@Marius, so far I agree with your answer in Comment 8.

One suggestion about the notation (but not the content!) of your ideas: I found it helpful to write the ordering as \\( (a , b, c + n) \leq (a + 2n, b + n , c) \\), for all \\(a,b,c \in \mathbb{N}\\). Translating this statement using John's meaning for \\( \leq\\) would give us "We can get \\(a\\) Hs, \\(b\\) Os, and \\((c +n )\\) H2Os from \\((a + 2n)\\) Hs, \\((b +n)\\) Os, and \\(c \\) H2Os". This jives with my understanding of the problem, and it gets rid of having to worry about \\(a \geq 2n\\) etc! Also reflexivity follows directly from the case where \\(n = 0\\).

Other than that I couldn't find a way of streamlining the proofs of the different properties we need to prove. I also did a lot of equation manipulation using the rules of addition. Did anyone else find a simplification?

One suggestion about the notation (but not the content!) of your ideas: I found it helpful to write the ordering as \\( (a , b, c + n) \leq (a + 2n, b + n , c) \\), for all \\(a,b,c \in \mathbb{N}\\). Translating this statement using John's meaning for \\( \leq\\) would give us "We can get \\(a\\) Hs, \\(b\\) Os, and \\((c +n )\\) H2Os from \\((a + 2n)\\) Hs, \\((b +n)\\) Os, and \\(c \\) H2Os". This jives with my understanding of the problem, and it gets rid of having to worry about \\(a \geq 2n\\) etc! Also reflexivity follows directly from the case where \\(n = 0\\).

Other than that I couldn't find a way of streamlining the proofs of the different properties we need to prove. I also did a lot of equation manipulation using the rules of addition. Did anyone else find a simplification?