author  huffman 
Tue, 14 Aug 2007 23:05:55 +0200  
changeset 24269  4b2aac7669b3 
parent 23710  a8ac2305eaf2 
child 25535  4975b7529a14 
permissions  rwrr 
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(* Title: HOL/Typedef.thy 
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ID: $Id$ 

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Author: Markus Wenzel, TU Munich 

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*) 
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header {* HOL type definitions *} 
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theory Typedef 
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imports Set 
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uses 
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("Tools/typedef_package.ML") 

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("Tools/typecopy_package.ML") 

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("Tools/typedef_codegen.ML") 

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begin 
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ML {* 
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structure HOL = struct val thy = theory "HOL" end; 

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*}  "belongs to theory HOL" 

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locale type_definition = 
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fixes Rep and Abs and A 

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assumes Rep: "Rep x \<in> A" 

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and Rep_inverse: "Abs (Rep x) = x" 

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and Abs_inverse: "y \<in> A ==> Rep (Abs y) = y" 

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 {* This will be axiomatized for each typedef! *} 

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begin 
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lemma Rep_inject: 
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"(Rep x = Rep y) = (x = y)" 
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proof 

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assume "Rep x = Rep y" 

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then have "Abs (Rep x) = Abs (Rep y)" by (simp only:) 
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removed proof dependency on transitivity theorems
haftmann
parents:
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moreover have "Abs (Rep x) = x" by (rule Rep_inverse) 
a8ac2305eaf2
removed proof dependency on transitivity theorems
haftmann
parents:
23433
diff
changeset

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moreover have "Abs (Rep y) = y" by (rule Rep_inverse) 
a8ac2305eaf2
removed proof dependency on transitivity theorems
haftmann
parents:
23433
diff
changeset

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ultimately show "x = y" by simp 
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next 
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assume "x = y" 

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thus "Rep x = Rep y" by (simp only:) 

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qed 

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lemma Abs_inject: 
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assumes x: "x \<in> A" and y: "y \<in> A" 
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shows "(Abs x = Abs y) = (x = y)" 

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proof 

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assume "Abs x = Abs y" 

23710
a8ac2305eaf2
removed proof dependency on transitivity theorems
haftmann
parents:
23433
diff
changeset

46 
then have "Rep (Abs x) = Rep (Abs y)" by (simp only:) 
a8ac2305eaf2
removed proof dependency on transitivity theorems
haftmann
parents:
23433
diff
changeset

47 
moreover from x have "Rep (Abs x) = x" by (rule Abs_inverse) 
a8ac2305eaf2
removed proof dependency on transitivity theorems
haftmann
parents:
23433
diff
changeset

48 
moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse) 
a8ac2305eaf2
removed proof dependency on transitivity theorems
haftmann
parents:
23433
diff
changeset

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ultimately show "x = y" by simp 
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next 
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assume "x = y" 

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thus "Abs x = Abs y" by (simp only:) 

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qed 
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lemma Rep_cases [cases set]: 
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assumes y: "y \<in> A" 
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and hyp: "!!x. y = Rep x ==> P" 

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shows P 

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proof (rule hyp) 

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from y have "Rep (Abs y) = y" by (rule Abs_inverse) 

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thus "y = Rep (Abs y)" .. 

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qed 
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lemma Abs_cases [cases type]: 
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assumes r: "!!y. x = Abs y ==> y \<in> A ==> P" 
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shows P 

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proof (rule r) 

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have "Abs (Rep x) = x" by (rule Rep_inverse) 

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thus "x = Abs (Rep x)" .. 

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show "Rep x \<in> A" by (rule Rep) 

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qed 
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lemma Rep_induct [induct set]: 
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assumes y: "y \<in> A" 
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and hyp: "!!x. P (Rep x)" 

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shows "P y" 

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proof  
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have "P (Rep (Abs y))" by (rule hyp) 
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haftmann
parents:
23433
diff
changeset

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moreover from y have "Rep (Abs y) = y" by (rule Abs_inverse) 
a8ac2305eaf2
removed proof dependency on transitivity theorems
haftmann
parents:
23433
diff
changeset

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ultimately show "P y" by simp 
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qed 
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lemma Abs_induct [induct type]: 
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assumes r: "!!y. y \<in> A ==> P (Abs y)" 
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shows "P x" 

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proof  
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have "Rep x \<in> A" by (rule Rep) 
23710
a8ac2305eaf2
removed proof dependency on transitivity theorems
haftmann
parents:
23433
diff
changeset

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then have "P (Abs (Rep x))" by (rule r) 
a8ac2305eaf2
removed proof dependency on transitivity theorems
haftmann
parents:
23433
diff
changeset

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moreover have "Abs (Rep x) = x" by (rule Rep_inverse) 
a8ac2305eaf2
removed proof dependency on transitivity theorems
haftmann
parents:
23433
diff
changeset

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ultimately show "P x" by simp 
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qed 
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lemma Rep_range: 
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shows "range Rep = A" 
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proof 

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show "range Rep <= A" using Rep by (auto simp add: image_def) 

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show "A <= range Rep" 

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proof 
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fix x assume "x : A" 

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hence "x = Rep (Abs x)" by (rule Abs_inverse [symmetric]) 
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thus "x : range Rep" by (rule range_eqI) 

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qed 
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qed 

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end 
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use "Tools/typedef_package.ML" 
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use "Tools/typecopy_package.ML" 
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use "Tools/typedef_codegen.ML" 
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setup {* 
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TypecopyPackage.setup 
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#> TypedefCodegen.setup 
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*} 

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end 