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Rearrange some of Naturals (#48)

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Patrick Stevens
2019-10-03 06:53:13 +01:00
committed by GitHub
parent 21ee0f899d
commit 7ed41b0c09
10 changed files with 135 additions and 166 deletions
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20
Sets/CantorBijection/Proofs.agda
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@@ -55,13 +55,13 @@ module Sets.CantorBijection.Proofs where
cantorInverseOrderPreserving zero (succ y) x<y | inl (inl bl) = bl
cantorInverseOrderPreserving zero (succ y) x<y | inl (inr bl) = exFalso (leastElement bl)
cantorInverseOrderPreserving zero (succ y) x<y | inr x = exFalso (leastElement {cantorInverse y} (identityOfIndiscernablesRight _ _ _ order (cantorInverseLemmaIncreases (cantorInverse y)) (equalityCommutative x)))
cantorInverseOrderPreserving (succ x) (succ y) x<y with cantorInverseOrderPreserving x y (subtractOneFromInequality x<y)
cantorInverseOrderPreserving (succ x) (succ y) x<y with cantorInverseOrderPreserving x y (canRemoveSuccFrom<N x<y)
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr with cantorInverse x
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | x1 ,, x2 with cantorInverse y
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | zero ,, succ y2 = inl (succPreservesInequality (succIsPositive (y2 +N 0)))
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | succ y1 ,, zero with TotalOrder.totality ℕTotalOrder 1 (y1 +N 1)
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | succ y1 ,, zero | inl (inl pr1) = inl pr1
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | succ y1 ,, zero | inl (inr pr1) rewrite Semiring.commutative ℕSemiring y1 1 = exFalso (zeroNeverGreater (subtractOneFromInequality pr1))
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | succ y1 ,, zero | inl (inr pr1) rewrite Semiring.commutative ℕSemiring y1 1 = exFalso (zeroNeverGreater (canRemoveSuccFrom<N pr1))
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | succ y1 ,, zero | inr pr1 = inr (pr1 ,, le 0 refl)
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | succ y1 ,, succ y2 rewrite Semiring.commutative ℕSemiring y1 (succ (succ y2)) = inl (succPreservesInequality (succIsPositive (y2 +N y1)))
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | succ x1 ,, zero | zero ,, succ y2 rewrite Semiring.commutative ℕSemiring x1 1 | Semiring.sumZeroRight ℕSemiring y2 | Semiring.sumZeroRight ℕSemiring x1 = inl (TotalOrder.transitive ℕTotalOrder pr (a<SuccA _))
@@ -106,23 +106,23 @@ module Sets.CantorBijection.Proofs where
cantorInverseDiscrete : (a : ℕ) → (c : ℕ && ℕ) → order (cantorInverse a) c → order c (cantorInverse (succ a)) → False
cantorInverseDiscrete zero (zero ,, zero) (inl ()) c<sa
cantorInverseDiscrete zero (zero ,, zero) (inr ()) c<sa
cantorInverseDiscrete zero (zero ,, succ c) a<c (inl x) = zeroNeverGreater (subtractOneFromInequality x)
cantorInverseDiscrete zero (succ b ,, zero) a<c (inl x) = zeroNeverGreater (subtractOneFromInequality x)
cantorInverseDiscrete zero (succ b ,, succ c) a<c (inl x) = zeroNeverGreater (subtractOneFromInequality x)
cantorInverseDiscrete zero (zero ,, succ c) a<c (inl x) = zeroNeverGreater (canRemoveSuccFrom<N x)
cantorInverseDiscrete zero (succ b ,, zero) a<c (inl x) = zeroNeverGreater (canRemoveSuccFrom<N x)
cantorInverseDiscrete zero (succ b ,, succ c) a<c (inl x) = zeroNeverGreater (canRemoveSuccFrom<N x)
cantorInverseDiscrete (succ a) (b ,, c) a<c c<sa with cantorInverse a
cantorInverseDiscrete (succ a) (zero ,, zero) (inl ()) (inl x) | zero ,, zero
cantorInverseDiscrete (succ a) (zero ,, zero) (inr ()) (inl x) | zero ,, zero
cantorInverseDiscrete (succ a) (zero ,, succ zero) a<c (inl x) | zero ,, zero = TotalOrder.irreflexive ℕTotalOrder x
cantorInverseDiscrete (succ a) (zero ,, succ (succ c)) a<c (inl x) | zero ,, zero = zeroNeverGreater (subtractOneFromInequality x)
cantorInverseDiscrete (succ a) (succ b ,, c) a<c (inl x) | zero ,, zero = zeroNeverGreater (subtractOneFromInequality x)
cantorInverseDiscrete (succ a) (zero ,, succ (succ c)) a<c (inl x) | zero ,, zero = zeroNeverGreater (canRemoveSuccFrom<N x)
cantorInverseDiscrete (succ a) (succ b ,, c) a<c (inl x) | zero ,, zero = zeroNeverGreater (canRemoveSuccFrom<N x)
cantorInverseDiscrete (succ a) (b ,, zero) (inl x) (inr (fst ,, snd)) | zero ,, zero = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesRight _ _ _ _<N_ x fst)
cantorInverseDiscrete (succ a) (b ,, succ c) a<c (inr (fst ,, snd)) | zero ,, zero = zeroNeverGreater (subtractOneFromInequality snd)
cantorInverseDiscrete (succ a) (b ,, succ c) a<c (inr (fst ,, snd)) | zero ,, zero = zeroNeverGreater (canRemoveSuccFrom<N snd)
cantorInverseDiscrete (succ a) (b ,, c) (inl y) (inl x) | zero ,, succ snd rewrite Semiring.commutative ℕSemiring snd 1 | Semiring.sumZeroRight ℕSemiring snd = TotalOrder.irreflexive ℕTotalOrder (TotalOrder.transitive ℕTotalOrder x y)
cantorInverseDiscrete (succ a) (b ,, succ c) (inr (fst ,, _)) (inl x) | zero ,, succ snd rewrite Semiring.commutative ℕSemiring snd 1 | Semiring.sumZeroRight ℕSemiring snd | fst = TotalOrder.irreflexive ℕTotalOrder x
cantorInverseDiscrete (succ a) (b ,, zero) (inl x) (inr (fst ,, snd₁)) | zero ,, succ snd rewrite fst | Semiring.commutative ℕSemiring snd 1 | Semiring.sumZeroRight ℕSemiring snd = TotalOrder.irreflexive ℕTotalOrder x
cantorInverseDiscrete (succ a) (b ,, succ c) (inl x) (inr (fst ,, snd1)) | zero ,, succ snd = zeroNeverGreater (subtractOneFromInequality snd1)
cantorInverseDiscrete (succ a) (b ,, succ c) (inl x) (inr (fst ,, snd1)) | zero ,, succ snd = zeroNeverGreater (canRemoveSuccFrom<N snd1)
cantorInverseDiscrete (succ a) (b ,, succ zero) (inr (fst ,, _)) (inr (fst₁ ,, bad)) | zero ,, succ snd = TotalOrder.irreflexive ℕTotalOrder bad
cantorInverseDiscrete (succ a) (b ,, succ (succ c)) (inr (fst ,, _)) (inr (fst₁ ,, bad)) | zero ,, succ snd = zeroNeverGreater (subtractOneFromInequality bad)
cantorInverseDiscrete (succ a) (b ,, succ (succ c)) (inr (fst ,, _)) (inr (fst₁ ,, bad)) | zero ,, succ snd = zeroNeverGreater (canRemoveSuccFrom<N bad)
cantorInverseDiscrete (succ a) (b ,, c) (inl x1) (inl x) | succ zero ,, zero = noIntegersBetweenXAndSuccX 1 x1 x
cantorInverseDiscrete (succ a) (zero ,, succ zero) (inr (fst ,, snd)) (inl x) | succ zero ,, zero = TotalOrder.irreflexive ℕTotalOrder snd
cantorInverseDiscrete (succ a) (succ b ,, succ c) (inr (fst ,, snd)) (inl x) | succ zero ,, zero rewrite Semiring.commutative ℕSemiring b (succ c) = bad fst