(* Title: HOL/HOLCF/Sprod.thy
Author: Franz Regensburger
Author: Brian Huffman
*)
section \<open>The type of strict products\<close>
theory Sprod
imports Cfun
begin
default_sort pcpo
subsection \<open>Definition of strict product type\<close>
definition "sprod = {p::'a \ 'b. p = \ \ (fst p \ \ \ snd p \ \)}"
pcpodef ('a, 'b) sprod ("(_ \/ _)" [21,20] 20) = "sprod :: ('a \ 'b) set"
by (simp_all add: sprod_def)
instance sprod :: ("{chfin,pcpo}", "{chfin,pcpo}") chfin
by (rule typedef_chfin [OF type_definition_sprod below_sprod_def])
type_notation (ASCII)
sprod (infixr "**" 20)
subsection \<open>Definitions of constants\<close>
definition sfst :: "('a ** 'b) \ 'a"
where "sfst = (\ p. fst (Rep_sprod p))"
definition ssnd :: "('a ** 'b) \ 'b"
where "ssnd = (\ p. snd (Rep_sprod p))"
definition spair :: "'a \ 'b \ ('a ** 'b)"
where "spair = (\ a b. Abs_sprod (seq\b\a, seq\a\b))"
definition ssplit :: "('a \ 'b \ 'c) \ ('a ** 'b) \ 'c"
where "ssplit = (\ f p. seq\p\(f\(sfst\p)\(ssnd\p)))"
syntax "_stuple" :: "[logic, args] \ logic" ("(1'(:_,/ _:'))")
translations
"(:x, y, z:)" \<rightleftharpoons> "(:x, (:y, z:):)"
"(:x, y:)" \<rightleftharpoons> "CONST spair\<cdot>x\<cdot>y"
translations
"\(CONST spair\x\y). t" \ "CONST ssplit\(\ x y. t)"
subsection \<open>Case analysis\<close>
lemma spair_sprod: "(seq\b\a, seq\a\b) \ sprod"
by (simp add: sprod_def seq_conv_if)
lemma Rep_sprod_spair: "Rep_sprod (:a, b:) = (seq\b\a, seq\a\b)"
by (simp add: spair_def cont_Abs_sprod Abs_sprod_inverse spair_sprod)
lemmas Rep_sprod_simps =
Rep_sprod_inject [symmetric] below_sprod_def
prod_eq_iff below_prod_def
Rep_sprod_strict Rep_sprod_spair
lemma sprodE [case_names bottom spair, cases type: sprod]:
obtains "p = \" | x y where "p = (:x, y:)" and "x \ \" and "y \ \"
using Rep_sprod [of p] by (auto simp add: sprod_def Rep_sprod_simps)
lemma sprod_induct [case_names bottom spair, induct type: sprod]:
"\P \; \x y. \x \ \; y \ \\ \ P (:x, y:)\ \ P x"
by (cases x) simp_all
subsection \<open>Properties of \emph{spair}\<close>
lemma spair_strict1 [simp]: "(:\, y:) = \"
by (simp add: Rep_sprod_simps)
lemma spair_strict2 [simp]: "(:x, \:) = \"
by (simp add: Rep_sprod_simps)
lemma spair_bottom_iff [simp]: "(:x, y:) = \ \ x = \ \ y = \"
by (simp add: Rep_sprod_simps seq_conv_if)
lemma spair_below_iff: "(:a, b:) \ (:c, d:) \ a = \ \ b = \ \ (a \ c \ b \ d)"
by (simp add: Rep_sprod_simps seq_conv_if)
lemma spair_eq_iff: "(:a, b:) = (:c, d:) \ a = c \ b = d \ (a = \ \ b = \) \ (c = \ \ d = \)"
by (simp add: Rep_sprod_simps seq_conv_if)
lemma spair_strict: "x = \ \ y = \ \ (:x, y:) = \"
by simp
lemma spair_strict_rev: "(:x, y:) \ \ \ x \ \ \ y \ \"
by simp
lemma spair_defined: "\x \ \; y \ \\ \ (:x, y:) \ \"
by simp
lemma spair_defined_rev: "(:x, y:) = \ \ x = \ \ y = \"
by simp
lemma spair_below: "x \ \ \ y \ \ \ (:x, y:) \ (:a, b:) \ x \ a \ y \ b"
by (simp add: spair_below_iff)
lemma spair_eq: "x \ \ \ y \ \ \ (:x, y:) = (:a, b:) \ x = a \ y = b"
by (simp add: spair_eq_iff)
lemma spair_inject: "x \ \ \ y \ \ \ (:x, y:) = (:a, b:) \ x = a \ y = b"
by (rule spair_eq [THEN iffD1])
lemma inst_sprod_pcpo2: "\ = (:\, \:)"
by simp
lemma sprodE2: "(\x y. p = (:x, y:) \ Q) \ Q"
by (cases p) (simp only: inst_sprod_pcpo2, simp)
subsection \<open>Properties of \emph{sfst} and \emph{ssnd}\<close>
lemma sfst_strict [simp]: "sfst\\ = \"
by (simp add: sfst_def cont_Rep_sprod Rep_sprod_strict)
lemma ssnd_strict [simp]: "ssnd\\ = \"
by (simp add: ssnd_def cont_Rep_sprod Rep_sprod_strict)
lemma sfst_spair [simp]: "y \ \ \ sfst\(:x, y:) = x"
by (simp add: sfst_def cont_Rep_sprod Rep_sprod_spair)
lemma ssnd_spair [simp]: "x \ \ \ ssnd\(:x, y:) = y"
by (simp add: ssnd_def cont_Rep_sprod Rep_sprod_spair)
lemma sfst_bottom_iff [simp]: "sfst\p = \ \ p = \"
by (cases p) simp_all
lemma ssnd_bottom_iff [simp]: "ssnd\p = \ \ p = \"
by (cases p) simp_all
lemma sfst_defined: "p \ \ \ sfst\p \ \"
by simp
lemma ssnd_defined: "p \ \ \ ssnd\p \ \"
by simp
lemma spair_sfst_ssnd: "(:sfst\p, ssnd\p:) = p"
by (cases p) simp_all
lemma below_sprod: "x \ y \ sfst\x \ sfst\y \ ssnd\x \ ssnd\y"
by (simp add: Rep_sprod_simps sfst_def ssnd_def cont_Rep_sprod)
lemma eq_sprod: "x = y \ sfst\x = sfst\y \ ssnd\x = ssnd\y"
by (auto simp add: po_eq_conv below_sprod)
lemma sfst_below_iff: "sfst\x \ y \ x \ (:y, ssnd\x:)"
by (cases "x = \", simp, cases "y = \", simp, simp add: below_sprod)
lemma ssnd_below_iff: "ssnd\x \ y \ x \ (:sfst\x, y:)"
by (cases "x = \", simp, cases "y = \", simp, simp add: below_sprod)
subsection \<open>Compactness\<close>
lemma compact_sfst: "compact x \ compact (sfst\x)"
by (rule compactI) (simp add: sfst_below_iff)
lemma compact_ssnd: "compact x \ compact (ssnd\x)"
by (rule compactI) (simp add: ssnd_below_iff)
lemma compact_spair: "compact x \ compact y \ compact (:x, y:)"
by (rule compact_sprod) (simp add: Rep_sprod_spair seq_conv_if)
lemma compact_spair_iff: "compact (:x, y:) \ x = \ \ y = \ \ (compact x \ compact y)"
apply (safe elim!: compact_spair)
apply (drule compact_sfst, simp)
apply (drule compact_ssnd, simp)
apply simp
apply simp
done
subsection \<open>Properties of \emph{ssplit}\<close>
lemma ssplit1 [simp]: "ssplit\f\\ = \"
by (simp add: ssplit_def)
lemma ssplit2 [simp]: "x \ \ \ y \ \ \ ssplit\f\(:x, y:) = f\x\y"
by (simp add: ssplit_def)
lemma ssplit3 [simp]: "ssplit\spair\z = z"
by (cases z) simp_all
subsection \<open>Strict product preserves flatness\<close>
instance sprod :: (flat, flat) flat
proof
fix x y :: "'a \ 'b"
assume "x \ y"
then show "x = \ \ x = y"
apply (induct x, simp)
apply (induct y, simp)
apply (simp add: spair_below_iff flat_below_iff)
done
qed
end
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