author | wenzelm |
Mon, 25 Mar 2019 17:21:26 +0100 | |
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parent 69597 | ff784d5a5bfb |
permissions | -rw-r--r-- |
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(* Title: HOL/ex/Records.thy |
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Author: Wolfgang Naraschewski, Norbert Schirmer and Markus Wenzel, |
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TU Muenchen |
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*) |
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section \<open>Using extensible records in HOL -- points and coloured points\<close> |
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theory Records |
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imports Main |
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begin |
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subsection \<open>Points\<close> |
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record point = |
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xpos :: nat |
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ypos :: nat |
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text \<open> |
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Apart many other things, above record declaration produces the |
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following theorems: |
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\<close> |
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thm point.simps |
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thm point.iffs |
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thm point.defs |
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text \<open> |
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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 types and type abbreviations: |
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@{text [display] |
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\<open>point = \<lparr>xpos :: nat, ypos :: nat\<rparr> = () point_ext_type |
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'a point_scheme = \<lparr>xpos :: nat, ypos :: nat, ... :: 'a\<rparr> = 'a point_ext_type\<close>} |
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\<close> |
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consts foo2 :: "(| xpos :: nat, ypos :: nat |)" |
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consts foo4 :: "'a => (| xpos :: nat, ypos :: nat, ... :: 'a |)" |
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subsubsection \<open>Introducing concrete records and record schemes\<close> |
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definition foo1 :: point |
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where "foo1 = (| xpos = 1, ypos = 0 |)" |
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definition foo3 :: "'a => 'a point_scheme" |
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where "foo3 ext = (| xpos = 1, ypos = 0, ... = ext |)" |
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subsubsection \<open>Record selection and record update\<close> |
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definition getX :: "'a point_scheme => nat" |
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where "getX r = xpos r" |
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definition setX :: "'a point_scheme => nat => 'a point_scheme" |
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where "setX r n = r (| xpos := n |)" |
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subsubsection \<open>Some lemmas about records\<close> |
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text \<open>Basic simplifications.\<close> |
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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 \<open>\medskip Equality of records.\<close> |
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lemma "n = n' ==> p = p' ==> (| xpos = n, ypos = p |) = (| xpos = n', ypos = p' |)" |
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\<comment> \<open>introduction of concrete record equality\<close> |
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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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\<comment> \<open>elimination of concrete record equality\<close> |
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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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\<comment> \<open>introduction of abstract record equality\<close> |
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by simp |
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lemma "r (| xpos := n |) = r (| xpos := n' |) ==> n = n'" |
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\<comment> \<open>elimination of abstract record equality (manual proof)\<close> |
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proof - |
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assume "r (| xpos := n |) = r (| xpos := n' |)" (is "?lhs = ?rhs") |
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then have "xpos ?lhs = xpos ?rhs" by simp |
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then show ?thesis by simp |
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qed |
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text \<open>\medskip Surjective pairing\<close> |
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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, ... = point.more r |)" |
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by simp |
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text \<open> |
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\medskip Representation of records by cases or (degenerate) |
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induction. |
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\<close> |
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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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then show ?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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then show ?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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then show ?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 \<open> |
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\medskip Concrete records are type instances of record schemes. |
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\<close> |
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definition foo5 :: nat |
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where "foo5 = getX (| xpos = 1, ypos = 0 |)" |
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text \<open>\medskip Manipulating the ``\<open>...\<close>'' (more) part.\<close> |
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definition incX :: "'a point_scheme => 'a point_scheme" |
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where "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 \<open>An alternative definition.\<close> |
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definition incX' :: "'a point_scheme => 'a point_scheme" |
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where "incX' r = r (| xpos := xpos r + 1 |)" |
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subsection \<open>Coloured points: record extension\<close> |
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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 \<open> |
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The record declaration defines a new type constructor and abbreviations: |
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@{text [display] |
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\<open>cpoint = (| xpos :: nat, ypos :: nat, colour :: colour |) = |
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() cpoint_ext_type point_ext_type |
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'a cpoint_scheme = (| xpos :: nat, ypos :: nat, colour :: colour, ... :: 'a |) = |
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'a cpoint_ext_type point_ext_type\<close>} |
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\<close> |
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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 \<open> |
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Functions on \<open>point\<close> schemes work for \<open>cpoints\<close> as well. |
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\<close> |
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definition foo10 :: nat |
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where "foo10 = getX (| xpos = 2, ypos = 0, colour = Blue |)" |
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subsubsection \<open>Non-coercive structural subtyping\<close> |
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text \<open> |
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Term \<^term>\<open>foo11\<close> has type \<^typ>\<open>cpoint\<close>, not type \<^typ>\<open>point\<close> --- |
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Great! |
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\<close> |
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definition foo11 :: cpoint |
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where "foo11 = setX (| xpos = 2, ypos = 0, colour = Blue |) 0" |
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subsection \<open>Other features\<close> |
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text \<open>Field names contribute to record identity.\<close> |
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record point' = |
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xpos' :: nat |
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ypos' :: nat |
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text \<open> |
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\noindent May not apply \<^term>\<open>getX\<close> to @{term [source] "(| xpos' = |
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2, ypos' = 0 |)"} -- type error. |
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\<close> |
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text \<open>\medskip Polymorphic records.\<close> |
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record 'a point'' = point + |
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content :: 'a |
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type_synonym cpoint'' = "colour point''" |
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text \<open>Updating a record field with an identical value is simplified.\<close> |
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lemma "r (| xpos := xpos r |) = r" |
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by simp |
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text \<open>Only the most recent update to a component survives simplification.\<close> |
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lemma "r (| xpos := x, ypos := y, xpos := x' |) = r (| ypos := y, xpos := x' |)" |
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by simp |
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text \<open>In some cases its convenient to automatically split |
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(quantified) records. For this purpose there is the simproc @{ML [source] |
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"Record.split_simproc"} and the tactic @{ML [source] |
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"Record.split_simp_tac"}. The simplification procedure |
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only splits the records, whereas the tactic also simplifies the |
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resulting goal with the standard record simplification rules. A |
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(generalized) predicate on the record is passed as parameter that |
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decides whether or how `deep' to split the record. It can peek on the |
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subterm starting at the quantified occurrence of the record (including |
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the quantifier). The value \<^ML>\<open>0\<close> indicates no split, a value |
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greater \<^ML>\<open>0\<close> splits up to the given bound of record extension and |
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finally the value \<^ML>\<open>~1\<close> completely splits the record. |
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@{ML [source] "Record.split_simp_tac"} additionally takes a list of |
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equations for simplification and can also split fixed record variables. |
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\<close> |
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lemma "(\<forall>r. P (xpos r)) \<longrightarrow> (\<forall>x. P x)" |
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apply (tactic \<open>simp_tac (put_simpset HOL_basic_ss \<^context> |
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addsimprocs [Record.split_simproc (K ~1)]) 1\<close>) |
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apply simp |
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done |
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lemma "(\<forall>r. P (xpos r)) \<longrightarrow> (\<forall>x. P x)" |
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apply (tactic \<open>Record.split_simp_tac \<^context> [] (K ~1) 1\<close>) |
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apply simp |
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done |
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lemma "(\<exists>r. P (xpos r)) \<longrightarrow> (\<exists>x. P x)" |
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apply (tactic \<open>simp_tac (put_simpset HOL_basic_ss \<^context> |
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addsimprocs [Record.split_simproc (K ~1)]) 1\<close>) |
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apply simp |
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done |
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lemma "(\<exists>r. P (xpos r)) \<longrightarrow> (\<exists>x. P x)" |
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apply (tactic \<open>Record.split_simp_tac \<^context> [] (K ~1) 1\<close>) |
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apply simp |
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done |
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lemma "\<And>r. P (xpos r) \<Longrightarrow> (\<exists>x. P x)" |
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apply (tactic \<open>simp_tac (put_simpset HOL_basic_ss \<^context> |
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addsimprocs [Record.split_simproc (K ~1)]) 1\<close>) |
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apply auto |
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done |
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lemma "\<And>r. P (xpos r) \<Longrightarrow> (\<exists>x. P x)" |
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apply (tactic \<open>Record.split_simp_tac \<^context> [] (K ~1) 1\<close>) |
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apply auto |
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done |
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lemma "P (xpos r) \<Longrightarrow> (\<exists>x. P x)" |
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apply (tactic \<open>Record.split_simp_tac \<^context> [] (K ~1) 1\<close>) |
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apply auto |
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done |
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lemma True |
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proof - |
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{ |
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fix P r |
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assume pre: "P (xpos r)" |
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then have "\<exists>x. P x" |
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apply - |
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apply (tactic \<open>Record.split_simp_tac \<^context> [] (K ~1) 1\<close>) |
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apply auto |
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done |
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} |
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show ?thesis .. |
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qed |
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text \<open>The effect of simproc @{ML [source] Record.ex_sel_eq_simproc} is |
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illustrated by the following lemma.\<close> |
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lemma "\<exists>r. xpos r = x" |
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apply (tactic \<open>simp_tac (put_simpset HOL_basic_ss \<^context> |
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addsimprocs [Record.ex_sel_eq_simproc]) 1\<close>) |
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done |
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subsection \<open>A more complex record expression\<close> |
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record ('a, 'b, 'c) bar = bar1 :: 'a |
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bar2 :: 'b |
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bar3 :: 'c |
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bar21 :: "'b \<times> 'a" |
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bar32 :: "'c \<times> 'b" |
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bar31 :: "'c \<times> 'a" |
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print_record "('a,'b,'c) bar" |
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subsection \<open>Some code generation\<close> |
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export_code foo1 foo3 foo5 foo10 checking SML |
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text \<open> |
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Code generation can also be switched off, for instance for very large records |
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\<close> |
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declare [[record_codegen = false]] |
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record not_so_large_record = |
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bar520 :: nat |
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bar521 :: "nat * nat" |
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declare [[record_codegen = true]] |
bfc08ce7b7b9
provide [[record_codegen]] option for skipping codegen setup for records
Gerwin Klein <gerwin.klein@nicta.com.au>
parents:
46231
diff
changeset
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343 |
|
10052 | 344 |
end |