chore(order/*): golf 2 proofs (#17095)

src/order/atoms.lean

@@ -156,11 +156,11 @@ section atomic
 variables [partial_order α] (α)
 
 /-- A lattice is atomic iff every element other than `⊥` has an atom below it. -/
-class is_atomic [order_bot α] : Prop :=
+@[mk_iff] class is_atomic [order_bot α] : Prop :=
 (eq_bot_or_exists_atom_le : ∀ (b : α), b = ⊥ ∨ ∃ (a : α), is_atom a ∧ a ≤ b)
 
 /-- A lattice is coatomic iff every element other than `⊤` has a coatom above it. -/
-class is_coatomic [order_top α] : Prop :=
+@[mk_iff] class is_coatomic [order_top α] : Prop :=
 (eq_top_or_exists_le_coatom : ∀ (b : α), b = ⊤ ∨ ∃ (a : α), is_coatom a ∧ b ≤ a)
 
 export is_atomic (eq_bot_or_exists_atom_le) is_coatomic (eq_top_or_exists_le_coatom)
@@ -678,25 +678,14 @@ lemma is_simple_order [bounded_order α] [bounded_order β] [h : is_simple_order
   is_simple_order α :=
 f.is_simple_order_iff.mpr h
 
-lemma is_atomic_iff [order_bot α] [order_bot β] (f : α ≃o β) : is_atomic α ↔ is_atomic β :=
-begin
-  suffices : (∀ b : α, b = ⊥ ∨ ∃ (a : α), is_atom a ∧ a ≤ b) ↔
-    (∀ b : β, b = ⊥ ∨ ∃ (a : β), is_atom a ∧ a ≤ b),
-  from ⟨λ ⟨p⟩, ⟨this.mp p⟩, λ ⟨p⟩, ⟨this.mpr p⟩⟩,
-  apply f.to_equiv.forall_congr,
-  simp_rw [rel_iso.coe_fn_to_equiv],
-  intro b, apply or_congr,
-  { rw [f.apply_eq_iff_eq_symm_apply, map_bot], },
-  { split,
-    { exact λ ⟨a, ha⟩, ⟨f a, ⟨(f.is_atom_iff a).mpr ha.1, f.le_iff_le.mpr ha.2⟩⟩, },
-    { rintros ⟨b, ⟨hb1, hb2⟩⟩,
-      refine ⟨f.symm b, ⟨(f.symm.is_atom_iff b).mpr hb1, _⟩⟩,
-      rwa [←f.le_iff_le, f.apply_symm_apply], }, },
-end
+protected lemma is_atomic_iff [order_bot α] [order_bot β] (f : α ≃o β) :
+  is_atomic α ↔ is_atomic β :=
+by simp only [is_atomic_iff, f.surjective.forall, f.surjective.exists, ← map_bot f, f.eq_iff_eq,
+  f.le_iff_le, f.is_atom_iff]
 
-lemma is_coatomic_iff [order_top α] [order_top β] (f : α ≃o β) : is_coatomic α ↔ is_coatomic β :=
-by { rw [←is_atomic_dual_iff_is_coatomic, ←is_atomic_dual_iff_is_coatomic],
-  exact f.dual.is_atomic_iff }
+protected lemma is_coatomic_iff [order_top α] [order_top β] (f : α ≃o β) :
+  is_coatomic α ↔ is_coatomic β :=
+by simp only [← is_atomic_dual_iff_is_coatomic, f.dual.is_atomic_iff]
 
 end order_iso
 

src/order/filter/bases.lean

@@ -205,12 +205,7 @@ variables {l l' : filter α} {p : ι → Prop} {s : ι → set α} {t : set α}
 
 lemma has_basis_generate (s : set (set α)) :
   (generate s).has_basis (λ t, set.finite t ∧ t ⊆ s) (λ t, ⋂₀ t) :=
-⟨begin
-  intro U,
-  rw mem_generate_iff,
-  apply exists_congr,
-  tauto
-end⟩
+⟨λ U, by simp only [mem_generate_iff, exists_prop, and.assoc, and.left_comm]⟩
 
 /-- The smallest filter basis containing a given collection of sets. -/
 def filter_basis.of_sets (s : set (set α)) : filter_basis α :=