def le_div

bend-collections-laws-math@1.0.0.0 · proofs/math/natural/arith.bend:82

import Base

82# x <= n / d is x d <= n   (Mathlib Nat.le_div_iff_mul_le)
83def le_div(+bp: Nat, +x: Nat, +n: Nat) -> {Nat.is_le(x, Nat.div(n, 1n+bp)) == Nat.is_le(Nat.mul(x, 1n+bp), n) : Bool}:
84  +e = dm_eq(bp, n)
85  %Equal.sym(Nat, n, Nat.add(Nat.mul(Nat.div(n, 1n+bp), 1n+bp), Nat.mod(n, 1n+bp)), e) : {Nat.is_le(x, Nat.div(n, 1n+bp)) == Nat.is_le(Nat.mul(x, 1n+bp), _) : Bool}
86  le_div_g(x, Nat.div(n, 1n+bp), bp, Nat.mod(n, 1n+bp), dm_lt(bp, n))