Last active
February 23, 2021 03:40
-
-
Save SekiT/bf9f60b2539e5be85f149c7402efdc9a to your computer and use it in GitHub Desktop.
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
import ZZ | |
%default total | |
integralDomainN : Not (m = Z) -> Not (n = Z) -> Not (m * n = Z) | |
integralDomainN mNotZ nNotZ mnEqZ {m = Z} = mNotZ Refl | |
integralDomainN mNotZ nNotZ mnEqZ {n = Z} = nNotZ Refl | |
integralDomainN mNotZ nNotZ mnEqZ {m = S a} {n = S b} = uninhabited mnEqZ | |
infixl 9 |> | |
(|>) : a -> (a -> b) -> b | |
(|>) a f = f a | |
intergalDomainZBaseCase : {a, c, i, j : Nat} -> a * c + (a + S i) * (c + S j) = a * (c + S j) + (a + S i) * c -> Void | |
intergalDomainZBaseCase {a} {c} {i} {j} prf = | |
let | |
rhs1 = a * (c + S j) + (a + S i) * c | |
lhs1 = a * c + ((a + S i) * c + (a * S j + S i * S j)) | |
lhs2 = (a + S i) * c + (a * S j + S i * S j) | |
in | |
prf | |
|> replace {P = \z => a * c + z = rhs1} (multDistributesOverPlusRight (a + S i) c (S j)) | |
|> replace {P = \z => a * c + ((a + S i) * c + z) = rhs1} (multDistributesOverPlusLeft a (S i) (S j)) | |
|> replace {P = \z => lhs1 = z + (a + S i) * c} (multDistributesOverPlusRight a c (S j)) | |
|> replace {P = \z => lhs1 = z} (sym $ plusAssociative (a * c) (a * S j) ((a + S i) * c)) | |
|> plusLeftCancel (a * c) lhs2 (a * S j + (a + S i) * c) | |
|> replace {P = \z => lhs2 = z} (plusCommutative (a * S j) ((a + S i) * c)) | |
|> plusLeftCancel ((a + S i) * c) (a * S j + S i * S j) (a * S j) | |
|> replace {P = \z => a * S j + S i * S j = z} (sym $ plusZeroRightNeutral (a * S j)) | |
|> plusLeftCancel (a * S j) (S i * S j) Z | |
|> uninhabited | |
integralDomainZ : Not (ZZEquiv x 0) -> Not (ZZEquiv y 0) -> Not (ZZEquiv (x * y) 0) | |
integralDomainZ xNeq0 yNeq0 (IsZZEquiv prf) {x = Minus a b} {y = Minus c d} with (diff a b) | |
integralDomainZ xNeq0 yNeq0 (IsZZEquiv prf) {x = Minus b b} {y = Minus c d} | DiffZ = xNeq0 (IsZZEquiv Refl) | |
integralDomainZ xNeq0 yNeq0 (IsZZEquiv prf) {x = Minus a (a + S i)} {y = Minus c d} | (LTByS i) with (diff c d) | |
integralDomainZ xNeq0 yNeq0 (IsZZEquiv prf) {x = Minus a (a + S i)} {y = Minus d d} | (LTByS i) | DiffZ = yNeq0 (IsZZEquiv Refl) | |
integralDomainZ xNeq0 yNeq0 (IsZZEquiv prf) {x = Minus a (a + S i)} {y = Minus c (c + S j)} | (LTByS i) | (LTByS j) = | |
prf | |
|> replace {P = \z => z = a * (c + S j) + (a + S i) * c + 0} (plusZeroRightNeutral (a * c + (a + S i) * (c + S j))) | |
|> replace {P = \z => a * c + (a + S i) * (c + S j) = z} (plusZeroRightNeutral (a * (c + S j) + (a + S i) * c)) | |
|> intergalDomainZBaseCase | |
integralDomainZ xNeq0 yNeq0 (IsZZEquiv prf) {x = Minus a (a + S i)} {y = Minus (d + S j) d} | (LTByS i) | (GTByS j) = | |
prf | |
|> replace {P = \z => z = a * d + (a + S i) * (d + S j) + 0} (plusZeroRightNeutral (a * (d + S j) + (a + S i) * d)) | |
|> replace {P = \z => a * (d + S j) + (a + S i) * d = z} (plusZeroRightNeutral (a * d + (a + S i) * (d + S j))) | |
|> sym | |
|> intergalDomainZBaseCase | |
integralDomainZ xNeq0 yNeq0 (IsZZEquiv prf) {x = Minus (b + S i) b} {y = Minus c d} | (GTByS i) with (diff c d) | |
integralDomainZ xNeq0 yNeq0 (IsZZEquiv prf) {x = Minus (b + S i) b} {y = Minus d d} | (GTByS i) | DiffZ = yNeq0 (IsZZEquiv Refl) | |
integralDomainZ xNeq0 yNeq0 (IsZZEquiv prf) {x = Minus (b + S i) b} {y = Minus c (c + S j)} | (GTByS i) | (LTByS j) = | |
prf | |
|> replace {P = \z => z = (b + S i) * (c + S j) + b * c + 0} (plusZeroRightNeutral ((b + S i) * c + b * (c + S j))) | |
|> replace {P = \z => (b + S i) * c + b * (c + S j) = z} (plusZeroRightNeutral ((b + S i) * (c + S j) + b * c)) | |
|> sym | |
|> replace {P = \z => z = (b + S i) * c + b * (c + S j)} (plusCommutative ((b + S i) * (c + S j)) (b * c)) | |
|> replace {P = \z => b * c + (b + S i) * (c + S j) = z} (plusCommutative ((b + S i) * c) (b * (c + S j))) | |
|> intergalDomainZBaseCase | |
integralDomainZ xNeq0 yNeq0 (IsZZEquiv prf) {x = Minus (b + S i) b} {y = Minus (d + S j) d} | (GTByS i) | (GTByS j) = | |
prf | |
|> replace {P = \z => z = (b + S i) * d + b * (d + S j) + 0} (plusZeroRightNeutral ((b + S i) * (d + S j) + b * d)) | |
|> replace {P = \z => (b + S i) * (d + S j) + b * d = z} (plusZeroRightNeutral ((b + S i) * d + b * (d + S j))) | |
|> replace {P = \z => z = (b + S i) * d + b * (d + S j)} (plusCommutative ((b + S i) * (d + S j)) (b * d)) | |
|> replace {P = \z => b * d + (b + S i) * (d + S j) = z} (plusCommutative ((b + S i) * d) (b * (d + S j))) | |
|> intergalDomainZBaseCase | |
factorZeroN : {m, n : Nat} -> m * n = 0 -> Either (m = 0) (n = 0) | |
factorZeroN prf {m = Z} {n = n} = Left Refl | |
factorZeroN prf {m = S k} {n = Z} = Right Refl | |
factorZeroN prf {m = S k} {n = S j} = absurd (uninhabited prf) | |
factorZeroZ : {m, n : ZZ} -> ZZEquiv (m * n) 0 -> Either (ZZEquiv m 0) (ZZEquiv n 0) | |
factorZeroZ prf {m} {n} = case ((decEquiv m 0, decEquiv n 0)) of | |
(Yes p, _ ) => Left p | |
(No c1, Yes p) => Right p | |
(No c1, No c2) => absurd $ integralDomainZ c1 c2 prf |
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment