PROOF.pow_elt: def of bend-osdev-gf@0.1.1.0
Import: import bend-osdev-gf@0.1.1.0/PROOF.bend as PROOF
Lines 1862-1867 of bend-osdev-gf@0.1.1.0/PROOF.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-gf@0.1.1.0/PROOF.bend&start_line=1862&end_line=1867

import Base
import bend-osdev-gf@0.1.1.0/gf.bend as G

def pow_elt(+n: Nat, +f: G.Gf8Field, +a: U32) -> {G.Gf8.pow(f, a, n) == G.Gf8.pow(f, G.Gf8.elt(a), n) : U32}:
  match n:
    case 0n:
      {==}
    case 1n+k:
      Equal.trans(U32, G.Gf8.mul(f, a, G.Gf8.pow(f, a, k)), G.Gf8.mul(f, G.Gf8.elt(a), G.Gf8.pow(f, a, k)), G.Gf8.mul(f, G.Gf8.elt(a), G.Gf8.pow(f, G.Gf8.elt(a), k)), mul_elt_l(f, a, G.Gf8.pow(f, a, k)), Equal.cong(U32, U32, z => G.Gf8.mul(f, G.Gf8.elt(a), z), G.Gf8.pow(f, a, k), G.Gf8.pow(f, G.Gf8.elt(a), k), pow_elt(k, f, a)))

Uses (9):
  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.trans
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-gf@0.1.1.0/PROOF.bend&name=mul_elt_l
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-gf@0.1.1.0/gf.bend&name=Gf8.elt
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-gf@0.1.1.0/gf.bend&name=Gf8.mul
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-gf@0.1.1.0/gf.bend&name=Gf8.pow
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-gf@0.1.1.0/gf.bend&name=Gf8Field