some cleanup

homotopy/LES_applications.hlean

@@ -43,7 +43,7 @@ namespace is_conn
     { /- k = 0 -/
       change (is_equiv (trunc_functor 0 f)), apply is_equiv_trunc_functor_of_is_conn_fun,
       refine is_conn_fun_of_le f (zero_le_of_nat n)},
-    { /- k > 0 even -/
+    { /- k > 0 -/
      have H2' : k ≤ n, from le.trans !self_le_succ H2,
       exact
       @is_equiv_of_trivial _
@@ -84,12 +84,6 @@ namespace hopf
 
   open sphere.ops susp circle sphere_index
 
-  -- definition complex_phopf : S. 3 →* S. 2 :=
-  -- proof pmap.mk complex_hopf idp qed
-
-  -- variable (A : Type)
-  -- variables [h_space A] [is_conn 0 A]
-
   attribute hopf [unfold 4]
   -- definition phopf (x : psusp A) : Type* :=
   -- pointed.MK (hopf A x)
@@ -120,13 +114,16 @@ namespace hopf
   -- definition pjoin_pspheres (n m : ℕ) : join (S. n) (S. m) ≃ (S. (n + m + 1)) :=
   -- join.spheres n m ⬝e begin esimp, apply equiv_of_eq, apply ap S, apply add_plus_one_of_nat end
 
---  set_option pp.all true
+  definition part_of_complex_hopf : S (of_nat 3) → sigma (hopf S¹) :=
-  -- definition pjoin_spheres_inv_base (n m : ℕ)
+  begin
-  --   : (join.spheres 1 1)⁻¹ (cast proof idp qed (@sphere.base 3)) = inl base :=
+    intro x, apply inv (hopf.total S¹), apply inv (join.spheres 1 1), exact x
-  -- begin
+  end
-  --   exact sorry
-  -- end
 
+  definition part_of_complex_hopf_base2
+    : (part_of_complex_hopf (@sphere.base 3)).2 = circle.base :=
+  begin
+    exact sorry
+  end
 
   definition pfiber_complex_phopf : pfiber complex_phopf ≃* S. 1 :=
   begin
@@ -136,7 +133,7 @@ namespace hopf
                (join.spheres 1 1)⁻¹ᵉ _ ⬝e _,
       refine fiber.equiv_precompose (hopf.total S¹)⁻¹ᵉ ⬝e _,
       apply fiber_pr1},
-    { exact sorry}
+    { esimp, refine _ ⬝ part_of_complex_hopf_base2, apply fiber_pr1_fun}
   end
 
   open int

homotopy/chain_complex.hlean

@@ -43,13 +43,6 @@ notation `-ℤ` := sint'
 
 definition stratified_type [reducible] (N : succ_str) (n : ℕ) : Type₀ := N × fin (succ n)
 
--- definition stratified_succ' {N : succ_str}  : Π{n : ℕ}, N → ifin n → stratified_type N n
--- | (succ k) n (fz k) := (S n, ifin.max k)
--- | (succ k) n (fs x) := (n, ifin.incl x)
-
--- definition stratified_succ {N : succ_str} {n : ℕ} (x : stratified_type N n) : stratified_type N n :=
--- stratified_succ' (pr1 x) (pr2 x)
-
 definition stratified_succ {N : succ_str} {n : ℕ} (x : stratified_type N n)
   : stratified_type N n :=
 (if val (pr2 x) = n then S (pr1 x) else pr1 x, my_succ (pr2 x))
@@ -57,8 +50,6 @@ definition stratified_succ {N : succ_str} {n : ℕ} (x : stratified_type N n)
 definition stratified [reducible] [constructor] (N : succ_str) (n : ℕ) : succ_str :=
 succ_str.mk (stratified_type N n) stratified_succ
 
---example (n : ℕ) : flatten (n, (2 : ifin (nat.succ (nat.succ 4)))) = 6*n+2 := proof rfl qed
-
 notation `+3ℕ` := stratified +ℕ 2
 notation `-3ℕ` := stratified -ℕ 2
 notation `+3ℤ` := stratified +ℤ 2
@@ -68,20 +59,6 @@ notation `-6ℕ` := stratified -ℕ 5
 notation `+6ℤ` := stratified +ℤ 5
 notation `-6ℤ` := stratified -ℤ 5
 
--- definition triple_type (N : succ_str) : Type₀ := N ⊎ N ⊎ N
--- definition triple_succ {N : succ_str} : triple_type N → triple_type N
--- | (inl n)       := inr (inl n)
--- | (inr (inl n)) := inr (inr n)
--- | (inr (inr n)) := inl (S n)
-
--- definition triple [reducible] [constructor] (N : succ_str) : succ_str :=
--- succ_str.mk (triple_type N) triple_succ
-
--- notation `+3ℕ` := triple +ℕ
--- notation `-3ℕ` := triple -ℕ
--- notation `+3ℤ` := triple +ℤ
--- notation `-3ℤ` := triple -ℤ
-
 namespace chain_complex
 
   -- are chain complexes with the "set"-requirement removed interesting?
@@ -510,7 +487,12 @@ namespace chain_complex
     : is_trunc n (ptrunctype.to_pType X) :=
   trunctype.struct X
 
-  /- a group where the point in the pointed corresponds with 1 in the group -/
+  /-
+    A group where the point in the pointed type corresponds with 1 in the group.
+    We need this structure because a chain complex is a sequence of pointed types, and we need
+    to assume for some of the theorems below that some of these pointed types are groups, where the
+    unit for multiplication is the point.
+  -/
   structure pgroup [class] (X : Type*) extends semigroup X, has_inv X :=
     (pt_mul : Πa, mul pt a = a)
     (mul_pt : Πa, mul a pt = a)
@@ -533,7 +515,13 @@ namespace chain_complex
             mul_left_inv_pt := mul.left_inv⦄
   end
 
-  -- the following theorems would also be true of the replace "is_contr" by "is_prop"
+  /-
+    The following theorems state that in a chain complex, if certain types are contractible, and
+    the chain complex is exact at the right spots, a map in the chain complex is an
+    embedding/surjection/equivalence. For the first and third we also need to assume that
+    the map is a group homomorphism (and hence that the two types around it are groups).
+  -/
+
   definition is_embedding_of_trivial (X : chain_complex N) {n : N}
     (H : is_exact_at X n) [HX : is_contr (X (S (S n)))]
     [pgroup (X n)] [pgroup (X (S n))] [is_homomorphism (cc_to_fn X n)]