Corollary to_Z_bounded : forall x, (0 <= φ  x  < wB)%Z.

(* Specifications: *)
Theorem Zmod_distr: forall a b r t, 0 <= a <= b -> 0 <= r -> 0 <= t < 2 ^a -> (2 ^a * r + t) mod (2 ^ b) = (2 ^a * r) mod (2 ^ b) + t.
Lemma pow2_pos n : 0 <= n → 2 ^ n > 0.
Lemma pow2_nz n : 0 <= n → 2 ^ n ≠ 0.
Lemma wB_diff_0 : wB <> 0.
Lemma wB_pos : 0 < wB.
Lemma to_Z_0 : φ 0 = 0.
Lemma to_Z_1 : φ 1 = 1.