Modpow.pow_double: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/modpow.bend as Modpow
Lines 29-37 of bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/modpow.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/modpow.bend&start_line=29&end_line=37

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as A

# x^(2 y) == (x x)^y
def pow_double(+x: Nat, +y: Nat) -> {Nat.pow(x, Nat.double(y)) == Nat.pow(Nat.mul(x, x), y) : Nat}:
  match y:
    case 0n:
      {==}
    case 1n+ +yp:
      Equal.trans(Nat, Nat.mul(x, Nat.mul(x, Nat.pow(x, Nat.double(yp)))), Nat.mul(Nat.mul(x, x), Nat.pow(x, Nat.double(yp))), Nat.mul(Nat.mul(x, x), Nat.pow(Nat.mul(x, x), yp)),
        Equal.sym(Nat, Nat.mul(Nat.mul(x, x), Nat.pow(x, Nat.double(yp))), Nat.mul(x, Nat.mul(x, Nat.pow(x, Nat.double(yp)))), A.mul_assoc(x, x, Nat.pow(x, Nat.double(yp)))),
        Equal.cong(Nat, Nat, z => Nat.mul(Nat.mul(x, x), z), Nat.pow(x, Nat.double(yp)), Nat.pow(Nat.mul(x, x), yp), pow_double(x, yp)))

Uses (8):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.trans
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.double
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.pow
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_assoc