feat(library/*): add various theorems

library/data/set/basic.lean

@@ -301,6 +301,9 @@ ext (take x, iff.intro
 theorem mem_sep_iff {s : set X} {P : X → Prop} {x : X} : x ∈ {x ∈ s | P x} ↔ x ∈ s ∧ P x :=
 !iff.refl
 
+theorem sep_subset (s : set X) (P : X → Prop) : {x ∈ s | P x} ⊆ s :=
+take x, assume H, and.left H
+
 /- complement -/
 
 definition complement (s : set X) : set X := {x | x ∉ s}
@@ -310,7 +313,15 @@ theorem mem_comp {s : set X} {x : X} (H : x ∉ s) : x ∈ -s := H
 
 theorem not_mem_of_mem_comp {s : set X} {x : X} (H : x ∈ -s) : x ∉ s := H
 
-theorem mem_comp_iff {s : set X} {x : X} : x ∈ -s ↔ x ∉ s := !iff.refl
+theorem mem_comp_iff (s : set X) (x : X) : x ∈ -s ↔ x ∉ s := !iff.refl
+
+theorem inter_comp_self (s : set X) : s ∩ -s = ∅ :=
+ext (take x, !and_not_self_iff)
+
+theorem comp_inter_self (s : set X) : -s ∩ s = ∅ :=
+ext (take x, !not_and_self_iff)
+
+/- some classical identities -/
 
 section
   open classical
@@ -320,6 +331,12 @@ section
 
   theorem inter_eq_comp_comp_union_comp (s t : set X) : s ∩ t = -(-s ∪ -t) :=
   ext (take x, !and_iff_not_or_not)
+
+  theorem union_comp_self (s : set X) : s ∪ -s = univ :=
+  ext (take x, !or_not_self_iff)
+
+  theorem comp_union_self (s : set X) : -s ∪ s = univ :=
+  ext (take x, !not_or_self_iff)
 end
 
 /- set difference -/
@@ -350,6 +367,8 @@ ext (take x, iff.intro
       (suppose x ∈ s, or.inl this)
       (suppose x ∉ s, or.inr (and.intro H1 this))))
 
+theorem diff_subset (s t : set X) : s \ t ⊆ s := inter_subset_left s _
+
 /- powerset -/
 
 definition powerset (s : set X) : set (set X) := {x : set X | x ⊆ s}
@@ -377,6 +396,12 @@ section
   definition sUnion : set X := {x : X | ∃₀ c ∈ C, x ∈ c}
 
   -- TODO: need notation for these
+
+  theorem Union_subset {b : I → set X} {c : set X} (H : ∀ i, b i ⊆ c) : Union b ⊆ c :=
+  take x,
+  suppose x ∈ Union b,
+  obtain i (Hi : x ∈ b i), from this,
+  show x ∈ c, from H i Hi
 end
 
 end set

library/data/set/function.lean

@@ -100,6 +100,16 @@ take x, assume H : x ∈ a, H1 (H2 H)
 theorem maps_to_univ_univ (f : X → Y) : maps_to f univ univ :=
 take x, assume H, trivial
 
+theorem image_subset_of_maps_to {f : X → Y} {a : set X} {b : set Y} (mfab : maps_to f a b)
+    {c : set X} (csuba : c ⊆ a) :
+  f '[c] ⊆ b :=
+take y,
+suppose y ∈ f '[c],
+obtain x [(xc : x ∈ c) (yeq : f x = y)], from this,
+have x ∈ a, from csuba `x ∈ c`,
+have f x ∈ b, from mfab this,
+show y ∈ b, from yeq ▸ this
+
 /- injectivity -/
 
 definition inj_on [reducible] (f : X → Y) (a : set X) : Prop :=

library/logic/identities.lean

@@ -27,6 +27,18 @@ calc
     ... ↔ a ∧ (c ∧ b)       : {and.comm}
      ... ↔ (a ∧ c) ∧ b      : iff.symm and.assoc
 
+theorem or_not_self_iff {a : Prop} [D : decidable a] : a ∨ ¬ a ↔ true :=
+iff.intro (assume H, trivial) (assume H, em a)
+
+theorem not_or_self_iff {a : Prop} [D : decidable a] : ¬ a ∨ a ↔ true :=
+!or.comm ▸ !or_not_self_iff
+
+theorem and_not_self_iff {a : Prop} : a ∧ ¬ a ↔ false :=
+iff.intro (assume H, (and.right H) (and.left H)) (assume H, false.elim H)
+
+theorem not_and_self_iff {a : Prop} : ¬ a ∧ a ↔ false :=
+!and.comm ▸ !and_not_self_iff
+
 theorem and.left_comm [simp] (a b c : Prop) : a ∧ (b ∧ c) ↔ b ∧ (a ∧ c) :=
 calc
   a ∧ (b ∧ c) ↔ (a ∧ b) ∧ c : iff.symm and.assoc