src/HOL/SetInterval.thy
author kleing
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(*  Title:      HOL/SetInterval.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow and Clemens Ballarin
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                Additions by Jeremy Avigad in March 2004
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    Copyright   2000  TU Muenchen
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lessThan, greaterThan, atLeast, atMost and two-sided intervals
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*)
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header {* Set intervals *}
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theory SetInterval
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imports IntArith
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begin
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constdefs
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  lessThan    :: "('a::ord) => 'a set"	("(1{..<_})")
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  "{..<u} == {x. x<u}"
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  atMost      :: "('a::ord) => 'a set"	("(1{.._})")
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  "{..u} == {x. x<=u}"
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  greaterThan :: "('a::ord) => 'a set"	("(1{_<..})")
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  "{l<..} == {x. l<x}"
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  atLeast     :: "('a::ord) => 'a set"	("(1{_..})")
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  "{l..} == {x. l<=x}"
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  greaterThanLessThan :: "['a::ord, 'a] => 'a set"  ("(1{_<..<_})")
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  "{l<..<u} == {l<..} Int {..<u}"
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  atLeastLessThan :: "['a::ord, 'a] => 'a set"      ("(1{_..<_})")
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  "{l..<u} == {l..} Int {..<u}"
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  greaterThanAtMost :: "['a::ord, 'a] => 'a set"    ("(1{_<.._})")
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  "{l<..u} == {l<..} Int {..u}"
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  atLeastAtMost :: "['a::ord, 'a] => 'a set"        ("(1{_.._})")
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  "{l..u} == {l..} Int {..u}"
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(* Old syntax, no longer supported:
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syntax
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  "_lessThan"    :: "('a::ord) => 'a set"	("(1{.._'(})")
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  "_greaterThan" :: "('a::ord) => 'a set"	("(1{')_..})")
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  "_greaterThanLessThan" :: "['a::ord, 'a] => 'a set"  ("(1{')_.._'(})")
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  "_atLeastLessThan" :: "['a::ord, 'a] => 'a set"      ("(1{_.._'(})")
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  "_greaterThanAtMost" :: "['a::ord, 'a] => 'a set"    ("(1{')_.._})")
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translations
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  "{..m(}" => "{..<m}"
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  "{)m..}" => "{m<..}"
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  "{)m..n(}" => "{m<..<n}"
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  "{m..n(}" => "{m..<n}"
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  "{)m..n}" => "{m<..n}"
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*)
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text{* A note of warning when using @{term"{..<n}"} on type @{typ
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nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
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@{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
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syntax
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  "@UNION_le"   :: "nat => nat => 'b set => 'b set"       ("(3UN _<=_./ _)" 10)
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  "@UNION_less" :: "nat => nat => 'b set => 'b set"       ("(3UN _<_./ _)" 10)
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  "@INTER_le"   :: "nat => nat => 'b set => 'b set"       ("(3INT _<=_./ _)" 10)
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  "@INTER_less" :: "nat => nat => 'b set => 'b set"       ("(3INT _<_./ _)" 10)
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syntax (input)
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  "@UNION_le"   :: "nat => nat => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" 10)
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  "@UNION_less" :: "nat => nat => 'b set => 'b set"       ("(3\<Union> _<_./ _)" 10)
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  "@INTER_le"   :: "nat => nat => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" 10)
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  "@INTER_less" :: "nat => nat => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" 10)
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syntax (xsymbols)
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  "@UNION_le"   :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Union>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
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  "@UNION_less" :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Union>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
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  "@INTER_le"   :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Inter>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
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  "@INTER_less" :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Inter>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
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translations
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  "UN i<=n. A"  == "UN i:{..n}. A"
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  "UN i<n. A"   == "UN i:{..<n}. A"
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  "INT i<=n. A" == "INT i:{..n}. A"
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  "INT i<n. A"  == "INT i:{..<n}. A"
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subsection {* Various equivalences *}
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lemma lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
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by (simp add: lessThan_def)
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lemma Compl_lessThan [simp]:
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    "!!k:: 'a::linorder. -lessThan k = atLeast k"
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apply (auto simp add: lessThan_def atLeast_def)
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done
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lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
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by auto
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lemma greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
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by (simp add: greaterThan_def)
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lemma Compl_greaterThan [simp]:
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    "!!k:: 'a::linorder. -greaterThan k = atMost k"
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apply (simp add: greaterThan_def atMost_def le_def, auto)
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done
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lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
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apply (subst Compl_greaterThan [symmetric])
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apply (rule double_complement)
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done
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lemma atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
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by (simp add: atLeast_def)
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lemma Compl_atLeast [simp]:
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    "!!k:: 'a::linorder. -atLeast k = lessThan k"
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apply (simp add: lessThan_def atLeast_def le_def, auto)
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done
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lemma atMost_iff [iff]: "(i: atMost k) = (i<=k)"
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by (simp add: atMost_def)
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lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
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by (blast intro: order_antisym)
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subsection {* Logical Equivalences for Set Inclusion and Equality *}
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lemma atLeast_subset_iff [iff]:
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     "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
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by (blast intro: order_trans)
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lemma atLeast_eq_iff [iff]:
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     "(atLeast x = atLeast y) = (x = (y::'a::linorder))"
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by (blast intro: order_antisym order_trans)
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lemma greaterThan_subset_iff [iff]:
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     "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
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apply (auto simp add: greaterThan_def)
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 apply (subst linorder_not_less [symmetric], blast)
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done
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lemma greaterThan_eq_iff [iff]:
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     "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
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apply (rule iffI)
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 apply (erule equalityE)
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 apply (simp_all add: greaterThan_subset_iff)
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done
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lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))"
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by (blast intro: order_trans)
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lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"
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by (blast intro: order_antisym order_trans)
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lemma lessThan_subset_iff [iff]:
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     "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
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apply (auto simp add: lessThan_def)
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 apply (subst linorder_not_less [symmetric], blast)
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done
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lemma lessThan_eq_iff [iff]:
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     "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
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apply (rule iffI)
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 apply (erule equalityE)
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 apply (simp_all add: lessThan_subset_iff)
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done
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subsection {*Two-sided intervals*}
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lemma greaterThanLessThan_iff [simp]:
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  "(i : {l<..<u}) = (l < i & i < u)"
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by (simp add: greaterThanLessThan_def)
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lemma atLeastLessThan_iff [simp]:
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  "(i : {l..<u}) = (l <= i & i < u)"
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by (simp add: atLeastLessThan_def)
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lemma greaterThanAtMost_iff [simp]:
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  "(i : {l<..u}) = (l < i & i <= u)"
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by (simp add: greaterThanAtMost_def)
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lemma atLeastAtMost_iff [simp]:
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  "(i : {l..u}) = (l <= i & i <= u)"
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by (simp add: atLeastAtMost_def)
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text {* The above four lemmas could be declared as iffs.
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  If we do so, a call to blast in Hyperreal/Star.ML, lemma @{text STAR_Int}
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  seems to take forever (more than one hour). *}
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subsubsection{* Emptyness and singletons *}
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lemma atLeastAtMost_empty [simp]: "n < m ==> {m::'a::order..n} = {}";
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  by (auto simp add: atLeastAtMost_def atMost_def atLeast_def);
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lemma atLeastLessThan_empty[simp]: "n \<le> m ==> {m..<n::'a::order} = {}"
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by (auto simp add: atLeastLessThan_def)
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lemma greaterThanAtMost_empty[simp]:"l \<le> k ==> {k<..(l::'a::order)} = {}"
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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::'a::order)} = {}"
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by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
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lemma atLeastAtMost_singleton [simp]: "{a::'a::order..a} = {a}";
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by (auto simp add: atLeastAtMost_def atMost_def atLeast_def);
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subsection {* Intervals of natural numbers *}
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subsubsection {* The Constant @{term lessThan} *}
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lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
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by (simp add: lessThan_def)
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lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
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by (simp add: lessThan_def less_Suc_eq, blast)
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lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
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by (simp add: lessThan_def atMost_def less_Suc_eq_le)
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lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
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by blast
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subsubsection {* The Constant @{term greaterThan} *}
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lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
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apply (simp add: greaterThan_def)
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apply (blast dest: gr0_conv_Suc [THEN iffD1])
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done
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lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
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apply (simp add: greaterThan_def)
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apply (auto elim: linorder_neqE)
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done
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lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
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by blast
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subsubsection {* The Constant @{term atLeast} *}
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lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
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by (unfold atLeast_def UNIV_def, simp)
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lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
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apply (simp add: atLeast_def)
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apply (simp add: Suc_le_eq)
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apply (simp add: order_le_less, blast)
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done
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lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
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  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
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lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
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by blast
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subsubsection {* The Constant @{term atMost} *}
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lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
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by (simp add: atMost_def)
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lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
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apply (simp add: atMost_def)
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apply (simp add: less_Suc_eq order_le_less, blast)
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done
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lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
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by blast
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subsubsection {* The Constant @{term atLeastLessThan} *}
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text{*But not a simprule because some concepts are better left in terms
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  of @{term atLeastLessThan}*}
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lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
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by(simp add:lessThan_def atLeastLessThan_def)
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(*
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lemma atLeastLessThan0 [simp]: "{m..<0::nat} = {}"
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by (simp add: atLeastLessThan_def)
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*)
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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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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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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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  proof
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    fix n assume a: "n : ?B"
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    hence "n - k : {i..j}" by auto arith+
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    moreover have "n = (n - k) + k" using a by auto
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    ultimately show "n : ?A" by blast
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  qed
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qed
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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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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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  proof
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    fix n assume a: "n : ?B"
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    hence "n - k : {i..<j}" by auto arith+
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    moreover have "n = (n - k) + k" using a by auto
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    ultimately show "n : ?A" by blast
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  qed
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qed
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corollary image_Suc_atLeastAtMost[simp]:
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  "Suc ` {i..j} = {Suc i..Suc j}"
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using image_add_atLeastAtMost[where k=1] by simp
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corollary image_Suc_atLeastLessThan[simp]:
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  "Suc ` {i..<j} = {Suc i..<Suc j}"
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using image_add_atLeastLessThan[where k=1] by simp
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   346
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lemma image_add_int_atLeastLessThan:
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    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
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  apply (auto simp add: image_def)
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  apply (rule_tac x = "x - l" in bexI)
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  apply auto
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   352
  done
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   353
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   354
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subsubsection {* Finiteness *}
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   356
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lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
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  by (induct k) (simp_all add: lessThan_Suc)
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   359
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   360
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
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   361
  by (induct k) (simp_all add: atMost_Suc)
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parents: 14478
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   362
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   363
lemma finite_greaterThanLessThan [iff]:
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   364
  fixes l :: nat shows "finite {l<..<u}"
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   365
by (simp add: greaterThanLessThan_def)
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diff changeset
   366
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   367
lemma finite_atLeastLessThan [iff]:
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   368
  fixes l :: nat shows "finite {l..<u}"
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   369
by (simp add: atLeastLessThan_def)
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   370
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   371
lemma finite_greaterThanAtMost [iff]:
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   372
  fixes l :: nat shows "finite {l<..u}"
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   373
by (simp add: greaterThanAtMost_def)
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diff changeset
   374
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   375
lemma finite_atLeastAtMost [iff]:
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   376
  fixes l :: nat shows "finite {l..u}"
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parents: 14478
diff changeset
   377
by (simp add: atLeastAtMost_def)
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diff changeset
   378
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   379
lemma bounded_nat_set_is_finite:
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   380
    "(ALL i:N. i < (n::nat)) ==> finite N"
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   381
  -- {* A bounded set of natural numbers is finite. *}
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   382
  apply (rule finite_subset)
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parents: 14478
diff changeset
   383
   apply (rule_tac [2] finite_lessThan, auto)
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paulson
parents: 14478
diff changeset
   384
  done
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diff changeset
   385
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   386
subsubsection {* Cardinality *}
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diff changeset
   387
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   388
lemma card_lessThan [simp]: "card {..<u} = u"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   389
  by (induct u, simp_all add: lessThan_Suc)
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parents: 14478
diff changeset
   390
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parents: 14478
diff changeset
   391
lemma card_atMost [simp]: "card {..u} = Suc u"
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parents: 14478
diff changeset
   392
  by (simp add: lessThan_Suc_atMost [THEN sym])
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paulson
parents: 14478
diff changeset
   393
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
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diff changeset
   394
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
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diff changeset
   395
  apply (subgoal_tac "card {l..<u} = card {..<u-l}")
14485
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parents: 14478
diff changeset
   396
  apply (erule ssubst, rule card_lessThan)
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
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parents: 15042
diff changeset
   397
  apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
14485
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parents: 14478
diff changeset
   398
  apply (erule subst)
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paulson
parents: 14478
diff changeset
   399
  apply (rule card_image)
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paulson
parents: 14478
diff changeset
   400
  apply (simp add: inj_on_def)
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paulson
parents: 14478
diff changeset
   401
  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
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paulson
parents: 14478
diff changeset
   402
  apply arith
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paulson
parents: 14478
diff changeset
   403
  apply (rule_tac x = "x - l" in exI)
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paulson
parents: 14478
diff changeset
   404
  apply arith
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paulson
parents: 14478
diff changeset
   405
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   406
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   407
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
14485
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paulson
parents: 14478
diff changeset
   408
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
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paulson
parents: 14478
diff changeset
   409
15418
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paulson
parents: 15402
diff changeset
   410
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
14485
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paulson
parents: 14478
diff changeset
   411
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
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paulson
parents: 14478
diff changeset
   412
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   413
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
14485
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paulson
parents: 14478
diff changeset
   414
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
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paulson
parents: 14478
diff changeset
   415
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   416
subsection {* Intervals of integers *}
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paulson
parents: 14478
diff changeset
   417
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   418
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
14485
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paulson
parents: 14478
diff changeset
   419
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
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paulson
parents: 14478
diff changeset
   420
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   421
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   422
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
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paulson
parents: 14478
diff changeset
   423
15418
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paulson
parents: 15402
diff changeset
   424
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   425
    "{l+1..<u} = {l<..<u::int}"
14485
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paulson
parents: 14478
diff changeset
   426
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
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paulson
parents: 14478
diff changeset
   427
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   428
subsubsection {* Finiteness *}
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   429
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   430
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   431
    {(0::int)..<u} = int ` {..<nat u}"
14485
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paulson
parents: 14478
diff changeset
   432
  apply (unfold image_def lessThan_def)
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paulson
parents: 14478
diff changeset
   433
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   434
  apply (rule_tac x = "nat x" in exI)
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paulson
parents: 14478
diff changeset
   435
  apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   436
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   437
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   438
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
14485
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paulson
parents: 14478
diff changeset
   439
  apply (case_tac "0 \<le> u")
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paulson
parents: 14478
diff changeset
   440
  apply (subst image_atLeastZeroLessThan_int, assumption)
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paulson
parents: 14478
diff changeset
   441
  apply (rule finite_imageI)
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paulson
parents: 14478
diff changeset
   442
  apply auto
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   443
  done
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 finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   446
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
14485
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paulson
parents: 14478
diff changeset
   447
  apply (erule subst)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   448
  apply (rule finite_imageI)
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paulson
parents: 14478
diff changeset
   449
  apply (rule finite_atLeastZeroLessThan_int)
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   450
  apply (rule image_add_int_atLeastLessThan)
14485
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paulson
parents: 14478
diff changeset
   451
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   452
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   453
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   454
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   455
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   456
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   457
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   458
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   459
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   460
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   461
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   462
subsubsection {* Cardinality *}
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   463
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   464
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   465
  apply (case_tac "0 \<le> u")
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   466
  apply (subst image_atLeastZeroLessThan_int, assumption)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   467
  apply (subst card_image)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   468
  apply (auto simp add: inj_on_def)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   469
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   470
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   471
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
   472
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   473
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   474
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   475
  apply (erule subst)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   476
  apply (rule card_image)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   477
  apply (simp add: inj_on_def)
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   478
  apply (rule image_add_int_atLeastLessThan)
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   479
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   480
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   481
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   482
  apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   483
  apply (auto simp add: compare_rls)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   484
  done
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   485
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   486
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   487
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   488
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   489
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
14485
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   490
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   491
ea2707645af8 new material from Avigad
paulson
parents: 14478
diff changeset
   492
13850
6d1bb3059818 new logical equivalences
paulson
parents: 13735
diff changeset
   493
subsection {*Lemmas useful with the summation operator setsum*}
6d1bb3059818 new logical equivalences
paulson
parents: 13735
diff changeset
   494
16102
c5f6726d9bb1 Locale expressions: rename with optional mixfix syntax.
ballarin
parents: 16052
diff changeset
   495
text {* For examples, see Algebra/poly/UnivPoly2.thy *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   496
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   497
subsubsection {* Disjoint Unions *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   498
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   499
text {* Singletons and open intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   500
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   501
lemma ivl_disj_un_singleton:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   502
  "{l::'a::linorder} Un {l<..} = {l..}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   503
  "{..<u} Un {u::'a::linorder} = {..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   504
  "(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
   505
  "(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
   506
  "(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
   507
  "(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
   508
by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   509
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   510
text {* One- and two-sided intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   511
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   512
lemma ivl_disj_un_one:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   513
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   514
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   515
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   516
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   517
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   518
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   519
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   520
  "(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
   521
by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   522
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   523
text {* Two- and two-sided intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   524
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   525
lemma ivl_disj_un_two:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   526
  "[| (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
   527
  "[| (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
   528
  "[| (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
   529
  "[| (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
   530
  "[| (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
   531
  "[| (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
   532
  "[| (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
   533
  "[| (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
   534
by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   535
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   536
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
   537
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   538
subsubsection {* Disjoint Intersections *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   539
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   540
text {* Singletons and open intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   541
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   542
lemma ivl_disj_int_singleton:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   543
  "{l::'a::order} Int {l<..} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   544
  "{..<u} Int {u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   545
  "{l} Int {l<..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   546
  "{l<..<u} Int {u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   547
  "{l} Int {l<..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   548
  "{l..<u} Int {u} = {}"
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   549
  by simp+
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   550
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   551
text {* One- and two-sided intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   552
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   553
lemma ivl_disj_int_one:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   554
  "{..l::'a::order} Int {l<..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   555
  "{..<l} Int {l..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   556
  "{..l} Int {l<..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   557
  "{..<l} Int {l..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   558
  "{l<..u} Int {u<..} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   559
  "{l<..<u} Int {u..} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   560
  "{l..u} Int {u<..} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   561
  "{l..<u} Int {u..} = {}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
   562
  by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   563
14577
dbb95b825244 tuned document;
wenzelm
parents: 14485
diff changeset
   564
text {* Two- and two-sided intervals *}
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   565
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   566
lemma ivl_disj_int_two:
15045
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   567
  "{l::'a::order<..<m} Int {m..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   568
  "{l<..m} Int {m<..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   569
  "{l..<m} Int {m..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   570
  "{l..m} Int {m<..<u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   571
  "{l<..<m} Int {m..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   572
  "{l<..m} Int {m<..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   573
  "{l..<m} Int {m..u} = {}"
d59f7e2e18d3 Moved to new m<..<n syntax for set intervals.
nipkow
parents: 15042
diff changeset
   574
  "{l..m} Int {m<..u} = {}"
14398
c5c47703f763 Efficient, graph-based reasoner for linear and partial orders.
ballarin
parents: 13850
diff changeset
   575
  by auto
13735
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   576
7de9342aca7a HOL-Algebra partially ported to Isar.
ballarin
parents: 11609
diff changeset
   577
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
   578
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   579
subsubsection {* Some Differences *}
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   580
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   581
lemma ivl_diff[simp]:
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   582
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   583
by(auto)
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   584
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   585
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   586
subsubsection {* Some Subset Conditions *}
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   587
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   588
lemma ivl_subset[simp]:
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   589
 "({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
   590
apply(auto simp:linorder_not_le)
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   591
apply(rule ccontr)
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   592
apply(insert linorder_le_less_linear[of i n])
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   593
apply(clarsimp simp:linorder_not_le)
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   594
apply(fastsimp)
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   595
done
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   596
15041
a6b1f0cef7b3 Got rid of Summation and made it a translation into setsum instead.
nipkow
parents: 14846
diff changeset
   597
15042
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
   598
subsection {* Summation indexed over intervals *}
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
   599
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
   600
syntax
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
   601
  "_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
   602
  "_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
   603
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   604
  "_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
   605
syntax (xsymbols)
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
   606
  "_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
   607
  "_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
   608
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   609
  "_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
   610
syntax (HTML output)
fa7d27ef7e59 added {0::nat..n(} = {..n(}
nipkow
parents: 15041
diff changeset
   611
  "_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
   612
  "_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
   613
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   614
  "_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
   615
syntax (latex_sum output)
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   616
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   617
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   618
  "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   619
 ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   620
  "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   621
 ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   622
  "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   623
 ("(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
   624
15048
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
   625
translations
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
   626
  "\<Sum>x=a..b. t" == "setsum (%x. t) {a..b}"
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
   627
  "\<Sum>x=a..<b. t" == "setsum (%x. t) {a..<b}"
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   628
  "\<Sum>i\<le>n. t" == "setsum (\<lambda>i. t) {..n}"
15048
11b4dce71d73 more syntax
nipkow
parents: 15047
diff changeset
   629
  "\<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
   630
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   631
text{* The above introduces some pretty alternative syntaxes for
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   632
summation over intervals:
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   633
\begin{center}
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   634
\begin{tabular}{lll}
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   635
Old & New & \LaTeX\\
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   636
@{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
   637
@{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
   638
@{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
   639
@{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
   640
\end{tabular}
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   641
\end{center}
15056
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   642
The left column shows the term before introduction of the new syntax,
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   643
the middle column shows the new (default) syntax, and the right column
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   644
shows a special syntax. The latter is only meaningful for latex output
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   645
and has to be activated explicitly by setting the print mode to
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   646
\texttt{latex\_sum} (e.g.\ via \texttt{mode=latex\_sum} in
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   647
antiquotations). It is not the default \LaTeX\ output because it only
b75073d90bff Fine-tuned sum syntax.
nipkow
parents: 15052
diff changeset
   648
works well with italic-style formulae, not tt-style.
15052
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   649
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   650
Note that for uniformity on @{typ nat} it is better to use
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   651
@{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
   652
not provide all lemmas available for @{term"{m..<n}"} also in the
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   653
special form for @{term"{..<n}"}. *}
cc562a263609 Added nice latex syntax.
nipkow
parents: 15048
diff changeset
   654
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   655
text{* This congruence rule should be used for sums over intervals as
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   656
the standard theorem @{text[source]setsum_cong} does not work well
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   657
with the simplifier who adds the unsimplified premise @{term"x:B"} to
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   658
the context. *}
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   659
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   660
lemma setsum_ivl_cong:
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   661
 "\<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
   662
 setsum f {a..<b} = setsum g {c..<d}"
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   663
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
   664
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   665
(* FIXME why are the following simp rules but the corresponding eqns
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   666
on intervals are not? *)
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   667
16052
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   668
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
   669
by (simp add:atMost_Suc add_ac)
880b0e786c1b tuned setsum rewrites
nipkow
parents: 16041
diff changeset
   670
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   671
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
   672
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
   673
15911
b730b0edc085 turned 2 lemmas into simp rules
nipkow
parents: 15561
diff changeset
   674
lemma setsum_cl_ivl_Suc[simp]:
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
   675
  "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
   676
by (auto simp:add_ac atLeastAtMostSuc_conv)
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
   677
15911
b730b0edc085 turned 2 lemmas into simp rules
nipkow
parents: 15561
diff changeset
   678
lemma setsum_op_ivl_Suc[simp]:
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
   679
  "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
   680
by (auto simp:add_ac atLeastLessThanSuc)
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   681
(*
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
   682
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15554
diff changeset
   683
    (\<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
   684
by (auto simp:add_ac atLeastAtMostSuc_conv)
16041
5a8736668ced simplifier trace info; Suc-intervals
nipkow
parents: 15911
diff changeset
   685
*)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   686
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
   687
  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
   688
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
   689
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   690
lemma setsum_diff_nat_ivl:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   691
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   692
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   693
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   694
using setsum_add_nat_ivl [of m n p f,symmetric]
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   695
apply (simp add: add_ac)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   696
done
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   697
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   698
subsection{* Shifting bounds *}
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   699
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   700
lemma setsum_shift_bounds_nat_ivl:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   701
  "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
   702
by (induct "n", auto simp:atLeastLessThanSuc)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15418
diff changeset
   703
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   704
lemma setsum_shift_bounds_cl_nat_ivl:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   705
  "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
   706
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
   707
apply (simp add:image_add_atLeastAtMost o_def)
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   708
done
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   709
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   710
corollary setsum_shift_bounds_cl_Suc_ivl:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   711
  "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
   712
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
   713
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   714
corollary setsum_shift_bounds_Suc_ivl:
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 16102
diff changeset
   715
  "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
   716
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
   717
19106
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   718
lemma setsum_head:
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   719
  fixes n :: nat
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   720
  assumes mn: "m <= n" 
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   721
  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
   722
proof -
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   723
  from mn
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   724
  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
   725
    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
   726
  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
   727
    by (simp add: atLeast0LessThan)
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   728
  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
   729
  finally show ?thesis .
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   730
qed
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   731
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   732
lemma setsum_head_upt:
19022
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   733
  fixes m::nat
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   734
  assumes m: "0 < m"
19106
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   735
  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
   736
proof -
19106
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   737
  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
   738
    by (simp add: atLeast0LessThan)
19106
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   739
  also 
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   740
  from m 
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   741
  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
   742
    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
   743
  also
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   744
  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
   745
    by (simp add: setsum_head)
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   746
  also 
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   747
  from m 
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 19022
diff changeset
   748
  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
   749
    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
   750
  finally show ?thesis .
19022
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   751
qed
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   752
17149
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   753
subsection {* The formula for geometric sums *}
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   754
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   755
lemma geometric_sum:
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   756
  "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   757
  (x ^ n - 1) / (x - 1::'a::{field, recpower, division_by_zero})"
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   758
  apply (induct "n", auto)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   759
  apply (rule_tac c = "x - 1" in field_mult_cancel_right_lemma)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   760
  apply (auto simp add: mult_assoc left_distrib)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   761
  apply (simp add: right_distrib diff_minus mult_commute power_Suc)
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   762
  done
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   763
e2b19c92ef51 Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents: 16733
diff changeset
   764
19469
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   765
subsection {* The formula for arithmetic sums *}
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   766
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   767
lemma gauss_sum:
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   768
  "((1::'a::comm_semiring_1_cancel) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   769
   of_nat n*((of_nat n)+1)"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   770
proof (induct n)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   771
  case 0
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   772
  show ?case by simp
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   773
next
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   774
  case (Suc n)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   775
  then show ?case by (simp add: right_distrib add_assoc mult_ac)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   776
qed
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   777
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   778
theorem arith_series_general:
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   779
  "((1::'a::comm_semiring_1_cancel) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   780
  of_nat n * (a + (a + of_nat(n - 1)*d))"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   781
proof cases
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   782
  assume ngt1: "n > 1"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   783
  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
   784
  have
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   785
    "(\<Sum>i\<in>{..<n}. a+?I i*d) =
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   786
     ((\<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
   787
    by (rule setsum_addf)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   788
  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
   789
  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
   790
    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
   791
  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
   792
    by (simp add: left_distrib right_distrib)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   793
  also from ngt1 have "{1..<n} = {1..n - 1}"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   794
    by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)    
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   795
  also from ngt1 
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   796
  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
   797
    by (simp only: mult_ac gauss_sum [of "n - 1"])
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   798
       (simp add:  mult_ac of_nat_Suc [symmetric])
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   799
  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
   800
next
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   801
  assume "\<not>(n > 1)"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   802
  hence "n = 1 \<or> n = 0" by auto
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   803
  thus ?thesis by (auto simp: mult_ac right_distrib)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   804
qed
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   805
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   806
lemma arith_series_nat:
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   807
  "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
   808
proof -
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   809
  have
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   810
    "((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
   811
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   812
    by (rule arith_series_general)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   813
  thus ?thesis by (auto simp add: of_nat_id)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   814
qed
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   815
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   816
lemma arith_series_int:
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   817
  "(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
   818
  of_nat n * (a + (a + of_nat(n - 1)*d))"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   819
proof -
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   820
  have
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   821
    "((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
   822
    of_nat(n) * (a + (a + of_nat(n - 1)*d))"
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   823
    by (rule arith_series_general)
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   824
  thus ?thesis by simp
958d2f2dd8d4 moved arithmetic series to geometric series in SetInterval
kleing
parents: 19376
diff changeset
   825
qed
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   826
19022
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   827
lemma sum_diff_distrib:
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   828
  fixes P::"nat\<Rightarrow>nat"
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   829
  shows
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   830
  "\<forall>x. Q x \<le> P x  \<Longrightarrow>
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   831
  (\<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
   832
proof (induct n)
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   833
  case 0 show ?case by simp
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   834
next
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   835
  case (Suc n)
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   836
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   837
  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
   838
  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
   839
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   840
  from Suc have "?lhs = ?rhs" by simp
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   841
  moreover
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   842
  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
   843
  moreover
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   844
  from Suc have
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   845
    "(\<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
   846
    by (subst diff_diff_left[symmetric],
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   847
        subst diff_add_assoc2)
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   848
       (auto simp: diff_add_assoc2 intro: setsum_mono)
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   849
  ultimately
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   850
  show ?case by simp
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   851
qed
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   852
0e6ec4fd204c * moved ThreeDivides from Isar_examples to better suited HOL/ex
kleing
parents: 17719
diff changeset
   853
15418
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   854
ML
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   855
{*
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   856
val Compl_atLeast = thm "Compl_atLeast";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   857
val Compl_atMost = thm "Compl_atMost";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   858
val Compl_greaterThan = thm "Compl_greaterThan";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   859
val Compl_lessThan = thm "Compl_lessThan";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   860
val INT_greaterThan_UNIV = thm "INT_greaterThan_UNIV";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   861
val UN_atLeast_UNIV = thm "UN_atLeast_UNIV";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   862
val UN_atMost_UNIV = thm "UN_atMost_UNIV";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   863
val UN_lessThan_UNIV = thm "UN_lessThan_UNIV";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   864
val atLeastAtMost_def = thm "atLeastAtMost_def";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   865
val atLeastAtMost_iff = thm "atLeastAtMost_iff";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   866
val atLeastLessThan_def  = thm "atLeastLessThan_def";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   867
val atLeastLessThan_iff = thm "atLeastLessThan_iff";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   868
val atLeast_0 = thm "atLeast_0";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   869
val atLeast_Suc = thm "atLeast_Suc";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   870
val atLeast_def      = thm "atLeast_def";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   871
val atLeast_iff = thm "atLeast_iff";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   872
val atMost_0 = thm "atMost_0";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   873
val atMost_Int_atLeast = thm "atMost_Int_atLeast";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   874
val atMost_Suc = thm "atMost_Suc";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   875
val atMost_def       = thm "atMost_def";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   876
val atMost_iff = thm "atMost_iff";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   877
val greaterThanAtMost_def  = thm "greaterThanAtMost_def";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   878
val greaterThanAtMost_iff = thm "greaterThanAtMost_iff";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   879
val greaterThanLessThan_def  = thm "greaterThanLessThan_def";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   880
val greaterThanLessThan_iff = thm "greaterThanLessThan_iff";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   881
val greaterThan_0 = thm "greaterThan_0";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   882
val greaterThan_Suc = thm "greaterThan_Suc";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   883
val greaterThan_def  = thm "greaterThan_def";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   884
val greaterThan_iff = thm "greaterThan_iff";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   885
val ivl_disj_int = thms "ivl_disj_int";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   886
val ivl_disj_int_one = thms "ivl_disj_int_one";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   887
val ivl_disj_int_singleton = thms "ivl_disj_int_singleton";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   888
val ivl_disj_int_two = thms "ivl_disj_int_two";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   889
val ivl_disj_un = thms "ivl_disj_un";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   890
val ivl_disj_un_one = thms "ivl_disj_un_one";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   891
val ivl_disj_un_singleton = thms "ivl_disj_un_singleton";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   892
val ivl_disj_un_two = thms "ivl_disj_un_two";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   893
val lessThan_0 = thm "lessThan_0";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   894
val lessThan_Suc = thm "lessThan_Suc";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   895
val lessThan_Suc_atMost = thm "lessThan_Suc_atMost";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   896
val lessThan_def     = thm "lessThan_def";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   897
val lessThan_iff = thm "lessThan_iff";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   898
val single_Diff_lessThan = thm "single_Diff_lessThan";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   899
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   900
val bounded_nat_set_is_finite = thm "bounded_nat_set_is_finite";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   901
val finite_atMost = thm "finite_atMost";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   902
val finite_lessThan = thm "finite_lessThan";
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   903
*}
e28853da5df5 removed two looping simplifications in SetInterval.thy; deleted the .ML file
paulson
parents: 15402
diff changeset
   904
8924
c434283b4cfa Added SetInterval
nipkow
parents:
diff changeset
   905
end