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Comparison on the reals (#55)

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Patrick Stevens
2019-10-26 10:36:24 +01:00
committed by GitHub
parent 4e56b68024
commit 0d68919127
18 changed files with 1343 additions and 691 deletions
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56
Setoids/Orders.agda
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@@ -9,34 +9,34 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Setoids.Orders where
record SetoidPartialOrder {a b c : _} {A : Set a} (S : Setoid {a} {b} A) (_<_ : Rel {a} {c} A) : Set (a ⊔ b ⊔ c) where
open Setoid S
field
<WellDefined : {a b c d : A} → (a ∼ b) → (c ∼ d) → a < c → b < d
irreflexive : {x : A} → (x < x) → False
transitive : {a b c : A} → (a < b) → (b < c) → (a < c)
record SetoidPartialOrder {a b c : _} {A : Set a} (S : Setoid {a} {b} A) (_<_ : Rel {a} {c} A) : Set (a ⊔ b ⊔ c) where
open Setoid S
field
<WellDefined : {a b c d : A} → (a ∼ b) → (c ∼ d) → a < c → b < d
irreflexive : {x : A} → (x < x) → False
transitive : {a b c : A} → (a < b) → (b < c) → (a < c)
record SetoidTotalOrder {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_<_ : Rel {a} {c} A} (P : SetoidPartialOrder S _<_) : Set (a ⊔ b ⊔ c) where
open Setoid S
field
totality : (a b : A) → ((a < b) || (b < a)) || (a ∼ b)
partial : SetoidPartialOrder S _<_
partial = P
min : A → A → A
min a b with totality a b
min a b | inl (inl a<b) = a
min a b | inl (inr b<a) = b
min a b | inr a=b = a
max : A → A → A
max a b with totality a b
max a b | inl (inl a<b) = b
max a b | inl (inr b<a) = a
max a b | inr a=b = b
record SetoidTotalOrder {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_<_ : Rel {a} {c} A} (P : SetoidPartialOrder S _<_) : Set (a ⊔ b ⊔ c) where
open Setoid S
field
totality : (a b : A) → ((a < b) || (b < a)) || (a ∼ b)
partial : SetoidPartialOrder S _<_
partial = P
min : A → A → A
min a b with totality a b
min a b | inl (inl a<b) = a
min a b | inl (inr b<a) = b
min a b | inr a=b = a
max : A → A → A
max a b with totality a b
max a b | inl (inl a<b) = b
max a b | inl (inr b<a) = a
max a b | inr a=b = b
partialOrderToSetoidPartialOrder : {a b : _} {A : Set a} (P : PartialOrder {a} {b} A) → SetoidPartialOrder (reflSetoid A) (PartialOrder._<_ P)
SetoidPartialOrder.<WellDefined (partialOrderToSetoidPartialOrder P) a=b c=d a<c rewrite a=b | c=d = a<c
SetoidPartialOrder.irreflexive (partialOrderToSetoidPartialOrder P) = PartialOrder.irreflexive P
SetoidPartialOrder.transitive (partialOrderToSetoidPartialOrder P) = PartialOrder.transitive P
partialOrderToSetoidPartialOrder : {a b : _} {A : Set a} (P : PartialOrder {a} {b} A) → SetoidPartialOrder (reflSetoid A) (PartialOrder._<_ P)
SetoidPartialOrder.<WellDefined (partialOrderToSetoidPartialOrder P) a=b c=d a<c rewrite a=b | c=d = a<c
SetoidPartialOrder.irreflexive (partialOrderToSetoidPartialOrder P) = PartialOrder.irreflexive P
SetoidPartialOrder.transitive (partialOrderToSetoidPartialOrder P) = PartialOrder.transitive P
totalOrderToSetoidTotalOrder : {a b : _} {A : Set a} (T : TotalOrder {a} {b} A) → SetoidTotalOrder (partialOrderToSetoidPartialOrder (TotalOrder.order T))
SetoidTotalOrder.totality (totalOrderToSetoidTotalOrder T) = TotalOrder.totality T
totalOrderToSetoidTotalOrder : {a b : _} {A : Set a} (T : TotalOrder {a} {b} A) → SetoidTotalOrder (partialOrderToSetoidPartialOrder (TotalOrder.order T))
SetoidTotalOrder.totality (totalOrderToSetoidTotalOrder T) = TotalOrder.totality T