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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.Lemmas
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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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@@ -12,137 +14,140 @@ open import WellFoundedInduction
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open import Sets.CantorBijection.CantorBijection
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module Sets.Cardinality where
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record CountablyInfiniteSet {a : _} (A : Set a) : Set a where
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field
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counting : A → ℕ
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countingIsBij : Bijection counting
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data Countable {a : _} (A : Set a) : Set a where
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finite : FiniteSet A → Countable A
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infinite : CountablyInfiniteSet A → Countable A
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open import Semirings.Lemmas ℕSemiring
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ℕCountable : CountablyInfiniteSet ℕ
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ℕCountable = record { counting = id ; countingIsBij = invertibleImpliesBijection (record { inverse = id ; isLeft = λ b → refl ; isRight = λ a → refl}) }
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record CountablyInfiniteSet {a : _} (A : Set a) : Set a where
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field
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counting : A → ℕ
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countingIsBij : Bijection counting
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doubleLemma : (a b : ℕ) → 2 *N a ≡ 2 *N b → a ≡ b
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doubleLemma a b pr = productCancelsLeft 2 a b (le 1 refl) pr
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data Countable {a : _} (A : Set a) : Set a where
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finite : FiniteSet A → Countable A
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infinite : CountablyInfiniteSet A → Countable A
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evenCannotBeOdd : (a b : ℕ) → 2 *N a ≡ succ (2 *N b) → False
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evenCannotBeOdd zero b ()
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evenCannotBeOdd (succ a) zero pr rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring a (succ a) = naughtE (equalityCommutative (succInjective pr))
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evenCannotBeOdd (succ a) (succ b) pr = evenCannotBeOdd a b pr''
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where
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pr' : a +N a ≡ (b +N succ b)
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pr' rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring b 0 | Semiring.commutative ℕSemiring a (succ a) = succInjective (succInjective pr)
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pr'' : 2 *N a ≡ succ (2 *N b)
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pr'' rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring b 0 | Semiring.commutative ℕSemiring (succ b) b = pr'
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ℕCountable : CountablyInfiniteSet ℕ
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ℕCountable = record { counting = id ; countingIsBij = invertibleImpliesBijection (record { inverse = id ; isLeft = λ b → refl ; isRight = λ a → refl}) }
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aMod2 : (a : ℕ) → Sg ℕ (λ i → (2 *N i ≡ a) || (succ (2 *N i) ≡ a))
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aMod2 zero = (0 , inl refl)
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aMod2 (succ a) with aMod2 a
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aMod2 (succ a) | b , inl x = b , inr (applyEquality succ x)
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aMod2 (succ a) | b , inr x = (succ b) , inl pr
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where
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pr : succ (b +N succ (b +N 0)) ≡ succ a
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pr rewrite Semiring.commutative ℕSemiring b (succ (b +N 0)) | Semiring.commutative ℕSemiring (b +N 0) b = applyEquality succ x
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doubleLemma : (a b : ℕ) → 2 *N a ≡ 2 *N b → a ≡ b
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doubleLemma a b pr = productCancelsLeft 2 a b (le 1 refl) pr
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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))
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sqrtFloor zero = (0 ,, 0) , (refl ,, le zero refl)
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sqrtFloor (succ n) with sqrtFloor n
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sqrtFloor (succ n) | (a ,, b) , pr with orderIsTotal b (2 *N a)
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sqrtFloor (succ n) | (a ,, b) , pr | inl (inl x) = (a ,, succ b) , (p ,, q)
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where
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p : a *N a +N succ b ≡ succ n
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p rewrite Semiring.commutative ℕSemiring (a *N a) (succ b) | Semiring.commutative ℕSemiring b (a *N a) = applyEquality succ (_&&_.fst pr)
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q : succ b <N (a +N (a +N 0)) +N 1
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q rewrite Semiring.commutative ℕSemiring (a +N (a +N 0)) (succ 0) | Semiring.commutative ℕSemiring a 0 = succPreservesInequality x
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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)
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sqrtFloor (succ n) | (a ,, b) , pr | inr x = (succ a ,, 0) , (q ,, succIsPositive _)
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where
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p : a +N a *N succ a ≡ n
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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
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q : succ ((a +N a *N succ a) +N 0) ≡ succ n
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q rewrite Semiring.commutative ℕSemiring (a +N a *N succ a) 0 = applyEquality succ p
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evenCannotBeOdd : (a b : ℕ) → 2 *N a ≡ succ (2 *N b) → False
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evenCannotBeOdd zero b ()
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evenCannotBeOdd (succ a) zero pr rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring a (succ a) = naughtE (equalityCommutative (succInjective pr))
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evenCannotBeOdd (succ a) (succ b) pr = evenCannotBeOdd a b pr''
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where
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pr' : a +N a ≡ (b +N succ b)
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pr' rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring b 0 | Semiring.commutative ℕSemiring a (succ a) = succInjective (succInjective pr)
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pr'' : 2 *N a ≡ succ (2 *N b)
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pr'' rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring b 0 | Semiring.commutative ℕSemiring (succ b) b = pr'
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pairUnionIsCountable : {a b : _} {A : Set a} {B : Set b} → (X : CountablyInfiniteSet A) → (Y : CountablyInfiniteSet B) → CountablyInfiniteSet (A || B)
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CountablyInfiniteSet.counting (pairUnionIsCountable X Y) (inl r) = 2 *N (CountablyInfiniteSet.counting X r)
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CountablyInfiniteSet.counting (pairUnionIsCountable X Y) (inr s) = succ (2 *N (CountablyInfiniteSet.counting Y s))
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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)
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where
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inter : CountablyInfiniteSet.counting X x ≡ CountablyInfiniteSet.counting X y
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inter = doubleLemma (CountablyInfiniteSet.counting X x) (CountablyInfiniteSet.counting X y) pr
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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)
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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))
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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) ))
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Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b with aMod2 b
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Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inl x with Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij X)) a
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Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inl x | r , pr = inl r , ans
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where
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ans : 2 *N CountablyInfiniteSet.counting X r ≡ b
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ans rewrite pr = x
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Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inr x with Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij Y)) a
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Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inr x | r , pr = inr r , ans
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where
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ans : succ (2 *N CountablyInfiniteSet.counting Y r) ≡ b
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ans rewrite pr = x
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aMod2 : (a : ℕ) → Sg ℕ (λ i → (2 *N i ≡ a) || (succ (2 *N i) ≡ a))
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aMod2 zero = (0 , inl refl)
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aMod2 (succ a) with aMod2 a
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aMod2 (succ a) | b , inl x = b , inr (applyEquality succ x)
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aMod2 (succ a) | b , inr x = (succ b) , inl pr
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where
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pr : succ (b +N succ (b +N 0)) ≡ succ a
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pr rewrite Semiring.commutative ℕSemiring b (succ (b +N 0)) | Semiring.commutative ℕSemiring (b +N 0) b = applyEquality succ x
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firstEqualityOfPair : {a b : _} {A : Set a} {B : Set b} → {x1 x2 : A} → {y1 y2 : B} → (x1 ,, y1) ≡ (x2 ,, y2) → x1 ≡ x2
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firstEqualityOfPair {x1} {x2} {y1} {y2} refl = refl
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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))
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sqrtFloor zero = (0 ,, 0) , (refl ,, le zero refl)
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sqrtFloor (succ n) with sqrtFloor n
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sqrtFloor (succ n) | (a ,, b) , pr with TotalOrder.totality ℕTotalOrder b (2 *N a)
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sqrtFloor (succ n) | (a ,, b) , pr | inl (inl x) = (a ,, succ b) , (p ,, q)
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where
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p : a *N a +N succ b ≡ succ n
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p rewrite Semiring.commutative ℕSemiring (a *N a) (succ b) | Semiring.commutative ℕSemiring b (a *N a) = applyEquality succ (_&&_.fst pr)
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q : succ b <N (a +N (a +N 0)) +N 1
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q rewrite Semiring.commutative ℕSemiring (a +N (a +N 0)) (succ 0) | Semiring.commutative ℕSemiring a 0 = succPreservesInequality x
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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)
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sqrtFloor (succ n) | (a ,, b) , pr | inr x = (succ a ,, 0) , (q ,, succIsPositive _)
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where
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p : a +N a *N succ a ≡ n
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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
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q : succ ((a +N a *N succ a) +N 0) ≡ succ n
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q rewrite Semiring.commutative ℕSemiring (a +N a *N succ a) 0 = applyEquality succ p
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secondEqualityOfPair : {a b : _} {A : Set a} {B : Set b} → {x1 x2 : A} → {y1 y2 : B} → (x1 ,, y1) ≡ (x2 ,, y2) → y1 ≡ y2
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secondEqualityOfPair {x1} {x2} {y1} {y2} refl = refl
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pairUnionIsCountable : {a b : _} {A : Set a} {B : Set b} → (X : CountablyInfiniteSet A) → (Y : CountablyInfiniteSet B) → CountablyInfiniteSet (A || B)
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CountablyInfiniteSet.counting (pairUnionIsCountable X Y) (inl r) = 2 *N (CountablyInfiniteSet.counting X r)
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CountablyInfiniteSet.counting (pairUnionIsCountable X Y) (inr s) = succ (2 *N (CountablyInfiniteSet.counting Y s))
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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)
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where
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inter : CountablyInfiniteSet.counting X x ≡ CountablyInfiniteSet.counting X y
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inter = doubleLemma (CountablyInfiniteSet.counting X x) (CountablyInfiniteSet.counting X y) pr
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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)
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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))
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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) ))
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Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b with aMod2 b
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Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inl x with Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij X)) a
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Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inl x | r , pr = inl r , ans
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where
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ans : 2 *N CountablyInfiniteSet.counting X r ≡ b
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ans rewrite pr = x
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Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inr x with Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij Y)) a
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Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (pairUnionIsCountable X Y))) b | a , inr x | r , pr = inr r , ans
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where
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ans : succ (2 *N CountablyInfiniteSet.counting Y r) ≡ b
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ans rewrite pr = x
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reflPair : {a b : _} {A : Set a} {B : Set b} {x1 x2 : A} {y1 y2 : B} → (x1 ≡ x2) → (y1 ≡ y2) → (x1 ,, y1) ≡ x2 ,, y2
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reflPair refl refl = refl
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firstEqualityOfPair : {a b : _} {A : Set a} {B : Set b} → {x1 x2 : A} → {y1 y2 : B} → (x1 ,, y1) ≡ (x2 ,, y2) → x1 ≡ x2
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firstEqualityOfPair {x1} {x2} {y1} {y2} refl = refl
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bijectionOfCountableSetIsCountable : {a b : _} {A : Set a} {B : Set b} → (X : CountablyInfiniteSet A) {f : A → B} → (bij : Bijection f) → CountablyInfiniteSet B
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CountablyInfiniteSet.counting (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }) b with Surjection.property surj b
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CountablyInfiniteSet.counting (bijectionOfCountableSetIsCountable record { counting = counting ; countingIsBij = countingIsBij } {f} record { inj = inj ; surj = surj }) b | a , pr = counting a
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Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }))) {m} {n} m=n with Surjection.property surj m
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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
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... | c , d = transitivity (equalityCommutative b) (transitivity (applyEquality f (Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij X)) m=n)) d)
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Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }))) b with Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij X)) b
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Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }))) b | a , pr = f a , s
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where
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s : CountablyInfiniteSet.counting (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }) (f a) ≡ b
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s with Surjection.property surj (f a)
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s | t , u = transitivity (applyEquality (CountablyInfiniteSet.counting X) (Injection.property inj u)) pr
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secondEqualityOfPair : {a b : _} {A : Set a} {B : Set b} → {x1 x2 : A} → {y1 y2 : B} → (x1 ,, y1) ≡ (x2 ,, y2) → y1 ≡ y2
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secondEqualityOfPair {x1} {x2} {y1} {y2} refl = refl
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N^2Countable : CountablyInfiniteSet (ℕ && ℕ)
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CountablyInfiniteSet.counting N^2Countable x = Invertible.inverse (bijectionImpliesInvertible (cantorBijection)) x
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CountablyInfiniteSet.countingIsBij N^2Countable = invertibleImpliesBijection (inverseIsInvertible (bijectionImpliesInvertible cantorBijection))
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reflPair : {a b : _} {A : Set a} {B : Set b} {x1 x2 : A} {y1 y2 : B} → (x1 ≡ x2) → (y1 ≡ y2) → (x1 ,, y1) ≡ x2 ,, y2
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reflPair refl refl = refl
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tupleRmap : {a b c : _} {A : Set a} {B : Set b} {C : Set c} → (f : B → C) → (A && B) → (A && C)
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tupleRmap f (a ,, b) = (a ,, f b)
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bijectionOfCountableSetIsCountable : {a b : _} {A : Set a} {B : Set b} → (X : CountablyInfiniteSet A) {f : A → B} → (bij : Bijection f) → CountablyInfiniteSet B
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CountablyInfiniteSet.counting (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }) b with Surjection.property surj b
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CountablyInfiniteSet.counting (bijectionOfCountableSetIsCountable record { counting = counting ; countingIsBij = countingIsBij } {f} record { inj = inj ; surj = surj }) b | a , pr = counting a
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Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }))) {m} {n} m=n with Surjection.property surj m
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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
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... | c , d = transitivity (equalityCommutative b) (transitivity (applyEquality f (Injection.property (Bijection.inj (CountablyInfiniteSet.countingIsBij X)) m=n)) d)
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Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }))) b with Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij X)) b
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Surjection.property (Bijection.surj (CountablyInfiniteSet.countingIsBij (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }))) b | a , pr = f a , s
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where
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s : CountablyInfiniteSet.counting (bijectionOfCountableSetIsCountable X {f} record { inj = inj ; surj = surj }) (f a) ≡ b
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s with Surjection.property surj (f a)
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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 } = {!!}
|
||||
|
||||
Reference in New Issue
Block a user