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(* Title: HOL/ex/Records.thy
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ID: $Id$
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Author: Wolfgang Naraschewski and Markus Wenzel, TU Muenchen
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License: GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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header {* Using extensible records in HOL -- points and coloured points *}
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theory Records = Main:
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subsection {* Points *}
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record point =
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xpos :: nat
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ypos :: nat
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text {*
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Apart many other things, above record declaration produces the
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following theorems:
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*}
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thm "point.simps"
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thm "point.iffs"
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thm "point.derived_defs"
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text {*
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The set of theorems @{thm [source] point.simps} is added
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automatically to the standard simpset, @{thm [source] point.iffs} is
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added to the Classical Reasoner and Simplifier context.
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\medskip Record declarations define new type abbreviations:
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@{text [display]
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" point = (| xpos :: nat, ypos :: nat |)
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'a point_scheme = (| xpos :: nat, ypos :: nat, ... :: 'a |)"}
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*}
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consts foo1 :: point
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consts foo2 :: "(| xpos :: nat, ypos :: nat |)"
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consts foo3 :: "'a => 'a point_scheme"
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consts foo4 :: "'a => (| xpos :: nat, ypos :: nat, ... :: 'a |)"
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subsubsection {* Introducing concrete records and record schemes *}
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defs
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foo1_def: "foo1 == (| xpos = 1, ypos = 0 |)"
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foo3_def: "foo3 ext == (| xpos = 1, ypos = 0, ... = ext |)"
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subsubsection {* Record selection and record update *}
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constdefs
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getX :: "'a point_scheme => nat"
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"getX r == xpos r"
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setX :: "'a point_scheme => nat => 'a point_scheme"
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"setX r n == r (| xpos := n |)"
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subsubsection {* Some lemmas about records *}
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text {* Basic simplifications. *}
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lemma "point.make n p = (| xpos = n, ypos = p |)"
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by (simp only: point.make_def)
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lemma "xpos (| xpos = m, ypos = n, ... = p |) = m"
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by simp
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lemma "(| xpos = m, ypos = n, ... = p |) (| xpos:= 0 |) = (| xpos = 0, ypos = n, ... = p |)"
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by simp
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text {* \medskip Equality of records. *}
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lemma "n = n' ==> p = p' ==> (| xpos = n, ypos = p |) = (| xpos = n', ypos = p' |)"
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-- "introduction of concrete record equality"
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by simp
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lemma "(| xpos = n, ypos = p |) = (| xpos = n', ypos = p' |) ==> n = n'"
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-- "elimination of concrete record equality"
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by simp
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lemma "r (| xpos := n |) (| ypos := m |) = r (| ypos := m |) (| xpos := n |)"
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-- "introduction of abstract record equality"
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by simp
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lemma "r (| xpos := n |) = r (| xpos := n' |) ==> n = n'"
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-- "elimination of abstract record equality (manual proof)"
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proof -
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assume "r (| xpos := n |) = r (| xpos := n' |)" (is "?lhs = ?rhs")
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hence "xpos ?lhs = xpos ?rhs" by simp
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thus ?thesis by simp
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qed
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text {* \medskip Surjective pairing *}
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lemma "r = (| xpos = xpos r, ypos = ypos r |)"
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by simp
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lemma "r = (| xpos = xpos r, ypos = ypos r, ... = more r |)"
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by simp
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text {*
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\medskip Representation of records by cases or (degenerate)
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induction.
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*}
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lemma "r (| xpos := n |) (| ypos := m |) = r (| ypos := m |) (| xpos := n |)"
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proof (cases r)
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fix xpos ypos more
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assume "r = (| xpos = xpos, ypos = ypos, ... = more |)"
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thus ?thesis by simp
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qed
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lemma "r (| xpos := n |) (| ypos := m |) = r (| ypos := m |) (| xpos := n |)"
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proof (induct r)
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fix xpos ypos more
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show "(| xpos = xpos, ypos = ypos, ... = more |) (| xpos := n, ypos := m |) =
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(| xpos = xpos, ypos = ypos, ... = more |) (| ypos := m, xpos := n |)"
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by simp
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qed
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lemma "r (| xpos := n |) (| xpos := m |) = r (| xpos := m |)"
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proof (cases r)
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fix xpos ypos more
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assume "r = \<lparr>xpos = xpos, ypos = ypos, \<dots> = more\<rparr>"
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thus ?thesis by simp
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qed
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lemma "r (| xpos := n |) (| xpos := m |) = r (| xpos := m |)"
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proof (cases r)
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case fields
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thus ?thesis by simp
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qed
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lemma "r (| xpos := n |) (| xpos := m |) = r (| xpos := m |)"
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by (cases r) simp
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text {*
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\medskip Concrete records are type instances of record schemes.
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*}
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constdefs
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foo5 :: nat
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"foo5 == getX (| xpos = 1, ypos = 0 |)"
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text {* \medskip Manipulating the ``@{text "..."}'' (more) part. *}
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constdefs
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incX :: "'a point_scheme => 'a point_scheme"
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"incX r == (| xpos = xpos r + 1, ypos = ypos r, ... = point.more r |)"
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lemma "incX r = setX r (Suc (getX r))"
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by (simp add: getX_def setX_def incX_def)
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text {* An alternative definition. *}
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constdefs
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incX' :: "'a point_scheme => 'a point_scheme"
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"incX' r == r (| xpos := xpos r + 1 |)"
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subsection {* Coloured points: record extension *}
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datatype colour = Red | Green | Blue
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record cpoint = point +
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colour :: colour
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text {*
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The record declaration defines new type constructors:
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@{text [display]
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" cpoint = (| xpos :: nat, ypos :: nat, colour :: colour |)
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'a cpoint_scheme = (| xpos :: nat, ypos :: nat, colour :: colour, ... :: 'a |)"}
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*}
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consts foo6 :: cpoint
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consts foo7 :: "(| xpos :: nat, ypos :: nat, colour :: colour |)"
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consts foo8 :: "'a cpoint_scheme"
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consts foo9 :: "(| xpos :: nat, ypos :: nat, colour :: colour, ... :: 'a |)"
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text {*
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Functions on @{text point} schemes work for @{text cpoints} as well.
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*}
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constdefs
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foo10 :: nat
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"foo10 == getX (| xpos = 2, ypos = 0, colour = Blue |)"
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subsubsection {* Non-coercive structural subtyping *}
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text {*
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Term @{term foo11} has type @{typ cpoint}, not type @{typ point} ---
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Great!
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*}
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constdefs
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foo11 :: cpoint
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"foo11 == setX (| xpos = 2, ypos = 0, colour = Blue |) 0"
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subsection {* Other features *}
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text {* Field names contribute to record identity. *}
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record point' =
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xpos' :: nat
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ypos' :: nat
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text {*
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\noindent May not apply @{term getX} to @{term [source] "(| xpos' =
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2, ypos' = 0 |)"} -- type error.
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*}
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text {* \medskip Polymorphic records. *}
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record 'a point'' = point +
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content :: 'a
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types cpoint'' = "colour point''"
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end
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