fix(types.W): clean-up W file, remove 'exit'

hott/types/W.hlean

@@ -11,11 +11,7 @@ Theorems about W-types (well-founded trees)
 import .sigma .pi
 open eq sigma sigma.ops equiv is_equiv
 
--- TODO fix universe levels
-/-
-exit
-
-inductive Wtype.{l k} {A : Type.{l}} (B : A → Type.{k}) :=
+inductive Wtype.{l k} {A : Type.{l}} (B : A → Type.{k}) : Type.{max l k} :=
 sup : Π (a : A), (B a → Wtype.{l k} B) → Wtype.{l k} B
 
 namespace Wtype
@@ -25,11 +21,11 @@ namespace Wtype
   variables {A A' : Type.{u}} {B B' : A → Type.{v}} {C : Π(a : A), B a → Type}
             {a a' : A} {f : B a → W a, B a} {f' : B a' → W a, B a} {w w' : W(a : A), B a}
 
-  protected definition pr1 (w : W(a : A), B a) : A :=
-  Wtype.rec_on w (λa f IH, a)
+  protected definition pr1 [unfold-c 3] (w : W(a : A), B a) : A :=
+  by cases w with a f; exact a
 
-  protected definition pr2 (w : W(a : A), B a) : B (pr1 w) → W(a : A), B a :=
-  Wtype.rec_on w (λa f IH, f)
+  protected definition pr2 [unfold-c 3] (w : W(a : A), B a) : B (pr1 w) → W(a : A), B a :=
+  by cases w with a f; exact f
 
   namespace ops
   postfix `.1`:(max+1) := Wtype.pr1
@@ -39,88 +35,69 @@ namespace Wtype
   open ops
 
   protected definition eta (w : W a, B a) : ⟨w.1 , w.2⟩ = w :=
-  cases_on w (λa f, idp)
+  by cases w; exact idp
 
   definition sup_eq_sup (p : a = a') (q : p ▸ f = f') : ⟨a, f⟩ = ⟨a', f'⟩ :=
-  path.rec_on p (λf' q, path.rec_on q idp) f' q
+  by cases p; cases q; exact idp
 
   protected definition Wtype_eq (p : w.1 = w'.1) (q : p ▸ w.2 = w'.2) : w = w' :=
-  cases_on w
-           (λw1 w2, cases_on w' (λ w1' w2', sup_eq_sup))
-           p q
+  by cases w; cases w';exact (sup_eq_sup p q)
 
   protected definition Wtype_eq_pr1 (p : w = w') : w.1 = w'.1 :=
-  path.rec_on p idp
+  by cases p;exact idp
 
   protected definition Wtype_eq_pr2 (p : w = w') : Wtype_eq_pr1 p ▸ w.2 = w'.2 :=
-  path.rec_on p idp
+  by cases p;exact idp
 
   namespace ops
   postfix `..1`:(max+1) := Wtype_eq_pr1
   postfix `..2`:(max+1) := Wtype_eq_pr2
-  end ops
-  open ops
+  end ops open ops open sigma
 
   definition sup_path_W (p : w.1 = w'.1) (q : p ▸ w.2 = w'.2)
-      :  dpair (Wtype_eq p q)..1 (Wtype_eq p q)..2 = dpair p q :=
-  begin
-    revert p q,
-    apply (cases_on w), intro w1 w2,
-    apply (cases_on w'), intro w1' w2' p, generalize w2', --change to revert
-    apply (path.rec_on p), intro w2' q,
-    apply (path.rec_on q), apply idp
-  end
+      :  ⟨(Wtype_eq p q)..1,(Wtype_eq p q)..2⟩ = ⟨p, q⟩ :=
+  by cases w; cases w'; cases p; cases q; exact idp
 
   definition pr1_path_W (p : w.1 = w'.1) (q : p ▸ w.2 = w'.2) : (Wtype_eq p q)..1 = p :=
-  (!sup_path_W)..1
+  !sup_path_W..1
 
   definition pr2_path_W (p : w.1 = w'.1) (q : p ▸ w.2 = w'.2)
       : pr1_path_W p q ▸ (Wtype_eq p q)..2 = q :=
-  (!sup_path_W)..2
+  !sup_path_W..2
 
   definition eta_path_W (p : w = w') : Wtype_eq (p..1) (p..2) = p :=
-  begin
-    apply (path.rec_on p),
-    apply (cases_on w), intro w1 w2,
-    apply idp
-  end
+  by cases p; cases w; exact idp
 
   definition transport_pr1_path_W {B' : A → Type} (p : w.1 = w'.1) (q : p ▸ w.2 = w'.2)
       : transport (λx, B' x.1) (Wtype_eq p q) = transport B' p :=
-  begin
-    revert p q,
-    apply (cases_on w), intro w1 w2,
-    apply (cases_on w'), intro w1' w2' p, generalize w2',
-    apply (path.rec_on p), intro w2' q,
-    apply (path.rec_on q), apply idp
-  end
+  by cases w; cases w'; cases p; cases q; exact idp
 
   definition path_W_uncurried (pq : Σ(p : w.1 = w'.1), p ▸ w.2 = w'.2) : w = w' :=
-  destruct pq Wtype_eq
+  by cases pq with p q; exact (Wtype_eq p q)
 
   definition sup_path_W_uncurried (pq : Σ(p : w.1 = w'.1), p ▸ w.2 = w'.2)
-      :  dpair (path_W_uncurried pq)..1 (path_W_uncurried pq)..2 = pq :=
-  destruct pq sup_path_W
+      :  ⟨(path_W_uncurried pq)..1, (path_W_uncurried pq)..2⟩ = pq :=
+  by cases pq with p q; exact (sup_path_W p q)
 
   definition pr1_path_W_uncurried (pq : Σ(p : w.1 = w'.1), p ▸ w.2 = w'.2)
       : (path_W_uncurried pq)..1 = pq.1 :=
-  (!sup_path_W_uncurried)..1
+  !sup_path_W_uncurried..1
 
   definition pr2_path_W_uncurried (pq : Σ(p : w.1 = w'.1), p ▸ w.2 = w'.2)
       : (pr1_path_W_uncurried pq) ▸ (path_W_uncurried pq)..2 = pq.2 :=
-  (!sup_path_W_uncurried)..2
+  !sup_path_W_uncurried..2
 
-  definition eta_path_W_uncurried (p : w = w') : path_W_uncurried (dpair p..1 p..2) = p :=
+  definition eta_path_W_uncurried (p : w = w') : path_W_uncurried ⟨p..1, p..2⟩ = p :=
   !eta_path_W
 
   definition transport_pr1_path_W_uncurried {B' : A → Type} (pq : Σ(p : w.1 = w'.1), p ▸ w.2 = w'.2)
       : transport (λx, B' x.1) (@path_W_uncurried A B w w' pq) = transport B' pq.1 :=
-    destruct pq transport_pr1_path_W
+  by cases pq with p q; exact (transport_pr1_path_W p q)
 
   definition isequiv_path_W /-[instance]-/ (w w' : W a, B a)
       : is_equiv (@path_W_uncurried A B w w') :=
   adjointify path_W_uncurried
-             (λp, dpair (p..1) (p..2))
+             (λp, ⟨p..1, p..2⟩)
              eta_path_W_uncurried
              sup_path_W_uncurried
 
@@ -132,28 +109,24 @@ namespace Wtype
       (∀ (b : B a) (b' : B a'), P (f b) (f' b')) → P (sup a f) (sup a' f')) : P w w' :=
   begin
     revert w',
-    apply (rec_on w), intro a f IH w',
-    apply (cases_on w'), intro a' f',
+    eapply (Wtype.rec_on w), intro a f IH w',
+    cases w' with a' f',
     apply H, intro b b',
     apply IH
   end
 
   /- truncatedness -/
-  open truncation
-  definition trunc_W [instance] [FUN : funext.{v (max 1 u v)}] (n : trunc_index)
+  open is_trunc
+  definition trunc_W [instance] (n : trunc_index)
     [HA : is_trunc (n.+1) A] : is_trunc (n.+1) (W a, B a) :=
   begin
-  fapply is_trunc_succ, intro w w',
-  apply (double_induction_on w w'), intro a a' f f' IH,
+  fapply is_trunc_succ_intro, intro w w',
+  eapply (double_induction_on w w'), intro a a' f f' IH,
   fapply is_trunc_equiv_closed,
-    apply equiv_path_W,
-    apply is_trunc_sigma,
-      fapply (is_trunc_eq n),
-      intro p, revert IH, generalize f', --change to revert after simpl
-      apply (path.rec_on p), intro f' IH,
-      apply pi.is_trunc_eq_pi, intro b,
-      apply IH
+  { apply equiv_path_W},
+  { fapply is_trunc_sigma,
+    intro p, cases p, esimp,
+    apply pi.is_trunc_eq_pi}
   end
 
 end Wtype
--/