def dvd_mod

bend-collections-laws-math@1.0.0.0 · proofs/math/natural/gcd.bend:15

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/lemmas/proofs/nat_algebra.bend as A
import bend-collections-laws-math@1.0.0.0/proofs/math/natural/arith.bend as R

15# d | a and d | n give d | a mod n: a mod n == (ka - q kn) d, q = a / n
16def dvd_mod(+bp: Nat, +a: Nat, +d: Nat, +ka: Nat, +kn: Nat, +ea: {a == Nat.mul(ka, d) : Nat}, +en: {1n+bp == Nat.mul(kn, d) : Nat}) -> {Nat.mod(a, 1n+bp) == Nat.mul(Nat.sub(ka, Nat.mul(Nat.div(a, 1n+bp), kn)), d) : Nat}:
17  +q = Nat.div(a, 1n+bp)
18  %Equal.sym(Nat, Nat.mul(Nat.sub(ka, Nat.mul(q, kn)), d), Nat.sub(Nat.mul(ka, d), Nat.mul(Nat.mul(q, kn), d)), R.mul_sub(ka, Nat.mul(q, kn), d)) : {Nat.mod(a, 1n+bp) == _ : Nat}
19  %Equal.sym(Nat, Nat.mul(Nat.mul(q, kn), d), Nat.mul(q, Nat.mul(kn, d)), A.mul_assoc(q, kn, d)) : {Nat.mod(a, 1n+bp) == Nat.sub(Nat.mul(ka, d), _) : Nat}
20  %Equal.sym(Nat, Nat.mul(kn, d), 1n+bp, Equal.sym(Nat, 1n+bp, Nat.mul(kn, d), en)) : {Nat.mod(a, 1n+bp) == Nat.sub(Nat.mul(ka, d), Nat.mul(q, _)) : Nat}
21  %Equal.sym(Nat, Nat.mul(ka, d), a, Equal.sym(Nat, a, Nat.mul(ka, d), ea)) : {Nat.mod(a, 1n+bp) == Nat.sub(_, Nat.mul(q, 1n+bp)) : Nat}
22  %Equal.sym(Nat, a, Nat.add(Nat.mul(q, 1n+bp), Nat.mod(a, 1n+bp)), R.dm_eq(bp, a)) : {Nat.mod(a, 1n+bp) == Nat.sub(_, Nat.mul(q, 1n+bp)) : Nat}
23  Equal.sym(Nat, Nat.sub(Nat.add(Nat.mul(q, 1n+bp), Nat.mod(a, 1n+bp)), Nat.mul(q, 1n+bp)), Nat.mod(a, 1n+bp), N.add_sub_cancel(Nat.mul(q, 1n+bp), Nat.mod(a, 1n+bp)))