GhostTT.Univ

From Coq Require Import Utf8 List.
From GhostTT.autosubst Require Import GAST unscoped.
From GhostTT Require Import Util BasicAST SubstNotations ContextDecl CastRemoval
  TermMode Scoping.

Import ListNotations.

Set Default Goal Selector "!".

(*** Equality of universe levels **)
Definition ueq (m : mode) (i j : level) :=
  m = mProp ∨ i = j.

Lemma ueq_eq :
  ∀ m i j,
    isProp m = false →
    ueq m i j →
    i = j.
Proof.
  intros m i j hm [-> | ->]. 1: discriminate.
  reflexivity.
Qed.

Lemma ueq_Kind_eq :
  ∀ i j,
    ueq mKind i j →
    i = j.
Proof.
  intros i j e.
  eapply ueq_eq. 2: eassumption.
  reflexivity.
Qed.

Lemma ueq_Type_eq :
  ∀ i j,
    ueq mType i j →
    i = j.
Proof.
  intros i j e.
  eapply ueq_eq. 2: eassumption.
  reflexivity.
Qed.

Lemma ueq_Ghost_eq :
  ∀ i j,
    ueq mGhost i j →
    i = j.
Proof.
  intros i j e.
  eapply ueq_eq. 2: eassumption.
  reflexivity.
Qed.

Ltac ueq_subst :=
  repeat lazymatch goal with
  | e : ueq mKind ?i ?j |- _ ⇒ eapply ueq_Kind_eq in e ; try subst i
  | e : ueq mType ?i ?j |- _ ⇒ eapply ueq_Type_eq in e ; try subst i
  | e : ueq mGhost ?i ?j |- _ ⇒ eapply ueq_Ghost_eq in e ; try subst i
  end.

Definition usup m i :=
  if isProp m then 0 else S i.

Definition umax mx m i j :=
  if isProp m then 0
  else if isProp mx then j
  else max i j.