Lists.g_dvd: 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 107-117 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=107&end_line=117

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/gcd.bend as G

def g_dvd(+d: Nat, +xs: List<&2, Nat>, +ks: List<&2, Nat>, +acc: Nat, +ka: Nat, +ea: {acc == Nat.mul(ka, d) : Nat}, h: dvd_with(d, xs, ks)) -> {M.gcd_all_go(xs, acc) == Nat.mul(gw_dvd(xs, ks, acc, ka), d) : Nat}:
  match xs ks:
    case Nil{} Nil{}:
      ea
    case Nil{} Con{k, kt}:
      ea
    case Con{x, t} Nil{}:
      Empty.absurd({M.gcd_all_go(Con{x, t}, acc) == Nat.mul(gw_dvd(Con{x, t}, Nil{}, acc, ka), d) : Nat}, h)
    case Con{+x, +t} Con{+k, +kt}:
      (ex, ht) = h
      g_dvd(d, t, kt, M.gcd(acc, x), G.dw(x, acc, x, ka, k), G.dvd_gcd(acc, x, d, 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/gcd.bend&name=dvd_gcd
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/gcd.bend&name=dw
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/lists.bend&name=dvd_with
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/natural/lists.bend&name=gw_dvd
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/natural.bend&name=gcd
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/src/math/natural.bend&name=gcd_all_go