Lists.l_mul: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/natural/lists.bend as Lists
Lines 172-182 of bend-collections-laws-math@1.0.0.0/proofs/math/natural/lists.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/lists.bend&start_line=172&end_line=182

import Base
import bend-collections-laws-math@1.0.0.0/src/math/natural.bend as M
import bend-collections-laws-math@1.0.0.0/proofs/math/natural/lcm.bend as LC

def l_mul(+m: Nat, +xs: List<&2, Nat>, +ks: List<&2, Nat>, +acc: Nat, +ka: Nat, +ea: {m == Nat.mul(ka, acc) : Nat}, h: mul_with(m, xs, ks)) -> {m == Nat.mul(lw_mul(m, xs, ks, acc, ka), M.lcm_all_go(xs, acc)) : Nat}:
  match xs ks:
    case Nil{} Nil{}:
      ea
    case Nil{} Con{k, kt}:
      ea
    case Con{x, t} Nil{}:
      Empty.absurd({m == Nat.mul(lw_mul(m, Con{x, t}, Nil{}, acc, ka), M.lcm_all_go(Con{x, t}, acc)) : Nat}, h)
    case Con{+x, +t} Con{+k, +kt}:
      (ex, ht) = h
      l_mul(m, t, kt, M.lcm(acc, x), LC.lcm_dw(acc, x, m, ka, k), LC.lcm_dvd(acc, x, m, ka, k, ea, ex), ht)

Uses (11):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Empty.absurd
  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.mul
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Sigma
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/lcm.bend&name=lcm_dvd
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/lcm.bend&name=lcm_dw
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/lists.bend&name=lw_mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/lists.bend&name=mul_with
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/natural.bend&name=lcm
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/natural.bend&name=lcm_all_go