Added new files for proof terms.
(* Title: HOL/subset.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Subset lemmas and HOL type definitions.
*)
theory subset = Set
files "Tools/induct_attrib.ML" ("Tools/typedef_package.ML"):
(** belongs to theory Ord **)
theorems linorder_cases [case_names less equal greater] =
linorder_less_split
(* Courtesy of Stephan Merz *)
lemma Least_mono:
"[| mono (f::'a::order => 'b::order); EX x:S. ALL y:S. x <= y |]
==> (LEAST y. y : f`S) = f(LEAST x. x : S)"
apply clarify
apply (erule_tac P = "%x. x : S" in LeastI2)
apply fast
apply (rule LeastI2)
apply (auto elim: monoD intro!: order_antisym)
done
(*belongs to theory Set*)
setup Rulify.setup
section {* HOL type definitions *}
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! *}
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
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
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
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
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
qed
theorem Abs_cases:
"type_definition Rep Abs A ==> (!!y. x = Abs y ==> y \\<in> A ==> P) ==> P"
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
qed
theorem Rep_induct:
"type_definition Rep Abs A ==> y \\<in> A ==> (!!x. P (Rep x)) ==> P y"
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])
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:)
qed
setup InductAttrib.setup
use "Tools/typedef_package.ML"
end