def pow2_add

bend-collections-laws-math@1.0.0.0 · proofs/math/natural/roots.bend:182

import Base
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as SC
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/natural_products.bend as PR
import bend-collections-laws-math@1.0.0.0/proofs/math/natural/lcm.bend as LC

182# 2^(x + y) == 2^x 2^y
183def pow2_add(+x: Nat, +y: Nat) -> {SC.pow2(Nat.add(x, y)) == Nat.mul(SC.pow2(x), SC.pow2(y)) : Nat}:
184  match x:
185    case 0n:
186      Equal.sym(Nat, Nat.mul(1n, SC.pow2(y)), SC.pow2(y), LC.one_mul(SC.pow2(y)))
187    case 1n+ +xp:
188      Equal.trans(Nat, Nat.double(SC.pow2(Nat.add(xp, y))), Nat.double(Nat.mul(SC.pow2(xp), SC.pow2(y))), Nat.mul(Nat.double(SC.pow2(xp)), SC.pow2(y)),
189        Equal.cong(Nat, Nat, z => Nat.double(z), SC.pow2(Nat.add(xp, y)), Nat.mul(SC.pow2(xp), SC.pow2(y)), pow2_add(xp, y)),
190        Equal.sym(Nat, Nat.mul(Nat.double(SC.pow2(xp)), SC.pow2(y)), Nat.double(Nat.mul(SC.pow2(xp), SC.pow2(y))), PR.double_product(SC.pow2(xp), SC.pow2(y))))