feat(data/list/min_max): add minimum (#892)

Author: minchaowu

src/data/list/min_max.lean

@@ -11,22 +11,40 @@ variables {α : Type u} [inhabited α] [decidable_linear_order α]
 
 @[simp] def maximum (l : list α) : α := l.foldl max l.head
 
+@[simp] def minimum (l : list α) : α := l.foldl min l.head
+
 def maximum_aux (l : list α) : α := l.foldr max l.head
 
+def minimum_aux (l : list α) : α := l.foldr min l.head
+
 @[simp] def maximum_singleton {a : α} : maximum [a] = a := by simp
 
+@[simp] def minimum_singleton {a : α} : minimum [a] = a := by simp
+
 theorem le_of_foldr_max : Π {a b : α} {l}, a ∈ l → a ≤ foldr max b l
 | a b [] h := absurd h $ not_mem_nil _
 | a b (hd::tl) h := 
 begin 
   cases h, 
-  { simp [h, le_max_left] }, 
+  { simp [h, le_refl] }, 
   { simp [le_max_right_of_le, le_of_foldr_max h] } 
 end
 
+theorem le_of_foldr_min : Π {a b : α} {l}, a ∈ l → foldr min b l ≤ a
+| a b [] h := absurd h $ not_mem_nil _
+| a b (hd::tl) h := 
+begin 
+  cases h, 
+  { simp [h, le_refl] }, 
+  { simp [min_le_left_of_le, le_of_foldr_min h] } 
+end
+
 theorem le_of_foldl_max {a b : α} {l} (h : a ∈ l) : a ≤ foldl max b l := 
 by { rw foldl_eq_foldr max_comm max_assoc, apply le_of_foldr_max h }
 
+theorem le_of_foldl_min {a b : α} {l} (h : a ∈ l) : foldl min b l ≤ a := 
+by { rw foldl_eq_foldr min_comm min_assoc, apply le_of_foldr_min h }
+
 theorem mem_foldr_max : Π {a : α} {l}, foldr max a l ∈ a :: l
 | a [] := by simp
 | a (hd::tl) := 
@@ -39,9 +57,24 @@ begin
     cases hmem, { simp [hmem] }, { right, right, exact hmem } }
 end
 
+theorem mem_foldr_min : Π {a : α} {l}, foldr min a l ∈ a :: l
+| a [] := by simp
+| a (hd::tl) := 
+begin
+  simp only [foldr_cons],
+  cases (@min_choice _ _ hd (foldr min a tl)), 
+  { simp [h] }, 
+  { rw h, 
+    have hmem := @mem_foldr_min a tl, 
+    cases hmem, { simp [hmem] }, { right, right, exact hmem } }
+end
+
 theorem mem_foldl_max {a : α} {l} : foldl max a l ∈ a :: l :=
 by { rw foldl_eq_foldr max_comm max_assoc, apply mem_foldr_max }
 
+theorem mem_foldl_min {a : α} {l} : foldl min a l ∈ a :: l :=
+by { rw foldl_eq_foldr min_comm min_assoc, apply mem_foldr_min }
+
 theorem mem_maximum_aux : Π {l : list α}, l ≠ [] →  maximum_aux l ∈ l
 | [] h := by contradiction
 | (hd::tl) h := 
@@ -51,9 +84,21 @@ begin
   cases hc, { simp [hc] }, { simp [hc, mem_foldr_max] }
 end
 
+theorem mem_minimum_aux : Π {l : list α}, l ≠ [] →  minimum_aux l ∈ l
+| [] h := by contradiction
+| (hd::tl) h := 
+begin
+  dsimp [minimum_aux],
+  have hc := @min_choice _ _ hd (foldr min hd tl),
+  cases hc, { simp [hc] }, { simp [hc, mem_foldr_min] }
+end
+
 theorem mem_maximum {l : list α} (h : l ≠ []) : maximum l ∈ l :=
 by { dsimp, rw foldl_eq_foldr max_comm max_assoc, apply mem_maximum_aux h }
 
+theorem mem_minimum {l : list α} (h : l ≠ []) : minimum l ∈ l :=
+by { dsimp, rw foldl_eq_foldr min_comm min_assoc, apply mem_minimum_aux h }
+
 theorem le_maximum_aux_of_mem : Π {a : α} {l}, a ∈ l → a ≤ maximum_aux l
 | a [] h := absurd h $ not_mem_nil _
 | a (hd::tl) h := 
@@ -63,9 +108,21 @@ begin
   { dsimp [maximum_aux], apply le_max_right_of_le, apply le_of_foldr_max h }
 end
 
+theorem le_minimum_aux_of_mem : Π {a : α} {l}, a ∈ l → minimum_aux l ≤ a
+| a [] h := absurd h $ not_mem_nil _
+| a (hd::tl) h := 
+begin
+  cases h,
+  { rw h, apply le_of_foldr_min, simp },
+  { dsimp [minimum_aux], apply min_le_right_of_le, apply le_of_foldr_min h }
+end
+
 theorem le_maximum_of_mem {a : α} {l} (h : a ∈ l) : a ≤ maximum l :=
 by { dsimp, rw foldl_eq_foldr max_comm max_assoc, apply le_maximum_aux_of_mem h }
 
+theorem le_minimum_of_mem {a : α} {l} (h : a ∈ l) : minimum l ≤ a :=
+by { dsimp, rw foldl_eq_foldr min_comm min_assoc, apply le_minimum_aux_of_mem h }
+
 def maximum_aux_cons : Π {a : α} {l}, l ≠ [] → maximum_aux (a :: l) = max a (maximum_aux l)
 | a [] h := by contradiction
 | a (hd::tl) h := 
@@ -82,13 +139,38 @@ begin
     { simp [hc, le_maximum_aux_of_mem, mem_maximum_aux h] } }
 end
 
+def minimum_aux_cons : Π {a : α} {l}, l ≠ [] → minimum_aux (a :: l) = min a (minimum_aux l)
+| a [] h := by contradiction
+| a (hd::tl) h := 
+begin
+  apply le_antisymm,
+  { have hc := @min_choice _ _ a (minimum_aux $ hd :: tl),
+    cases hc, 
+    { simp [hc, le_minimum_aux_of_mem] }, 
+    { simp [hc, le_minimum_aux_of_mem, mem_minimum_aux h] } },
+  { have : a :: hd :: tl ≠ [], { simp [h] },
+    have hle := mem_minimum_aux this, 
+    cases hle, 
+    { simp [hle, min_le_left] }, 
+    { apply min_le_right_of_le, apply le_minimum_aux_of_mem, exact hle } }
+end
+
 def maximum_cons {a : α} {l} (h : l ≠ []) : maximum (a :: l) = max a (maximum l) :=
 begin 
   dsimp only [maximum], 
-  repeat {rw foldl_eq_foldr max_comm max_assoc}, 
+  repeat { rw foldl_eq_foldr max_comm max_assoc }, 
   have := maximum_aux_cons h, 
   dsimp only [maximum_aux] at this, 
   exact this
 end
 
+def minimum_cons {a : α} {l} (h : l ≠ []) : minimum (a :: l) = min a (minimum l) :=
+begin 
+  dsimp only [minimum], 
+  repeat { rw foldl_eq_foldr min_comm min_assoc }, 
+  have := minimum_aux_cons h, 
+  dsimp only [minimum_aux] at this, 
+  exact this
+end
+
 end list