Limbs.pos_mul: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/limbs.bend as Limbs
Lines 128-135 of bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/limbs.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/limbs.bend&start_line=128&end_line=135

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/width.bend as WW

def pos_mul(n: Nat, +x: Nat, +y: Nat) -> {pos(n, Nat.mul(x, y)) == Nat.mul(pos(n, x), y) : Nat}:
  match n:
    case 0n:
      {==}
    case 1n+ +k:
      %Equal.sym(Nat, pos(k, Nat.mul(x, y)), Nat.mul(pos(k, x), y), pos_mul(k, x, y)) : {C.shift(16n, _) == Nat.mul(C.shift(16n, pos(k, x)), y) : Nat}
      %Equal.sym(Nat, Nat.mul(C.shift(16n, pos(k, x)), y), C.shift(16n, Nat.mul(pos(k, x), y)), WW.shift_mul_l(16n, pos(k, x), y)) : {C.shift(16n, Nat.mul(pos(k, x), y)) == _ : Nat}
      {==}

Uses (6):
  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.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/limbs.bend&name=pos
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/width.bend&name=shift_mul_l
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=shift