patrick/agdaproofs
1
0
Fork 0
mirror of https://github.com/Smaug123/agdaproofs synced 2025-10-12 23:28:39 +00:00
Code Issues Packages Projects Releases Wiki Activity

Split out much more structure (#37)

Browse Source
This commit is contained in:
Patrick Stevens
2019-08-18 14:57:41 +01:00
committed by GitHub
parent 69389bb889
commit 1f26064502
52 changed files with 2137 additions and 2014 deletions
Download Patch File Download Diff File Expand all files Collapse all files

11
Numbers/BinaryNaturals/Multiplication.agda
View File

@@ -3,10 +3,11 @@
open import LogicalFormulae
open import Functions
open import Lists.Lists
open import Numbers.Naturals
open import Groups.GroupDefinition
open import Numbers.Naturals.Naturals
open import Groups.Definition
open import Numbers.BinaryNaturals.Definition
open import Numbers.BinaryNaturals.Addition
open import Semirings.Definition
module Numbers.BinaryNaturals.Multiplication where
@@ -92,7 +93,7 @@ module Numbers.BinaryNaturals.Multiplication where
binNatToNDistributesPlus a b = transitivity (equalityCommutative (binNatToNIsCanonical (a +B b))) (transitivity (applyEquality binNatToN (equalityCommutative (+BIsInherited' a b))) (nToN (binNatToN a +N binNatToN b)))
+BAssociative : (a b c : BinNat) → canonical (a +B (b +B c)) ≡ canonical ((a +B b) +B c)
+BAssociative a b c rewrite equalityCommutative (+BIsInherited' a (b +B c)) | equalityCommutative (+BIsInherited' (a +B b) c) | binNatToNDistributesPlus a b | binNatToNDistributesPlus b c | additionNIsAssociative (binNatToN a) (binNatToN b) (binNatToN c) = refl
+BAssociative a b c rewrite equalityCommutative (+BIsInherited' a (b +B c)) | equalityCommutative (+BIsInherited' (a +B b) c) | binNatToNDistributesPlus a b | binNatToNDistributesPlus b c | equalityCommutative (Semiring.+Associative ℕSemiring (binNatToN a) (binNatToN b) (binNatToN c)) = refl
timesOne : (a b : BinNat) → (canonical b) ≡ (one :: []) → canonical (a *B b) ≡ canonical a
timesOne [] b pr = refl
@@ -173,7 +174,7 @@ module Numbers.BinaryNaturals.Multiplication where
where
ans : (a b : BinNat) → binNatToN a *N binNatToN b ≡ binNatToN (a *B b)
ans [] b = refl
ans (zero :: a) b rewrite equalityCommutative (ans a b) = equalityCommutative (multiplicationNIsAssociative 2 (binNatToN a) (binNatToN b))
ans (one :: a) b rewrite binNatToNDistributesPlus (zero :: (a *B b)) b | additionNIsCommutative (binNatToN b) ((2 *N (binNatToN a)) *N (binNatToN b)) | equalityCommutative (ans a b) = applyEquality (_+N binNatToN b) (equalityCommutative (multiplicationNIsAssociative 2 (binNatToN a) (binNatToN b)))
ans (zero :: a) b rewrite equalityCommutative (ans a b) = equalityCommutative (Semiring.*Associative ℕSemiring 2 (binNatToN a) (binNatToN b))
ans (one :: a) b rewrite binNatToNDistributesPlus (zero :: (a *B b)) b | Semiring.commutative ℕSemiring (binNatToN b) ((2 *N (binNatToN a)) *N (binNatToN b)) | equalityCommutative (ans a b) = applyEquality (_+N binNatToN b) (equalityCommutative (Semiring.*Associative ℕSemiring 2 (binNatToN a) (binNatToN b)))
t : (binNatToN a *N binNatToN b) ≡ binNatToN (canonical (a *B b))
t = transitivity (ans a b) (equalityCommutative (binNatToNIsCanonical (a *B b)))