feat(order/filter): prod_assoc (#12002)

map (prod_assoc α β γ) ((f ×ᶠ g) ×ᶠ h) = f ×ᶠ (g ×ᶠ h)
with two tiny supporting lemmas

Author: PatrickMassot

src/order/filter/bases.lean

@@ -338,6 +338,13 @@ begin
   exact forall_congr (λ s, ⟨λ h, h.1, λ h, ⟨h, λ ⟨t, hl, hP, hts⟩, mem_of_superset hl hts⟩⟩)
 end
 
+lemma has_basis.comp_of_surjective (h : l.has_basis p s) {g : ι' → ι} (hg : function.surjective g) :
+  l.has_basis (p ∘ g) (s ∘ g) :=
+⟨λ t, h.mem_iff.trans hg.exists⟩
+
+lemma has_basis.comp_equiv (h : l.has_basis p s) (e : ι' ≃ ι) : l.has_basis (p ∘ e) (s ∘ e) :=
+h.comp_of_surjective e.surjective
+
 /-- If `{s i | p i}` is a basis of a filter `l` and each `s i` includes `s j` such that
 `p j ∧ q j`, then `{s j | p j ∧ q j}` is a basis of `l`. -/
 lemma has_basis.restrict (h : l.has_basis p s) {q : ι → Prop}
@@ -688,6 +695,16 @@ end
 
 end two_types
 
+open equiv
+
+lemma prod_assoc (f : filter α) (g : filter β) (h : filter γ) :
+  map (prod_assoc α β γ) ((f ×ᶠ g) ×ᶠ h) = f ×ᶠ (g ×ᶠ h) :=
+begin
+  apply ((((basis_sets f).prod $ basis_sets g).prod $ basis_sets h).map _).eq_of_same_basis,
+  simpa only [prod_assoc_image, function.comp, and_assoc] using
+    ((basis_sets f).prod $ (basis_sets g).prod $ basis_sets h).comp_equiv (prod_assoc _ _ _)
+end
+
 end filter
 
 end sort

src/order/filter/basic.lean

@@ -2000,6 +2000,14 @@ le_antisymm
   (λ b (hb : preimage m b ∈ f),
     ⟨preimage m b, hb, show preimage (m ∘ n) b ⊆ b, by simp only [h₁]; apply subset.refl⟩)
 
+lemma map_equiv_symm (e : α ≃ β) (f : filter β) :
+  map e.symm f = comap e f :=
+map_eq_comap_of_inverse e.symm_comp_self e.self_comp_symm
+
+lemma comap_equiv_symm (e : α ≃ β) (f : filter α) :
+  comap e.symm f = map e f :=
+(map_eq_comap_of_inverse e.self_comp_symm e.symm_comp_self).symm
+
 lemma map_swap_eq_comap_swap {f : filter (α × β)} : prod.swap <$> f = comap prod.swap f :=
 map_eq_comap_of_inverse prod.swap_swap_eq prod.swap_swap_eq