def half_bound

bend-collections-laws-math@1.0.0.0 · proofs/math/natural/bits.bend:63

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/lib/lemmas/proofs/nat_algebra.bend as A

63# 2 h + r < 2 (1 + h) for r < 2
64def half_bound(+np: Nat) -> {Nat.is_lt(1n+np, Nat.double(1n+Nat.div(1n+np, 2n))) == True{} : Bool}:
65  +h = Nat.div(1n+np, 2n)
66  %Equal.sym(Nat, 1n+np, Nat.add(Nat.double(h), Nat.mod(1n+np, 2n)), half_eq(1n+np)) : {Nat.is_lt(_, Nat.double(1n+h)) == True{} : Bool}
67  %Equal.sym(Nat, Nat.add(2n, Nat.double(h)), Nat.add(Nat.double(h), 2n), A.add_comm(2n, Nat.double(h))) : {Nat.is_lt(Nat.add(Nat.double(h), Nat.mod(1n+np, 2n)), _) == True{} : Bool}
68  N.lt_add_left(Nat.mod(1n+np, 2n), 2n, Nat.double(h), half_rem(1n+np))