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Rename order transitivity (#62)

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Patrick Stevens
2019-11-02 19:05:52 +00:00
committed by GitHub
parent 763ddb8dbb
commit 1325236359
20 changed files with 220 additions and 220 deletions
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10
Sets/CantorBijection/Order.agda
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@@ -19,10 +19,10 @@ totalOrder : TotalOrder (ℕ && ℕ)
PartialOrder._<_ (TotalOrder.order totalOrder) = order
PartialOrder.irreflexive (TotalOrder.order totalOrder) {fst ,, snd} (inl x) = exFalso (TotalOrder.irreflexive ℕTotalOrder x)
PartialOrder.irreflexive (TotalOrder.order totalOrder) {fst ,, snd} (inr (_ ,, p)) = exFalso (TotalOrder.irreflexive ℕTotalOrder p)
PartialOrder.transitive (TotalOrder.order totalOrder) {x1 ,, x2} {y1 ,, y2} {z1 ,, z2} (inl pr1) (inl pr2) = inl (TotalOrder.transitive ℕTotalOrder pr1 pr2)
PartialOrder.transitive (TotalOrder.order totalOrder) {x1 ,, x2} {y1 ,, y2} {z1 ,, z2} (inl pr1) (inr (f1 ,, f2)) = inl (identityOfIndiscernablesRight _<N_ pr1 f1)
PartialOrder.transitive (TotalOrder.order totalOrder) {x1 ,, x2} {y1 ,, y2} {z1 ,, z2} (inr (f1 ,, f2)) (inl x) = inl (identityOfIndiscernablesLeft _<N_ x (equalityCommutative f1))
PartialOrder.transitive (TotalOrder.order totalOrder) {x1 ,, x2} {y1 ,, y2} {z1 ,, z2} (inr (f1 ,, f2)) (inr (fst ,, snd)) = inr (transitivity f1 fst ,, TotalOrder.transitive ℕTotalOrder f2 snd)
PartialOrder.<Transitive (TotalOrder.order totalOrder) {x1 ,, x2} {y1 ,, y2} {z1 ,, z2} (inl pr1) (inl pr2) = inl (TotalOrder.<Transitive ℕTotalOrder pr1 pr2)
PartialOrder.<Transitive (TotalOrder.order totalOrder) {x1 ,, x2} {y1 ,, y2} {z1 ,, z2} (inl pr1) (inr (f1 ,, f2)) = inl (identityOfIndiscernablesRight _<N_ pr1 f1)
PartialOrder.<Transitive (TotalOrder.order totalOrder) {x1 ,, x2} {y1 ,, y2} {z1 ,, z2} (inr (f1 ,, f2)) (inl x) = inl (identityOfIndiscernablesLeft _<N_ x (equalityCommutative f1))
PartialOrder.<Transitive (TotalOrder.order totalOrder) {x1 ,, x2} {y1 ,, y2} {z1 ,, z2} (inr (f1 ,, f2)) (inr (fst ,, snd)) = inr (transitivity f1 fst ,, TotalOrder.<Transitive ℕTotalOrder f2 snd)
TotalOrder.totality totalOrder (a ,, b) (c ,, d) with TotalOrder.totality ℕTotalOrder (a +N b) (c +N d)
TotalOrder.totality totalOrder (a ,, b) (c ,, d) | inl (inl x) = inl (inl (inl x))
TotalOrder.totality totalOrder (a ,, b) (c ,, d) | inl (inr x) = inl (inr (inl x))
@@ -83,7 +83,7 @@ orderWellfounded (a ,, b) = access (go b a)
where
bad : (x y1 y2 : ℕ) → y1 +N y2 ≡ x → x <N y2 → False
bad x zero y2 y1+y2=x x<y2 = TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesRight _<N_ x<y2 y1+y2=x)
bad x (succ y1) y2 y1+y2=x x<y2 rewrite equalityCommutative y1+y2=x = TotalOrder.irreflexive ℕTotalOrder (TotalOrder.transitive ℕTotalOrder x<y2 (identityOfIndiscernablesRight _<N_ (addingIncreases y2 y1) (Semiring.commutative ℕSemiring y2 (succ y1))))
bad x (succ y1) y2 y1+y2=x x<y2 rewrite equalityCommutative y1+y2=x = TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive ℕTotalOrder x<y2 (identityOfIndiscernablesRight _<N_ (addingIncreases y2 y1) (Semiring.commutative ℕSemiring y2 (succ y1))))
ans (succ fst ,, snd) (inr (a1 ,, a2)) rewrite y1+y2=x = inr (a1 ,, le fst a1)
g (succ x) pr zero (zero ,, y2) (inr (fst ,, snd)) = exFalso (TotalOrder.irreflexive ℕTotalOrder (identityOfIndiscernablesLeft _<N_ snd fst))
g (succ x) pr zero (succ y1 ,, y2) (inr (fst ,, snd)) = pr y2 snd (succ (succ y1)) (succ y1 ,, y2) (inl (le 0 refl))

14
Sets/CantorBijection/Proofs.agda
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@@ -64,7 +64,7 @@ module Sets.CantorBijection.Proofs where
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | succ y1 ,, zero | inl (inr pr1) rewrite Semiring.commutative ℕSemiring y1 1 = exFalso (zeroNeverGreater (canRemoveSuccFrom<N pr1))
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | succ y1 ,, zero | inr pr1 = inr (pr1 ,, le 0 refl)
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | zero ,, zero | succ y1 ,, succ y2 rewrite Semiring.commutative ℕSemiring y1 (succ (succ y2)) = inl (succPreservesInequality (succIsPositive (y2 +N y1)))
cantorInverseOrderPreserving (succ x) (succ y) x<y | inl pr | succ x1 ,, zero | zero ,, succ y2 rewrite Semiring.commutative ℕSemiring x1 1 | Semiring.sumZeroRight ℕSemiring y2 | Semiring.sumZeroRight ℕSemiring x1 = inl (TotalOrder.transitive ℕTotalOrder pr (a<SuccA _))
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 _))
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
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)))
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)
@@ -82,7 +82,7 @@ module Sets.CantorBijection.Proofs where
ans | inl (inl x) = inl x
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)))))
ans | inr x = inr (x ,, succIsPositive (succ y2))
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 | 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
@@ -90,7 +90,7 @@ module Sets.CantorBijection.Proofs where
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 (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))
@@ -117,7 +117,7 @@ module Sets.CantorBijection.Proofs where
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 ,, 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)
@@ -129,14 +129,14 @@ module Sets.CantorBijection.Proofs where
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) (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) (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