You are an expert Lean4 and formal verification agent proving a Lean4 theorem. Your output **must** strictly adhere to the specified format.

### Input Information
You will receive the following components, relevant to the target theorem:
* <used_lib_defs>...</used_lib_defs>: Imported library definitions directly used by the theorem.
* <used_repo_defs>...</used_repo_defs>: Definitions from the local project (but outside the current file), directly or transitively used by the theorem.
* <lib_lemmas>...</lib_lemmas>: Imported library lemmas/theorems that may assist the proof.
* <repo_lemmas>...</repo_lemmas>: Lemmas/theorems defined in the local project outside this file.
* <local_ctx>...</local_ctx>: The local file context (imports, namespaces, definitions, comments, etc.) from the top of the file up to—but not including—the target theorem or any previously defined local lemmas/theorems.
* <target_theorem>...</target_theorem>: The target theorem statement, ending with `:=`.

**Notes**
1. Definitions include their full bodies, but for definitions whose bodies contain proofs, the proof parts are replaced by `admit` (i.e., admit /- proof elided -/), while the rest of the definition is kept unchanged; lemmas include only their statements.
2. You may freely reference any definition or lemma contained in these tags without redefining them.
3. The tags for used definitions and helpful lemmas (used_lib_defs, used_repo_defs, lib_lemmas, repo_lemmas) may be omitted if empty.

### Output Format & Requirements
Your output MUST contain a complete, syntactically correct Lean 4 proof for the target theorem.

**Strict Output Structure:**
<reasoning> 
...
</reasoning> 

<lean4_invented_lemmas>
-- each lemma is a full Lean 4 declaration with a complete proof
...
</lean4_invented_lemmas>

<lean4_proof>
-- Lean 4 proof body of the target theorem
...
</lean4_proof>

**Requirements:**
1. <reasoning> and <lean4_invented_lemmas> are OPTIONAL. Include them only if they aid proof construction.
2. <lean4_proof> MUST contain only the proof body (tactics/terms), starting by "by", NOT the theorem declaration.
3. <lean4_invented_lemmas> (if present) MUST contain syntactically correct lemma declarations that type checks AND complete proof bodies. Defining a lemma using `axiom` is NOT allowed.
4. <reasoning> contains high-level proof planning.
5. If you create a lemma under <lean4_invented_lemmas> whose name is namespaced (e.g., X.Y.Z), you MUST always refer to it using its fully qualified name. Do NOT rely on shortened names. For example, write rw [X.Y.Z], not rw [Z] for invented lemmas.
6. All Lean 4 code MUST be syntactically correct. No `sorry`, `admit`, or non-Lean text. Comments allowed using `--` or `/- ... -/`.
7. Follow the tags and structure EXACTLY. No extra text before, between, or after the sections.
8. Do NOT wrap invented lemmas in a `mutual` block unless they have genuine mutual recursion. Declare each lemma independently.

### Examples are provided here (example input are under "User Prompt", and example assistant response are under "Agent Response"):
Example 1 ------------------- 
User Prompt:
<used_lib_defs>
structure BitVec (w : Nat) where
  /-- Construct a `BitVec w` from a number less than `2^w`.
  O(1), because we use `Fin` as the internal representation of a bitvector. -/
  ofFin ::

  /-- Interpret a bitvector as a number less than `2^w`.
  O(1), because we use `Fin` as the internal representation of a bitvector. -/
  toFin : Fin (hPow 2 w)
</used_lib_defs>

<repo_lemmas>
theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\n(l₁ ++ l₂)[i] = l₁[i]
</repo_lemmas>

<local_ctx>
import Mathlib.Computability.NFA
import Mathlib.Data.FinEnum
import Mathlib.Data.Rel
import Mathlib.Data.Vector.Basic
import Blase.AutoStructs.ForLean

open Set
open Mathlib
open SetRel

-- this is better because card is defeq to n
instance (α : Type) : Inter (Language α) := ⟨Set.inter⟩
instance (α : Type) : Union (Language α) := ⟨Set.union⟩

/--
The set of `n`-tuples of bit vectors of an arbitrary width.
-/
structure BitVecs (n : Nat) where
  w : Nat
  bvs : List.Vector (BitVec w) n

def BitVecs.cast (bvs : BitVecs n) (h : n = n') : BitVecs n' :=
  { w := bvs.w, bvs := h ▸ bvs.bvs }

abbrev BitVecs.empty : BitVecs n := ⟨0, List.Vector.replicate n .nil⟩
abbrev BitVecs.singleton {w : Nat} (bv : BitVec w) : BitVecs 1 := ⟨w, bv ::ᵥ .nil⟩
abbrev BitVecs.pair {w : Nat} (bv1 bv2 : BitVec w) : BitVecs 2 := ⟨w, bv1 ::ᵥ bv2 ::ᵥ .nil⟩

/--
The set of `n`-tuples of bit vectors of an arbitrary width, encoded as a list of
bit vectors of width `n`. The width of the encoded bit vectors is the length of
the list.
-/
abbrev BitVecs' (n : Nat) := List (BitVec n)

@[simps]
def dec (bvs' : BitVecs' n) : BitVecs n where
  w := bvs'.length
  bvs := List.Vector.ofFn fun k => BitVec.ofFn fun i => bvs'[i].getLsbD k

@[simp]
</local_ctx>

<target_theorem>
lemma dec_snoc n (bvs' : BitVecs' n) (a : BitVec n) :
  dec (bvs' ++ [a]) =
    { w := bvs'.length + 1
      bvs := List.Vector.ofFn fun k =>
        BitVec.cons (a.getLsbD k) ((dec bvs').bvs.get k) } :=
</target_theorem>

Agent Response:
<reasoning> 
The goal is to understand what `dec` does after appending one more bitvector `a` to `bvs'`. By definition, `dec` builds a width `= length` and a vector of bitvectors whose `i`-th bit at position `k` is `bvs'[i].getLsbD k`. So for `bvs' ++ [a]`, we expect the width to increase by 1 and each resulting bitvector to be `BitVec.cons` of the new bit coming from `a`.

First, I use `ext` twice: once on the record to focus on the `bvs` field, and once on the resulting bitvector. This reduces everything to a pointwise statement:
for fixed `k` and `i` with `i < bvs'.length + 1`,
the bit `(bvs' ++ [a])[i].getLsbD k` should match the bit at index `i` of `BitVec.cons (a.getLsbD k) ((dec bvs').bvs.get k)`.

At this point I need a clean description of indexing into `BitVec.cons`: if we have `x : BitVec n`, then `(cons b x)[i]` should give `b` at the new “last” index `n` and `x[i]` otherwise. That motivates an auxiliary lemma `getElem_cons` expressing exactly this behavior, proved via the `testBit`/`toNat` characterization and a case split on `i < n`, `i = n`, `i > n`.

Using `getElem_cons`, the RHS becomes an `if` on whether `i = bvs'.length`. So I split into cases:
- If `i = bvs'.length`, then `(bvs' ++ [a])[i]` is the appended `a`, so both sides reduce to `a.getLsbD k`.
- If `i ≠ bvs'.length`, the bound `i < bvs'.length + 1` forces `i < bvs'.length`. Then `(bvs' ++ [a])[i] = bvs'[i]` by `List.getElem_append_left`, and the RHS takes the “else” branch and likewise gives the bit from `bvs'[i]`. In both subcases, `simp` (plus the definition of `dec`) finishes.

So the whole proof is: `ext` to bit level, unfold `dec` with `simp_all`, rewrite with `getElem_cons`, split on the boundary index `i = length bvs'`, and in the “< length” branch use `List.getElem_append_left` and bounds to line up both sides.
</reasoning> 

<lean4_invented_lemmas>
@[grind =]
theorem getElem_cons {b : Bool} {n} {x : BitVec n} {i : Nat} (h : i < n + 1) :
    (cons b x)[i] = if h : i = n then b else x[i] := by
  simp only [BitVec.getElem_eq_testBit_toNat, BitVec.toNat_cons, Nat.testBit_or]
  rw [Nat.testBit_shiftLeft]
  rcases Nat.lt_trichotomy i n with i_lt_n | i_eq_n | n_lt_i
  · have p1 : ¬(n ≤ i) := by omega
    have p2 : i ≠ n := by omega
    simp [p1, p2]
  · simp only [i_eq_n, ge_iff_le, Nat.le_refl, decide_true, Nat.sub_self, Nat.testBit_zero,
    Bool.true_and, BitVec.testBit_toNat, BitVec.getLsbD_of_ge, Bool.or_false]
    cases b <;> trivial
  · have p1 : i ≠ n := by omega
    have p2 : i - n ≠ 0 := by omega
    simp [p1, p2, Nat.testBit_bool_toNat]
</lean4_invented_lemmas>

<lean4_proof>
by
  ext k i hi <;> simp_all
  rw [getElem_cons]
  split
  next heq => simp_all
  next h =>
    have hlt : i < List.length bvs' := by simp at hi; omega
    rw [List.getElem_append_left hlt]
    simp
</lean4_proof>

### Your task:
<used_lib_defs>
Nat in Init.Prelude
Fin in Init.Prelude
Fin.succ in Init.Data.Fin.Basic
</used_lib_defs>

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

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

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

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

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

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

notation:60 "□" T => SType.box T

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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)

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

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

inductive SType.Dealias : Context n m k -> SType n m k -> SType n m k -> Prop where
| refl :
  Dealias Γ S S
| step :
  Context.TBound Γ X (TBinding.inst S) ->
  Dealias Γ S S' ->
  Dealias Γ (SType.tvar X) S'

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

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

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)

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)

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

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)

inductive Context.IsTight : Context n m k -> Prop where
| empty : Context.IsTight Context.empty
| var :
  Context.IsTight Γ ->
  Context.IsTight (Γ.var T)
| tvar :
  Context.IsTight Γ ->
  Context.IsTight (Γ.tvar (TBinding.inst S))
| cvar :
  Context.IsTight Γ ->
  Context.IsTight (Γ.cvar (CBinding.inst C))
| label :
  Context.IsTight Γ ->
  Context.IsTight (Γ.label S)

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

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

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

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

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

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

@[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)

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

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)

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)

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

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)

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

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

def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken

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)

def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=
  b.trename FinFun.weaken

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)

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

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

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)

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)

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

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)

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)

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)

@[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}

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

def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken

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)

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

def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=
  b.crename FinFun.weaken

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

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

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

def Cont.cweaken : Cont n m k -> Cont n m (k+1)
| Cont.none => Cont.none
| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken
| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken
| Cont.scope x cont => Cont.scope x cont.cweaken

def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=
  t.crename FinFun.weaken.ext

def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=
  T.crename FinFun.weaken.ext

def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=
  S.crename FinFun.weaken.ext

def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=
  C.crename FinFun.weaken.ext

def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=
  E.crename FinFun.weaken.ext

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

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)

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)

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)

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)

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

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)

@[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}

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

notation "⊤" => SType.top

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

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

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

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

notation:40 "∃c." T => EType.ex T

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

notation:60 "□" T => SType.box T

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

<lib_lemmas>
refl in Mathlib.Order.Defs.Unbundled
</lib_lemmas>

<repo_lemmas>
theorem Context.tbound_inj
  (h1 : Context.TBound Γ X b1)
  (h2 : Context.TBound Γ X b2) : b1 = b2

theorem Context.tvar_tbound_succ_inv
  (hb : Context.TBound (Γ.tvar p) (Fin.succ X) b) :
  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.tweaken

theorem Context.tvar_tbound_succ_inv'
  (he1 : Γ0 = Γ.tvar p) (he2 : X0 = Fin.succ X)
  (hb : Context.TBound Γ0 X0 b) :
  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.tweaken

theorem Context.tight_bound_tvar_absurd
  (ht : Context.IsTight Γ)
  (hb : Context.TBound Γ X (TBinding.bound S)) : False

theorem Context.cvar_tbound_inv_bound
  (hb : Context.TBound (Γ.cvar p) X (TBinding.bound S)) :
  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = S0.cweaken

theorem Context.cvar_tbound_inv
  (hb : Context.TBound (Γ.cvar p) X b) :
  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.cweaken

theorem Context.cvar_tbound_inv'
  (he : Γ0 = Γ.cvar p)
  (hb : Context.TBound Γ0 X b) :
  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.cweaken

theorem Context.label_tbound_inv_bound
  (hb : Context.TBound (Γ.label l) X (TBinding.bound S)) :
  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = SType.weaken S0

theorem Context.label_tbound_inv
  (hb : Context.TBound (Γ.label l) X b) :
  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken

theorem Context.label_tbound_inv'
  (he : Γ0 = Γ.label l)
  (hb : Context.TBound Γ0 X b) :
  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken

theorem Context.tinst_tbound_bound_inv
  (hb : Context.TBound (Γ.tvar (TBinding.inst P)) X (TBinding.bound S)) :
  ∃ X0 S0, Context.TBound Γ X0 (TBinding.bound S0)
    ∧ S = SType.tweaken S0
    ∧ X = X0.succ

theorem Context.tinst_tbound_bound_inv'
  (he1 : Γ0 = Γ.tvar (TBinding.inst P))
  (he2 : b0 = TBinding.bound S)
  (hb : Context.TBound Γ0 X b0) :
  ∃ X0 S0, Context.TBound Γ X0 (TBinding.bound S0)
    ∧ S = SType.tweaken S0
    ∧ X = X0.succ

theorem Context.var_tbound_inv_bound
  (hb : Context.TBound (Γ.var P) X (TBinding.bound S)) :
  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = SType.weaken S0

theorem Context.var_tbound_inv
  (hb : Context.TBound (Γ.var P) X b) :
  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken

theorem Context.var_tbound_inv'
  (he : Γ0 = Γ.var P)
  (hb : Context.TBound Γ0 X b) :
  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken

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

theorem CSubtyp.refl :
  CSubtyp Γ T T
</repo_lemmas>

<local_ctx>
import Capless.Subtyping

import Capless.Store

import Capless.Inversion.Basic

import Capless.Inversion.Context

import Capless.Subtyping.Basic

import Capless.Narrowing

namespace Capless

def SSubtyp.dealias_right_boxed.emotive
  (Γ : Context n m k)
  (E1 : EType n m k)
  (E2 : EType n m k)
  : Prop := True

def SSubtyp.dealias_right_boxed.cmotive
  (Γ : Context n m k)
  (C1 : CType n m k)
  (C2 : CType n m k)
  : Prop := True

def SSubtyp.dealias_right_boxed.smotive
  (Γ : Context n m k)
  (S1 : SType n m k)
  (S2 : SType n m k)
  : Prop :=
  ∀ {T2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.box T2)),
    ∃ T1, SType.Dealias Γ S1 (SType.box T1)

theorem SSubtyp.dealias_right_boxed
  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)
  (hd : SType.Dealias Γ S2 (SType.box T2)) :
  ∃ T1, SType.Dealias Γ S1 (SType.box T1)

def SSubtyp.dealias_boxed_inv.emotive
  (Γ : Context n m k)
  (E1 : EType n m k)
  (E2 : EType n m k)
  : Prop := True

def SSubtyp.dealias_boxed_inv.cmotive
  (Γ : Context n m k)
  (C1 : CType n m k)
  (C2 : CType n m k)
  : Prop := True

def SSubtyp.dealias_boxed_inv.smotive
  (Γ : Context n m k)
  (S1 : SType n m k)
  (S2 : SType n m k)
  : Prop :=
  ∀ {T1 T2}
    (ht : Γ.IsTight)
    (h1 : SType.Dealias Γ S1 (SType.box T1))
    (h2 : SType.Dealias Γ S2 (SType.box T2)),
    CSubtyp Γ T1 T2
</local_ctx>

<target_theorem>
theorem SSubtyp.sub_dealias_boxed_inv
  (ht : Γ.IsTight)
  (h1 : SType.Dealias Γ S1 (SType.box T1))
  (h2 : SType.Dealias Γ S2 (SType.box T2))
  (hs : SSubtyp Γ S1 S2) :
  CSubtyp Γ T1 T2 :=
</target_theorem>

