def mod_identify

bend-collections-laws-crypto@1.0.0.0 · proofs/lib/u32div.bend:202

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/natural_division.bend as ND

202def mod_identify(+q: Nat, +d: Nat, +r: Nat, +bound: {Nat.is_lt(r, d) == True{} : Bool}) -> {Nat.mod(Nat.add(Nat.mul(q, d), r), d) == r : Nat}:
203  match d:
204    case 0n:
205      Empty.absurd({Nat.mod(Nat.add(Nat.mul(q, 0n), r), 0n) == r : Nat}, N.lt_zero_absurd(r, bound))
206    case 1n+p:
207      ND.remainder(q, p, r, N.lt_succ_le(r, p, bound))