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guarantee proof terminates

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This commit is contained in:
Rígille S. B. Menezes 2021-10-15 14:06:10 -03:00
parent 6953d0e360
commit 41a0703a14
2 changed files with 23 additions and 20 deletions
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38
base/Nat/div_mod/recover.kind
View File

@ -1,10 +1,12 @@
// (n % m) + (n / m) * m = n
Nat.div_mod.recover(
n: Nat, m: Nat
Hyp: Equal<Bool>(Nat.lte(1, m), true)
): 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 lemma = Nat.div_mod.recover.go(n, m, 0, n, Hyp, 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
@ -20,7 +22,8 @@ Nat.div_mod.recover.go(
n: Nat
d: Nat
r: Nat
Hyp: Equal<Nat>(Nat.add(Nat.mul(d, n), r), m)
Hyp0: Equal<Bool>(Nat.lte(1, n), true)
Hyp1: Equal<Nat>(Nat.add(Nat.mul(d, n), r), m)
): def p = Nat.div_mod.go(n, d, r)
def d = Pair.fst<Nat, Nat>(p)
def r = Pair.snd<Nat, Nat>(p)
@ -30,24 +33,21 @@ Nat.div_mod.recover.go(
)
def cmp = Nat.lte(n, r)
def cmp_eq = refl :: cmp == cmp
(case cmp {
case cmp with cmp_eq : cmp^ == cmp {
true:
(cmp_eq)
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
let comm = Nat.add.comm(Nat.mul(d,n), r)
let lemma = Hyp1 :: 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), Hyp0, lemma)
ind
false:
(cmp_eq)
Hyp
}: (cmp_eq: Equal<Bool>(Nat.lte(n, r), cmp)) ->
Equal<Nat>(
Hyp1
}! :: Equal<Nat>(
def p =
cmp(
() Pair(Nat,Nat)
@ -62,6 +62,4 @@ Nat.div_mod.recover.go(
Pair.snd<Nat, Nat>(p)
)
m
))(
cmp_eq
)

5
base/Nat/mod/small.kind Normal file
View File

@ -0,0 +1,5 @@
Nat.mod.small(
n: Nat
m: Nat
): ?mod.small.type
?mod.small.body