--- a/src/HOL/ex/MonoidGroup.thy Thu Oct 25 22:42:50 2001 +0200
+++ b/src/HOL/ex/MonoidGroup.thy Thu Oct 25 22:43:05 2001 +0200
@@ -18,15 +18,15 @@
inv :: "'a => 'a"
constdefs
- monoid :: "(| times :: 'a => 'a => 'a, one :: 'a, ... :: 'b::more |) => bool"
+ monoid :: "(| times :: 'a => 'a => 'a, one :: 'a, ... :: 'b |) => bool"
"monoid M == \<forall>x y z.
times M (times M x y) z = times M x (times M y z) \<and>
times M (one M) x = x \<and> times M x (one M) = x"
- group :: "(| times :: 'a => 'a => 'a, one :: 'a, inv :: 'a => 'a, ... :: 'b::more |) => bool"
+ group :: "(| times :: 'a => 'a => 'a, one :: 'a, inv :: 'a => 'a, ... :: 'b |) => bool"
"group G == monoid G \<and> (\<forall>x. times G (inv G x) x = one G)"
- reverse :: "(| times :: 'a => 'a => 'a, one :: 'a, ... :: 'b::more |) =>
+ reverse :: "(| times :: 'a => 'a => 'a, one :: 'a, ... :: 'b |) =>
(| times :: 'a => 'a => 'a, one :: 'a, ... :: 'b |)"
"reverse M == M (| times := \<lambda>x y. times M y x |)"
--- a/src/HOL/ex/Records.thy Thu Oct 25 22:42:50 2001 +0200
+++ b/src/HOL/ex/Records.thy Thu Oct 25 22:43:05 2001 +0200
@@ -11,121 +11,133 @@
subsection {* Points *}
record point =
- x :: nat
- y :: nat
+ xpos :: nat
+ ypos :: nat
text {*
- Apart many other things, above record declaration produces the
- following theorems:
+ Apart many other things, above record declaration produces the
+ following theorems:
*}
thm "point.simps"
thm "point.iffs"
-thm "point.update_defs"
+thm "point.derived_defs"
text {*
- The set of theorems @{thm [source] point.simps} is added
- automatically to the standard simpset, @{thm [source] point.iffs} is
- added to the Classical Reasoner and Simplifier context.
-*}
+ The set of theorems @{thm [source] point.simps} is added
+ automatically to the standard simpset, @{thm [source] point.iffs} is
+ added to the Classical Reasoner and Simplifier context.
-text {*
- Record declarations define new type abbreviations:
+ \medskip Record declarations define new type abbreviations:
@{text [display]
-" point = (| x :: nat, y :: nat |)
- 'a point_scheme = (| x :: nat, y :: nat, ... :: 'a |)"}
- Extensions `...' must be in type class @{text more}.
+" point = (| xpos :: nat, ypos :: nat |)
+ 'a point_scheme = (| xpos :: nat, ypos :: nat, ... :: 'a |)"}
*}
consts foo1 :: point
-consts foo2 :: "(| x :: nat, y :: nat |)"
-consts foo3 :: "'a => ('a::more) point_scheme"
-consts foo4 :: "'a => (| x :: nat, y :: nat, ... :: 'a |)"
+consts foo2 :: "(| xpos :: nat, ypos :: nat |)"
+consts foo3 :: "'a => 'a point_scheme"
+consts foo4 :: "'a => (| xpos :: nat, ypos :: nat, ... :: 'a |)"
subsubsection {* Introducing concrete records and record schemes *}
defs
- foo1_def: "foo1 == (| x = 1, y = 0 |)"
- foo3_def: "foo3 ext == (| x = 1, y = 0, ... = ext |)"
+ foo1_def: "foo1 == (| xpos = 1, ypos = 0 |)"
+ foo3_def: "foo3 ext == (| xpos = 1, ypos = 0, ... = ext |)"
subsubsection {* 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 |)"
+ getX :: "'a point_scheme => nat"
+ "getX r == xpos r"
+ setX :: "'a point_scheme => nat => 'a point_scheme"
+ "setX r n == r (| xpos := n |)"
subsubsection {* Some lemmas about records *}
text {* Basic simplifications. *}
-lemma "point.make n p = (| x = n, y = p |)"
- by (simp add: point.make_def)
+lemma "point.make n p = (| xpos = n, ypos = p |)"
+ by (simp only: point.make_def)
-lemma "x (| x = m, y = n, ... = p |) = m"
+lemma "xpos (| xpos = m, ypos = n, ... = p |) = m"
by simp
-lemma "(| x = m, y = n, ... = p |) (| x:= 0 |) = (| x = 0, y = n, ... = p |)"
+lemma "(| xpos = m, ypos = n, ... = p |) (| xpos:= 0 |) = (| xpos = 0, ypos = n, ... = p |)"
by simp
text {* \medskip Equality of records. *}
-lemma "n = n' ==> p = p' ==> (| x = n, y = p |) = (| x = n', y = p' |)"
+lemma "n = n' ==> p = p' ==> (| xpos = n, ypos = p |) = (| xpos = n', ypos = p' |)"
-- "introduction of concrete record equality"
by simp
-lemma "(| x = n, y = p |) = (| x = n', y = p' |) ==> n = n'"
+lemma "(| xpos = n, ypos = p |) = (| xpos = n', ypos = p' |) ==> n = n'"
-- "elimination of concrete record equality"
by simp
-lemma "r (| x := n |) (| y := m |) = r (| y := m |) (| x := n |)"
+lemma "r (| xpos := n |) (| ypos := m |) = r (| ypos := m |) (| xpos := n |)"
-- "introduction of abstract record equality"
by simp
-lemma "r (| x := n |) = r (| x := n' |) ==> n = n'"
+lemma "r (| xpos := n |) = r (| xpos := n' |) ==> n = n'"
-- "elimination of abstract record equality (manual proof)"
proof -
- assume "r (| x := n |) = r (| x := n' |)" (is "?lhs = ?rhs")
- hence "x ?lhs = x ?rhs" by simp
+ assume "r (| xpos := n |) = r (| xpos := n' |)" (is "?lhs = ?rhs")
+ hence "xpos ?lhs = xpos ?rhs" by simp
thus ?thesis by simp
qed
text {* \medskip Surjective pairing *}
-lemma "r = (| x = x r, y = y r |)"
+lemma "r = (| xpos = xpos r, ypos = ypos r |)"
by simp
-lemma "r = (| x = x r, y = y r, ... = more r |)"
+lemma "r = (| xpos = xpos r, ypos = ypos r, ... = more r |)"
by simp
text {*
- \medskip Splitting quantifiers: the @{text "!!r"} is \emph{necessary}
- here!
+ \medskip Representation of records by cases or (degenerate)
+ induction.
*}
-lemma "!!r. r (| x := n |) (| y := m |) = r (| y := m |) (| x := n |)"
-proof record_split
- fix x y more
- show "(| x = x, y = y, ... = more |)(| x := n, y := m |) =
- (| x = x, y = y, ... = more |)(| y := m, x := n |)"
+lemma "r (| xpos := n |) (| ypos := m |) = r (| ypos := m |) (| xpos := n |)"
+proof (cases r)
+ fix xpos ypos more
+ assume "r = (| xpos = xpos, ypos = ypos, ... = more |)"
+ thus ?thesis by simp
+qed
+
+lemma "r (| xpos := n |) (| ypos := m |) = r (| ypos := m |) (| xpos := n |)"
+proof (induct r)
+ fix xpos ypos more
+ show "(| xpos = xpos, ypos = ypos, ... = more |) (| xpos := n, ypos := m |) =
+ (| xpos = xpos, ypos = ypos, ... = more |) (| ypos := m, xpos := n |)"
by simp
qed
-lemma "!!r. r (| x := n |) (| x := m |) = r (| x := m |)"
-proof record_split
- fix x y more
- show "(| x = x, y = y, ... = more |)(| x := n, x := m |) =
- (| x = x, y = y, ... = more |)(| x := m |)"
- by simp
+lemma "r (| xpos := n |) (| xpos := m |) = r (| xpos := m |)"
+proof (cases r)
+ fix xpos ypos more
+ assume "r = \<lparr>xpos = xpos, ypos = ypos, \<dots> = more\<rparr>"
+ thus ?thesis by simp
qed
+lemma "r (| xpos := n |) (| xpos := m |) = r (| xpos := m |)"
+proof (cases r)
+ case fields
+ thus ?thesis by simp
+qed
+
+lemma "r (| xpos := n |) (| xpos := m |) = r (| xpos := m |)"
+ by (cases r) simp
+
text {*
\medskip Concrete records are type instances of record schemes.
@@ -133,26 +145,24 @@
constdefs
foo5 :: nat
- "foo5 == getX (| x = 1, y = 0 |)"
+ "foo5 == getX (| xpos = 1, ypos = 0 |)"
-text {* \medskip Manipulating the `...' (more) part. *}
+text {* \medskip Manipulating the ``@{text "..."}'' (more) part. *}
constdefs
- incX :: "('a::more) point_scheme => 'a point_scheme"
- "incX r == (| x = Suc (x r), y = y r, ... = point.more r |)"
+ incX :: "'a point_scheme => 'a point_scheme"
+ "incX r == (| xpos = xpos r + 1, ypos = ypos 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
+lemma "incX r = setX r (Suc (getX r))"
+ by (simp add: getX_def setX_def incX_def)
+
text {* An alternative definition. *}
constdefs
- incX' :: "('a::more) point_scheme => 'a point_scheme"
- "incX' r == r (| x := Suc (x r) |)"
+ incX' :: "'a point_scheme => 'a point_scheme"
+ "incX' r == r (| xpos := xpos r + 1 |)"
subsection {* Coloured points: record extension *}
@@ -166,14 +176,14 @@
text {*
The record declaration defines new type constructors:
@{text [display]
-" cpoint = (| x :: nat, y :: nat, colour :: colour |)
- 'a cpoint_scheme = (| x :: nat, y :: nat, colour :: colour, ... :: 'a |)"}
+" cpoint = (| xpos :: nat, ypos :: nat, colour :: colour |)
+ 'a cpoint_scheme = (| xpos :: nat, ypos :: 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 |)"
+consts foo7 :: "(| xpos :: nat, ypos :: nat, colour :: colour |)"
+consts foo8 :: "'a cpoint_scheme"
+consts foo9 :: "(| xpos :: nat, ypos :: nat, colour :: colour, ... :: 'a |)"
text {*
@@ -182,7 +192,7 @@
constdefs
foo10 :: nat
- "foo10 == getX (| x = 2, y = 0, colour = Blue |)"
+ "foo10 == getX (| xpos = 2, ypos = 0, colour = Blue |)"
subsubsection {* Non-coercive structural subtyping *}
@@ -194,7 +204,7 @@
constdefs
foo11 :: cpoint
- "foo11 == setX (| x = 2, y = 0, colour = Blue |) 0"
+ "foo11 == setX (| xpos = 2, ypos = 0, colour = Blue |) 0"
subsection {* Other features *}
@@ -202,12 +212,12 @@
text {* Field names contribute to record identity. *}
record point' =
- x' :: nat
- y' :: nat
+ xpos' :: nat
+ ypos' :: nat
text {*
- \noindent May not apply @{term getX} to
- @{term [source] "(| x' = 2, y' = 0 |)"}.
+ \noindent May not apply @{term getX} to @{term [source] "(| xpos' =
+ 2, ypos' = 0 |)"} -- type error.
*}
text {* \medskip Polymorphic records. *}