src/HOL/SetInterval.thy
author nipkow
Mon May 02 18:59:50 2005 +0200 (2005-05-02)
changeset 15911 b730b0edc085
parent 15561 045a07ac35a7
child 16041 5a8736668ced
permissions -rw-r--r--
turned 2 lemmas into simp rules
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(*  Title:      HOL/SetInterval.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow and Clemens Ballarin
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                Additions by Jeremy Avigad in March 2004
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    Copyright   2000  TU Muenchen
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lessThan, greaterThan, atLeast, atMost and two-sided intervals
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*)
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header {* Set intervals *}
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theory SetInterval
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imports IntArith
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begin
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constdefs
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  lessThan    :: "('a::ord) => 'a set"	("(1{..<_})")
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  "{..<u} == {x. x<u}"
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  atMost      :: "('a::ord) => 'a set"	("(1{.._})")
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  "{..u} == {x. x<=u}"
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  greaterThan :: "('a::ord) => 'a set"	("(1{_<..})")
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  "{l<..} == {x. l<x}"
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  atLeast     :: "('a::ord) => 'a set"	("(1{_..})")
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  "{l..} == {x. l<=x}"
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  greaterThanLessThan :: "['a::ord, 'a] => 'a set"  ("(1{_<..<_})")
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  "{l<..<u} == {l<..} Int {..<u}"
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  atLeastLessThan :: "['a::ord, 'a] => 'a set"      ("(1{_..<_})")
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  "{l..<u} == {l..} Int {..<u}"
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  greaterThanAtMost :: "['a::ord, 'a] => 'a set"    ("(1{_<.._})")
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  "{l<..u} == {l<..} Int {..u}"
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  atLeastAtMost :: "['a::ord, 'a] => 'a set"        ("(1{_.._})")
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  "{l..u} == {l..} Int {..u}"
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(* Old syntax, will disappear! *)
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syntax
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  "_lessThan"    :: "('a::ord) => 'a set"	("(1{.._'(})")
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  "_greaterThan" :: "('a::ord) => 'a set"	("(1{')_..})")
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  "_greaterThanLessThan" :: "['a::ord, 'a] => 'a set"  ("(1{')_.._'(})")
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  "_atLeastLessThan" :: "['a::ord, 'a] => 'a set"      ("(1{_.._'(})")
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  "_greaterThanAtMost" :: "['a::ord, 'a] => 'a set"    ("(1{')_.._})")
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translations
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  "{..m(}" => "{..<m}"
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  "{)m..}" => "{m<..}"
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  "{)m..n(}" => "{m<..<n}"
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  "{m..n(}" => "{m..<n}"
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  "{)m..n}" => "{m<..n}"
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text{* A note of warning when using @{term"{..<n}"} on type @{typ
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nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
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@{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
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syntax
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  "@UNION_le"   :: "nat => nat => 'b set => 'b set"       ("(3UN _<=_./ _)" 10)
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  "@UNION_less" :: "nat => nat => 'b set => 'b set"       ("(3UN _<_./ _)" 10)
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  "@INTER_le"   :: "nat => nat => 'b set => 'b set"       ("(3INT _<=_./ _)" 10)
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  "@INTER_less" :: "nat => nat => 'b set => 'b set"       ("(3INT _<_./ _)" 10)
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syntax (input)
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  "@UNION_le"   :: "nat => nat => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" 10)
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  "@UNION_less" :: "nat => nat => 'b set => 'b set"       ("(3\<Union> _<_./ _)" 10)
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  "@INTER_le"   :: "nat => nat => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" 10)
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  "@INTER_less" :: "nat => nat => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" 10)
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syntax (xsymbols)
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  "@UNION_le"   :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Union>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
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  "@UNION_less" :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Union>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
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  "@INTER_le"   :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Inter>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
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  "@INTER_less" :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Inter>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
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translations
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  "UN i<=n. A"  == "UN i:{..n}. A"
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  "UN i<n. A"   == "UN i:{..<n}. A"
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  "INT i<=n. A" == "INT i:{..n}. A"
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  "INT i<n. A"  == "INT i:{..<n}. A"
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subsection {* Various equivalences *}
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lemma 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 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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apply (simp add: greaterThan_def atMost_def le_def, auto)
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done
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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 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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apply (simp add: lessThan_def atLeast_def le_def, auto)
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done
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lemma atMost_iff [iff]: "(i: atMost k) = (i<=k)"
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by (simp add: atMost_def)
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lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
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by (blast intro: order_antisym)
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subsection {* Logical Equivalences for Set Inclusion and Equality *}
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lemma atLeast_subset_iff [iff]:
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     "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
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by (blast intro: order_trans)
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lemma atLeast_eq_iff [iff]:
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     "(atLeast x = atLeast y) = (x = (y::'a::linorder))"
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by (blast intro: order_antisym order_trans)
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lemma greaterThan_subset_iff [iff]:
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     "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
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apply (auto simp add: greaterThan_def)
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 apply (subst linorder_not_less [symmetric], blast)
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done
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lemma greaterThan_eq_iff [iff]:
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     "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
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apply (rule iffI)
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 apply (erule equalityE)
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 apply (simp_all add: greaterThan_subset_iff)
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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 add: lessThan_subset_iff)
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done
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subsection {*Two-sided intervals*}
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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 {* The above four lemmas could be declared as iffs.
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  If we do so, a call to blast in Hyperreal/Star.ML, lemma @{text STAR_Int}
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  seems to take forever (more than one hour). *}
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subsubsection{* Emptyness and singletons *}
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lemma atLeastAtMost_empty [simp]: "n < m ==> {m::'a::order..n} = {}";
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  by (auto simp add: atLeastAtMost_def atMost_def atLeast_def);
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lemma atLeastLessThan_empty[simp]: "n \<le> m ==> {m..<n::'a::order} = {}"
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by (auto simp add: atLeastLessThan_def)
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lemma atLeastAtMost_singleton [simp]: "{a::'a::order..a} = {a}";
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  by (auto simp add: atLeastAtMost_def atMost_def atLeast_def);
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subsection {* Intervals of natural numbers *}
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subsubsection {* The Constant @{term lessThan} *}
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lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
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by (simp add: lessThan_def)
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lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
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by (simp add: lessThan_def less_Suc_eq, blast)
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lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
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by (simp add: lessThan_def atMost_def less_Suc_eq_le)
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lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
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by blast
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subsubsection {* The Constant @{term greaterThan} *}
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lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
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apply (simp add: greaterThan_def)
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apply (blast dest: gr0_conv_Suc [THEN iffD1])
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done
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lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
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apply (simp add: greaterThan_def)
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apply (auto elim: linorder_neqE)
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done
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lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
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by blast
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subsubsection {* The Constant @{term atLeast} *}
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lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
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by (unfold atLeast_def UNIV_def, simp)
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lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
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apply (simp add: atLeast_def)
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apply (simp add: Suc_le_eq)
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apply (simp add: order_le_less, blast)
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done
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lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
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  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
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lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
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by blast
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subsubsection {* The Constant @{term atMost} *}
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lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
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by (simp add: atMost_def)
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lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
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apply (simp add: atMost_def)
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apply (simp add: less_Suc_eq order_le_less, blast)
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done
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lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
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by blast
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subsubsection {* The Constant @{term atLeastLessThan} *}
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text{*But not a simprule because some concepts are better left in terms
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  of @{term atLeastLessThan}*}
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lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
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by(simp add:lessThan_def atLeastLessThan_def)
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lemma atLeastLessThan0 [simp]: "{m..<0::nat} = {}"
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by (simp add: atLeastLessThan_def)
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subsubsection {* Intervals of nats with @{term Suc} *}
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text{*Not a simprule because the RHS is too messy.*}
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lemma atLeastLessThanSuc:
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    "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
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by (auto simp add: atLeastLessThan_def)
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lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
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by (auto simp add: atLeastLessThan_def)
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lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
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by (induct k, simp_all add: atLeastLessThanSuc)
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lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
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by (auto simp add: atLeastLessThan_def)
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lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
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  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
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lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
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  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
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    greaterThanAtMost_def)
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lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
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  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
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    greaterThanLessThan_def)
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lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
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by (auto simp add: atLeastAtMost_def)
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subsubsection {* Finiteness *}
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lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
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  by (induct k) (simp_all add: lessThan_Suc)
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lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
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  by (induct k) (simp_all add: atMost_Suc)
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lemma finite_greaterThanLessThan [iff]:
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  fixes l :: nat shows "finite {l<..<u}"
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by (simp add: greaterThanLessThan_def)
paulson@14485
   314
paulson@14485
   315
lemma finite_atLeastLessThan [iff]:
nipkow@15045
   316
  fixes l :: nat shows "finite {l..<u}"
paulson@14485
   317
by (simp add: atLeastLessThan_def)
paulson@14485
   318
paulson@14485
   319
lemma finite_greaterThanAtMost [iff]:
nipkow@15045
   320
  fixes l :: nat shows "finite {l<..u}"
paulson@14485
   321
by (simp add: greaterThanAtMost_def)
paulson@14485
   322
paulson@14485
   323
lemma finite_atLeastAtMost [iff]:
paulson@14485
   324
  fixes l :: nat shows "finite {l..u}"
paulson@14485
   325
by (simp add: atLeastAtMost_def)
paulson@14485
   326
paulson@14485
   327
lemma bounded_nat_set_is_finite:
paulson@14485
   328
    "(ALL i:N. i < (n::nat)) ==> finite N"
paulson@14485
   329
  -- {* A bounded set of natural numbers is finite. *}
paulson@14485
   330
  apply (rule finite_subset)
paulson@14485
   331
   apply (rule_tac [2] finite_lessThan, auto)
paulson@14485
   332
  done
paulson@14485
   333
paulson@14485
   334
subsubsection {* Cardinality *}
paulson@14485
   335
nipkow@15045
   336
lemma card_lessThan [simp]: "card {..<u} = u"
paulson@15251
   337
  by (induct u, simp_all add: lessThan_Suc)
paulson@14485
   338
paulson@14485
   339
lemma card_atMost [simp]: "card {..u} = Suc u"
paulson@14485
   340
  by (simp add: lessThan_Suc_atMost [THEN sym])
paulson@14485
   341
nipkow@15045
   342
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
nipkow@15045
   343
  apply (subgoal_tac "card {l..<u} = card {..<u-l}")
paulson@14485
   344
  apply (erule ssubst, rule card_lessThan)
nipkow@15045
   345
  apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
paulson@14485
   346
  apply (erule subst)
paulson@14485
   347
  apply (rule card_image)
paulson@14485
   348
  apply (simp add: inj_on_def)
paulson@14485
   349
  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
paulson@14485
   350
  apply arith
paulson@14485
   351
  apply (rule_tac x = "x - l" in exI)
paulson@14485
   352
  apply arith
paulson@14485
   353
  done
paulson@14485
   354
paulson@15418
   355
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
paulson@14485
   356
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
paulson@14485
   357
paulson@15418
   358
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
paulson@14485
   359
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
paulson@14485
   360
nipkow@15045
   361
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
paulson@14485
   362
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
paulson@14485
   363
paulson@14485
   364
subsection {* Intervals of integers *}
paulson@14485
   365
nipkow@15045
   366
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
paulson@14485
   367
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   368
paulson@15418
   369
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
paulson@14485
   370
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
paulson@14485
   371
paulson@15418
   372
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
paulson@15418
   373
    "{l+1..<u} = {l<..<u::int}"
paulson@14485
   374
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
paulson@14485
   375
paulson@14485
   376
subsubsection {* Finiteness *}
paulson@14485
   377
paulson@15418
   378
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
nipkow@15045
   379
    {(0::int)..<u} = int ` {..<nat u}"
paulson@14485
   380
  apply (unfold image_def lessThan_def)
paulson@14485
   381
  apply auto
paulson@14485
   382
  apply (rule_tac x = "nat x" in exI)
paulson@14485
   383
  apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
paulson@14485
   384
  done
paulson@14485
   385
nipkow@15045
   386
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
paulson@14485
   387
  apply (case_tac "0 \<le> u")
paulson@14485
   388
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   389
  apply (rule finite_imageI)
paulson@14485
   390
  apply auto
paulson@14485
   391
  done
paulson@14485
   392
paulson@15418
   393
lemma image_atLeastLessThan_int_shift:
nipkow@15045
   394
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
nipkow@15561
   395
  apply (auto simp add: image_def)
paulson@14485
   396
  apply (rule_tac x = "x - l" in bexI)
paulson@14485
   397
  apply auto
paulson@14485
   398
  done
paulson@14485
   399
nipkow@15045
   400
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
nipkow@15045
   401
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   402
  apply (erule subst)
paulson@14485
   403
  apply (rule finite_imageI)
paulson@14485
   404
  apply (rule finite_atLeastZeroLessThan_int)
paulson@14485
   405
  apply (rule image_atLeastLessThan_int_shift)
paulson@14485
   406
  done
paulson@14485
   407
paulson@15418
   408
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
paulson@14485
   409
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
paulson@14485
   410
paulson@15418
   411
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
paulson@14485
   412
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   413
paulson@15418
   414
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
paulson@14485
   415
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   416
paulson@14485
   417
subsubsection {* Cardinality *}
paulson@14485
   418
nipkow@15045
   419
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
paulson@14485
   420
  apply (case_tac "0 \<le> u")
paulson@14485
   421
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   422
  apply (subst card_image)
paulson@14485
   423
  apply (auto simp add: inj_on_def)
paulson@14485
   424
  done
paulson@14485
   425
nipkow@15045
   426
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
nipkow@15045
   427
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
paulson@14485
   428
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
nipkow@15045
   429
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   430
  apply (erule subst)
paulson@14485
   431
  apply (rule card_image)
paulson@14485
   432
  apply (simp add: inj_on_def)
paulson@14485
   433
  apply (rule image_atLeastLessThan_int_shift)
paulson@14485
   434
  done
paulson@14485
   435
paulson@14485
   436
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
paulson@14485
   437
  apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
paulson@14485
   438
  apply (auto simp add: compare_rls)
paulson@14485
   439
  done
paulson@14485
   440
paulson@15418
   441
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
paulson@14485
   442
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   443
nipkow@15045
   444
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
paulson@14485
   445
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   446
paulson@14485
   447
paulson@13850
   448
subsection {*Lemmas useful with the summation operator setsum*}
paulson@13850
   449
wenzelm@14577
   450
text {* For examples, see Algebra/poly/UnivPoly.thy *}
ballarin@13735
   451
wenzelm@14577
   452
subsubsection {* Disjoint Unions *}
ballarin@13735
   453
wenzelm@14577
   454
text {* Singletons and open intervals *}
ballarin@13735
   455
ballarin@13735
   456
lemma ivl_disj_un_singleton:
nipkow@15045
   457
  "{l::'a::linorder} Un {l<..} = {l..}"
nipkow@15045
   458
  "{..<u} Un {u::'a::linorder} = {..u}"
nipkow@15045
   459
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
nipkow@15045
   460
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
nipkow@15045
   461
  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
nipkow@15045
   462
  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
ballarin@14398
   463
by auto
ballarin@13735
   464
wenzelm@14577
   465
text {* One- and two-sided intervals *}
ballarin@13735
   466
ballarin@13735
   467
lemma ivl_disj_un_one:
nipkow@15045
   468
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
nipkow@15045
   469
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
nipkow@15045
   470
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
nipkow@15045
   471
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
nipkow@15045
   472
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
nipkow@15045
   473
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
nipkow@15045
   474
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
nipkow@15045
   475
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
ballarin@14398
   476
by auto
ballarin@13735
   477
wenzelm@14577
   478
text {* Two- and two-sided intervals *}
ballarin@13735
   479
ballarin@13735
   480
lemma ivl_disj_un_two:
nipkow@15045
   481
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
nipkow@15045
   482
  "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
nipkow@15045
   483
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
nipkow@15045
   484
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
nipkow@15045
   485
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
nipkow@15045
   486
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
nipkow@15045
   487
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
nipkow@15045
   488
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
ballarin@14398
   489
by auto
ballarin@13735
   490
ballarin@13735
   491
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
ballarin@13735
   492
wenzelm@14577
   493
subsubsection {* Disjoint Intersections *}
ballarin@13735
   494
wenzelm@14577
   495
text {* Singletons and open intervals *}
ballarin@13735
   496
ballarin@13735
   497
lemma ivl_disj_int_singleton:
nipkow@15045
   498
  "{l::'a::order} Int {l<..} = {}"
nipkow@15045
   499
  "{..<u} Int {u} = {}"
nipkow@15045
   500
  "{l} Int {l<..<u} = {}"
nipkow@15045
   501
  "{l<..<u} Int {u} = {}"
nipkow@15045
   502
  "{l} Int {l<..u} = {}"
nipkow@15045
   503
  "{l..<u} Int {u} = {}"
ballarin@13735
   504
  by simp+
ballarin@13735
   505
wenzelm@14577
   506
text {* One- and two-sided intervals *}
ballarin@13735
   507
ballarin@13735
   508
lemma ivl_disj_int_one:
nipkow@15045
   509
  "{..l::'a::order} Int {l<..<u} = {}"
nipkow@15045
   510
  "{..<l} Int {l..<u} = {}"
nipkow@15045
   511
  "{..l} Int {l<..u} = {}"
nipkow@15045
   512
  "{..<l} Int {l..u} = {}"
nipkow@15045
   513
  "{l<..u} Int {u<..} = {}"
nipkow@15045
   514
  "{l<..<u} Int {u..} = {}"
nipkow@15045
   515
  "{l..u} Int {u<..} = {}"
nipkow@15045
   516
  "{l..<u} Int {u..} = {}"
ballarin@14398
   517
  by auto
ballarin@13735
   518
wenzelm@14577
   519
text {* Two- and two-sided intervals *}
ballarin@13735
   520
ballarin@13735
   521
lemma ivl_disj_int_two:
nipkow@15045
   522
  "{l::'a::order<..<m} Int {m..<u} = {}"
nipkow@15045
   523
  "{l<..m} Int {m<..<u} = {}"
nipkow@15045
   524
  "{l..<m} Int {m..<u} = {}"
nipkow@15045
   525
  "{l..m} Int {m<..<u} = {}"
nipkow@15045
   526
  "{l<..<m} Int {m..u} = {}"
nipkow@15045
   527
  "{l<..m} Int {m<..u} = {}"
nipkow@15045
   528
  "{l..<m} Int {m..u} = {}"
nipkow@15045
   529
  "{l..m} Int {m<..u} = {}"
ballarin@14398
   530
  by auto
ballarin@13735
   531
ballarin@13735
   532
lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
ballarin@13735
   533
nipkow@15542
   534
subsubsection {* Some Differences *}
nipkow@15542
   535
nipkow@15542
   536
lemma ivl_diff[simp]:
nipkow@15542
   537
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
nipkow@15542
   538
by(auto)
nipkow@15542
   539
nipkow@15542
   540
nipkow@15542
   541
subsubsection {* Some Subset Conditions *}
nipkow@15542
   542
nipkow@15542
   543
lemma ivl_subset[simp]:
nipkow@15542
   544
 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
nipkow@15542
   545
apply(auto simp:linorder_not_le)
nipkow@15542
   546
apply(rule ccontr)
nipkow@15542
   547
apply(insert linorder_le_less_linear[of i n])
nipkow@15542
   548
apply(clarsimp simp:linorder_not_le)
nipkow@15542
   549
apply(fastsimp)
nipkow@15542
   550
done
nipkow@15542
   551
nipkow@15041
   552
nipkow@15042
   553
subsection {* Summation indexed over intervals *}
nipkow@15042
   554
nipkow@15042
   555
syntax
nipkow@15042
   556
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   557
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
nipkow@15048
   558
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
nipkow@15042
   559
syntax (xsymbols)
nipkow@15042
   560
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   561
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@15048
   562
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@15042
   563
syntax (HTML output)
nipkow@15042
   564
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   565
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@15048
   566
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@15056
   567
syntax (latex_sum output)
nipkow@15052
   568
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   569
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052
   570
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   571
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052
   572
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   573
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15041
   574
nipkow@15048
   575
translations
nipkow@15048
   576
  "\<Sum>x=a..b. t" == "setsum (%x. t) {a..b}"
nipkow@15048
   577
  "\<Sum>x=a..<b. t" == "setsum (%x. t) {a..<b}"
nipkow@15048
   578
  "\<Sum>i<n. t" == "setsum (\<lambda>i. t) {..<n}"
nipkow@15041
   579
nipkow@15052
   580
text{* The above introduces some pretty alternative syntaxes for
nipkow@15056
   581
summation over intervals:
nipkow@15052
   582
\begin{center}
nipkow@15052
   583
\begin{tabular}{lll}
nipkow@15056
   584
Old & New & \LaTeX\\
nipkow@15056
   585
@{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
nipkow@15056
   586
@{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
nipkow@15056
   587
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
nipkow@15052
   588
\end{tabular}
nipkow@15052
   589
\end{center}
nipkow@15056
   590
The left column shows the term before introduction of the new syntax,
nipkow@15056
   591
the middle column shows the new (default) syntax, and the right column
nipkow@15056
   592
shows a special syntax. The latter is only meaningful for latex output
nipkow@15056
   593
and has to be activated explicitly by setting the print mode to
nipkow@15056
   594
\texttt{latex\_sum} (e.g.\ via \texttt{mode=latex\_sum} in
nipkow@15056
   595
antiquotations). It is not the default \LaTeX\ output because it only
nipkow@15056
   596
works well with italic-style formulae, not tt-style.
nipkow@15052
   597
nipkow@15052
   598
Note that for uniformity on @{typ nat} it is better to use
nipkow@15052
   599
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
nipkow@15052
   600
not provide all lemmas available for @{term"{m..<n}"} also in the
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   601
special form for @{term"{..<n}"}. *}
nipkow@15052
   602
nipkow@15542
   603
(* FIXME change the simplifier's treatment of congruence rules?? *)
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   604
nipkow@15542
   605
text{* This congruence rule should be used for sums over intervals as
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   606
the standard theorem @{text[source]setsum_cong} does not work well
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   607
with the simplifier who adds the unsimplified premise @{term"x:B"} to
nipkow@15542
   608
the context. *}
nipkow@15542
   609
nipkow@15542
   610
lemma setsum_ivl_cong:
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   611
 "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
nipkow@15542
   612
 setsum f {a..<b} = setsum g {c..<d}"
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   613
by(rule setsum_cong, simp_all)
nipkow@15041
   614
nipkow@15554
   615
(* FIXME delete
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   616
lemma Summation_Suc[simp]: "(\<Sum>i < Suc n. b i) = b n + (\<Sum>i < n. b i)"
nipkow@15041
   617
by (simp add:lessThan_Suc)
nipkow@15554
   618
*)
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   619
nipkow@15911
   620
lemma setsum_cl_ivl_Suc[simp]:
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   621
  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
nipkow@15561
   622
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@15561
   623
nipkow@15911
   624
lemma setsum_op_ivl_Suc[simp]:
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   625
  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
nipkow@15561
   626
by (auto simp:add_ac atLeastLessThanSuc)
nipkow@15561
   627
nipkow@15561
   628
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
nipkow@15561
   629
    (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
nipkow@15561
   630
by (auto simp:add_ac atLeastAtMostSuc_conv)
nipkow@15561
   631
nipkow@15539
   632
lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539
   633
  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
nipkow@15539
   634
by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
nipkow@15539
   635
nipkow@15539
   636
lemma setsum_diff_nat_ivl:
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   637
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
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   638
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
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   639
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
nipkow@15539
   640
using setsum_add_nat_ivl [of m n p f,symmetric]
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   641
apply (simp add: add_ac)
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   642
done
nipkow@15539
   643
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   644
lemma setsum_shift_bounds_nat_ivl:
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   645
  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
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   646
by (induct "n", auto simp:atLeastLessThanSuc)
nipkow@15539
   647
paulson@15418
   648
paulson@15418
   649
ML
paulson@15418
   650
{*
paulson@15418
   651
val Compl_atLeast = thm "Compl_atLeast";
paulson@15418
   652
val Compl_atMost = thm "Compl_atMost";
paulson@15418
   653
val Compl_greaterThan = thm "Compl_greaterThan";
paulson@15418
   654
val Compl_lessThan = thm "Compl_lessThan";
paulson@15418
   655
val INT_greaterThan_UNIV = thm "INT_greaterThan_UNIV";
paulson@15418
   656
val UN_atLeast_UNIV = thm "UN_atLeast_UNIV";
paulson@15418
   657
val UN_atMost_UNIV = thm "UN_atMost_UNIV";
paulson@15418
   658
val UN_lessThan_UNIV = thm "UN_lessThan_UNIV";
paulson@15418
   659
val atLeastAtMost_def = thm "atLeastAtMost_def";
paulson@15418
   660
val atLeastAtMost_iff = thm "atLeastAtMost_iff";
paulson@15418
   661
val atLeastLessThan_def  = thm "atLeastLessThan_def";
paulson@15418
   662
val atLeastLessThan_iff = thm "atLeastLessThan_iff";
paulson@15418
   663
val atLeast_0 = thm "atLeast_0";
paulson@15418
   664
val atLeast_Suc = thm "atLeast_Suc";
paulson@15418
   665
val atLeast_def      = thm "atLeast_def";
paulson@15418
   666
val atLeast_iff = thm "atLeast_iff";
paulson@15418
   667
val atMost_0 = thm "atMost_0";
paulson@15418
   668
val atMost_Int_atLeast = thm "atMost_Int_atLeast";
paulson@15418
   669
val atMost_Suc = thm "atMost_Suc";
paulson@15418
   670
val atMost_def       = thm "atMost_def";
paulson@15418
   671
val atMost_iff = thm "atMost_iff";
paulson@15418
   672
val greaterThanAtMost_def  = thm "greaterThanAtMost_def";
paulson@15418
   673
val greaterThanAtMost_iff = thm "greaterThanAtMost_iff";
paulson@15418
   674
val greaterThanLessThan_def  = thm "greaterThanLessThan_def";
paulson@15418
   675
val greaterThanLessThan_iff = thm "greaterThanLessThan_iff";
paulson@15418
   676
val greaterThan_0 = thm "greaterThan_0";
paulson@15418
   677
val greaterThan_Suc = thm "greaterThan_Suc";
paulson@15418
   678
val greaterThan_def  = thm "greaterThan_def";
paulson@15418
   679
val greaterThan_iff = thm "greaterThan_iff";
paulson@15418
   680
val ivl_disj_int = thms "ivl_disj_int";
paulson@15418
   681
val ivl_disj_int_one = thms "ivl_disj_int_one";
paulson@15418
   682
val ivl_disj_int_singleton = thms "ivl_disj_int_singleton";
paulson@15418
   683
val ivl_disj_int_two = thms "ivl_disj_int_two";
paulson@15418
   684
val ivl_disj_un = thms "ivl_disj_un";
paulson@15418
   685
val ivl_disj_un_one = thms "ivl_disj_un_one";
paulson@15418
   686
val ivl_disj_un_singleton = thms "ivl_disj_un_singleton";
paulson@15418
   687
val ivl_disj_un_two = thms "ivl_disj_un_two";
paulson@15418
   688
val lessThan_0 = thm "lessThan_0";
paulson@15418
   689
val lessThan_Suc = thm "lessThan_Suc";
paulson@15418
   690
val lessThan_Suc_atMost = thm "lessThan_Suc_atMost";
paulson@15418
   691
val lessThan_def     = thm "lessThan_def";
paulson@15418
   692
val lessThan_iff = thm "lessThan_iff";
paulson@15418
   693
val single_Diff_lessThan = thm "single_Diff_lessThan";
paulson@15418
   694
paulson@15418
   695
val bounded_nat_set_is_finite = thm "bounded_nat_set_is_finite";
paulson@15418
   696
val finite_atMost = thm "finite_atMost";
paulson@15418
   697
val finite_lessThan = thm "finite_lessThan";
paulson@15418
   698
*}
paulson@15418
   699
nipkow@8924
   700
end