Letter Lines

The letters A-I represent the digits 1-9 in some order. The letters are placed in a line from left to right. Determine which digit is represented by each letter and the order of the letters from left to right, given the following clues. ("A _ 6" means "A is two positions to the left of 6." "_ _ 8" means "There are at least two positions to the left of 8.")
Puzzle 1
A _ _ _ 5
4 _ _ B
C
D _ _ _ 2
6 _ _ _ E
7 _ _ _ _ F
5 _ G _
_ _ 8 _ H
1 _ I _ 9
 Puzzle 2
A _ 6
B _ _ _ 4
C _ _ _ _ 7
1 _ D _ 7
1 _ _ _ _ E
9 _ F
G _ _ 8
2 _ _ H
5 _ _ _ I

Source: Original. [Has anyone seen this sort of puzzle before? It seems new to me.]

Solution
Mail to Ken