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proof how to recover a number after division

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This commit is contained in:
Rígille S. B. Menezes 2021-10-15 13:45:19 -03:00
parent 50516d41de
commit 6953d0e360
3 changed files with 40 additions and 16 deletions
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2
base/Nat/div_mod.kind
View File

@ -1,2 +1,2 @@
Nat.div_mod(n: Nat, m: Nat): Pair<Nat, Nat>
Nat.div_mod.go(m, n, 0)
Nat.div_mod.go(m, 0, n)

52
base/Nat/div_mod/recover.kind
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@ -1,10 +1,13 @@
//Nat.div_mod.recover(
// n: Nat, m: Nat
//): Equal<Nat>(
// Nat.add(Nat.mod(n, m), Nat.mul(Nat.div(n, m), m))
// m
//)
// ?recover
Nat.div_mod.recover(
n: Nat, m: Nat
): Equal<Nat>(
Nat.add(Nat.mod(n, m), Nat.mul(Nat.div(n, m), m))
n
)
let lemma = Nat.div_mod.recover.go(n, m, 0, n, refl)
let comm = Nat.add.comm(Nat.mul(Nat.div(n, m), m), Nat.mod(n, m))
let qed = lemma :: rewrite X in Equal<Nat>(X, n) with comm
qed
//Nat.div_mod.go(2, 0, 7) Nat.div_mod.recover.go(7, 2, 0, 7, 0*2 + 7 == 7)
//Nat.div_mod.go(2, 1, 5) Nat.div_mod.recover.go(7, 2, 1, 5, 1*2 + 5 == 7)
@ -30,14 +33,35 @@ Nat.div_mod.recover.go(
(case cmp {
true:
(cmp_eq)
let ind = Nat.div_mod.recover.go(m, n,Nat.succ(d), Nat.sub(r,n), ?lemma)
let comm = Nat.add.comm(Nat.mul(d,n), r)
let lemma = Hyp :: rewrite X in Equal<Nat>(X, m) with comm
let sub = mirror(Nat.sub.add!!(cmp_eq))
let lemma = lemma :: rewrite X in Equal<Nat>(Nat.add(X,Nat.mul(d,n)), m) with sub
let assoc = mirror(Nat.add.assoc(Nat.sub(r,n), n, Nat.mul(d,n)))
let lemma = chain(assoc, lemma)
let comm = Nat.add.comm(Nat.add(n,Nat.mul(d,n)), Nat.sub(r,n))
let lemma = chain(comm, lemma)
let ind = Nat.div_mod.recover.go(m, n,Nat.succ(d), Nat.sub(r,n), lemma)
ind
false:
(cmp_eq)
Hyp
}: (cmp_eq: Equal<Bool>(Nat.lte(n, r), cmp)) -> Equal<Nat>(
Nat.add(Nat.mul(Pair.fst(Nat,Nat,cmp(() Pair(Nat,Nat),Nat.div_mod.go(n,Nat.succ(d),Nat.sub(r,n)),Pair.new(Nat,Nat,d,r))),n),Pair.snd(Nat,Nat,cmp(() Pair(Nat,Nat),Nat.div_mod.go(n,Nat.succ(d),Nat.sub(r,n)),Pair.new(Nat,Nat,d,r))))
m
))(
cmp_eq
)
}: (cmp_eq: Equal<Bool>(Nat.lte(n, r), cmp)) ->
Equal<Nat>(
def p =
cmp(
() Pair(Nat,Nat)
Nat.div_mod.go(n,Nat.succ(d),Nat.sub(r,n))
Pair.new(Nat,Nat,d,r)
)
Nat.add(
Nat.mul(
Pair.fst<Nat,Nat>(p)
n
)
Pair.snd<Nat, Nat>(p)
)
m
))(
cmp_eq
)

2
base/Nat/mod.kind
View File

@ -1,3 +1,3 @@
// Finds the remainder of division of n by m
Nat.mod(n: Nat, m: Nat): Nat
Nat.mod.go(n, m, 0)
Pair.snd!!(Nat.div_mod(n, m))