def cm0

bend-collections-laws-crypto@1.0.0.0 · proofs/math/random/lemire.bend:444

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

444# n = 1: every output draws 0
445def cm0(+w: Nat, +N: Nat) -> {cm(w, 0n, 0n, N) == N : Nat}:
446  match N:
447    case 0n:
448      {==}
449    case 1n+ +x:
450      %Equal.sym(Nat, SR.and_bits(w, x, 0n), 0n, and0(w, x)) : {Nat.add(cm(w, 0n, 0n, x), b2n(Nat.is_eq(_, 0n))) == 1n+x : Nat}
451      %Equal.sym(Nat, cm(w, 0n, 0n, x), x, cm0(w, x)) : {Nat.add(_, 1n) == 1n+x : Nat}
452      Equal.trans(Nat, Nat.add(x, 1n), 1n+Nat.add(x, 0n), 1n+x, N.add_succ(x, 0n), Equal.cong(Nat, Nat, z => 1n+z, Nat.add(x, 0n), x, N.add_zero(x)))