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Lots of speedups (#116)

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This commit is contained in:
Patrick Stevens
2020-04-16 13:41:51 +01:00
committed by GitHub
parent 1bcb3f8537
commit 9b80058157
63 changed files with 1082 additions and 564 deletions
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19
Setoids/Orders/Orders.agda Normal file
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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Orders.Total.Definition
open import Orders.Partial.Definition
open import Setoids.Setoids
open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Setoids.Orders where
partialOrderToSetoidPartialOrder : {a b : _} {A : Set a} (P : PartialOrder {a} A {b}) → 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} A {b}) → SetoidTotalOrder (partialOrderToSetoidPartialOrder (TotalOrder.order T))
SetoidTotalOrder.totality (totalOrderToSetoidTotalOrder T) = TotalOrder.totality T

25
Setoids/Orders/Partial/Definition.agda Normal file
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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Orders.Total.Definition
open import Orders.Partial.Definition
open import Setoids.Setoids
open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Setoids.Orders.Partial.Definition 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)
_<=_ : Rel {a} {b ⊔ c} A
a <= b = (a < b) || (a ∼ b)
partialOrderToSetoidPartialOrder : {a b : _} {A : Set a} (P : PartialOrder {a} A {b}) → 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

33
Setoids/Orders/Total/Definition.agda Normal file
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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Orders.Total.Definition
open import Orders.Partial.Definition
open import Setoids.Setoids
open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Setoids.Orders.Total.Definition where
open import Setoids.Orders.Partial.Definition
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
totalOrderToSetoidTotalOrder : {a b : _} {A : Set a} (T : TotalOrder {a} A {b}) → SetoidTotalOrder (partialOrderToSetoidPartialOrder (TotalOrder.order T))
SetoidTotalOrder.totality (totalOrderToSetoidTotalOrder T) = TotalOrder.totality T

67
Setoids/Orders/Total/Lemmas.agda Normal file
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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Orders.Total.Definition
open import Orders.Partial.Definition
open import Setoids.Setoids
open import Setoids.Orders.Partial.Definition
open import Setoids.Orders.Total.Definition
open import Functions
open import Sets.EquivalenceRelations
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Setoids.Orders.Total.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {c : _} {_<_ : A → A → Set c} {P : SetoidPartialOrder S _<_} (T : SetoidTotalOrder P) where
open SetoidTotalOrder T
open SetoidPartialOrder P
open Setoid S
open Equivalence eq
maxInequalitiesR : {a b c : A} → (a < b) → (a < c) → (a < max b c)
maxInequalitiesR {a} {b} {c} a<b a<c with totality b c
... | inl (inl x) = a<c
... | inl (inr x) = a<b
... | inr x = a<c
minInequalitiesR : {a b c : A} → (a < b) → (a < c) → (a < min b c)
minInequalitiesR {a} {b} {c} a<b a<c with totality b c
... | inl (inl x) = a<b
... | inl (inr x) = a<c
... | inr x = a<b
maxInequalitiesL : {a b c : A} → (a < c) → (b < c) → (max a b < c)
maxInequalitiesL {a} {b} {c} a<b a<c with totality a b
... | inl (inl x) = a<c
... | inl (inr x) = a<b
... | inr x = a<c
minInequalitiesL : {a b c : A} → (a < c) → (b < c) → (min a b < c)
minInequalitiesL {a} {b} {c} a<b a<c with totality a b
... | inl (inl x) = a<b
... | inl (inr x) = a<c
... | inr x = a<b
minLessL : (a b : A) → min a b <= a
minLessL a b with totality a b
... | inl (inl x) = inr reflexive
... | inl (inr x) = inl x
... | inr x = inr reflexive
minLessR : (a b : A) → min a b <= b
minLessR a b with totality a b
... | inl (inl x) = inl x
... | inl (inr x) = inr reflexive
... | inr x = inr x
maxGreaterL : (a b : A) → a <= max a b
maxGreaterL a b with totality a b
... | inl (inl x) = inl x
... | inl (inr x) = inr reflexive
... | inr x = inr x
maxGreaterR : (a b : A) → b <= max a b
maxGreaterR a b with totality a b
... | inl (inl x) = inr reflexive
... | inl (inr x) = inl x
... | inr x = inr reflexive