feat(hit.suspension): add definition of spheres and the circle

hott/hit/pushout.hlean

@@ -44,7 +44,7 @@ parameters {TL BL TR : Type.{u}} (f : TL → BL) (g : TL → TR)
   definition pushout := colimit pushout_diag
   local attribute pushout_diag [instance]
 
-  definition inl (x : BL) : pushout :=
+ definition inl (x : BL) : pushout :=
   @ι _ _ x
 
   definition inr (x : TR) : pushout :=

hott/hit/suspension.hlean

@@ -5,7 +5,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Module: hit.suspension
 Authors: Floris van Doorn
 
-Declaration of suspension
+Declaration of suspension and spheres
 -/
 
 import .pushout
@@ -26,9 +26,9 @@ namespace suspension
   glue _ _ a
 
   protected definition rec {A : Type} {P : suspension A → Type} (PN : P !north) (PS : P !south)
-    (Pmerid : Π(a : A), merid a ▹ PN = PS) (y : suspension A) : P y :=
+    (Pmerid : Π(a : A), merid a ▹ PN = PS) (x : suspension A) : P x :=
   begin
-    fapply (pushout.rec_on _ _ y),
+    fapply (pushout.rec_on _ _ x),
     { intro u, cases u, exact PN},
     { intro u, cases u, exact PS},
     { exact Pmerid},
@@ -39,3 +39,59 @@ namespace suspension
   rec PN PS Pmerid y
 
 end suspension
+
+open nat suspension bool
+
+definition sphere (n : ℕ) := nat.rec_on n bool (λk Sk, suspension Sk)
+definition circle [reducible] := sphere 1
+
+namespace circle
+
+  definition base : circle := !north
+  definition loop : base = base := merid tt ⬝ (merid ff)⁻¹
+
+  protected definition rec2 {P : circle → Type} (PN : P !north) (PS : P !south)
+    (Pff : merid ff ▹ PN = PS) (Ptt : merid tt ▹ PN = PS) (x : circle) : P x :=
+  begin
+    fapply (suspension.rec_on x),
+    { exact PN},
+    { exact PS},
+    { intro b, cases b, exact Pff, exact Ptt},
+  end
+
+  protected definition rec2_on {P : circle → Type} (x : circle) (PN : P !north) (PS : P !south)
+    (Pff : merid ff ▹ PN = PS) (Ptt : merid tt ▹ PN = PS) : P x :=
+  circle.rec2 PN PS Pff Ptt x
+
+  protected definition rec {P : circle → Type} (Pbase : P base) (Ploop : loop ▹ Pbase = Pbase)
+    (x : circle) : P x :=
+  begin
+    fapply (rec2_on x),
+    { exact Pbase},
+    { sexact (merid ff ▹ Pbase)},
+    { apply idp},
+    { apply eq_tr_of_inv_tr_eq, rewrite -tr_con, apply Ploop},
+  end
+
+  protected definition rec_on {P : circle → Type} (x : circle) (Pbase : P base)
+    (Ploop : loop ▹ Pbase = Pbase) : P x :=
+  circle.rec Pbase Ploop x
+
+  protected definition rec_constant {P : Type} (Pbase : P) (Ploop : Pbase = Pbase)
+    (x : circle) : P :=
+  circle.rec Pbase (tr_constant loop Pbase ⬝ Ploop) x
+
+  definition comp_loop {P : circle → Type} (Pbase : P base) (Ploop : loop ▹ Pbase = Pbase) :
+    ap (circle.rec Pbase Ploop) loop = sorry ⬝ Ploop ⬝ sorry :=
+  sorry
+
+  definition comp_constant_loop {P : Type} (Pbase : P) (Ploop : Pbase = Pbase) :
+    ap (circle.rec_constant Pbase Ploop) loop = sorry ⬝ Ploop ⬝ sorry :=
+  sorry
+
+
+  protected definition rec_on_constant {P : Type} (x : circle) (Pbase : P) (Ploop : Pbase = Pbase)
+     : P :=
+  rec_constant Pbase Ploop x
+
+end circle