# HG changeset patch
# User wenzelm
# Date 1027462252 -7200
# Node ID 666137b488a4728d06050f460eca06940a024eb6
# Parent  181a293aa37a9aae8a2aa4afd8f8a1727f08dc6c
predicate defs via locales;

diff -r 181a293aa37a -r 666137b488a4 src/HOL/HOL.thy
--- a/src/HOL/HOL.thy	Wed Jul 24 00:09:44 2002 +0200
+++ b/src/HOL/HOL.thy	Wed Jul 24 00:10:52 2002 +0200
@@ -601,15 +601,17 @@
 
 subsubsection {* Monotonicity *}
 
-constdefs
-  mono :: "['a::ord => 'b::ord] => bool"
-  "mono f == ALL A B. A <= B --> f A <= f B"
+locale mono =
+  fixes f
+  assumes mono: "A <= B ==> f A <= f B"
 
-lemma monoI [intro?]: "(!!A B. A <= B ==> f A <= f B) ==> mono f"
-  by (unfold mono_def) rules
+lemmas monoI [intro?] = mono.intro [OF mono_axioms.intro]
+  and monoD [dest?] = mono.mono
 
-lemma monoD [dest?]: "mono f ==> A <= B ==> f A <= f B"
-  by (unfold mono_def) rules
+lemma mono_def: "mono f == ALL A B. A <= B --> f A <= f B"
+  -- compatibility
+  by (simp only: atomize_eq mono_def mono_axioms_def)
+
 
 constdefs
   min :: "['a::ord, 'a] => 'a"
diff -r 181a293aa37a -r 666137b488a4 src/HOL/Record.thy
--- a/src/HOL/Record.thy	Wed Jul 24 00:09:44 2002 +0200
+++ b/src/HOL/Record.thy	Wed Jul 24 00:10:52 2002 +0200
@@ -11,118 +11,57 @@
 
 subsection {* Abstract product types *}
 
-constdefs
-  product_type :: "('p => 'a * 'b) => ('a * 'b => 'p) =>
-    ('a => 'b => 'p) => ('p => 'a) => ('p => 'b) => bool"
-  "product_type Rep Abs pair dest1 dest2 ==
-    type_definition Rep Abs UNIV \<and>
-    pair = (\<lambda>a b. Abs (a, b)) \<and>
-    dest1 = (\<lambda>p. fst (Rep p)) \<and>
-    dest2 = (\<lambda>p. snd (Rep p))"
-
-lemma product_typeI:
-  "type_definition Rep Abs UNIV ==>
-    pair == \<lambda>a b. Abs (a, b) ==>
-    dest1 == (\<lambda>p. fst (Rep p)) ==>
-    dest2 == (\<lambda>p. snd (Rep p)) ==>
-    product_type Rep Abs pair dest1 dest2"
-  by (simp add: product_type_def)
+locale product_type =
+  fixes Rep and Abs and pair and dest1 and dest2
+  assumes "typedef": "type_definition Rep Abs UNIV"
+    and pair: "pair == (\<lambda>a b. Abs (a, b))"
+    and dest1: "dest1 == (\<lambda>p. fst (Rep p))"
+    and dest2: "dest2 == (\<lambda>p. snd (Rep p))"
 
-lemma product_type_typedef:
-    "product_type Rep Abs pair dest1 dest2 ==> type_definition Rep Abs UNIV"
-  by (unfold product_type_def) blast
-
-lemma product_type_pair:
-    "product_type Rep Abs pair dest1 dest2 ==> pair a b = Abs (a, b)"
-  by (unfold product_type_def) blast
+lemmas product_typeI =
+  product_type.intro [OF product_type_axioms.intro]
 
-lemma product_type_dest1:
-    "product_type Rep Abs pair dest1 dest2 ==> dest1 p = fst (Rep p)"
-  by (unfold product_type_def) blast
-
-lemma product_type_dest2:
-    "product_type Rep Abs pair dest1 dest2 ==> dest2 p = snd (Rep p)"
-  by (unfold product_type_def) blast
-
+lemma (in product_type)
+    "inject": "(pair x y = pair x' y') = (x = x' \<and> y = y')"
+  by (simp add: pair type_definition.Abs_inject [OF "typedef"])
 
-theorem product_type_inject:
-  "product_type Rep Abs pair dest1 dest2 ==>
-    (pair x y = pair x' y') = (x = x' \<and> y = y')"
-proof -
-  case rule_context
-  show ?thesis
-    by (simp add: product_type_pair [OF rule_context]
-      Abs_inject [OF product_type_typedef [OF rule_context]])
-qed
+lemma (in product_type) conv1: "dest1 (pair x y) = x"
+  by (simp add: pair dest1 type_definition.Abs_inverse [OF "typedef"])
 
-theorem product_type_conv1:
-  "product_type Rep Abs pair dest1 dest2 ==> dest1 (pair x y) = x"
-proof -
-  case rule_context
-  show ?thesis
-    by (simp add: product_type_pair [OF rule_context]
-      product_type_dest1 [OF rule_context]
-      Abs_inverse [OF product_type_typedef [OF rule_context]])
-qed
+lemma (in product_type) conv2: "dest2 (pair x y) = y"
+  by (simp add: pair dest2 type_definition.Abs_inverse [OF "typedef"])
 
-theorem product_type_conv2:
-  "product_type Rep Abs pair dest1 dest2 ==> dest2 (pair x y) = y"
-proof -
-  case rule_context
-  show ?thesis
-    by (simp add: product_type_pair [OF rule_context]
-      product_type_dest2 [OF rule_context]
-      Abs_inverse [OF product_type_typedef [OF rule_context]])
-qed
-
-theorem product_type_induct [induct set: product_type]:
-  "product_type Rep Abs pair dest1 dest2 ==>
-    (!!x y. P (pair x y)) ==> P p"
-proof -
-  assume hyp: "!!x y. P (pair x y)"
-  assume prod_type: "product_type Rep Abs pair dest1 dest2"
-  show "P p"
-  proof (rule Abs_induct [OF product_type_typedef [OF prod_type]])
-    fix pair show "P (Abs pair)"
-    proof (rule prod_induct)
-      fix x y from hyp show "P (Abs (x, y))"
-        by (simp only: product_type_pair [OF prod_type])
-    qed
+lemma (in product_type) induct [induct type]:
+  assumes hyp: "!!x y. P (pair x y)"
+  shows "P p"
+proof (rule type_definition.Abs_induct [OF "typedef"])
+  fix q show "P (Abs q)"
+  proof (induct q)
+    fix x y have "P (pair x y)" by (rule hyp)
+    also have "pair x y = Abs (x, y)" by (simp only: pair)
+    finally show "P (Abs (x, y))" .
   qed
 qed
 
-theorem product_type_cases [cases set: product_type]:
-  "product_type Rep Abs pair dest1 dest2 ==>
-    (!!x y. p = pair x y ==> C) ==> C"
-proof -
-  assume prod_type: "product_type Rep Abs pair dest1 dest2"
-  assume "!!x y. p = pair x y ==> C"
-  with prod_type show C
-    by (induct p) (simp only: product_type_inject [OF prod_type], blast)
-qed
+lemma (in product_type) cases [cases type]:
+    "(!!x y. p = pair x y ==> C) ==> C"
+  by (induct p) (auto simp add: "inject")
 
-theorem product_type_surjective_pairing:
-  "product_type Rep Abs pair dest1 dest2 ==>
-    p = pair (dest1 p) (dest2 p)"
-proof -
-  case rule_context
-  thus ?thesis by (induct p)
-    (simp add: product_type_conv1 [OF rule_context] product_type_conv2 [OF rule_context])
-qed
+lemma (in product_type) surjective_pairing:
+    "p = pair (dest1 p) (dest2 p)"
+  by (induct p) (simp only: conv1 conv2)
 
-theorem product_type_split_paired_all:
-  "product_type Rep Abs pair dest1 dest2 ==>
-  (!!x. PROP P x) == (!!a b. PROP P (pair a b))"
+lemma (in product_type) split_paired_all:
+  "(!!x. PROP P x) == (!!a b. PROP P (pair a b))"
 proof
   fix a b
   assume "!!x. PROP P x"
   thus "PROP P (pair a b)" .
 next
-  case rule_context
   fix x
   assume "!!a b. PROP P (pair a b)"
   hence "PROP P (pair (dest1 x) (dest2 x))" .
-  thus "PROP P x" by (simp only: product_type_surjective_pairing [OF rule_context, symmetric])
+  thus "PROP P x" by (simp only: surjective_pairing [symmetric])
 qed
 
 
diff -r 181a293aa37a -r 666137b488a4 src/HOL/Typedef.thy
--- a/src/HOL/Typedef.thy	Wed Jul 24 00:09:44 2002 +0200
+++ b/src/HOL/Typedef.thy	Wed Jul 24 00:10:52 2002 +0200
@@ -8,105 +8,79 @@
 theory Typedef = Set
 files ("Tools/typedef_package.ML"):
 
-constdefs
-  type_definition :: "('a => 'b) => ('b => 'a) => 'b set => bool"
-  "type_definition Rep Abs A ==
-    (\<forall>x. Rep x \<in> A) \<and>
-    (\<forall>x. Abs (Rep x) = x) \<and>
-    (\<forall>y \<in> A. Rep (Abs y) = y)"
-  -- {* This will be stated as an axiom for each typedef! *}
+locale type_definition =
+  fixes Rep and Abs and A
+  assumes Rep: "Rep x \<in> A"
+    and Rep_inverse: "Abs (Rep x) = x"
+    and Abs_inverse: "y \<in> A ==> Rep (Abs y) = y"
+  -- {* This will be axiomatized for each typedef! *}
 
-lemma type_definitionI [intro]:
-  "(!!x. Rep x \<in> A) ==>
-    (!!x. Abs (Rep x) = x) ==>
-    (!!y. y \<in> A ==> Rep (Abs y) = y) ==>
-    type_definition Rep Abs A"
-  by (unfold type_definition_def) blast
-
-theorem Rep: "type_definition Rep Abs A ==> Rep x \<in> A"
-  by (unfold type_definition_def) blast
+lemmas type_definitionI [intro] =
+  type_definition.intro [OF type_definition_axioms.intro]
 
-theorem Rep_inverse: "type_definition Rep Abs A ==> Abs (Rep x) = x"
-  by (unfold type_definition_def) blast
-
-theorem Abs_inverse: "type_definition Rep Abs A ==> y \<in> A ==> Rep (Abs y) = y"
-  by (unfold type_definition_def) blast
+lemma (in type_definition) Rep_inject:
+  "(Rep x = Rep y) = (x = y)"
+proof
+  assume "Rep x = Rep y"
+  hence "Abs (Rep x) = Abs (Rep y)" by (simp only:)
+  also have "Abs (Rep x) = x" by (rule Rep_inverse)
+  also have "Abs (Rep y) = y" by (rule Rep_inverse)
+  finally show "x = y" .
+next
+  assume "x = y"
+  thus "Rep x = Rep y" by (simp only:)
+qed
 
-theorem Rep_inject: "type_definition Rep Abs A ==> (Rep x = Rep y) = (x = y)"
-proof -
-  assume tydef: "type_definition Rep Abs A"
-  show ?thesis
-  proof
-    assume "Rep x = Rep y"
-    hence "Abs (Rep x) = Abs (Rep y)" by (simp only:)
-    thus "x = y" by (simp only: Rep_inverse [OF tydef])
-  next
-    assume "x = y"
-    thus "Rep x = Rep y" by simp
-  qed
+lemma (in type_definition) Abs_inject:
+  assumes x: "x \<in> A" and y: "y \<in> A"
+  shows "(Abs x = Abs y) = (x = y)"
+proof
+  assume "Abs x = Abs y"
+  hence "Rep (Abs x) = Rep (Abs y)" by (simp only:)
+  also from x have "Rep (Abs x) = x" by (rule Abs_inverse)
+  also from y have "Rep (Abs y) = y" by (rule Abs_inverse)
+  finally show "x = y" .
+next
+  assume "x = y"
+  thus "Abs x = Abs y" by (simp only:)
 qed
 
-theorem Abs_inject:
-  "type_definition Rep Abs A ==> x \<in> A ==> y \<in> A ==> (Abs x = Abs y) = (x = y)"
-proof -
-  assume tydef: "type_definition Rep Abs A"
-  assume x: "x \<in> A" and y: "y \<in> A"
-  show ?thesis
-  proof
-    assume "Abs x = Abs y"
-    hence "Rep (Abs x) = Rep (Abs y)" by simp
-    moreover from x have "Rep (Abs x) = x" by (rule Abs_inverse [OF tydef])
-    moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef])
-    ultimately show "x = y" by (simp only:)
-  next
-    assume "x = y"
-    thus "Abs x = Abs y" by simp
-  qed
+lemma (in type_definition) Rep_cases [cases set]:
+  assumes y: "y \<in> A"
+    and hyp: "!!x. y = Rep x ==> P"
+  shows P
+proof (rule hyp)
+  from y have "Rep (Abs y) = y" by (rule Abs_inverse)
+  thus "y = Rep (Abs y)" ..
 qed
 
-theorem Rep_cases:
-  "type_definition Rep Abs A ==> y \<in> A ==> (!!x. y = Rep x ==> P) ==> P"
-proof -
-  assume tydef: "type_definition Rep Abs A"
-  assume y: "y \<in> A" and r: "(!!x. y = Rep x ==> P)"
-  show P
-  proof (rule r)
-    from y have "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef])
-    thus "y = Rep (Abs y)" ..
-  qed
+lemma (in type_definition) Abs_cases [cases type]:
+  assumes r: "!!y. x = Abs y ==> y \<in> A ==> P"
+  shows P
+proof (rule r)
+  have "Abs (Rep x) = x" by (rule Rep_inverse)
+  thus "x = Abs (Rep x)" ..
+  show "Rep x \<in> A" by (rule Rep)
 qed
 
-theorem Abs_cases:
-  "type_definition Rep Abs A ==> (!!y. x = Abs y ==> y \<in> A ==> P) ==> P"
+lemma (in type_definition) Rep_induct [induct set]:
+  assumes y: "y \<in> A"
+    and hyp: "!!x. P (Rep x)"
+  shows "P y"
 proof -
-  assume tydef: "type_definition Rep Abs A"
-  assume r: "!!y. x = Abs y ==> y \<in> A ==> P"
-  show P
-  proof (rule r)
-    have "Abs (Rep x) = x" by (rule Rep_inverse [OF tydef])
-    thus "x = Abs (Rep x)" ..
-    show "Rep x \<in> A" by (rule Rep [OF tydef])
-  qed
+  have "P (Rep (Abs y))" by (rule hyp)
+  also from y have "Rep (Abs y) = y" by (rule Abs_inverse)
+  finally show "P y" .
 qed
 
-theorem Rep_induct:
-  "type_definition Rep Abs A ==> y \<in> A ==> (!!x. P (Rep x)) ==> P y"
+lemma (in type_definition) Abs_induct [induct type]:
+  assumes r: "!!y. y \<in> A ==> P (Abs y)"
+  shows "P x"
 proof -
-  assume tydef: "type_definition Rep Abs A"
-  assume "!!x. P (Rep x)" hence "P (Rep (Abs y))" .
-  moreover assume "y \<in> A" hence "Rep (Abs y) = y" by (rule Abs_inverse [OF tydef])
-  ultimately show "P y" by (simp only:)
-qed
-
-theorem Abs_induct:
-  "type_definition Rep Abs A ==> (!!y. y \<in> A ==> P (Abs y)) ==> P x"
-proof -
-  assume tydef: "type_definition Rep Abs A"
-  assume r: "!!y. y \<in> A ==> P (Abs y)"
-  have "Rep x \<in> A" by (rule Rep [OF tydef])
+  have "Rep x \<in> A" by (rule Rep)
   hence "P (Abs (Rep x))" by (rule r)
-  moreover have "Abs (Rep x) = x" by (rule Rep_inverse [OF tydef])
-  ultimately show "P x" by (simp only:)
+  also have "Abs (Rep x) = x" by (rule Rep_inverse)
+  finally show "P x" .
 qed
 
 use "Tools/typedef_package.ML"