feat(hott): add theory about pointed pushouts

hott/hit/pointed_pushout.hlean

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+/-
+Copyright (c) 2016 Jakob von Raumer. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jakob von Raumer
+
+Pointed Pushouts
+-/
+import .pushout types.pointed
+
+open eq pushout
+
+namespace pointed
+  
+  definition pointed_pushout [instance] [constructor] {TL BL TR : Type} [HTL : pointed TL]
+    [HBL : pointed BL] [HTR : pointed TR] (f : TL → BL) (g : TL → TR) : pointed (pushout f g) :=
+  pointed.mk (inl (point _))
+
+end pointed
+
+open pointed
+
+namespace pushout
+  section
+  parameters {TL BL TR : Type*} (f : TL →* BL) (g : TL →* TR)
+
+  definition Pushout [constructor] : Type* :=
+  pointed.mk' (pushout f g)
+  
+  parameters {f g}
+  definition pinl : BL →* Pushout :=
+  pmap.mk inl idp
+
+  definition pinr : TR →* Pushout :=
+  pmap.mk inr ((ap inr (respect_pt g))⁻¹ ⬝ !glue⁻¹ ⬝ (ap inl (respect_pt f)))
+
+  definition pglue (x : TL) : pinl (f x) = pinr (g x) := -- TODO do we need this?
+  !glue
+
+  definition prec {P : Pushout → Type} (Pinl : Π x, P (pinl x)) (Pinr : Π x, P (pinr x))
+    (H : Π x, Pinl (f x) =[pglue x] Pinr (g x)) : (Π y, P y) :=
+  pushout.rec Pinl Pinr H
+
+  end
+end pushout