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(* first part: prop2 and lambda2 and their connection *)
(* via the Curry-Howard-De Bruijn isomorphism *)
(* the paradigmatic example: the polymorphic identity *)
Lemma example1a : forall a:Prop, a->a.
Proof.
intro a.
intro x.
exact x.
Qed.
Print example1a.
(* \a:Prop. \x:a. x *)
Lemma example1b : forall a:Set, a -> a.
Proof.
intro a.
intro x.
exact x.
Qed.
Print example1b.
(* \a:Set. \x:a. x *)
(* another example from the course *)
Lemma example2: forall a:Prop, a -> (forall b:Prop, ((a -> b) -> b)).
Proof.
intro a.
intro x.
intro b.
intro y.
apply y.
exact x.
Qed.
Print example2.
(* \a:Prop. \x:a. \b:Prop. \y:a->b. (y x) *)
(* the following "lemma" is not valid *)
(*
Lemma example_wrong : forall a:Prop, a -> (forall b:Prop, b).
Proof.
intro a.
intro x.
intro b.
Abort.
*)
(* the polymorphic identity *)
Definition polyid : forall a:Set, forall x:a, a :=
fun a:Set => fun x:a => x.
(* instantiation by application *)
(* Check gives the types *)
Check polyid.
Check polyid nat.
Check polyid nat O.
(* Eval compute computes normal forms *)
Eval compute in (polyid nat).
Eval compute in (polyid nat O).
(* exercise 1 *)
(* give inhabitants of the following types *)
(* forall a : Prop, a -> a *)
(* forall a b : Prop, (a -> b) -> a -> b *)
(* forall a : Prop, a -> forall b : Prop, (a -> b) -> b*)
(* exercises with negation *)
Lemma exercise2 : forall a:Prop, a -> ~~a.
Proof.
intros a H H0.
apply H0.
exact H.
Qed.
Lemma exercise3: forall a:Prop, ~~~a -> ~a.
Proof.
intros a H H0.
apply H.
apply exercise2.
exact H0.
Qed.
Lemma exercise4: forall a:Prop, ~~(~~a -> a).
Proof.
intros a H.
apply H.
intro.
elimtype False.
apply H0.
intro.
apply H.
intro.
exact H1.
Qed.
(* exercises with full intuitionistic prop2 *)
Lemma exercise5 : forall a:Prop, (exists b:Prop, a) -> a.
Proof.
intros a H.
elim H.
intros b Ha.
exact Ha.
Qed.
Lemma exercise6 : forall a:Prop, exists b:Prop, ((a -> b) \/ (b -> a)).
Proof.
intro.
exists a.
left.
intro.
assumption.
Qed.
(* exercise with classical prop2 *)
Definition em:= forall a:Prop, a \/ ~a.
Lemma exercise7 : em -> forall a b:Prop, ((a -> b) \/ (b -> a)).
Proof.
unfold em.
intros em' a b.
assert (a \/ ~a).
apply em'.
inversion_clear H.
right. intro. assumption.
left. intro. elimtype False. apply H0. assumption.
Qed.
(* expressibility of prop2 *)
(* we will represent logical connectives in prop2 *)
(* definition of false *)
Definition new_false := forall a:Prop, a.
(* False implies new_false *)
Lemma exercise8 : False -> new_false.
Proof.
intro.
elimtype False.
exact H.
Qed.
(* new_false implies False *)
Lemma exercise9 : new_false -> False.
Proof.
unfold new_false.
intro.
apply H.
Qed.
(* from new_false we can prove anything *)
Lemma exercise10 : forall a:Prop, new_false -> a.
Proof.
unfold new_false.
intros a H.
apply H.
Qed.
(* definition of and *)
(* "a and b" is valid is everything that can be
derived from {a,b} is valid *)
Definition new_and (a b : Prop) := forall c : Prop, (a -> b -> c) -> c.
(* and implies new_and *)
Lemma exercise11 : forall a b : Prop,a /\ b -> new_and a b.
Proof.
unfold new_and.
intros a b H c H'.
elim H.
intros Ha Hb.
apply H'.
apply Ha.
apply Hb.
Qed.
(* new_and implies and *)
Lemma exercise12 : forall a b : Prop ,
new_and a b -> a /\ b.
Proof.
unfold new_and.
intros a b H.
split ; apply H ; intros Ha Hb ; assumption.
Qed.
(* exercise 13 *)
(* assume an inhabitant P of new_and A B with A and B in Prop *)
(* give an inhabitant of A and an inhabitant of B *)
Parameters A B : Prop.
Parameter P : new_and A B.
(* "a or b" is valid if everything that follows form
{a} and follows from {b} is valid *)
Definition new_or (a b : Prop) := forall c : Prop, (a -> c) -> (b -> c) -> c.
Lemma exercise14 : forall a b , a \/ b -> new_or a b.
Proof.
unfold new_or.
intros a b H c Hac Hbc.
inversion_clear H.
apply Hac. assumption.
apply Hbc. assumption.
Qed.
Lemma exercise15 : forall a b , new_or a b -> a \/ b.
Proof.
unfold new_or.
intros a b H.
apply H.
intro. left. assumption.
intro. right. assumption.
Qed.
(* exercise 16 *)
(* given an inhabitant Q:A *)
(* give an inhabitant of new_or A B with A and B in Prop *)
Parameter Q:A.
Check (fun (c : Prop) (Ha : A -> c) (Hb : B -> c) => Ha Q) : new_or A B.
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