def sc_mul

bend-collections-laws-math@1.0.0.0 · proofs/lib/arith.bend:26

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

26def sc_mul(+k: Nat, +q: Nat, +d: Nat) -> {sc(k, Nat.mul(q, d)) == Nat.mul(sc(k, q), d) : Nat}:
27  match k:
28    case 0n:
29      {==}
30    case 1n+j:
31      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)))