def unbiased_c

bend-collections-laws-math@1.0.0.0 · proofs/math/random/lemire.bend:589

import Base
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-math@1.0.0.0/spec/math/random/rand.bend as SR
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N

589def unbiased_c(+w: Nat, +m: Nat, +k: Nat, +hw: {C.fits(w, 1n+m) == True{} : Bool}, +hk: {Nat.is_lt(k, 1n+m) == True{} : Bool}, +c: Bool, +hc: {Nat.is_eq(SR.and_bits(w, 1n+m, m), 0n) == c : Bool}) -> {SR.count(w, 1n+m, k, C.shift(w, 1n)) == Nat.div(C.shift(w, 1n), 1n+m) : Nat}:
590  match c:
591    case False{}:
592      lemire_count(w, m, k, hc, hw, hk)
593    case True{}:
594      Equal.trans(Nat, SR.count(w, 1n+m, k, C.shift(w, 1n)), cm(w, m, k, C.shift(w, 1n)), Nat.div(C.shift(w, 1n), 1n+m),
595        count_mask(w, m, k, C.shift(w, 1n), hc),
596        mask(w, m, k, C.bit(m), C.half(m), {==}, {==}, N.eq_from_is_eq(SR.and_bits(w, 1n+m, m), 0n, hc), hw, hk))