def g_dvd

bend-collections-laws-math@1.0.0.0 · proofs/math/natural/lists.bend:107

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

107def 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}:
108  match xs ks:
109    case Nil{} Nil{}:
110      ea
111    case Nil{} Con{k, kt}:
112      ea
113    case Con{x, t} Nil{}:
114      Empty.absurd({M.gcd_all_go(Con{x, t}, acc) == Nat.mul(gw_dvd(Con{x, t}, Nil{}, acc, ka), d) : Nat}, h)
115    case Con{+x, +t} Con{+k, +kt}:
116      (ex, ht) = h
117      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)