\import Algebra.Group
\import Algebra.Group.Fin
\import Algebra.Monoid
\import Algebra.Monoid.Category
\import Algebra.Ring
\import Algebra.Ring.RingHom
\import Arith.Fin
\import Arith.Nat
\import Combinatorics.Factorial
\import Data.Array
\import Data.Fin \hiding (Index)
\import Data.Or
\import Equiv
\import Equiv.Sigma
\import Equiv.Univalence
\import Function
\import Function.Meta
\import HLevel
\import Logic
\import Logic.Meta
\import Meta
\import Order.LinearOrder
\import Order.StrictOrder
\import Paths
\import Paths.Meta
\import Set
\import Set.Fin
\import Set.Fin.Pigeonhole
\open FinLinearOrder (FinLinearOrderInst)

\type Sym (n : Nat) => Equiv {Fin n} {Fin n}
\where {
\func fromEquiv {n : Nat} (e : Equiv {Fin n} {Fin n}) : Sym n => e

\lemma equals {n : Nat} {x y : Sym n} (p : \Pi (j : Fin n) -> x j = y j) : x = y
=> pmap fromEquiv (Equiv.equals (ext p))

\func reduce {n : Nat} (e : Sym (suc n)) : Sym n
=> PigeonholeSet.isEquiv (\lam j => FinSet.skip (e 0) (e (suc j)) \lam p => \case Equiv.isInj p) \lam p => unfsuc $Equiv.isInj (FinSet.skip-isInj p) \lemma reduce_id {n : Nat} : reduce {n} idEquiv = idEquiv => Sym.equals \lam j => cases (n,j) idp \func Support {n : Nat} (e : Sym n) (i : Fin n) : \Prop => e i /= i \func SupportFin {n : Nat} (e : Sym n) : FinSet (\Sigma (i : Fin n) (Support e i)) => SigmaFin (FinFin n) (\lam i => DecFin {Support e i} decide) \lemma Support_inverse {n : Nat} {e : Sym n} {i : Fin n} (s : Support e i) : Support (symQEquiv e) i => \lam p => s$ inv (pmap e p) *> e.f_ret i

\lemma Support_ret {n : Nat} {e : Sym n} {i : Fin n} (s : Support e i) : Support e (e.ret i)
=> \lam p => s $pmap e (inv (e.f_ret i) *> p) *> e.f_ret i \lemma Support_Not {n : Nat} {e : Sym n} {i : Fin n} (s : Not (Support e i)) : e i = i => \case decideEq (e i) i \with { | yes r => r | no q => absurd (s q) } } \instance SymmetricGroup (n : Nat) : Group (Sym n) | ide => idEquiv | * e1 e2 => transEquiv e1 e2 | ide-left => Sym.equals \lam j => idp | ide-right => Sym.equals \lam j => idp | *-assoc => Sym.equals \lam j => idp | inverse e => symQEquiv e | inverse-left => Sym.equals Equiv.f_ret | inverse-right => Sym.equals ret_f \where { \func inv-isEquiv {n : Nat} : QEquiv {Sym n} {Sym n} \cowith | f e => symQEquiv e | ret e => symQEquiv e | ret_f e => Sym.equals (\lam j => idp) | f_sec e => Sym.equals (\lam j => idp) } \open Sym \hiding (equals) \type isCycle {n : Nat} (e : Sym n) : \Prop => ∃ (i : Support e) (∀ (j : Support e) (∃ (k : Nat) (Monoid.pow e k i = j))) \lemma independent_* {n : Nat} {e e' : Sym n} (p : \Pi {i : Fin n} -> Support e i -> Support e' i -> Empty) : e * e' = e' * e => Sym.equals \lam i => unfold \case decideEq (e i) i, decideEq (e' i) i \with { | yes q1, yes q2 => pmap e' q1 *> q2 *> inv (pmap e q2 *> q1) | yes q1, no q2 => \case decideEq (e (e' i)) (e' i) \with { | yes q3 => pmap e' q1 *> inv q3 | no q3 => absurd$ p q3 \lam s => q2 (e'.isInj s)
}
| no q1, yes q2 => \case decideEq (e' (e i)) (e i) \with {
| yes q3 => q3 *> pmap e (inv q2)
| no q3 => absurd $p (\lam s => q1$ e.isInj s) q3
}
| no q1, no q2 => absurd (p q1 q2)
}

\lemma independent_Support-right {n : Nat} {e e' : Sym n} (p : \Pi {i : Fin n} -> Support e i -> Support e' i -> Empty) {i : Fin n} (s : Support e' i) : Support (e * e') i
=> \lam q => \case decideEq (e i) i \with {
| yes r => s $pmap e' (inv r) *> q | no r => p r s } \lemma independent_Support-left {n : Nat} {e e' : Sym n} (p : \Pi {i : Fin n} -> Support e i -> Support e' i -> Empty) {i : Fin n} (s : Support e i) : Support (e * e') i => rewrite (independent_* p)$ independent_Support-right (\lam x y => p y x) s

\lemma independent_Support-conv {n : Nat} {e e' : Sym n} (p : \Pi {i : Fin n} -> Support e i -> Support e' i -> Empty) {i : Fin n} (s : Support (e * e') i) : Support e i || Support e' i
=> \case decideEq (e i) i \with {
| yes r => byRight \lam q => s $pmap e' r *> q | no r => byLeft r } \lemma independent_Support_BigProd {n : Nat} {l : Array (Sym n)} (p : \Pi (i1 i2 : Fin l.len) {i : Fin n} -> Support (l i1) i -> Support (l i2) i -> i1 = i2) {i : Fin n} {j : Fin l.len} (s : Support (l j) i) : Support (Monoid.BigProd l) i \elim l, j | e :: l, 0 => independent_Support-left (\lam s1 s2 => \case conv (\lam i1 i2 s1 s2 => unfsuc$ p (suc i1) (suc i2) s1 s2) s2 \with {
| inP r => \case p 0 (suc r.1) s1 r.2
}) s
| e :: l, suc j => independent_Support-right (\lam s1 s2 => \case conv (\lam i1 i2 s1 s2 => unfsuc $p (suc i1) (suc i2) s1 s2) s2 \with { | inP r => \case p 0 (suc r.1) s1 r.2 })$ independent_Support_BigProd (\lam i1 i2 s1 s2 => unfsuc $p (suc i1) (suc i2) s1 s2) s \where \lemma conv {n : Nat} {l : Array (Sym n)} (p : \Pi (i1 i2 : Fin l.len) {i : Fin n} -> Support (l i1) i -> Support (l i2) i -> i1 = i2) {i : Fin n} (s : Support (Monoid.BigProd l) i) : ∃ (j : Fin l.len) (Support (l j) i) \elim l | nil => absurd (s idp) | a :: l => \case independent_Support-conv (\lam s1 s2 => \case conv (\lam i1 i2 s1 s2 => unfsuc$ p (suc i1) (suc i2) s1 s2) s2 \with {
| inP r => \case p 0 (suc r.1) s1 r.2
}) s \with {
| byLeft s1 => inP (0,s1)
| byRight s2 => \case conv (\lam i1 i2 s1 s2 => unfsuc $p (suc i1) (suc i2) s1 s2) s2 \with { | inP r => inP (suc r.1, r.2) } } \func cycle {n k : Nat} (l : Array (Fin n) k) (inj : isInj l) : Sym n \elim k | 0 => 1 | suc k => PigeonholeSet.isEquiv (func l) (func-inj inj) \where { \func func {n k : Nat} (l : Array (Fin n) (suc k)) (j : Fin n) : Fin n => \case index-dec l j \with { | inl p => l (suc p.1 Nat.mod suc k) | inr _ => j } \lemma func-inj {n k : Nat} {l : Array (Fin n) (suc k)} (inj : isInj l) {i j : Fin n} (q : func l i = func l j) : i = j => unfold func at q$ cases (index-dec l i, index-dec l j, q) \with {
| inl p1, inl p2, q => inv p1.2 *> pmap l (suc-inj $inj q) *> p2.2 | inl _, inr p, q => absurd$ p (_, q)
| inr p, inl _, q => absurd $p (_, inv q) | inr p, inr _, q => q } \lemma mod_/= {k : Nat} {x : Fin (suc k)} (p : x /= {Nat} k) : suc x Nat.mod suc k = {Nat} suc x => mod_<$ NatSemiring.suc<suc $LinearOrder.<=_/= (suc<=suc.conv$ suc_<_<= $fin_< x) p \lemma suc-inj {k : Nat} {x y : Fin (suc k)} (p : suc x Nat.mod suc k = suc y Nat.mod suc k) : x = y => \case decideEq (x : Nat) k, decideEq (y : Nat) k \with { | yes e, yes e' => nat_fin_= (e *> inv e') | yes e, no q => \case (inv (later$ rewrite e id_mod) *> p *> mod_/= q : 0 = suc y)
| no q, yes e => \case (inv (mod_/= q) *> p *> rewrite e id_mod : suc x = {Nat} 0)
| no q, no q' => nat_fin_= $pmap pred$ inv (mod_/= q) *> p *> mod_/= q'
}

\func step {n k : Nat} {l : Array (Fin n) k} (k>0 : k > 0) (inj : isInj l) (i : Fin k) : cycle l inj (l i) = l (mod_Fin (suc i) k>0) \elim k
| suc k => unfold $unfold func$ cases (index-dec l (l i)) \with {
| inl (j,lj=li) => rewrite (inj lj=li) idp
| inr r => absurd $r (i,idp) } \lemma steps {n k : Nat} {m : Fin (suc k)} {l : Array (Fin n) (suc k)} {inj : isInj l} : Monoid.pow {SymmetricGroup n} (PigeonholeSet.isEquiv (func l) (func-inj inj)) m (l 0) = l m \elim k, m | k, 0 => idp | suc k, suc m => unfold$ rewrite steps $step NatSemiring.zero<suc inj m *> pmap l (nat_fin_=$ mod_< $NatSemiring.suc<suc$ fin_< m)
}

\sfunc transposition {n : Nat} {a b : Fin n} (a/=b : a /= b) : Sym n
=> cycle (a :: b :: nil) inj
\where {
\protected \func inj {i j : Fin 2} (p : (a :: b :: nil) i = (a :: b :: nil) j) : i = j \elim i, j
| 0, 0 => idp
| 0, 1 => absurd (a/=b p)
| 1, 0 => absurd $a/=b (inv p) | 1, 1 => idp \lemma transposition-left : transposition a/=b a = b => pmap (__ a) (\peval (transposition a/=b)) *> cycle.step NatSemiring.zero<suc inj 0 \lemma transposition-right : transposition a/=b b = a => pmap (__ b) (\peval (transposition a/=b)) *> cycle.step NatSemiring.zero<suc inj 1 \lemma transposition_/= {i : Fin n} (i/=a : i /= a) (i/=b : i /= b) : transposition a/=b i = i => pmap (__ i) (\peval (transposition a/=b)) *> mcases \with { | inl p => \case \elim p \with { | (0,q) => absurd$ i/=a (inv q)
| (1,q) => absurd $i/=b (inv q) } | inr _ => idp } \lemma transposition-isInv {i : Fin n} : transposition a/=b (transposition a/=b i) = i => \case decideEq i a, decideEq i b \with { | yes p, _ => rewrite p$ pmap (transposition a/=b) transposition-left *> transposition-right
| _, yes p => rewrite p $pmap (transposition a/=b) transposition-right *> transposition-left | no q, no q' => pmap (transposition a/=b) (transposition_/= q q') *> transposition_/= q q' } } \func transposition1 {n : Nat} (a : Fin n) : Sym (suc n) => transposition {suc n} {a} (\lam p => id/=suc p) \where { \open transposition \lemma transposition1-left {n : Nat} {a : Fin n} : transposition1 a a = suc a => transposition-left \lemma transposition1-right {n : Nat} {a : Fin n} : transposition1 a (suc a) = a => transposition-right \lemma transposition1_/= {n : Nat} {a : Fin n} {i : Fin (suc n)} (i/=a : i /= a) (i/=b : i /= suc a) : transposition1 a i = i => transposition_/= i/=a i/=b \lemma transposition1_> {n : Nat} {a : Fin n} {i : Fin (suc n)} (p : (suc a : Nat) < i) : transposition1 a i = i => transposition1_/= (\lam q => NatSemiring.<_/= (id<suc <∘ p) (inv q)) (\lam q => NatSemiring.<_/= p (inv q)) \lemma transposition1_< {n : Nat} {a : Fin n} {i : Fin (suc n)} (p : (i : Nat) < a) : transposition1 a i = i => transposition1_/= (NatSemiring.<_/= p __) (NatSemiring.<_/= (p <∘ id<suc) __) \lemma transposition1-isInv {n : Nat} {a : Fin n} {i : Fin (suc n)} : transposition1 a (transposition1 a i) = i => transposition-isInv } \func Support_cycle {n k : Nat} {l : Array (Fin n) k} (inj : isInj l) {i : Fin n} (s : Support (cycle l inj) i) : \Sigma (j : Fin k) (l j = i) \elim k | 0 => absurd (s idp) | suc k => (unfold Support, unfold, unfold cycle.func) at s$ cases (index-dec l i, s) \with {
| inl r, _ => r
| inr _, s => absurd (s idp)
}

\lemma cycle_Support {n k : Nat} {l : Array (Fin n) k} (k>1 : k > 1) {inj : isInj l} {j : Fin k} : Support (cycle l inj) (l j) \elim k, k>1
| suc k, NatSemiring.suc<suc k>0 => unfold Support $unfold$ unfold cycle.func $cases (index-dec l (l j)) \with { | inl r => \lam q => \have t => inj q *> inv (inj r.2) \in \case decideEq (r.1 : Nat) k \with { | yes e => \case rewriteI (inv id_mod *> inv (pmap (suc __ Nat.mod suc k) e) *> t *> e) k>0 | no s => \case NatSemiring.cancel-left r.1 {0} {1} (inv t *> cycle.mod_/= s) } | inr r => absurd$ r (j,idp)
}

\lemma cycle-isCycle {n k : Nat} {l : Array (Fin n) k} {inj : isInj l} (k>1 : k > 1) : isCycle (cycle l inj) \elim k, k>1
| suc k, NatSemiring.suc<suc k>0 => inP (l 0, unfold Support $unfold$ unfold cycle.func $cases (index-dec l (l 0)) \with { | inl r => \lam q => \case inv (rewrite (inj r.2) in inj q) *> {Nat} mod_< (NatSemiring.suc<suc k>0) | inr r => absurd$ r (0,idp)
}, \lam j s => \have t => Support_cycle inj s \in inP (t.1, cycle.steps *> t.2))

\func symmetric-rec {n : Nat} : QEquiv {Sym (suc n)} {\Sigma (Fin (suc n)) (Sym n)} \cowith
| f e => (e 0, Sym.reduce e)
| ret s => PigeonholeSet.isEquiv {FinFin (suc n)}
(\case __ \with {
| 0 => s.1
| suc j => sface s.1 (s.2 j)
}) (\lam {j} {j'} => \case \elim j, \elim j' \with {
| 0, 0 => \lam _ => idp
| 0, suc j' => \lam p => absurd $sface-skip (inv p) | suc j, 0 => \lam p => absurd$ sface-skip p
| suc j, suc j' => \lam p => pmap fsuc $Equiv.isInj {s.2} (sface-inj p) }) | ret_f e => Sym.equals \case \elim __ \with { | 0 => idp | suc j => FinSet.sface_skip } | f_sec s => pmap (s.1,__)$ Sym.equals \lam j => FinSet.skip_sface

\instance SymFin (n : Nat) : FinSet (Sym n) (fac n)
| finEq => inP fin_equiv
\where {
\func fin_equiv {n : Nat} : Equiv {Fin (fac n)} {Sym n} \elim n
| 0 => \new QEquiv {
| f _ => idEquiv
| ret _ => 0
| ret_f (0) => idp
| f_sec e => Sym.equals (\case __)
}
| suc n => ProdFin.prod_equiv transEquiv sigma-right (\lam _ => fin_equiv) transEquiv symQEquiv symmetric-rec
}

\instance EquivFin (A B : FinSet) : FinSet (Equiv {A} {B})
=> \case decideEq A.finCard B.finCard, A.finEq, B.finEq \with {
| yes e, inP Ae, inP Be => transport FinSet (QEquiv-to-= (later \new QEquiv {
| f e => e
| ret e => e
| ret_f => idpe
| f_sec => idpe
}) *> pmap2 (Equiv {__} {__}) (Equiv-to-= $transport (\lam x => Equiv {Fin x} {A}) e Ae) (Equiv-to-= Be)) (SymFin B.finCard) | no q, _, _ => \new FinSet { | finCard => 0 | finEq => inP$ emptyEquiv (later \case __) \lam e => q \case A.finEq, B.finEq \with {
| inP p1, inP p2 => FinSet.FinCardBij $p1 transEquiv e transEquiv symQEquiv p2 } } } \where { \sfunc equiv_fin {n m : Nat} : \Sigma (k : Nat) (Equiv {Equiv {Fin n} {Fin m}} {Fin k}) => \case decideEq n m \with { | yes p => (fac m, rewrite p$ later \new QEquiv {
| f e => e
| ret e => e
| ret_f => idpe
| f_sec => idpe
} transEquiv symQEquiv SymFin.fin_equiv)
| no q => (0, emptyEquiv (\lam e => q $FinSet.FinCardBij e) (later \case __)) } } \func lift {n : Nat} (e : Sym n) : Sym (suc n) => aux e \where \func aux {n m : Nat} (e : Equiv {Fin n} {Fin m}) : Equiv {Fin (suc n)} {Fin (suc m)} => \new QEquiv { | f => \case __ \with { | 0 => 0 | suc j => suc (e j) } | ret => \case __ \with { | 0 => 0 | suc j => suc (e.ret j) } | ret_f => \case \elim __ \with { | 0 => idp | suc j => pmap fsuc (e.ret_f j) } | f_sec => \case \elim __ \with { | 0 => idp | suc j => pmap fsuc (e.f_ret j) } } \lemma lift_ide {n : Nat} : lift {n} 1 = 1 => Sym.equals (\lam j => cases j idp) \lemma lift_cycle {n k : Nat} {l : Array (Fin n) k} {inj : isInj l} : lift (cycle l inj) = cycle (map fsuc l) (\lam {i} {j} p => inj (unfsuc p)) \elim k | 0 => lift_ide | suc k => Sym.equals \case \elim __ \with { | 0 => unfold$ unfold cycle.func $mcases \with { | inl p => \case p.2 | inr _ => idp } | suc j => unfold$ unfold cycle.func $unfold$ mcases \with {
| inl p, inl q => rewrite (inj $p.2 *> inv (unfsuc q.2)) idp | inl p, inr q => absurd$ q (p.1, pmap fsuc p.2)
| inr p, inl q => absurd $p (q.1, unfsuc q.2) | inr p, inr q => idp } } \lemma lift_transposition {n : Nat} {a b : Fin n} {a/=b : a /= b} : lift (transposition a/=b) = transposition (fsuc/= a/=b) => pmap lift (\peval transposition a/=b) *> lift_cycle {n} {2} *> inv (\peval transposition (fsuc/= a/=b)) \lemma lift_transposition1 {n : Nat} {a : Fin n} : lift (transposition1 a) = transposition1 (suc a) => lift_transposition \lemma lift_* {n : Nat} {e e' : Sym n} : lift (e * e') = lift e * lift e' => Sym.equals \lam j => cases j idp \lemma lift_BigProd {n : Nat} {l : Array (Sym n)} : lift (Monoid.BigProd l) = Monoid.BigProd (map lift l) \elim l | nil => lift_ide | e :: l => lift_* *> pmap (lift e *) lift_BigProd \func transposition1-decomp {n : Nat} (e : Sym (suc n)) : \Sigma (l : Array (Fin n)) (e = BigProd (map transposition1 l)) \elim n | 0 => (nil, Sym.equals \lam (0) => cases (e 0) idp) | suc n => \have r => transposition1-decomp (reduce e) \in (map fsuc r.1 ++ \new Array (Fin (suc n)) (e 0) (\lam j => j Nat.mod suc n), aux NatSemiring.zero<suc *> pmap (* _) {lift (reduce e)} (pmap lift r.2 *> lift_BigProd *> path (\lam i => BigProd (map (\lam a => lift_transposition1 {n} {a} i) r.1))) *> inv (pmap BigProd (map_++ transposition1) *> BigProd_++)) \where { \open Monoid \open NatSemiring(zero<suc,suc<suc) \lemma BigProd-fixed {n : Nat} {l : Array (Sym n)} {i : Fin n} (p : ∀ (e : l) (e i = i)) : BigProd l i = i \elim l | nil => idp | e :: l => pmap (BigProd l) (p 0) *> BigProd-fixed (\lam j => p (suc j)) \lemma BigProd-unique {n : Nat} {l : Array (Sym n)} {x : Fin n} (i : Fin l.len) (p : \Pi (j : Fin l.len) -> (j : Nat) < i -> l j x = x) (q : \Pi (j : Fin l.len) -> (i : Nat) < j -> l j (l i x) = l i x) : BigProd l x = l i x \elim l, i | e :: l, 0 => BigProd-fixed \lam j => q (suc j) zero<suc | e :: l, suc i => pmap (BigProd l) (p 0 zero<suc) *> BigProd-unique i (\lam j s => p (suc j) (suc<suc s)) (\lam j s => q (suc j) (suc<suc s)) \lemma aux {n : Nat} (n>0 : n > 0) {e : Sym (suc n)} : e = lift (reduce e) * BigProd (\new Array (Sym (suc n)) (e 0) (\lam j => transposition1 (mod_Fin j n>0))) => Sym.equals \case \elim __ \with { | 0 => inv (aux1 n>0 e) | suc j => \case LinearOrder.trichotomy (e 0) (e (suc j)) \with { | less s => \have t1 => pmap suc (FinSet.skip-right s) *> suc_pred (/=-sym$ NatSemiring.<_/= $zero<=_ <∘r s) \in inv$ BigProd-fixed (\lam i => transposition1.transposition1_> $transportInv (suc __ < _) (mod_Fin_<$ fin_< i <∘l <_suc_<= (fin_< (e 0))) $suc_<_<= (fin_< i) NatSemiring.<∘r transportInv (_ <) t1 s) *> nat_fin_= t1 | Tri.equals s => \case e.isInj (nat_fin_= s) | greater s => \have | r => pmap (transposition1 __ _) (nat_fin_=$ mod_Fin_< (transportInv (< n) (toFin'=id s) $s NatSemiring.<∘l <_suc_<= (fin_< (e 0))) *> toFin'=id s *> inv (FinSet.skip-left s)) *> transposition1.transposition1-right *> nat_fin_= (FinSet.skip-left s) | t i => inv$ mod_Fin_< $fin_< i <∘l <_suc_<= (fin_< (e 0)) \in inv$ BigProd-unique (toFin' s) (\lam i p => transposition1.transposition1_> $later$ suc<suc $transport2 (<) (t i) (toFin'=id s *> inv (FinSet.skip-left s)) p) (\lam i p => pmap (transposition1 _) r *> transposition1.transposition1_< (transport2 (<) (toFin'=id s) (t i) p) *> inv r) *> r } } \lemma aux1 {n : Nat} (n>0 : n > 0) {k : Fin (suc n)} (e : Sym (suc n)) : BigProd (\new Array (Sym (suc n)) k (\lam j => transposition1 (mod_Fin j n>0))) 0 = k \elim k | 0 => idp | suc k => pmap (__ 0) (BigProd_suc {_} {k} {\lam j => transposition1 (mod_Fin j n>0)}) *> pmap2 (\lam x => transposition1 (mod_Fin x n>0)) (mod_< id<suc) (aux1 n>0 e) *> pmap (transposition1 __ k) mod_Fin=id *> transposition1.transposition1-left } \func sign {R : Ring} {n : Nat} (e : Sym n) : R => R.pow -1 (inversions e) \where { \func inversions {n : Nat} (e : Sym n) : Nat \elim n | 0 => 0 | suc n => e 0 Nat.+ inversions (reduce e) \lemma inversions_ide {n : Nat} : inversions {n} idEquiv = 0 \elim n | 0 => idp | suc n => rewrite reduce_id inversions_ide \open NatSemiring (<, suc<suc, <-irreflexive) \func Inversions {n : Nat} (e : Sym n) => \Sigma (i j : Fin n) (i < j) (e j < e i) \func InversionsFin {n : Nat} (e : Sym n) : FinSet (Inversions e) (inversions e) \cowith | finEq => inP (aux e) \where \func aux {n : Nat} (e : Sym n) : Equiv {Fin (inversions e)} {Inversions e} \elim n | 0 => \new QEquiv { | f => \case __ | ret t => t.1 | ret_f => \case __ | f_sec t => \case t.1 } | suc n => OrFin.aux transEquiv Or.Or_Equiv idEquiv (aux (reduce e)) transEquiv later ( \have lem (a : Fin (e 0)) : e (e.ret (a Nat.mod suc n)) = {Nat} a => e.f_ret _ *> mod_< (fin_< a <∘ fin_< (e 0)) \in \new QEquiv { | f => \case __ \with { | inl a => (0, e.ret (a Nat.mod suc n), nonZero>0 \lam p => <-irreflexive$ rewrite (inv (lem a) *> pmap e (nat_fin_= p)) in fin_< a, rewrite lem (fin_< a))
| inr b => (suc b.1, suc b.2, suc<suc b.3, FinSet.skip_< b.4)
}
| ret => \case __ \with {
| (0, j, j>0, ej<e0) => inl (toFin' ej<e0)
| (suc i, 0, (), _)
| (suc i, suc j, suc<suc i<j, ej+1<ei+1) => inr (i, j, i<j, FinSet.<_skip ej+1<ei+1)
}
| ret_f => \case \elim __ \with {
| inl a => pmap inl $nat_fin_= (toFin'=id _ *> lem a) | inr b => idp } | f_sec => \case \elim __ \with { | (0, j, j>0, ej<e0) => ext (idp, pmap e.ret (nat_fin_=$ later $rewrite (toFin'=id ej<e0)$ mod_< $fin_< (e j)) *> e.ret_f j) | (suc i, 0, (), _) | (suc i, suc j, suc<suc i<j, ej+1<ei+1) => ext (idp, idp) } }) \lemma sign_hom (f : RingHom) {n : Nat} {e : Sym n} : f (sign e) = sign e => f.func-pow *> pmap (Monoid.pow __ _) (f.func-negative *> pmap negative f.func-ide) } \lemma sign_ide {R : Ring} {n : Nat} : sign {R} {n} 1 = 1 => unfold sign$ rewrite sign.inversions_ide idp

\lemma sign_* {R : Ring} {n : Nat} {e e' : Sym n} : sign (e * e') = sign e R.* sign e'
=> R.pow_-1_mod2 *> rewrite inversions_* (inv R.pow_-1_mod2 *> R.pow_+)
\where {
\open sign
\open transposition1
\open NatSemiring (<-irreflexive)

\sfunc Inverstions_transposition_> {n : Nat} {e : Sym (suc n)} {a : Fin n} (ea<ea+1 : e a < e (suc a))
: Equiv {Or (Inversions e) (Fin 1)} {Inversions (transposition1 a * e)}
=> \have | lem1 {i j : Fin (suc n)} (i<j : i < j) (p : i /= a) : transposition1 a i < transposition1 a j
=> \case decideEq i (suc a) \with {
| yes q => rewrite (q, transposition1-right, transposition1_/= (\lam r => <-irreflexive $id<suc <∘ transport2 {Nat} {Nat} (<) q r i<j) \lam r => <-irreflexive$ transport2 {Nat} {Nat} (<) q r i<j) $NatSemiring.<-transitive (rewrite q id<suc) i<j | no q => rewrite (transposition1_/= p q) \case decideEq j a, decideEq j (suc a) \with { | yes s, _ => rewrite (s,transposition1-left)$ (rewrite s in i<j) <∘ id<suc
| _, yes s => rewrite (s,transposition1-right) \case LinearOrder.trichotomy i a \with {
| less i<a => i<a
| Tri.equals i=a => absurd (p i=a)
| greater a<i => absurd $<-irreflexive$ a<i <∘l <_suc_<= (rewrite s in i<j)
}
| no s1, no s2 => rewrite (transposition1_/= s1 s2) i<j
}
}
| lem2 {i j : Fin (suc n)} (i<j : i < j) (p : j /= suc a) : transposition1 a i < transposition1 a j
=> \case decideEq i a \with {
| yes q => rewrite (q, transposition1-left, transposition1_/= (\lam q' => <-irreflexive $transport2 {Nat} {Nat} (<) q q' i<j) p) \case LinearOrder.trichotomy (suc a) j \with { | less s => s | Tri.equals s => absurd$ p (inv s)
| greater s => absurd $<-irreflexive$ (rewrite q in i<j) <∘l <_suc_<= s
}
| no q => lem1 i<j q
}
\in Equiv.fromInjSurj (later \case __ \with {
| inl r => (transposition1 a r.1, transposition1 a r.2, \case decideEq r.1 a, decideEq r.2 (suc a) \with {
| yes e1, yes e2 => absurd $<-irreflexive$ ea<ea+1 <∘ transport2 {Nat} {Nat} (<) (pmap e e2) (pmap e e1) r.4
| no q, _ => lem1 r.3 q
| _, no q => lem2 r.3 q
}, transport2 (e __ < e __) (inv transposition1-isInv) (inv transposition1-isInv) r.4)
| inr _ => (a, suc a, id<suc, transport2 (e __ < e __) (inv transposition1-right) (inv transposition1-left) ea<ea+1)
}) (\lam {x} {y} =>
\have lem3 {i j : Fin (suc n)} (i<j : i < j) (p : transposition1 a i = a) (q : transposition1 a j = suc a) : Empty
=> <-irreflexive $id<suc <∘ transport2 {Nat} {Nat} (<) (inv transposition1-isInv *> pmap (transposition1 a) p *> transposition1-left) (inv transposition1-isInv *> pmap (transposition1 a) q *> transposition1-right) i<j \in \case \elim x, \elim y \with { | inl r, inl r' => \lam s => pmap inl$ ext (Equiv.isInj {transposition1 a} $pmap __.1 s, Equiv.isInj {transposition1 a}$ pmap __.2 s)
| inl r, inr _ => \lam s => absurd $lem3 r.3 (pmap __.1 s) (pmap __.2 s) | inr _, inl r => \lam s => absurd$ lem3 r.3 (inv $pmap __.1 s) (inv$ pmap __.2 s)
| inr (0), inr (0) => \lam _ => idp
}) \lam r => inP \case decideEq (a : Fin (suc n)) r.1, decideEq (suc a) r.2 \with {
| yes e1, yes e2 => (inr 0, ext (e1, e2))
| no q, _ => (inl (transposition1 a r.1, transposition1 a r.2, lem1 r.3 $/=-sym q, r.4), ext (transposition1-isInv, transposition1-isInv)) | _, no q => (inl (transposition1 a r.1, transposition1 a r.2, lem2 r.3$ /=-sym q, r.4), ext (transposition1-isInv, transposition1-isInv))
}

\sfunc Inverstions_transposition_< {n : Nat} {e : Sym (suc n)} {a : Fin n} (ea+1<ea : e (suc a) < e a)
: Equiv {Or (Inversions (transposition1 a * e)) (Fin 1)} {Inversions e}
=> transEquiv (Inverstions_transposition_> {n} {transposition1 a * e} (unfold $rewrite (transposition1-left, transposition1-right) ea+1<ea))$ later
\new QEquiv {
| f s => (s.1, s.2, s.3, transport2 (e __ < e __) transposition1-isInv transposition1-isInv s.4)
| ret s => (s.1, s.2, s.3, transport2 (e __ < e __) (inv transposition1-isInv) (inv transposition1-isInv) s.4)
| ret_f s => ext (idp,idp)
| f_sec s => ext (idp,idp)
}

\lemma inverstions_transposition_> {n : Nat} {e : Sym (suc n)} {a : Fin n} (ea<ea+1 : e a < e (suc a))
: suc (inversions e) = inversions (transposition1 a * e)
=> finCard_Equiv {OrFin (InversionsFin e) (FinFin 1)} {InversionsFin (transposition1 a * e)} (Inverstions_transposition_> ea<ea+1)

\lemma inverstions_transposition_< {n : Nat} {e : Sym (suc n)} {a : Fin n} (ea+1<ea : e (suc a) < e a)
: suc (inversions (transposition1 a * e)) = inversions e
=> finCard_Equiv {OrFin (InversionsFin (transposition1 a * e)) (FinFin 1)} {InversionsFin e} (Inverstions_transposition_< ea+1<ea)

\open Nat(mod)

\lemma inversions_transposition {n : Nat} {e : Sym (suc n)} {a : Fin n} : inversions (transposition1 a * e) mod 2 = suc (inversions e) mod 2
=> \case LinearOrder.trichotomy (e a) (e (suc a)) \with {
| less p => inv $pmap (__ mod 2) (inverstions_transposition_> p) | equals p => absurd$ id/=suc {a} (e.isInj p)
| greater p => inv (n*_+_mod_n=mod {1} {1}) *> pmap (suc __ mod 2) (inverstions_transposition_< p)
}

\lemma inversions_* {n : Nat} {e e' : Sym n} : inversions (e * e') mod 2 = (inversions e + inversions e') mod 2 \elim n
| 0 => idp
| suc n => rewrite (transposition1-decomp e).2 aux
\where {
\lemma aux {n : Nat} {l : Array (Fin n)} {e : Sym (suc n)} : inversions (Monoid.BigProd (map transposition1 l) * e) mod 2 = (inversions (Monoid.BigProd (map transposition1 l)) + inversions e) mod 2 \elim l
| nil => pmap (inversions {suc n} __ mod 2) ide-left *> inv (pmap (\lam x => (x + inversions e) mod 2) (inversions_ide {suc n}))
| a :: l => pmap (inversions {suc n} __ mod 2) *-assoc *> inversions_transposition {n} {Monoid.BigProd (map transposition1 l) * e} *> mod_+-cong-left {1} {1} aux *> inv mod_+-left *> pmap (\lam x => (x + inversions e) mod 2) (inv inversions_transposition) *> mod_+-left
}
}

\lemma sign_inverse {R : Ring} {n : Nat} {e : Sym n} : sign (inverse e) = {R} sign e
=> Monoid.Inv.inv-isUnique (\new Monoid.Inv (sign e) (sign (inverse e)) {
}) (\have t => inv R.pow_+ *> R.pow_-1_even
\in \new Monoid.Inv (sign e) (sign e) t t) idp

\func signHom {R : Ring} {n : Nat} : MonoidHom (SymmetricGroup n) R \cowith
| func => sign
| func-ide => sign_ide
| func-* => sign_*

\sfunc cycle-decomp {n : Nat} (e : Sym n) => aux e id<suc
\where {
\open Monoid(pow,BigProd)

\func aux {k n : Nat} (e : Sym n) (p : finCard {Sym.SupportFin e} < k)
: \Sigma (l : Array (Sym n))
(∀ (e' : l) (isCycle e'))
(\Pi (i1 i2 : Fin l.len) {i : Fin n} -> Support (l i1) i -> Support (l i2) i -> i1 = i2)
(e = BigProd l)
\elim k
| suc k => \case FinSet.searchFin (\lam i => e i /= i) (\lam i => decide) \with {
| inl d =>
\let
| t => Contr.center {search (\lam n => \Sigma (n > 0) (pow e n d.1 = d.1)) (\lam m => \have t => LinearOrder.<-dec 0 m \in decide) (TruncP.map (hasFiniteOrder e) \lam s => (s.1, (s.2, pmap (__ d.1) s.3)))}
| l => \new Array (Fin n) t.1 \lam j => pow e j d.1
\in \have
| t1>1 : t.1 > 1 => cases (t.1 arg addPath) \with {
| 0, q => \case rewrite q in t.2.1
| 1, q => absurd $d.2$ later (rewrite q idp) *> t.2.2
| suc (suc x), q => NatSemiring.suc<suc NatSemiring.zero<suc
}
| inj : isInj l => \lam {i} {j} q => nat_fin_= \case LinearOrder.trichotomy i j \with {
| less i<j => absurd $<-irreflexive$ t.3 (j -' i) (<_-' i<j, pow-lem i<j q) <∘r -'<=id <∘r fin_< j
| equals i=j => i=j
| greater j<i => absurd $<-irreflexive$ t.3 (i -' j) (<_-' j<i, pow-lem j<i (inv q)) <∘r -'<=id <∘r fin_< i
}
\in \let c => cycle l inj
\in \have
| c-supp {i} (s : Support c i) => Support_cycle inj s
| c=e {i : Fin n} (s : Support c i) : c i = e i => inv (pmap c (c-supp s).2) *> cycle.step t.2.1 inj _ *> pmap (pow e __ d.1) mod_Fin=mod *> pow_mod {n} {suc (c-supp s).1} {t.1} {e} {d.1} t.2.2 *> pmap e (c-supp s).2
| r-supp {k : Fin t.1} (s : Support (inverse c * e) (pow e k d.1)) => s $inv (c=e$ Support_ret $cycle_Support t1>1) *> Equiv.f_ret _ \in \let supp-f (x : Or (Fin t.1) (\Sigma (i : Fin n) (Support (inverse c * e) i))) : \Sigma (i : Fin n) (Support e i) => \case \elim x \with { | inl j => (pow e j d.1, \lam p => d.2$ Equiv.isInj (inv pow-rotate *> p))
| inr p => (p.1, \lam q => p.2 $unfold$ pmap e (inv $Equiv.adjoint \case decideEq (c p.1) p.1 \with { | yes r => r | no s => c=e s *> q }) *> q) } \in \have | supp-f-inj : isInj supp-f => \lam {x} {y} => \case \elim x, \elim y \with { | inl j, inl j' => \lam s => pmap inl$ inj (pmap __.1 s)
| inl j, inr p => \lam s => absurd $r-supp$ rewrite (pmap __.1 s) p.2
| inr p, inl j => \lam s => absurd $r-supp$ rewriteI (pmap __.1 s) p.2
| inr p, inr p' => \lam s => pmap inr $ext (pmap __.1 s) } | r => aux (inverse c * e)$ NatSemiring.<_+-left _ t.2.1 <∘l finCard_inj {OrFin (FinFin t.1) (SupportFin (inverse c * e))} {SupportFin e} supp-f supp-f-inj <∘l <_suc_<= p
\in (c :: r.1, \case \elim __ \with {
| 0 => cycle-isCycle t1>1
| suc j => r.2 j
}, \case \elim __, \elim  __ \with {
| 0, 0 => \lam _ _ => idp
| 0, suc j2 => \lam s1 s2 => absurd $r-supp$ rewrite (inv r.4, inv (c-supp s1).2) in independent_Support_BigProd r.3 s2
| suc j1, 0 => \lam s1 s2 => absurd $r-supp$ rewrite (inv r.4, inv (c-supp s2).2) in independent_Support_BigProd r.3 s1
| suc j1, suc j2 => \lam s1 s2 => pmap fsuc (r.3 j1 j2 s1 s2)
}, inv (pmap (* e) inverse-right *> ide-left) *> *-assoc {SymmetricGroup n} {c} {inverse c} *> pmap (c *) r.4)
| inr r => (nil, (\case __), (\case __), Sym.equals \lam j => tightness (r j))
}

\lemma pow-lem {N M : Nat} {n k : Fin M} {e : Sym N} {x : Fin N} (n<k : n < k) (p : pow e n x = pow e k x) : pow e (k -' n) x = x \elim M, n, k, n<k
| suc M, 0, suc k, NatSemiring.zero<suc => inv p
| suc M, suc n, suc k, NatSemiring.suc<suc n<k => pow-lem n<k (e.isInj p)

\lemma pow_mod {n k m : Nat} {e : Sym n} {i : Fin n} (p : pow e m i = i) : pow e (k Nat.mod m) i = pow e k i
=> later (rewrite Monoid.pow_+ $pmap (pow e _)$ later $rewrite Monoid.pow_*$ inv \$ fixPoint-iter p) *> pmap (pow e __ i) (Nat.divModProp k m)

\lemma fixPoint-iter {n k : Nat} {e : Sym n} {i : Fin n} (p : e i = i) : pow e k i = i \elim k
| 0 => idp
| suc k => pmap e (fixPoint-iter p) *> p

\lemma pow-rotate {n : Nat} {e : Sym n} {k : Nat} {i : Fin n} : e (pow e k i) = pow e k (e i) \elim k
| 0 => idp
| suc k => pmap e pow-rotate
}`