feat(hott/homotopy): add commutativity proof for join

hott/homotopy/join.hlean

@@ -8,7 +8,7 @@ Declaration of a join as a special case of a pushout
 
 import hit.pushout .susp
 
-open eq prod equiv pushout is_trunc bool
+open eq function prod equiv pushout is_trunc bool
 
 namespace join
 
@@ -60,12 +60,38 @@ namespace join
       apply square_of_eq_top, rewrite idp_con, apply !con.right_inv⁻¹,   
   end
 
-  set_option unifier.max_steps 40000
-  protected definition assoc (A B C : Type) :
-    join (join A B) C ≃ join A (join B C) :=
+  protected definition swap (A B : Type) :
+    join A B → join B A :=
   begin
-    fapply equiv.MK,
-    { intro x, induction x with ab c, induction ab with a b,
+    intro x, induction x with a b, exact inr a, exact inl b,
+    apply !jglue⁻¹
+  end
+
+  protected definition swap_involutive (A B : Type) (x : join A B) :
+    join.swap B A (join.swap A B x) = x :=
+  begin
+    induction x with a b, do 2 reflexivity,
+    induction x with a b, esimp,
+    apply eq_pathover, rewrite ap_id,
+    apply hdeg_square, esimp[join.swap],
+    apply concat, apply ap_compose' (pushout.elim _ _ _),
+    krewrite [elim_glue, ap_inv, elim_glue], apply inv_inv,
+  end
+
+  protected definition symm (A B : Type) : join A B ≃ join B A :=
+  begin
+    fapply equiv.MK, do 2 apply join.swap,
+    do 2 apply join.swap_involutive,
+  end
+
+exit
+  section
+  parameters (A B C : Type)
+
+  private definition assoc_fun [reducible] :
+    join (join A B) C → join A (join B C) :=
+  begin
+    intro x, induction x with ab c, induction ab with a b,
     exact inl a, exact inr (inl b),
     induction x with a b, apply jglue, exact inr (inr c),
     induction x with ab c, induction ab with a b, apply jglue,
@@ -76,8 +102,12 @@ namespace join
     krewrite [elim_glue, ap_constant], esimp,
     apply square_of_eq, apply concat, rotate 1, exact eq_of_square H',
     rewrite [con_idp, idp_con],
-    },
-    { intro x, induction x with a bc, exact inl (inl a),
+  end
+
+  private definition assoc_inv [reducible] :
+    join A (join B C) → join (join A B) C :=
+  begin
+    intro x, induction x with a bc, exact inl (inl a),
     induction bc with b c, exact inl (inr b), exact inr c,
     induction x with b c, apply jglue, esimp,
     induction x with a bc, induction bc with b c,
@@ -88,27 +118,44 @@ namespace join
     krewrite [elim_glue, ap_constant], esimp,
     apply square_of_eq, apply concat, exact eq_of_square H',
     rewrite [con_idp, idp_con],
-    },
-    { intro x, induction x with a bc, reflexivity,
+  end
+
+  private definition assoc_right_inv (x : join A (join B C)) :
+    assoc_fun (assoc_inv x) = x :=
+  begin
+    induction x with a bc, reflexivity,
     induction bc with b c, reflexivity, reflexivity,
     induction x with b c, esimp, apply eq_pathover,
     apply hdeg_square, esimp,
     apply concat, apply ap_compose' (pushout.elim _ _ _),
-      apply concat, apply ap (ap _), apply elim_glue, esimp,
-      apply concat, apply elim_glue, esimp,
+    apply concat, apply ap (ap _), unfold assoc_inv, apply elim_glue, esimp,
+    krewrite elim_glue,
     induction x with a bc, induction bc with b c, esimp,
-      apply eq_pathover, apply hdeg_square, esimp,
+    { apply eq_pathover, apply hdeg_square, esimp,
       apply concat, apply ap_compose' (pushout.elim _ _ _),
-      apply concat, apply ap (ap _), krewrite elim_glue, reflexivity,
-      apply inverse, apply concat, apply ap_id,
-      apply inverse, apply concat, apply inverse, 
-      apply ap_compose' (pushout.elim _ _ _), apply elim_glue,
-      esimp, apply eq_pathover, apply hdeg_square,
+      krewrite elim_glue,
+      apply concat, apply !(ap_compose' (pushout.elim _ _ _))⁻¹,
+      esimp, krewrite [elim_glue, ap_id],
+    },
+    { esimp, apply eq_pathover, apply hdeg_square, esimp,
       apply concat, apply ap_compose' (pushout.elim _ _ _),
-      apply concat, apply ap (ap _), krewrite elim_glue, reflexivity,
-      apply inverse, apply concat, apply ap_id,
-      apply inverse, apply concat,
-      --apply ap_compose' (pushout.elim _ _ _),
+      krewrite elim_glue,
+      esimp[jglue], apply concat, apply (refl (ap _ (glue (inl a, c)))),
+      esimp, krewrite [elim_glue, ap_id],
+    },
+    { esimp, induction x with b c, esimp,
+      apply eq_pathover,
+    },
+  end
+
+exit
+
+  protected definition assoc (A B C : Type) :
+    join (join A B) C ≃ join A (join B C) :=
+  begin
+    fapply equiv.MK,
+    {     },
+    { 
     },
   end
   check elim_glue