Cong.pow_pow: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/cong.bend as Cong
Lines 74-84 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/cong.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/cong.bend&start_line=74&end_line=84

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend as NR

def pow_pow(+x: Nat, +a: Nat, +b: Nat) -> {Nat.pow(Nat.pow(x, a), b) == Nat.pow(x, Nat.mul(a, b)) : Nat}:
  match b:
    case 0n:
      %Equal.sym(Nat, Nat.mul(a, 0n), 0n, NA.mul_zero(a)) : {1n == Nat.pow(x, _) : Nat}
      {==}
    case 1n+ +q:
      +ih = pow_pow(x, a, q)
      +e1 = Equal.cong(Nat, Nat, t => Nat.mul(Nat.pow(x, a), t), Nat.pow(Nat.pow(x, a), q), Nat.pow(x, Nat.mul(a, q)), ih)
      +e2 = Equal.sym(Nat, Nat.pow(x, Nat.add(a, Nat.mul(a, q))), Nat.mul(Nat.pow(x, a), Nat.pow(x, Nat.mul(a, q))), NR.pow_add(x, a, Nat.mul(a, q)))
      +e3 = Equal.cong(Nat, Nat, t => Nat.pow(x, t), Nat.add(a, Nat.mul(a, q)), Nat.mul(a, 1n+q), Equal.sym(Nat, Nat.mul(a, 1n+q), Nat.add(a, Nat.mul(a, q)), NA.mul_succ(a, q)))
      Equal.trans(Nat, Nat.mul(Nat.pow(x, a), Nat.pow(Nat.pow(x, a), q)), Nat.mul(Nat.pow(x, a), Nat.pow(x, Nat.mul(a, q))), Nat.pow(x, Nat.mul(a, 1n+q)), e1, Equal.trans(Nat, Nat.mul(Nat.pow(x, a), Nat.pow(x, Nat.mul(a, q))), Nat.pow(x, Nat.add(a, Nat.mul(a, q))), Nat.pow(x, Nat.mul(a, 1n+q)), e2, e3))

Uses (10):
  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.add
  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_succ
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&name=pow_add