-- Some isomorphisms between ℕ and ℕ×ℕ. ||| An isomorphism between ℕ and ℕ×ℕ, counting off "by squares", like this: ||| ||| 0 3 8 ||| 1 2 7 ||| 4 5 6 ||| ||| and so on, where the first column contains the square numbers. ||| sqIso' is the inverse. !!! ∀ n : Nat. sqIso (sqIso' n) == n sqIso : ℕ×ℕ → ℕ sqIso (x,y) = {? y^2 + x if x <= y, (x+1)^2 .- 1 .- y otherwise ?} ||| Inverse direction of the square isomorphism. !!! ∀ p : Nat*Nat. sqIso' (sqIso p) == p sqIso' : ℕ → ℕ×ℕ sqIso' n = let r = sqrt n in {? (n .- r^2, r) if n <= r^2 + r, (r, (r+1)^2 .- 1 .- n) otherwise ?} ||| The classic "diagonal" isomorphism: ||| ||| 0 2 5 ||| 1 4 ||| 3 ||| ||| where the first column contains the triangular numbers. !!! ∀ n : Nat. diagIso (diagIso' n) == n diagIso : ℕ×ℕ → ℕ diagIso (x,y) = (x+y)*(x+y+1)//2 + x diagIso' : ℕ → ℕ×ℕ diagIso' n = let d = (sqrt(1 + 8n) .- 1)//2 : N in let t = d*(d+1)//2 in (n .- t, d .- (n .- t)) ||| Every POSITIVE n can be decomposed into a power of two times an ||| odd number, n = 2^x (2y + 1). This creates an isomorphism n <-> ||| (x,y). This is actually an isomorphism between {n | n : ℕ, n > 0} ||| and ℕ×ℕ, but at the moment the disco type system doesn't let us ||| say that (and it likely never will). -- We have to be careful not to call powerIso' 0 because it gets stuck -- in infinite recursion! One way that would work is to write the -- test as follows: !!! forall n:Nat. powerIso (powerIso' (n+1)) == (n+1) -- Alternatively, since 'implies' is lazy (i.e. "short-circuiting"), -- we can write !!! ∀ n : Nat. (n > 0) ==> powerIso (powerIso' n) == n powerIso : ℕ×ℕ → ℕ powerIso (x,y) = 2^x * (2y + 1) powerIso' : ℕ → ℕ×ℕ powerIso' n = {? (0, n//2) if not (2 divides n), (x+1,y) when powerIso' (n//2) is (x,y) ?}