Formalisation of "Axiomatic Set Theory in Type Theory" by Gert Smolka

- Version: 16 May 2015
- Author: Dominik Kirst, Saarland University
- This file shows how the general linearity proof for towers is instantiated
- Acknowlegments: This instantiation is based on the library of the general proof given by Gert Smolka, Steven SchÃ¤fer and Christian Doczkal (https://www.ps.uni-saarland.de/extras/itp15/)

# Instantiation of Linearity Proof

Definition increasing f :=

forall x, x <<= f x.

Variable f: set -> set.

Variable FI: increasing f.

Inductive Tower: set -> Prop :=

| TU x: subsc x Tower -> Tower (union x)

| TS x: Tower x -> Tower (f x).

2. To define a join operator for classes over sets (= realizable union),
we have to assume an empty set and define the general description operator.

Axiom empty: set.

Axiom Empty: ~ inhab empty.

Lemma ninhab_union x:

~ inhab x -> union x = empty.

Proof.

intros H. apply sub_anti; intros y Y.

- apply Union in Y as [z[Z _]]. contradict H. now exists z.

- exfalso. apply Empty. now exists y.

Qed.

Definition des p :=

union (rep (sing empty) (fun y => p)).

Lemma des_correct p x:

unique p x -> p (des p).

Proof.

intros [P1 P2]. cutrewrite (des p = x); trivial.

apply sub_anti; intros y Y.

- apply Union in Y as [z[Z1 Z2]]. apply Rep in Z1 as [u[U1 U2]].

cutrewrite (x = z); trivial. apply P2. apply U2.

- apply Union. exists x. split; trivial. apply Rep.

exists empty. split; firstorder using el_sing.

Qed.

Lemma des_unique p x:

unique p x -> x = des p.

Proof.

intros H. apply H. apply (des_correct H).

Qed.

3. Since the realizes predicate acts uniquely on sets,
we can derive the wished join operator.

Lemma realizes_unique M x:

realizes x M -> unique (fun x => realizes x M) x.

Proof.

intros XM. split; trivial. intros y YM. apply Ext. firstorder.

Qed.

Definition join M :=

union (des (fun x => realizes x M)).

Lemma join_union x:

join (class x) = union x.

Proof.

unfold join. rewrite <- des_unique with (x:=x); trivial. firstorder using Ext.

Qed.

4. Altogether, the type of sets with the inclusion ordering
implements the abstract definition of a complete partial order.

Definition sets: CompletePartialOrder.CompletePartialOrder.

exists set realizable sub join; auto using sub_anti.

intros M y [x X]. apply realizes_unique in X. split; intros H.

- apply union_incl. rewrite <- (des_unique X). firstorder.

- unfold join in H. rewrite <- (des_unique X) in H.

destruct (union_incl y x) as [I _]. firstorder.

Defined.

5. Finally, since our predicate Tower corresponds to Reach,
the predicate that describes transfinite reachability in the library,
we can easily instantiate the linearity proof for towers.

Lemma tower_reach x:

Tower x -> Reach (T:=sets) f empty x.

Proof.

induction 1 as [x H IH|x H IH].

- destruct (classic (inhab x)) as [NE|E].

+ rewrite <- (join_union x). constructor; firstorder.

+ rewrite (ninhab_union E). constructor.

- now constructor.

Qed.

Lemma linearity x y:

Tower x -> Tower y -> x <<= y \/ y <<= x.

Proof.

intros TX TY. apply (@linearity sets f FI) with (c:=empty).

- intros p q PQ [z Z]. exists (sep z p). apply sep_realizes. firstorder.

- now apply tower_reach.

- now apply tower_reach.

Qed.