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Polynomial ring (#78)
This commit is contained in:
Patrick Stevens
GitHub
@@ -2,6 +2,7 @@
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open import LogicalFormulae
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open import Groups.Groups
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open import Groups.Abelian.Definition
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open import Groups.Homomorphisms.Definition
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open import Groups.Definition
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open import Numbers.Naturals.Definition
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@@ -25,37 +26,88 @@ open Group G
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_+P_ : NaivePoly → NaivePoly → NaivePoly
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_+P_ = listZip _+_ id id
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PidentRight : {m : NaivePoly} → polysEqual (m +P []) m
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PidentRight {[]} = record {}
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PidentRight {x :: m} = reflexive ,, identityOfIndiscernablesLeft polysEqual (Equivalence.reflexive (Setoid.eq naivePolySetoid)) (equalityCommutative (mapId m))
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inverse' : NaivePoly → NaivePoly
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inverse' = map (Group.inverse G)
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+PwellDefined : {m n x y : NaivePoly} → polysEqual m x → polysEqual n y → polysEqual (m +P n) (x +P y)
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+PwellDefined {[]} {[]} {[]} {[]} m=x n=y = record {}
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+PwellDefined {[]} {[]} {[]} {x :: y} m=x (fst ,, snd) = fst ,, identityOfIndiscernablesRight polysEqual snd (equalityCommutative (mapId y))
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+PwellDefined {[]} {[]} {x :: xs} {[]} (fst ,, snd) n=y = fst ,, identityOfIndiscernablesRight polysEqual snd (equalityCommutative (mapId xs))
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+PwellDefined {[]} {[]} {x :: xs} {y :: ys} (fst ,, snd) (fst2 ,, snd2) = transitive (+WellDefined fst fst2) identRight ,, +PwellDefined {[]} {[]} {xs} {ys} snd snd2
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+PwellDefined {[]} {n :: ns} {[]} {[]} m=x (fst ,, snd) = fst ,, identityOfIndiscernablesLeft polysEqual snd (equalityCommutative (mapId ns))
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+PwellDefined {[]} {n :: ns} {[]} {y :: ys} m=x (fst ,, snd) = fst ,, +PwellDefined m=x snd
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+PwellDefined {[]} {n :: ns} {x :: xs} {[]} (fst ,, snd) (fst2 ,, snd2) = transitive fst2 (symmetric fst) ,, ans
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where
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ans : polysEqual (map (λ z → z) ns) (map (λ z → z) xs)
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ans rewrite mapId ns | mapId xs = Equivalence.transitive (Setoid.eq naivePolySetoid) snd2 snd
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+PwellDefined {[]} {n :: ns} {x :: xs} {y :: ys} (fst ,, snd) (fst2 ,, snd2) = transitive (symmetric identLeft) (+WellDefined (symmetric fst) fst2) ,, (+PwellDefined snd snd2)
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+PwellDefined {m :: ms} {[]} {[]} {[]} (fst ,, snd) _ = fst ,, identityOfIndiscernablesLeft polysEqual snd (equalityCommutative (mapId ms))
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+PwellDefined {m :: ms} {[]} {[]} {x :: ys} (fst ,, snd) (fst2 ,, snd2) = transitive fst (symmetric fst2) ,, ans
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where
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ans : polysEqual (map (λ z → z) ms) (map (λ z → z) ys)
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ans rewrite mapId ms | mapId ys = Equivalence.transitive (Setoid.eq naivePolySetoid) snd snd2
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+PwellDefined {m :: ms} {[]} {x :: xs} {[]} (fst ,, snd) n=y = fst ,, ans
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where
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ans : polysEqual (map (λ z → z) ms) (map (λ z → z) xs)
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ans rewrite mapId ms | mapId xs = snd
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+PwellDefined {m :: ms} {[]} {x :: xs} {y :: ys} (fst ,, snd) (fst2 ,, snd2) = transitive (symmetric identRight) (+WellDefined fst (symmetric fst2)) ,, identityOfIndiscernablesLeft polysEqual (Equivalence.transitive (Setoid.eq naivePolySetoid) (Equivalence.symmetric (Setoid.eq naivePolySetoid) PidentRight) (+PwellDefined snd snd2)) (equalityCommutative (mapId ms))
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+PwellDefined {m :: ms} {n :: ns} {[]} {[]} (fst ,, snd) (fst2 ,, snd2) = transitive (+WellDefined fst fst2) identLeft ,, +PwellDefined snd snd2
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+PwellDefined {m :: ms} {n :: ns} {[]} {y :: ys} (fst ,, snd) (fst2 ,, snd2) = transitive (+WellDefined fst fst2) identLeft ,, +PwellDefined snd snd2
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+PwellDefined {m :: ms} {n :: ns} {x :: xs} {[]} (fst ,, snd) (fst2 ,, snd2) = transitive (+WellDefined fst fst2) identRight ,, identityOfIndiscernablesRight polysEqual (Equivalence.transitive (Setoid.eq naivePolySetoid) (+PwellDefined snd snd2) PidentRight) (equalityCommutative (mapId xs))
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+PwellDefined {m :: ms} {n :: ns} {x :: xs} {y :: ys} (fst ,, snd) (fst2 ,, snd2) = +WellDefined fst fst2 ,, +PwellDefined snd snd2
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abstract
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PidentRight : {m : NaivePoly} → polysEqual (m +P []) m
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PidentRight {[]} = record {}
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PidentRight {x :: m} = reflexive ,, identityOfIndiscernablesLeft polysEqual (Equivalence.reflexive (Setoid.eq naivePolySetoid)) (equalityCommutative (mapId m))
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PidentLeft : {m : NaivePoly} → polysEqual ([] +P m) m
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PidentLeft {[]} = record {}
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PidentLeft {x :: m} = reflexive ,, identityOfIndiscernablesLeft polysEqual (Equivalence.reflexive (Setoid.eq naivePolySetoid)) (equalityCommutative (mapId m))
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+PwellDefined : {m n x y : NaivePoly} → polysEqual m x → polysEqual n y → polysEqual (m +P n) (x +P y)
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+PwellDefined {[]} {[]} {[]} {[]} m=x n=y = record {}
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+PwellDefined {[]} {[]} {[]} {x :: y} m=x (fst ,, snd) = fst ,, identityOfIndiscernablesRight polysEqual snd (equalityCommutative (mapId y))
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+PwellDefined {[]} {[]} {x :: xs} {[]} (fst ,, snd) n=y = fst ,, identityOfIndiscernablesRight polysEqual snd (equalityCommutative (mapId xs))
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+PwellDefined {[]} {[]} {x :: xs} {y :: ys} (fst ,, snd) (fst2 ,, snd2) = transitive (+WellDefined fst fst2) identRight ,, +PwellDefined {[]} {[]} {xs} {ys} snd snd2
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+PwellDefined {[]} {n :: ns} {[]} {[]} m=x (fst ,, snd) = fst ,, identityOfIndiscernablesLeft polysEqual snd (equalityCommutative (mapId ns))
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+PwellDefined {[]} {n :: ns} {[]} {y :: ys} m=x (fst ,, snd) = fst ,, +PwellDefined m=x snd
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+PwellDefined {[]} {n :: ns} {x :: xs} {[]} (fst ,, snd) (fst2 ,, snd2) = transitive fst2 (symmetric fst) ,, ans
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where
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ans : polysEqual (map (λ z → z) ns) (map (λ z → z) xs)
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ans rewrite mapId ns | mapId xs = Equivalence.transitive (Setoid.eq naivePolySetoid) snd2 snd
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+PwellDefined {[]} {n :: ns} {x :: xs} {y :: ys} (fst ,, snd) (fst2 ,, snd2) = transitive (symmetric identLeft) (+WellDefined (symmetric fst) fst2) ,, (+PwellDefined snd snd2)
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+PwellDefined {m :: ms} {[]} {[]} {[]} (fst ,, snd) _ = fst ,, identityOfIndiscernablesLeft polysEqual snd (equalityCommutative (mapId ms))
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+PwellDefined {m :: ms} {[]} {[]} {x :: ys} (fst ,, snd) (fst2 ,, snd2) = transitive fst (symmetric fst2) ,, ans
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where
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ans : polysEqual (map (λ z → z) ms) (map (λ z → z) ys)
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ans rewrite mapId ms | mapId ys = Equivalence.transitive (Setoid.eq naivePolySetoid) snd snd2
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+PwellDefined {m :: ms} {[]} {x :: xs} {[]} (fst ,, snd) n=y = fst ,, ans
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where
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ans : polysEqual (map (λ z → z) ms) (map (λ z → z) xs)
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ans rewrite mapId ms | mapId xs = snd
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+PwellDefined {m :: ms} {[]} {x :: xs} {y :: ys} (fst ,, snd) (fst2 ,, snd2) = transitive (symmetric identRight) (+WellDefined fst (symmetric fst2)) ,, identityOfIndiscernablesLeft polysEqual (Equivalence.transitive (Setoid.eq naivePolySetoid) (Equivalence.symmetric (Setoid.eq naivePolySetoid) PidentRight) (+PwellDefined snd snd2)) (equalityCommutative (mapId ms))
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+PwellDefined {m :: ms} {n :: ns} {[]} {[]} (fst ,, snd) (fst2 ,, snd2) = transitive (+WellDefined fst fst2) identLeft ,, +PwellDefined snd snd2
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+PwellDefined {m :: ms} {n :: ns} {[]} {y :: ys} (fst ,, snd) (fst2 ,, snd2) = transitive (+WellDefined fst fst2) identLeft ,, +PwellDefined snd snd2
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+PwellDefined {m :: ms} {n :: ns} {x :: xs} {[]} (fst ,, snd) (fst2 ,, snd2) = transitive (+WellDefined fst fst2) identRight ,, identityOfIndiscernablesRight polysEqual (Equivalence.transitive (Setoid.eq naivePolySetoid) (+PwellDefined snd snd2) PidentRight) (equalityCommutative (mapId xs))
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+PwellDefined {m :: ms} {n :: ns} {x :: xs} {y :: ys} (fst ,, snd) (fst2 ,, snd2) = +WellDefined fst fst2 ,, +PwellDefined snd snd2
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PidentLeft : {m : NaivePoly} → polysEqual ([] +P m) m
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PidentLeft {[]} = record {}
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PidentLeft {x :: m} = reflexive ,, identityOfIndiscernablesLeft polysEqual (Equivalence.reflexive (Setoid.eq naivePolySetoid)) (equalityCommutative (mapId m))
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invLeft' : {a : NaivePoly} → polysEqual ((inverse' a) +P a) []
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invLeft' {[]} = record {}
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invLeft' {x :: a} = Group.invLeft G ,, invLeft' {a}
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invRight' : {a : NaivePoly} → polysEqual (a +P (inverse' a)) []
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invRight' {[]} = record {}
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invRight' {x :: a} = Group.invRight G ,, invRight' {a}
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assoc : {a b c : NaivePoly} → polysEqual (a +P (b +P c)) ((a +P b) +P c)
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assoc {[]} {[]} {[]} = record {}
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assoc {[]} {[]} {x :: c} = reflexive ,, ans
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where
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ans : polysEqual (map (λ z → z) (map (λ z → z) c)) (map (λ z → z) c)
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ans rewrite mapId c | mapId c = Equivalence.reflexive (Setoid.eq naivePolySetoid)
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assoc {[]} {b :: bs} {[]} = reflexive ,, ans
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where
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ans : polysEqual (map (λ z → z) (map (λ z → z) bs)) (map (λ z → z) (map (λ z → z) bs))
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ans rewrite mapId bs | mapId bs = Equivalence.reflexive (Setoid.eq naivePolySetoid)
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assoc {[]} {b :: bs} {c :: cs} = reflexive ,, ans
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where
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ans : polysEqual (map (λ z → z) (listZip _+_ (λ z → z) (λ z → z) bs cs)) (listZip _+_ (λ z → z) (λ z → z) (map (λ z → z) bs) cs)
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ans rewrite mapId bs | mapId (listZip _+_ id id bs cs) = Equivalence.reflexive (Setoid.eq naivePolySetoid)
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assoc {a :: as} {[]} {[]} = reflexive ,, ans
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where
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ans : polysEqual (map (λ z → z) as) (map (λ z → z) (map (λ z → z) as))
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ans rewrite mapId as | mapId as = Equivalence.reflexive (Setoid.eq naivePolySetoid)
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assoc {a :: as} {[]} {c :: cs} = reflexive ,, ans
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where
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ans : polysEqual (listZip _+_ (λ z → z) (λ z → z) as (map (λ z → z) cs)) (listZip _+_ (λ z → z) (λ z → z) (map (λ z → z) as) cs)
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ans rewrite mapId cs | mapId as = Equivalence.reflexive (Setoid.eq naivePolySetoid)
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assoc {a :: as} {b :: bs} {[]} = reflexive ,, ans
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where
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ans : polysEqual (listZip _+_ (λ z → z) (λ z → z) as (map (λ z → z) bs)) (map (λ z → z) (listZip _+_ (λ z → z) (λ z → z) as bs))
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ans rewrite mapId (listZip _+_ id id as bs) | mapId bs = Equivalence.reflexive (Setoid.eq naivePolySetoid)
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assoc {a :: as} {b :: bs} {c :: cs} = Group.+Associative G ,, assoc {as} {bs} {cs}
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comm : AbelianGroup G → {x y : NaivePoly} → polysEqual (x +P y) (y +P x)
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comm com {[]} {y} = Equivalence.transitive (Setoid.eq naivePolySetoid) PidentLeft (Equivalence.symmetric (Setoid.eq naivePolySetoid) PidentRight)
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comm com {x :: xs} {[]} = reflexive ,, Equivalence.reflexive (Setoid.eq naivePolySetoid)
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comm com {x :: xs} {y :: ys} = AbelianGroup.commutative com ,, comm com {xs} {ys}
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mapDist : (f : A → A) (fDist : {x y : A} → f (x + y) ∼ (f x) + (f y)) (xs ys : NaivePoly) → polysEqual (map f (xs +P ys)) ((map f xs) +P (map f ys))
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mapDist f fDist [] [] = record {}
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mapDist f fDist [] (x :: ys) rewrite mapId ys | mapId (map f ys) = reflexive ,, Equivalence.reflexive (Setoid.eq naivePolySetoid)
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mapDist f fDist (x :: xs) [] rewrite mapId xs | mapId (map f xs) = reflexive ,, Equivalence.reflexive (Setoid.eq naivePolySetoid)
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mapDist f fDist (x :: xs) (y :: ys) = fDist {x} {y} ,, mapDist f fDist xs ys
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@@ -23,59 +23,15 @@ open import Groups.Polynomials.Addition G public
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open Setoid S
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open Equivalence eq
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inverse : NaivePoly → NaivePoly
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inverse = map (Group.inverse G)
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invLeft : {a : NaivePoly} → polysEqual ((inverse a) +P a) []
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invLeft {[]} = record {}
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invLeft {x :: a} = Group.invLeft G ,, invLeft {a}
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invRight : {a : NaivePoly} → polysEqual (a +P (inverse a)) []
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invRight {[]} = record {}
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invRight {x :: a} = Group.invRight G ,, invRight {a}
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assoc : {a b c : NaivePoly} → polysEqual (a +P (b +P c)) ((a +P b) +P c)
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assoc {[]} {[]} {[]} = record {}
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assoc {[]} {[]} {x :: c} = reflexive ,, ans
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where
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ans : polysEqual (map (λ z → z) (map (λ z → z) c)) (map (λ z → z) c)
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ans rewrite mapId c | mapId c = Equivalence.reflexive (Setoid.eq naivePolySetoid)
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assoc {[]} {b :: bs} {[]} = reflexive ,, ans
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where
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ans : polysEqual (map (λ z → z) (map (λ z → z) bs)) (map (λ z → z) (map (λ z → z) bs))
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ans rewrite mapId bs | mapId bs = Equivalence.reflexive (Setoid.eq naivePolySetoid)
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assoc {[]} {b :: bs} {c :: cs} = reflexive ,, ans
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where
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ans : polysEqual (map (λ z → z) (listZip _+_ (λ z → z) (λ z → z) bs cs)) (listZip _+_ (λ z → z) (λ z → z) (map (λ z → z) bs) cs)
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ans rewrite mapId bs | mapId (listZip _+_ id id bs cs) = Equivalence.reflexive (Setoid.eq naivePolySetoid)
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assoc {a :: as} {[]} {[]} = reflexive ,, ans
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where
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ans : polysEqual (map (λ z → z) as) (map (λ z → z) (map (λ z → z) as))
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ans rewrite mapId as | mapId as = Equivalence.reflexive (Setoid.eq naivePolySetoid)
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assoc {a :: as} {[]} {c :: cs} = reflexive ,, ans
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where
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ans : polysEqual (listZip _+_ (λ z → z) (λ z → z) as (map (λ z → z) cs)) (listZip _+_ (λ z → z) (λ z → z) (map (λ z → z) as) cs)
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ans rewrite mapId cs | mapId as = Equivalence.reflexive (Setoid.eq naivePolySetoid)
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assoc {a :: as} {b :: bs} {[]} = reflexive ,, ans
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where
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ans : polysEqual (listZip _+_ (λ z → z) (λ z → z) as (map (λ z → z) bs)) (map (λ z → z) (listZip _+_ (λ z → z) (λ z → z) as bs))
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ans rewrite mapId (listZip _+_ id id as bs) | mapId bs = Equivalence.reflexive (Setoid.eq naivePolySetoid)
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assoc {a :: as} {b :: bs} {c :: cs} = Group.+Associative G ,, assoc {as} {bs} {cs}
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polyGroup : Group naivePolySetoid _+P_
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Group.+WellDefined polyGroup = +PwellDefined
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Group.0G polyGroup = []
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Group.inverse polyGroup = inverse
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Group.inverse polyGroup = inverse'
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Group.+Associative polyGroup {a} {b} {c} = assoc {a} {b} {c}
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Group.identRight polyGroup = PidentRight
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Group.identLeft polyGroup = PidentLeft
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Group.invLeft polyGroup {a} = invLeft {a}
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Group.invRight polyGroup {a} = invRight {a}
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comm : AbelianGroup G → {x y : NaivePoly} → polysEqual (x +P y) (y +P x)
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comm com {[]} {y} = Equivalence.transitive (Setoid.eq naivePolySetoid) (Group.identLeft polyGroup) (Equivalence.symmetric (Setoid.eq naivePolySetoid) (Group.identRight polyGroup))
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comm com {x :: xs} {[]} = reflexive ,, Equivalence.reflexive (Setoid.eq naivePolySetoid)
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comm com {x :: xs} {y :: ys} = AbelianGroup.commutative com ,, comm com {xs} {ys}
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Group.invLeft polyGroup {a} = invLeft' {a}
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Group.invRight polyGroup {a} = invRight' {a}
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abelian : AbelianGroup G → AbelianGroup polyGroup
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AbelianGroup.commutative (abelian ab) {x} {y} = comm ab {x} {y}
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