_all
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   904
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   905
lemma
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   906
  fixes x :: 'a
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   907
  shows image_uminus_lessThan[simp]: "uminus ` {..<x} = {-x<..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   908
  and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   909
proof -
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   910
  have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   911
    and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   912
  thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   913
    by (simp_all add: image_image
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   914
        del: image_uminus_greaterThan image_uminus_atLeast)
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   915
qed
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   916
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   917
lemma
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   918
  fixes x :: 'a
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   919
  shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   920
  and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   921
  and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   922
  and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   923
  by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   924
      greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35216
diff changeset
   925
end
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   926
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   927
subsubsection \<open>Finiteness\<close>
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   928
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   929
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   930
  by (induct k) (simp_all add: lessThan_Suc)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   931
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   932
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   933
  by (induct k) (simp_all add: atMost_Suc)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   934
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   935
lemma finite_greaterThanLessThan [iff]:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   936
  fixes l :: nat shows "finite {l<..<u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   937
by (simp add: greaterThanLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   938
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   939
lemma finite_atLeastLessThan [iff]:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   940
  fixes l :: nat shows "finite {l..<u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   941
by (simp add: atLeastLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   942
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   943
lemma finite_greaterThanAtMost [iff]:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   944
  fixes l :: nat shows "finite {l<..u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   945
by (simp add: greaterThanAtMost_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   946
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   947
lemma finite_atLeastAtMost [iff]:
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   948
  fixes l :: nat shows "finite {l..u}"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   949
by (simp add: atLeastAtMost_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   950
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   951
text \<open>A bounded set of natural numbers is finite.\<close>
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   952
lemma bounded_nat_set_is_finite:
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   953
  "(ALL i:N. i < (n::nat)) ==> finite N"
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   954
apply (rule finite_subset)
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   955
 apply (rule_tac [2] finite_lessThan, auto)
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   956
done
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   957
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   958
text \<open>A set of natural numbers is finite iff it is bounded.\<close>
31044
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   959
lemma finite_nat_set_iff_bounded:
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   960
  "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   961
proof
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   962
  assume f:?F  show ?B
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   963
    using Max_ge[OF \<open>?F\<close>, simplified less_Suc_eq_le[symmetric]] by blast
31044
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   964
next
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   965
  assume ?B show ?F using \<open>?B\<close> by(blast intro:bounded_nat_set_is_finite)
31044
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   966
qed
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   967
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   968
lemma finite_nat_set_iff_bounded_le:
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   969
  "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   970
apply(simp add:finite_nat_set_iff_bounded)
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   971
apply(blast dest:less_imp_le_nat le_imp_less_Suc)
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   972
done
6896c2498ac0 new lemmas
nipkow
parents: 31017
diff changeset
   973
28068
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   974
lemma finite_less_ub:
f6b2d1995171 cleaned up code generation for {.._} and {..<_}
nipkow
parents: 27656
diff changeset
   975
     "!!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
   976
by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   977
56328
b3551501424e add rules about infinity of intervals
hoelzl
parents: 56238
diff changeset
   978
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   979
text\<open>Any subset of an interval of natural numbers the size of the
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
   980
subset is exactly that interval.\<close>
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   981
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   982
lemma subset_card_intvl_is_intvl:
55085
0e8e4dc55866 moved 'fundef_cong' attribute (and other basic 'fun' stuff) up the dependency chain
blanchet
parents: 54606
diff changeset
   983
  assumes "A \<subseteq> {k..<k + card A}"
0e8e4dc55866 moved 'fundef_cong' attribute (and other basic 'fun' stuff) up the dependency chain
blanchet
parents: 54606
diff changeset
   984
  shows "A = {k..<k + card A}"
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
   985
proof (cases "finite A")
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
   986
  case True
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
   987
  from this and assms show ?thesis
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
   988
  proof (induct A rule: finite_linorder_max_induct)
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   989
    case empty thus ?case by auto
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   990
  next
33434
e9de8d69c1b9 fixed order of parameters in induction rules
nipkow
parents: 33318
diff changeset
   991
    case (insert b A)
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
   992
    hence *: "b \<notin> A" by auto
55085
0e8e4dc55866 moved 'fundef_cong' attribute (and other basic 'fun' stuff) up the dependency chain
blanchet
parents: 54606
diff changeset
   993
    with insert have "A <= {k..<k + card A}" and "b = k + card A"
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
   994
      by fastforce+
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
   995
    with insert * show ?case by auto
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   996
  qed
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   997
next
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
   998
  case False
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 53216
diff changeset
   999
  with assms show ?thesis by simp
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
  1000
qed
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
  1001
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
  1002
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1003
subsubsection \<open>Proving Inclusions and Equalities between Unions\<close>
32596
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1004
36755
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1005
lemma UN_le_eq_Un0:
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1006
  "(\<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
  1007
proof
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1008
  show "?A <= ?B"
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1009
  proof
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1010
    fix x assume "x : ?A"
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1011
    then obtain i where i: "i\<le>n" "x : M i" by auto
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1012
    show "x : ?B"
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1013
    proof(cases i)
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1014
      case 0 with i show ?thesis by simp
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1015
    next
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1016
      case (Suc j) with i show ?thesis by auto
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1017
    qed
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1018
  qed
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1019
next
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1020
  show "?B <= ?A" by auto
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1021
qed
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1022
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1023
lemma UN_le_add_shift:
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1024
  "(\<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
  1025
proof
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44008
diff changeset
  1026
  show "?A <= ?B" by fastforce
36755
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1027
next
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1028
  show "?B <= ?A"
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1029
  proof
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1030
    fix x assume "x : ?B"
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1031
    then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1032
    hence "i-k\<le>n & x : M((i-k)+k)" by auto
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1033
    thus "x : ?A" by blast
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1034
  qed
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1035
qed
d1b498f2f50b added lemmas
nipkow
parents: 36365
diff changeset
  1036
32596
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1037
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
  1038
  by (auto simp add: atLeast0LessThan) 
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1039
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1040
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
  1041
  by (subst UN_UN_finite_eq [symmetric]) blast
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1042
33044
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
  1043
lemma UN_finite2_subset: 
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
  1044
     "(!!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
  1045
  apply (rule UN_finite_subset)
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
  1046
  apply (subst UN_UN_finite_eq [symmetric, of B]) 
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
  1047
  apply blast
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
  1048
  done
32596
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1049
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1050
lemma UN_finite2_eq:
33044
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
  1051
  "(!!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
  1052
  apply (rule subset_antisym)
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
  1053
   apply (rule UN_finite2_subset, blast)
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
  1054
 apply (rule UN_finite2_subset [where k=k])
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
  1055
 apply (force simp add: atLeastLessThan_add_Un [of 0])
33044
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32960
diff changeset
  1056
 done
32596
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1057
bd68c04dace1 New theorems for proving equalities and inclusions involving unions
paulson
parents: 32456
diff changeset
  1058
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1059
subsubsection \<open>Cardinality\<close>
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1060
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1061
lemma card_lessThan [simp]: "card {..<u} = u"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
  1062
  by (induct u, simp_all add: lessThan_Suc)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1063
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1064
lemma card_atMost [simp]: "card {..u} = Suc u"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1065
  by (simp add: lessThan_Suc_atMost [THEN sym])
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1066
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1067
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
57113
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1068
proof -
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1069
  have "{l..<u} = (%x. x + l) ` {..<u-l}"
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1070
    apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1071
    apply (rule_tac x = "x - l" in exI)
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1072
    apply arith
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1073
    done
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1074
  then have "card {l..<u} = card {..<u-l}"
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1075
    by (simp add: card_image inj_on_def)
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1076
  then show ?thesis
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1077
    by simp
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1078
qed
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1079
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1080
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1081
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1082
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1083
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1084
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1085
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1086
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1087
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1088
26105
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
  1089
lemma ex_bij_betw_nat_finite:
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
  1090
  "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
  1091
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
  1092
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
  1093
done
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
  1094
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
  1095
lemma ex_bij_betw_finite_nat:
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
  1096
  "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
  1097
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
  1098
31438
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
  1099
lemma finite_same_card_bij:
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
  1100
  "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
  1101
apply(drule ex_bij_betw_finite_nat)
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
  1102
apply(drule ex_bij_betw_nat_finite)
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
  1103
apply(auto intro!:bij_betw_trans)
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
  1104
done
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
  1105
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
  1106
lemma ex_bij_betw_nat_finite_1:
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
  1107
  "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
  1108
by (rule finite_same_card_bij) auto
a1c4c1500abe A few finite lemmas
nipkow
parents: 31044
diff changeset
  1109
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
  1110
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
  1111
  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
  1112
  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
  1113
using assms
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
  1114
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
  1115
  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
  1116
  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
  1117
  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
  1118
  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
  1119
  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
  1120
  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
  1121
  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
  1122
  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
  1123
qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
  1124
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
  1125
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
  1126
  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
  1127
  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
  1128
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
  1129
  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
  1130
  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
  1131
  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
  1132
  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
  1133
  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
  1134
  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
  1135
  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
  1136
  moreover
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
  1137
  {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
  1138
   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
  1139
   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
  1140
  }
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 39302
diff changeset
  1141
  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
  1142
qed (insert assms, auto)
26105
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
  1143
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1144
subsection \<open>Intervals of integers\<close>
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1145
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1146
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1147
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1148
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1149
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1150
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1151
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1152
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1153
    "{l+1..<u} = {l<..<u::int}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1154
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1155
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1156
subsubsection \<open>Finiteness\<close>
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1157
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1158
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1159
    {(0::int)..<u} = int ` {..<nat u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1160
  apply (unfold image_def lessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1161
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1162
  apply (rule_tac x = "nat x" in exI)
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35171
diff changeset
  1163
  apply (auto simp add: zless_nat_eq_int_zless [THEN sym])
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1164
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1165
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1166
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
47988
e4b69e10b990 tuned proofs;
wenzelm
parents: 47317
diff changeset
  1167
  apply (cases "0 \<le> u")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1168
  apply (subst image_atLeastZeroLessThan_int, assumption)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1169
  apply (rule finite_imageI)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1170
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1171
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1172
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1173
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1174
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1175
  apply (erule subst)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1176
  apply (rule finite_imageI)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1177
  apply (rule finite_atLeastZeroLessThan_int)
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1178
  apply (rule image_add_int_atLeastLessThan)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1179
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1180
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1181
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1182
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1183
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1184
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1185
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1186
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1187
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1188
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1189
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
  1190
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1191
subsubsection \<open>Cardinality\<close>
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1192
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1193
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
47988
e4b69e10b990 tuned proofs;
wenzelm
parents: 47317
diff changeset
  1194
  apply (cases "0 \<le> u")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1195
  apply (subst image_atLeastZeroLessThan_int, assumption)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1196
  apply (subst card_image)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1197
  apply (auto simp add: inj_on_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1198
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1199
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1200
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
  1201
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1202
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1203
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1204
  apply (erule subst)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1205
  apply (rule card_image)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1206
  apply (simp add: inj_on_def)
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
  1207
  apply (rule image_add_int_atLeastLessThan)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1208
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1209
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1210
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
  1211
apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28853
diff changeset
  1212
apply (auto simp add: algebra_simps)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28853
diff changeset
  1213
done
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1214
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
  1215
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
  1216
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1217
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1218
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
  1219
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1220
27656
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1221
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
  1222
proof -
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1223
  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
  1224
  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
  1225
qed
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1226
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1227
lemma card_less:
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1228
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
  1229
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
  1230
proof -
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1231
  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
  1232
  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
  1233
qed
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1234
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1235
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
  1236
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
  1237
apply auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1238
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
  1239
apply auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1240
apply (case_tac x)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1241
apply auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1242
apply (case_tac xa)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1243
apply auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1244
apply (case_tac xa)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1245
apply auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1246
done
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1247
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1248
lemma card_less_Suc:
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1249
  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
  1250
    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
  1251
proof -
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1252
  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
  1253
  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
  1254
    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
  1255
  have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
57113
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1256
  from finite_M_bounded_by_nat[of "\<lambda>x. x \<in> M" "Suc i"] 
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 56949
diff changeset
  1257
  have "Suc (card {k. Suc k \<in> M \<and> k < i}) = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"
27656
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1258
    apply (subst card_insert)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1259
    apply simp_all
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1260
    apply (subst b)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1261
    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
  1262
    apply simp_all
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1263
    done
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1264
  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
  1265
qed
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
  1266
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
  1267
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1268
subsection \<open>Lemmas useful with the summation operator setsum\<close>
13850
6d1bb3059818 new logical equivalences
paulson
parents: 13735
diff changeset
  1269
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1270
text \<open>For examples, see Algebra/poly/UnivPoly2.thy\<close>
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1271
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1272
subsubsection \<open>Disjoint Unions\<close>
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1273
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1274
text \<open>Singletons and open intervals\<close>
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1275
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1276
lemma ivl_disj_un_singleton:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1277
  "{l::'a::linorder} Un {l<..} = {l..}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1278
  "{..<u} Un {u::'a::linorder} = {..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1279
  "(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
  1280
  "(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
  1281
  "(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
  1282
  "(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
  1283
by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1284
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60615
diff changeset
  1285
text \<open>One- and two-sided intervals\<close>
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1286
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
  1287
lemma ivl_disj_un_one:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
  1288
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"