mirror of
https://github.com/Smaug123/agdaproofs
synced 2025-10-05 20:08:41 +00:00
32 lines
1.6 KiB
Agda
32 lines
1.6 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Setoids.Setoids
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import Groups.Definition
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module Groups.Subgroups.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} (G : Group S _+_) where
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open import Setoids.Subset S
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open Group G
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record Subgroup {c : _} (pred : A → Set c) : Set (a ⊔ b ⊔ c) where
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field
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isSubset : subset pred
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closedUnderPlus : {g h : A} → (pred g) → (pred h) → pred (g + h)
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containsIdentity : pred 0G
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closedUnderInverse : ({g : A} → (pred g) → (pred (inverse g)))
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subgroupOp : {c : _} {pred : A → Set c} → (s : Subgroup pred) → Sg A pred → Sg A pred → Sg A pred
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subgroupOp {pred = pred} record { closedUnderPlus = one } (a , prA) (b , prB) = (a + b) , one prA prB
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subgroupIsGroup : {c : _} {pred : A → Set c} → (s : Subgroup pred) → Group (subsetSetoid (Subgroup.isSubset s)) (subgroupOp s)
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Group.+WellDefined (subgroupIsGroup s) {m , prM} {n , prN} {x , prX} {y , prY} m=x n=y = +WellDefined m=x n=y
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Group.0G (subgroupIsGroup record { containsIdentity = two }) = 0G , two
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Group.inverse (subgroupIsGroup record { closedUnderInverse = three }) (a , prA) = (inverse a) , three prA
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Group.+Associative (subgroupIsGroup s) {a , prA} {b , prB} {c , prC} = +Associative
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Group.identRight (subgroupIsGroup s) {a , prA} = identRight
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Group.identLeft (subgroupIsGroup s) {a , prA} = identLeft
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Group.invLeft (subgroupIsGroup s) {a , prA} = invLeft
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Group.invRight (subgroupIsGroup s) {a , prA} = invRight
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