PROOF.dot_scale: def of bendtorch@1.3.0.0
Import: import bendtorch@1.3.0.0/PROOF.bend as PROOF
Lines 37-50 of bendtorch@1.3.0.0/PROOF.bend: https://jend.leeeet.dev/read.txt?path=bendtorch@1.3.0.0/PROOF.bend&start_line=37&end_line=50

import Base
import bendtorch@1.3.0.0/spec.bend as Spec
import bend-ml-nat-lemmas@0.1.2.0/main.bend as NL

# dot(k·r, v) == k · dot(r, v): at each entry (k·x)·z == k·(x·z) (associativity), then k factors out of the sum
def dot_scale(+k: Nat, r: List<&2, Nat>, v: List<&2, Nat>) -> {Spec.dotn(Spec.vscale(k, r), v) == Nat.mul(k, Spec.dotn(r, v)) : Nat}:
  match r v:
    case Nil{} _:
      Equal.sym(Nat, Nat.mul(k, 0n), 0n, NL.mul_zero(k))
    case Con{x, xt} Nil{}:
      Equal.sym(Nat, Nat.mul(k, 0n), 0n, NL.mul_zero(k))
    case Con{+x, +xt} Con{+z, +zt}:
      +d = Spec.dotn(xt, zt)
      Equal.trans(Nat, Nat.add(Nat.mul(Nat.mul(k, x), z), Spec.dotn(Spec.vscale(k, xt), zt)), Nat.add(Nat.mul(Nat.mul(k, x), z), Nat.mul(k, d)), Nat.mul(k, Nat.add(Nat.mul(x, z), d)),
        Equal.cong(Nat, Nat, s => Nat.add(Nat.mul(Nat.mul(k, x), z), s), Spec.dotn(Spec.vscale(k, xt), zt), Nat.mul(k, d), dot_scale(k, xt, zt)),
        Equal.trans(Nat, Nat.add(Nat.mul(Nat.mul(k, x), z), Nat.mul(k, d)), Nat.add(Nat.mul(k, Nat.mul(x, z)), Nat.mul(k, d)), Nat.mul(k, Nat.add(Nat.mul(x, z), d)),
          Equal.cong(Nat, Nat, s => Nat.add(s, Nat.mul(k, d)), Nat.mul(Nat.mul(k, x), z), Nat.mul(k, Nat.mul(x, z)), Equal.sym(Nat, Nat.mul(k, Nat.mul(x, z)), Nat.mul(Nat.mul(k, x), z), NL.mul_assoc(k, x, z))),
          Equal.sym(Nat, Nat.mul(k, Nat.add(Nat.mul(x, z), d)), Nat.add(Nat.mul(k, Nat.mul(x, z)), Nat.mul(k, d)), NL.mul_dist_l(k, Nat.mul(x, z), d))))

Uses (12):
  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=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=bend-ml-nat-lemmas@0.1.2.0/main.bend&name=mul_assoc
  https://jend.leeeet.dev/definition.txt?path=bend-ml-nat-lemmas@0.1.2.0/main.bend&name=mul_dist_l
  https://jend.leeeet.dev/definition.txt?path=bend-ml-nat-lemmas@0.1.2.0/main.bend&name=mul_zero
  https://jend.leeeet.dev/definition.txt?path=bendtorch@1.3.0.0/spec.bend&name=dotn
  https://jend.leeeet.dev/definition.txt?path=bendtorch@1.3.0.0/spec.bend&name=vscale