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Diffstat (limited to 'Common/Nat.lean')
| -rw-r--r-- | Common/Nat.lean | 13 |
1 files changed, 13 insertions, 0 deletions
diff --git a/Common/Nat.lean b/Common/Nat.lean new file mode 100644 index 0000000..4c25315 --- /dev/null +++ b/Common/Nat.lean @@ -0,0 +1,13 @@ +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 |