src/HOL/SetInterval.thy
author paulson
Thu Jul 15 15:32:32 2004 +0200 (2004-07-15)
changeset 15047 fa62de5862b9
parent 15045 d59f7e2e18d3
child 15048 11b4dce71d73
permissions -rw-r--r--
redefining sumr to be a translation to setsum
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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 = IntArith:
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constdefs
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  lessThan    :: "('a::ord) => 'a set"	("(1{..<_})")
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  "{..<u} == {x. x<u}"
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  atMost      :: "('a::ord) => 'a set"	("(1{.._})")
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  "{..u} == {x. x<=u}"
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  greaterThan :: "('a::ord) => 'a set"	("(1{_<..})")
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  "{l<..} == {x. l<x}"
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  atLeast     :: "('a::ord) => 'a set"	("(1{_..})")
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  "{l..} == {x. l<=x}"
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  greaterThanLessThan :: "['a::ord, 'a] => 'a set"  ("(1{_<..<_})")
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  "{l<..<u} == {l<..} Int {..<u}"
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  atLeastLessThan :: "['a::ord, 'a] => 'a set"      ("(1{_..<_})")
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  "{l..<u} == {l..} Int {..<u}"
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  greaterThanAtMost :: "['a::ord, 'a] => 'a set"    ("(1{_<.._})")
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  "{l<..u} == {l<..} Int {..u}"
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  atLeastAtMost :: "['a::ord, 'a] => 'a set"        ("(1{_.._})")
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  "{l..u} == {l..} Int {..u}"
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(* Old syntax, will disappear! *)
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syntax
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  "_lessThan"    :: "('a::ord) => 'a set"	("(1{.._'(})")
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  "_greaterThan" :: "('a::ord) => 'a set"	("(1{')_..})")
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  "_greaterThanLessThan" :: "['a::ord, 'a] => 'a set"  ("(1{')_.._'(})")
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  "_atLeastLessThan" :: "['a::ord, 'a] => 'a set"      ("(1{_.._'(})")
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  "_greaterThanAtMost" :: "['a::ord, 'a] => 'a set"    ("(1{')_.._})")
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translations
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  "{..m(}" => "{..<m}"
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  "{)m..}" => "{m<..}"
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  "{)m..n(}" => "{m<..<n}"
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  "{m..n(}" => "{m..<n}"
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  "{)m..n}" => "{m<..n}"
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syntax
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  "@UNION_le"   :: "nat => nat => 'b set => 'b set"       ("(3UN _<=_./ _)" 10)
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  "@UNION_less" :: "nat => nat => 'b set => 'b set"       ("(3UN _<_./ _)" 10)
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  "@INTER_le"   :: "nat => nat => 'b set => 'b set"       ("(3INT _<=_./ _)" 10)
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  "@INTER_less" :: "nat => nat => 'b set => 'b set"       ("(3INT _<_./ _)" 10)
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syntax (input)
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  "@UNION_le"   :: "nat => nat => 'b set => 'b set"       ("(3\<Union> _\<le>_./ _)" 10)
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  "@UNION_less" :: "nat => nat => 'b set => 'b set"       ("(3\<Union> _<_./ _)" 10)
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  "@INTER_le"   :: "nat => nat => 'b set => 'b set"       ("(3\<Inter> _\<le>_./ _)" 10)
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  "@INTER_less" :: "nat => nat => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" 10)
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syntax (xsymbols)
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  "@UNION_le"   :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Union>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
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  "@UNION_less" :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Union>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
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  "@INTER_le"   :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Inter>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
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  "@INTER_less" :: "nat \<Rightarrow> nat => 'b set => 'b set"       ("(3\<Inter>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
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translations
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  "UN i<=n. A"  == "UN i:{..n}. A"
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  "UN i<n. A"   == "UN i:{..<n}. A"
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  "INT i<=n. A" == "INT i:{..n}. A"
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  "INT i<n. A"  == "INT i:{..<n}. A"
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subsection {* Various equivalences *}
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lemma lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
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by (simp add: lessThan_def)
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lemma Compl_lessThan [simp]: 
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    "!!k:: 'a::linorder. -lessThan k = atLeast k"
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apply (auto simp add: lessThan_def atLeast_def)
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done
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lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
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by auto
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lemma greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
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by (simp add: greaterThan_def)
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lemma Compl_greaterThan [simp]: 
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    "!!k:: 'a::linorder. -greaterThan k = atMost k"
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apply (simp add: greaterThan_def atMost_def le_def, auto)
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done
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lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
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apply (subst Compl_greaterThan [symmetric])
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apply (rule double_complement) 
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done
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lemma atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
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by (simp add: atLeast_def)
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lemma Compl_atLeast [simp]: 
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    "!!k:: 'a::linorder. -atLeast k = lessThan k"
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apply (simp add: lessThan_def atLeast_def le_def, auto)
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done
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lemma atMost_iff [iff]: "(i: atMost k) = (i<=k)"
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by (simp add: atMost_def)
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lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
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by (blast intro: order_antisym)
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subsection {* Logical Equivalences for Set Inclusion and Equality *}
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lemma atLeast_subset_iff [iff]:
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     "(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))" 
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by (blast intro: order_trans) 
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lemma atLeast_eq_iff [iff]:
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     "(atLeast x = atLeast y) = (x = (y::'a::linorder))" 
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by (blast intro: order_antisym order_trans)
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lemma greaterThan_subset_iff [iff]:
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     "(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))" 
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apply (auto simp add: greaterThan_def) 
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 apply (subst linorder_not_less [symmetric], blast) 
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done
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lemma greaterThan_eq_iff [iff]:
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     "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))" 
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apply (rule iffI) 
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 apply (erule equalityE) 
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 apply (simp add: greaterThan_subset_iff order_antisym, simp) 
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done
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lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))" 
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by (blast intro: order_trans)
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lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))" 
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by (blast intro: order_antisym order_trans)
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lemma lessThan_subset_iff [iff]:
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     "(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))" 
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apply (auto simp add: lessThan_def) 
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 apply (subst linorder_not_less [symmetric], blast) 
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done
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lemma lessThan_eq_iff [iff]:
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     "(lessThan x = lessThan y) = (x = (y::'a::linorder))" 
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apply (rule iffI) 
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 apply (erule equalityE) 
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 apply (simp add: lessThan_subset_iff order_antisym, simp) 
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done
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subsection {*Two-sided intervals*}
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text {* @{text greaterThanLessThan} *}
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lemma greaterThanLessThan_iff [simp]:
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  "(i : {l<..<u}) = (l < i & i < u)"
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by (simp add: greaterThanLessThan_def)
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text {* @{text atLeastLessThan} *}
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lemma atLeastLessThan_iff [simp]:
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  "(i : {l..<u}) = (l <= i & i < u)"
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by (simp add: atLeastLessThan_def)
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text {* @{text greaterThanAtMost} *}
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lemma greaterThanAtMost_iff [simp]:
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  "(i : {l<..u}) = (l < i & i <= u)"
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by (simp add: greaterThanAtMost_def)
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text {* @{text atLeastAtMost} *}
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lemma atLeastAtMost_iff [simp]:
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  "(i : {l..u}) = (l <= i & i <= u)"
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by (simp add: atLeastAtMost_def)
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text {* The above four lemmas could be declared as iffs.
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  If we do so, a call to blast in Hyperreal/Star.ML, lemma @{text STAR_Int}
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  seems to take forever (more than one hour). *}
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subsection {* Intervals of natural numbers *}
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subsubsection {* The Constant @{term lessThan} *}
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lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
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by (simp add: lessThan_def)
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lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
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by (simp add: lessThan_def less_Suc_eq, blast)
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lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
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by (simp add: lessThan_def atMost_def less_Suc_eq_le)
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lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
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by blast
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subsubsection {* The Constant @{term greaterThan} *}
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lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
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apply (simp add: greaterThan_def)
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apply (blast dest: gr0_conv_Suc [THEN iffD1])
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done
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lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
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apply (simp add: greaterThan_def)
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apply (auto elim: linorder_neqE)
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done
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lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
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by blast
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subsubsection {* The Constant @{term atLeast} *}
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lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
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by (unfold atLeast_def UNIV_def, simp)
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lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
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apply (simp add: atLeast_def)
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apply (simp add: Suc_le_eq)
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apply (simp add: order_le_less, blast)
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done
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lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
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  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
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lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
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by blast
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subsubsection {* The Constant @{term atMost} *}
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lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
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by (simp add: atMost_def)
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lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
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apply (simp add: atMost_def)
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apply (simp add: less_Suc_eq order_le_less, blast)
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done
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lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
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by blast
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subsubsection {* The Constant @{term atLeastLessThan} *}
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text{*But not a simprule because some concepts are better left in terms
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  of @{term atLeastLessThan}*}
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lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
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by(simp add:lessThan_def atLeastLessThan_def)
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lemma atLeastLessThan0 [simp]: "{m..<0::nat} = {}"
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by (simp add: atLeastLessThan_def)
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lemma atLeastLessThan_self [simp]: "{n::'a::order..<n} = {}"
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by (auto simp add: atLeastLessThan_def)
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lemma atLeastLessThan_empty: "n \<le> m ==> {m..<n::'a::order} = {}"
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by (auto simp add: atLeastLessThan_def)
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subsubsection {* Intervals of nats with @{term Suc} *}
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text{*Not a simprule because the RHS is too messy.*}
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lemma atLeastLessThanSuc:
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    "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
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by (auto simp add: atLeastLessThan_def) 
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lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}" 
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by (auto simp add: atLeastLessThan_def)
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lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
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by (induct k, simp_all add: atLeastLessThanSuc)
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lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
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by (auto simp add: atLeastLessThan_def)
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lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
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  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
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lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"  
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  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def 
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    greaterThanAtMost_def)
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lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"  
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  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def 
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    greaterThanLessThan_def)
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subsubsection {* Finiteness *}
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lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
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  by (induct k) (simp_all add: lessThan_Suc)
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lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
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  by (induct k) (simp_all add: atMost_Suc)
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lemma finite_greaterThanLessThan [iff]:
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  fixes l :: nat shows "finite {l<..<u}"
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by (simp add: greaterThanLessThan_def)
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lemma finite_atLeastLessThan [iff]:
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  fixes l :: nat shows "finite {l..<u}"
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by (simp add: atLeastLessThan_def)
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lemma finite_greaterThanAtMost [iff]:
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  fixes l :: nat shows "finite {l<..u}"
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by (simp add: greaterThanAtMost_def)
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lemma finite_atLeastAtMost [iff]:
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  fixes l :: nat shows "finite {l..u}"
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   319
by (simp add: atLeastAtMost_def)
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lemma bounded_nat_set_is_finite:
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   322
    "(ALL i:N. i < (n::nat)) ==> finite N"
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  -- {* A bounded set of natural numbers is finite. *}
paulson@14485
   324
  apply (rule finite_subset)
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   325
   apply (rule_tac [2] finite_lessThan, auto)
paulson@14485
   326
  done
paulson@14485
   327
paulson@14485
   328
subsubsection {* Cardinality *}
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   329
nipkow@15045
   330
lemma card_lessThan [simp]: "card {..<u} = u"
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   331
  by (induct_tac u, simp_all add: lessThan_Suc)
paulson@14485
   332
paulson@14485
   333
lemma card_atMost [simp]: "card {..u} = Suc u"
paulson@14485
   334
  by (simp add: lessThan_Suc_atMost [THEN sym])
paulson@14485
   335
nipkow@15045
   336
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
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   337
  apply (subgoal_tac "card {l..<u} = card {..<u-l}")
paulson@14485
   338
  apply (erule ssubst, rule card_lessThan)
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   339
  apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
paulson@14485
   340
  apply (erule subst)
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   341
  apply (rule card_image)
paulson@14485
   342
  apply (rule finite_lessThan)
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   343
  apply (simp add: inj_on_def)
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   344
  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
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   345
  apply arith
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   346
  apply (rule_tac x = "x - l" in exI)
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   347
  apply arith
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   348
  done
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   349
paulson@15047
   350
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"  
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   351
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
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   352
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   353
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l" 
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   354
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
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   355
nipkow@15045
   356
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
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   357
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
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   358
paulson@14485
   359
subsection {* Intervals of integers *}
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   360
nipkow@15045
   361
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
paulson@14485
   362
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
paulson@14485
   363
nipkow@15045
   364
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"  
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   365
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
paulson@14485
   366
paulson@14485
   367
lemma atLeastPlusOneLessThan_greaterThanLessThan_int: 
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   368
    "{l+1..<u} = {l<..<u::int}"  
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   369
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
paulson@14485
   370
paulson@14485
   371
subsubsection {* Finiteness *}
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   372
paulson@14485
   373
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==> 
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   374
    {(0::int)..<u} = int ` {..<nat u}"
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   375
  apply (unfold image_def lessThan_def)
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   376
  apply auto
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   377
  apply (rule_tac x = "nat x" in exI)
paulson@14485
   378
  apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
paulson@14485
   379
  done
paulson@14485
   380
nipkow@15045
   381
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
paulson@14485
   382
  apply (case_tac "0 \<le> u")
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   383
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   384
  apply (rule finite_imageI)
paulson@14485
   385
  apply auto
nipkow@15045
   386
  apply (subgoal_tac "{0..<u} = {}")
paulson@14485
   387
  apply auto
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   388
  done
paulson@14485
   389
paulson@14485
   390
lemma image_atLeastLessThan_int_shift: 
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   391
    "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
paulson@14485
   392
  apply (auto simp add: image_def atLeastLessThan_iff)
paulson@14485
   393
  apply (rule_tac x = "x - l" in bexI)
paulson@14485
   394
  apply auto
paulson@14485
   395
  done
paulson@14485
   396
nipkow@15045
   397
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
nipkow@15045
   398
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   399
  apply (erule subst)
paulson@14485
   400
  apply (rule finite_imageI)
paulson@14485
   401
  apply (rule finite_atLeastZeroLessThan_int)
paulson@14485
   402
  apply (rule image_atLeastLessThan_int_shift)
paulson@14485
   403
  done
paulson@14485
   404
paulson@14485
   405
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}" 
paulson@14485
   406
  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
paulson@14485
   407
nipkow@15045
   408
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}" 
paulson@14485
   409
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   410
nipkow@15045
   411
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}" 
paulson@14485
   412
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   413
paulson@14485
   414
subsubsection {* Cardinality *}
paulson@14485
   415
nipkow@15045
   416
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
paulson@14485
   417
  apply (case_tac "0 \<le> u")
paulson@14485
   418
  apply (subst image_atLeastZeroLessThan_int, assumption)
paulson@14485
   419
  apply (subst card_image)
paulson@14485
   420
  apply (auto simp add: inj_on_def)
paulson@14485
   421
  done
paulson@14485
   422
nipkow@15045
   423
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
nipkow@15045
   424
  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
paulson@14485
   425
  apply (erule ssubst, rule card_atLeastZeroLessThan_int)
nipkow@15045
   426
  apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485
   427
  apply (erule subst)
paulson@14485
   428
  apply (rule card_image)
paulson@14485
   429
  apply (rule finite_atLeastZeroLessThan_int)
paulson@14485
   430
  apply (simp add: inj_on_def)
paulson@14485
   431
  apply (rule image_atLeastLessThan_int_shift)
paulson@14485
   432
  done
paulson@14485
   433
paulson@14485
   434
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
paulson@14485
   435
  apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
paulson@14485
   436
  apply (auto simp add: compare_rls)
paulson@14485
   437
  done
paulson@14485
   438
nipkow@15045
   439
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)" 
paulson@14485
   440
  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485
   441
nipkow@15045
   442
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
paulson@14485
   443
  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485
   444
paulson@14485
   445
paulson@13850
   446
subsection {*Lemmas useful with the summation operator setsum*}
paulson@13850
   447
wenzelm@14577
   448
text {* For examples, see Algebra/poly/UnivPoly.thy *}
ballarin@13735
   449
wenzelm@14577
   450
subsubsection {* Disjoint Unions *}
ballarin@13735
   451
wenzelm@14577
   452
text {* Singletons and open intervals *}
ballarin@13735
   453
ballarin@13735
   454
lemma ivl_disj_un_singleton:
nipkow@15045
   455
  "{l::'a::linorder} Un {l<..} = {l..}"
nipkow@15045
   456
  "{..<u} Un {u::'a::linorder} = {..u}"
nipkow@15045
   457
  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
nipkow@15045
   458
  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
nipkow@15045
   459
  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
nipkow@15045
   460
  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
ballarin@14398
   461
by auto
ballarin@13735
   462
wenzelm@14577
   463
text {* One- and two-sided intervals *}
ballarin@13735
   464
ballarin@13735
   465
lemma ivl_disj_un_one:
nipkow@15045
   466
  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
nipkow@15045
   467
  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
nipkow@15045
   468
  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
nipkow@15045
   469
  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
nipkow@15045
   470
  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
nipkow@15045
   471
  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
nipkow@15045
   472
  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
nipkow@15045
   473
  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
ballarin@14398
   474
by auto
ballarin@13735
   475
wenzelm@14577
   476
text {* Two- and two-sided intervals *}
ballarin@13735
   477
ballarin@13735
   478
lemma ivl_disj_un_two:
nipkow@15045
   479
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
nipkow@15045
   480
  "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
nipkow@15045
   481
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
nipkow@15045
   482
  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
nipkow@15045
   483
  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
nipkow@15045
   484
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
nipkow@15045
   485
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
nipkow@15045
   486
  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
ballarin@14398
   487
by auto
ballarin@13735
   488
ballarin@13735
   489
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
ballarin@13735
   490
wenzelm@14577
   491
subsubsection {* Disjoint Intersections *}
ballarin@13735
   492
wenzelm@14577
   493
text {* Singletons and open intervals *}
ballarin@13735
   494
ballarin@13735
   495
lemma ivl_disj_int_singleton:
nipkow@15045
   496
  "{l::'a::order} Int {l<..} = {}"
nipkow@15045
   497
  "{..<u} Int {u} = {}"
nipkow@15045
   498
  "{l} Int {l<..<u} = {}"
nipkow@15045
   499
  "{l<..<u} Int {u} = {}"
nipkow@15045
   500
  "{l} Int {l<..u} = {}"
nipkow@15045
   501
  "{l..<u} Int {u} = {}"
ballarin@13735
   502
  by simp+
ballarin@13735
   503
wenzelm@14577
   504
text {* One- and two-sided intervals *}
ballarin@13735
   505
ballarin@13735
   506
lemma ivl_disj_int_one:
nipkow@15045
   507
  "{..l::'a::order} Int {l<..<u} = {}"
nipkow@15045
   508
  "{..<l} Int {l..<u} = {}"
nipkow@15045
   509
  "{..l} Int {l<..u} = {}"
nipkow@15045
   510
  "{..<l} Int {l..u} = {}"
nipkow@15045
   511
  "{l<..u} Int {u<..} = {}"
nipkow@15045
   512
  "{l<..<u} Int {u..} = {}"
nipkow@15045
   513
  "{l..u} Int {u<..} = {}"
nipkow@15045
   514
  "{l..<u} Int {u..} = {}"
ballarin@14398
   515
  by auto
ballarin@13735
   516
wenzelm@14577
   517
text {* Two- and two-sided intervals *}
ballarin@13735
   518
ballarin@13735
   519
lemma ivl_disj_int_two:
nipkow@15045
   520
  "{l::'a::order<..<m} Int {m..<u} = {}"
nipkow@15045
   521
  "{l<..m} Int {m<..<u} = {}"
nipkow@15045
   522
  "{l..<m} Int {m..<u} = {}"
nipkow@15045
   523
  "{l..m} Int {m<..<u} = {}"
nipkow@15045
   524
  "{l<..<m} Int {m..u} = {}"
nipkow@15045
   525
  "{l<..m} Int {m<..u} = {}"
nipkow@15045
   526
  "{l..<m} Int {m..u} = {}"
nipkow@15045
   527
  "{l..m} Int {m<..u} = {}"
ballarin@14398
   528
  by auto
ballarin@13735
   529
ballarin@13735
   530
lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
ballarin@13735
   531
nipkow@15041
   532
nipkow@15042
   533
subsection {* Summation indexed over intervals *}
nipkow@15042
   534
nipkow@15042
   535
text{* We introduce the obvious syntax @{text"\<Sum>x=a..b. e"} for
nipkow@15042
   536
@{term"\<Sum>x\<in>{a..b}. e"}. *}
nipkow@15042
   537
nipkow@15042
   538
syntax
nipkow@15042
   539
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
nipkow@15042
   540
syntax (xsymbols)
nipkow@15042
   541
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15042
   542
syntax (HTML output)
nipkow@15042
   543
  "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15042
   544
nipkow@15042
   545
translations "\<Sum>x=a..b. t" == "setsum (%x. t) {a..b}"
nipkow@15042
   546
nipkow@15042
   547
nipkow@15042
   548
subsection {* Summation up to *}
nipkow@15041
   549
nipkow@15041
   550
text{* Legacy, only used in HoareParallel and Isar-Examples. Really
nipkow@15042
   551
needed? Probably better to replace it with above syntax. *}
nipkow@15041
   552
nipkow@15041
   553
syntax
nipkow@15042
   554
  "_Summation" :: "idt => 'a => 'b => 'b"    ("\<Sum>_<_. _" [0, 51, 10] 10)
nipkow@15041
   555
translations
nipkow@15045
   556
  "\<Sum>i < n. b" == "setsum (\<lambda>i. b) {..<n}"
nipkow@15041
   557
nipkow@15041
   558
lemma Summation_Suc[simp]: "(\<Sum>i < Suc n. b i) = b n + (\<Sum>i < n. b i)"
nipkow@15041
   559
by (simp add:lessThan_Suc)
nipkow@15041
   560
nipkow@8924
   561
end