def lt_ceil

bend-collections-laws-math@1.0.0.0 · proofs/math/random/lemire.bend:190

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/proofs/math/natural/arith.bend as AR

190# x n < c is x < ceil(c / n) = (c + n - 1) / n, n = 1 + bp
191def lt_ceil(+bp: Nat, +x: Nat, +c: Nat) -> {Nat.is_lt(Nat.mul(x, 1n+bp), c) == Nat.is_lt(x, Nat.div(Nat.add(c, bp), 1n+bp)) : Bool}:
192  +X = Nat.mul(x, 1n+bp)
193  +D = Nat.div(Nat.add(c, bp), 1n+bp)
194  Equal.trans(Bool, Nat.is_lt(X, c), Nat.is_le(1n+X, c), Nat.is_lt(x, D),
195    Equal.sym(Bool, Nat.is_le(1n+X, c), Nat.is_lt(X, c), le_succ_lt(X, c)),
196    Equal.trans(Bool, Nat.is_le(1n+X, c), Nat.is_le(Nat.add(bp, 1n+X), Nat.add(bp, c)), Nat.is_lt(x, D),
197      Equal.sym(Bool, Nat.is_le(Nat.add(bp, 1n+X), Nat.add(bp, c)), Nat.is_le(1n+X, c), AR.le_add_cancel(bp, 1n+X, c)),
198      Equal.trans(Bool, Nat.is_le(Nat.add(bp, 1n+X), Nat.add(bp, c)), Nat.is_le(1n+Nat.add(bp, X), Nat.add(c, bp)), Nat.is_lt(x, D),
199        %N.add_succ(bp, X) : {Nat.is_le(Nat.add(bp, 1n+X), Nat.add(bp, c)) == Nat.is_le(_, Nat.add(c, bp)) : Bool}
200        %N.add_comm(bp, c) : {Nat.is_le(Nat.add(bp, 1n+X), Nat.add(bp, c)) == Nat.is_le(Nat.add(bp, 1n+X), _) : Bool}
201        {==},
202        Equal.trans(Bool, Nat.is_le(1n+Nat.add(bp, X), Nat.add(c, bp)), Nat.is_le(1n+x, D), Nat.is_lt(x, D),
203          Equal.sym(Bool, Nat.is_le(1n+x, D), Nat.is_le(1n+Nat.add(bp, X), Nat.add(c, bp)), AR.le_div(bp, 1n+x, Nat.add(c, bp))),
204          le_succ_lt(x, D)))))