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Lots of without-K (#110)

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This commit is contained in:
Patrick Stevens
2020-04-11 12:14:03 +01:00
committed by GitHub
parent 412edaf4c7
commit e9aa1bcc05
30 changed files with 424 additions and 98 deletions
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14
Numbers/Naturals/Definition.agda
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@@ -1,6 +1,7 @@
{-# OPTIONS --warning=error --safe --without-K #-}
open import LogicalFormulae
open import Decidable.Lemmas
module Numbers.Naturals.Definition where
@@ -34,3 +35,16 @@ aIsNotSuccA (succ a) pr = aIsNotSuccA a (succInjective pr)
ℕDecideEquality' a b with ℕDecideEquality a b
ℕDecideEquality' a b | inl x = BoolTrue
ℕDecideEquality' a b | inr x = BoolFalse
record _=N'_ (a b : ℕ) : Set where
field
.eq : a ≡ b
squashN : {a b : ℕ} → a =N' b → a ≡ b
squashN record { eq = eq } = squash ℕDecideEquality eq
collapseN : {a b : ℕ} → a ≡ b → a =N' b
collapseN refl = record { eq = refl }
=N'Refl : {a b : ℕ} → (p1 p2 : a =N' b) → p1 ≡ p2
=N'Refl p1 p2 = refl

55
Numbers/Naturals/EuclideanAlgorithm.agda
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@@ -2,6 +2,7 @@
open import LogicalFormulae
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Definition
open import Numbers.Naturals.Naturals
open import Numbers.Naturals.Order
open import Numbers.Naturals.Order.Lemmas
@@ -14,6 +15,7 @@ module Numbers.Naturals.EuclideanAlgorithm where
open TotalOrder ℕTotalOrder
open Semiring ℕSemiring
open import Decidable.Lemmas ℕDecideEquality
record divisionAlgResult (a : ℕ) (b : ℕ) : Set where
field
@@ -23,6 +25,50 @@ record divisionAlgResult (a : ℕ) (b : ℕ) : Set where
remIsSmall : (rem <N a) || (a ≡ 0)
quotSmall : (0 <N a) || ((0 ≡ a) && (quot ≡ 0))
record divisionAlgResult' (a : ℕ) (b : ℕ) : Set where
field
quot : ℕ
rem : ℕ
.pr : a *N quot +N rem ≡ b
remIsSmall : (rem <N' a) || (a =N' 0)
quotSmall : (0 <N' a) || ((0 =N' a) && (quot =N' 0))
collapseDivAlgResult : {a b : ℕ} → (divisionAlgResult a b) → divisionAlgResult' a b
collapseDivAlgResult record { quot = q ; rem = r ; pr = pr ; remIsSmall = (inl x) ; quotSmall = (inl y) } = record { quot = q ; rem = r ; pr = pr ; remIsSmall = inl (<NTo<N' x) ; quotSmall = inl (<NTo<N' y) }
collapseDivAlgResult record { quot = q ; rem = r ; pr = pr ; remIsSmall = (inl x) ; quotSmall = (inr y) } = record { quot = q ; rem = r ; pr = pr ; remIsSmall = inl (<NTo<N' x) ; quotSmall = inr (collapseN (_&&_.fst y) ,, collapseN (_&&_.snd y)) }
collapseDivAlgResult record { quot = q ; rem = r ; pr = pr ; remIsSmall = (inr x) ; quotSmall = inl y } = record { quot = q ; rem = r ; pr = pr ; remIsSmall = inr (collapseN x) ; quotSmall = inl (<NTo<N' y) }
collapseDivAlgResult record { quot = q ; rem = r ; pr = pr ; remIsSmall = (inr x) ; quotSmall = inr y } = record { quot = q ; rem = r ; pr = pr ; remIsSmall = inr (collapseN x) ; quotSmall = inr (collapseN (_&&_.fst y) ,, collapseN (_&&_.snd y)) }
squashDivAlgResult : {a b : ℕ} → divisionAlgResult' a b → divisionAlgResult a b
squashDivAlgResult record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inl x) ; quotSmall = (inl y) } = record { quot = quot ; rem = rem ; pr = squash pr ; remIsSmall = inl (<N'To<N x) ; quotSmall = inl (<N'To<N y) }
squashDivAlgResult record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inl x) ; quotSmall = (inr y) } = record { quot = quot ; rem = rem ; pr = squash pr ; remIsSmall = inl (<N'To<N x) ; quotSmall = inr (squashN (_&&_.fst y) ,, squashN (_&&_.snd y)) }
squashDivAlgResult record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inr x) ; quotSmall = (inl y) } = record { quot = quot ; rem = rem ; pr = squash pr ; remIsSmall = inr (squashN x) ; quotSmall = inl (<N'To<N y) }
squashDivAlgResult record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inr x) ; quotSmall = (inr y) } = record { quot = quot ; rem = rem ; pr = squash pr ; remIsSmall = inr (squashN x) ; quotSmall = inr (squashN (_&&_.fst y) ,, squashN (_&&_.snd y)) }
squashPreservesRem : {a b : ℕ} → (p : divisionAlgResult' a b) → divisionAlgResult.rem (squashDivAlgResult p) ≡ divisionAlgResult'.rem p
squashPreservesRem record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inl x) ; quotSmall = (inl x₁) } = refl
squashPreservesRem record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inl x) ; quotSmall = (inr x₁) } = refl
squashPreservesRem record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inr x) ; quotSmall = (inl x₁) } = refl
squashPreservesRem record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inr x) ; quotSmall = (inr x₁) } = refl
squashPreservesQuot : {a b : ℕ} → (p : divisionAlgResult' a b) → divisionAlgResult.quot (squashDivAlgResult p) ≡ divisionAlgResult'.quot p
squashPreservesQuot record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inl x) ; quotSmall = (inl x₁) } = refl
squashPreservesQuot record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inl x) ; quotSmall = (inr x₁) } = refl
squashPreservesQuot record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inr x) ; quotSmall = (inl x₁) } = refl
squashPreservesQuot record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inr x) ; quotSmall = (inr x₁) } = refl
collapsePreservesRem : {a b : ℕ} → (p : divisionAlgResult a b) → divisionAlgResult'.rem (collapseDivAlgResult p) ≡ divisionAlgResult.rem p
collapsePreservesRem record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inl x) ; quotSmall = (inl x₁) } = refl
collapsePreservesRem record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inl x) ; quotSmall = (inr x₁) } = refl
collapsePreservesRem record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inr x) ; quotSmall = (inl x₁) } = refl
collapsePreservesRem record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inr x) ; quotSmall = (inr x₁) } = refl
collapsePreservesQuot : {a b : ℕ} → (p : divisionAlgResult a b) → divisionAlgResult'.quot (collapseDivAlgResult p) ≡ divisionAlgResult.quot p
collapsePreservesQuot record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inl x) ; quotSmall = (inl x₁) } = refl
collapsePreservesQuot record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inl x) ; quotSmall = (inr x₁) } = refl
collapsePreservesQuot record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inr x) ; quotSmall = (inl x₁) } = refl
collapsePreservesQuot record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = (inr x) ; quotSmall = (inr x₁) } = refl
divAlgLessLemma : (a b : ℕ) → (0 <N a) → (r : divisionAlgResult a b) → (divisionAlgResult.quot r ≡ 0) || (divisionAlgResult.rem r <N b)
divAlgLessLemma zero b pr _ = exFalso (TotalOrder.irreflexive ℕTotalOrder pr)
divAlgLessLemma (succ a) b _ record { quot = zero ; rem = a%b ; pr = pr ; remIsSmall = remIsSmall } = inl refl
@@ -97,6 +143,15 @@ divisionAlg (succ a) = rec <NWellfounded (λ n → divisionAlgResult (succ a) n)
data _∣_ : ℕ → ℕ → Set where
divides : {a b : ℕ} → (res : divisionAlgResult a b) → divisionAlgResult.rem res ≡ zero → a ∣ b
data _∣'_ : ℕ → ℕ → Set where
divides' : {a b : ℕ} → (res : divisionAlgResult' a b) → .(divisionAlgResult'.rem res ≡ zero) → a ∣' b
divToDiv' : {a b : ℕ} → a ∣ b → a ∣' b
divToDiv' (divides res x) = divides' (collapseDivAlgResult res) (transitivity (collapsePreservesRem res) x)
div'ToDiv : {a b : ℕ} → a ∣' b → a ∣ b
div'ToDiv (divides' res x) = divides (squashDivAlgResult res) (transitivity (squashPreservesRem res) (squashN record { eq = x }))
zeroDividesNothing : (a : ℕ) → zero ∣ succ a → False
zeroDividesNothing a (divides record { quot = quot ; rem = rem ; pr = pr } x) = naughtE p
where

19
Numbers/Naturals/Order.agda
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@@ -2,12 +2,14 @@
open import LogicalFormulae
open import Semirings.Definition
open import Numbers.Naturals.Definition
open import Numbers.Naturals.Semiring
open import Orders.Total.Definition
open import Orders.Partial.Definition
module Numbers.Naturals.Order where
open Semiring ℕSemiring
open import Decidable.Lemmas ℕDecideEquality
private
infix 5 _<NLogical_
@@ -24,6 +26,13 @@ record _<N_ (a : ℕ) (b : ℕ) : Set where
x : ℕ
proof : (succ x) +N a ≡ b
infix 5 _<N'_
record _<N'_ (a : ℕ) (b : ℕ) : Set where
constructor le'
field
x : ℕ
.proof : (succ x) +N a ≡ b
infix 5 _≤N_
_≤N_ : ℕ → ℕ → Set
a ≤N b = (a <N b) || (a ≡ b)
@@ -182,3 +191,13 @@ canFlipMultiplicationsInIneq {a} {b} {c} {d} pr = identityOfIndiscernablesRight
lessRespectsMultiplication : (a b c : ℕ) → (a <N b) → (zero <N c) → (a *N c <N b *N c)
lessRespectsMultiplication a b c prAB cPos = canFlipMultiplicationsInIneq {c} {a} {c} {b} (lessRespectsMultiplicationLeft a b c prAB cPos)
<NTo<N' : {a b : ℕ} → a <N b → a <N' b
<NTo<N' (le x proof) = le' x proof
<N'To<N : {a b : ℕ} → a <N' b → a <N b
<N'To<N {a} {b} (le' x proof) = le x (squash proof)
<N'Refl : {a b : ℕ} → (p1 p2 : a <N' b) → p1 ≡ p2
<N'Refl p1 p2 with <NWellDefined (<N'To<N p1) (<N'To<N p2)
... | refl = refl