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Set Implicit Arguments.
Notation "'fun' { x : A | P } => Q" :=
(fun x:{x:A|P} => Q)
(at level 200, x ident, right associativity).
Notation "( x & ? )" := (@exist _ _ x _) : core_scope.
Notation " ! " := (False_rect _ _).
Definition ex_pi1 (A : Prop) (P : A -> Prop) (t : ex P) : A.
intros.
induction t.
exact x.
Defined.
Lemma ex_pi2 : forall (A : Prop) (P : A -> Prop) (t : ex P),
P (ex_pi1 t).
intros A P.
dependent inversion t.
simpl.
exact p.
Defined.
Notation "` t" := (proj1_sig t) (at level 100) : core_scope.
Notation "'forall' { x : A | P } , Q" :=
(forall x:{x:A|P}, Q)
(at level 200, x ident, right associativity).
Lemma subset_simpl : forall (A : Set) (P : A -> Prop)
(t : sig P), P (` t).
Proof.
intros.
induction t.
simpl ; auto.
Qed.
Ltac destruct_one_pair :=
match goal with
| [H : (ex _) |- _] => destruct H
| [H : (ex2 _) |- _] => destruct H
| [H : (sig _) |- _] => destruct H
| [H : (_ /\ _) |- _] => destruct H
end.
Ltac destruct_exists := repeat (destruct_one_pair) .
Ltac subtac_simpl := simpl ; intros ; destruct_exists ; simpl in * ; try subst ; auto with arith.
(* Destructs calls to f in hypothesis or conclusion, useful if f creates a subset object *)
Ltac destruct_call f :=
match goal with
| H : ?T |- _ =>
match T with
context [f ?x ?y ?z] => destruct (f x y z)
| context [f ?x ?y] => destruct (f x y)
| context [f ?x] => destruct (f x)
end
| |- ?T =>
match T with
context [f ?x ?y ?z] => let n := fresh "H" in set (n:=f x y z); destruct n
| context [f ?x ?y] => let n := fresh "H" in set (n:=f x y); destruct n
| context [f ?x] => let n := fresh "H" in set (n:=f x); destruct n
end
end.
Extraction Inline proj1_sig.
Extract Inductive unit => "unit" [ "()" ].
Extract Inductive bool => "bool" [ "true" "false" ].
Extract Inductive sumbool => "bool" [ "true" "false" ].
Extract Inductive prod => "pair" [ "" ].
Extract Inductive sigT => "pair" [ "" ].
Require Export ProofIrrelevance.
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