Arith.pow_add: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend as Arith
Lines 169-176 of bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&start_line=169&end_line=176

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

# a^(x + y) == a^x a^y   (Mathlib pow_add)
def pow_add(+a: Nat, +x: Nat, +y: Nat) -> {Nat.pow(a, Nat.add(x, y)) == Nat.mul(Nat.pow(a, x), Nat.pow(a, y)) : Nat}:
  match x:
    case 0n:
      Equal.sym(Nat, Nat.add(Nat.pow(a, y), 0n), Nat.pow(a, y), A.add_zero(Nat.pow(a, y)))
    case 1n+ +xp:
      %Equal.sym(Nat, Nat.pow(a, Nat.add(xp, y)), Nat.mul(Nat.pow(a, xp), Nat.pow(a, y)), pow_add(a, xp, y)) : {Nat.mul(a, _) == Nat.mul(Nat.mul(a, Nat.pow(a, xp)), Nat.pow(a, y)) : Nat}
      Equal.sym(Nat, Nat.mul(Nat.mul(a, Nat.pow(a, xp)), Nat.pow(a, y)), Nat.mul(a, Nat.mul(Nat.pow(a, xp), Nat.pow(a, y))), A.mul_assoc(a, Nat.pow(a, xp), Nat.pow(a, y)))

Uses (7):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  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=add_zero
  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