def four

bend-collections-laws-math@1.0.0.0 · proofs/math/typed/f64sqk.bend:40

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

40def four(+x: Nat) -> {C.shift(2n, x) == Nat.add(Nat.add(x, x), Nat.add(x, x)) : Nat}:
41  Equal.trans(Nat, Nat.double(Nat.double(x)), Nat.add(Nat.double(x), Nat.double(x)), Nat.add(Nat.add(x, x), Nat.add(x, x)), NA.double_self(Nat.double(x)), Equal.trans(Nat, Nat.add(Nat.double(x), Nat.double(x)), Nat.add(Nat.add(x, x), Nat.double(x)), Nat.add(Nat.add(x, x), Nat.add(x, x)), Equal.cong(Nat, Nat, z => Nat.add(z, Nat.double(x)), Nat.double(x), Nat.add(x, x), NA.double_self(x)), Equal.cong(Nat, Nat, z => Nat.add(Nat.add(x, x), z), Nat.double(x), Nat.add(x, x), NA.double_self(x))))