reindex the AHSS and SSS

Now the 2nd page is the correct indexing.

algebra/spectral_sequence.hlean

@@ -292,7 +292,8 @@ namespace left_module
   structure is_bounded {R : Ring} {I : AddAbGroup} (X : exact_couple R I) : Type :=
   mk' :: (B B' : I → ℕ)
     (Dub : Π⦃x y⦄ ⦃s : ℕ⦄, (deg (i X))^[s] x = y → B x ≤ s → is_contr (D X y))
-    (Dlb : Π⦃x y z⦄ ⦃s : ℕ⦄ (p : deg (i X) x = y), (deg (i X))^[s] y = z → B' z ≤ s → is_surjective (i X ↘ p))
+    (Dlb : Π⦃x y z⦄ ⦃s : ℕ⦄ (p : deg (i X) x = y), (deg (i X))^[s] y = z → B' z ≤ s →
+      is_surjective (i X ↘ p))
     (Elb : Π⦃x y⦄ ⦃s : ℕ⦄, (deg (i X))⁻¹ᵉ^[s] x = y → B x ≤ s → is_contr (E X y))
 
 /- Note: Elb proves Dlb for some bound B', but we want tight control over when B' = 0 -/
@@ -495,22 +496,22 @@ namespace left_module
 
   parameters (x : I)
 
-  definition r (n : ℕ) : ℕ :=
+  definition rb (n : ℕ) : ℕ :=
   max (max (B (deg (j X) (deg (k X) x)) + n + 1) (B3 ((deg (i X))^[n] x)))
       (max (B' (deg (k X) ((deg (i X))^[n] x)))
            (max (B' (deg (k X) ((deg (i X))^[n+1] x))) (B ((deg (j X))⁻¹ ((deg (i X))^[n] x)))))
 
-  lemma rb0 (n : ℕ) : r n ≥ n + 1 :=
+  lemma rb0 (n : ℕ) : rb n ≥ n + 1 :=
   ge.trans !le_max_left (ge.trans !le_max_left !le_add_left)
-  lemma rb1 (n : ℕ) : B (deg (j X) (deg (k X) x)) ≤ r n - (n + 1) :=
+  lemma rb1 (n : ℕ) : B (deg (j X) (deg (k X) x)) ≤ rb n - (n + 1) :=
   nat.le_sub_of_add_le (le.trans !le_max_left !le_max_left)
-  lemma rb2 (n : ℕ) : B3 ((deg (i X))^[n] x) ≤ r n :=
+  lemma rb2 (n : ℕ) : B3 ((deg (i X))^[n] x) ≤ rb n :=
   le.trans !le_max_right !le_max_left
-  lemma rb3 (n : ℕ) : B' (deg (k X) ((deg (i X))^[n] x)) ≤ r n :=
+  lemma rb3 (n : ℕ) : B' (deg (k X) ((deg (i X))^[n] x)) ≤ rb n :=
   le.trans !le_max_left !le_max_right
-  lemma rb4 (n : ℕ) : B' (deg (k X) ((deg (i X))^[n+1] x)) ≤ r n :=
+  lemma rb4 (n : ℕ) : B' (deg (k X) ((deg (i X))^[n+1] x)) ≤ rb n :=
   le.trans (le.trans !le_max_left !le_max_right) !le_max_right
-  lemma rb5 (n : ℕ) : B ((deg (j X))⁻¹ ((deg (i X))^[n] x)) ≤ r n :=
+  lemma rb5 (n : ℕ) : B ((deg (j X))⁻¹ ((deg (i X))^[n] x)) ≤ rb n :=
   le.trans (le.trans !le_max_right !le_max_right) !le_max_right
 
   definition Einfdiag : graded_module R ℕ :=
@@ -520,24 +521,24 @@ namespace left_module
   λn, Dinf (deg (k X) ((deg (i X))^[n] x))
 
   definition short_exact_mod_page_r (n : ℕ) : short_exact_mod
-    (E (page (r n)) ((deg (i X))^[n] x))
+    (E (page (rb n)) ((deg (i X))^[n] x))
-    (D (page (r n)) (deg (k (page (r n))) ((deg (i X))^[n] x)))
+    (D (page (rb n)) (deg (k (page (rb n))) ((deg (i X))^[n] x)))
-    (D (page (r n)) (deg (i (page (r n))) (deg (k (page (r n))) ((deg (i X))^[n] x)))) :=
+    (D (page (rb n)) (deg (i (page (rb n))) (deg (k (page (rb n))) ((deg (i X))^[n] x)))) :=
   begin
     fapply short_exact_mod_of_is_exact,
-    { exact j (page (r n)) ← ((deg (i X))^[n] x) },
+    { exact j (page (rb n)) ← ((deg (i X))^[n] x) },
-    { exact k (page (r n)) ((deg (i X))^[n] x) },
+    { exact k (page (rb n)) ((deg (i X))^[n] x) },
-    { exact i (page (r n)) (deg (k (page (r n))) ((deg (i X))^[n] x)) },
+    { exact i (page (rb n)) (deg (k (page (rb n))) ((deg (i X))^[n] x)) },
-    { exact j (page (r n)) _ },
+    { exact j (page (rb n)) _ },
     { apply is_contr_D, refine Dub !deg_j_inv⁻¹ (rb5 n) },
     { apply is_contr_E, refine Elb _ (rb1 n),
       refine !deg_iterate_ij_commute ⬝ _,
       refine ap (deg (j X)) _ ⬝ !deg_j⁻¹,
-      refine iterate_sub _ !rb0 _ ⬝ _, apply ap (_^[r n]),
+      refine iterate_sub _ !rb0 _ ⬝ _, apply ap (_^[rb n]),
       exact ap (deg (i X)) (!deg_iterate_ik_commute ⬝ !deg_k⁻¹) ⬝ !deg_i⁻¹ },
-    { apply jk (page (r n)) },
+    { apply jk (page (rb n)) },
-    { apply ki (page (r n)) },
+    { apply ki (page (rb n)) },
-    { apply ij (page (r n)) }
+    { apply ij (page (rb n)) }
   end
 
   definition short_exact_mod_infpage (n : ℕ) :
@@ -573,6 +574,20 @@ namespace left_module
     (λs, Dinfdiag_stable)
 
   end
+
+  definition deg_d_reindex {R : Ring} {I J : AddAbGroup}
+    (e : AddGroup_of_AddAbGroup J ≃g AddGroup_of_AddAbGroup I) (X : exact_couple R I)
+    (r : ℕ) : deg (d (page (exact_couple_reindex e X) r)) ~
+    e⁻¹ᵍ ∘ deg (d (page X r)) ∘ e :=
+  begin
+    intro x, refine !deg_d ⬝ _ ⬝ ap e⁻¹ᵍ !deg_d⁻¹,
+    apply ap (e⁻¹ᵍ ∘ deg (j X)),
+    note z := @iterate_hsquare _ _ (deg (i (exact_couple_reindex e X)))⁻¹ᵉ (deg (i X))⁻¹ᵉ e
+      (λx, proof to_right_inv (group.equiv_of_isomorphism e) _ qed)⁻¹ʰᵗʸʰ r,
+    refine z _ ⬝ _, apply ap ((deg (i X))⁻¹ᵉ^[r]),
+    exact to_right_inv (group.equiv_of_isomorphism e) _
+  end
+
   end convergence_theorem
 
   open int group prod convergence_theorem prod.ops
@@ -592,12 +607,12 @@ namespace left_module
                                  (Dinf : agℤ → LeftModule.{u w} R) : Type.{max u (v+1) (w+1)} :=
     (X : exact_couple.{u 0 w v} R Z2)
     (HH : is_bounded X)
-    (s₀ : agℤ → agℤ)
+    (s₁ : agℤ → agℤ) (s₂ : agℤ → agℤ)
-    (HB : Π(n : agℤ), is_bounded.B' HH (deg (k X) (n, s₀ n)) = 0)
+    (HB : Π(n : agℤ), is_bounded.B' HH (deg (k X) (s₁ n, s₂ n)) = 0)
     (e : Π(x : Z2), exact_couple.E X x ≃lm E' x.1 x.2)
-    (f : Π(n : agℤ), exact_couple.D X (deg (k X) (n, s₀ n)) ≃lm Dinf n)
+    (f : Π(n : agℤ), exact_couple.D X (deg (k X) (s₁ n, s₂ n)) ≃lm Dinf n)
     (deg_d1 : ℕ → agℤ) (deg_d2 : ℕ → agℤ)
-    (deg_d_eq : Π(r : ℕ) (n s : agℤ), deg (d (page X r)) (n, s) = (n, s) + (deg_d1 r, deg_d2 r))
+    (deg_d_eq0 : Π(r : ℕ), deg (d (page X r)) 0 = (deg_d1 r, deg_d2 r))
 
   structure converging_spectral_sequence.{u v w} {R : Ring} (E' : agℤ → agℤ → LeftModule.{u v} R)
                                  (Dinf : agℤ → LeftModule.{u w} R) : Type.{max u (v+1) (w+1)} :=
@@ -622,9 +637,14 @@ namespace left_module
   local abbreviation X := X c
   local abbreviation i := i X
   local abbreviation HH := HH c
-  local abbreviation s₀ := s₀ c
+  local abbreviation s₁ := s₁ c
-  local abbreviation Dinfdiag (n : agℤ) (k : ℕ) := Dinfdiag HH (n, s₀ n) k
+  local abbreviation s₂ := s₂ c
-  local abbreviation Einfdiag (n : agℤ) (k : ℕ) := Einfdiag HH (n, s₀ n) k
+  local abbreviation Dinfdiag (n : agℤ) (k : ℕ) := Dinfdiag HH (s₁ n, s₂ n) k
+  local abbreviation Einfdiag (n : agℤ) (k : ℕ) := Einfdiag HH (s₁ n, s₂ n) k
+
+  definition deg_d_eq (r : ℕ) (n s : agℤ) :
+    (deg (d (page X r))) (n, s) = (n + deg_d1 c r, s + deg_d2 c r) :=
+  !deg_eq ⬝ ap (λx, _ + x) (deg_d_eq0 c r)
 
   definition deg_d_inv_eq (r : ℕ) (n s : agℤ) :
     (deg (d (page X r)))⁻¹ᵉ (n, s) = (n - deg_d1 c r, s - deg_d2 c r) :=
@@ -632,16 +652,38 @@ namespace left_module
 
   definition converges_to_isomorphism {E'' : agℤ → agℤ → LeftModule R} {Dinf' : graded_module R agℤ}
     (e' : Πn s, E' n s ≃lm E'' n s) (f' : Πn, Dinf n ≃lm Dinf' n) : E'' ⟹ Dinf' :=
-  converges_to.mk X HH s₀ (HB c)
+  converges_to.mk X HH s₁ s₂ (HB c)
     begin intro x, induction x with n s, exact e c (n, s) ⬝lm e' n s end
     (λn, f c n ⬝lm f' n)
-    (deg_d1 c) (deg_d2 c) (λr n s, deg_d_eq c r n s)
+    (deg_d1 c) (deg_d2 c) (λr, deg_d_eq0 c r)
+
+
+  include c
+  definition converges_to_reindex (i : agℤ ×ag agℤ ≃g agℤ ×ag agℤ) :
+    (λp q, E' (i (p, q)).1 (i (p, q)).2) ⟹ (λn, Dinf n) :=
+  begin
+    fapply converges_to.mk (exact_couple_reindex i X) (is_bounded_reindex i HH),
+    { exact λn, (i⁻¹ᵍ (s₁ n, s₂ n)).1 },
+    { exact λn, (i⁻¹ᵍ (s₁ n, s₂ n)).2 },
+    { intro n, refine ap (B' HH) _ ⬝ HB c n,
+      refine to_right_inv (group.equiv_of_isomorphism i) _ ⬝ _,
+      apply ap (deg (k X)), exact ap i !prod.eta ⬝ to_right_inv (group.equiv_of_isomorphism i) _ },
+    { intro x, induction x with p q, exact e c (i (p, q)) },
+    { intro n, refine _ ⬝lm f c n, refine isomorphism_of_eq (ap (D X) _),
+      refine to_right_inv (group.equiv_of_isomorphism i) _ ⬝ _,
+      apply ap (deg (k X)), exact ap i !prod.eta ⬝ to_right_inv (group.equiv_of_isomorphism i) _ },
+    { exact λn, (i⁻¹ᵍ (deg_d1 c n, deg_d2 c n)).1 },
+    { exact λn, (i⁻¹ᵍ (deg_d1 c n, deg_d2 c n)).2 },
+    { intro r, esimp, refine !deg_d_reindex ⬝ ap i⁻¹ᵍ (ap (deg (d _)) (group.to_respect_zero i) ⬝
+        deg_d_eq0 c r) ⬝ !prod.eta⁻¹ }
+  end
+  omit c
 
 /-  definition converges_to_reindex {E'' : agℤ → agℤ → LeftModule R} {Dinf' : graded_module R agℤ}
     (i : agℤ ×g agℤ ≃ agℤ × agℤ) (e' : Πp q, E' p q ≃lm E'' (i (p, q)).1 (i (p, q)).2)
     (i2 : agℤ ≃ agℤ) (f' : Πn, Dinf n ≃lm Dinf' (i2 n)) :
     (λp q, E'' p q) ⟹ λn, Dinf' n :=
-  converges_to.mk (exact_couple_reindex i X) (is_bounded_reindex i HH) s₀
+  converges_to.mk (exact_couple_reindex i X) (is_bounded_reindex i HH) s₁ s₂
     sorry --(λn, ap (B' HH) (to_right_inv i _ ⬝ begin end) ⬝ HB c n)
     sorry --begin intro x, induction x with p q, exact e c (p, q) ⬝lm e' p q end
     sorry-/
@@ -651,17 +693,17 @@ namespace left_module
     (f' : Πn, Dinf n ≃lm Dinf' (-n)) :
     (λp q, E'' p q) ⟹ λn, Dinf' n :=
   converges_to.mk (exact_couple_reindex (equiv_neg ×≃ equiv_neg) X) (is_bounded_reindex _ HH)
-    (λn, -s₀ (-n))
+    (λn, -s₂ (-n))
     (λn, ap (B' HH) (begin esimp, end) ⬝ HB c n)
     sorry --begin intro x, induction x with p q, exact e c (p, q) ⬝lm e' p q end
     sorry-/
 
   definition is_built_from_of_converges_to (n : ℤ) : is_built_from (Dinf n) (Einfdiag n) :=
-  is_built_from_isomorphism_left (f c n) (is_built_from_infpage HH (n, s₀ n) (HB c n))
+  is_built_from_isomorphism_left (f c n) (is_built_from_infpage HH (s₁ n, s₂ n) (HB c n))
 
   /- TODO: reprove this using is_built_of -/
   theorem is_contr_converges_to_precise (n : agℤ)
-  (H : Π(n : agℤ) (l : ℕ), is_contr (E' ((deg i)^[l] (n, s₀ n)).1 ((deg i)^[l] (n, s₀ n)).2)) :
+  (H : Π(n : agℤ) (l : ℕ), is_contr (E' ((deg i)^[l] (s₁ n, s₂ n)).1 ((deg i)^[l] (s₁ n, s₂ n)).2)) :
     is_contr (Dinf n) :=
   begin
     assert H2 : Π(l : ℕ), is_contr (Einfdiag n l),
@@ -669,13 +711,13 @@ namespace left_module
       refine is_trunc_equiv_closed_rev -2 (equiv_of_isomorphism (e c _)) (H n l) },
     assert H3 : is_contr (Dinfdiag n 0),
     { fapply nat.rec_down (λk, is_contr (Dinfdiag n k)),
-      { exact is_bounded.B HH (deg (k X) (n, s₀ n)) },
+      { exact is_bounded.B HH (deg (k X) (s₁ n, s₂ n)) },
       { apply Dinfdiag_stable, reflexivity },
       { intro l H,
-        exact is_contr_middle_of_short_exact_mod (short_exact_mod_infpage HH (n, s₀ n) l)
+        exact is_contr_middle_of_short_exact_mod (short_exact_mod_infpage HH (s₁ n, s₂ n) l)
                 (H2 l) H }},
     refine is_trunc_equiv_closed _ _ H3,
-    exact equiv_of_isomorphism (Dinfdiag0 HH (n, s₀ n) (HB c n) ⬝lm f c n)
+    exact equiv_of_isomorphism (Dinfdiag0 HH (s₁ n, s₂ n) (HB c n) ⬝lm f c n)
   end
 
   theorem is_contr_converges_to (n : agℤ) (H : Π(n s : agℤ), is_contr (E' n s)) : is_contr (Dinf n) :=
@@ -686,7 +728,7 @@ namespace left_module
     (H2 : Π⦃r⦄, r₁ ≤ r → r < r₂ → is_contr (E X (n + deg_d1 c r, s + deg_d2 c r))) :
     E (page X r₂) (n, s) ≃lm E (page X r₁) (n, s) :=
   E_isomorphism' X H (λr Hr₁ Hr₂, transport (is_contr ∘ E X) (deg_d_inv_eq r n s)⁻¹ᵖ (H1 Hr₁ Hr₂))
-                    (λr Hr₁ Hr₂, transport (is_contr ∘ E X) (deg_d_eq c r n s)⁻¹ᵖ (H2 Hr₁ Hr₂))
+                    (λr Hr₁ Hr₂, transport (is_contr ∘ E X) (deg_d_eq r n s)⁻¹ᵖ (H2 Hr₁ Hr₂))
 
   definition E_isomorphism0 {r : ℕ} {n s : agℤ}
     (H1 : Πr, is_contr (E X (n - deg_d1 c r, s - deg_d2 c r)))
@@ -699,7 +741,7 @@ namespace left_module
     (H2 : Π⦃r⦄, r₁ ≤ r → is_contr (E X (n + deg_d1 c r, s + deg_d2 c r))) :
     Einf HH (n, s) ≃lm E (page X r₁) (n, s) :=
   Einf_isomorphism' HH r₁ (λr Hr₁, transport (is_contr ∘ E X) (deg_d_inv_eq r n s)⁻¹ᵖ (H1 Hr₁))
-                         (λr Hr₁, transport (is_contr ∘ E X) (deg_d_eq c r n s)⁻¹ᵖ (H2 Hr₁))
+                         (λr Hr₁, transport (is_contr ∘ E X) (deg_d_eq r n s)⁻¹ᵖ (H2 Hr₁))
 
   definition Einf_isomorphism0 {n s : agℤ}
     (H1 : Π⦃r⦄, is_contr (E X (n - deg_d1 c r, s - deg_d2 c r)))
@@ -921,15 +963,16 @@ namespace spectrum
     fapply converges_to.mk,
     { exact exact_couple_sequence },
     { exact is_bounded_sequence },
+    { exact id },
     { exact ub },
     { intro n, change max0 (ub n - ub n) = 0, exact ap max0 (sub_self (ub n)) },
     { intro ns, reflexivity },
     { intro n, reflexivity },
     { intro r, exact - (1 : ℤ) },
     { intro r, exact r + (1 : ℤ) },
-    { intro r n s, refine !convergence_theorem.deg_d ⬝ _,
+    { intro r, refine !convergence_theorem.deg_d ⬝ _,
       refine ap (deg j_sequence) !iterate_deg_i_inv ⬝ _,
-      exact prod_eq idp (!add.assoc ⬝ ap (λx, s + (r + x)) !neg_neg) },
+      exact prod_eq idp !zero_add }
   end
 
   end

cohomology/serre.hlean

@@ -1,7 +1,7 @@
 import ..algebra.spectral_sequence ..spectrum.trunc .basic
 
 open eq spectrum trunc is_trunc pointed int EM algebra left_module fiber lift equiv is_equiv
-     cohomology group sigma unit is_conn
+     cohomology group sigma unit is_conn prod
 set_option pp.binder_types true
 
 /- Eilenberg MacLane spaces are the fibers of the Postnikov system of a type -/
@@ -171,14 +171,14 @@ section atiyah_hirzebruch
   definition is_bounded_atiyah_hirzebruch : is_bounded atiyah_hirzebruch_exact_couple :=
   is_bounded_sequence _ (λn, s₀) (λn, n - 1) atiyah_hirzebruch_lb atiyah_hirzebruch_ub
 
-  definition atiyah_hirzebruch_convergence' :
+  definition atiyah_hirzebruch_convergence1 :
     (λn s, πₛ[n] (sfiber (spi_compose_left (λx, postnikov_smap (Y x) s)))) ⟹ᵍ
     (λn, πₛ[n] (spi X (λx, strunc s₀ (Y x)))) :=
   converges_to_sequence _ (λn, s₀) (λn, n - 1) atiyah_hirzebruch_lb atiyah_hirzebruch_ub
 
-  definition atiyah_hirzebruch_convergence :
+  definition atiyah_hirzebruch_convergence2 :
     (λn s, opH^-(n-s)[(x : X), πₛ[s] (Y x)]) ⟹ᵍ (λn, pH^-n[(x : X), Y x]) :=
-  converges_to_g_isomorphism atiyah_hirzebruch_convergence'
+  converges_to_g_isomorphism atiyah_hirzebruch_convergence1
     begin
       intro n s,
       refine _ ⬝g (parametrized_cohomology_isomorphism_shomotopy_group_spi _ idp)⁻¹ᵍ,
@@ -194,11 +194,32 @@ section atiyah_hirzebruch
       exact ppi_pequiv_right (λx, ptrunc_pequiv (maxm2 (s₀ + k)) (Y x k)),
     end
 
--- set_option pp.metavar_args true
+  open prod.ops
---   definition atiyah_reindexed : (λp q, opH^p[(x : X), πₛ[q] (Y x)]) ⟹ᵍ (λn, pH^n[(x : X), Y x])
+  definition atiyah_hirzebruch_base_change [constructor] : agℤ ×ag agℤ ≃g agℤ ×ag agℤ :=
--- :=
+  begin
---   converges_to_reindex atiyah_hirzebruch_convergence (λp q, -(p - q)) (λp q, q) (λp q, by reflexivity)
+    fapply group.isomorphism.mk,
---     (λn, -n) (λn, by reflexivity)
+    { fapply group.homomorphism.mk, exact (λpq, (-(pq.1 + pq.2), -pq.2)),
+      intro pq pq',
+      induction pq with p q, induction pq' with p' q', esimp,
+      exact prod_eq (ap neg !add.comm4 ⬝ !neg_add) !neg_add },
+    { fapply adjointify,
+      { exact (λns, (ns.2 - ns.1, -ns.2)) },
+      { intro ns, esimp,
+        exact prod_eq (ap neg (!add.comm ⬝ !neg_add_cancel_left) ⬝ !neg_neg) !neg_neg },
+      { intro pq, esimp,
+        exact prod_eq (ap (λx, _ + x) !neg_neg ⬝ !add.comm ⬝ !add_neg_cancel_right) !neg_neg }}
+  end
+
+  definition atiyah_hirzebruch_convergence :
+    (λp q, opH^p[(x : X), πₛ[-q] (Y x)]) ⟹ᵍ (λn, pH^-n[(x : X), Y x]) :=
+  begin
+    note z := converges_to_reindex atiyah_hirzebruch_convergence2 atiyah_hirzebruch_base_change,
+    refine converges_to_g_isomorphism z _ (λn, by reflexivity),
+    intro p q,
+    apply parametrized_cohomology_change_int,
+    esimp,
+    refine !neg_neg_sub_neg ⬝ !add_neg_cancel_right
+  end
 
 end atiyah_hirzebruch
 
@@ -206,11 +227,11 @@ section unreduced_atiyah_hirzebruch
 
 definition unreduced_atiyah_hirzebruch_convergence {X : Type} (Y : X → spectrum) (s₀ : ℤ)
   (H : Πx, is_strunc s₀ (Y x)) :
-  (λn s, uopH^-(n-s)[(x : X), πₛ[s] (Y x)]) ⟹ᵍ (λn, upH^-n[(x : X), Y x]) :=
+  (λp q, uopH^p[(x : X), πₛ[-q] (Y x)]) ⟹ᵍ (λn, upH^-n[(x : X), Y x]) :=
 converges_to_g_isomorphism
   (@atiyah_hirzebruch_convergence X₊ (add_point_spectrum Y) s₀ (is_strunc_add_point_spectrum H))
   begin
-    intro n s, refine _ ⬝g !uopH_isomorphism_opH⁻¹ᵍ,
+    intro p q, refine _ ⬝g !uopH_isomorphism_opH⁻¹ᵍ,
     apply ordinary_parametrized_cohomology_isomorphism_right,
     intro x,
     apply shomotopy_group_add_point_spectrum
@@ -227,17 +248,17 @@ section serre
 
   include H
   definition serre_convergence :
-    (λn s, uopH^-(n-s)[(b : B), uH^-s[F b, Y]]) ⟹ᵍ (λn, uH^-n[Σ(b : B), F b, Y]) :=
+    (λp q, uopH^p[(b : B), uH^q[F b, Y]]) ⟹ᵍ (λn, uH^-n[Σ(b : B), F b, Y]) :=
   proof
   converges_to_g_isomorphism
     (unreduced_atiyah_hirzebruch_convergence
       (λx, sp_ucotensor (F x) Y) s₀
       (λx, is_strunc_sp_ucotensor s₀ (F x) H))
     begin
-      intro n s,
+      intro p q,
-      refine unreduced_ordinary_parametrized_cohomology_isomorphism_right _ (-(n-s)),
+      refine unreduced_ordinary_parametrized_cohomology_isomorphism_right _ p,
       intro x,
-      exact (unreduced_cohomology_isomorphism_shomotopy_group_sp_ucotensor _ _ idp)⁻¹ᵍ
+      exact (unreduced_cohomology_isomorphism_shomotopy_group_sp_ucotensor _ _ !neg_neg)⁻¹ᵍ
     end
     begin
      intro n,
@@ -249,29 +270,29 @@ section serre
   qed
 
   definition serre_convergence_map :
-    (λn s, uopH^-(n-s)[(b : B), uH^-s[fiber f b, Y]]) ⟹ᵍ (λn, uH^-n[X, Y]) :=
+    (λp q, uopH^p[(b : B), uH^q[fiber f b, Y]]) ⟹ᵍ (λn, uH^-n[X, Y]) :=
   proof
   converges_to_g_isomorphism
     (serre_convergence (fiber f) Y s₀ H)
-    begin intro n s, reflexivity end
+    begin intro p q, reflexivity end
     begin intro n, apply unreduced_cohomology_isomorphism, exact !sigma_fiber_equiv⁻¹ᵉ end
   qed
 
   definition serre_convergence_of_is_conn (H2 : is_conn 1 B) :
-    (λn s, uoH^-(n-s)[B, uH^-s[F b₀, Y]]) ⟹ᵍ (λn, uH^-n[Σ(b : B), F b, Y]) :=
+    (λp q, uoH^p[B, uH^q[F b₀, Y]]) ⟹ᵍ (λn, uH^-n[Σ(b : B), F b, Y]) :=
   proof
   converges_to_g_isomorphism
     (serre_convergence F Y s₀ H)
-    begin intro n s, exact @uopH_isomorphism_uoH_of_is_conn (pointed.MK B b₀) _ _ H2  end
+    begin intro p q, exact @uopH_isomorphism_uoH_of_is_conn (pointed.MK B b₀) _ _ H2  end
     begin intro n, reflexivity end
   qed
 
   definition serre_convergence_map_of_is_conn (H2 : is_conn 1 B) :
-    (λn s, uoH^-(n-s)[B, uH^-s[fiber f b₀, Y]]) ⟹ᵍ (λn, uH^-n[X, Y]) :=
+    (λp q, uoH^p[B, uH^q[fiber f b₀, Y]]) ⟹ᵍ (λn, uH^-n[X, Y]) :=
   proof
   converges_to_g_isomorphism
     (serre_convergence_of_is_conn b₀ (fiber f) Y s₀ H H2)
-    begin intro n s, reflexivity end
+    begin intro p q, reflexivity end
     begin intro n, apply unreduced_cohomology_isomorphism, exact !sigma_fiber_equiv⁻¹ᵉ end
   qed