diff --git a/library/data/finset/basic.lean b/library/data/finset/basic.lean
index d2a991af4..31eff6177 100644
--- a/library/data/finset/basic.lean
+++ b/library/data/finset/basic.lean
@@ -297,7 +297,7 @@ theorem mem_erase_eq (a b : A) (s : finset A) : a ∈ erase b s = (a ∈ s ∧ a
 propext !mem_erase_iff
 
 open decidable
-theorem erase_insert (a : A) (s : finset A) : a ∉ s → erase a (insert a s) = s :=
+theorem erase_insert {a : A} {s : finset A} : a ∉ s → erase a (insert a s) = s :=
 λ anins, finset.ext (λ b, by_cases
   (λ beqa : b = a, iff.intro
     (λ bin, by subst b; exact absurd bin !mem_erase)
@@ -481,16 +481,16 @@ variable [h : decidable_eq A]
 include h
 
 theorem inter.distrib_left (s t u : finset A) : s ∩ (t ∪ u) = (s ∩ t) ∪ (s ∩ u) :=
-ext (take x, by rewrite [mem_inter_eq, *mem_union_eq, *mem_inter_eq]; apply and.distrib_left)
+ext (take x, by rewrite [mem_inter_eq, *mem_union_eq, *mem_inter_eq]; apply and.left_distrib)
 
 theorem inter.distrib_right (s t u : finset A) : (s ∪ t) ∩ u = (s ∩ u) ∪ (t ∩ u) :=
-ext (take x, by rewrite [mem_inter_eq, *mem_union_eq, *mem_inter_eq]; apply and.distrib_right)
+ext (take x, by rewrite [mem_inter_eq, *mem_union_eq, *mem_inter_eq]; apply and.right_distrib)
 
 theorem union.distrib_left (s t u : finset A) : s ∪ (t ∩ u) = (s ∪ t) ∩ (s ∪ u) :=
-ext (take x, by rewrite [mem_union_eq, *mem_inter_eq, *mem_union_eq]; apply or.distrib_left)
+ext (take x, by rewrite [mem_union_eq, *mem_inter_eq, *mem_union_eq]; apply or.left_distrib)
 
 theorem union.distrib_right (s t u : finset A) : (s ∩ t) ∪ u = (s ∪ u) ∩ (t ∪ u) :=
-ext (take x, by rewrite [mem_union_eq, *mem_inter_eq, *mem_union_eq]; apply or.distrib_right)
+ext (take x, by rewrite [mem_union_eq, *mem_inter_eq, *mem_union_eq]; apply or.right_distrib)
 end inter
 
 /- disjoint -/
diff --git a/library/data/finset/comb.lean b/library/data/finset/comb.lean
index 260d09204..5a4a9b2b0 100644
--- a/library/data/finset/comb.lean
+++ b/library/data/finset/comb.lean
@@ -150,6 +150,11 @@ iff.intro
 theorem mem_filter_eq : x ∈ filter p s = (x ∈ s ∧ p x) :=
 propext !mem_filter_iff
 
+variable t : finset A
+
+theorem mem_filter_union_iff : x ∈ filter p (s ∪ t) ↔ x ∈ filter p s ∨ x ∈ filter p t :=
+by rewrite [*mem_filter_iff, mem_union_iff, and.right_distrib]
+
 end filter
 
 theorem mem_singleton_eq' {A : Type} [deceq : decidable_eq A] (x a : A) : x ∈ '{a} = (x = a) :=
@@ -423,6 +428,15 @@ begin
               (show x ⊆ s, by rewrite [(erase_eq_of_not_mem this) at H']; apply H'))))
 end
 
+theorem subset_of_mem_powerset {s t : finset A} (H : s ∈ powerset t) : s ⊆ t :=
+by rewrite mem_powerset_iff_subset at H; exact H
+
+theorem mem_powerset_of_subset {s t : finset A} (H : s ⊆ t) : s ∈ powerset t :=
+by rewrite -mem_powerset_iff_subset at H; exact H
+
+theorem empty_mem_powerset (s : finset A) : ∅ ∈ powerset s :=
+by rewrite mem_powerset_iff_subset; apply empty_subset
+
 end powerset
 
 end finset
diff --git a/library/data/set/basic.lean b/library/data/set/basic.lean
index 461587fa6..d868be40c 100644
--- a/library/data/set/basic.lean
+++ b/library/data/set/basic.lean
@@ -157,16 +157,16 @@ setext (take x, !true_and)
 /- distributivity laws -/
 
 theorem inter.distrib_left (s t u : set X) : s ∩ (t ∪ u) = (s ∩ t) ∪ (s ∩ u) :=
-setext (take x, !and.distrib_left)
+setext (take x, !and.left_distrib)
 
 theorem inter.distrib_right (s t u : set X) : (s ∪ t) ∩ u = (s ∩ u) ∪ (t ∩ u) :=
-setext (take x, !and.distrib_right)
+setext (take x, !and.right_distrib)
 
 theorem union.distrib_left (s t u : set X) : s ∪ (t ∩ u) = (s ∪ t) ∩ (s ∪ u) :=
-setext (take x, !or.distrib_left)
+setext (take x, !or.left_distrib)
 
 theorem union.distrib_right (s t u : set X) : (s ∩ t) ∪ u = (s ∪ u) ∩ (t ∪ u) :=
-setext (take x, !or.distrib_right)
+setext (take x, !or.right_distrib)
 
 /- set-builder notation -/
 
diff --git a/library/logic/connectives.lean b/library/logic/connectives.lean
index 2b9599aaf..b7ce97139 100644
--- a/library/logic/connectives.lean
+++ b/library/logic/connectives.lean
@@ -152,7 +152,7 @@ theorem and_imp_eq (a b c : Prop) : (a ∧ b → c) = (a → b → c) :=
 propext !and_imp_iff
 
 theorem and_iff_and (H1 : a ↔ c) (H2 : b ↔ d) : (a ∧ b) ↔ (c ∧ d) :=
-iff.intro (and.imp (iff.mp H1) (iff.mp H2)) (and.imp (iff.mpr H1) (iff.mpr H2)) 
+iff.intro (and.imp (iff.mp H1) (iff.mp H2)) (and.imp (iff.mpr H1) (iff.mpr H2))
 
 /- or -/
 
@@ -221,25 +221,25 @@ theorem or_self (a : Prop) : a ∨ a ↔ a :=
 iff.intro (or.rec imp.id imp.id) or.inl
 
 theorem or_iff_or (H1 : a ↔ c) (H2 : b ↔ d) : (a ∨ b) ↔ (c ∨ d) :=
-iff.intro (or.imp (iff.mp H1) (iff.mp H2)) (or.imp (iff.mpr H1) (iff.mpr H2)) 
+iff.intro (or.imp (iff.mp H1) (iff.mp H2)) (or.imp (iff.mpr H1) (iff.mpr H2))
 
 /- distributivity -/
 
-theorem and.distrib_left (a b c : Prop) : a ∧ (b ∨ c) ↔ (a ∧ b) ∨ (a ∧ c) :=
+theorem and.left_distrib (a b c : Prop) : a ∧ (b ∨ c) ↔ (a ∧ b) ∨ (a ∧ c) :=
 iff.intro
   (and.rec (λH, or.imp (and.intro H) (and.intro H)))
   (or.rec (and.imp_right or.inl) (and.imp_right or.inr))
 
-theorem and.distrib_right (a b c : Prop) : (a ∨ b) ∧ c ↔ (a ∧ c) ∨ (b ∧ c) :=
-iff.trans (iff.trans !and.comm !and.distrib_left) (or_iff_or !and.comm !and.comm)
+theorem and.right_distrib (a b c : Prop) : (a ∨ b) ∧ c ↔ (a ∧ c) ∨ (b ∧ c) :=
+iff.trans (iff.trans !and.comm !and.left_distrib) (or_iff_or !and.comm !and.comm)
 
-theorem or.distrib_left (a b c : Prop) : a ∨ (b ∧ c) ↔ (a ∨ b) ∧ (a ∨ c) :=
+theorem or.left_distrib (a b c : Prop) : a ∨ (b ∧ c) ↔ (a ∨ b) ∧ (a ∨ c) :=
 iff.intro
   (or.rec (λH, and.intro (or.inl H) (or.inl H)) (and.imp or.inr or.inr))
   (and.rec (or.rec (imp.syl imp.intro or.inl) (imp.syl or.imp_right and.intro)))
 
-theorem or.distrib_right (a b c : Prop) : (a ∧ b) ∨ c ↔ (a ∨ c) ∧ (b ∨ c) :=
-iff.trans (iff.trans !or.comm !or.distrib_left) (and_iff_and !or.comm !or.comm)
+theorem or.right_distrib (a b c : Prop) : (a ∧ b) ∨ c ↔ (a ∨ c) ∧ (b ∨ c) :=
+iff.trans (iff.trans !or.comm !or.left_distrib) (and_iff_and !or.comm !or.comm)
 
 /- iff -/