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.

Definition des p :=

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

Lemma des_correct p x:

unique p x -> p (des p).

Lemma des_unique p x:

unique p x -> x = des p.

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.

Definition join M :=

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

Lemma join_union x:

join (class x) = union x.

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

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.