Lcm.div_exact: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/lcm.bend as Lcm
Lines 112-118 of bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/lcm.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/lcm.bend&start_line=112&end_line=118

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend as L
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as A
import bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend as R

def div_exact(+x: Nat, +l: Nat, +g: Nat, +hg: {Nat.is_lt(0n, g) == True{} : Bool}, +e: {x == Nat.mul(l, g) : Nat}) -> {Nat.div(x, g) == l : Nat}:
  match g:
    case 0n:
      Empty.absurd({Nat.div(x, 0n) == l : Nat}, L.false_true(hg))
    case 1n+ +gp:
      %Equal.sym(Nat, x, Nat.add(Nat.mul(l, 1n+gp), 0n), Equal.trans(Nat, x, Nat.mul(l, 1n+gp), Nat.add(Nat.mul(l, 1n+gp), 0n), e, Equal.sym(Nat, Nat.add(Nat.mul(l, 1n+gp), 0n), Nat.mul(l, 1n+gp), A.add_zero(Nat.mul(l, 1n+gp))))) : {Nat.div(_, 1n+gp) == l : Nat}
      R.div_of(l, gp, 0n, {==})

Uses (12):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Empty.absurd
  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=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.div
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=add_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend&name=false_true
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/natural/arith.bend&name=div_of