patrick/agdaproofs
1
0
Fork 0
mirror of https://github.com/Smaug123/agdaproofs synced 2025-10-13 23:58:38 +00:00
Code Issues Packages Projects Releases Wiki Activity

Lots of without-K (#110)

Browse Source
This commit is contained in:
Patrick Stevens
2020-04-11 12:14:03 +01:00
committed by GitHub
parent 412edaf4c7
commit e9aa1bcc05
30 changed files with 424 additions and 98 deletions
Download Patch File Download Diff File Expand all files Collapse all files

11
Fields/Orders/Lemmas.agda
View File

@@ -120,16 +120,7 @@ abstract
IntegralDomain.intDom orderedFieldIsIntDom = decidedIntDom R orderedFieldIntDom
IntegralDomain.nontrivial orderedFieldIsIntDom pr = Field.nontrivial F (Equivalence.symmetric (Setoid.eq S) pr)
fromNIncreasing : (n : ℕ) → (fromN n) < (fromN (succ n))
fromNIncreasing zero = <WellDefined reflexive (symmetric identRight) (0<1 nontrivial)
fromNIncreasing (succ n) = <WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (fromNIncreasing n) 1R)
fromNPreservesOrder : {a b : ℕ} → (a <N b) → (fromN a) < (fromN b)
fromNPreservesOrder {zero} {succ zero} a<b = fromNIncreasing 0
fromNPreservesOrder {zero} {succ (succ b)} a<b = <Transitive (fromNPreservesOrder (succIsPositive b)) (fromNIncreasing (succ b))
fromNPreservesOrder {succ a} {succ b} a<b = <WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (fromNPreservesOrder (canRemoveSuccFrom<N a<b)) 1R)
charZero : (n : ℕ) → (0R ∼ (fromN (succ n))) → False
charZero n 0=sn = irreflexive (<WellDefined 0=sn reflexive (fromNPreservesOrder (succIsPositive n)))
charZero n 0=sn = irreflexive (<WellDefined 0=sn reflexive (fromNPreservesOrder nontrivial (succIsPositive n)))
charZero' : (n : ℕ) → ((fromN (succ n)) ∼ 0R) → False
charZero' n pr = charZero n (symmetric pr)