-- Capless/Renaming/Type/Typing.lean

import Capless.Typing

import Capless.Renaming.Basic

import Capless.Renaming.Type.Subtyping

namespace Capless

-- Repository definitions used:
content: notation:max "{x=" x "}" => CaptureSet.singleton x

content: notation:max "{c=" c "}" => CaptureSet.csingleton c

content: notation:50 Γ " ⊢ " E1 " <:e " E2 => ESubtyp Γ E1 E2

content: notation:50 Γ " ⊢ " T1 " <:s " T2 => SSubtyp Γ T1 T2

content: notation:50 Γ " ⊢ " T1 " <: " T2 => CSubtyp Γ T1 T2

content: notation:40 Γ " ⊢ " t:80 " : " E " @ " C => Typed Γ t E C

content: notation:50 Γ " ⊢ " C1 " <:c " C2 => Subcapt Γ C1 C2

content: notation:40 "Label[" S "]" => SType.label S

content: notation:60 "□" T => SType.box T

content: macro "easy" : tactic => `(tactic| assumption)

content: macro "apply!" e:term : tactic => `(tactic| apply $e <;> easy)

content: macro "apply?" e:term : tactic => `(tactic| apply $e <;> try easy)

content: inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where
| var :
  Context.Bound Γ x (S^C) ->
  Typed Γ (Term.var x) (S^{x=x}) {x=x}
| label :
  Context.LBound Γ x S ->
  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}
| pack :
  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->
  Typed Γ (Term.pack C x) (∃c.T) {}
| sub :
  Typed Γ t E1 C1 ->
  (Γ ⊢ C1 <:c C2) ->
  (Γ ⊢ E1 <:e E2) ->
  Typed Γ t E2 C2
| abs {C : CaptureSet n k} :
  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->
  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}
| tabs {C : CaptureSet n k} :
  Typed (Γ,X<:S) t E C ->
  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}
| cabs {C : CaptureSet n k} :
  Typed (Γ,c<:B) t E C.cweaken ->
  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}
| app :
  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->
  Typed Γ (Term.var y) T {x=y} ->
  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})
| invoke :
  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->
  Typed Γ (Term.var y) (S^{}) {x=y} ->
  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})
| tapp :
  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->
  Typed Γ (Term.tapp x X) (E.topen X) {x=x}
| capp :
  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->
  Typed Γ (Term.capp x c) (E.copen c) {x=x}
| letin :
  Typed Γ t (EType.type T) C ->
  Typed (Γ,x: T) u E.weaken C.weaken ->  
  Typed Γ (let x=t in u) E C
| letex :
  Typed Γ t (EType.ex T) C ->
  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->
  Typed Γ (let (c,x)=t in u) E C
| bindt :
  Typed (Γ,X:=S) t E.tweaken C ->
  Typed Γ (let X=S in t) E C
| bindc :
  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->
  Typed Γ (let c=C in t) E C0
| boundary {Γ : Context n m k} {S : SType n m k} :
  Typed
    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})
    t
    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->
  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C

content: def SType.trename : SType n m k -> FinFun m m' -> SType n m' k
| SType.top, _ => SType.top
| SType.tvar X, f => SType.tvar (f X)
| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)
| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)
| SType.cforall B E, f => SType.cforall B (E.trename f)
| SType.box T, f => SType.box (T.trename f)
| SType.label S, f => SType.label (S.trename f)

content: def CType.trename : CType n m k -> FinFun m m' -> CType n m' k
| CType.capt C S, f => CType.capt C (S.trename f)

content: inductive CType : Nat -> Nat -> Nat -> Type where
| capt : CaptureSet n k -> SType n m k -> CType n m k

content: def FinFun (n n' : Nat) : Type :=
  Fin n -> Fin n'

content: def EType.trename : EType n m k -> FinFun m m' -> EType n m' k
| EType.ex T, f => EType.ex (T.trename f)
| EType.type T, f => EType.type (T.trename f)

content: inductive EType : Nat -> Nat -> Nat -> Type where
| ex : CType n m (k+1) -> EType n m k
| type : CType n m k -> EType n m k

content: inductive SType : Nat -> Nat -> Nat -> Type where
| top : SType n m k
| tvar : Fin m -> SType n m k
| forall : CType n m k -> EType (n+1) m k -> SType n m k
| tforall : SType n m k -> EType n (m+1) k -> SType n m k
| cforall : CBound n k -> EType n m (k+1) -> SType n m k
| box : CType n m k -> SType n m k
| label : SType n m k -> SType n m k

content: def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :=

content: def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}
  (ρ : TVarMap Γ f Δ) (E : CType n m k) :
  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :=

content: def TVarMap.weaken {Γ : Context n m k} :
  TVarMap Γ FinFun.weaken (Γ.tvar b) :=

content: def TVarMap.weaken_ext {Γ : Context n m k} :
  TVarMap
    (Γ.var T)
    FinFun.weaken
    ((Γ.tvar b).var T.tweaken) :=

content: def TVarMap.weaken_cext_ext {Γ : Context n m k} :
  TVarMap
    ((Γ.cvar (CBinding.bound B)).var T)
    FinFun.weaken
    (((Γ.tvar b).cvar (CBinding.bound B)).var T.tweaken) :=
    
structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where
  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)
  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)
  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)
  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)

content: structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where
  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)
  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)
  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)
  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)

content: structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where
  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)
  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)
  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b
  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)

content: inductive Context : Nat -> Nat -> Nat -> Type where
| empty : Context 0 0 0
| var : Context n m k -> CType n m k -> Context (n+1) m k
| label : Context n m k -> SType n m k -> Context (n+1) m k
| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k
| cvar : Context n m k -> CBinding n k -> Context n m (k+1)

content: inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where
| here : LBound (label Γ0 S) 0 S.weaken
| there_var :
  LBound Γ x S ->
  LBound (var Γ E) x.succ S.weaken
| there_tvar :
  LBound Γ x S ->
  LBound (tvar Γ b) x S.tweaken
| there_cvar :
  LBound Γ x S ->
  LBound (cvar Γ b) x S.cweaken
| there_label :
  LBound Γ x S ->
  LBound (label Γ S') x.succ S.weaken

content: inductive CBound : Nat -> Nat -> Type where
| upper : CaptureSet n k -> CBound n k
| star : CBound n k

content: inductive CaptureSet : Nat -> Nat -> Type where
| empty : CaptureSet n k
| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k
| singleton : Fin n -> CaptureSet n k
| csingleton : Fin k -> CaptureSet n k

content: inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where
| empty : Subset {} C
| rfl : Subset C C
| union_l :
  Subset C1 C ->
  Subset C2 C ->
  Subset (C1 ∪ C2) C
| union_rl :
  Subset C C1 ->
  Subset C (C1 ∪ C2)
| union_rr :
  Subset C C2 ->
  Subset C (C1 ∪ C2)

content: inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where
| trans :
  Subcapt Γ C1 C2 ->
  Subcapt Γ C2 C3 ->
  Subcapt Γ C1 C3
| subset :
  C1 ⊆ C2 ->
  Subcapt Γ C1 C2
| union :
  Subcapt Γ C1 C3 ->
  Subcapt Γ C2 C3 ->
  Subcapt Γ (C1 ∪ C2) C3
| var :
  Context.Bound Γ x (CType.capt C S) ->
  Subcapt Γ {x=x} C
| cinstl :
  Context.CBound Γ c (CBinding.inst C) ->
  Subcapt Γ C {c=c}
| cinstr :
  Context.CBound Γ c (CBinding.inst C) ->
  Subcapt Γ {c=c} C
| cbound :
  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->
  Subcapt Γ {c=c} C

content: def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=
  match b with
  | bound S => bound (S.trename f)
  | inst S => inst (S.trename f)

content: inductive Term : Nat -> Nat -> Nat -> Type where
 
| var : Fin n -> Term n m k
 
| lam : CType n m k -> Term (n+1) m k -> Term n m k
 
| tlam : SType n m k -> Term n (m+1) k -> Term n m k
 
| clam : CBound n k -> Term n m (k+1) -> Term n m k
 
| pack : CaptureSet n k -> Fin n -> Term n m k
 
| app : Fin n -> Fin n -> Term n m k
 
| invoke : Fin n -> Fin n -> Term n m k
 
| tapp : Fin n -> Fin m -> Term n m k
 
| capp : Fin n -> Fin k -> Term n m k
 
| letin : Term n m k -> Term (n+1) m k -> Term n m k
 
| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k
 
| bindt : SType n m k -> Term n (m+1) k -> Term n m k
 
| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k
 
| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k

content: inductive CBinding : Nat -> Nat -> Type where
| bound : CBound n k -> CBinding n k
| inst : CaptureSet n k -> CBinding n k

content: inductive TBinding : Nat -> Nat -> Nat -> Type where
| bound : SType n m k -> TBinding n m k
| inst : SType n m k -> TBinding n m k

content: def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=
  match t with
  | Term.var x => Term.var x
  | Term.lam E t => Term.lam (E.trename f) (t.trename f)
  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)
  | Term.clam B t => Term.clam B (t.trename f)
  | Term.pack c x => Term.pack c x
  | Term.app x y => Term.app x y
  | Term.invoke x y => Term.invoke x y
  | Term.tapp x X => Term.tapp x (f X)
  | Term.capp x c => Term.capp x c
  | Term.letin t u => Term.letin (t.trename f) (u.trename f)
  | Term.letex t u => Term.letex (t.trename f) (u.trename f)
  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)
  | Term.bindc c t => Term.bindc c (t.trename f)
  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)

content: def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}
  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :
  TVarMap (Γ.cvar b) f (Δ.cvar b) :=

content: def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}
  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :
  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :=

content: def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=
  match b with
  | bound S => bound (S.crename f)
  | inst S => inst (S.crename f)

content: def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=
  match b with
  | bound b0 => bound (b0.crename f)
  | inst C => inst (C.crename f)

content: def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=
  match b with
  | upper C => upper (C.crename f)
  | star => star

content: def EType.crename : EType n m k -> FinFun k k' -> EType n m k'
| EType.ex T, f => EType.ex (T.crename f.ext)
| EType.type T, f => EType.type (T.crename f)

content: def CType.crename : CType n m k -> FinFun k k' -> CType n m k'
| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)

content: def SType.crename : SType n m k -> FinFun k k' -> SType n m k'
| SType.top, _ => SType.top
| SType.tvar X, _ => SType.tvar X
| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)
| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)
| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)
| SType.box T, f => SType.box (T.crename f)
| SType.label S, f => SType.label (S.crename f)

content: @[simp]
def CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=
  match C with
  | empty => empty
  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)
  | singleton x => {x=x}
  | csingleton c => {c=f c}

content: inductive SType.IsVar : SType n m k -> Prop where
| tvar : SType.IsVar (SType.tvar X)

content: inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where
| top :
  SSubtyp Γ S SType.top
| refl :
  SSubtyp Γ S S
| trans :
  SSubtyp Γ S1 S2 ->
  SSubtyp Γ S2 S3 ->
  SSubtyp Γ S1 S3
| tvar :
  Context.TBound Γ X (TBinding.bound S) ->
  SSubtyp Γ (SType.tvar X) S
| tinstl :
  Context.TBound Γ X (TBinding.inst S) ->
  SSubtyp Γ S (SType.tvar X)
| tinstr :
  Context.TBound Γ X (TBinding.inst S) ->
  SSubtyp Γ (SType.tvar X) S
| boxed :
  CSubtyp Γ T1 T2 ->
  SSubtyp Γ (□ T1) (□ T2)
| label :
  SSubtyp Γ S2 S1 ->
  SSubtyp Γ (Label[S1]) (Label[S2])
| xforall :
  CSubtyp Γ E2 E1 ->
  ESubtyp (Context.var Γ E2) F1 F2 ->
  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)
| tforall :
  SSubtyp Γ S2 S1 ->
  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->
  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)
| cforall :
  Subbound Γ B2 B1 ->
  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->
  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)

content: def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=
  match t with
  | Term.var x => Term.var x
  | Term.lam E t => Term.lam (E.crename f) (t.crename f)
  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)
  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)
  | Term.pack C x => Term.pack (C.crename f) x
  | Term.app x y => Term.app x y
  | Term.invoke x y => Term.invoke x y
  | Term.tapp x X => Term.tapp x X
  | Term.capp x c => Term.capp x (f c)
  | Term.letin t u => Term.letin (t.crename f) (u.crename f)
  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)
  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)
  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)
  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)

content: def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}
  (ρ : VarMap Γ f Δ) (E : CType n m k) :
  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :=

content: def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=
  match b with
  | bound b0 => bound (b0.rename f)
  | inst C => inst (C.rename f)

content: def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=
  match b with
  | bound S => bound (S.rename f)
  | inst S => inst (S.rename f)

content: def SType.rename : SType n m k -> FinFun n n' -> SType n' m k
| SType.top, _ => SType.top
| SType.tvar X, _ => SType.tvar X
| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)
| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)
| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)
| SType.box T, f => SType.box (T.rename f)
| SType.label S, f => SType.label (S.rename f)

content: def CType.rename : CType n m k -> FinFun n n' -> CType n' m k
| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)

content: def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=
  match b with
  | upper C => upper (C.rename f)
  | star => star

content: def EType.rename : EType n m k -> FinFun n n' -> EType n' m k
| EType.ex T, f => EType.ex (T.rename f)
| EType.type T, f => EType.type (T.rename f)

content: @[simp]
def CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=
  match C with
  | empty => empty
  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)
  | singleton x => {x=f x}
  | csingleton c => {c=c}

content: def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=
  match t with
  | Term.var x => Term.var (f x)
  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)
  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)
  | Term.clam B t => Term.clam (B.rename f) (t.rename f)
  | Term.pack C x => Term.pack (C.rename f) (f x)
  | Term.app x y => Term.app (f x) (f y)
  | Term.invoke x y => Term.invoke (f x) (f y)
  | Term.tapp x X => Term.tapp (f x) X
  | Term.capp x c => Term.capp (f x) c
  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)
  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)
  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)
  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)
  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)

content: def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}
  (ρ : CVarMap Γ f Δ) (E : CType n m k) :
  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :=

content: def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}
  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :
  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :=

content: inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where
| here : Bound (var Γ0 E) 0 E.weaken
| there_var :
  Bound Γ x E ->
  Bound (var Γ E') (Fin.succ x) E.weaken
| there_tvar :
  Bound Γ x E ->
  Bound (tvar Γ b) x E.tweaken
| there_cvar :
  Bound Γ x E ->
  Bound (cvar Γ b) x E.cweaken
| there_label :
  Bound Γ x E ->
  Bound (label Γ S) (Fin.succ x) E.weaken

content: inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where
| here : CBound (cvar Γ0 b) 0 b.cweaken
| there_var :
  CBound Γ x b ->
  CBound (var Γ E) x b.weaken
| there_tvar :
  CBound Γ x b ->
  CBound (tvar Γ b') x b
| there_cvar :
  CBound Γ x b ->
  CBound (cvar Γ b') (Fin.succ x) b.cweaken
| there_label :
  CBound Γ x b ->
  CBound (label Γ S) x b.weaken

content: inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where
| here : TBound (tvar Γ0 b) 0 b.tweaken
| there_var :
  TBound Γ x b ->
  TBound (var Γ E) x b.weaken
| there_tvar :
  TBound Γ x b ->
  TBound (tvar Γ b') (Fin.succ x) b.tweaken
| there_cvar :
  TBound Γ x b ->
  TBound (cvar Γ b') x b.cweaken
| there_label :
  TBound Γ x b ->
  TBound (label Γ S) x b.weaken

content: def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}
  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :
  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :=

content: def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}
  (ρ : VarMap Γ f Δ) (b : CBinding n k) :
  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :=

content: def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}
  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :
  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :=

content: def SType.topen (S : SType n (m+1) k) (X : Fin m) : SType n m k :=
  S.trename (FinFun.open X)

content: def FinFun.open (x : Fin n) : FinFun (n+1) n :=

content: def CType.topen (C : CType n (m+1) k) (X : Fin m) : CType n m k :=
  C.trename (FinFun.open X)

content: def EType.topen (E : EType n (m+1) k) (X : Fin m) : EType n m k :=
  E.trename (FinFun.open X)

content: inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where
| exist :
  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->
  ESubtyp Γ (EType.ex T1) (EType.ex T2)
| type :
  CSubtyp Γ T1 T2 ->
  ESubtyp Γ (EType.type T1) (EType.type T2)

content: inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where
| capt :
  (Γ ⊢ C1 <:c C2) ->
  SSubtyp Γ S1 S2 ->
  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)

content: def SSubtyp.trename_motive1
  (Γ : Context n m k)
  (E1 : EType n m k)
  (E2 : EType n m k)
  : Prop :=
  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),
  ESubtyp Δ (E1.trename f) (E2.trename f)

content: inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where
| set :
  (Γ ⊢ C1 <:c C2) ->
  Subbound Γ (CBound.upper C1) (CBound.upper C2)
| star :
  Subbound Γ B CBound.star

content: def SSubtyp.trename_motive3
  (Γ : Context n m k)
  (S1 : SType n m k)
  (S2 : SType n m k)
  : Prop :=
  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),
  SSubtyp Δ (S1.trename f) (S2.trename f)

content: def SSubtyp.trename_motive2
  (Γ : Context n m k)
  (T1 : CType n m k)
  (T2 : CType n m k)
  : Prop :=
  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),
  CSubtyp Δ (T1.trename f) (T2.trename f)

content: def SType.weaken (S : SType n m k) : SType (n+1) m k :=
  S.rename FinFun.weaken

content: def FinFun.weaken : FinFun n (n+1) :=
  Fin.succ

content: def CType.weaken (C : CType n m k) : CType (n+1) m k :=
  C.rename FinFun.weaken

content: def CBound.weaken (b : CBound n k) : CBound (n+1) k :=
  b.rename FinFun.weaken

content: def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=
  C.rename FinFun.weaken

content: def EType.weaken (E : EType n m k) : EType (n+1) m k :=
  E.rename FinFun.weaken

content: def CType.cweaken (C : CType n m k) : CType n m (k+1) :=
  C.crename FinFun.weaken

content: def EType.cweaken (E : EType n m k) : EType n m (k+1) :=
  E.crename FinFun.weaken

content: def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=
  C.crename FinFun.weaken

content: def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=
  b.crename FinFun.weaken

content: def SType.cweaken (S : SType n m k) : SType n m (k+1) :=
  S.crename FinFun.weaken

content: def CType.copen (C : CType n m (k+1)) (x : Fin k) : CType n m k :=
  C.crename (FinFun.open x)

content: def CaptureSet.copen (C : CaptureSet n (k+1)) (x : Fin k) : CaptureSet n k :=
  C.crename (FinFun.open x)

content: def SType.copen (S : SType n m (k+1)) (x : Fin k) : SType n m k :=
  S.crename (FinFun.open x)

content: def EType.copen (E : EType n m (k+1)) (x : Fin k) : EType n m k :=
  E.crename (FinFun.open x)

content: def SType.tweaken (S : SType n m k) : SType n (m+1) k :=
  S.trename FinFun.weaken

content: def EType.tweaken (E : EType n m k) : EType n (m+1) k :=
  E.trename FinFun.weaken

content: def CType.tweaken (C : CType n m k) : CType n (m+1) k :=
  C.trename FinFun.weaken

content: def CType.open (C : CType (n+1) m k) (x : Fin n) : CType n m k :=
  C.rename (FinFun.open x)

content: def EType.open (E : EType (n+1) m k) (x : Fin n) : EType n m k :=
  E.rename (FinFun.open x)

content: def CaptureSet.open (C : CaptureSet (n+1) k) (x : Fin n) : CaptureSet n k :=
  C.rename (FinFun.open x)

content: def SType.open (S : SType (n+1) m k) (x : Fin n) : SType n m k :=
  S.rename (FinFun.open x)

content: notation:50 "λ(x:" T ")" t => Term.lam T t

content: notation:50 "λ[X<:" S "]" t => Term.tlam S t

content: notation:50 "λ[c<:" B "]" t => Term.clam B t

content: notation:40 "let" "x=" t " in " u => Term.letin t u

content: notation:40 "let" "(c,x)=" t " in " u => Term.letex t u

content: notation:40 "let" "X=" S " in " t => Term.bindt S t

content: notation:40 "let" "c=" C " in " t => Term.bindc C t

content: notation:40 "boundary:" S " in " t => Term.boundary S t

content: notation:max "{x=" x "}" => CaptureSet.singleton x

content: notation:max "{c=" c "}" => CaptureSet.csingleton c

content: notation:50 Γ " ⊢ " E1 " <:e " E2 => ESubtyp Γ E1 E2

content: notation:50 Γ " ⊢ " T1 " <:s " T2 => SSubtyp Γ T1 T2

content: notation:50 Γ " ⊢ " T1 " <: " T2 => CSubtyp Γ T1 T2

content: notation:30 Γ ",x:" T => Context.var Γ T

content: notation:30 Γ ",X<:" T => Context.tvar Γ (TBinding.bound T)

content: notation:30 Γ ",X:=" T => Context.tvar Γ (TBinding.inst T)

content: notation:30 Γ ",c<:" B => Context.cvar Γ (CBinding.bound B)

content: notation:30 Γ ",c<:*" => Context.cvar Γ (CBinding.bound CBound.star)

content: notation:30 Γ ",c:=" C => Context.cvar Γ (CBinding.inst C)

content: notation:40 Γ " ⊢ " t:80 " : " E " @ " C => Typed Γ t E C

content: notation "⊤" => SType.top

content: notation:50 "∀(x:" T ")" U => SType.forall T U

content: notation:50 "∀[X<:" S "]" T => SType.tforall S T

content: notation:50 "∀[c<:" B "]" T => SType.cforall B T

content: notation:max S " ^ " C => CType.capt C S

content: notation:40 "∃c." T => EType.ex T

content: notation:40 "Label[" S "]" => SType.label S

content: notation:60 "□" T => SType.box T

content: notation:50 Γ " ⊢ " C1 " <:c " C2 => Subcapt Γ C1 C2

-- Repository lemmas used:
content: theorem EType.trename_topen {E : EType n (m+1) k} :
  (E.topen X).trename f = (E.trename f.ext).topen (f X)

content: theorem EType.trename_trename (E : EType n m k) (f : FinFun m m') (g : FinFun m' m'') :
  (E.trename f).trename g = E.trename (g ∘ f)

content: theorem CType.trename_trename (T : CType n m k) (f : FinFun m m') (g : FinFun m' m'') :
  (T.trename f).trename g = T.trename (g ∘ f)

content: theorem SType.trename_trename (S : SType n m k) (f : FinFun m m') (g : FinFun m' m'') :
  (S.trename f).trename g = S.trename (g ∘ f)

content: theorem ESubtyp.trename
  (h : ESubtyp Γ E1 E2)
  (ρ : TVarMap Γ f Δ) :
  ESubtyp Δ (E1.trename f) (E2.trename f)

content: theorem CSubtyp.trename
  (h : CSubtyp Γ T1 T2)
  (ρ : TVarMap Γ f Δ) :
  CSubtyp Δ (T1.trename f) (T2.trename f)

content: theorem SSubtyp.trename
  (h : SSubtyp Γ S1 S2)
  (ρ : TVarMap Γ f Δ) :
  SSubtyp Δ (S1.trename f) (S2.trename f)

content: theorem Subbound.trename
  (h : Subbound Γ T1 T2)
  (ρ : TVarMap Γ f Δ) :
  Subbound Δ T1 T2

content: theorem EType.weaken_trename {E : EType n m k} :
  (E.trename f).weaken = E.weaken.trename f

content: theorem CType.weaken_trename {C : CType n m k} :
  (C.trename f).weaken = C.weaken.trename f

content: theorem EType.trename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun m m') :
  (E.trename g).rename f = (E.rename f).trename g

content: theorem CType.trename_rename_comm (T : CType n m k) (f : FinFun n n') (g : FinFun m m') :
  (T.trename g).rename f = (T.rename f).trename g

content: theorem SType.trename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun m m') :
  (S.trename g).rename f = (S.rename f).trename g

content: theorem SType.weaken_trename {S : SType n m k} :
  (S.trename f).weaken = S.weaken.trename f

content: theorem TBinding.weaken_trename {b : TBinding n m k} :
  (b.trename f).weaken = b.weaken.trename f

content: theorem Subcapt.trename
  (h : Subcapt Γ C1 C2)
  (ρ : TVarMap Γ f Δ) :
  Subcapt Δ C1 C2

content: theorem EType.cweaken_trename {E : EType n m k} :
  (E.trename f).cweaken = E.cweaken.trename f

content: theorem EType.crename_trename_comm (E : EType n m k) (f : FinFun k k') (g : FinFun m m') :
  (E.crename f).trename g = (E.trename g).crename f

content: theorem CType.crename_trename_comm (T : CType n m k) (f : FinFun k k') (g : FinFun m m') :
  (T.crename f).trename g = (T.trename g).crename f

content: theorem SType.crename_trename_comm (S : SType n m k) (f : FinFun k k') (g : FinFun m m') :
  (S.crename f).trename g = (S.trename g).crename f

content: theorem EType.trename_copen {E : EType n m (k+1)} :
  (E.copen c).trename f = (E.trename f).copen c

content: theorem SType.trename_copen {S : SType n m (k+1)} :
  (S.copen c).trename f = (S.trename f).copen c

content: theorem CType.trename_copen {C : CType n m (k+1)} :
  (C.copen c).trename f = (C.trename f).copen c

content: theorem EType.tweaken_trename {E : EType n m k} :
  (E.trename f).tweaken = E.tweaken.trename f.ext

content: theorem SType.cweaken_trename {S : SType n m k} :
  (S.trename f).cweaken = S.cweaken.trename f

content: theorem EType.trename_open {E : EType (n+1) m k} :
  (E.open x).trename f = (E.trename f).open x

content: theorem SType.cweaken_copen_id {S : SType n m k} :
  S.cweaken.crename (FinFun.open x) = S

content: theorem CType.cweaken_copen_id {T : CType n m k} :
  T.cweaken.crename (FinFun.open x) = T

content: theorem TBinding.cweaken_copen_id {b : TBinding n m k} :
  b.cweaken.crename (FinFun.open x) = b

content: theorem Term.copen_cweaken_ext {t : Term n m (k+1)} :
  (t.crename (FinFun.weaken.ext)).crename (FinFun.open 0) = t

content: theorem EType.copen_cweaken_ext {E : EType n m (k+1)} :
  (E.crename (FinFun.weaken.ext)).crename (FinFun.open 0) = E

content: theorem CaptureSet.copen_cweaken_ext {C : CaptureSet n (k+1)} :
  (C.crename (FinFun.weaken.ext)).crename (FinFun.open 0) = C

content: theorem Term.open_weaken_ext {t : Term (n+1) m k} :
  (t.rename (FinFun.weaken.ext)).rename (FinFun.open 0) = t

content: theorem EType.open_weaken_ext {E : EType (n+1) m k} :
  (E.rename (FinFun.weaken.ext)).rename (FinFun.open 0) = E

content: theorem CaptureSet.open_weaken_ext {C : CaptureSet (n+1) k} :
  (C.rename (FinFun.weaken.ext)).rename (FinFun.open 0) = C

content: theorem Typed.boundary_body_typing {Γ : Context n m k} {S : SType n m k}
  (ht : Typed ((Γ,c<:*),x:(Label[S.cweaken])^{c=0}) t E Ct) :
  Typed ((Γ.label S),c:={x=0}) t E Ct

content: theorem Typed.narrow
  (h : Typed (Γ,x: T) t E Ct)
  (hs : CSubtyp Γ T' T) :
  Typed (Γ,x: T') t E Ct

content: theorem Typed.tnarrow
  (h : Typed (Γ,X<: S) t E Ct)
  (hs : SSubtyp Γ S' S) :
  Typed (Γ,X<: S') t E Ct

content: theorem Typed.cnarrow
  (h : Typed (Γ,c<:B) t E Ct)
  (hs : Subbound Γ B' B) :
  Typed (Γ,c<:B') t E Ct

content: theorem TypedCont.narrow
  (h : TypedCont Γ E1 cont E C0)
  (hsub : ESubtyp Γ E2 E1) :
  TypedCont Γ E2 cont E C0

-- Local definitions used:

-- Local lemmas used:

-- Target theorem and proof:
theorem Typed.trename
  {Γ : Context n m k} {Δ : Context n m' k}
  (h : Typed Γ t E Ct)
  (ρ : TVarMap Γ f Δ) :
  Typed Δ (t.trename f) (E.trename f) Ct
:= by
  induction h generalizing m'
  case var =>
    simp [Term.trename, EType.trename, CType.trename]
    apply var
    rename_i hb
    have hb1 := ρ.map _ _ hb
    simp [CType.trename] at hb1
    trivial
  case pack ih =>
    simp [Term.trename, EType.trename]
    apply pack
    have ih := ih (ρ.cext _)
    simp [Term.trename, EType.trename] at ih
    trivial
  case sub hsc hs ih =>
    apply sub
    aesop
    apply! hsc.trename
    apply! ESubtyp.trename
  case abs ih =>
    simp [Term.trename, EType.trename, CType.trename, SType.trename]
    apply abs
    apply? ih
    apply! TVarMap.ext
  case app ih1 ih2 =>
    simp [Term.trename]
    rw [EType.trename_open]
    apply app
    have ih1 := ih1 ρ
    simp [EType.trename, CType.trename, SType.trename, Term.trename] at ih1
    trivial
    have ih2 := ih2 ρ
    simp [Term.trename, EType.trename] at ih2
    trivial
  case tabs ih =>
    simp [Term.trename, EType.trename, CType.trename, SType.trename]
    apply tabs
    apply? ih
    apply! TVarMap.text
  case cabs ih =>
    simp [Term.trename, EType.trename, CType.trename, SType.trename]
    apply cabs
    have ih1 := ih (ρ.cext _)
    trivial
  case tapp ih =>
    simp [Term.trename]
    rw [EType.trename_topen]
    apply tapp
    have ih := ih ρ
    simp [Term.trename, EType.trename, CType.trename, SType.trename] at ih
    trivial
  case capp ih =>
    simp [Term.trename]
    rw [EType.trename_copen]
    apply capp
    have ih := ih ρ
    simp [Term.trename, EType.trename, CType.trename, SType.trename] at ih
    trivial
  case letin ih1 ih2 =>
    simp [Term.trename]
    apply letin
    simp [EType.trename] at ih1
    aesop
    have ih2 := ih2 (ρ.ext _)
    rw [<- EType.weaken_trename] at ih2
    trivial
  case letex ih1 ih2 =>
    simp [Term.trename]
    apply letex
    simp [EType.trename] at ih1
    aesop
    have ih2 := ih2 ((ρ.cext _).ext _)
    rw [<- EType.weaken_trename] at ih2
    rw [<- EType.cweaken_trename] at ih2
    trivial
  case bindt ih =>
    simp [Term.trename]
    apply bindt
    have ih := ih (ρ.text _)
    rw [EType.tweaken_trename]
    trivial
  case bindc ih =>
    simp [Term.trename]
    apply bindc
    have ih := ih (ρ.cext _)
    rw [EType.cweaken_trename]
    trivial
  case label =>
    simp [Term.trename, EType.trename, CType.trename, SType.trename]
    apply label
    have h := ρ.lmap
    aesop
  case invoke ih1 ih2 =>
    simp [Term.trename]
    apply invoke
    simp [Term.trename, EType.trename, CType.trename, SType.trename] at ih1
    apply ih1; trivial
    simp [Term.trename, EType.trename, CType.trename, SType.trename] at ih2
    apply ih2; trivial
  case boundary ih =>
    simp [Term.trename, EType.trename, CType.trename]
    apply boundary
    have ih := ih ((ρ.cext _).ext _)
    simp [FinFun.ext, CType.trename, SType.trename] at ih
    rw [ SType.cweaken_trename
       , SType.weaken_trename ]
    simp [EType.trename, CType.trename] at ih
    exact ih