Prime.snoc: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/number/prime.bend as Prime
Lines 87-95 of bend-collections-laws-math@1.0.0.0/proofs/math/number/prime.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/number/prime.bend&start_line=87&end_line=95

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as A
import bend-collections-laws-math@1.0.0.0/spec/math/number.bend as SN

def snoc(k: Nat, +n: Nat, +d: Nat) -> {SN.nodiv(1n+k, n, d) == Bool.and(SN.nodiv(k, n, d), Bool.not(Nat.is_eq(Nat.mod(n, Nat.add(d, k)), 0n))) : Bool}:
  match k:
    case 0n:
      %Equal.sym(Nat, Nat.add(d, 0n), d, A.add_zero(d)) : {SN.nodiv(1n, n, d) == Bool.and(True{}, Bool.not(Nat.is_eq(Nat.mod(n, _), 0n))) : Bool}
      and_true(Bool.not(Nat.is_eq(Nat.mod(n, d), 0n)))
    case 1n+ +m:
      %Equal.sym(Nat, Nat.add(d, 1n+m), 1n+Nat.add(d, m), A.add_succ(d, m)) : {SN.nodiv(2n+m, n, d) == Bool.and(SN.nodiv(1n+m, n, d), Bool.not(Nat.is_eq(Nat.mod(n, _), 0n))) : Bool}
      %Equal.sym(Bool, SN.nodiv(1n+m, n, 1n+d), Bool.and(SN.nodiv(m, n, 1n+d), Bool.not(Nat.is_eq(Nat.mod(n, Nat.add(1n+d, m)), 0n))), snoc(m, n, 1n+d)) : {Bool.and(Bool.not(Nat.is_eq(Nat.mod(n, d), 0n)), _) == Bool.and(SN.nodiv(1n+m, n, d), Bool.not(Nat.is_eq(Nat.mod(n, 1n+Nat.add(d, m)), 0n))) : Bool}
      and_assoc(Bool.not(Nat.is_eq(Nat.mod(n, d), 0n)), SN.nodiv(m, n, 1n+d), Bool.not(Nat.is_eq(Nat.mod(n, 1n+Nat.add(d, m)), 0n)))

Uses (13):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool.and
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool.not
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  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.is_eq
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mod
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_succ
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/number/prime.bend&name=and_assoc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/number/prime.bend&name=and_true
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/number.bend&name=nodiv