--- a/src/HOL/Library/Finite_Cartesian_Product.thy Tue Oct 27 12:59:57 2009 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,95 +0,0 @@
-(* Title: HOL/Library/Finite_Cartesian_Product
- Author: Amine Chaieb, University of Cambridge
-*)
-
-header {* Definition of finite Cartesian product types. *}
-
-theory Finite_Cartesian_Product
-imports Main (*FIXME: ATP_Linkup is only needed for metis at a few places. We could dispense of that by changing the proofs.*)
-begin
-
-definition hassize (infixr "hassize" 12) where
- "(S hassize n) = (finite S \<and> card S = n)"
-
-lemma hassize_image_inj: assumes f: "inj_on f S" and S: "S hassize n"
- shows "f ` S hassize n"
- using f S card_image[OF f]
- by (simp add: hassize_def inj_on_def)
-
-
-subsection {* Finite Cartesian products, with indexing and lambdas. *}
-
-typedef (open Cart)
- ('a, 'b) "^" (infixl "^" 15)
- = "UNIV :: ('b \<Rightarrow> 'a) set"
- morphisms Cart_nth Cart_lambda ..
-
-notation Cart_nth (infixl "$" 90)
-
-notation (xsymbols) Cart_lambda (binder "\<chi>" 10)
-
-lemma stupid_ext: "(\<forall>x. f x = g x) \<longleftrightarrow> (f = g)"
- apply auto
- apply (rule ext)
- apply auto
- done
-
-lemma Cart_eq: "((x:: 'a ^ 'b) = y) \<longleftrightarrow> (\<forall>i. x$i = y$i)"
- by (simp add: Cart_nth_inject [symmetric] expand_fun_eq)
-
-lemma Cart_lambda_beta [simp]: "Cart_lambda g $ i = g i"
- by (simp add: Cart_lambda_inverse)
-
-lemma Cart_lambda_unique:
- fixes f :: "'a ^ 'b"
- shows "(\<forall>i. f$i = g i) \<longleftrightarrow> Cart_lambda g = f"
- by (auto simp add: Cart_eq)
-
-lemma Cart_lambda_eta: "(\<chi> i. (g$i)) = g"
- by (simp add: Cart_eq)
-
-text{* A non-standard sum to "paste" Cartesian products. *}
-
-definition pastecart :: "'a ^ 'm \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ ('m + 'n)" where
- "pastecart f g = (\<chi> i. case i of Inl a \<Rightarrow> f$a | Inr b \<Rightarrow> g$b)"
-
-definition fstcart:: "'a ^('m + 'n) \<Rightarrow> 'a ^ 'm" where
- "fstcart f = (\<chi> i. (f$(Inl i)))"
-
-definition sndcart:: "'a ^('m + 'n) \<Rightarrow> 'a ^ 'n" where
- "sndcart f = (\<chi> i. (f$(Inr i)))"
-
-lemma nth_pastecart_Inl [simp]: "pastecart f g $ Inl a = f$a"
- unfolding pastecart_def by simp
-
-lemma nth_pastecart_Inr [simp]: "pastecart f g $ Inr b = g$b"
- unfolding pastecart_def by simp
-
-lemma nth_fstcart [simp]: "fstcart f $ i = f $ Inl i"
- unfolding fstcart_def by simp
-
-lemma nth_sndtcart [simp]: "sndcart f $ i = f $ Inr i"
- unfolding sndcart_def by simp
-
-lemma finite_sum_image: "(UNIV::('a + 'b) set) = range Inl \<union> range Inr"
-by (auto, case_tac x, auto)
-
-lemma fstcart_pastecart: "fstcart (pastecart (x::'a ^'m ) (y:: 'a ^ 'n)) = x"
- by (simp add: Cart_eq)
-
-lemma sndcart_pastecart: "sndcart (pastecart (x::'a ^'m ) (y:: 'a ^ 'n)) = y"
- by (simp add: Cart_eq)
-
-lemma pastecart_fst_snd: "pastecart (fstcart z) (sndcart z) = z"
- by (simp add: Cart_eq pastecart_def fstcart_def sndcart_def split: sum.split)
-
-lemma pastecart_eq: "(x = y) \<longleftrightarrow> (fstcart x = fstcart y) \<and> (sndcart x = sndcart y)"
- using pastecart_fst_snd[of x] pastecart_fst_snd[of y] by metis
-
-lemma forall_pastecart: "(\<forall>p. P p) \<longleftrightarrow> (\<forall>x y. P (pastecart x y))"
- by (metis pastecart_fst_snd fstcart_pastecart sndcart_pastecart)
-
-lemma exists_pastecart: "(\<exists>p. P p) \<longleftrightarrow> (\<exists>x y. P (pastecart x y))"
- by (metis pastecart_fst_snd fstcart_pastecart sndcart_pastecart)
-
-end