def bl_le

bend-collections-laws-math@1.0.0.0 · proofs/math/natural/roots.bend:202

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
import bend-collections-laws-math@1.0.0.0/proofs/math/natural/arith.bend as R

202# L <= k (1 + L / k)
203def bl_le(+kp: Nat, +bl: Nat) -> {Nat.is_le(bl, Nat.mul(1n+kp, 1n+Nat.div(bl, 1n+kp))) == True{} : Bool}:
204  +k = {1n+kp : Nat}
205  +q = Nat.div(bl, k)
206  +r = Nat.mod(bl, k)
207  %Equal.sym(Nat, bl, Nat.add(Nat.mul(q, k), r), R.dm_eq(kp, bl)) : {Nat.is_le(_, Nat.mul(k, 1n+q)) == True{} : Bool}
208  %Equal.sym(Nat, Nat.mul(k, 1n+q), Nat.add(k, Nat.mul(k, q)), A.mul_succ(k, q)) : {Nat.is_le(Nat.add(Nat.mul(q, k), r), _) == True{} : Bool}
209  %Equal.sym(Nat, Nat.add(k, Nat.mul(k, q)), Nat.add(Nat.mul(q, k), k), Equal.trans(Nat, Nat.add(k, Nat.mul(k, q)), Nat.add(Nat.mul(k, q), k), Nat.add(Nat.mul(q, k), k), A.add_comm(k, Nat.mul(k, q)), Equal.cong(Nat, Nat, z => Nat.add(z, k), Nat.mul(k, q), Nat.mul(q, k), A.mul_comm(k, q)))) : {Nat.is_le(Nat.add(Nat.mul(q, k), r), _) == True{} : Bool}
210  N.lt_le(Nat.add(Nat.mul(q, k), r), Nat.add(Nat.mul(q, k), k), N.lt_add_left(r, k, Nat.mul(q, k), R.dm_lt(kp, bl)))