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namespace Nat
theorem two_d_coordinate_to_index_lt_size {x y w h: Nat} (h₁ : x < w) (h₂ : y < h) : x + w*y < w*h :=
Nat.le_pred_of_lt h₂
|> Nat.mul_le_mul_left w
|> Nat.add_le_add_iff_right.mpr
|> (Nat.mul_pred w h).subst (motive :=λx↦w * y + w ≤ x + w)
|> (Nat.sub_add_cancel (Nat.le_mul_of_pos_right w (Nat.zero_lt_of_lt h₂))).subst
|> (Nat.add_comm _ _).subst (motive := λx↦x ≤ w*h)
|> Nat.le_sub_of_add_le
|> Nat.lt_of_lt_of_le h₁
|> λx↦(Nat.add_lt_add_right) x (w * y)
|> (Nat.sub_add_cancel (Nat.le_of_lt ((Nat.mul_lt_mul_left (Nat.zero_lt_of_lt h₁)).mpr h₂))).subst
theorem gt_of_sub_lt {a b c : Nat} (h₁ : a - b < a - c) : c < b := by omega
theorem sub_lt_of_gt {a b c : Nat} (h₁ : b ≤ a) (h₂ : c < b) : a - b < a - c := by omega
theorem lt_of_pred_lt {a b : Nat} (h₁ : a < b.pred) : (a < b) :=
match b with
| 0 => h₁
| _ + 1 => /-(Nat.pred_succ a).substr $-/ Nat.lt_succ_of_lt h₁
theorem lt_imp_pred_lt {a b : Nat} (h₁ : a < b) : (a.pred < b) :=
Nat.lt_of_le_of_lt (Nat.pred_le a) h₁
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