(* Title: HOL/ex/Points.thy
ID: $Id$
Author: Wolfgang Naraschewski and Markus Wenzel, TU Muenchen
*)
theory Points = Main:;
text {* Basic use of extensible records in HOL, including the famous points
and coloured points. *};
section {* Points *};
record point =
x :: nat
y :: nat;
text {*
Apart many other things, above record declaration produces the
following theorems:
thms "point.simps"
thms "point.iffs"
thms "point.update_defs"
The set of theorems "point.simps" is added automatically to the
standard simpset, "point.iffs" is added to the claset and simpset.
*};
text {*
Record declarations define new type abbreviations:
point = "(| x :: nat, y :: nat |)"
'a point_scheme = "(| x :: nat, y :: nat, ... :: 'a |)"
Extensions `...' must be in type class `more'!
*};
consts foo1 :: point;
consts foo2 :: "(| x :: nat, y :: nat |)";
consts foo3 :: "'a => ('a::more) point_scheme";
consts foo4 :: "'a => (| x :: nat, y :: nat, ... :: 'a |)";
subsection {* Introducing concrete records and record schemes *};
defs
foo1_def: "foo1 == (| x = 1, y = 0 |)"
foo3_def: "foo3 ext == (| x = 1, y = 0, ... = ext |)";
subsection {* Record selection and record update *};
constdefs
getX :: "('a::more) point_scheme => nat"
"getX r == x r"
setX :: "('a::more) point_scheme => nat => 'a point_scheme"
"setX r n == r (| x := n |)";
subsection {* Some lemmas about records *};
text {* Note: any of these goals may be solved with a single
stroke of the auto or force proof method. *};
text {* basic simplifications *};
lemma "x (| x = m, y = n, ... = p |) = m";
by simp;
lemma "(| x = m, y = n, ... = p |) (| x:= 0 |) = (| x = 0, y = n, ... = p |)";
by simp;
text {* splitting quantifiers: the !!r is NECESSARY *};
lemma "!!r. r (| x := n |) (| y := m |) = r (| y := m |) (| x := n |)";
proof record_split;
fix a b rest;
show "(| x = a, y = b, ... = rest |)(| x := n, y := m |) =
(| x = a, y = b, ... = rest |)(| y := m, x := n |)";
by simp;
qed;
lemma "!!r. r (| x := n |) (| x := m |) = r (| x := m |)";
proof record_split;
fix a b rest;
show "(| x = a, y = b, ... = rest |)(| x := n, x := m |) =
(| x = a, y = b, ... = rest |)(| x := m |)";
by simp;
qed;
text {* equality of records *};
lemma "(| x = n, y = p |) = (| x = n', y = p' |) --> n = n'" (is "?EQ --> _");
proof;
assume ?EQ;
thus "n = n'"; by simp;
qed;
text {* concrete records are type instances of record schemes *};
constdefs
foo5 :: nat
"foo5 == getX (| x = 1, y = 0 |)";
text {* manipulating the `...' (more) part *};
constdefs
incX :: "('a::more) point_scheme => 'a point_scheme"
"incX r == (| x = Suc (x r), y = y r, ... = point.more r |)";
lemma "!!r n. incX r = setX r (Suc (getX r))";
proof (unfold getX_def setX_def incX_def);
show "!! r n. (| x = Suc (x r), y = y r, ... = more r |) = r(| x := Suc (x r) |)";
by (record_split, simp);
qed;
text {* alternative definition *};
constdefs
incX' :: "('a::more) point_scheme => 'a point_scheme"
"incX' r == r (| x := Suc (x r) |)";
section {* coloured points: record extension *};
datatype colour = Red | Green | Blue;
record cpoint = point +
colour :: colour;
text {*
The record declaration defines new type constructors:
cpoint = (| x :: nat, y :: nat, colour :: colour |)
'a cpoint_scheme = (| x :: nat, y :: nat, colour :: colour, ... :: 'a |)
*};
consts foo6 :: cpoint;
consts foo7 :: "(| x :: nat, y :: nat, colour :: colour |)";
consts foo8 :: "('a::more) cpoint_scheme";
consts foo9 :: "(| x :: nat, y :: nat, colour :: colour, ... :: 'a |)";
text {* functions on point schemes work for cpoints as well *};
constdefs
foo10 :: nat
"foo10 == getX (| x = 2, y = 0, colour = Blue |)";
subsection {* Non-coercive structural subtyping *};
text {* foo11 has type cpoint, not type point --- Great! *};
constdefs
foo11 :: cpoint
"foo11 == setX (| x = 2, y = 0, colour = Blue |) 0";
section {* Other features *};
text {* field names contribute to record identity *};
record point' =
x' :: nat
y' :: nat;
consts
foo12 :: nat;
text {* Definition @{term "foo12 == getX (| x' = 2, y' = 0 |)"} is invalid *};
text {* polymorphic records *};
record 'a point'' = point +
content :: 'a;
types cpoint'' = "colour point''";
text {* Have a lot of fun! *};
end;