List of booleans #

In this file we prove lemmas about the number of falses and trues in a list of booleans. First we prove that the number of falses plus the number of true equals the length of the list. Then we prove that in a list with alternating trues and falses, the number of trues differs from the number of falses by at most one. We provide several versions of these statements.

@[simp]
theorem List.count_not_add_count (l : ) (b : Bool) :
List.count (!b) l + = l.length
@[simp]
theorem List.count_add_count_not (l : ) (b : Bool) :
+ List.count (!b) l = l.length
@[simp]
theorem List.count_false_add_count_true (l : ) :
= l.length
@[simp]
theorem List.count_true_add_count_false (l : ) :
= l.length
theorem List.Chain.count_not {b : Bool} {l : } :
List.Chain (fun (x1 x2 : Bool) => x1 x2) b lList.count (!b) l = + l.length % 2
theorem List.Chain'.count_not_eq_count {l : } (hl : List.Chain' (fun (x1 x2 : Bool) => x1 x2) l) (h2 : Even l.length) (b : Bool) :
List.count (!b) l =
theorem List.Chain'.count_false_eq_count_true {l : } (hl : List.Chain' (fun (x1 x2 : Bool) => x1 x2) l) (h2 : Even l.length) :
theorem List.Chain'.count_not_le_count_add_one {l : } (hl : List.Chain' (fun (x1 x2 : Bool) => x1 x2) l) (b : Bool) :
List.count (!b) l + 1
theorem List.Chain'.count_false_le_count_true_add_one {l : } (hl : List.Chain' (fun (x1 x2 : Bool) => x1 x2) l) :
+ 1
theorem List.Chain'.count_true_le_count_false_add_one {l : } (hl : List.Chain' (fun (x1 x2 : Bool) => x1 x2) l) :
+ 1
theorem List.Chain'.two_mul_count_bool_of_even {l : } (hl : List.Chain' (fun (x1 x2 : Bool) => x1 x2) l) (h2 : Even l.length) (b : Bool) :
2 * = l.length
theorem List.Chain'.two_mul_count_bool_eq_ite {l : } (hl : List.Chain' (fun (x1 x2 : Bool) => x1 x2) l) (b : Bool) :
2 * = if Even l.length then l.length else if (some b == l.head?) = true then l.length + 1 else l.length - 1
theorem List.Chain'.length_sub_one_le_two_mul_count_bool {l : } (hl : List.Chain' (fun (x1 x2 : Bool) => x1 x2) l) (b : Bool) :
l.length - 1 2 *
theorem List.Chain'.length_div_two_le_count_bool {l : } (hl : List.Chain' (fun (x1 x2 : Bool) => x1 x2) l) (b : Bool) :
l.length / 2
theorem List.Chain'.two_mul_count_bool_le_length_add_one {l : } (hl : List.Chain' (fun (x1 x2 : Bool) => x1 x2) l) (b : Bool) :
2 * l.length + 1