Pow.pb_val: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/pow.bend as Pow
Lines 176-184 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/pow.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/pow.bend&start_line=176&end_line=184

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/crypto/curve25519/field.bend as FS
import bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend as F
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend as WD
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/crypto/curve25519/limbs.bend as LM
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/mulw.bend as MW
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/cong.bend as G

def pb_val(+one: Nat, +h1: {one == 1n : Nat}, +pp: Nat, +hP: {Nat.add(1n+pp, 19n) == WD.sc(255n, one) : Nat}, +b: Bool, +x: List<&2, U32>, +acc: List<&2, U32>, +hx: {LM.okb(32n, x, 255n) == True{} : Bool}, +ha: {LM.okb(32n, acc, 255n) == True{} : Bool}) -> G.ceq(pp, FS.value(F.pow_bit(b, x, acc)), Nat.mul(Nat.mul(FS.value(acc), FS.value(acc)), Nat.pow(FS.value(x), bexp(b)))):
  match b:
    case True{}:
      +A = FS.value(acc)
      +X = FS.value(x)
      G.c_trans(pp, FS.value(F.mul(F.sq(acc), x)), Nat.mul(Nat.mul(A, A), X), Nat.mul(Nat.mul(A, A), Nat.pow(X, 1n)), step_c(one, h1, pp, hP, acc, x, ha, hx), G.c_eq(pp, Nat.mul(Nat.mul(A, A), X), Nat.mul(Nat.mul(A, A), Nat.pow(X, 1n)), Equal.cong(Nat, Nat, t => Nat.mul(Nat.mul(A, A), t), X, Nat.pow(X, 1n), Equal.sym(Nat, Nat.mul(X, 1n), X, NA.mul_one(X)))))
    case False{}:
      +A = FS.value(acc)
      G.c_trans(pp, FS.value(F.sq(acc)), Nat.mul(A, A), Nat.mul(Nat.mul(A, A), 1n), MW.sq_mod(one, h1, pp, hP, acc, ha), G.c_eq(pp, Nat.mul(A, A), Nat.mul(Nat.mul(A, A), 1n), Equal.sym(Nat, Nat.mul(Nat.mul(A, A), 1n), Nat.mul(A, A), NA.mul_one(Nat.mul(A, A)))))

Uses (22):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  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=List
  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=Base/base.bend&name=U32
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/cong.bend&name=c_eq
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/cong.bend&name=c_trans
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/cong.bend&name=ceq
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/limbs.bend&name=okb
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/mulw.bend&name=sq_mod
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/pow.bend&name=bexp
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/pow.bend&name=step_c
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_one
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend&name=sc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/crypto/curve25519/field.bend&name=value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend&name=mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend&name=pow_bit
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend&name=sq