(* Title: HOL/Import/MakeEqual.thy
ID: $Id$
Author: Sebastian Skalberg (TU Muenchen)
*)
theory MakeEqual imports Main
uses "shuffler.ML" begin
setup Shuffler.setup
lemma conj_norm [shuffle_rule]: "(A & B ==> PROP C) == ([| A ; B |] ==> PROP C)"
proof
assume "A & B ==> PROP C" A B
thus "PROP C"
by auto
next
assume "[| A; B |] ==> PROP C" "A & B"
thus "PROP C"
by auto
qed
lemma imp_norm [shuffle_rule]: "(Trueprop (A --> B)) == (A ==> B)"
proof
assume "A --> B" A
thus B ..
next
assume "A ==> B"
thus "A --> B"
by auto
qed
lemma all_norm [shuffle_rule]: "(Trueprop (ALL x. P x)) == (!!x. P x)"
proof
fix x
assume "ALL x. P x"
thus "P x" ..
next
assume "!!x. P x"
thus "ALL x. P x"
..
qed
lemma ex_norm [shuffle_rule]: "(EX x. P x ==> PROP Q) == (!!x. P x ==> PROP Q)"
proof
fix x
assume ex: "EX x. P x ==> PROP Q"
assume "P x"
hence "EX x. P x" ..
with ex show "PROP Q" .
next
assume allx: "!!x. P x ==> PROP Q"
assume "EX x. P x"
hence p: "P (SOME x. P x)"
..
from allx
have "P (SOME x. P x) ==> PROP Q"
.
with p
show "PROP Q"
by auto
qed
lemma eq_norm [shuffle_rule]: "Trueprop (t = u) == (t == u)"
proof
assume "t = u"
thus "t == u" by simp
next
assume "t == u"
thus "t = u"
by simp
qed
end