def nodiv_at

bend-collections-laws-math@1.0.0.0 · proofs/math/number/prime.bend:62

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend as L
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

62def nodiv_at(k: Nat, +n: Nat, +d: Nat, +e: Nat, +h: {SN.nodiv(k, n, d) == True{} : Bool}, +hlo: {Nat.is_le(d, e) == True{} : Bool}, +hhi: {Nat.is_lt(e, Nat.add(d, k)) == True{} : Bool}) -> {Nat.is_eq(Nat.mod(n, e), 0n) == False{} : Bool}:
63  match k:
64    case 0n:
65      +hh = Equal.trans(Bool, Nat.is_lt(e, d), Nat.is_lt(e, Nat.add(d, 0n)), True{}, Equal.cong(Nat, Bool, t => Nat.is_lt(e, t), d, Nat.add(d, 0n), Equal.sym(Nat, Nat.add(d, 0n), d, A.add_zero(d))), hhi)
66      Empty.absurd({Nat.is_eq(Nat.mod(n, e), 0n) == False{} : Bool}, L.true_not_false(Nat.is_lt(d, d), N.le_lt_trans(d, e, d, hlo, hh), N.lt_irrefl(d)))
67    case 1n+ +p:
68      +h1 = L.and_left(Bool.not(Nat.is_eq(Nat.mod(n, d), 0n)), SN.nodiv(p, n, 1n+d), h)
69      +h2 = L.and_right(Bool.not(Nat.is_eq(Nat.mod(n, d), 0n)), SN.nodiv(p, n, 1n+d), h)
70      at_c(p, n, d, e, h1, h2, hlo, hhi, Nat.is_eq(d, e), {==}, e2 => hl => hh => nodiv_at(p, n, 1n+d, e2, h2, hl, hh))