{-# OPTIONS --safe --warning=error #-}

open import LogicalFormulae
open import Setoids.Setoids
open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Numbers.Naturals.Naturals
open import Numbers.Integers.Integers
open import Numbers.Rationals
open import Sets.FinSet
open import Groups.Definition
open import Groups.Groups
open import Rings.Definition
open import Numbers.Modulo.IntegersModN
open import Semirings.Definition

module Groups.LectureNotes.Lecture1 where

  ℤIsGroup : _
  ℤIsGroup = ℤGroup

  ℚIsGroup : _
  ℚIsGroup = Ring.additiveGroup ℚRing

  -- TODO: R is a group with +

  integersMinusNotGroup : Group (reflSetoid ℤ) (_-Z_) → False
  integersMinusNotGroup record { +WellDefined = wellDefined ; 0G = identity ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with multAssoc {nonneg 3} {nonneg 2} {nonneg 1}
  integersMinusNotGroup record { +WellDefined = wellDefined ; 0G = identity ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | ()

  negSuccInjective : {a b : ℕ} → (negSucc a ≡ negSucc b) → a ≡ b
  negSuccInjective {a} {.a} refl = refl

  nonnegInjective : {a b : ℕ} → (nonneg a ≡ nonneg b) → a ≡ b
  nonnegInjective {a} {.a} refl = refl

  integersTimesNotGroup : Group (reflSetoid ℤ) (_*Z_) → False
  integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg zero) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with multIdentLeft {negSucc 1}
  ... | ()
  integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with invLeft {nonneg zero}
  ... | bl with inverse (nonneg zero)
  integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | () | nonneg zero
  integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | p | nonneg (succ x) = naughtE (nonnegInjective (transitivity (applyEquality nonneg (equalityCommutative (Semiring.productZeroRight ℕSemiring x))) p))
  integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | () | negSucc x
  integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ (succ x))) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with succInjective (negSuccInjective (multIdentLeft {negSucc 1}))
  ... | ()
  integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (negSucc x) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with multIdentLeft {nonneg 2}
  integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (negSucc x) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | ()

  -- TODO: Q is not a group with *Q
  -- TODO: Q without 0 is a group with *Q
  -- TODO: {1, -1} is a group with *

  ℤnIsGroup : (n : ℕ) → (0<n : 0 <N n) → _
  ℤnIsGroup n pr = ℤnGroup n pr

  -- Groups example 8.9 from lecture 1
  data Weird : Set where
    e : Weird
    a : Weird
    b : Weird
    c : Weird

  _+W_ : Weird → Weird → Weird
  e +W t = t
  a +W e = a
  a +W a = e
  a +W b = c
  a +W c = b
  b +W e = b
  b +W a = c
  b +W b = e
  b +W c = a
  c +W e = c
  c +W a = b
  c +W b = a
  c +W c = e

  +WAssoc : {x y z : Weird} → (x +W (y +W z)) ≡ (x +W y) +W z
  +WAssoc {e} {y} {z} = refl
  +WAssoc {a} {e} {z} = refl
  +WAssoc {a} {a} {e} = refl
  +WAssoc {a} {a} {a} = refl
  +WAssoc {a} {a} {b} = refl
  +WAssoc {a} {a} {c} = refl
  +WAssoc {a} {b} {e} = refl
  +WAssoc {a} {b} {a} = refl
  +WAssoc {a} {b} {b} = refl
  +WAssoc {a} {b} {c} = refl
  +WAssoc {a} {c} {e} = refl
  +WAssoc {a} {c} {a} = refl
  +WAssoc {a} {c} {b} = refl
  +WAssoc {a} {c} {c} = refl
  +WAssoc {b} {e} {z} = refl
  +WAssoc {b} {a} {e} = refl
  +WAssoc {b} {a} {a} = refl
  +WAssoc {b} {a} {b} = refl
  +WAssoc {b} {a} {c} = refl
  +WAssoc {b} {b} {e} = refl
  +WAssoc {b} {b} {a} = refl
  +WAssoc {b} {b} {b} = refl
  +WAssoc {b} {b} {c} = refl
  +WAssoc {b} {c} {e} = refl
  +WAssoc {b} {c} {a} = refl
  +WAssoc {b} {c} {b} = refl
  +WAssoc {b} {c} {c} = refl
  +WAssoc {c} {e} {z} = refl
  +WAssoc {c} {a} {e} = refl
  +WAssoc {c} {a} {a} = refl
  +WAssoc {c} {a} {b} = refl
  +WAssoc {c} {a} {c} = refl
  +WAssoc {c} {b} {e} = refl
  +WAssoc {c} {b} {a} = refl
  +WAssoc {c} {b} {b} = refl
  +WAssoc {c} {b} {c} = refl
  +WAssoc {c} {c} {e} = refl
  +WAssoc {c} {c} {a} = refl
  +WAssoc {c} {c} {b} = refl
  +WAssoc {c} {c} {c} = refl

  weirdGroup : Group (reflSetoid Weird) _+W_
  Group.+WellDefined weirdGroup = reflGroupWellDefined
  Group.0G weirdGroup = e
  Group.inverse weirdGroup t = t
  Group.+Associative weirdGroup {r} {s} {t} = +WAssoc {r} {s} {t}
  Group.identRight weirdGroup {e} = refl
  Group.identRight weirdGroup {a} = refl
  Group.identRight weirdGroup {b} = refl
  Group.identRight weirdGroup {c} = refl
  Group.identLeft weirdGroup {e} = refl
  Group.identLeft weirdGroup {a} = refl
  Group.identLeft weirdGroup {b} = refl
  Group.identLeft weirdGroup {c} = refl
  Group.invLeft weirdGroup {e} = refl
  Group.invLeft weirdGroup {a} = refl
  Group.invLeft weirdGroup {b} = refl
  Group.invLeft weirdGroup {c} = refl
  Group.invRight weirdGroup {e} = refl
  Group.invRight weirdGroup {a} = refl
  Group.invRight weirdGroup {b} = refl
  Group.invRight weirdGroup {c} = refl

  weirdAb : AbelianGroup weirdGroup
  AbelianGroup.commutative weirdAb {e} {e} = refl
  AbelianGroup.commutative weirdAb {e} {a} = refl
  AbelianGroup.commutative weirdAb {e} {b} = refl
  AbelianGroup.commutative weirdAb {e} {c} = refl
  AbelianGroup.commutative weirdAb {a} {e} = refl
  AbelianGroup.commutative weirdAb {a} {a} = refl
  AbelianGroup.commutative weirdAb {a} {b} = refl
  AbelianGroup.commutative weirdAb {a} {c} = refl
  AbelianGroup.commutative weirdAb {b} {e} = refl
  AbelianGroup.commutative weirdAb {b} {a} = refl
  AbelianGroup.commutative weirdAb {b} {b} = refl
  AbelianGroup.commutative weirdAb {b} {c} = refl
  AbelianGroup.commutative weirdAb {c} {e} = refl
  AbelianGroup.commutative weirdAb {c} {a} = refl
  AbelianGroup.commutative weirdAb {c} {b} = refl
  AbelianGroup.commutative weirdAb {c} {c} = refl

  weirdProjection : Weird → FinSet 4
  weirdProjection a = ofNat 0 (le 3 refl)
  weirdProjection b = ofNat 1 (le 2 refl)
  weirdProjection c = ofNat 2 (le 1 refl)
  weirdProjection e = ofNat 3 (le zero refl)

  weirdProjectionSurj : Surjection weirdProjection
  Surjection.property weirdProjectionSurj fzero = a , refl
  Surjection.property weirdProjectionSurj (fsucc fzero) = b , refl
  Surjection.property weirdProjectionSurj (fsucc (fsucc fzero)) = c , refl
  Surjection.property weirdProjectionSurj (fsucc (fsucc (fsucc fzero))) = e , refl
  Surjection.property weirdProjectionSurj (fsucc (fsucc (fsucc (fsucc ()))))

  weirdProjectionInj : (x y : Weird) → weirdProjection x ≡ weirdProjection y → Setoid._∼_ (reflSetoid Weird) x y
  weirdProjectionInj e e fx=fy = refl
  weirdProjectionInj e a ()
  weirdProjectionInj e b ()
  weirdProjectionInj e c ()
  weirdProjectionInj a e ()
  weirdProjectionInj a a fx=fy = refl
  weirdProjectionInj a b ()
  weirdProjectionInj a c ()
  weirdProjectionInj b e ()
  weirdProjectionInj b a ()
  weirdProjectionInj b b fx=fy = refl
  weirdProjectionInj b c ()
  weirdProjectionInj c e ()
  weirdProjectionInj c a ()
  weirdProjectionInj c b ()
  weirdProjectionInj c c fx=fy = refl

  weirdFinite : FiniteGroup weirdGroup (FinSet 4)
  SetoidToSet.project (FiniteGroup.toSet weirdFinite) = weirdProjection
  SetoidToSet.wellDefined (FiniteGroup.toSet weirdFinite) x y = applyEquality weirdProjection
  SetoidToSet.surj (FiniteGroup.toSet weirdFinite) = weirdProjectionSurj
  SetoidToSet.inj (FiniteGroup.toSet weirdFinite) = weirdProjectionInj
  FiniteSet.size (FiniteGroup.finite weirdFinite) = 4
  FiniteSet.mapping (FiniteGroup.finite weirdFinite) = id
  FiniteSet.bij (FiniteGroup.finite weirdFinite) = idIsBijective

  weirdOrder : groupOrder weirdFinite ≡ 4
  weirdOrder = refl

  -- TODO: dihedral groups
  -- TODO: matrix groups on R
  -- TODO: general linear groups on R