Arith.mul_sc: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/lib/arith.bend as Arith
Lines 16-21 of bend-collections-laws-math@1.0.0.0/proofs/lib/arith.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/arith.bend&start_line=16&end_line=21

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/natural_products.bend as PR

def mul_sc(+k: Nat, +x: Nat) -> {Nat.mul(x, sc(k, 1n)) == sc(k, x) : Nat}:
  match k:
    case 0n:
      NA.mul_one(x)
    case 1n+j:
      Equal.trans(Nat, Nat.mul(x, Nat.double(sc(j, 1n))), Nat.double(Nat.mul(x, sc(j, 1n))), Nat.double(sc(j, x)), PR.multiply_double(x, sc(j, 1n)), Equal.cong(Nat, Nat, Nat.double, Nat.mul(x, sc(j, 1n)), sc(j, x), mul_sc(j, x)))

Uses (8):
  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=Nat.double
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/arith.bend&name=sc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_one
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/natural_products.bend&name=multiply_double