feat(hott): the complex hopf fibration S3 to S2

hott/homotopy/complex_hopf.hlean

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+/-
+Copyright (c) 2016 Ulrik Buchholtz and Egbert Rijke. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Ulrik Buchholtz, Egbert Rijke
+
+The H-space structure on S¹ and the complex Hopf fibration
+ (the standard one).
+-/
+
+import .hopf .circle
+
+open eq equiv is_equiv circle is_conn trunc is_trunc sphere_index sphere susp
+
+namespace hopf
+
+  definition circle_h_space [instance] : h_space S¹ :=
+  ⦃ h_space, one := base, mul := circle_mul,
+    one_mul := circle_base_mul, mul_one := circle_mul_base ⦄
+
+  definition circle_assoc (x y z : S¹) : (x * y) * z = x * (y * z) :=
+  begin
+    induction x,
+    { reflexivity },
+    { apply eq_pathover, induction y,
+      { exact natural_square_tr
+          (λa : S¹, ap (λb : S¹, b * z) (circle_mul_base a))
+          loop },
+      { apply is_prop.elimo, apply is_trunc_square } }
+  end
+
+  open sphere.ops function
+
+  definition complex_hopf : S 3 → S 2 :=
+  begin
+    intro x, apply @sigma.pr1 (susp S¹) (hopf S¹),
+    apply inv (hopf.total S¹), apply inv (join.spheres 1 1), exact x
+  end
+
+end hopf