patrick/agdaproofs
1
0
Fork 0
mirror of https://github.com/Smaug123/agdaproofs synced 2025-10-20 18:48:39 +00:00
Code Issues Packages Projects Releases Wiki Activity

Lots of speedups (#116)

Browse Source
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
Download Patch File Download Diff File Expand all files Collapse all files

3
Numbers/Integers/Definition.agda
View File

@@ -8,6 +8,9 @@ module Numbers.Integers.Definition where
data ℤ : Set where
nonneg : ℕ → ℤ
negSucc : ℕ → ℤ
{-# BUILTIN INTEGER ℤ #-}
{-# BUILTIN INTEGERPOS nonneg #-}
{-# BUILTIN INTEGERNEGSUC negSucc #-}
nonnegInjective : {a b : ℕ} → nonneg a ≡ nonneg b → a ≡ b
nonnegInjective refl = refl

2
Numbers/Integers/Order.agda
View File

@@ -7,9 +7,9 @@ open import Numbers.Integers.RingStructure.Ring
open import Semirings.Definition
open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Setoids.Orders
open import Orders.Total.Definition
open import Orders.Partial.Definition
open import Setoids.Orders.Total.Definition
module Numbers.Integers.Order where

30
Numbers/Integers/RingStructure/Archimedean.agda Normal file
View File

@@ -0,0 +1,30 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import Semirings.Definition
open import LogicalFormulae
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
open import Numbers.Integers.RingStructure.Ring
open import Groups.Orders.Archimedean
open import Rings.Orders.Partial.Definition
open import Numbers.Integers.Order
module Numbers.Integers.RingStructure.Archimedean where
open import Groups.Cyclic.Definition ℤGroup
open import Semirings.Solver ℕSemiring multiplicationNIsCommutative
private
lemma : (x y : ℕ) → positiveEltPower (nonneg x) y ≡ nonneg (x *N y)
lemma x zero rewrite Semiring.productZeroRight ℕSemiring x = refl
lemma x (succ y) rewrite lemma x y | multiplicationNIsCommutative x (succ y) | multiplicationNIsCommutative y x = equalityCommutative (+Zinherits x (x *N y))
ℤArchimedean : Archimedean (toGroup ℤRing ℤPOrderedRing)
ℤArchimedean (nonneg (succ a)) (nonneg (succ b)) 0<a 0<b = succ (succ b) , t
where
v : b +N (a +N (a +N a *N b)) ≡ a +N (a +N (b +N a *N b))
v rewrite Semiring.+Associative ℕSemiring a a (a *N b) | Semiring.+Associative ℕSemiring b (a +N a) (a *N b) | Semiring.+Associative ℕSemiring a b (a *N b) | Semiring.+Associative ℕSemiring a (a +N b) (a *N b) | Semiring.commutative ℕSemiring b (a +N a) | Semiring.+Associative ℕSemiring a a b = refl
u : succ ((a +N (a +N a *N b)) +N b) ≡ a +N succ (a +N (b +N a *N b))
u = from (succ (plus (plus (const a) (plus (const a) (times (const a) (const b)))) (const b))) to (plus (const a) (succ (plus (const a) (plus (const b) (times (const a) (const b)))))) by applyEquality succ v
t : nonneg (succ b) <Z nonneg (succ a) +Z (nonneg (succ a) +Z positiveEltPower (nonneg (succ a)) b)
t rewrite lemma (succ a) b = lessInherits (succPreservesInequality (le (a +N (a +N a *N b)) u))