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:
* <all_available_defs>: Definitions available from imported modules (either from the current Lean repository or from Mathlib)
* <all_available_lemmas>: Lemmas/theorems available that may assist the proof.
* <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 available definitions and lemmas (all_available_defs, all_available_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:
<all_available_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)
</all_available_defs>

<all_available_lemmas>
theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) : (l₁ ++ l₂)[i] = l₁[i]
theorem getElem_eq_testBit_toNat (x : BitVec w) (i : Nat) (h : i < w) :
  x[i] = x.toNat.testBit i := rfl
@[simp, grind =] theorem toNat_cons (b : Bool) (x : BitVec w) :
    (cons b x).toNat = (b.toNat <<< w) ||| x.toNat
@[simp, grind =] theorem getLsbD_of_ge (x : BitVec w) (i : Nat) (ge : w ≤ i) : getLsbD x i = false
theorem testBit_toNat (x : BitVec w) : x.toNat.testBit i = x.getLsbD i
</all_available_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:
<all_available_defs>
def hProp := heap -> Prop

abbrev himpl (H1 H2 : hProp) : Prop :=
  forall h, H1 h -> H2 h

def qimpl {A} (Q1 Q2 : A → hProp) : Prop :=
  forall (v:A), Q1 v ==> Q2 v

def hempty : hProp :=
  fun h => (h = ∅)

def hexists {A} (J : A → hProp) : hProp :=
  fun h => exists x, J x h

class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where
   
  hStar : α → β → γ

macro "∃ʰ" xs:Lean.explicitBinders ", " b:term : term => Lean.expandExplicitBinders ``hexists xs b

macro "h∀" xs:Lean.explicitBinders ", " b:term : term => Lean.expandExplicitBinders ``hforall xs b

def hpure (P : Prop) : hProp :=
  hexists (fun (_ : P) => emp)

class HWand (α : Type u) (β : Type v) (γ : outParam (Type w)) where
   
  hWand : α → β → γ

abbrev loc := Nat

abbrev var := String

abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)

inductive prim where
  
  | val_get : prim
  | val_set : prim
  
  | val_neg : prim
  | val_opp : prim
  | val_eq : prim
  | val_add : prim
  | val_neq : prim
  | val_sub : prim
  | val_mul : prim
  | val_div : prim
  | val_mod : prim
  
  | val_le : prim
  | val_lt : prim
  | val_ge : prim
  | val_gt : prim
  | val_ptr_add : prim

inductive val : Type where
    | val_unit   : val
    | val_bool   : Bool → val
    | val_int    : Int → val
    | val_real   : ℝ → val
    | val_loc    : loc → val
    | val_prim   : prim → val
    | val_fun    : var -> trm -> val
    | val_fix    : var -> var -> trm -> val
    | val_uninit : val
    | val_error : val

inductive trm : Type where
    | trm_val   : val -> trm
    | trm_var   : var -> trm
    | trm_fun   : var -> trm -> trm
    | trm_fix   : var -> var -> trm -> trm
    | trm_app   : trm -> trm -> trm
    | trm_seq   : trm -> trm -> trm
    | trm_let   : var -> trm -> trm -> trm
    | trm_if    : trm -> trm -> trm -> trm
    | trm_for   : var -> trm -> trm -> trm -> trm
    | trm_while : trm -> trm -> trm
    | trm_ref   : var → trm → trm → trm
    | trm_alloc : var → trm → trm → trm

abbrev heap := Heap.heap val

abbrev RevExtState := List Ident

Inhabited in Init.Prelude

Lean.binderIdent in Init.Notation

inductive steps : state → trm → state → trm → Prop :=
  | steps_refl : forall s t,
      steps s t s t
  | steps_step : forall s1 s2 s3 t1 t2 t3,
      step s1 t1 s2 t2 →
      steps s2 t2 s3 t3 →
      steps s1 t1 s3 t3

OfNat.ofNat in Init.Prelude

def hstar (H1 H2 : hProp) : hProp :=
  fun h => exists h1 h2,
    H1 h1 ∧ H2 h2 ∧ Finmap.Disjoint h1 h2 ∧ h = h1 ∪ h2

Lean.pp.funBinderTypes in Lean.PrettyPrinter.Delaborator.Options

def default_set : val := [lang|
  fun p i v =>
      let i := val_abs i in
      let L := val_array_length p in
      let b := i < L in
      if b then
        val_array_set p i v
      else
        () ]

def getTripleFun (g : Expr) : MetaM Name := do
  let_expr triple t _ _ := g | throwError "goal is not a triple"
  let_expr trm_app t _ := t | throwError "triple term is not an application"
  let .some n := (getNestedApp t).constName? | throwError "nested application is not a constant"
  return n

def qstar {A} (Q : A → hProp) (H : hProp) : A → hProp :=
  fun x => hstar (Q x) H

def notstuck (s : state) (t : trm) : Prop :=
  trm_is_val t \/ reducible s t

Option.any in Init.Data.Option.Basic

def xchangeApply (l : Expr) : MVarId -> MetaM (List MVarId) := fun goal => do
  let g :: gs <- goal.applyConst `xchange_lemma | throwError "Expected a heap entailment"
  let gs' <- g.apply $ <- toHimp l
  return gs ++ gs'

String in Init.Prelude

@[simp]
def haddE : (· + ·) = (· ∗ ·) :=

AList.keys in Mathlib.Data.List.AList

fresh in Ssreflect.Utils

@[app_unexpander val_fun] def unexpandVFun : Lean.PrettyPrinter.Unexpander
  | `($(_) $x:str [lang| fun $xs* => $t]) =>
    let str := x.getString
    let name := Lean.mkIdent $ Lean.Name.mkSimple str
    `([lang| fun $name $xs* => $t])
  | `($(_) $x:str [lang| $t]) =>
    let str := x.getString
    let name := Lean.mkIdent $ Lean.Name.mkSimple str
    `([lang| fun $name => $t])
  | _ => throw ( )

t in Ssreflect.IntroPats

scase in Ssreflect.Elim

instance : Inhabited val where
  default := val.val_unit

partial def xsimp_step_r (xsimp : XSimpR) : TacticM Unit := do
  trace[xsimp] "RHS step"
  trace[xsimp] "hrt: {xsimp.hrt}"
  match xsimp.hlw, xsimp.hlt, xsimp.hrt with
  | `(hempty), `(hempty), `(@HStar.hStar hProp hProp $_ $_ $h $_) =>
     
    try
      trace[xsimp] "xsimp_r deals with: {h}"
      match h with
      | `(hempty) => {| apply xsimp_r_hempty |}
      | `(hpure $_) => {| apply xsimp_r_hpure |}
      | `(@HStar.hStar hProp hProp $_ $_ $h1 $h2) =>
        {| erw [@hstar_assoc $h1 $h2];
           simpNums |} 
                       
          
        xsimp_step_r (<- XSimpRIni)
      | `(@HWand.hWand hProp hProp $_ $_ $h1 $_) =>
        match h1 with
        | `(hpure $_) => {| apply xsimp_r_keep |}
        | _ => {| apply xsimp_r_hwand_same |}
      | `(htop) =>
        match xsimp.hrg with
        | `(hempty) =>
          {| apply xsimpl_r_hgc_or_htop |}
          repeat'' do
            xsimp_pick_same h xsimp.hla
            {| apply xsimp_lr_cancel_same |}
        | `(@HStar.hStar hProp hProp $_ $_ htop hempty) => {| apply xsimpl_r_htop_drop |}
        | _ => throwError "xsimp_step_r: @ unreachable"
      | `(@hexists $_ fun ($x:ident : $_) => $_) => xsimp_r_hexists_apply_hints x
      | `(protect $_) => {| apply xsimp_r_keep |}
      | `(hsingle $x $_) =>
        
        if x.isMVarStx then
          {| apply xsimp_r_keep |}
        else
          xsimp_pick_same_pointer x xsimp.hla
          
          
          {| apply xsimpl_lr_cancel_same_hsingle <;>
             xsimp_hsingle_discharge |}
      | _ =>
        if h.isMVarStx then
          {| apply xsimp_r_keep |}
        else
        xsimp_pick_same h xsimp.hla
        {| apply xsimp_lr_cancel_same |}
    catch ex =>
      trace[xsimp] ex.toMessageData
      {| apply xsimp_r_keep |}
  | _, _, _ => throwError "not implemented"

local instance : Coe val trm where
  coe v := trm.trm_val v

| evalbinop_ptr_add : forall (p1 : loc) (p2 : Int) n,
      p2 = p1 + n ->
      evalbinop val_ptr_add (val_loc p1) (val_int n)
        (fun v => v = val_loc (Int.natAbs p2))

def getPrim (t :Expr) : Expr :=
  let_expr val_prim v := t | t
  v

def htop : hProp :=
  hexists (fun (H : hProp) => H)

instance (priority := low) : Zero hProp := ⟨emp⟩

Lean.bracketedExplicitBinders in Init.NotationExtra

Set in Mathlib.Data.Set.Defs

List in Init.Prelude

semiOutParam in Init.Prelude

private def castToProp (e: Lean.Expr) : Lean.Elab.Tactic.TacticM (Option Q(Prop)) := do
  Qq.checkTypeQ (u := Lean.levelOne) e q(Prop)

@[app_unexpander hexists] def unexpandHExists : Lean.PrettyPrinter.Unexpander
  | `($(_) fun $x:ident => ∃ʰ $xs:binderIdent*, $b) => `(∃ʰ $x:ident $xs:binderIdent*, $b)
  | `($(_) fun $x:ident => $b)                     => `(∃ʰ $x:ident, $b)
  | `($(_) fun ($x:ident : $t) => $b)              => `(∃ʰ ($x:ident : $t), $b)
  | t                                              => pure t

Int in Init.Data.Int.Basic

Finmap.erase in Mathlib.Data.Finmap

def conseq {B : Type} (vs : List B) (l : Nat) : Finmap (fun _ : Nat ↦ B) :=
  match vs with
  | [] => ∅
  | v :: vs' => (Finmap.singleton l v) ∪ (conseq vs' (l + 1))

Lean.pp.universes in Lean.PrettyPrinter.Delaborator.Options

List.length in Init.Prelude

List.foldl in Init.Prelude

def hseg (L : List val) (p : loc) (j : Int) : hProp :=
  match L with
  | []      => emp
  | x :: L' => (hcell x p j) ∗ (hseg L' p (j + 1))

scoped instance : PartialCommMonoid val where
  add := add
  add_assoc := by admit /- proof elided -/

def wpgen (t : trm) : formula :=
  mkstruct (match t with
  | trm_val v          => wpgen_val v
  | trm_fun x t1       => wpgen_fun (fun v ↦ wp $ subst x v t1)
  | trm_fix f x t1     => wpgen_fix
      (fun vf v => wp $ subst x v $ subst f vf t1)
  | trm_if t0 t1 t2    => wpgen_if t0 (wp t1) (wp t2)
  | trm_seq t1 t2      => wpgen_seq (wp t1) (wp t2)
  | trm_let x t1 t2    => wpgen_let (wp t1) (fun v ↦ wp $ subst x v t2)
  | trm_app _ _        => wpgen_app t
  
  
  | trm_ref x t1 t2    => wpgen_ref x t1 t2
  | trm_alloc x t1 t2  => wpgen_alloc x t1 t2
  | _ => wp t
  )

allGoal in Ssreflect.Utils

def null : loc := 0

def delabPpAll :=
  (withOptions (fun _ => KVMap.empty.insert `pp.all true) $ PrettyPrinter.delab ·)

partial def dup (n : Nat) : TacticM Unit := do
  match n with
  | 0 => {|skip|}
  | _ => dup (n-1); {| apply dup_lemma|}

AddCommMonoid in Mathlib.Algebra.Group.Defs

instance (priority := high) : AddCommMonoidWRT hProp hadd where
  addE := by admit /- proof elided -/

Finset in Mathlib.Data.Finset.Defs

def val_fixs (f:var) (xs:List var) (t:trm) : val :=
  match xs with
  | .nil => val_uninit
  | x1::xs' => val_fix f x1 (trm_funs xs' t)

List.keys in Mathlib.Data.List.Sigma

abbrev default_get' := default_get (val_int 0)

def hforall {A} (J : A → hProp) : hProp :=
  fun h => forall x, J x h

def val_array_make : val := [lang|
  fun n v =>
    let m := n + 1 in
    alloc m as p in
    (val_set p) n ;
    (((val_array_fill p) 0) n) v ;
    p ]

scoped instance : PartialCommMonoid val where
  add := add
  add_assoc := by admit /- proof elided -/

def val_array_fill : val := [lang|
  fix f p i n v =>
    let b := n > 0 in
    if b then
      ((val_array_set p) i) v ;
      let m := n - 1 in
      let j := i + 1 in
      f p j m v
    end ]

def xsimp_pick (i : Nat) (last? : Bool) : TacticM Unit :=
   {| apply xsimp_pick_lemma
      · apply $(hstar_pick_lemma i last?) |}

def hstars_simp_post :=
  {| hsimp ; try apply himpl_refl |}

Lean.levelOne in Lean.Level

def xsimp_pick_applied (h hla : Term) : TacticM Unit := do
  let h <- Term.elabTerm h  none
  hstar_search hla fun i h' last? => do
    let h' <- Term.elabTerm h' none
    let numArgs + 1 := h'.getAppNumArgs' | throwError "not equal"
    let h' := h'.consumeMData.getAppPrefix numArgs
    let .true <-
      withAssignableSyntheticOpaque <| isDefEq h h' | throwError "not equal"
    xsimp_pick i last?

local macro "LabType" : term => `(ℕ)

inductive eval : state → trm → (val → state → Prop) -> Prop where
  | eval_val : forall s v Q,
      Q v s ->
      eval s (trm_val v) Q
  | eval_fun : forall s x t1 Q,
      Q (val_fun x t1) s ->
      eval s (trm_fun x t1) Q
  | eval_fix : forall s f x t1 Q,
      Q (val_fix f x t1) s ->
      eval s (trm_fix f x t1) Q
  | eval_app_arg1 : forall s1 t1 t2 Q1 Q,
      ¬ trm_is_val t1 ->
      eval s1 t1 Q1 ->
      (forall v1 s2, Q1 v1 s2 -> eval s2 (trm_app v1 t2) Q) ->
      eval s1 (trm_app t1 t2) Q
  | eval_app_arg2 : forall s1 (v1 : val) t2 Q1 Q,
      ¬ trm_is_val t2 ->
      eval s1 t2 Q1 ->
      (forall v2 s2, Q1 v2 s2 -> eval s2 (trm_app v1 v2) Q) ->
      eval s1 (trm_app v1 t2) Q
  | eval_app_fun : forall s1 v1 (v2 :val) x t1 Q,
      v1 = val_fun x t1 ->
      eval s1 (subst x v2 t1) Q ->
      eval s1 (trm_app v1 v2) Q
  | eval_app_fix : forall s (v1 v2 : val) f x t1 Q,
      v1 = val_fix f x t1 ->
      eval s (subst x v2 (subst f v1 t1)) Q ->
      eval s (trm_app v1 v2) Q
  | eval_seq : forall Q1 s t1 t2 Q,
      eval s t1 Q1 ->
      (forall v1 s2, Q1 v1 s2 -> eval s2 t2 Q) ->
      eval s (trm_seq t1 t2) Q
  | eval_let : forall Q1 s x t1 t2 Q,
      eval s t1 Q1 ->
      (forall v1 s2, Q1 v1 s2 -> eval s2 (subst x v1 t2) Q) ->
      eval s (trm_let x t1 t2) Q
  | eval_if : forall s (b : Bool) t1 t2 Q,
      eval s (if b then t1 else t2) Q ->
      eval s (trm_if (val_bool b) t1 t2) Q
  | eval_unop : forall op s v1 P Q,
      evalunop op v1 P ->
      purepostin s P Q ->
      eval s (trm_app op v1) Q
  | eval_binop : forall op s (v1 v2 : val) P Q,
      evalbinop op v1 v2 P ->
      purepostin s P Q ->
      eval s (trm_app (trm_app op v1) v2) Q
  | eval_ref : forall s x t1 t2 (Q Q₁ : val → state → Prop),
    eval s t1 Q₁ →
    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,
     eval (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →
    eval s (trm_ref x t1 t2) Q
  | eval_get : forall s p Q,
      p ∈ s ->
      Q (read_state p s) s ->
      eval s (trm_app val_get (val_loc p)) Q
  | eval_set : forall s p v Q,
      v = trm_val v' ->
      p ∈ s ->
      Q val_unit (Finmap.insert p v' s) ->
      eval s (trm_app (trm_app val_set (val_loc p)) v) Q
  | eval_alloc_arg : forall s Q₁ Q,
    ¬ trm_is_val t1 →
    eval s t1 Q₁ →
    (∀ v' s', Q₁ v' s' → eval s' (trm_alloc x v' t2) Q) →
    eval s (trm_alloc x t1 t2) Q
  | eval_alloc : forall (sa : state) (n : ℤ) Q,
    n ≥ 0 →
    (∀ (p : loc) (sb : state),
      sb = conseq (make_list n.natAbs val_uninit) p →
      p ≠ null →
      Finmap.Disjoint sa sb →
      eval (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \ sb)) →
    eval sa (trm_alloc x n t2) Q
  
  
  
  
  
  
  
  
  | eval_for (n₁ n₂ : Int) (Q : val -> state -> Prop) :
    eval s (if (n₁ < n₂) then
               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))
            else val_unit) Q ->
    eval s (trm_for x n₁ n₂ t₁) Q
  | eval_while (t₁ t₂ : trm) (Q : val -> state -> Prop) :
    eval s t₁ Q₁ ->
    (∀ s v, Q₁ v s -> eval s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->
    eval s (trm_while t₁ t₂) Q

tryGoal in Ssreflect.Utils

Lean.explicitBinders in Init.NotationExtra

def hwand (H1 H2 : hProp) : hProp :=
  hexists (fun (H0 : hProp) => H0 ∗ hpure ((H1 ∗ H0) ==> H2))

def default_get d := [lang|
  fun p i =>
    let n := val_abs i in
    let l := val_array_length p in
    let b := n < l in
    if b then
      val_array_get p n
    else
      d ]

Option.merge in Init.Data.Option.Basic

def intersect (s1 s2 : state) := s1 \ (s1 \ s2)

Bool in Init.Prelude

def hind : hProp :=
  hexists (fun b ↦ if b then emp else ⊤)

def eval_like (t1 t2:trm) : Prop :=
  forall s Q, eval s t1 Q -> eval s t2 Q

def reducible (s : state) (t : trm) : Prop :=
  exists s' t', step s t s' t'

AList.entries in Mathlib.Data.List.AList

@[app_unexpander val_fixs] def unexpandVFixs : Lean.PrettyPrinter.Unexpander
  | `($(_) $f:str [ $xs:str,* ] [lang| $t]) =>
    let xs := xs.getElems.map (Lean.mkIdent $ Lean.Name.mkSimple ·.getString)
    let f := Lean.mkIdent $ Lean.Name.mkSimple f.getString
    `([lang| fix $f $xs* => $t])
  | _ => throw ( )

Lean.Syntax in Init.Prelude

@[simp]
abbrev valid : val -> Prop
  | .val_bool _ => True
  | _ => False

Prod in Init.Prelude

Finmap.singleton in Mathlib.Data.Finmap

def val.is_loc : val -> Prop
  | val_loc _ => True
  | _ => False

def val_funs (xs:List var) (t:trm) : val :=
  match xs with
  | [] => panic! "function with zero argumets!"
  | x1::xs' => val_fun x1 (trm_funs xs' t)

Lean.Meta.MetaM in Lean.Meta.Basic

def trm_fixs (f:var) (xs:List var) (t:trm) : trm :=
  match xs with
  | [] => t
  | x1::xs' => trm_fix f x1 (trm_funs xs' t)

def toHimp (e : Expr) : MetaM Expr := do
  let eTy <- inferType e
  trace[xchange] "type of give hypothesis {eTy}"
  forallTelescope eTy fun xs eTy => do
    let himpl <- match_expr eTy with
      | himpl _ _ => return e
      | qimpl _ _ _ => return e
      | Eq _ _ _  =>
        mkAppOptM `himpl_of_eq #[.none, .none, eTy]
      | _        => throwError "Expected a heap entailment or an equality"
    mkForallFVars xs himpl

Not in Init.Prelude

def XSimpRIni : TacticM XSimpR := withMainContext do
  (<- getMainGoal).setTag `xsimp_goal
  let goal <- getGoalStxNN
  let `(XSimp $hl $hr) := goal | throwError "not a XSimp goal"
  let `(@Prod.mk hProp $_ $hla $hlwt) := hl | throwError "not a XSimp goal"
  let `(@Prod.mk hProp hProp $hlw $hlt) := hlwt | throwError "not a XSimp goal"
  let `(@Prod.mk hProp $_ $hra $hrgt) := hr | throwError "not a XSimp goal"
  let `(@Prod.mk hProp hProp $hrg $hrt) := hrgt | throwError "not a XSimp goal"
  return { hla := hla, hlw := hlw, hlt := hlt, hra := hra, hrg := hrg, hrt := hrt }

def purepost (s : state) (P : val → Prop) : val → state → Prop :=
  fun v s' => P v ∧ s' = s

def xcahngeCore (l : Expr) : MVarId -> MetaM (List MVarId) := fun goal => do
  let goal <-
    match_expr <- goal.getType with
    | himpl _ _ => pure goal
    | qimpl _ _ _ => pure (<- goal.intro1P).2
    | _ => throwError "Expected a heap entailment"
  xchangeApply l goal

partial def getNestedApp (t :Expr) : Expr :=
  let_expr trm_app t _ := t | getVal t
  getNestedApp t

instOfNat in Init.Data.Int.Basic

@[app_unexpander trm_seq] def unexpandSeq : Lean.PrettyPrinter.Unexpander
  | `($(_) [lang| $t1] [lang| $t2]) => `([lang| $t1 ; $t2])
  | _ => throw ( )

def wpgen_fix (Fof : val → val → formula) : formula :=
  fun Q ↦ h∀ vf, ⌜forall vx Q', Fof vf vx Q' ==>
    wp (trm_app vf vx) Q'⌝ -∗ Q vf

macro "lang_def" n:ident ":=" l:lang : command => do
  `(def $n:ident : val := [lang| $l])

partial def getList! (xs : Syntax) : (List Syntax) :=
  match xs with
  | `(List.nil) => []
  | `(List.cons $x $xs) => x::getList! xs
  | _ => panic! "getList!: {xs} is not a list"

def hstars_simp_step : TacticM Unit := withMainContext do
  let goal <- getGoalStxNN
  match goal with
    | `(himpl $Hl $Hr) =>
        match Hr with
          | `(@HStar.hStar hProp hProp $_ $_ $_ $hs) =>
              match hs with
              | `(@HStar.hStar hProp hProp $_ $_ $H $_) =>
                    try
                      xsimp_pick_same H Hl hstars_simp_pick ;
                      {| apply hstars_simp_cancel_lemma |}
                    catch ex =>
                      let s <- ex.toMessageData.toString
                      if s == "not equal" then
                        {| apply hstars_simp_keep_lemma |}
                      else
                        throw ex
              | _ => throwError "cannot simplify"
          | _ => throwError "cannot simplify"
    | _ => throwError "cannot simplify"

scoped instance (priority := low) : AddCommSemigroup val where
  add := add
  add_assoc := by admit /- proof elided -/

Unit in Init.Prelude

def hgc := htop

AList in Mathlib.Data.List.AList

local instance : HAdd ℤ ℤ val := ⟨fun x y => val.val_int (x + y)⟩

scase_if in Ssreflect.Elim

instance (α : Type) : HWand (α → hProp) (α → hProp) hProp where
  hWand := qwand

local instance : HAdd ℤ ℕ val := ⟨fun x y => (x + (y : Int))⟩

def wpgen_while (F1 F2 : formula) : formula := mkstruct fun Q =>
  h∀ R : formula,
    let F := wpgen_if_trm F1 (wpgen_seq F2 R) (wpgen_val val_unit)
    ⌜structural R ∧ F ===> R⌝ -∗ R Q

@[simp]
def trms_vals (vs : List var) : List trm :=
  vs.map trm_var

List.kinsert in Mathlib.Data.List.Sigma

Int.toNat in Init.Data.Int.Basic

List.zip in Init.Data.List.Basic

abbrev trm_is_val : trm → Prop
  | trm_val _ => true
  | _         => false

nsmulRec in Init.Data.Zero

class PartialCommMonoid (α : Type) extends AddCommSemigroup α where
  valid : α -> Prop
  valid_add : ∀ x, valid (x + y) -> valid x
  add_valid : ∀ x y, valid x -> valid y -> valid (x + y)

Lean.Expr.equal in Lean.Expr

move in Ssreflect.Basic

sdone in Ssreflect.Done

scoped instance inst : PartialCommMonoidWRT val add valid where
  validE := by admit /- proof elided -/

@[simp]
def wpgen_app (t : trm) : formula :=
  fun Q ↦ ∃ʰ H, H ∗ ⌜triple t H Q⌝

HAdd in Init.Prelude

@[app_unexpander trm_var] def unexpandVar : Lean.PrettyPrinter.Unexpander
  | `($(_) $x:str) =>
    let str := x.getString
    let name := Lean.mkIdent $ Lean.Name.mkSimple str
    `([lang| $name:ident])
  | _ => throw ( )

instance (A : Type) : HStar (A → hProp) hProp (A → hProp) where
  hStar := qstar

def wpgen_for (v₁ v₂ : trm) (F1 : val -> formula) : formula :=
  mkstruct fun Q =>
    ∃ʰ n₁ n₂ : Int, ⌜v₁ = n₁⌝ ∗ ⌜v₂ = n₂⌝ ∗
      h∀ (S : Int -> formula),
        (let F i :=
          if i < n₂ then
            wpgen_seq (F1 (val_int i)) (S (i + 1))
          else wpgen_val val_unit
        ⌜structural_pred S /\ ∀ i, F i ===> S i⌝ -∗ S n₁ Q )

def xsimp_r_hexists_apply_hints (x : Ident) : TacticM Unit := do
  let hints <- hintExt.getSSR
  match hints with
  | [] =>
    trace[xsimp] "no hints"
    eApplyAndName `xsimp_r_hexists $ `xsimp ++ x.getId
  | h :: hs =>
    hintExt.setSSR hs
    match h with
    | `(hint| ?)       => eApplyAndName `xsimp_r_hexists $ `xsimp ++ x.getId
    | `(hint| $t:term) => {| apply (@xsimp_r_hexists _ $t) |}
    | _ => throwError "xsimp_r_hexists_apply_hints: @ unreachable"

abbrev tohProp (h : heap -> Prop) : hProp := h

def structural_pred (S : α -> formula) :=
  ∀ x, structural $ S x

Or in Init.Prelude

def wpgen_seq (F1 F2 : formula) : formula :=
  fun Q ↦ F1 (fun _ ↦ F2 Q)

| step_ref : forall s v p,
      ¬ p ∈ s →
      v = trm_val v' →
      step s (trm_app val_ref v) (Finmap.insert p v' s) (val_loc p)
  | step_get : forall s p,
      p ∈ s →
      step s (trm_app val_get (val_loc p)) s (read_state p s)
  | step_set : forall s p v,
      p ∈ s ->
      v = trm_val v' →
      step s (trm_app (trm_app val_set (val_loc p)) v)
        (Finmap.insert p v' s) val_unit
  | step_free : forall s p,
      p ∈ s →
      step s (trm_app val_free (val_loc p)) (Finmap.erase p s) val_unit

Nat in Init.Prelude

def st1 : state := (Finmap.singleton 0 1).insert 1 1

def hadd [PartialCommMonoid val] (H₁ H₂ : hProp) : hProp :=
  fun h => exists h1 h2, H₁ h1 ∧ H₂ h2 ∧ h1 ⊥ʰ h2 ∧ h = h1 +ʰ h2

def getGoalProp : Lean.Elab.Tactic.TacticM Q(Prop) := do
  let goal ← Lean.Elab.Tactic.getMainTarget
  match ← castToProp goal with
  | some qType => return qType
  | none => throwError "goal is not a proposition"

List.zipWith in Init.Data.List.Basic

noncomputable def is_true (P : Prop) : Bool :=
  if P then true else false

Lean.trace in Lean.Util.Trace

sby in Ssreflect.Done
</all_available_defs>

<all_available_lemmas>
lemma himpl_trans H2 H1 H3 :
  (H1 ==> H2) → (H2 ==> H3) → (H1 ==> H3)

lemma update_middle (A : Type) (_ : Inhabited A) (n : Nat)
  (l1 l2 : List A) (x v : A) :
  n = l1.length → (l1 ++ x :: l2).set n v = (l1.concat v) ++ l2

lemma himpl_lr_qwand_unify (Q : α -> hProp):
  XSimp (Hla, emp, emp) ((Q -∗ (Q ∗ Hla)) ∗ emp, emp, emp)

lemma nth_app_r {A : Type} (_ : Inhabited A) n (l1 l2 : List A) :
  n ≥ l1.length →
  (l1 ++ l2)[n]! = l2[n - l1.length]!

lemma natabs_nonneg (p : Nat) (n : Int) :
  n ≥ 0 → (p + n).natAbs = p + n.natAbs

lemma hpure_intr :
  (P -> H s) -> (⌜P⌝ -∗ H) s

lemma add_Int.natAbs i j :
  0 <= i - j → j + Int.natAbs (i - j) = i

lemma hempty_eq_hpure_true :
  emp = ⌜True⌝

lemma triple_arrayFun_length (p : loc) :
  triple (trm_app val_array_length p)
    (harrayFun f n p)
    (fun r ↦ ⌜r = val_int n⌝ ∗ harrayFun f n p)

lemma hstar_hgc_hgc : /GC ∗ /GC = /GC

@[simp]
lemma labSet_inter (l : ℕ) (s s' : Set α) :
  ⟪l, s⟫ ∩ ⟪l, s'⟫ = ⟪l, s ∩ s'⟫

lemma hcell_nonneg v p i :
  hcell v p i ==> hcell v p i ∗ ⌜i >= 0⌝

lemma read_state_single p v :
  read_state p (Finmap.singleton p v) = v

lemma erase_of_non_mem (h : state) :
  p ∉ h →
  h.erase p = h

lemma hstar_pick_1 :
  h1 ∗ h = h1 ∗ h

@[simp]
lemma AList.merge_empty_r (h : AList (fun _ : loc => val)) : AList.merge h ∅ = h

lemma List.kerase_noterased {α : Type u} {β : α → Type v} [DecidableEq α] (l : List (Sigma β))
  (a a' : α) (hneq : a ≠ a') (b : β a) : ⟨a, b⟩ ∈ l ↔ ⟨a, b⟩ ∈ List.kerase a' l

lemma eval_like_trm_fun_val_fun x t : eval_like (trm_fun x t) (val_fun x t)

lemma sum_single' (v : β -> Int) (fs : Finset β) (x : ℤ) :
  (p ~~> x) + ∑ i in fs, (p ~~> v i) = p ~~> val.val_int (x + ∑ i in fs, v i)

lemma xsimp_flip_acc_r_lemma :
  hra = hra' →
  XSimp (hla, emp, emp) (hra', hrg, emp) →
  XSimp (hla, emp, emp) (hra, hrg, emp)

lemma AList.erase_empty {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) :
  AList.erase a (∅ : AList β) = ∅

lemma hstar_hsingle_same_loc p v1 v2 :
  (p ~~> v1) ∗ (p ~~> v2) ==> ⌜False⌝

lemma triple_alloc (n : Int) :
  n ≥ 0 →
  (∀ (p : loc), triple (subst x p t)
    (H ∗ ⌜p ≠ null⌝ ∗ hrange (make_list n.natAbs val_uninit) p)
    (Q ∗ ⌜p ≠ null⌝  ∗ ∃ʰ L, ⌜L.length = n⌝ ∗ hrange L p) ) →
  triple (trm_alloc x n t) H Q

lemma eval_like_val_fun_trm_fun_app_left (x : var) (t1 t2 : trm) :
  eval_like (trm_app (val_fun x t1) t2) (trm_app (trm_fun x t1) t2)

lemma wpgen_app_sound t :
  formula_sound t (wpgen_app t)

lemma sum_single (v : β -> Bool) (fs : Finset β) :
  (p ~~> false) + ∑ i in fs, (p ~~> v i) =
  p ~~> val.val_bool (∃ i ∈ fs, v i)

lemma hstar_flip_6 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ emp =
  h6 ∗ h5 ∗ h4 ∗ h3 ∗ h2 ∗ h1 ∗ emp

lemma hstar_intro (H1 H2 : hProp) h1 h2 :
  H1 h1 →
  H2 h2 →
  Finmap.Disjoint h1 h2 →
  (H1 ∗ H2) (h1 ∪ h2)

lemma set_nth_same (A : Type) (IA : Inhabited A) (n : Nat) (l : List A) :
  n < l.length → l.set n l[n]! = l

lemma trm_apps2 :
  trm_apps (trm_app t1 t2) ts = trm_apps t1 (t2::ts)

@[simp]
lemma star_post_empty (Q : α -> hProp) :
  Q ∗ emp = Q

@[simp]
lemma Heap.add_lookup (h₁ h₂ : heap) (l : loc) :
  (h₁ +ʰ h₂).lookup l = Option.merge (·+·) (h₁.lookup l) (h₂.lookup l)

lemma conseq_cons B (l : Nat) (v : B) (vs : List B) :
  conseq (v :: vs) l = (Finmap.singleton l v) ∪ (conseq vs (l + 1))

lemma xsimp_l_keep_wand :
  XSimp (H ∗ Hla, Hlw, Hlt) HR →
  XSimp (Hla, H ∗ Hlw, Hlt) HR

lemma Option.merge_none_l (a : Option α) : Option.merge f none a = a

lemma triple_ptr_add (p : loc) (n : ℤ) :
  p + n >= 0 →
  triple (trm_app val_ptr_add p n)
    emp
    (fun r ↦ ⌜r = val_loc ((p + n).natAbs)⌝)

lemma haffine_hany : forall H,
  haffine H

lemma xsimp_l_acc_wand :
  XSimp (hla, h ∗ hlw, hlt) hr ->
  XSimp (hla, hlw, h ∗ hlt) hr

lemma keys_eq_not_mem_r (h1 h2 : state) :
  h1.keys = h2.keys →
  p ∉ h2 →
  p ∉ h1

lemma validInter_hop_distr_r (h₁ h₂ h₃ : heap) :
  h₁ ⊥ʰ (h₂ +ʰ h₃) -> (h₁ ⊥ʰ h₂ ∧ h₁ ⊥ʰ h₃)

lemma hadd_single (v v' : Int) :
  (p ~~> v) + (p ~~> v') = p ~~> (v + v')

lemma hstar_hpure_l P H h :
  (⌜P⌝ ∗ H) h = (P ∧ H h)

lemma haffine_hgc : haffine /GC

lemma isubst_insert (al : ctx) x v t :
  isubst (al.insert x v) t = subst x v (isubst (al.erase x) t)

lemma harray_eq p L :
  harray L p = ∃ʰ n, ⌜n = L.length⌝ ∗ hheader n p ∗ hseg L p 0

lemma xref_lemma x v t2 H Q :
  H ==> (h∀ p, (p ~~> v) -∗
    protect (wp (subst x p t2) (fun hv ↦ Q hv ∗ ∃ʰ u, p ~~> u))) →
  H ==> wpgen_ref x (trm_val v) t2 Q

lemma Int.Ico_succ_right : ⟦i, j+1⟧ = Finset.Icc (α := ℤ) i j

@[trans]
lemma eval_like_trans {t1 t2 t3 : trm} : eval_like t1 t2 → eval_like t2 t3 → eval_like t1 t3

lemma triple_div n1 n2 :
  n2 ≠ 0 →
  triple (trm_app val_div (val_int n1) (val_int n2))
    emp
    (fun r ↦ ⌜r = val_int (n1 / n2)⌝)

lemma triple_hpure' t (P : Prop) Q :
  (P → triple t emp Q) →
  triple t ⌜P⌝ Q

lemma himpl_hforall_l A (x : A) (J : A → hProp) H :
  (J x ==> H) →
  (hforall J) ==> H

lemma hwand_inv h1 h2 H1 H2 :
  (H1 -∗ H2) h2 →
  H1 h1 →
  Finmap.Disjoint h1 h2 →
  H2 (h1 ∪ h2)

lemma trm_funs_app :
  trm_funs (xs ++ xs') t1 = trm_funs xs (trm_funs xs' t1)

lemma xsimp_start_lemma :
  XSimp (emp, emp, h1 ∗ emp) (emp, emp, h2 ∗ emp) ->
  h1 ==> h2

lemma triple_get v (p : loc) :
  triple (trm_app val_get p)
    (p ~~> v)
    (fun r ↦ ⌜r = v⌝ ∗ (p ~~> v))

lemma triple_hforall t A (x : A) (J : A → hProp) Q:
  triple t (J x) Q →
  triple t (hforall J) Q

lemma hwand_hpure_l P H :
  P → (⌜P⌝ -∗ H) = H

lemma wp_if_case (b : Bool) t1 t2 Q :
  (if b then wp t1 Q else wp t2 Q) ==> wp (trm_if b t1 t2) Q

lemma wp_ramified_trans t H Q1 Q2 :
  H ==> (wp t Q1) ∗ (Q1 -∗ Q2) →
  H ==> (wp t Q2)

lemma finite_state' n (s : state) :
  ∃ p, p ≠ null ∧
    Finmap.Disjoint s (conseq (make_list n val_uninit) p)

lemma himpl_hempty_hwand_same H :
  emp ==> (H -∗ H)

lemma triple_eval_like t1 t2 H Q :
  eval_like t1 t2 →
  triple t1 H Q →
  triple t2 H Q

lemma triple_ger r1 r2 :
  triple (trm_app val_ge (val_real r1) (val_real r2))
    emp
    (fun r ↦ ⌜r = val_bool (r1 >= r2)⌝)

lemma insert_neq_of_non_mem (h : state) :
  x ∉ h →
  x ≠ p →
  x ∉ h.insert p v

lemma sP_strongest :
  eval h t Q -> sP h t ===> Q

lemma evalExact_WellAlloc :
  evalExact s t Q →
  Q v s' →
  s'.keys = s.keys

lemma hstar_pick_5 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h = h5 ∗ h1 ∗ h2 ∗ h3 ∗ h4 ∗ h

lemma triple_alloc_arg :
  ¬trm_is_val t1 →
  triple t1 H Q1 →
  (∀ v, triple (trm_alloc x (trm_val v) t2) (Q1 v) Q ) →
  triple (trm_alloc x t1 t2) H Q

lemma xsimp_l_hpure :
  (p -> XSimp (hla, hlw, hlt) hr) ->
  XSimp (hla, hlw, ⌜p⌝ ∗ hlt) hr

lemma foo''' (n : ℕ) : (@OfNat.ofNat ℕ n _) = n

lemma insert_eq_union_single (h : state) :
  p ∉ h →
  h.insert p v = h ∪ (Finmap.singleton p v)

lemma xsimp_r_hpure :
  XSimp hl (hra, hrg, hrt) -> p ->
  XSimp hl (hra, hrg, hpure p ∗ hrt)

lemma conseq_nil B (l : Nat) :
  conseq ([] : List B) l = ∅

lemma triple_app_fix v1 v2 f x t1 H Q :
  v1 = val_fix f x t1 →
  triple (subst x v2 (subst f v1 t1)) H Q →
  triple (trm_app v1 v2) H Q

lemma hpure_intro P :
  P → ⌜P⌝  ∅

lemma wp_ref x v t Q :
  (h∀ p, (p ~~> v) -∗ wp (subst x p t) (Q ∗ ∃ʰ v', (p ~~> v'))) ==>
  wp (trm_ref x v t) Q

lemma wp_sound t :
  formula_sound t (wp t)

lemma of_nat_nat n :
  OfNat.ofNat n = n

lemma trms_to_vals_some_equiv ts vs : trms_to_vals ts = some vs → ts = vs.map trm_val

lemma validInter_assoc_r (h₁ h₂ h₃ : heap) :
  h₂ ⊥ʰ h₃ -> h₁ ⊥ʰ (h₂ +ʰ h₃) -> (h₁ +ʰ h₂) ⊥ʰ h₃

lemma evalExact_sP' :
  evalExact h t Q → evalExact h t (sP' h t)

lemma hstar_pick_last_10 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ h8 ∗ h9 ∗ h10 = h10 ∗ h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ h8 ∗ h9

lemma triple_gt n1 n2 :
  triple (trm_app val_gt (val_int n1) (val_int n2))
    emp
    (fun r ↦ ⌜r = val_bool (n1 > n2)⌝)

lemma triple_mul n1 n2 :
  triple (trm_app val_mul (val_int n1) (val_int n2))
    emp
    (fun r ↦ ⌜r = val_int (n1 * n2)⌝)

lemma triple_array_get L (p : loc) (i : Int) : --(v : 0 <= i ∧ i < L.length) :
   0 <= i ∧ i < L.length →
   triple (trm_app val_array_get p i)
    (harray L p)
    (fun r ↦ ⌜r = L[i.natAbs]!⌝ ∗ harray L p)

lemma xsimp_lr_exit :
  Hla ==> Hra ∗ Hrg ->
  XSimp (Hla, emp, emp) (Hra, Hrg, emp)

lemma triple_ptr_add_nat p (f : ℕ) :
  triple (trm_app val_ptr_add (val_loc p) (val_int (Int.ofNat f)))
    emp
    (fun r ↦ ⌜r = val_loc (p + f)⌝)

lemma keys_eq_not_mem_l (h1 h2 : state) :
  h1.keys = h2.keys →
  p ∉ h1 →
  p ∉ h2

lemma triple_binop op v1 v2 (P : val → Prop) :
  evalbinop op v1 v2 P →
  triple (trm_app op v1 v2) emp (fun r ↦ ⌜P r⌝)

lemma himpl_wpgen_wp t Q :
  wpgen t Q ==> wp t Q

lemma triple_ler r1 r2 :
  triple (trm_app val_le (val_real r1) (val_real r2))
    emp
    (fun r ↦ ⌜r = val_bool (r1 <= r2)⌝)

lemma intersect_disjoint_cancel (s₁ s₂ s₃ : state) :
  s₁.Disjoint s₃ →
  (s₁ ∪ intersect s₂ s₃) \ s₃ = s₁

lemma xsimpl_l_cancel_hwand_hstar_hempty :
  XSimp (Hla, Hlw, ((H2 -∗ H3) ∗ Hlt)) HR ->
  XSimp (Hla, (((emp ∗ H2) -∗ H3) ∗ Hlw), Hlt) HR

lemma hwandE :
  H1 -∗ H2 = hexists (fun H0 => H0 ∗ hpure ((H1 ∗ H0) ==> H2))

lemma non_mem_diff_helper1 (h₁ : state) (l : @AList loc fun _ ↦ val) :
  a ∉ l →
  p ∈ List.foldl (fun d s ↦ Finmap.erase s.fst d) (Finmap.erase a h₁) l.entries →
  p ≠ a

lemma erase_disjoint (h1 h2 : state) (p : loc) :
  h1.Disjoint h2 →
  (h1.erase p).Disjoint h2

lemma lookup_intersect (s₁ s₂ : state) :
  p ∈ s₁ ∧ p ∈ s₂ →
  (intersect s₁ s₂).lookup p = s₁.lookup p

lemma val_int_congr :
  n1 = n2 →
  val.val_int n1 = val.val_int n2

lemma eval_sat :
  eval h t Q -> ∃ h v, Q h v

lemma triple_ge n1 n2 :
  triple (trm_app val_ge (val_int n1) (val_int n2))
    emp
    (fun r ↦ ⌜r = val_bool (n1 >= n2)⌝)

lemma union_singleton_eq_insert (h : state) :
  Finmap.singleton p v ∪ h = h.insert p v

lemma val_funs_snoc (xs : List var) (x : var) (h : xs ≠ []) (t : trm) :
  val_funs (xs ++ [x]) t = val_funs xs (trm_fun x t)

lemma hstar_flip_8 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ h8 ∗ emp =
  h8 ∗ h7 ∗ h6 ∗ h5 ∗ h4 ∗ h3 ∗ h2 ∗ h1 ∗ emp

lemma hstars_simp_cancel_lemma :
  H1 ==> Ha ∗ Ht →
  H ∗ H1 ==> Ha ∗ H ∗ Ht

lemma himpl_lr_htop :
  XSimp (emp, emp, emp) (emp, Hrg, emp) ->
  XSimp (Hla, emp, emp) (emp, (⊤ ∗ Hrg), emp)

lemma single_disjoint :
  p ≠ p' ->
  (p ~~> v).Disjoint (p' ~~> v')

lemma triple_subr r1 r2 :
  triple (trm_app val_sub (val_real r1) (val_real r2))
    emp
    (fun r ↦ ⌜r = val_real (r1 - r2)⌝)

lemma wpgen_fail_sound t :
  formula_sound t wpgen_fail

lemma himpl_hexists_r A (x : A) H (J : A → hProp) :
  (H ==> J x) →
  H ==> (hexists J)

lemma hrange_intro L p :
  (hrange L p) (conseq L p)

lemma eval_like_val_funs_trm_funs xs t : xs ≠ [] → eval_like (val_funs xs t) (trm_funs xs t)

lemma hfalse_hstar_any H :
  ⌜False⌝ ∗ H = ⌜False⌝

lemma haffine_hpure P :
  haffine ⌜P⌝

lemma hexists_intro A (x : A) (J : A → hProp) h :
  J x h → (hexists J) h

lemma conseq_ind (n : ℕ) (v : val) (p : loc) :
  x ∈ conseq (make_list n v) p → x ≥ p

lemma AList.erase_twice {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : AList β) :
    (AList.erase a (AList.erase a l)).entries.Perm (AList.erase a l).entries

lemma wp_alloc x (n : ℤ) t Q :
  n ≥ 0 →
  (h∀ p, (hrange (make_list n.natAbs val_uninit) p) -∗
    wp (subst x p t) (Q ∗ ⌜p ≠ null⌝ ∗ ∃ʰ L, ⌜L.length = n⌝ ∗ hrange L p)) ==>
  wp (trm_alloc x n t) Q

@[simp]
theorem mem_diff_iff (h₁ h₂ : state) :
  p ∈ h₁ \ h₂ ↔ p ∈ h₁ ∧ p ∉ h₂

lemma hempty_inv h :
  emp h → h = ∅

lemma hstars_simp_pick_lemma :
  H1 = H1' →
  H1' ==> H2 →
  H1 ==> H2

lemma wp_ramified t (Q1 Q2 : val -> hProp) :
  (wp t Q1) ∗ (Q1 -∗ Q2) ==> (wp t Q2)

lemma xsimpl_lr_qwand_unit (Q1 Q2 : Unit -> hProp) :
  XSimp (emp, emp, (Q1 ( ) ∗ Hla)) (emp, emp, (Q2 ( ) ∗ emp)) ->
  XSimp (Hla, emp, emp) ((Q1 -∗ Q2) ∗ emp, emp, emp)

lemma hind_any h : ⊤⊤ h

lemma himpl_hforall A (J1 J2 : A → hProp) :
  J1 ===> J2 →
  hforall J1 ==> hforall J2

lemma disjoint_insert_l (h1 h2 : state) (x : loc) (v : val) :
  Finmap.Disjoint h1 h2 →
  x ∈ h1 →
  Finmap.Disjoint (Finmap.insert x v h1) h2

lemma hseg_eq_hrange L p (k : Nat) :
  hseg L p k = hrange L (p + 1 + k)

lemma wp_conseq_frame t H (Q1 Q2 : val -> hProp) :
  Q1 ∗ H ===> Q2 →
  (wp t Q1) ∗ H ==> (wp t Q2)

lemma xfun_spec_lemma (S:val->Prop) H Q Fof :
  (forall (vf : val),
    (forall (vx : val) H' Q', (H' ==> Fof vx Q') → triple (trm_app vf vx) H' Q') →
    S vf) →
  (forall vf, S vf -> (H ==> Q vf)) →
  H ==> wpgen_fun Fof Q

lemma hstar_hforall A (J : A → hProp) H :
  (hforall J) ∗ H ==> hforall (J ∗ H)

lemma triple_app_fun_direct x v2 t1 H Q :
  triple (subst x v2 t1) H Q →
  triple (trm_app (val_fun x t1) v2) H Q

lemma mkstruct_ramified Q1 Q2 F :
  (mkstruct F Q1) ∗ (Q1 -∗ Q2) ==> (mkstruct F Q2)

lemma union_difference_id (h₁ h₂ : state) :
  h₁.Disjoint h₂ →
  (h₁ ∪ h₂) \ h₂ = h₁

lemma validInter_hop_distr_l (h₁ h₂ h₃ : heap) :
  (h₁ +ʰ h₂) ⊥ʰ h₃ -> (h₁ ⊥ʰ h₃ ∧ h₂ ⊥ʰ h₃)

lemma kmerge_NodupKeys (l₁ l₂ : List (Sigma (fun _ : loc => val))) (nd₁ : l₁.NodupKeys) (nd₂ : l₂.NodupKeys) : (kmerge l₁ l₂).NodupKeys

lemma isubst_empty t : isubst ∅ t = t

lemma List.not_nil_snoc {α : Type u} (l : List α) : l ≠ [] → ∃ l' x, l = l' ++ [x]

lemma val_loc_congr :
  n1 = n2 →
  val.val_loc n1 = val.val_loc n2

lemma wpgen_sound t :
  formula_sound t (wpgen t)

lemma hrange_eq_conseq (L : List val) (n : ℤ) (p : loc) (s : state) :
  L.length = n →
  hrange L p s →
  s.keys = (conseq (make_list n.natAbs val_uninit) p).keys

lemma hwand_hempty_l H :
  (emp -∗ H) = H

lemma triple_let x t1 t2 Q1 H Q :
  triple t1 H Q1 →
  (forall v1, triple (subst x v1 t2) (Q1 v1) Q) →
  triple (trm_let x t1 t2) H Q

@[simp]
lemma kmerge_empty_r (l : List (Sigma (fun _ : loc => val))) : kmerge l [] = l

lemma xsimp_flip_acc_l_lemma :
  hla = hla' →
  XSimp (hla', emp, emp) (hra, hrg, emp) →
  XSimp (hla, emp, emp) (hra, hrg, emp)

lemma xsimp_r_hempty :
  XSimp hl (hra, hrg, hrt) ->
  XSimp hl (hra, hrg, emp ∗ hrt)

lemma hsingle_intro p v :
  (p ~~> v) (Finmap.singleton p v)

lemma wpgen_val_sound v :
  formula_sound (trm_val v) (wpgen_val v)

lemma Alist_insert_delete_id (l : @AList loc (fun _ ↦ val)) :
  p ∉ l →
  (l.insert p v).erase p = l

lemma eval_divr s r1 r2 Q :
  r2 ≠ 0 ->
  Q (val_real (r1 / r2)) s ->
  eval s (trm_app (trm_app val_div (val_real r1)) (val_real r2)) Q

lemma get_out_of_bounds {_ : Inhabited α} (L : List α) (i : Int) :
  L.length ≤ i.natAbs →
  L[i]! = default

lemma wpgen_ref_sound x t1 t2 :
  formula_sound (trm_ref x t1 t2) (wpgen_ref x t1 t2)

lemma himpl_hwand_r H1 H2 H3 :
  H2 ∗ H1 ==> H3 →
  H1 ==> (H2 -∗ H3)

theorem Perm.kmerge {l₁ l₂ l₃ l₄ : List (Sigma (fun _ : loc => val))} (nd₁ : l₁.NodupKeys) /- nd₁ is necessary -/ (nd₃ : l₃.NodupKeys)
    (p₁₂ : l₁.Perm l₂) (p₃₄ : l₃.Perm l₄) : (kmerge l₁ l₃).Perm $ kmerge l₂ l₄

@[heapSimp]
lemma foo : (@OfNat.ofNat ℕ n _) = (n : ℤ)

lemma xsimp_l_hexists :
  (∀ x, XSimp (hla, hlw, j x ∗ hlt) hr) ->
  XSimp (hla, hlw, (hexists j) ∗ hlt) hr

lemma disjoint_single (h : state) :
  p ∉ h →
  h.Disjoint (Finmap.singleton p v)

lemma list_nil_length (A : Type) :
  ([] : List A).length = 0

lemma hstar_flip_7 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ emp =
  h7 ∗ h6 ∗ h5 ∗ h4 ∗ h3 ∗ h2 ∗ h1 ∗ emp

lemma himpl_refl H : H ==> H

lemma lookup_diff (h₁ h₂ : state) :
  p ∉ h₂ →
  (h₁ \ h₂).lookup p = h₁.lookup p

lemma validInter_empty_r (h : heap) : h ⊥ʰ ∅

lemma disjoint_keys (h₁ h₂ : state) :
  h₁.Disjoint h₂ →
  Disjoint h₁.keys h₂.keys

lemma hsingl_inv p v h :
  (p ~~> v) h →
  h = Finmap.singleton p v

lemma hstar_pick_9 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ h8 ∗ h9 ∗ h = h9 ∗ h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ h8 ∗ h

lemma htop_eq :
  ⊤ = ∃ʰ H, H

lemma ofnat_plusone (n : Nat) :
  Int.ofNat (n + 1) = (Int.ofNat n) + 1

lemma haffine_hstar H1 H2 :
  haffine H1 → haffine H2 → haffine (H1 ∗ H2)

lemma AList.erase_noop {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : AList β) :
    a ∉ l → (AList.erase a l).entries.Perm l.entries

lemma xsimp_l_cancel_qwand α (Q1 Q2 : α -> hProp) x :
  XSimp (emp, Hlw, (Hla ∗ Q2 x ∗ Hlt)) HR ->
  XSimp ((Q1 x ∗ Hla), ((Q1 -∗ Q2) ∗ Hlw), Hlt) HR

lemma read_state_frame :
  s.Disjoint h2 →
  p ∈ s →
  (fun v' s' ↦ v' = read_state p (s ∪ h2) ∧ s' = s ∪ h2 ) =
  (qstar (fun v' s' ↦ v' = read_state p s ∧ s' = s) (tohProp fun h ↦ h = h2))

lemma Option.merge_none_r (a : Option α) : Option.merge f a none = a

lemma set_keep_length (L : List val) i v :
  L.length = (L.set i v).length

lemma himpl_of_eq H1 H2 : H1 = H2 -> H1 ==> H2

lemma hwand_cancel H1 H2 :
  H1 ∗ (H1 -∗ H2) ==> H2

lemma triple_ramified_frame H1 Q1 t H Q :
  triple t H1 Q1 →
  H ==> H1 ∗ (Q1 -∗ Q) →
  triple t H Q

lemma update_app_r {A : Type} (l1 l2 : List A) n v :
  n ≥ l1.length →
  (l1 ++ l2).set n v = l1 ++ l2.set (n - l1.length) v

lemma eval_addr s r1 r2 Q :
  Q (val_real (r1 + r2)) s →
  eval s (trm_app (trm_app val_add (val_real r1)) (val_real r2)) Q

lemma list_inc_natabs {α : Type} (L : List α) :
  ((L.length : ℤ) + 1).natAbs = (L.length : ℤ).natAbs + 1

lemma himpl_hexists_l A H (J : A → hProp) :
  (forall x, J x ==> H) → (hexists J) ==> H

lemma haffine_hastar_hpure (P : Prop) H :
  (P → haffine H) →
  haffine (⌜P⌝ ∗ H)

lemma Heap.add_comm (h₁ h₂ : heap) : h₁ +ʰ h₂ = h₂ +ʰ h₁

lemma xsimp_l_cancel_hwand :
  XSimp (emp, Hlw, Hla ∗ H2 ∗ Hlt) HR →
  XSimp ((H1 ∗ Hla), ((H1 -∗ H2) ∗ Hlw), Hlt) HR

lemma nth_middle (A : Type) (IA : Inhabited A) (n : Nat)
  (l1 l2 : List A) (x : A) :
  n = l1.length → (l1 ++ x :: l2)[n]! = x

lemma eval_app_arg2' : forall s1 (v1 : val) t2 Q1 Q,
  eval s1 t2 Q1 ->
  (forall v2 s2, eval s2 (trm_app v1 v2) Q) ->
  eval s1 (trm_app v1 t2) Q

lemma himpl_lr_refl :
  XSimp (hla, emp, emp) (hla, emp, emp)

lemma wp_if (b : Bool) t1 t2 Q :
  wp (if b then t1 else t2) Q ==> wp (trm_if b t1 t2) Q

lemma wpgen_alloc_sound x t1 t2 :
  formula_sound (trm_alloc x t1 t2) (wpgen_alloc x t1 t2)

lemma diff_disjoint (h₁ h₂ : state) :
  h₂.Disjoint (h₁ \ h₂)

lemma triple_opp n1 :
  triple (trm_app val_opp (val_int n1))
    emp
    (fun r ↦ ⌜r = val_int (-n1)⌝)

lemma hstar_pick_7 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ h = h7 ∗ h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h

lemma triple_hexists t A (J : A → hProp) Q :
  (forall x, triple t (J x) Q) →
  triple t (hexists J) Q

lemma harray_focus i L p (v : 0 <= i ∧ i < L.length) :
  0 <= i ∧ i < L.length →
  harray L p ==>
  hcell L[Int.natAbs i]! p i
    ∗ (h∀ w, hcell w p i -∗ harray (L.set (Int.natAbs i) w) p)

lemma triple_array_set L (p : loc) (i : Int) (v : val) :
  0 <= i ∧ i < L.length →
  triple (trm_app val_array_set p i v)
    (harray L p)
    (fun _ ↦ harray (L.set (Int.natAbs i) v) p)

lemma disjoint_update_not_r (h1 h2 : state) (x : loc) (v: val) :
  Finmap.Disjoint h1 h2 →
  x ∉ h2 →
  Finmap.Disjoint (Finmap.insert x v h1) h2

lemma triple_gtr r1 r2 :
  triple (trm_app val_gt (val_real r1) (val_real r2))
    emp
    (fun r ↦ ⌜r = val_bool (r1 > r2)⌝)

lemma sum_Ico_predr {_ : AddCommMonoid M} (f : Int -> M) (i j : Int) :
  i < j ->
  ∑ i in ⟦i, j⟧, f i = (∑ i in ⟦i, j - 1⟧, f i) + f (j -1)

lemma himpl_frame_lr H1 H1' H2 H2' :
  H1 ==> H1' →
  H2 ==> H2' →
  (H1 ∗ H2) ==> (H1' ∗ H2')

lemma hwand_curry H1 H2 H3 :
  (H1 ∗ H2) -∗ H3 ==> H1 -∗ (H2 -∗ H3)

lemma xsimpl_lr_cancel_same_hsingle :
  XSimp (Hla, Hlw, Hlt) (Hra, Hrg, Hrt) →
  v1 = v2 →
  XSimp (p ~~> v1 ∗ Hla, Hlw, Hlt) (Hra, Hrg, p ~~> v2 ∗ Hrt)

lemma neg_mul_abs (n : Int) :
  n < 0 → -1 * n = n.natAbs

lemma hstar_flip_0 :
  emp = emp

lemma mem_conseq :
  x ∈ conseq L p → p ≤ x

lemma hforall_intro A (J : A → hProp) h :
  (forall x, J x h) → (hforall J) h

lemma eval_like_trm_apps_funs_pre (heqv0 : v0 = trm_funs xs t1) :
  eval_like t (trm_apps (val_funs xs t1) ts) ∧  -- NOTE: this part do not require `xs.Nodup`, but anyway
  eval_like (isubst (xs.mkAlist vs) t1) t

lemma hstar_flip_1 :
  h1 ∗ emp = h1 ∗ emp

lemma xwp_lemma_funs (xs : List _) (vs : List val) :
  t = trm_apps v0 ts ->
  v0 = val_funs xs t1 ->
  trms_to_vals ts = vs ->
  var_funs xs vs.length ->
  H ==> wpgen (isubst (xs.mkAlist vs) t1) Q ->
  triple t H Q

lemma triple_harrayFun_get (p : loc) (i : Int) :
   0 <= i ∧ i < (n : ℕ) →
   triple (trm_app val_array_get p i)
    (harrayFun f n p)
    (fun r ↦ ⌜r = f i⌝ ∗ harrayFun f n p)
</all_available_lemmas>

<local_ctx>
import Lean

import Mathlib.Data.Finmap

import Mathlib.Data.List.Indexes

import SPLean.Common.State

import SPLean.Common.Util

import SPLean.Theories.HProp

import SPLean.Theories.XSimp

import SPLean.Theories.XChange

import SPLean.Theories.SepLog

import SPLean.Theories.WPUtil

open trm val prim

namespace Theories

abbrev formula := (val → hProp) → hProp

def mkstruct (F : formula) :=
  fun (Q : val -> hProp) ↦ ∃ʰ Q', F Q' ∗ (Q' -∗ Q)

def structural (F : formula) :=
  forall Q, mkstruct F Q ==> F Q

def structural_pred (S : α -> formula) :=
  ∀ x, structural $ S x

def wpgen_val (v : val) : formula :=
  fun Q ↦ Q v

def wpgen_seq (F1 F2 : formula) : formula :=
  fun Q ↦ F1 (fun _ ↦ F2 Q)

def wpgen_for (v₁ v₂ : trm) (F1 : val -> formula) : formula :=
  mkstruct fun Q =>
    ∃ʰ n₁ n₂ : Int, ⌜v₁ = n₁⌝ ∗ ⌜v₂ = n₂⌝ ∗
      h∀ (S : Int -> formula),
        (let F i :=
          if i < n₂ then
            wpgen_seq (F1 (val_int i)) (S (i + 1))
          else wpgen_val val_unit
        ⌜structural_pred S /\ ∀ i, F i ===> S i⌝ -∗ S n₁ Q )

section tactics

open Lean Elab Tactic

section xapp

end xapp

end tactics

open AList

section funs_fixs_eval_like

variable (xs : List var) (vs : List val) (t : trm) (v0 : trm)
  (heqt : t = trm_apps v0 ts)
  (hconv : trms_to_vals ts = vs)
  (hform : var_funs xs vs.length)   -- NOTE: can be relaxed to `vs.length ≤ xs.length`

variable (f : var) (hf : f ∉ xs)

end funs_fixs_eval_like

end Theories

open Theories
</local_ctx>

<target_theorem>
lemma xfor_inv_lemma (I : Int -> hProp) (a b : Int)
  (F : val -> formula)
  (Q : val -> hProp) :
  structural_pred F ->
  a <= b ->
    (∃ H',
      H ==> I a ∗ H' ∧
      (∀ i, a <= i ∧ i < b -> I i ==> F i fun _ => I (i + 1)) ∧
      (fun _ => I b ∗ H') ===> Q) ->
    H ==> wpgen_for a b F Q :=
</target_theorem>

