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https://github.com/Smaug123/agdaproofs
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Reshuffle in preparation to break the dependency on N's implementation (#75)
This commit is contained in:
Patrick Stevens
GitHub
@@ -4,7 +4,9 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import LogicalFormulae
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open import Functions
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open import Numbers.Naturals.Naturals
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open import Numbers.Naturals.Semiring
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open import Numbers.Naturals.Order
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open import Numbers.Naturals.Order.WellFounded
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open import Sets.FinSet
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open import Semirings.Definition
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open import Orders
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@@ -4,7 +4,8 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import LogicalFormulae
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open import Functions
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open import Numbers.Naturals.Naturals
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open import Numbers.Naturals.Semiring
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open import Numbers.Naturals.Order
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open import Sets.FinSet
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open import Semirings.Definition
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open import Orders
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@@ -13,146 +14,146 @@ open import Sets.CantorBijection.Order
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module Sets.CantorBijection.Proofs where
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open Sets.CantorBijection.Order using (order ; totalOrder ; orderWellfounded)
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open Sets.CantorBijection.Order using (order ; totalOrder ; orderWellfounded)
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cantorInverseLemma : (ℕ && ℕ) → (ℕ && ℕ)
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cantorInverseLemma (0 ,, s) = (succ s) ,, 0
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cantorInverseLemma (succ r ,, 0) = r ,, 1
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cantorInverseLemma (succ r ,, succ s) = (r ,, succ (succ s))
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cantorInverseLemma : (ℕ && ℕ) → (ℕ && ℕ)
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cantorInverseLemma (0 ,, s) = (succ s) ,, 0
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cantorInverseLemma (succ r ,, 0) = r ,, 1
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cantorInverseLemma (succ r ,, succ s) = (r ,, succ (succ s))
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cantorInverse : ℕ → (ℕ && ℕ)
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cantorInverse zero = (0 ,, 0)
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cantorInverse (succ z) = cantorInverseLemma (cantorInverse z)
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cantorInverse : ℕ → (ℕ && ℕ)
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cantorInverse zero = (0 ,, 0)
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cantorInverse (succ z) = cantorInverseLemma (cantorInverse z)
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cantorInverseLemmaInjective : (a b : ℕ && ℕ) → cantorInverseLemma a ≡ cantorInverseLemma b → a ≡ b
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cantorInverseLemmaInjective (zero ,, b) (zero ,, .b) refl = refl
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cantorInverseLemmaInjective (zero ,, b) (succ c ,, zero) ()
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cantorInverseLemmaInjective (zero ,, b) (succ c ,, succ d) ()
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cantorInverseLemmaInjective (succ a ,, zero) (zero ,, d) ()
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cantorInverseLemmaInjective (succ a ,, succ b) (zero ,, d) ()
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cantorInverseLemmaInjective (succ a ,, zero) (succ .a ,, zero) refl = refl
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cantorInverseLemmaInjective (succ a ,, succ b) (succ .a ,, succ .b) refl = refl
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cantorInverseLemmaInjective : (a b : ℕ && ℕ) → cantorInverseLemma a ≡ cantorInverseLemma b → a ≡ b
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cantorInverseLemmaInjective (zero ,, b) (zero ,, .b) refl = refl
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cantorInverseLemmaInjective (zero ,, b) (succ c ,, zero) ()
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cantorInverseLemmaInjective (zero ,, b) (succ c ,, succ d) ()
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cantorInverseLemmaInjective (succ a ,, zero) (zero ,, d) ()
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cantorInverseLemmaInjective (succ a ,, succ b) (zero ,, d) ()
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cantorInverseLemmaInjective (succ a ,, zero) (succ .a ,, zero) refl = refl
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cantorInverseLemmaInjective (succ a ,, succ b) (succ .a ,, succ .b) refl = refl
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cantorInverseLemmaSurjective : (a : ℕ && ℕ) → (a ≡ (0 ,, 0)) || (Sg (ℕ && ℕ) (λ b → cantorInverseLemma b ≡ a))
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cantorInverseLemmaSurjective (zero ,, zero) = inl refl
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cantorInverseLemmaSurjective (zero ,, succ zero) = inr ((1 ,, 0) , refl)
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cantorInverseLemmaSurjective (zero ,, succ (succ b)) = inr ((1 ,, succ b) , refl)
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cantorInverseLemmaSurjective (succ a ,, zero) = inr ((0 ,, a) , refl)
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cantorInverseLemmaSurjective (succ a ,, succ zero) = inr ((succ (succ a) ,, 0) , refl)
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cantorInverseLemmaSurjective (succ a ,, succ (succ b)) = inr ((succ (succ a) ,, succ b) , refl)
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cantorInverseLemmaSurjective : (a : ℕ && ℕ) → (a ≡ (0 ,, 0)) || (Sg (ℕ && ℕ) (λ b → cantorInverseLemma b ≡ a))
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cantorInverseLemmaSurjective (zero ,, zero) = inl refl
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cantorInverseLemmaSurjective (zero ,, succ zero) = inr ((1 ,, 0) , refl)
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cantorInverseLemmaSurjective (zero ,, succ (succ b)) = inr ((1 ,, succ b) , refl)
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cantorInverseLemmaSurjective (succ a ,, zero) = inr ((0 ,, a) , refl)
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cantorInverseLemmaSurjective (succ a ,, succ zero) = inr ((succ (succ a) ,, 0) , refl)
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cantorInverseLemmaSurjective (succ a ,, succ (succ b)) = inr ((succ (succ a) ,, succ b) , refl)
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cantorInverseLemmaNotZero : (a : ℕ && ℕ) → (cantorInverseLemma a ≡ (0 ,, 0)) → False
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cantorInverseLemmaNotZero (succ a ,, zero) ()
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cantorInverseLemmaNotZero (succ a ,, succ b) ()
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cantorInverseLemmaNotZero : (a : ℕ && ℕ) → (cantorInverseLemma a ≡ (0 ,, 0)) → False
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cantorInverseLemmaNotZero (succ a ,, zero) ()
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cantorInverseLemmaNotZero (succ a ,, succ b) ()
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cantorInverseLemmaIncreases : (x : ℕ && ℕ) → order x (cantorInverseLemma x)
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cantorInverseLemmaIncreases (zero ,, b) = inl (identityOfIndiscernablesRight _<N_ (a<SuccA b) (applyEquality succ (equalityCommutative (Semiring.sumZeroRight ℕSemiring b))))
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cantorInverseLemmaIncreases (succ a ,, zero) = inr (transitivity (applyEquality succ (Semiring.sumZeroRight ℕSemiring a)) (Semiring.commutative ℕSemiring 1 a) ,, (le 0 refl))
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cantorInverseLemmaIncreases (succ a ,, succ b) = inr (transitivity (applyEquality succ (Semiring.commutative ℕSemiring a (succ b))) (Semiring.commutative ℕSemiring (succ (succ b)) a) ,, (le 0 refl))
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cantorInverseLemmaIncreases : (x : ℕ && ℕ) → order x (cantorInverseLemma x)
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cantorInverseLemmaIncreases (zero ,, b) = inl (identityOfIndiscernablesRight _<N_ (a<SuccA b) (applyEquality succ (equalityCommutative (Semiring.sumZeroRight ℕSemiring b))))
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cantorInverseLemmaIncreases (succ a ,, zero) = inr (transitivity (applyEquality succ (Semiring.sumZeroRight ℕSemiring a)) (Semiring.commutative ℕSemiring 1 a) ,, (le 0 refl))
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cantorInverseLemmaIncreases (succ a ,, succ b) = inr (transitivity (applyEquality succ (Semiring.commutative ℕSemiring a (succ b))) (Semiring.commutative ℕSemiring (succ (succ b)) a) ,, (le 0 refl))
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cantorInverseOrderPreserving : (x y : ℕ) → (x <N y) → order (cantorInverse x) (cantorInverse y)
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cantorInverseOrderPreserving zero (succ y) x<y with TotalOrder.totality totalOrder (0 ,, 0) (cantorInverseLemma (cantorInverse y))
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cantorInverseOrderPreserving zero (succ y) x<y | inl (inl bl) = bl
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cantorInverseOrderPreserving zero (succ y) x<y | inl (inr bl) = exFalso (leastElement bl)
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cantorInverseOrderPreserving zero (succ y) x<y | inr x = exFalso (leastElement {cantorInverse y} (identityOfIndiscernablesRight order (cantorInverseLemmaIncreases (cantorInverse y)) (equalityCommutative x)))
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cantorInverseOrderPreserving (succ x) (succ y) x<y with cantorInverseOrderPreserving x y (canRemoveSuccFrom<N x<y)
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr with cantorInverse x
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | x1 ,, x2 with cantorInverse y
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | zero ,, succ y2 = inl (succPreservesInequality (succIsPositive (y2 +N 0)))
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | succ y1 ,, zero with TotalOrder.totality ℕTotalOrder 1 (y1 +N 1)
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | succ y1 ,, zero | inl (inl pr1) = inl pr1
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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))
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | succ y1 ,, zero | inr pr1 = inr (pr1 ,, le 0 refl)
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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)))
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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 _))
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | succ x1 ,, zero | succ y1 ,, zero rewrite Semiring.commutative ℕSemiring x1 1 | Semiring.commutative ℕSemiring y1 1 | Semiring.sumZeroRight ℕSemiring x1 | Semiring.sumZeroRight ℕSemiring y1 = inl pr
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | succ x1 ,, zero | succ y1 ,, succ y2 rewrite Semiring.commutative ℕSemiring x1 1 | Semiring.sumZeroRight ℕSemiring x1 = inl (identityOfIndiscernablesRight _<N_ pr (transitivity (applyEquality succ (Semiring.commutative ℕSemiring y1 (succ y2))) (Semiring.commutative ℕSemiring (succ (succ y2)) y1)))
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, succ x2 | zero ,, succ y2 rewrite Semiring.sumZeroRight ℕSemiring x2 | Semiring.sumZeroRight ℕSemiring y2 = inl (succPreservesInequality pr)
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, succ x2 | succ y1 ,, zero rewrite Semiring.commutative ℕSemiring y1 1 | Semiring.sumZeroRight ℕSemiring y1 | Semiring.sumZeroRight ℕSemiring x2 = ans
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where
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ans : (succ (succ x2) <N succ y1) || (succ (succ x2) ≡ succ y1) && (zero <N 1)
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ans with TotalOrder.totality ℕTotalOrder (succ (succ x2)) (succ y1)
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ans | inl (inl x) = inl x
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ans | inl (inr x) = exFalso (noIntegersBetweenXAndSuccX (succ x2) pr x)
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ans | inr x = inr (x ,, le zero refl)
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, succ x2 | succ y1 ,, succ y2 rewrite Semiring.commutative ℕSemiring y1 1 | Semiring.sumZeroRight ℕSemiring y1 | Semiring.sumZeroRight ℕSemiring x2 = ans
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where
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ans : (succ (succ x2) <N y1 +N succ (succ y2)) || (succ (succ x2) ≡ y1 +N succ (succ y2)) && (zero <N succ (succ y2))
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ans with TotalOrder.totality ℕTotalOrder (succ (succ x2)) (y1 +N succ (succ y2))
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ans | inl (inl x) = inl x
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ans | inl (inr x) = exFalso (noIntegersBetweenXAndSuccX (succ x2) pr (identityOfIndiscernablesLeft _<N_ x (transitivity (Semiring.commutative ℕSemiring y1 (succ (succ y2))) (applyEquality succ (Semiring.commutative ℕSemiring (succ y2) y1)))))
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ans | inr x = inr (x ,, succIsPositive (succ y2))
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | succ x1 ,, succ x2 | zero ,, succ y2 rewrite Semiring.sumZeroRight ℕSemiring y2 = inl (TotalOrder.<Transitive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ pr (transitivity (applyEquality succ (Semiring.commutative ℕSemiring x1 (succ x2))) (Semiring.commutative ℕSemiring (succ (succ x2)) x1))) (a<SuccA (succ y2)))
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | succ x1 ,, succ x2 | succ y1 ,, zero rewrite Semiring.commutative ℕSemiring y1 1 | Semiring.sumZeroRight ℕSemiring y1 | Semiring.commutative ℕSemiring x1 (succ x2) | Semiring.commutative ℕSemiring x1 (succ (succ x2)) = inl pr
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | succ x1 ,, succ x2 | succ y1 ,, succ y2 rewrite Semiring.commutative ℕSemiring x1 (succ x2) | Semiring.commutative ℕSemiring y1 (succ y2) | Semiring.commutative ℕSemiring x1 (succ (succ x2)) | Semiring.commutative ℕSemiring y1 (succ (succ y2)) = inl pr
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (fst ,, snd) with cantorInverse x
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (fst ,, snd) | x1 ,, x2 with cantorInverse y
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (fst ,, ()) | zero ,, zero | zero ,, zero
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (() ,, snd) | zero ,, zero | zero ,, succ y2
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (refl ,, snd) | zero ,, succ .y2 | zero ,, succ y2 = exFalso (TotalOrder.irreflexive ℕTotalOrder snd)
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (refl ,, snd) | zero ,, succ .(y1 +N succ y2) | succ y1 ,, succ y2 = exFalso (TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive ℕTotalOrder snd (identityOfIndiscernablesRight _<N_ (addingIncreases (succ y2) y1) (Semiring.commutative ℕSemiring (succ y2) (succ y1)))))
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (fst ,, snd) | succ x1 ,, zero | zero ,, succ y2 rewrite Semiring.sumZeroRight ℕSemiring x1 | succInjective fst | Semiring.commutative ℕSemiring y2 1 | Semiring.sumZeroRight ℕSemiring y2 = inl (le zero refl)
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (fst ,, snd) | succ x1 ,, zero | succ y1 ,, succ y2 rewrite Semiring.commutative ℕSemiring x1 1 | Semiring.sumZeroRight ℕSemiring x1 = inr (transitivity fst (transitivity (applyEquality succ (Semiring.commutative ℕSemiring y1 (succ y2))) (Semiring.commutative ℕSemiring (succ (succ y2)) y1)) ,, succPreservesInequality snd)
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (fst ,, snd) | succ x1 ,, succ x2 | zero ,, succ y2 rewrite Semiring.sumZeroRight ℕSemiring y2 | Semiring.commutative ℕSemiring x1 (succ x2) | Semiring.commutative ℕSemiring (succ (succ x2)) x1 | fst = inl (succPreservesInequality (le zero refl))
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (fst ,, snd) | succ x1 ,, succ x2 | succ y1 ,, succ y2 = inr (transitivity (transitivity (Semiring.commutative ℕSemiring x1 (succ (succ x2))) (applyEquality succ (Semiring.commutative ℕSemiring (succ x2) x1))) (transitivity fst (transitivity (applyEquality succ (Semiring.commutative ℕSemiring y1 (succ y2))) (Semiring.commutative ℕSemiring (succ (succ y2)) y1))) ,, succPreservesInequality snd)
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cantorInverseOrderPreserving : (x y : ℕ) → (x <N y) → order (cantorInverse x) (cantorInverse y)
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cantorInverseOrderPreserving zero (succ y) x<y with TotalOrder.totality totalOrder (0 ,, 0) (cantorInverseLemma (cantorInverse y))
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cantorInverseOrderPreserving zero (succ y) x<y | inl (inl bl) = bl
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cantorInverseOrderPreserving zero (succ y) x<y | inl (inr bl) = exFalso (leastElement bl)
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cantorInverseOrderPreserving zero (succ y) x<y | inr x = exFalso (leastElement {cantorInverse y} (identityOfIndiscernablesRight order (cantorInverseLemmaIncreases (cantorInverse y)) (equalityCommutative x)))
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cantorInverseOrderPreserving (succ x) (succ y) x<y with cantorInverseOrderPreserving x y (canRemoveSuccFrom<N x<y)
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr with cantorInverse x
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | x1 ,, x2 with cantorInverse y
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | zero ,, succ y2 = inl (succPreservesInequality (succIsPositive (y2 +N 0)))
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | succ y1 ,, zero with TotalOrder.totality ℕTotalOrder 1 (y1 +N 1)
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | succ y1 ,, zero | inl (inl pr1) = inl pr1
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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))
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | succ y1 ,, zero | inr pr1 = inr (pr1 ,, le 0 refl)
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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)))
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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 _))
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | succ x1 ,, zero | succ y1 ,, zero rewrite Semiring.commutative ℕSemiring x1 1 | Semiring.commutative ℕSemiring y1 1 | Semiring.sumZeroRight ℕSemiring x1 | Semiring.sumZeroRight ℕSemiring y1 = inl pr
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | succ x1 ,, zero | succ y1 ,, succ y2 rewrite Semiring.commutative ℕSemiring x1 1 | Semiring.sumZeroRight ℕSemiring x1 = inl (identityOfIndiscernablesRight _<N_ pr (transitivity (applyEquality succ (Semiring.commutative ℕSemiring y1 (succ y2))) (Semiring.commutative ℕSemiring (succ (succ y2)) y1)))
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, succ x2 | zero ,, succ y2 rewrite Semiring.sumZeroRight ℕSemiring x2 | Semiring.sumZeroRight ℕSemiring y2 = inl (succPreservesInequality pr)
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, succ x2 | succ y1 ,, zero rewrite Semiring.commutative ℕSemiring y1 1 | Semiring.sumZeroRight ℕSemiring y1 | Semiring.sumZeroRight ℕSemiring x2 = ans
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where
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ans : (succ (succ x2) <N succ y1) || (succ (succ x2) ≡ succ y1) && (zero <N 1)
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ans with TotalOrder.totality ℕTotalOrder (succ (succ x2)) (succ y1)
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ans | inl (inl x) = inl x
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ans | inl (inr x) = exFalso (noIntegersBetweenXAndSuccX (succ x2) pr x)
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ans | inr x = inr (x ,, le zero refl)
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, succ x2 | succ y1 ,, succ y2 rewrite Semiring.commutative ℕSemiring y1 1 | Semiring.sumZeroRight ℕSemiring y1 | Semiring.sumZeroRight ℕSemiring x2 = ans
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where
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ans : (succ (succ x2) <N y1 +N succ (succ y2)) || (succ (succ x2) ≡ y1 +N succ (succ y2)) && (zero <N succ (succ y2))
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ans with TotalOrder.totality ℕTotalOrder (succ (succ x2)) (y1 +N succ (succ y2))
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ans | inl (inl x) = inl x
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ans | inl (inr x) = exFalso (noIntegersBetweenXAndSuccX (succ x2) pr (identityOfIndiscernablesLeft _<N_ x (transitivity (Semiring.commutative ℕSemiring y1 (succ (succ y2))) (applyEquality succ (Semiring.commutative ℕSemiring (succ y2) y1)))))
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ans | inr x = inr (x ,, succIsPositive (succ y2))
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cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | succ x1 ,, succ x2 | zero ,, succ y2 rewrite Semiring.sumZeroRight ℕSemiring y2 = inl (TotalOrder.<Transitive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ pr (transitivity (applyEquality succ (Semiring.commutative ℕSemiring x1 (succ x2))) (Semiring.commutative ℕSemiring (succ (succ x2)) x1))) (a<SuccA (succ y2)))
|
||||
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | succ x1 ,, succ x2 | succ y1 ,, zero rewrite Semiring.commutative ℕSemiring y1 1 | Semiring.sumZeroRight ℕSemiring y1 | Semiring.commutative ℕSemiring x1 (succ x2) | Semiring.commutative ℕSemiring x1 (succ (succ x2)) = inl pr
|
||||
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | succ x1 ,, succ x2 | succ y1 ,, succ y2 rewrite Semiring.commutative ℕSemiring x1 (succ x2) | Semiring.commutative ℕSemiring y1 (succ y2) | Semiring.commutative ℕSemiring x1 (succ (succ x2)) | Semiring.commutative ℕSemiring y1 (succ (succ y2)) = inl pr
|
||||
cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (fst ,, snd) with cantorInverse x
|
||||
cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (fst ,, snd) | x1 ,, x2 with cantorInverse y
|
||||
cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (fst ,, ()) | zero ,, zero | zero ,, zero
|
||||
cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (() ,, snd) | zero ,, zero | zero ,, succ y2
|
||||
cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (refl ,, snd) | zero ,, succ .y2 | zero ,, succ y2 = exFalso (TotalOrder.irreflexive ℕTotalOrder snd)
|
||||
cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (refl ,, snd) | zero ,, succ .(y1 +N succ y2) | succ y1 ,, succ y2 = exFalso (TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive ℕTotalOrder snd (identityOfIndiscernablesRight _<N_ (addingIncreases (succ y2) y1) (Semiring.commutative ℕSemiring (succ y2) (succ y1)))))
|
||||
cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (fst ,, snd) | succ x1 ,, zero | zero ,, succ y2 rewrite Semiring.sumZeroRight ℕSemiring x1 | succInjective fst | Semiring.commutative ℕSemiring y2 1 | Semiring.sumZeroRight ℕSemiring y2 = inl (le zero refl)
|
||||
cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (fst ,, snd) | succ x1 ,, zero | succ y1 ,, succ y2 rewrite Semiring.commutative ℕSemiring x1 1 | Semiring.sumZeroRight ℕSemiring x1 = inr (transitivity fst (transitivity (applyEquality succ (Semiring.commutative ℕSemiring y1 (succ y2))) (Semiring.commutative ℕSemiring (succ (succ y2)) y1)) ,, succPreservesInequality snd)
|
||||
cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (fst ,, snd) | succ x1 ,, succ x2 | zero ,, succ y2 rewrite Semiring.sumZeroRight ℕSemiring y2 | Semiring.commutative ℕSemiring x1 (succ x2) | Semiring.commutative ℕSemiring (succ (succ x2)) x1 | fst = inl (succPreservesInequality (le zero refl))
|
||||
cantorInverseOrderPreserving (succ x) (succ y) x<y | inr (fst ,, snd) | succ x1 ,, succ x2 | succ y1 ,, succ y2 = inr (transitivity (transitivity (Semiring.commutative ℕSemiring x1 (succ (succ x2))) (applyEquality succ (Semiring.commutative ℕSemiring (succ x2) x1))) (transitivity fst (transitivity (applyEquality succ (Semiring.commutative ℕSemiring y1 (succ y2))) (Semiring.commutative ℕSemiring (succ (succ y2)) y1))) ,, succPreservesInequality snd)
|
||||
|
||||
cantorInverseInjective : (a b : ℕ) → cantorInverse a ≡ cantorInverse b → a ≡ b
|
||||
cantorInverseInjective zero zero pr = refl
|
||||
cantorInverseInjective zero (succ b) pr = exFalso (cantorInverseLemmaNotZero _ (equalityCommutative pr))
|
||||
cantorInverseInjective (succ a) zero pr = exFalso (cantorInverseLemmaNotZero _ pr)
|
||||
cantorInverseInjective (succ a) (succ b) pr = applyEquality succ (cantorInverseInjective a b (cantorInverseLemmaInjective (cantorInverse a) (cantorInverse b) pr))
|
||||
cantorInverseInjective : (a b : ℕ) → cantorInverse a ≡ cantorInverse b → a ≡ b
|
||||
cantorInverseInjective zero zero pr = refl
|
||||
cantorInverseInjective zero (succ b) pr = exFalso (cantorInverseLemmaNotZero _ (equalityCommutative pr))
|
||||
cantorInverseInjective (succ a) zero pr = exFalso (cantorInverseLemmaNotZero _ pr)
|
||||
cantorInverseInjective (succ a) (succ b) pr = applyEquality succ (cantorInverseInjective a b (cantorInverseLemmaInjective (cantorInverse a) (cantorInverse b) pr))
|
||||
|
||||
-- Some unnecessary things on the way to the final proof
|
||||
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 (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 (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 (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 (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 (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
|
||||
where
|
||||
bad : {a : ℕ} → 1 ≡ succ (succ a) → False
|
||||
bad ()
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inl y) (inl x) | succ (succ fst) ,, zero rewrite Semiring.commutative ℕSemiring fst 2 | Semiring.commutative ℕSemiring fst 1 = TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive ℕTotalOrder y x)
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inr (bad ,, _)) (inl x) | succ (succ fst) ,, zero rewrite Semiring.commutative ℕSemiring fst 2 | Semiring.commutative ℕSemiring fst 1 = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesRight _<N_ x bad)
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inl x) (inr (bad ,, snd)) | succ (succ fst) ,, zero rewrite Semiring.commutative ℕSemiring fst 2 | Semiring.commutative ℕSemiring fst 1 = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesRight _<N_ x bad)
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inr (_ ,, bad)) (inr (fst₁ ,, snd)) | succ (succ fst) ,, zero rewrite Semiring.commutative ℕSemiring fst 2 | Semiring.commutative ℕSemiring fst 1 = noIntegersBetweenXAndSuccX 1 bad snd
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inl x) (inl y) | succ zero ,, succ snd rewrite Semiring.sumZeroRight ℕSemiring snd = noIntegersBetweenXAndSuccX (succ (succ snd)) x y
|
||||
cantorInverseDiscrete (succ a) (zero ,, c) (inr (fst ,, snd1)) (inl y) | succ zero ,, succ snd rewrite Semiring.sumZeroRight ℕSemiring snd = noIntegersBetweenXAndSuccX (succ (succ snd)) snd1 y
|
||||
cantorInverseDiscrete (succ a) (succ b ,, c) (inr (fst ,, bad)) (inl y) | succ zero ,, succ snd rewrite Semiring.sumZeroRight ℕSemiring snd | fst = TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive ℕTotalOrder bad (identityOfIndiscernablesRight _<N_ (addingIncreases c b) (Semiring.commutative ℕSemiring c (succ b))))
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inl x) (inl y) | succ (succ fst) ,, succ snd rewrite Semiring.commutative ℕSemiring fst (succ (succ (succ snd))) | Semiring.commutative ℕSemiring (succ (succ snd)) fst = TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive ℕTotalOrder y x)
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inl x) (inr (y ,, z)) | succ (succ fst) ,, succ snd rewrite y | Semiring.commutative ℕSemiring fst (succ (succ (succ snd))) | Semiring.commutative ℕSemiring (succ (succ snd)) fst = TotalOrder.irreflexive ℕTotalOrder x
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inr (x ,, y)) (inl z) | succ (succ fst) ,, succ snd rewrite equalityCommutative x | Semiring.commutative ℕSemiring fst (succ (succ (succ snd))) | Semiring.commutative ℕSemiring (succ (succ snd)) fst = TotalOrder.irreflexive ℕTotalOrder z
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inr (x ,, y)) (inr (m ,, n)) | succ (succ fst) ,, succ snd = noIntegersBetweenXAndSuccX (succ (succ snd)) y n
|
||||
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 (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 (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 (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 (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 (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
|
||||
where
|
||||
bad : {a : ℕ} → 1 ≡ succ (succ a) → False
|
||||
bad ()
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inl y) (inl x) | succ (succ fst) ,, zero rewrite Semiring.commutative ℕSemiring fst 2 | Semiring.commutative ℕSemiring fst 1 = TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive ℕTotalOrder y x)
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inr (bad ,, _)) (inl x) | succ (succ fst) ,, zero rewrite Semiring.commutative ℕSemiring fst 2 | Semiring.commutative ℕSemiring fst 1 = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesRight _<N_ x bad)
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inl x) (inr (bad ,, snd)) | succ (succ fst) ,, zero rewrite Semiring.commutative ℕSemiring fst 2 | Semiring.commutative ℕSemiring fst 1 = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesRight _<N_ x bad)
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inr (_ ,, bad)) (inr (fst₁ ,, snd)) | succ (succ fst) ,, zero rewrite Semiring.commutative ℕSemiring fst 2 | Semiring.commutative ℕSemiring fst 1 = noIntegersBetweenXAndSuccX 1 bad snd
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inl x) (inl y) | succ zero ,, succ snd rewrite Semiring.sumZeroRight ℕSemiring snd = noIntegersBetweenXAndSuccX (succ (succ snd)) x y
|
||||
cantorInverseDiscrete (succ a) (zero ,, c) (inr (fst ,, snd1)) (inl y) | succ zero ,, succ snd rewrite Semiring.sumZeroRight ℕSemiring snd = noIntegersBetweenXAndSuccX (succ (succ snd)) snd1 y
|
||||
cantorInverseDiscrete (succ a) (succ b ,, c) (inr (fst ,, bad)) (inl y) | succ zero ,, succ snd rewrite Semiring.sumZeroRight ℕSemiring snd | fst = TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive ℕTotalOrder bad (identityOfIndiscernablesRight _<N_ (addingIncreases c b) (Semiring.commutative ℕSemiring c (succ b))))
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inl x) (inl y) | succ (succ fst) ,, succ snd rewrite Semiring.commutative ℕSemiring fst (succ (succ (succ snd))) | Semiring.commutative ℕSemiring (succ (succ snd)) fst = TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive ℕTotalOrder y x)
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inl x) (inr (y ,, z)) | succ (succ fst) ,, succ snd rewrite y | Semiring.commutative ℕSemiring fst (succ (succ (succ snd))) | Semiring.commutative ℕSemiring (succ (succ snd)) fst = TotalOrder.irreflexive ℕTotalOrder x
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inr (x ,, y)) (inl z) | succ (succ fst) ,, succ snd rewrite equalityCommutative x | Semiring.commutative ℕSemiring fst (succ (succ (succ snd))) | Semiring.commutative ℕSemiring (succ (succ snd)) fst = TotalOrder.irreflexive ℕTotalOrder z
|
||||
cantorInverseDiscrete (succ a) (b ,, c) (inr (x ,, y)) (inr (m ,, n)) | succ (succ fst) ,, succ snd = noIntegersBetweenXAndSuccX (succ (succ snd)) y n
|
||||
|
||||
boundedInversesExist : (a : ℕ && ℕ) (s : ℕ) → order a (cantorInverse s) → Sg ℕ (λ i → cantorInverse i ≡ a)
|
||||
boundedInversesExist a zero a<s = exFalso (leastElement a<s)
|
||||
boundedInversesExist a (succ s) a<s with TotalOrder.totality totalOrder a (cantorInverse s)
|
||||
boundedInversesExist a (succ s) _ | inl (inl a<s) = boundedInversesExist a s a<s
|
||||
boundedInversesExist a (succ s) a<s | inl (inr s<a) = exFalso (cantorInverseDiscrete s a s<a a<s)
|
||||
boundedInversesExist a (succ s) a<s | inr a=s = s , equalityCommutative a=s
|
||||
boundedInversesExist : (a : ℕ && ℕ) (s : ℕ) → order a (cantorInverse s) → Sg ℕ (λ i → cantorInverse i ≡ a)
|
||||
boundedInversesExist a zero a<s = exFalso (leastElement a<s)
|
||||
boundedInversesExist a (succ s) a<s with TotalOrder.totality totalOrder a (cantorInverse s)
|
||||
boundedInversesExist a (succ s) _ | inl (inl a<s) = boundedInversesExist a s a<s
|
||||
boundedInversesExist a (succ s) a<s | inl (inr s<a) = exFalso (cantorInverseDiscrete s a s<a a<s)
|
||||
boundedInversesExist a (succ s) a<s | inr a=s = s , equalityCommutative a=s
|
||||
|
||||
cantorInverseSurjective : (x : ℕ && ℕ) → Sg ℕ (λ i → (cantorInverse i) ≡ x)
|
||||
cantorInverseSurjective = rec orderWellfounded (λ z → Sg ℕ (λ z₁ → (cantorInverse z₁) ≡ z)) go
|
||||
where
|
||||
go : (a : ℕ && ℕ) → (pr : (x : ℕ && ℕ) (x₁ : order x a) → Sg ℕ (λ z → (cantorInverse z) ≡ x)) → Sg ℕ (λ i → (cantorInverse i) ≡ a)
|
||||
go a pr with cantorInverseLemmaSurjective a
|
||||
go .(0 ,, 0) pr | inl refl = 0 , refl
|
||||
go a ind | inr (decr , proof) with ind decr (identityOfIndiscernablesRight order (cantorInverseLemmaIncreases decr) proof)
|
||||
go a ind | inr (decr , proof) | boundForDecr , boundIsBound = succ boundForDecr , transitivity (applyEquality cantorInverseLemma boundIsBound) proof
|
||||
cantorInverseSurjective : (x : ℕ && ℕ) → Sg ℕ (λ i → (cantorInverse i) ≡ x)
|
||||
cantorInverseSurjective = rec orderWellfounded (λ z → Sg ℕ (λ z₁ → (cantorInverse z₁) ≡ z)) go
|
||||
where
|
||||
go : (a : ℕ && ℕ) → (pr : (x : ℕ && ℕ) (x₁ : order x a) → Sg ℕ (λ z → (cantorInverse z) ≡ x)) → Sg ℕ (λ i → (cantorInverse i) ≡ a)
|
||||
go a pr with cantorInverseLemmaSurjective a
|
||||
go .(0 ,, 0) pr | inl refl = 0 , refl
|
||||
go a ind | inr (decr , proof) with ind decr (identityOfIndiscernablesRight order (cantorInverseLemmaIncreases decr) proof)
|
||||
go a ind | inr (decr , proof) | boundForDecr , boundIsBound = succ boundForDecr , transitivity (applyEquality cantorInverseLemma boundIsBound) proof
|
||||
|
||||
@@ -4,7 +4,9 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Functions
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Numbers.Naturals.Semiring
|
||||
open import Numbers.Naturals.Order
|
||||
open import Numbers.Naturals.Order.Lemmas
|
||||
open import Sets.FinSet
|
||||
open import Semirings.Definition
|
||||
open import Orders
|
||||
@@ -12,137 +14,140 @@ open import WellFoundedInduction
|
||||
open import Sets.CantorBijection.CantorBijection
|
||||
|
||||
module Sets.Cardinality where
|
||||
record CountablyInfiniteSet {a : _} (A : Set a) : Set a where
|
||||
field
|
||||
counting : A → ℕ
|
||||
countingIsBij : Bijection counting
|
||||
|
||||
data Countable {a : _} (A : Set a) : Set a where
|
||||
finite : FiniteSet A → Countable A
|
||||
infinite : CountablyInfiniteSet A → Countable A
|
||||
open import Semirings.Lemmas ℕSemiring
|
||||
|
||||
ℕCountable : CountablyInfiniteSet ℕ
|
||||
ℕCountable = record { counting = id ; countingIsBij = invertibleImpliesBijection (record { inverse = id ; isLeft = λ b → refl ; isRight = λ a → refl}) }
|
||||
record CountablyInfiniteSet {a : _} (A : Set a) : Set a where
|
||||
field
|
||||
counting : A → ℕ
|
||||
countingIsBij : Bijection counting
|
||||
|
||||
doubleLemma : (a b : ℕ) → 2 *N a ≡ 2 *N b → a ≡ b
|
||||
doubleLemma a b pr = productCancelsLeft 2 a b (le 1 refl) pr
|
||||
data Countable {a : _} (A : Set a) : Set a where
|
||||
finite : FiniteSet A → Countable A
|
||||
infinite : CountablyInfiniteSet A → Countable A
|
||||
|
||||
evenCannotBeOdd : (a b : ℕ) → 2 *N a ≡ succ (2 *N b) → False
|
||||
evenCannotBeOdd zero b ()
|
||||
evenCannotBeOdd (succ a) zero pr rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring a (succ a) = naughtE (equalityCommutative (succInjective pr))
|
||||
evenCannotBeOdd (succ a) (succ b) pr = evenCannotBeOdd a b pr''
|
||||
where
|
||||
pr' : a +N a ≡ (b +N succ b)
|
||||
pr' rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring b 0 | Semiring.commutative ℕSemiring a (succ a) = succInjective (succInjective pr)
|
||||
pr'' : 2 *N a ≡ succ (2 *N b)
|
||||
pr'' rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring b 0 | Semiring.commutative ℕSemiring (succ b) b = pr'
|
||||
ℕCountable : CountablyInfiniteSet ℕ
|
||||
ℕCountable = record { counting = id ; countingIsBij = invertibleImpliesBijection (record { inverse = id ; isLeft = λ b → refl ; isRight = λ a → refl}) }
|
||||
|
||||
aMod2 : (a : ℕ) → Sg ℕ (λ i → (2 *N i ≡ a) || (succ (2 *N i) ≡ a))
|
||||
aMod2 zero = (0 , inl refl)
|
||||
aMod2 (succ a) with aMod2 a
|
||||
aMod2 (succ a) | b , inl x = b , inr (applyEquality succ x)
|
||||
aMod2 (succ a) | b , inr x = (succ b) , inl pr
|
||||
where
|
||||
pr : succ (b +N succ (b +N 0)) ≡ succ a
|
||||
pr rewrite Semiring.commutative ℕSemiring b (succ (b +N 0)) | Semiring.commutative ℕSemiring (b +N 0) b = applyEquality succ x
|
||||
doubleLemma : (a b : ℕ) → 2 *N a ≡ 2 *N b → a ≡ b
|
||||
doubleLemma a b pr = productCancelsLeft 2 a b (le 1 refl) pr
|
||||
|
||||
sqrtFloor : (a : ℕ) → Sg (ℕ && ℕ) (λ pair → ((_&&_.fst pair) *N (_&&_.fst pair) +N (_&&_.snd pair) ≡ a) && ((_&&_.snd pair) <N 2 *N (_&&_.fst pair) +N 1))
|
||||
sqrtFloor zero = (0 ,, 0) , (refl ,, le zero refl)
|
||||
sqrtFloor (succ n) with sqrtFloor n
|
||||
sqrtFloor (succ n) | (a ,, b) , pr with orderIsTotal b (2 *N a)
|
||||
sqrtFloor (succ n) | (a ,, b) , pr | inl (inl x) = (a ,, succ b) , (p ,, q)
|
||||
where
|
||||
p : a *N a +N succ b ≡ succ n
|
||||
p rewrite Semiring.commutative ℕSemiring (a *N a) (succ b) | Semiring.commutative ℕSemiring b (a *N a) = applyEquality succ (_&&_.fst pr)
|
||||
q : succ b <N (a +N (a +N 0)) +N 1
|
||||
q rewrite Semiring.commutative ℕSemiring (a +N (a +N 0)) (succ 0) | Semiring.commutative ℕSemiring a 0 = succPreservesInequality x
|
||||
sqrtFloor (succ n) | (a ,, b) , (_ ,, pr) | inl (inr x) rewrite Semiring.commutative ℕSemiring (a +N (a +N 0)) (succ 0) = exFalso (noIntegersBetweenXAndSuccX (a +N (a +N 0)) x pr)
|
||||
sqrtFloor (succ n) | (a ,, b) , pr | inr x = (succ a ,, 0) , (q ,, succIsPositive _)
|
||||
where
|
||||
p : a +N a *N succ a ≡ n
|
||||
p rewrite x | Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring (a +N a) (succ 0) | Semiring.commutative ℕSemiring (a *N a) (succ a +N a) | multiplicationNIsCommutative a (succ a) | Semiring.commutative ℕSemiring (a *N a) (a +N a) | Semiring.+Associative ℕSemiring a a (a *N a) = _&&_.fst pr
|
||||
q : succ ((a +N a *N succ a) +N 0) ≡ succ n
|
||||
q rewrite Semiring.commutative ℕSemiring (a +N a *N succ a) 0 = applyEquality succ p
|
||||
evenCannotBeOdd : (a b : ℕ) → 2 *N a ≡ succ (2 *N b) → False
|
||||
evenCannotBeOdd zero b ()
|
||||
evenCannotBeOdd (succ a) zero pr rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring a (succ a) = naughtE (equalityCommutative (succInjective pr))
|
||||
evenCannotBeOdd (succ a) (succ b) pr = evenCannotBeOdd a b pr''
|
||||
where
|
||||
pr' : a +N a ≡ (b +N succ b)
|
||||
pr' rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring b 0 | Semiring.commutative ℕSemiring a (succ a) = succInjective (succInjective pr)
|
||||
pr'' : 2 *N a ≡ succ (2 *N b)
|
||||
pr'' rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring b 0 | Semiring.commutative ℕSemiring (succ b) b = pr'
|
||||
|
||||
pairUnionIsCountable : {a b : _} {A : Set a} {B : Set b} → (X : CountablyInfiniteSet A) → (Y : CountablyInfiniteSet B) → CountablyInfiniteSet (A || B)
|
||||
CountablyInfiniteSet.counting (pairUnionIsCountable X Y) (inl r) = 2 *N (CountablyInfiniteSet.counting X r)
|
||||
CountablyInfiniteSet.counting (pairUnionIsCountable X Y) (inr s) = succ (2 *N (CountablyInfiniteSet.counting Y s))
|
||||
Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) {inl x} {inl y} pr rewrite Semiring.commutative ℕSemiring (CountablyInfiniteSet.counting X x) 0 | Semiring.commutative ℕSemiring (CountablyInfiniteSet.counting X y) 0 | doubleIsAddTwo (CountablyInfiniteSet.counting X x) | doubleIsAddTwo (CountablyInfiniteSet.counting X y) = applyEquality inl (Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij X)) inter)
|
||||
where
|
||||
inter : CountablyInfiniteSet.counting X x ≡ CountablyInfiniteSet.counting X y
|
||||
inter = doubleLemma (CountablyInfiniteSet.counting X x) (CountablyInfiniteSet.counting X y) pr
|
||||
Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) {inl x} {inr y} pr = exFalso (evenCannotBeOdd (CountablyInfiniteSet.counting X x) (CountablyInfiniteSet.counting Y y) pr)
|
||||
Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) {inr x} {inl y} pr = exFalso (evenCannotBeOdd (CountablyInfiniteSet.counting X y) (CountablyInfiniteSet.counting Y x) (equalityCommutative pr))
|
||||
Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) {inr x} {inr y} pr = applyEquality inr (Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij Y)) (doubleLemma (CountablyInfiniteSet.counting Y x) (CountablyInfiniteSet.counting Y y) (succInjective pr) ))
|
||||
Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b with aMod2 b
|
||||
Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inl x with Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij X)) a
|
||||
Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inl x | r , pr = inl r , ans
|
||||
where
|
||||
ans : 2 *N CountablyInfiniteSet.counting X r ≡ b
|
||||
ans rewrite pr = x
|
||||
Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inr x with Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij Y)) a
|
||||
Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inr x | r , pr = inr r , ans
|
||||
where
|
||||
ans : succ (2 *N CountablyInfiniteSet.counting Y r) ≡ b
|
||||
ans rewrite pr = x
|
||||
aMod2 : (a : ℕ) → Sg ℕ (λ i → (2 *N i ≡ a) || (succ (2 *N i) ≡ a))
|
||||
aMod2 zero = (0 , inl refl)
|
||||
aMod2 (succ a) with aMod2 a
|
||||
aMod2 (succ a) | b , inl x = b , inr (applyEquality succ x)
|
||||
aMod2 (succ a) | b , inr x = (succ b) , inl pr
|
||||
where
|
||||
pr : succ (b +N succ (b +N 0)) ≡ succ a
|
||||
pr rewrite Semiring.commutative ℕSemiring b (succ (b +N 0)) | Semiring.commutative ℕSemiring (b +N 0) b = applyEquality succ x
|
||||
|
||||
firstEqualityOfPair : {a b : _} {A : Set a} {B : Set b} → {x1 x2 : A} → {y1 y2 : B} → (x1 ,, y1) ≡ (x2 ,, y2) → x1 ≡ x2
|
||||
firstEqualityOfPair {x1} {x2} {y1} {y2} refl = refl
|
||||
sqrtFloor : (a : ℕ) → Sg (ℕ && ℕ) (λ pair → ((_&&_.fst pair) *N (_&&_.fst pair) +N (_&&_.snd pair) ≡ a) && ((_&&_.snd pair) <N 2 *N (_&&_.fst pair) +N 1))
|
||||
sqrtFloor zero = (0 ,, 0) , (refl ,, le zero refl)
|
||||
sqrtFloor (succ n) with sqrtFloor n
|
||||
sqrtFloor (succ n) | (a ,, b) , pr with TotalOrder.totality ℕTotalOrder b (2 *N a)
|
||||
sqrtFloor (succ n) | (a ,, b) , pr | inl (inl x) = (a ,, succ b) , (p ,, q)
|
||||
where
|
||||
p : a *N a +N succ b ≡ succ n
|
||||
p rewrite Semiring.commutative ℕSemiring (a *N a) (succ b) | Semiring.commutative ℕSemiring b (a *N a) = applyEquality succ (_&&_.fst pr)
|
||||
q : succ b <N (a +N (a +N 0)) +N 1
|
||||
q rewrite Semiring.commutative ℕSemiring (a +N (a +N 0)) (succ 0) | Semiring.commutative ℕSemiring a 0 = succPreservesInequality x
|
||||
sqrtFloor (succ n) | (a ,, b) , (_ ,, pr) | inl (inr x) rewrite Semiring.commutative ℕSemiring (a +N (a +N 0)) (succ 0) = exFalso (noIntegersBetweenXAndSuccX (a +N (a +N 0)) x pr)
|
||||
sqrtFloor (succ n) | (a ,, b) , pr | inr x = (succ a ,, 0) , (q ,, succIsPositive _)
|
||||
where
|
||||
p : a +N a *N succ a ≡ n
|
||||
p rewrite x | Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring (a +N a) (succ 0) | Semiring.commutative ℕSemiring (a *N a) (succ a +N a) | multiplicationNIsCommutative a (succ a) | Semiring.commutative ℕSemiring (a *N a) (a +N a) | Semiring.+Associative ℕSemiring a a (a *N a) = _&&_.fst pr
|
||||
q : succ ((a +N a *N succ a) +N 0) ≡ succ n
|
||||
q rewrite Semiring.commutative ℕSemiring (a +N a *N succ a) 0 = applyEquality succ p
|
||||
|
||||
secondEqualityOfPair : {a b : _} {A : Set a} {B : Set b} → {x1 x2 : A} → {y1 y2 : B} → (x1 ,, y1) ≡ (x2 ,, y2) → y1 ≡ y2
|
||||
secondEqualityOfPair {x1} {x2} {y1} {y2} refl = refl
|
||||
pairUnionIsCountable : {a b : _} {A : Set a} {B : Set b} → (X : CountablyInfiniteSet A) → (Y : CountablyInfiniteSet B) → CountablyInfiniteSet (A || B)
|
||||
CountablyInfiniteSet.counting (pairUnionIsCountable X Y) (inl r) = 2 *N (CountablyInfiniteSet.counting X r)
|
||||
CountablyInfiniteSet.counting (pairUnionIsCountable X Y) (inr s) = succ (2 *N (CountablyInfiniteSet.counting Y s))
|
||||
Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) {inl x} {inl y} pr rewrite Semiring.commutative ℕSemiring (CountablyInfiniteSet.counting X x) 0 | Semiring.commutative ℕSemiring (CountablyInfiniteSet.counting X y) 0 | doubleIsAddTwo (CountablyInfiniteSet.counting X x) | doubleIsAddTwo (CountablyInfiniteSet.counting X y) = applyEquality inl (Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij X)) inter)
|
||||
where
|
||||
inter : CountablyInfiniteSet.counting X x ≡ CountablyInfiniteSet.counting X y
|
||||
inter = doubleLemma (CountablyInfiniteSet.counting X x) (CountablyInfiniteSet.counting X y) pr
|
||||
Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) {inl x} {inr y} pr = exFalso (evenCannotBeOdd (CountablyInfiniteSet.counting X x) (CountablyInfiniteSet.counting Y y) pr)
|
||||
Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) {inr x} {inl y} pr = exFalso (evenCannotBeOdd (CountablyInfiniteSet.counting X y) (CountablyInfiniteSet.counting Y x) (equalityCommutative pr))
|
||||
Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) {inr x} {inr y} pr = applyEquality inr (Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij Y)) (doubleLemma (CountablyInfiniteSet.counting Y x) (CountablyInfiniteSet.counting Y y) (succInjective pr) ))
|
||||
Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b with aMod2 b
|
||||
Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inl x with Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij X)) a
|
||||
Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inl x | r , pr = inl r , ans
|
||||
where
|
||||
ans : 2 *N CountablyInfiniteSet.counting X r ≡ b
|
||||
ans rewrite pr = x
|
||||
Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inr x with Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij Y)) a
|
||||
Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inr x | r , pr = inr r , ans
|
||||
where
|
||||
ans : succ (2 *N CountablyInfiniteSet.counting Y r) ≡ b
|
||||
ans rewrite pr = x
|
||||
|
||||
reflPair : {a b : _} {A : Set a} {B : Set b} {x1 x2 : A} {y1 y2 : B} → (x1 ≡ x2) → (y1 ≡ y2) → (x1 ,, y1) ≡ x2 ,, y2
|
||||
reflPair refl refl = refl
|
||||
firstEqualityOfPair : {a b : _} {A : Set a} {B : Set b} → {x1 x2 : A} → {y1 y2 : B} → (x1 ,, y1) ≡ (x2 ,, y2) → x1 ≡ x2
|
||||
firstEqualityOfPair {x1} {x2} {y1} {y2} refl = refl
|
||||
|
||||
bijectionOfCountableSetIsCountable : {a b : _} {A : Set a} {B : Set b} → (X : CountablyInfiniteSet A) {f : A → B} → (bij : Bijection f) → CountablyInfiniteSet B
|
||||
CountablyInfiniteSet.counting (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }) b with Surjection.property surj b
|
||||
CountablyInfiniteSet.counting (bijectionOfCountableSetIsCountable record { counting = counting ; countingIsBij = countingIsBij } {f} record { inj = inj ; surj = surj }) b | a , pr = counting a
|
||||
Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }))) {m} {n} m=n with Surjection.property surj m
|
||||
Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }))) {m} {n} m=n | a , b with Surjection.property surj n
|
||||
... | c , d = transitivity (equalityCommutative b) (transitivity (applyEquality f (Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij X)) m=n)) d)
|
||||
Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }))) b with Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij X)) b
|
||||
Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }))) b | a , pr = f a , s
|
||||
where
|
||||
s : CountablyInfiniteSet.counting (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }) (f a) ≡ b
|
||||
s with Surjection.property surj (f a)
|
||||
s | t , u = transitivity (applyEquality (CountablyInfiniteSet.counting X) (Injection.property inj u)) pr
|
||||
secondEqualityOfPair : {a b : _} {A : Set a} {B : Set b} → {x1 x2 : A} → {y1 y2 : B} → (x1 ,, y1) ≡ (x2 ,, y2) → y1 ≡ y2
|
||||
secondEqualityOfPair {x1} {x2} {y1} {y2} refl = refl
|
||||
|
||||
N^2Countable : CountablyInfiniteSet (ℕ && ℕ)
|
||||
CountablyInfiniteSet.counting N^2Countable x = Invertible.inverse (bijectionImpliesInvertible (cantorBijection)) x
|
||||
CountablyInfiniteSet.countingIsBij N^2Countable = invertibleImpliesBijection (inverseIsInvertible (bijectionImpliesInvertible cantorBijection))
|
||||
reflPair : {a b : _} {A : Set a} {B : Set b} {x1 x2 : A} {y1 y2 : B} → (x1 ≡ x2) → (y1 ≡ y2) → (x1 ,, y1) ≡ x2 ,, y2
|
||||
reflPair refl refl = refl
|
||||
|
||||
tupleRmap : {a b c : _} {A : Set a} {B : Set b} {C : Set c} → (f : B → C) → (A && B) → (A && C)
|
||||
tupleRmap f (a ,, b) = (a ,, f b)
|
||||
bijectionOfCountableSetIsCountable : {a b : _} {A : Set a} {B : Set b} → (X : CountablyInfiniteSet A) {f : A → B} → (bij : Bijection f) → CountablyInfiniteSet B
|
||||
CountablyInfiniteSet.counting (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }) b with Surjection.property surj b
|
||||
CountablyInfiniteSet.counting (bijectionOfCountableSetIsCountable record { counting = counting ; countingIsBij = countingIsBij } {f} record { inj = inj ; surj = surj }) b | a , pr = counting a
|
||||
Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }))) {m} {n} m=n with Surjection.property surj m
|
||||
Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }))) {m} {n} m=n | a , b with Surjection.property surj n
|
||||
... | c , d = transitivity (equalityCommutative b) (transitivity (applyEquality f (Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij X)) m=n)) d)
|
||||
Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }))) b with Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij X)) b
|
||||
Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }))) b | a , pr = f a , s
|
||||
where
|
||||
s : CountablyInfiniteSet.counting (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }) (f a) ≡ b
|
||||
s with Surjection.property surj (f a)
|
||||
s | t , u = transitivity (applyEquality (CountablyInfiniteSet.counting X) (Injection.property inj u)) pr
|
||||
|
||||
tupleLmap : {a b c : _} {A : Set a} {B : Set b} {C : Set c} → (f : A → C) → (A && B) → (C && B)
|
||||
tupleLmap f (a ,, b) = (f a ,, b)
|
||||
N^2Countable : CountablyInfiniteSet (ℕ && ℕ)
|
||||
CountablyInfiniteSet.counting N^2Countable x = Invertible.inverse (bijectionImpliesInvertible (cantorBijection)) x
|
||||
CountablyInfiniteSet.countingIsBij N^2Countable = invertibleImpliesBijection (inverseIsInvertible (bijectionImpliesInvertible cantorBijection))
|
||||
|
||||
bijectionOnRightOfProduct : {a b c : _} {A : Set a} {B : Set b} {C : Set c} → {f : B → C} → Bijection f → Bijection (tupleRmap {A = A} f)
|
||||
Injection.property (Bijection.inj (bijectionOnRightOfProduct {A = A} {B} {C} {f} bij)) {fst ,, snd} {fst₁ ,, snd₁} gx=gy rewrite firstEqualityOfPair gx=gy | Injection.property (Bijection.inj bij) (secondEqualityOfPair gx=gy) = refl
|
||||
Surjection.property (Bijection.surj (bijectionOnRightOfProduct {A = A} {B} {C} {f} bij)) (fst ,, snd) with Surjection.property (Bijection.surj bij) snd
|
||||
Surjection.property (Bijection.surj (bijectionOnRightOfProduct {A = A} {B} {C} {f} bij)) (fst ,, snd) | b , pr = (fst ,, b) , reflPair refl pr
|
||||
tupleRmap : {a b c : _} {A : Set a} {B : Set b} {C : Set c} → (f : B → C) → (A && B) → (A && C)
|
||||
tupleRmap f (a ,, b) = (a ,, f b)
|
||||
|
||||
bijectionOnLeftOfProduct : {a b c : _} {A : Set a} {B : Set b} {C : Set c} → {f : A → C} → Bijection f → Bijection (tupleLmap {B = B} f)
|
||||
Injection.property (Bijection.inj (bijectionOnLeftOfProduct {A = A} {B} {C} {f} bij)) {a ,, b} {c ,, d} gx=gy rewrite secondEqualityOfPair gx=gy | Injection.property (Bijection.inj bij) (firstEqualityOfPair gx=gy) = refl
|
||||
Surjection.property (Bijection.surj (bijectionOnLeftOfProduct {A = A} {B} {C} {f} bij)) (fst ,, snd) with Surjection.property (Bijection.surj bij) fst
|
||||
Surjection.property (Bijection.surj (bijectionOnLeftOfProduct {A = A} {B} {C} {f} bij)) (fst ,, snd) | a , b = (a ,, snd) , reflPair b refl
|
||||
tupleLmap : {a b c : _} {A : Set a} {B : Set b} {C : Set c} → (f : A → C) → (A && B) → (C && B)
|
||||
tupleLmap f (a ,, b) = (f a ,, b)
|
||||
|
||||
pairProductIsCountable : {a b : _} {A : Set a} {B : Set b} → (X : CountablyInfiniteSet A) → (Y : CountablyInfiniteSet B) → CountablyInfiniteSet (A && B)
|
||||
pairProductIsCountable {A = A} {B = B} X Y = bijectionOfCountableSetIsCountable N^2Countable {(tupleLmap m) ∘ (tupleRmap n)} bijF
|
||||
where
|
||||
bijM = (bijectionImpliesInvertible (CountablyInfiniteSet.countingIsBij X))
|
||||
bijN = (bijectionImpliesInvertible (CountablyInfiniteSet.countingIsBij Y))
|
||||
m : ℕ → A
|
||||
m = Invertible.inverse bijM
|
||||
n : ℕ → B
|
||||
n = Invertible.inverse bijN
|
||||
bM : Bijection m
|
||||
bM = invertibleImpliesBijection (record { inverse = CountablyInfiniteSet.counting X ; isLeft = Invertible.isRight bijM ; isRight = Invertible.isLeft bijM })
|
||||
bN : Bijection n
|
||||
bN = invertibleImpliesBijection (record { inverse = CountablyInfiniteSet.counting Y ; isLeft = Invertible.isRight bijN ; isRight = Invertible.isLeft bijN })
|
||||
bijF : Bijection ((tupleLmap {A = ℕ} {B = B} {C = A} m) ∘ (tupleRmap {A = ℕ} {B = ℕ} {C = B} n))
|
||||
bijF = bijectionComp {f = tupleRmap {A = ℕ} {B = ℕ} {C = B} n} {g = tupleLmap {A = ℕ} {B = B} {C = A} m} (bijectionOnRightOfProduct (invertibleImpliesBijection (bijectionImpliesInvertible bN))) (bijectionOnLeftOfProduct bM)
|
||||
bijectionOnRightOfProduct : {a b c : _} {A : Set a} {B : Set b} {C : Set c} → {f : B → C} → Bijection f → Bijection (tupleRmap {A = A} f)
|
||||
Injection.property (Bijection.inj (bijectionOnRightOfProduct {A = A} {B} {C} {f} bij)) {fst ,, snd} {fst₁ ,, snd₁} gx=gy rewrite firstEqualityOfPair gx=gy | Injection.property (Bijection.inj bij) (secondEqualityOfPair gx=gy) = refl
|
||||
Surjection.property (Bijection.surj (bijectionOnRightOfProduct {A = A} {B} {C} {f} bij)) (fst ,, snd) with Surjection.property (Bijection.surj bij) snd
|
||||
Surjection.property (Bijection.surj (bijectionOnRightOfProduct {A = A} {B} {C} {f} bij)) (fst ,, snd) | b , pr = (fst ,, b) , reflPair refl pr
|
||||
|
||||
bijectionOnLeftOfProduct : {a b c : _} {A : Set a} {B : Set b} {C : Set c} → {f : A → C} → Bijection f → Bijection (tupleLmap {B = B} f)
|
||||
Injection.property (Bijection.inj (bijectionOnLeftOfProduct {A = A} {B} {C} {f} bij)) {a ,, b} {c ,, d} gx=gy rewrite secondEqualityOfPair gx=gy | Injection.property (Bijection.inj bij) (firstEqualityOfPair gx=gy) = refl
|
||||
Surjection.property (Bijection.surj (bijectionOnLeftOfProduct {A = A} {B} {C} {f} bij)) (fst ,, snd) with Surjection.property (Bijection.surj bij) fst
|
||||
Surjection.property (Bijection.surj (bijectionOnLeftOfProduct {A = A} {B} {C} {f} bij)) (fst ,, snd) | a , b = (a ,, snd) , reflPair b refl
|
||||
|
||||
pairProductIsCountable : {a b : _} {A : Set a} {B : Set b} → (X : CountablyInfiniteSet A) → (Y : CountablyInfiniteSet B) → CountablyInfiniteSet (A && B)
|
||||
pairProductIsCountable {A = A} {B = B} X Y = bijectionOfCountableSetIsCountable N^2Countable {(tupleLmap m) ∘ (tupleRmap n)} bijF
|
||||
where
|
||||
bijM = (bijectionImpliesInvertible (CountablyInfiniteSet.countingIsBij X))
|
||||
bijN = (bijectionImpliesInvertible (CountablyInfiniteSet.countingIsBij Y))
|
||||
m : ℕ → A
|
||||
m = Invertible.inverse bijM
|
||||
n : ℕ → B
|
||||
n = Invertible.inverse bijN
|
||||
bM : Bijection m
|
||||
bM = invertibleImpliesBijection (record { inverse = CountablyInfiniteSet.counting X ; isLeft = Invertible.isRight bijM ; isRight = Invertible.isLeft bijM })
|
||||
bN : Bijection n
|
||||
bN = invertibleImpliesBijection (record { inverse = CountablyInfiniteSet.counting Y ; isLeft = Invertible.isRight bijN ; isRight = Invertible.isLeft bijN })
|
||||
bijF : Bijection ((tupleLmap {A = ℕ} {B = B} {C = A} m) ∘ (tupleRmap {A = ℕ} {B = ℕ} {C = B} n))
|
||||
bijF = bijectionComp {f = tupleRmap {A = ℕ} {B = ℕ} {C = B} n} {g = tupleLmap {A = ℕ} {B = B} {C = A} m} (bijectionOnRightOfProduct (invertibleImpliesBijection (bijectionImpliesInvertible bN))) (bijectionOnLeftOfProduct bM)
|
||||
|
||||
-- injectionToNMeansCountable : {a : _} {A : Set a} {f : A → ℕ} → Injection f → InfiniteSet A → Countable A
|
||||
-- injectionToNMeansCountable {f = f} inj record { isInfinite = isInfinite } = {!!}
|
||||
|
||||
@@ -1,6 +1,6 @@
|
||||
{-# OPTIONS --safe --warning=error #-}
|
||||
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Numbers.Naturals.Semiring
|
||||
open import Numbers.Naturals.Order
|
||||
open import Sets.FinSet
|
||||
open import LogicalFormulae
|
||||
|
||||
Reference in New Issue
Block a user