Arith.sc_mul: 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 26-31 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=26&end_line=31

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

def sc_mul(+k: Nat, +q: Nat, +d: Nat) -> {sc(k, Nat.mul(q, d)) == Nat.mul(sc(k, q), d) : Nat}:
  match k:
    case 0n:
      {==}
    case 1n+j:
      Equal.trans(Nat, Nat.double(sc(j, Nat.mul(q, d))), Nat.double(Nat.mul(sc(j, q), d)), Nat.mul(Nat.double(sc(j, q)), d), Equal.cong(Nat, Nat, Nat.double, sc(j, Nat.mul(q, d)), Nat.mul(sc(j, q), d), sc_mul(j, q, d)), Equal.sym(Nat, Nat.mul(Nat.double(sc(j, q)), d), Nat.double(Nat.mul(sc(j, q), d)), PR.double_product(sc(j, q), d)))

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.sym
  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/natural_products.bend&name=double_product