Library ZFord2

Require Import ZFnats ZFord ZFfix.
Import IZF.

Definition isPOrd x :=
  exists2 o, isOrd o & x == VN o.

Instance isPOrd_morph : Proper (eq_set ==> iff) isPOrd.

Qed.

Lemma isPOrd_zero : isPOrd zero.

Definition osucc := power.

Lemma isPOrd_succ : forall o, isPOrd o -> isPOrd (osucc o).

Definition HF := VN N.

Lemma isPOrd_HF : isPOrd HF.

Definition olt x y := isPOrd x /\ isPOrd y /\ x \in y.

Instance olt_morph : Proper (eq_set ==> eq_set ==> iff) olt.
Qed.

Lemma olt_ord1 : forall x y, olt x y -> isPOrd x.

Lemma olt_ord2 : forall x y, olt x y -> isPOrd y.

Lemma olt_succ : forall x y,
  olt x y -> olt (osucc x) (osucc y).

Lemma olt_trans : forall x y z, olt x y -> olt y z -> olt x z.

Lemma HF_limit : forall o,
  olt o HF ->
  olt (osucc o) HF.

Definition ole x y := isPOrd x /\ isPOrd y /\ x \incl y.

Instance ole_morph : Proper (eq_set ==> eq_set ==> iff) ole.
Qed.

Lemma olt_weak : forall x y, olt x y -> ole x y.

Lemma ole_refl : forall x, isPOrd x -> ole x x.

Hint Resolve olt_weak ole_refl.

Lemma ole_olt_trans :
  forall x y z, ole x y -> olt y z -> olt x z.

Lemma olt_ole_trans :
  forall x y z, olt x y -> ole y z -> olt x z.


Lemma isPOrd_ind : forall x (P:set->Prop),
  (forall y y', ole y x -> y == y' -> P y -> P y') ->
  (forall y, ole y x ->
   (forall z, olt z y -> P z) -> P y) ->
  isPOrd x -> P x.

Lemma isOrd_intro : forall x,
  (forall a b, a \in b -> b \in x -> a \in x) ->
  (forall y, y \in x -> isOrd y) ->
  isOrd x.

Lemma isOrd_elim : forall x (P:set->Prop),
  (forall x x', isOrd x -> x == x' -> P x -> P x') ->
  (forall y,
   (forall a b, a \in b -> b \in y -> a \in y) ->
   (forall z, z \in y -> isOrd z) ->
   (forall z, z \in y -> P z) -> P y) ->
  isOrd x -> P x.

Lemma isOrd_inv : forall x y,
  isOrd x -> lt y x -> isOrd y.

Lemma isOrd_le : forall x y,
  isOrd x -> le y x -> isOrd y.

Lemma isOrd_trans : forall x y z,
  isOrd x -> lt z y -> lt y x -> lt z x.

Lemma isOrd_ind : forall x (P:set->Prop),
  (forall y y', isOrd y -> y \incl x -> y == y' -> P y -> P y') ->
  (forall y, isOrd y ->
   y \incl x ->
   (forall z, lt z y -> P z) -> P y) ->
  isOrd x -> P x.

Lemma isOrd_zero : isOrd zero.

Lemma isOrd_succ : forall n, isOrd n -> isOrd (succ n).

Lemma le_lt_trans : forall x y z, isOrd z -> le x y -> lt y z -> lt x z.

Lemma lt_le_trans : forall x y z, isOrd z -> lt x y -> le y z -> lt x z.

Lemma lt_antirefl : forall x, isOrd x -> ~ lt x x.

Lemma isOrd_eq : forall o, isOrd o -> o == sup o succ.

Module ClassicOrdinal.

Axiom EM : forall A, A \/ ~A.

Lemma ord_total : forall x y,
  isOrd x -> isOrd y -> le x y \/ lt y x.

Lemma ord_incl_le : forall x y, isOrd x -> isOrd y -> x \incl y -> le x y.

End ClassicOrdinal.

Lemma ord_le_incl : forall x y, isOrd x -> isOrd y -> le x y -> x \incl y.

Definition increasing F :=
  forall x y, isOrd x -> isOrd y -> y \incl x -> F y \incl F x.

Lemma increasing_le : forall F x y,
  increasing F -> isOrd x -> le y x -> F y \incl F x.

Lemma isOrd_N : isOrd N.

Lemma natOrd : forall n, n \in N -> isOrd n.

Hint Resolve natOrd isOrd_N.

Definition succOrd o := exists2 o', isOrd o' & o == succ o'.


Definition limitOrd o := isOrd o /\ union o == o.
Definition limitOrd' o := isOrd o /\ (forall x, lt x o -> lt (succ x) o).

Lemma limit_equiv : forall o, limitOrd' o -> limitOrd o.

Lemma N_limit_ord : limitOrd' N.

Lemma limit_is_ord : forall o, limitOrd o -> isOrd o.
Hint Resolve limit_is_ord.
Lemma limit'_is_ord : forall o, limitOrd' o -> isOrd o.
Hint Resolve limit'_is_ord.

Lemma discr_lim_succ : forall o, limitOrd o -> succOrd o -> False.


Definition least_ord o (P:set->Prop) :=
  union (subset o (fun y => P y /\ forall x, isOrd x -> P x -> le y x)).

Lemma least_ord_morph : forall o o' (P P':set->Prop),
  isOrd o -> o == o' ->
  (forall x x', isOrd x -> x == x' -> (P x <-> P' x')) ->
  least_ord o P == least_ord o' P'.

Lemma least_ord1 : forall o (P:set->Prop),
  (forall x x', isOrd x -> x == x' -> P x -> P x') ->
  isOrd o ->
  forall x,
  lt x o ->
  P x ->
  P (least_ord o P) /\ isOrd (least_ord o P) /\ le (least_ord o P) x /\
  forall y, lt y (least_ord o P) -> ~ P y.
Import ClassicOrdinal.


Require Import ZFrepl.

Section LimOrd.

  Variable f : nat -> set.
  Variable ford : forall n, isOrd (f n).

  Definition ord_sup :=
    union (repl N (fun x y => exists2 n, x == nat2set n & f n == y)).

  Lemma repl_sup :
    repl_rel N (fun x y => exists2 n, x == nat2set n & f n == y).

  Lemma isOrd_sup_intro : forall n, f n \incl ord_sup.

  Lemma isOrd_sup_elim : forall x, lt x ord_sup -> exists n, lt x (f n).

  Lemma isOrd_sup : isOrd ord_sup.

End LimOrd.

Section LimOrdRel.

  Variable R : nat -> set -> Prop.
  Variable Rmorph : forall n x x', isOrd x -> x == x' -> R n x -> R n x'.
  Variable Rtot : forall n, exists x, R n x.
  Variable Rfun : forall n x x',
    isOrd x -> isOrd x' -> R n x -> R n x' -> x == x'.
  Variable Rord : forall n x, R n x -> isOrd x.

  Definition sup_rel :=
    union (repl N (fun x y => exists2 n, x == nat2set n & R n y)).

  Lemma repl_sup_rel :
    repl_rel N (fun x y => exists2 n, x == nat2set n & R n y).

  Lemma isOrd_sup_rel_intro2 : forall n y, R n y -> y \incl sup_rel.

  Lemma isOrd_sup_rel_intro : fkrAll n,
    exists2 y, R n y & y \incl sup_rel.

  Lemma isOrd_sup_rel_elim :
    forall x, lt x sup_rel -> exists n, exists2 y, R n y & lt x y.

  Lemma isOrd_sup_rel : isOrd sup_rel.

End LimOrdRel.

Inductive ord : Set := OO | OS (o:ord) | OL (f:nat->ord).

Fixpoint ord2set (o:ord) : set :=
  match o with
  | OO => zero
  | OS k => succ (ord2set k)
  | OL f => ord_sup (fun k => ord2set (f k))
  end.

Lemma ord2set_typ : forall o, isOrd (ord2set o).

Definition iter_w (f:set->set) o :=
  ord_sup(nat_rect(fun _=>set) o (fun _ => f)).

Lemma isOrd_iter_w : forall f o,
  (forall x, isOrd x -> isOrd (f x)) ->
  isOrd o ->
  isOrd (iter_w f o).

Definition plus_w := iter_w succ.
Definition mult_w := iter_w plus_w.
Definition pow_w := iter_w mult_w.
Definition epsilon0 : set := iter_w pow_w N.

Lemma isOrd_epsilon0: isOrd epsilon0.

Lemma zero_typ_e0 : zero \in epsilon0.

Section TransfiniteRecursion.

  Variable F : (set -> set) -> set -> set.
  Hypothesis Fmorph : forall o o' f f',
    o == o' -> eq_fun o f f' -> F f o == F f' o'.

  Definition TR_rel o y :=
    forall P,
    Proper (eq_set ==> eq_set ==> iff) P ->
    (forall o' f, ext_fun o' f ->
     (forall n, lt n o' -> P n (f n)) ->
     P o' (F f o')) ->
    P o y.

  Instance TR_rel_morph : Proper (eq_set ==> eq_set ==> iff) TR_rel.

Qed.

  Lemma TR_rel_intro : forall x f,
    ext_fun x f ->
    (forall y, y \in x -> TR_rel y (f y)) ->
    TR_rel x (F f x).

  Lemma TR_rel_inv : forall x y,
    TR_rel x y ->
    exists2 f,
      ext_fun x f /\ (forall y, y \in x -> TR_rel y (f y)) &
      y == F f x.

  Lemma TR_rel_fun :
    forall x y, TR_rel x y -> forall y', TR_rel x y' -> y == y'.

  Lemma TR_rel_repl_rel :
    forall o, isOrd o -> repl_rel o TR_rel.

  Lemma TR_rel_def : forall o, isOrd o -> exists y, TR_rel o y.

  Lemma TR_rel_choice_pred : forall o, isOrd o ->
    uchoice_pred (fun y => TR_rel o y).

  Definition TR o := uchoice (fun y => TR_rel o y).

  Lemma TR_morph0 : forall o o', isOrd o -> o == o' -> TR o == TR o'.

  Lemma TR_eqn : forall o, isOrd o -> TR o == F TR o.

  Lemma TR_ind : forall o (P:set->set->Prop),
    (forall x x', isOrd x -> x \incl o -> x == x' ->
     forall y y', y == y' -> P x y -> P x' y') ->
    isOrd o ->
    (forall y, isOrd y -> y \incl o ->
     (forall x, lt x y -> P x (TR x)) ->
     P y (F TR y)) ->
    P o (TR o).

  Lemma TR_typ : forall n X,
    isOrd n ->
    (forall y f, isOrd y -> y \incl n ->
     (forall z, lt z y -> f z \in X) -> F f y \in X) ->
    TR n \in X.

End TransfiniteRecursion.

Section TransfiniteIteration.

  Variable F : set -> set.
  Hypothesis Fmorph : Proper (eq_set ==> eq_set) F.

Let G f o := sup o (fun o' => F (f o')).

Lemma Gmorph : forall o o' f f',
    o == o' -> eq_fun o f f' -> G f o == G f' o'.
Hint Resolve Gmorph.

  Definition TI := TR G.

  Lemma TI_morph0 : forall x x', isOrd x -> x == x' ->
    TI x == TI x'.

  Lemma TI_fun_ext : forall o, isOrd o ->
    ext_fun o (fun y => F (TI y)).
Hint Resolve TI_fun_ext.

  Lemma TI_eq : forall o,
    isOrd o ->
    TI o == sup o (fun o' => F (TI o')).

  Lemma TI_intro : forall o o' x,
    isOrd o ->
    lt o' o ->
    x \in F (TI o') ->
    x \in TI o.

  Lemma TI_elim : forall o x,
    isOrd o ->
    x \in TI o ->
    exists2 o', lt o' o & x \in F (TI o').

  Lemma TI_initial : TI zero == empty.

  Lemma TI_typ : forall n X,
    (forall a, a \in X -> F a \in X) ->
    isOrd n ->
    (forall m G, le m n ->
     ext_fun m G ->
     (forall x, lt x m -> G x \in X) -> sup m G \in X) ->
    TI n \in X.

End TransfiniteIteration.
Hint Resolve TI_fun_ext.