-- Capless/Inversion/Typing.lean

import Capless.Tactics

import Capless.Typing

import Capless.Subtyping.Basic

import Capless.Subcapturing.Basic

import Capless.Narrowing

import Capless.Weakening.Subcapturing

import Capless.Inversion.Context

import Capless.Inversion.Subtyping

namespace Capless

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

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

content: notation:50 "λ[X<:" S "]" t => Term.tlam S 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:50 "λ(x:" T ")" t => Term.lam T t

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

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

content: notation:max "⟨" σ " | " cont " | " t "⟩" => State.mk σ cont t

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 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 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: 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 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: 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: 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: inductive CBound : Nat -> Nat -> Type where
| upper : CaptureSet n k -> CBound n k
| star : CBound 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 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 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: 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: inductive CType : Nat -> Nat -> Nat -> Type where
| capt : CaptureSet n k -> SType n m k -> CType n m k

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 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: 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 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 Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where
| here :
  Store.LBound (Store.label σ S) 0 S.weaken
| there_val :
  Store.LBound σ x S ->
  Store.LBound (Store.val σ t hv) x.succ S.weaken
| there_tval :
  Store.LBound σ x S ->
  Store.LBound (Store.tval σ S') x S.tweaken
| there_cval :
  Store.LBound σ x S ->
  Store.LBound (Store.cval σ C) x S.cweaken
| there_label :
  Store.LBound σ x S ->
  Store.LBound (Store.label σ S') x.succ S.weaken

content: def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken

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

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

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

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

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

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

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

content: def Cont.weaken : Cont n m k -> Cont (n+1) m k
| Cont.none => Cont.none
| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken
| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken
| Cont.scope x cont => Cont.scope x.succ cont.weaken

content: inductive Cont : Nat -> Nat -> Nat -> Type where
| none : Cont n m k
| cons :
  (t : Term (n+1) m k) ->
  (cont : Cont n m k) ->
  Cont n m k
| conse :
  (t : Term (n+1) m (k+1)) ->
  (cont : Cont n m k) ->
  Cont n m k
| scope :
  (l : Fin n) ->
  Cont n m k ->
  Cont n m k

content: inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where
| none :
  ESubtyp Γ E E' ->
  TypedCont Γ E Cont.none E' {}
| cons {Ct : CaptureSet n k} :
  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->
  WellScoped Γ cont Ct ->
  TypedCont Γ E cont E' C ->
  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)
| conse {Ct : CaptureSet n k} :
  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->
  WellScoped Γ cont Ct ->
  TypedCont Γ E cont E' C ->
  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)
| scope :
  Context.LBound Γ x S ->
  TypedCont Γ (S^{}) cont E' C ->
  (Γ ⊢ T0 <: S^{}) ->
  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C

content: structure State (n : Nat) (m : Nat) (k : Nat) where
  σ : Store n m k
  cont : Cont n m k
  t : Term n m k

content: @[aesop safe constructors]
inductive Term.IsValue : Term n m k -> Prop where
| lam : Term.IsValue (lam E t)
| tlam : Term.IsValue (tlam S t)
| clam : Term.IsValue (clam B t)
| pack : Term.IsValue (pack c x)

content: def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=
  t.rename FinFun.weaken.ext

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

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 CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=
  b.rename FinFun.weaken

content: def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=
  C.rename FinFun.weaken.ext

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

content: def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=
  E.rename FinFun.weaken.ext

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: 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 ESubtyp.sub_type_inv'
  (he : E2 = EType.type T2)
  (h : ESubtyp Γ E1 E2) :
  ∃ T1, E1 = EType.type T1 ∧ CSubtyp Γ T1 T2

content: theorem Context.bound_lbound_absurd
  (hb1 : Context.Bound Γ x T)
  (hb2 : Context.LBound Γ x S) : False

content: theorem Context.label_lbound_succ_inv
  (hb : Context.LBound (Γ.label l) x.succ S) :
  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken

content: theorem Context.label_lbound_succ_inv'
  (he1 : Γ0 = Γ.label l) (he2 : x0 = x.succ)
  (hb : Context.LBound Γ0 x0 S) :
  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken

content: theorem Context.var_lbound_succ_inv
  (hb : Context.LBound (Γ.var T) x.succ S) :
  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken

content: theorem Context.var_lbound_succ_inv'
  (he1 : Γ0 = Γ.var T) (he2 : x0 = x.succ)
  (hb : Context.LBound Γ0 x0 S) :
  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken

content: theorem CSubtyp.trans
  (h1 : CSubtyp Γ T1 T2)
  (h2 : CSubtyp Γ T2 T3) :
  CSubtyp Γ T1 T3

content: theorem CSubtyp.refl :
  CSubtyp Γ T T

-- Local definitions used:

-- Local lemmas used:

-- Target theorem and proof:
theorem Typed.var_inv'
  (he1 : t0 = Term.var x)
  (he2 : E0 = EType.type T)
  (h : Typed Γ t0 E0 Ct0) (hb : Γ.Bound x T0) :
  ∃ C0 S0, Γ.Bound x (S0^C0) ∧ (Γ ⊢ (S0^{x=x}) <: T)
:= by
  induction h <;> try (solve | cases he1 | cases he2)
  case var C0 S0 hb =>
    cases he1; cases he2
    apply Exists.intro C0
    apply Exists.intro S0
    constructor
    { trivial }
    { apply CSubtyp.refl }
  case label hbl =>
    exfalso
    cases he1
    apply Context.bound_lbound_absurd hb hbl
  case sub hs ih =>
    have h := ESubtyp.sub_type_inv' he2 hs
    have ⟨T1, he, hs1⟩ := h
    have ih := ih he1 he hb
    have ⟨C0, S0, hb, hs0⟩ := ih
    apply Exists.intro C0
    apply Exists.intro S0
    constructor
    { assumption }
    { apply CSubtyp.trans <;> assumption }