def hb

bend-collections-laws-crypto@1.0.0.0 · proofs/math/typed/width.bend:45

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA

45def hb(+n: Nat) -> {n == Nat.add(C.bit(n), Nat.double(C.half(n))) : Nat}:
46  match n:
47    case 0n:
48      {==}
49    case 1n:
50      {==}
51    case 2n+ +q:
52      +e = hb(q)
53      Equal.trans(Nat, 2n+q, 2n+Nat.add(C.bit(q), Nat.double(C.half(q))), Nat.add(C.bit(q), 2n+Nat.double(C.half(q))), Equal.cong(Nat, Nat, t => 2n+t, q, Nat.add(C.bit(q), Nat.double(C.half(q))), e), Equal.sym(Nat, Nat.add(C.bit(q), 2n+Nat.double(C.half(q))), 2n+Nat.add(C.bit(q), Nat.double(C.half(q))), Equal.trans(Nat, Nat.add(C.bit(q), 2n+Nat.double(C.half(q))), 1n+Nat.add(C.bit(q), 1n+Nat.double(C.half(q))), 2n+Nat.add(C.bit(q), Nat.double(C.half(q))), NA.add_succ(C.bit(q), 1n+Nat.double(C.half(q))), Equal.cong(Nat, Nat, t => 1n+t, Nat.add(C.bit(q), 1n+Nat.double(C.half(q))), 1n+Nat.add(C.bit(q), Nat.double(C.half(q))), NA.add_succ(C.bit(q), Nat.double(C.half(q)))))))