src/HOL/SetInterval.thy
author nipkow
Mon Aug 16 14:22:27 2004 +0200 (2004-08-16)
changeset 15131 c69542757a4d
parent 15056 b75073d90bff
child 15140 322485b816ac
permissions -rw-r--r--
New theory header syntax.
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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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import 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 add: greaterThan_subset_iff order_antisym, simp) 
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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 add: lessThan_subset_iff order_antisym, simp) 
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done
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subsection {*Two-sided intervals*}
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text {* @{text greaterThanLessThan} *}
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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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text {* @{text atLeastLessThan} *}
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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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text {* @{text greaterThanAtMost} *}
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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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text {* @{text atLeastAtMost} *}
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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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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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lemma atLeastLessThan_self [simp]: "{n::'a::order..<n} = {}"
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by (auto simp add: atLeastLessThan_def)
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lemma atLeastLessThan_empty: "n \<le> m ==> {m..<n::'a::order} = {}"
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by (auto 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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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)
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lemma finite_atLeastLessThan [iff]:
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  fixes l :: nat shows "finite {l..<u}"
paulson@14485
   318
by (simp add: atLeastLessThan_def)
paulson@14485
   319
paulson@14485
   320
lemma finite_greaterThanAtMost [iff]:
nipkow@15045
   321
  fixes l :: nat shows "finite {l<..u}"
paulson@14485
   322
by (simp add: greaterThanAtMost_def)
paulson@14485
   323
paulson@14485
   324
lemma finite_atLeastAtMost [iff]:
paulson@14485
   325
  fixes l :: nat shows "finite {l..u}"
paulson@14485
   326
by (simp add: atLeastAtMost_def)
paulson@14485
   327
paulson@14485
   328
lemma bounded_nat_set_is_finite:
paulson@14485
   329
    "(ALL i:N. i < (n::nat)) ==> finite N"
paulson@14485
   330
  -- {* A bounded set of natural numbers is finite. *}
paulson@14485
   331
  apply (rule finite_subset)
paulson@14485
   332
   apply (rule_tac [2] finite_lessThan, auto)
paulson@14485
   333
  done
paulson@14485
   334
paulson@14485
   335
subsubsection {* Cardinality *}
paulson@14485
   336
nipkow@15045
   337
lemma card_lessThan [simp]: "card {..<u} = u"
paulson@14485
   338
  by (induct_tac u, simp_all add: lessThan_Suc)
paulson@14485
   339
paulson@14485
   340
lemma card_atMost [simp]: "card {..u} = Suc u"
paulson@14485
   341
  by (simp add: lessThan_Suc_atMost [THEN sym])
paulson@14485
   342
nipkow@15045
   343
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
nipkow@15045
   344
  apply (subgoal_tac "card {l..<u} = card {..<u-l}")
paulson@14485
   345
  apply (erule ssubst, rule card_lessThan)
nipkow@15045
   346
  apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
paulson@14485
   347
  apply (erule subst)
paulson@14485
   348
  apply (rule card_image)
paulson@14485
   349
  apply (rule finite_lessThan)
paulson@14485
   350
  apply (simp add: inj_on_def)
paulson@14485
   351
  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
paulson@14485
   352
  apply arith
paulson@14485
   353
  apply (rule_tac x = "x - l" in exI)
paulson@14485
   354
  apply arith
paulson@14485
   355
  done
paulson@14485
   356
paulson@15047
   357
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"  
paulson@14485
   358
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
paulson@14485
   359
nipkow@15045
   360
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l" 
paulson@14485
   361
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
paulson@14485
   362
nipkow@15045
   363
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
paulson@14485
   364
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
paulson@14485
   365
paulson@14485
   366
subsection {* Intervals of integers *}
paulson@14485
   367
nipkow@15045
   368
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
paulson@14485
   369
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   370
nipkow@15045
   371
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"  
paulson@14485
   372
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
paulson@14485
   373
paulson@14485
   374
lemma atLeastPlusOneLessThan_greaterThanLessThan_int: 
nipkow@15045
   375
    "{l+1..<u} = {l<..<u::int}"  
paulson@14485
   376
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
paulson@14485
   377
paulson@14485
   378
subsubsection {* Finiteness *}
paulson@14485
   379
paulson@14485
   380
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==> 
nipkow@15045
   381
    {(0::int)..<u} = int ` {..<nat u}"
paulson@14485
   382
  apply (unfold image_def lessThan_def)
paulson@14485
   383
  apply auto
paulson@14485
   384
  apply (rule_tac x = "nat x" in exI)
paulson@14485
   385
  apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
paulson@14485
   386
  done
paulson@14485
   387
nipkow@15045
   388
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
paulson@14485
   389
  apply (case_tac "0 \<le> u")
paulson@14485
   390
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   391
  apply (rule finite_imageI)
paulson@14485
   392
  apply auto
nipkow@15045
   393
  apply (subgoal_tac "{0..<u} = {}")
paulson@14485
   394
  apply auto
paulson@14485
   395
  done
paulson@14485
   396
paulson@14485
   397
lemma image_atLeastLessThan_int_shift: 
nipkow@15045
   398
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
paulson@14485
   399
  apply (auto simp add: image_def atLeastLessThan_iff)
paulson@14485
   400
  apply (rule_tac x = "x - l" in bexI)
paulson@14485
   401
  apply auto
paulson@14485
   402
  done
paulson@14485
   403
nipkow@15045
   404
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
nipkow@15045
   405
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   406
  apply (erule subst)
paulson@14485
   407
  apply (rule finite_imageI)
paulson@14485
   408
  apply (rule finite_atLeastZeroLessThan_int)
paulson@14485
   409
  apply (rule image_atLeastLessThan_int_shift)
paulson@14485
   410
  done
paulson@14485
   411
paulson@14485
   412
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}" 
paulson@14485
   413
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
paulson@14485
   414
nipkow@15045
   415
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}" 
paulson@14485
   416
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   417
nipkow@15045
   418
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}" 
paulson@14485
   419
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   420
paulson@14485
   421
subsubsection {* Cardinality *}
paulson@14485
   422
nipkow@15045
   423
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
paulson@14485
   424
  apply (case_tac "0 \<le> u")
paulson@14485
   425
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   426
  apply (subst card_image)
paulson@14485
   427
  apply (auto simp add: inj_on_def)
paulson@14485
   428
  done
paulson@14485
   429
nipkow@15045
   430
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
nipkow@15045
   431
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
paulson@14485
   432
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
nipkow@15045
   433
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   434
  apply (erule subst)
paulson@14485
   435
  apply (rule card_image)
paulson@14485
   436
  apply (rule finite_atLeastZeroLessThan_int)
paulson@14485
   437
  apply (simp add: inj_on_def)
paulson@14485
   438
  apply (rule image_atLeastLessThan_int_shift)
paulson@14485
   439
  done
paulson@14485
   440
paulson@14485
   441
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
paulson@14485
   442
  apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
paulson@14485
   443
  apply (auto simp add: compare_rls)
paulson@14485
   444
  done
paulson@14485
   445
nipkow@15045
   446
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)" 
paulson@14485
   447
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   448
nipkow@15045
   449
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
paulson@14485
   450
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   451
paulson@14485
   452
paulson@13850
   453
subsection {*Lemmas useful with the summation operator setsum*}
paulson@13850
   454
wenzelm@14577
   455
text {* For examples, see Algebra/poly/UnivPoly.thy *}
ballarin@13735
   456
wenzelm@14577
   457
subsubsection {* Disjoint Unions *}
ballarin@13735
   458
wenzelm@14577
   459
text {* Singletons and open intervals *}
ballarin@13735
   460
ballarin@13735
   461
lemma ivl_disj_un_singleton:
nipkow@15045
   462
  "{l::'a::linorder} Un {l<..} = {l..}"
nipkow@15045
   463
  "{..<u} Un {u::'a::linorder} = {..u}"
nipkow@15045
   464
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
nipkow@15045
   465
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
nipkow@15045
   466
  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
nipkow@15045
   467
  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
ballarin@14398
   468
by auto
ballarin@13735
   469
wenzelm@14577
   470
text {* One- and two-sided intervals *}
ballarin@13735
   471
ballarin@13735
   472
lemma ivl_disj_un_one:
nipkow@15045
   473
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
nipkow@15045
   474
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
nipkow@15045
   475
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
nipkow@15045
   476
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
nipkow@15045
   477
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
nipkow@15045
   478
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
nipkow@15045
   479
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
nipkow@15045
   480
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
ballarin@14398
   481
by auto
ballarin@13735
   482
wenzelm@14577
   483
text {* Two- and two-sided intervals *}
ballarin@13735
   484
ballarin@13735
   485
lemma ivl_disj_un_two:
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}"
nipkow@15045
   489
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
nipkow@15045
   490
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
nipkow@15045
   491
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
nipkow@15045
   492
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
nipkow@15045
   493
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
ballarin@14398
   494
by auto
ballarin@13735
   495
ballarin@13735
   496
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
ballarin@13735
   497
wenzelm@14577
   498
subsubsection {* Disjoint Intersections *}
ballarin@13735
   499
wenzelm@14577
   500
text {* Singletons and open intervals *}
ballarin@13735
   501
ballarin@13735
   502
lemma ivl_disj_int_singleton:
nipkow@15045
   503
  "{l::'a::order} Int {l<..} = {}"
nipkow@15045
   504
  "{..<u} Int {u} = {}"
nipkow@15045
   505
  "{l} Int {l<..<u} = {}"
nipkow@15045
   506
  "{l<..<u} Int {u} = {}"
nipkow@15045
   507
  "{l} Int {l<..u} = {}"
nipkow@15045
   508
  "{l..<u} Int {u} = {}"
ballarin@13735
   509
  by simp+
ballarin@13735
   510
wenzelm@14577
   511
text {* One- and two-sided intervals *}
ballarin@13735
   512
ballarin@13735
   513
lemma ivl_disj_int_one:
nipkow@15045
   514
  "{..l::'a::order} Int {l<..<u} = {}"
nipkow@15045
   515
  "{..<l} Int {l..<u} = {}"
nipkow@15045
   516
  "{..l} Int {l<..u} = {}"
nipkow@15045
   517
  "{..<l} Int {l..u} = {}"
nipkow@15045
   518
  "{l<..u} Int {u<..} = {}"
nipkow@15045
   519
  "{l<..<u} Int {u..} = {}"
nipkow@15045
   520
  "{l..u} Int {u<..} = {}"
nipkow@15045
   521
  "{l..<u} Int {u..} = {}"
ballarin@14398
   522
  by auto
ballarin@13735
   523
wenzelm@14577
   524
text {* Two- and two-sided intervals *}
ballarin@13735
   525
ballarin@13735
   526
lemma ivl_disj_int_two:
nipkow@15045
   527
  "{l::'a::order<..<m} Int {m..<u} = {}"
nipkow@15045
   528
  "{l<..m} Int {m<..<u} = {}"
nipkow@15045
   529
  "{l..<m} Int {m..<u} = {}"
nipkow@15045
   530
  "{l..m} Int {m<..<u} = {}"
nipkow@15045
   531
  "{l<..<m} Int {m..u} = {}"
nipkow@15045
   532
  "{l<..m} Int {m<..u} = {}"
nipkow@15045
   533
  "{l..<m} Int {m..u} = {}"
nipkow@15045
   534
  "{l..m} Int {m<..u} = {}"
ballarin@14398
   535
  by auto
ballarin@13735
   536
ballarin@13735
   537
lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
ballarin@13735
   538
nipkow@15041
   539
nipkow@15042
   540
subsection {* Summation indexed over intervals *}
nipkow@15042
   541
nipkow@15042
   542
syntax
nipkow@15042
   543
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   544
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
nipkow@15048
   545
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
nipkow@15042
   546
syntax (xsymbols)
nipkow@15042
   547
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   548
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@15048
   549
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@15042
   550
syntax (HTML output)
nipkow@15042
   551
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048
   552
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@15048
   553
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@15056
   554
syntax (latex_sum output)
nipkow@15052
   555
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   556
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052
   557
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   558
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052
   559
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052
   560
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15041
   561
nipkow@15048
   562
translations
nipkow@15048
   563
  "\<Sum>x=a..b. t" == "setsum (%x. t) {a..b}"
nipkow@15048
   564
  "\<Sum>x=a..<b. t" == "setsum (%x. t) {a..<b}"
nipkow@15048
   565
  "\<Sum>i<n. t" == "setsum (\<lambda>i. t) {..<n}"
nipkow@15041
   566
nipkow@15052
   567
text{* The above introduces some pretty alternative syntaxes for
nipkow@15056
   568
summation over intervals:
nipkow@15052
   569
\begin{center}
nipkow@15052
   570
\begin{tabular}{lll}
nipkow@15056
   571
Old & New & \LaTeX\\
nipkow@15056
   572
@{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
   573
@{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
   574
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
nipkow@15052
   575
\end{tabular}
nipkow@15052
   576
\end{center}
nipkow@15056
   577
The left column shows the term before introduction of the new syntax,
nipkow@15056
   578
the middle column shows the new (default) syntax, and the right column
nipkow@15056
   579
shows a special syntax. The latter is only meaningful for latex output
nipkow@15056
   580
and has to be activated explicitly by setting the print mode to
nipkow@15056
   581
\texttt{latex\_sum} (e.g.\ via \texttt{mode=latex\_sum} in
nipkow@15056
   582
antiquotations). It is not the default \LaTeX\ output because it only
nipkow@15056
   583
works well with italic-style formulae, not tt-style.
nipkow@15052
   584
nipkow@15052
   585
Note that for uniformity on @{typ nat} it is better to use
nipkow@15052
   586
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
nipkow@15052
   587
not provide all lemmas available for @{term"{m..<n}"} also in the
nipkow@15052
   588
special form for @{term"{..<n}"}. *}
nipkow@15052
   589
nipkow@15041
   590
nipkow@15041
   591
lemma Summation_Suc[simp]: "(\<Sum>i < Suc n. b i) = b n + (\<Sum>i < n. b i)"
nipkow@15041
   592
by (simp add:lessThan_Suc)
nipkow@15041
   593
nipkow@8924
   594
end