src/HOL/SetInterval.thy
author bulwahn
Sat, 19 Jul 2008 19:27:13 +0200
changeset 27656 d4f6e64ee7cc
parent 26105 ae06618225ec
child 28068 f6b2d1995171
permissions -rw-r--r--
added verification framework for the HeapMonad and quicksort as example for this framework
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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 Int
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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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(*
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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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*)
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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 (in ord) lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
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by (simp add: lessThan_def)
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lemma Compl_lessThan [simp]:
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    "!!k:: 'a::linorder. -lessThan k = atLeast k"
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apply (auto simp add: lessThan_def atLeast_def)
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done
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lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
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by auto
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lemma (in ord) greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
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by (simp add: greaterThan_def)
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lemma Compl_greaterThan [simp]:
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    "!!k:: 'a::linorder. -greaterThan k = atMost k"
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  by (auto simp add: greaterThan_def atMost_def)
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lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
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apply (subst Compl_greaterThan [symmetric])
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apply (rule double_complement)
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done
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lemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
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by (simp add: atLeast_def)
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lemma Compl_atLeast [simp]:
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    "!!k:: 'a::linorder. -atLeast k = lessThan k"
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  by (auto simp add: lessThan_def atLeast_def)
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lemma (in ord) atMost_iff [iff]: "(i: atMost k) = (i<=k)"
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by (simp add: atMost_def)
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lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
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by (blast intro: order_antisym)
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subsection {* Logical Equivalences for Set Inclusion and Equality *}
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lemma atLeast_subset_iff [iff]:
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     "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
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by (blast intro: order_trans)
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lemma atLeast_eq_iff [iff]:
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     "(atLeast x = atLeast y) = (x = (y::'a::linorder))"
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by (blast intro: order_antisym order_trans)
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lemma greaterThan_subset_iff [iff]:
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     "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
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apply (auto simp add: greaterThan_def)
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 apply (subst linorder_not_less [symmetric], blast)
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done
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lemma greaterThan_eq_iff [iff]:
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     "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
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apply (rule iffI)
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 apply (erule equalityE)
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 apply (simp_all 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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context ord
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begin
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lemma greaterThanLessThan_iff [simp,noatp]:
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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,noatp]:
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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,noatp]:
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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,noatp]:
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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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end
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subsubsection{* Emptyness and singletons *}
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context order
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begin
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lemma atLeastAtMost_empty [simp]: "n < m ==> {m..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} = {}"
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by (auto simp add: atLeastLessThan_def)
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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 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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end
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subsection {* Intervals of natural numbers *}
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subsubsection {* The Constant @{term lessThan} *}
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lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
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by (simp add: lessThan_def)
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lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
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by (simp add: lessThan_def less_Suc_eq, blast)
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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{*The orientation of the following rule is tricky. The lhs is
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defined in terms of the rhs.  Hence the chosen orientation makes sense
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in this theory --- the reverse orientation complicates proofs (eg
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nontermination). But outside, when the definition of the lhs is rarely
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used, the opposite orientation seems preferable because it reduces a
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specific concept to a more general one. *}
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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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declare atLeast0LessThan[symmetric, code unfold]
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lemma atLeastLessThan0: "{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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(*
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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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   326
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lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
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by (auto simp add: atLeastLessThan_def)
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*)
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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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   340
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03d4347b071d integrated Jeremy's FiniteLib
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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 {* Image *}
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lemma image_add_atLeastAtMost:
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  "(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
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proof
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  show "?A \<subseteq> ?B" by auto
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next
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  show "?B \<subseteq> ?A"
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   352
  proof
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    fix n assume a: "n : ?B"
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25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
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   354
    hence "n - k : {i..j}" by auto
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   355
    moreover have "n = (n - k) + k" using a by auto
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   356
    ultimately show "n : ?A" by blast
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  qed
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qed
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   359
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   360
lemma image_add_atLeastLessThan:
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  "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
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   362
proof
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   363
  show "?A \<subseteq> ?B" by auto
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   364
next
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
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   365
  show "?B \<subseteq> ?A"
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   366
  proof
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   367
    fix n assume a: "n : ?B"
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19538
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   368
    hence "n - k : {i..<j}" by auto
16733
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parents: 16102
diff changeset
   369
    moreover have "n = (n - k) + k" using a by auto
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   370
    ultimately show "n : ?A" by blast
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   371
  qed
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   372
qed
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   373
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   374
corollary image_Suc_atLeastAtMost[simp]:
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   375
  "Suc ` {i..j} = {Suc i..Suc j}"
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   376
using image_add_atLeastAtMost[where k=1] by simp
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   377
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   378
corollary image_Suc_atLeastLessThan[simp]:
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   379
  "Suc ` {i..<j} = {Suc i..<Suc j}"
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parents: 16102
diff changeset
   380
using image_add_atLeastLessThan[where k=1] by simp
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parents: 16102
diff changeset
   381
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   382
lemma image_add_int_atLeastLessThan:
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   383
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
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nipkow
parents: 16102
diff changeset
   384
  apply (auto simp add: image_def)
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nipkow
parents: 16102
diff changeset
   385
  apply (rule_tac x = "x - l" in bexI)
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   386
  apply auto
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parents: 16102
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   387
  done
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
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parents: 16102
diff changeset
   388
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
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parents: 16102
diff changeset
   389
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   390
subsubsection {* Finiteness *}
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paulson
parents: 14478
diff changeset
   391
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
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diff changeset
   392
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
14485
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paulson
parents: 14478
diff changeset
   393
  by (induct k) (simp_all add: lessThan_Suc)
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paulson
parents: 14478
diff changeset
   394
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   395
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   396
  by (induct k) (simp_all add: atMost_Suc)
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paulson
parents: 14478
diff changeset
   397
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   398
lemma finite_greaterThanLessThan [iff]:
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d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   399
  fixes l :: nat shows "finite {l<..<u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   400
by (simp add: greaterThanLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   401
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   402
lemma finite_atLeastLessThan [iff]:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   403
  fixes l :: nat shows "finite {l..<u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   404
by (simp add: atLeastLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   405
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   406
lemma finite_greaterThanAtMost [iff]:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   407
  fixes l :: nat shows "finite {l<..u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   408
by (simp add: greaterThanAtMost_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   409
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   410
lemma finite_atLeastAtMost [iff]:
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   411
  fixes l :: nat shows "finite {l..u}"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   412
by (simp add: atLeastAtMost_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   413
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   414
lemma bounded_nat_set_is_finite:
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   415
  "(ALL i:N. i < (n::nat)) ==> finite N"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   416
  -- {* A bounded set of natural numbers is finite. *}
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   417
  apply (rule finite_subset)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   418
   apply (rule_tac [2] finite_lessThan, auto)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   419
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   420
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   421
text{* Any subset of an interval of natural numbers the size of the
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   422
subset is exactly that interval. *}
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   423
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   424
lemma subset_card_intvl_is_intvl:
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   425
  "A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   426
proof cases
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   427
  assume "finite A"
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   428
  thus "PROP ?P"
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   429
  proof(induct A rule:finite_linorder_induct)
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   430
    case empty thus ?case by auto
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   431
  next
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   432
    case (insert A b)
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   433
    moreover hence "b ~: A" by auto
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   434
    moreover have "A <= {k..<k+card A}" and "b = k+card A"
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   435
      using `b ~: A` insert by fastsimp+
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   436
    ultimately show ?case by auto
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   437
  qed
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   438
next
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   439
  assume "~finite A" thus "PROP ?P" by simp
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   440
qed
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   441
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   442
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   443
subsubsection {* Cardinality *}
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   444
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   445
lemma card_lessThan [simp]: "card {..<u} = u"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   446
  by (induct u, simp_all add: lessThan_Suc)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   447
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   448
lemma card_atMost [simp]: "card {..u} = Suc u"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   449
  by (simp add: lessThan_Suc_atMost [THEN sym])
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   450
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   451
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   452
  apply (subgoal_tac "card {l..<u} = card {..<u-l}")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   453
  apply (erule ssubst, rule card_lessThan)
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   454
  apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   455
  apply (erule subst)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   456
  apply (rule card_image)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   457
  apply (simp add: inj_on_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   458
  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   459
  apply (rule_tac x = "x - l" in exI)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   460
  apply arith
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   461
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   462
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   463
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   464
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   465
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   466
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   467
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   468
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   469
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   470
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   471
26105
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   472
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   473
lemma ex_bij_betw_nat_finite:
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   474
  "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
   475
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
   476
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
   477
done
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   478
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   479
lemma ex_bij_betw_finite_nat:
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   480
  "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
   481
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
   482
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 26072
diff changeset
   483
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   484
subsection {* Intervals of integers *}
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   485
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   486
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   487
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   488
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   489
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   490
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   491
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   492
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   493
    "{l+1..<u} = {l<..<u::int}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   494
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   495
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   496
subsubsection {* Finiteness *}
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   497
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   498
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   499
    {(0::int)..<u} = int ` {..<nat u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   500
  apply (unfold image_def lessThan_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   501
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   502
  apply (rule_tac x = "nat x" in exI)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   503
  apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   504
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   505
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   506
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   507
  apply (case_tac "0 \<le> u")
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   508
  apply (subst image_atLeastZeroLessThan_int, assumption)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   509
  apply (rule finite_imageI)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   510
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   511
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   512
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   513
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   514
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   515
  apply (erule subst)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   516
  apply (rule finite_imageI)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   517
  apply (rule finite_atLeastZeroLessThan_int)
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   518
  apply (rule image_add_int_atLeastLessThan)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   519
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   520
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   521
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   522
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   523
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   524
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   525
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   526
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   527
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   528
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   529
24853
aab5798e5a33 added lemmas
nipkow
parents: 24748
diff changeset
   530
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   531
subsubsection {* Cardinality *}
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   532
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   533
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   534
  apply (case_tac "0 \<le> u")
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   535
  apply (subst image_atLeastZeroLessThan_int, assumption)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   536
  apply (subst card_image)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   537
  apply (auto simp add: inj_on_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   538
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   539
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   540
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
   541
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   542
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   543
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   544
  apply (erule subst)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   545
  apply (rule card_image)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   546
  apply (simp add: inj_on_def)
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   547
  apply (rule image_add_int_atLeastLessThan)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   548
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   549
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   550
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   551
  apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   552
  apply (auto simp add: compare_rls)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   553
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   554
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   555
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   556
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   557
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   558
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   559
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   560
27656
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   561
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
   562
proof -
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   563
  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
   564
  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
   565
qed
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   566
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   567
lemma card_less:
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   568
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
   569
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
   570
proof -
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   571
  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
   572
  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
   573
qed
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   574
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   575
lemma card_less_Suc2: "0 \<notin> M \<Longrightarrow> 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
   576
apply (rule card_bij_eq [of "Suc" _ _ "\<lambda>x. x - 1"])
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   577
apply simp
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   578
apply fastsimp
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   579
apply auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   580
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
   581
apply auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   582
apply (case_tac x)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   583
apply auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   584
apply (case_tac xa)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   585
apply auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   586
apply (case_tac xa)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   587
apply auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   588
apply (auto simp add: finite_M_bounded_by_nat)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   589
done
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   590
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   591
lemma card_less_Suc:
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   592
  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
   593
    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
   594
proof -
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   595
  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
   596
  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
   597
    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
   598
  have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   599
  from finite_M_bounded_by_nat[of "\<lambda>x. x \<in> M" "Suc i"] have "Suc (card {k. Suc k \<in> M \<and> k < i}) = card (insert 0 ({k \<in> M. k < Suc i} - {0}))"
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   600
    apply (subst card_insert)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   601
    apply simp_all
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   602
    apply (subst b)
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   603
    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
   604
    apply simp_all
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   605
    done
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   606
  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
   607
qed
d4f6e64ee7cc added verification framework for the HeapMonad and quicksort as example for this framework
bulwahn
parents: 26105
diff changeset
   608
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   609
13850
6d1bb3059818 new logical equivalences
paulson
parents: 13735
diff changeset
   610
subsection {*Lemmas useful with the summation operator setsum*}
6d1bb3059818 new logical equivalences
paulson
parents: 13735
diff changeset
   611
16102
c5f6726d9bb1 Locale expressions: rename with optional mixfix syntax.
ballarin
parents: 16052
diff changeset
   612
text {* For examples, see Algebra/poly/UnivPoly2.thy *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   613
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   614
subsubsection {* Disjoint Unions *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   615
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   616
text {* Singletons and open intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   617
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   618
lemma ivl_disj_un_singleton:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   619
  "{l::'a::linorder} Un {l<..} = {l..}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   620
  "{..<u} Un {u::'a::linorder} = {..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   621
  "(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
   622
  "(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
   623
  "(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
   624
  "(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
   625
by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   626
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   627
text {* One- and two-sided intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   628
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   629
lemma ivl_disj_un_one:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   630
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   631
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   632
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   633
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   634
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   635
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   636
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   637
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
   638
by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   639
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   640
text {* Two- and two-sided intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   641
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   642
lemma ivl_disj_un_two:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   643
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   644
  "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   645
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   646
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   647
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   648
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   649
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   650
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
   651
by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   652
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   653
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   654
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   655
subsubsection {* Disjoint Intersections *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   656
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   657
text {* Singletons and open intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   658
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   659
lemma ivl_disj_int_singleton:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   660
  "{l::'a::order} Int {l<..} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   661
  "{..<u} Int {u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   662
  "{l} Int {l<..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   663
  "{l<..<u} Int {u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   664
  "{l} Int {l<..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   665
  "{l..<u} Int {u} = {}"
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   666
  by simp+
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   667
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   668
text {* One- and two-sided intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   669
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   670
lemma ivl_disj_int_one:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   671
  "{..l::'a::order} Int {l<..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   672
  "{..<l} Int {l..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   673
  "{..l} Int {l<..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   674
  "{..<l} Int {l..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   675
  "{l<..u} Int {u<..} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   676
  "{l<..<u} Int {u..} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   677
  "{l..u} Int {u<..} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   678
  "{l..<u} Int {u..} = {}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
   679
  by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   680
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   681
text {* Two- and two-sided intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   682
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   683
lemma ivl_disj_int_two:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   684
  "{l::'a::order<..<m} Int {m..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   685
  "{l<..m} Int {m<..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   686
  "{l..<m} Int {m..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   687
  "{l..m} Int {m<..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   688
  "{l<..<m} Int {m..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   689
  "{l<..m} Int {m<..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   690
  "{l..<m} Int {m..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   691
  "{l..m} Int {m<..u} = {}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
   692
  by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   693
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   694
lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   695
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   696
subsubsection {* Some Differences *}
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   697
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   698
lemma ivl_diff[simp]:
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   699
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   700
by(auto)
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   701
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   702
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   703
subsubsection {* Some Subset Conditions *}
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   704
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23496
diff changeset
   705
lemma ivl_subset [simp,noatp]:
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   706
 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   707
apply(auto simp:linorder_not_le)
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   708
apply(rule ccontr)
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   709
apply(insert linorder_le_less_linear[of i n])
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   710
apply(clarsimp simp:linorder_not_le)
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   711
apply(fastsimp)
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   712
done
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   713
15041
a6b1f0cef7b3 Got rid of Summation and made it a translation into setsum instead.
nipkow
parents: 14846
diff changeset
   714
15042
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
   715
subsection {* Summation indexed over intervals *}
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
   716
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
   717
syntax
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
   718
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
15048
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
   719
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   720
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   721
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
15042
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
   722
syntax (xsymbols)
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
   723
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
15048
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
   724
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   725
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   726
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
15042
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
   727
syntax (HTML output)
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
   728
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
15048
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
   729
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   730
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   731
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   732
syntax (latex_sum output)
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   733
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   734
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   735
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   736
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   737
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   738
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   739
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   740
 ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
15041
a6b1f0cef7b3 Got rid of Summation and made it a translation into setsum instead.
nipkow
parents: 14846
diff changeset
   741
15048
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
   742
translations
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
   743
  "\<Sum>x=a..b. t" == "setsum (%x. t) {a..b}"
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
   744
  "\<Sum>x=a..<b. t" == "setsum (%x. t) {a..<b}"
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   745
  "\<Sum>i\<le>n. t" == "setsum (\<lambda>i. t) {..n}"
15048
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
   746
  "\<Sum>i<n. t" == "setsum (\<lambda>i. t) {..<n}"
15041
a6b1f0cef7b3 Got rid of Summation and made it a translation into setsum instead.
nipkow
parents: 14846
diff changeset
   747
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   748
text{* The above introduces some pretty alternative syntaxes for
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   749
summation over intervals:
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   750
\begin{center}
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   751
\begin{tabular}{lll}
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   752
Old & New & \LaTeX\\
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   753
@{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   754
@{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   755
@{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   756
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   757
\end{tabular}
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   758
\end{center}
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   759
The left column shows the term before introduction of the new syntax,
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   760
the middle column shows the new (default) syntax, and the right column
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   761
shows a special syntax. The latter is only meaningful for latex output
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   762
and has to be activated explicitly by setting the print mode to
21502
7f3ea2b3bab6 prefer antiquotations over LaTeX macros;
wenzelm
parents: 20217
diff changeset
   763
@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   764
antiquotations). It is not the default \LaTeX\ output because it only
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   765
works well with italic-style formulae, not tt-style.
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   766
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   767
Note that for uniformity on @{typ nat} it is better to use
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   768
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   769
not provide all lemmas available for @{term"{m..<n}"} also in the
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   770
special form for @{term"{..<n}"}. *}
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   771
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   772
text{* This congruence rule should be used for sums over intervals as
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   773
the standard theorem @{text[source]setsum_cong} does not work well
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   774
with the simplifier who adds the unsimplified premise @{term"x:B"} to
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   775
the context. *}
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   776
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   777
lemma setsum_ivl_cong:
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   778
 "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   779
 setsum f {a..<b} = setsum g {c..<d}"
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   780
by(rule setsum_cong, simp_all)
15041
a6b1f0cef7b3 Got rid of Summation and made it a translation into setsum instead.
nipkow
parents: 14846
diff changeset
   781
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   782
(* FIXME why are the following simp rules but the corresponding eqns
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   783
on intervals are not? *)
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   784
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   785
lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   786
by (simp add:atMost_Suc add_ac)
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   787
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   788
lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   789
by (simp add:lessThan_Suc add_ac)
15041
a6b1f0cef7b3 Got rid of Summation and made it a translation into setsum instead.
nipkow
parents: 14846
diff changeset
   790
15911
b730b0edc085 turned 2 lemmas into simp rules
nipkow
parents: 15561
diff changeset
   791
lemma setsum_cl_ivl_Suc[simp]:
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
   792
  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
   793
by (auto simp:add_ac atLeastAtMostSuc_conv)
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
   794
15911
b730b0edc085 turned 2 lemmas into simp rules
nipkow
parents: 15561
diff changeset
   795
lemma setsum_op_ivl_Suc[simp]:
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
   796
  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
   797
by (auto simp:add_ac atLeastLessThanSuc)
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   798
(*
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
   799
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
   800
    (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
   801
by (auto simp:add_ac atLeastAtMostSuc_conv)
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   802
*)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   803
lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   804
  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   805
by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   806
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   807
lemma setsum_diff_nat_ivl:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   808
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   809
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   810
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   811
using setsum_add_nat_ivl [of m n p f,symmetric]
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   812
apply (simp add: add_ac)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   813
done
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   814
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   815
subsection{* Shifting bounds *}
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   816
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   817
lemma setsum_shift_bounds_nat_ivl:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   818
  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   819
by (induct "n", auto simp:atLeastLessThanSuc)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   820
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   821
lemma setsum_shift_bounds_cl_nat_ivl:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   822
  "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   823
apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   824
apply (simp add:image_add_atLeastAtMost o_def)
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   825
done
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   826
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   827
corollary setsum_shift_bounds_cl_Suc_ivl:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   828
  "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   829
by (simp add:setsum_shift_bounds_cl_nat_ivl[where k=1,simplified])
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   830
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   831
corollary setsum_shift_bounds_Suc_ivl:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   832
  "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   833
by (simp add:setsum_shift_bounds_nat_ivl[where k=1,simplified])
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   834
19106
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   835
lemma setsum_head:
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   836
  fixes n :: nat
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   837
  assumes mn: "m <= n" 
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   838
  shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   839
proof -
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   840
  from mn
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   841
  have "{m..n} = {m} \<union> {m<..n}"
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   842
    by (auto intro: ivl_disj_un_singleton)
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   843
  hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   844
    by (simp add: atLeast0LessThan)
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   845
  also have "\<dots> = ?rhs" by simp
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   846
  finally show ?thesis .
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   847
qed
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   848
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   849
lemma setsum_head_upt:
19022
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   850
  fixes m::nat
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   851
  assumes m: "0 < m"
19106
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   852
  shows "(\<Sum>x<m. P x) = P 0 + (\<Sum>x\<in>{1..<m}. P x)"
19022
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   853
proof -
19106
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   854
  have "(\<Sum>x<m. P x) = (\<Sum>x\<in>{0..<m}. P x)" 
19022
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   855
    by (simp add: atLeast0LessThan)
19106
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   856
  also 
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   857
  from m 
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   858
  have "\<dots> = (\<Sum>x\<in>{0..m - 1}. P x)"
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   859
    by (cases m) (auto simp add: atLeastLessThanSuc_atLeastAtMost)
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   860
  also
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   861
  have "\<dots> = P 0 + (\<Sum>x\<in>{0<..m - 1}. P x)"
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   862
    by (simp add: setsum_head)
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   863
  also 
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   864
  from m 
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   865
  have "{0<..m - 1} = {1..<m}" 
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   866
    by (cases m) (auto simp add: atLeastLessThanSuc_atLeastAtMost)
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   867
  finally show ?thesis .
19022
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   868
qed
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   869
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   870
subsection {* The formula for geometric sums *}
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   871
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   872
lemma geometric_sum:
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   873
  "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =
22713
3ea6c1cb3dab generalized type of lemma geometric_sum
huffman
parents: 21502
diff changeset
   874
  (x ^ n - 1) / (x - 1::'a::{field, recpower})"
23496
84e9216a6d0e removed redundant lemmas
nipkow
parents: 23477
diff changeset
   875
by (induct "n") (simp_all add:field_simps power_Suc)
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   876
19469
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   877
subsection {* The formula for arithmetic sums *}
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   878
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   879
lemma gauss_sum:
23277
aa158e145ea3 generalize class constraints on some lemmas
huffman
parents: 22713
diff changeset
   880
  "((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
19469
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   881
   of_nat n*((of_nat n)+1)"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   882
proof (induct n)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   883
  case 0
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   884
  show ?case by simp
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   885
next
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   886
  case (Suc n)
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23431
diff changeset
   887
  then show ?case by (simp add: ring_simps)
19469
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   888
qed
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   889
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   890
theorem arith_series_general:
23277
aa158e145ea3 generalize class constraints on some lemmas
huffman
parents: 22713
diff changeset
   891
  "((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
19469
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   892
  of_nat n * (a + (a + of_nat(n - 1)*d))"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   893
proof cases
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   894
  assume ngt1: "n > 1"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   895
  let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   896
  have
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   897
    "(\<Sum>i\<in>{..<n}. a+?I i*d) =
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   898
     ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   899
    by (rule setsum_addf)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   900
  also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   901
  also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   902
    by (simp add: setsum_right_distrib setsum_head_upt mult_ac)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   903
  also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   904
    by (simp add: left_distrib right_distrib)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   905
  also from ngt1 have "{1..<n} = {1..n - 1}"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   906
    by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)    
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   907
  also from ngt1 
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   908
  have "(1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..n - 1}. ?I i) = ((1+1)*?n*a + d*?I (n - 1)*?I n)"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   909
    by (simp only: mult_ac gauss_sum [of "n - 1"])
23431
25ca91279a9b change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents: 23413
diff changeset
   910
       (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
19469
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   911
  finally show ?thesis by (simp add: mult_ac add_ac right_distrib)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   912
next
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   913
  assume "\<not>(n > 1)"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   914
  hence "n = 1 \<or> n = 0" by auto
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   915
  thus ?thesis by (auto simp: mult_ac right_distrib)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   916
qed
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   917
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   918
lemma arith_series_nat:
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   919
  "Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   920
proof -
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   921
  have
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   922
    "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   923
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   924
    by (rule arith_series_general)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   925
  thus ?thesis by (auto simp add: of_nat_id)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   926
qed
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   927
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   928
lemma arith_series_int:
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   929
  "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   930
  of_nat n * (a + (a + of_nat(n - 1)*d))"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   931
proof -
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   932
  have
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   933
    "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   934
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   935
    by (rule arith_series_general)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   936
  thus ?thesis by simp
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   937
qed
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   938
19022
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   939
lemma sum_diff_distrib:
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   940
  fixes P::"nat\<Rightarrow>nat"
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   941
  shows
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   942
  "\<forall>x. Q x \<le> P x  \<Longrightarrow>
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   943
  (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   944
proof (induct n)
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   945
  case 0 show ?case by simp
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   946
next
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   947
  case (Suc n)
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   948
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   949
  let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   950
  let ?rhs = "\<Sum>x<n. P x - Q x"
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   951
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   952
  from Suc have "?lhs = ?rhs" by simp
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   953
  moreover
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   954
  from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   955
  moreover
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   956
  from Suc have
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   957
    "(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   958
    by (subst diff_diff_left[symmetric],
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   959
        subst diff_add_assoc2)
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   960
       (auto simp: diff_add_assoc2 intro: setsum_mono)
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   961
  ultimately
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   962
  show ?case by simp
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   963
qed
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   964
8924
c434283b4cfa Added SetInterval
nipkow
parents:
diff changeset
   965
end