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 = ∅)

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

def null : loc := 0

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 state := Finmap (λ _ : loc ↦ val)

abbrev heap := Heap.heap val

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

def subst (y : var) (v' : val) (t : trm) : trm :=
  
  let if_y_eq x t1 t2 := if x = y then t1 else t2
  match t with
  | trm_val v          => trm_val v
  | trm_var x          => if_y_eq x (trm_val v') t
  | trm_fun x t1       => trm_fun x (if_y_eq x t1 (subst y v' t1))
  | trm_fix f x t1     => trm_fix f x (if_y_eq f t1 (if_y_eq x t1 (subst y v' t1)))
  | trm_app t1 t2      => trm_app (subst y v' t1) (subst y v' t2)
  | trm_seq t1 t2      => trm_seq (subst y v' t1) (subst y v' t2)
  | trm_let x t1 t2    => trm_let x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))
  | trm_if t0 t1 t2    => trm_if (subst y v' t0) (subst y v' t1) (subst y v' t2)
  | trm_for x t1 t2 t3 => trm_for x (subst y v' t1) (subst y v' t2) (if_y_eq x t3 (subst y v' t3))
  | trm_while t1 t2    => trm_while (subst y v' t1) (subst y v' t2)
  | trm_ref x t1 t2    => trm_ref x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))
  | trm_alloc x t1 t2  => trm_alloc x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))

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

def read_state (p : loc) (h : state) :=
  match Finmap.lookup p h with
  | some v => v
  | none   => default

inductive evalunop : prim → val → (val → Prop) → Prop where
  | evalunop_neg : forall b1,
      evalunop val_neg (val_bool b1) (fun v => v = val_bool (¬ b1))
  | evalunop_opp : forall n1,
      evalunop val_opp (val_int n1) (fun v => v = val_int (- n1))
  | evalunop_oppr : forall r1,
      evalunop val_opp (val_real r1) (fun v => v = val_real (- r1))

inductive evalbinop : val → val → val → (val->Prop) → Prop where
  | evalbinop_eq : forall v1 v2,
      evalbinop val_eq v1 v2 (fun v => v = val_bool (is_true (v1 = v2)))
  | evalbinop_neq : forall v1 v2,
      evalbinop val_neq v1 v2 (fun v => v = val_bool (is_true (v1 ≠ v2)))
  | evalbinop_add : forall n1 n2,
      evalbinop val_add (val_int n1) (val_int n2)
        (fun v => v = val_int (n1 + n2))
  | evalbinop_addr : forall r₁ r₂,
      evalbinop val_add (val_real r₁) (val_real r₂)
        (fun v => v = val_real (r₁ + r₂))
  | evalbinop_sub : forall n1 n2,
      evalbinop val_sub (val_int n1) (val_int n2)
        (fun v => v = val_int (n1 - n2))
  | evalbinop_subr : forall r1 r2,
      evalbinop val_sub (val_real r1) (val_real r2)
        (fun v => v = val_real (r1 - r2))
  | evalbinop_mul : forall n1 n2,
      evalbinop val_mul (val_int n1) (val_int n2)
        (fun v => v = val_int (n1 * n2))
  | evalbinop_mulr : forall r1 r2,
      evalbinop val_mul (val_real r1) (val_real r2)
        (fun v => v = val_real (r1 * r2))
  | evalbinop_div : forall n1 n2,
      ¬(n2 = 0) →
      evalbinop val_div (val_int n1) (val_int n2)
        (fun v => v = val_int (n1 / n2))
  | evalbinop_divr : forall r1 r2,
      ¬(r2 = 0) →
      evalbinop val_div (val_real r1) (val_real r2)
        (fun v => v = val_real (r1 / r2))
  | evalbinop_mod : forall n1 n2,
      ¬(n2 = 0) →
      evalbinop val_mod (val_int n1) (val_int n2)
        (fun v => v = val_int (n1 % n2))
  | evalbinop_le : forall n1 n2,
      evalbinop val_le (val_int n1) (val_int n2)
        (fun v => v = val_bool (n1 <= n2))
  | evalbinop_ler : forall r1 r2,
      evalbinop val_le (val_real r1) (val_real r2)
        (fun v => v = val_bool (r1 <= r2))
  | evalbinop_lt : forall n1 n2,
      evalbinop val_lt (val_int n1) (val_int n2)
        (fun v => v = val_bool (n1 < n2))
  | evalbinop_ltr : forall r1 r2,
      evalbinop val_lt (val_real r1) (val_real r2)
        (fun v => v = val_bool (r1 < r2))
  | evalbinop_ge : forall n1 n2,
      evalbinop val_ge (val_int n1) (val_int n2)
        (fun v => v = val_bool (n1 >= n2))
  | evalbinop_ger : forall r1 r2,
      evalbinop val_ge (val_real r1) (val_real r2)
        (fun v => v = val_bool (r1 >= r2))
  | evalbinop_gt : forall n1 n2,
      evalbinop val_gt (val_int n1) (val_int n2)
        (fun v => v = val_bool (n1 > n2))
  | evalbinop_gtr : forall r1 r2,
      evalbinop val_gt (val_real r1) (val_real r2)
        (fun v => v = val_bool (r1 > r2))

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

def purepostin (s : state) (P : val → Prop) (Q : val → state → Prop) : Prop :=
  
  forall v, P v → Q v s

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))

def make_list {A} (n : Nat) (v : A) : List A :=
  match n with
  | 0      => []
  | n' + 1 => v :: make_list n' v

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

inductive evalExact : state → trm → (val → state → Prop) -> Prop where
  | val : forall s v,
      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)
  | fun : forall s x t1,
      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)
  | fix : forall s f x t1,
      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)
  | app_arg1 : forall s1 t1 t2 Q1 Q,
      ¬ trm_is_val t1 ->
      evalExact s1 t1 Q1 ->
      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->
      evalExact s1 (trm_app t1 t2) Q
  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,
      ¬ trm_is_val t2 ->
      evalExact s1 t2 Q1 ->
      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->
      evalExact s1 (trm_app v1 t2) Q
  | app_fun : forall s1 v1 (v2 :val) x t1 Q,
      v1 = val_fun x t1 ->
      evalExact s1 (subst x v2 t1) Q ->
      evalExact s1 (trm_app v1 v2) Q
  | app_fix : forall s (v1 v2 : val) f x t1 Q,
      v1 = val_fix f x t1 ->
      evalExact s (subst x v2 (subst f v1 t1)) Q ->
      evalExact s (trm_app v1 v2) Q
  | seq : forall Q1 s t1 t2 Q,
      evalExact s t1 Q1 ->
      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->
      evalExact s (trm_seq t1 t2) Q
  | let : forall Q1 s x t1 t2 Q,
      evalExact s t1 Q1 ->
      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->
      evalExact s (trm_let x t1 t2) Q
  | if : forall s (b : Bool) t1 t2 Q,
      evalExact s (if b then t1 else t2) Q ->
      evalExact s (trm_if (val_bool b) t1 t2) Q
  | unop : forall op s v1 P,
      evalunop op v1 P ->
      evalExact s (trm_app op v1) (purepost s P)
  | binop : forall op s (v1 v2 : val) P,
      evalbinop op v1 v2 P ->
      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)
  | ref : forall s x t1 t2 Q Q₁,
    evalExact s t1 Q₁ →
    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,
     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →
      evalExact s (trm_ref x t1 t2) Q
  | get : forall s p,
      p ∈ s ->
      evalExact s (trm_app val_get (val_loc p))
        (fun v' s' ↦ v' = read_state p s ∧ s' = s)
  | set : forall s p v,
      v = trm_val v' ->
      p ∈ s ->
      evalExact s (trm_app (trm_app val_set (val_loc p)) v)
        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')
  | alloc_arg : forall s Q₁ Q,
    ¬ trm_is_val t1 →
    evalExact s t1 Q₁ →
    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →
    evalExact s (trm_alloc x t1 t2) Q
  | 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 →
      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \ sb)) →
    evalExact sa (trm_alloc x n t2) Q
  
  
  
  
  
  
  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :
    evalExact s (if (n₁ < n₂) then
               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))
            else val_unit) Q ->
    evalExact s (trm_for x n₁ n₂ t₁) Q
  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :
    evalExact s t₁ Q₁ ->
    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->
    evalExact s (trm_while t₁ t₂) Q

Option.any in Init.Data.Option.Basic

Lean.binderIdent in Init.Notation

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"

def evalExact_ref_nonpositive s x t1 t2 (Q : val → state → Prop) :=
  ∃ Q₁, evalExact s t1 Q₁ ∧
  (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,
    (evalExact (s1.insert p v1) (subst x p t2) (fun v s ↦ Q v (s.erase p))))

local instance : HWand (hProp) (heap → Prop) hProp where
  hWand := hwand

def harray_int (L : List Int) : loc → hProp :=
  harray (L.map val_int)

instance : Coe ℝ val := ⟨val_real⟩

@[app_unexpander trm_app] def unexpandApp : Lean.PrettyPrinter.Unexpander := fun x => do
  
  match x with
  | `($(_) [uop|$uop] [lang| $t]) => `([lang| $uop:uop$t])
  | `($(_) $app [lang| $t2]) =>
    
    match app with
    | `([bop| $bop].trm_app [lang| $t1]) => `([lang| $t1 $bop $t2])
    | `([lang| $t1]) => `([lang| $t1 $t2])
    | `(val_array_get) => return x
    | `(trm_app val_array_get [lang| $t1]) => `([lang| $t1[$t2] ])
    | `(val_array_set) => return x
    | `(trm_app val_array_set [lang| $_]) => return x
    | `(trm_app $app [lang| $t1]) =>
      match app with
      | `(trm_app val_array_set [lang| $t0]) => `([lang| $t0[$t1] := $t2])
      
      
      
      
      
      | _ => throw ( )
    | _ => throw ( )
  | `($(_) [bop| $bop] [lang| $t1]) => return x
  
  
  
  
  | _ => throw ( )

scase in Ssreflect.Elim

Bool in Init.Prelude

@[simp]
abbrev add : val -> val -> val
  | .val_real i, .val_real j => val.val_real (i + j)
  | _, _ => val.val_unit

Set in Mathlib.Data.Set.Defs

Lean.Name in Init.Prelude

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

semiOutParam in Init.Prelude

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 ]

noncomputable def Heap.add (h₁ h₂ : heap) : heap :=
  Finmap.liftOn₂ h₁ h₂ (fun h₁ h₂ => (h₁.merge h₂).toFinmap) (by admit /- proof elided -/
    )

@[app_unexpander trm_fix] def unexpandTFix : Lean.PrettyPrinter.Unexpander
  | `($(_) $f:str $x:str [lang| fun $xs* => $t]) =>
    let f := f.getString
    let x := x.getString
    let nameF := Lean.mkIdent $ Lean.Name.mkSimple f
    let nameX := Lean.mkIdent $ Lean.Name.mkSimple x
    `([lang| fix $nameF $nameX $xs* => $t])
  | `($(_) $f:str $x:str [lang| $t]) =>
    let f := f.getString
    let x := x.getString
    let nameF := Lean.mkIdent $ Lean.Name.mkSimple f
    let nameX := Lean.mkIdent $ Lean.Name.mkSimple x
    `([lang| fix $nameF $nameX => $t])
  | _ => throw ( )

Lean.bracketedExplicitBinders in Init.NotationExtra

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

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

def val_array_get : val := [lang|
  fun p i =>
    let p1 := p ++ 1 in
    let q := p1 ++ i in
    !q ]

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

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

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

def hstar_pick_lemma (i : Nat) (pickLast : Bool) : Ident :=
  if pickLast then
    mkIdent s!"hstar_pick_last_{i}".toName
  else
    mkIdent s!"hstar_pick_{i}".toName

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

class PartialCommMonoidWRT (α : Type) (add' : semiOutParam (α -> α -> α)) (valid' : semiOutParam (α -> Prop)) extends PartialCommMonoid α where
  addE : (· + ·) = add'
  validE : PartialCommMonoid.valid = valid'

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?

Prod.mk in Init.Prelude

def test1 : Finmap (fun _ : ℕ ↦ ℕ) := Finmap.singleton 0 1

And in Init.Prelude

propext in Init.Core

srw in Ssreflect.Rewrite

List.Perm in Init.Data.List.Basic

@[app_unexpander val_fix] def unexpandVFix : Lean.PrettyPrinter.Unexpander
  | `($(_) $f:str $x:str [lang| fun $xs* => $t]) =>
    let f := f.getString
    let x := x.getString
    let nameF := Lean.mkIdent $ Lean.Name.mkSimple f
    let nameX := Lean.mkIdent $ Lean.Name.mkSimple x
    `([lang| fix $nameF $nameX $xs* => $t])
  | `($(_) $f:str $x:str [lang| $t]) =>
    let f := f.getString
    let x := x.getString
    let nameF := Lean.mkIdent $ Lean.Name.mkSimple f
    let nameX := Lean.mkIdent $ Lean.Name.mkSimple x
    `([lang| fix $nameF $nameX => $t])
  | _ => throw ( )

Finmap.insert in Mathlib.Data.Finmap

def xsimp_flip_acc_lr (hla hra : Term) : TacticM Unit :=
  {| xsimp_flip_acc_l $hla ; xsimp_flip_acc_r $hra |}

Option.merge in Init.Data.Option.Basic

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

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

Iff in Init.Core

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

Lean.Elab.Tactic.TacticM in Lean.Elab.Tactic.Basic

Finset in Mathlib.Data.Finset.Defs

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

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

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"

List.length 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))

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

@[app_unexpander val_prim] def unexpandPrim : Lean.PrettyPrinter.Unexpander
  
  | `($(_) val_free) => `([uop| free])
  | `($(_) val_not) => `([uop| not])
  | `($(_) val_opp) => `([uop| -])
  | `($(_) val_get) => `([uop| !])
  | `($(_) val_add) => `([bop| +])
  | `($(_) val_mul) => `([bop| *])
  | `($(_) val_sub) => `([bop| -])
  | `($(_) val_div) => `([bop| /])
  | `($(_) val_lt) => `([bop| <])
  | `($(_) val_gt) => `([bop| >])
  | `($(_) val_le) => `([bop| <=])
  | `($(_) val_ge) => `([bop| >=])
  | `($(_) val_eq) => `([bop| =])
  | `($(_) val_neq) => `([bop| !=])
  | `($(_) val_mod) => `([bop| mod])
  | `($(_) val_set) => `([bop| :=])
  | `($(_) val_ptr_add) => `([bop| ++])
  | _ => throw ( )

Finmap.liftOn₂ in Mathlib.Data.Finmap

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

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

sapply in Ssreflect.ApplyIn

@[app_unexpander wp] def unexpandWP : Lean.PrettyPrinter.Unexpander
  | `($(_) [lang| $t] fun $v ↦ $Q) => `(WP[$t:lang]{$v, $Q})
  | _ => throw ( )

AList.lookup in Mathlib.Data.List.AList

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

String in Init.Prelude

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

Lean.Meta.mkAppOptM in Lean.Meta.AppBuilder

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

scoped instance (priority := 1) EPCM : PartialCommMonoid val where
  valid := valid
  valid_add := by admit /- proof elided -/

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"

def qwand {A} (Q1 Q2 : A → hProp) : hProp :=
  hforall (fun (x : A) => hwand (Q1 x) (Q2 x))

t in Ssreflect.IntroPats

skip in Talk.DemoLeanSSR

Unit in Init.Prelude

instance : Coe (List var) (List trm) where
  coe := trms_vals

scoped instance (priority := 0) EPCM' : PartialCommMonoidWRT val add valid where
  addE := by admit /- proof elided -/

AList.entries in Mathlib.Data.List.AList

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 ]

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

Finmap.Disjoint in Mathlib.Data.Finmap

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

abbrev ctx := AList (fun _ : var ↦ val)

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

def val_abs : val := [lang|
  fun i =>
    let c := i < 0 in
    let m := 0 - 1 in
    let j := m * i in
    if c then j else i ]

sby in Ssreflect.Done

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

@[app_unexpander val_unit] def unexpandUnitL : Lean.PrettyPrinter.Unexpander
  | `($(_)) => `([lang| ()])

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

List.NodupKeys in Mathlib.Data.List.Sigma

List.foldl in Init.Prelude

move in Ssreflect.Basic

@[default_instance]
instance : HStar hProp hProp hProp where
  hStar := hstar

instOfNatNat in Init.Prelude

List.cons in Init.Prelude

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

def wpgen_ref (x : var) (t1 t2 : trm) : formula :=
  fun Q ↦ ∃ʰ v, ⌜t1 = trm_val v⌝ ∗
    h∀ p, (p ~~> v) -∗ protect (wp (subst x p t2) (fun hv ↦ Q hv ∗ ∃ʰ u, p ~~> u))

def haffine (_ : hProp) :=
  True

@[simp]
abbrev add : val -> val -> val
  | .val_int i, .val_int j => val.val_int (i + j)
  | _, _ => val.val_unit

@[app_unexpander trm_val] def unexpandVal : Lean.PrettyPrinter.Unexpander := fun x =>
  match x with
  | `($(_) $x:ident) =>
    match x with
    | `(val_array_get) => `($x)
    | `(val_array_set) => `($x)
    | `(val_unit) => `([lang| ()])
    | `(val_array_make) => `([uop| mkarr])
    | _ => `([lang| $x:ident])
  | `($(_) $x) =>
    match x with
    | `($n:num) => `([lang| $n:num])
    | `($n:ident) => `([lang| $n:ident])
    | `(lang|⟨$n:term⟩) => `([lang| ⟨$n⟩])
    | `([lang| $_]) => return x
    | `([uop| $_]) => return x
    | `([bop| $_]) => return x
    | t => `([lang| ⟨$t⟩])
  | _ => throw ( )

fresh in Ssreflect.Utils

abbrev default_get' := default_get (val_int 0)

AList.erase in Mathlib.Data.List.AList

Int.natAbs in Init.Data.Int.Basic

Lean.levelOne in Lean.Level
</all_available_defs>

<all_available_lemmas>
lemma insert_disjoint_l (h1 h2 : state) (x : loc) (v : val) :
  h2.Disjoint (h1.insert x v) →
  x ∉ h2 ∧ h2.Disjoint h1

lemma remove_not_in_l (h1 h2 : state) (p : loc) :
  p ∉ h1 →
  (h1 ∪ h2).erase p = h1 ∪ h2.erase p

lemma non_mem_union (h1 h2 : state) :
  a ∉ h1 ∪ h2 ↔ a ∉ h1 ∧ a ∉ h2

lemma insert_delete_id (h : state) (p : loc) :
  p ∉ h →
  h = (h.insert p v).erase p

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

lemma evalunop_unique :
  evalunop op v P → evalunop op v P' → P = P'

lemma evalbinop_unique :
  evalbinop op v1 v2 P → evalbinop op v1 v2 P' → P = 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 hstar_flip_0 :
  emp = emp

lemma wpgen_sound t :
  formula_sound t (wpgen t)

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

lemma wp_app_fun x v1 v2 t1 Q :
  v1 = val_fun x t1 →
  wp (subst x v2 t1) Q ==> wp (trm_app v1 v2) Q

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

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

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

lemma in_read_union_l (h1 h2 : state) (x : loc) :
  x ∈ h1 → read_state x (h1 ∪ h2) = read_state x h1

lemma xsimp_lr_hwand :
  XSimp (emp, emp, (H1 ∗ Hla)) (emp, emp, H2 ∗ emp) ->
  XSimp (Hla, emp, emp) ((H1 -∗ H2) ∗ emp, emp, emp)

lemma mem_erase_right (s : state) :
  p ∈ s.erase x → p ∈ s

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

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

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

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_pick_last_3 :
  h1 ∗ h2 ∗ h3 = h3 ∗ h1 ∗ h2

@[simp]
lemma intersect_mem_iff (s₁ s₂ : state) :
  p ∈ (intersect s₁ s₂) ↔ p ∈ s₁ ∧ p ∈ s₂

lemma xsimp_l_cancel_hwand_hstar :
  XSimp (Hla, Hlw, (H2 -∗ H3) ∗ Hlt) HR →
  XSimp ((H1 ∗ Hla), (((H1 ∗ H2) -∗ H3) ∗ Hlw), Hlt) HR

lemma xsimpl_lr_cancel_htop :
  XSimp (Hla, Hlw, Hlt) (Hra, (⊤ ∗ Hrg), Hrt) ->
  XSimp ((H ∗ Hla), Hlw, Hlt) (Hra, (⊤ ∗ Hrg), Hrt)

lemma hstar_flip_2 :
  h1 ∗ h2 ∗ emp = h2 ∗ h1 ∗ emp

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

lemma xwhile_inv_measure_lemma_up (Xtop : Int) (I : Bool -> Int -> hProp) :
  (H ==> H' ∗ ∃ʰ b a, I b a) ->
  (∀ b X, I b X ==> wp F1 (fun bv => I b X ∗ ⌜bv = b⌝)) ->
  (∀ X, I true X ==> wp F2 (fun _ => ∃ʰ b X', ⌜X < X' ∧ X' <= Xtop⌝ ∗ I b X')) ->
  ((fun _ => H' ∗ ∃ʰ a, I false a) ===> Q) ->
  H ==> wp (trm_while F1 F2) Q

lemma of_nat_nat n :
  OfNat.ofNat n = n

lemma int_index_eq {_ : Inhabited α} (L : List α) (i : Int) :
  L[i.natAbs]! = L[i]!

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

@[xapp] -- Another way to add a triple lemma to a hint database
lemma incr_spec (p : loc) (n : Int) :
  { p ~~> n }
  [ ++p ]
  { p ~~> val_int (n + 1) }

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

lemma qstar_simp (Q1 : α -> hProp) :
  (Q1 ∗ H) x = Q1 x ∗ H

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

lemma hstar_hempty_r H :
  H ∗ emp = H

lemma triple_abs (i : Int) :
  triple [lang| val_abs i]
    emp
    (fun r ↦ ⌜r = val_int i.natAbs⌝)

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

lemma hstar_pick_last_6 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 = h6 ∗ h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5

lemma hstar_flip_1 :
  h1 ∗ emp = h1 ∗ emp

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

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

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

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

lemma hstar_pick_4 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h = h4 ∗ h1 ∗ h2 ∗ h3 ∗ h

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

lemma hseg_focus_relative (k : Nat) L p j (v : 0 <= k ∧ k < L.length):
  hseg L p j ==>
  hcell L[k]! p (j + k)
    ∗ (h∀ w, hcell w p (j + k) -∗ hseg (L.set k w) p j)

lemma exact_imp_eval :
  evalExact s t Q → eval s t Q

lemma himpl_antisym H1 H2:
  (H1 ==> H2) → (H2 ==> H1) → (H1 = H2)

lemma wpgen_fun_sound x t1 Fof :
  (forall vx, formula_sound (subst x vx t1) (Fof vx)) →
  formula_sound (trm_fun x t1) (wpgen_fun Fof)

lemma wpgen_fail_sound t :
  formula_sound t wpgen_fail

lemma xsimp_r_keep :
  XSimp hl (h ∗ hra, hrg, hrt) ->
  XSimp hl (hra, hrg, h ∗ hrt)

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

lemma hstar_pick_last_7 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 = h7 ∗ h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6

lemma himpl_hstar_hpure_r P H H' :
   P →
   (H ==> H') →
   H ==> ⌜P⌝ ∗ H'

lemma hstar_flip_10 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ h8 ∗ h9 ∗ h10 ∗ emp =
  h10 ∗ h9 ∗ h8 ∗ h7 ∗ h6 ∗ h5 ∗ h4 ∗ h3 ∗ h2 ∗ h1 ∗ emp

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

lemma xseq_lemma H F1 F2 Q :
  H ==> F1 (fun _ => F2 Q) →
  H ==> wpgen_seq F1 F2 Q

lemma set_out_of_bounds (L : List val) i v :
  i < 0 ∨ i ≥ L.length →
  L.set i v = L

lemma union_same_eq_r (s₁ s₂ s₃ : state) :
  s₁.Disjoint s₃ →
  s₂.Disjoint s₃ →
  s₁ ∪ s₃ = s₂ ∪ s₃ →
  s₁ = s₂

lemma insert_same_eq (h1 h2 : state) :
  p ∉ h1 → p ∉ h2 →
  h1.insert p v = h2.insert p v →
  h1 = h2

lemma isubst_perm {al al'} t (hp : al.entries.Perm al'.entries) :
  isubst al t = isubst al' t

lemma himpl_qwand_r A (Q1 Q2 : A → hProp) H :
  Q1 ∗ H ===> Q2 →
  H ==> (Q1 -∗ Q2)

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

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

lemma keys_eq_mem_r (h1 h2 : state) :
  h1.keys = h2.keys →
  p ∈ h2 →
  p ∈ h1

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

lemma hstar_hgc_hgc : /GC ∗ /GC = /GC

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

lemma haffine_hpure P :
  haffine ⌜P⌝

lemma hstar_pick_6 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h = h6 ∗ h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h

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

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

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

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

lemma hstar_pick_last_9 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ h8 ∗ h9 = h9 ∗ h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ h8

lemma wp_equiv t H Q :
  (H ==> wp t Q) ↔ triple t H Q

lemma sum_disjoint_hstar (H : β -> hProp) (fs : Finset β) [inst : PartialCommMonoid val] :
  (∀ᵉ (i ∈ fs) (j ∈ fs), i ≠ j -> (H i).Disjoint (H j)) ->
  (∑ b in fs, H b) =
  (@Finset.sum β hProp (@instAddCommMonoidHPropOfPartialCommMonoidVal (PartialCommMonoidWRT.toPartialCommMonoid add valid)) fs H)

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

lemma isubst_empty t : isubst ∅ t = t

lemma mkstruct_sound t F :
  formula_sound t F →
  formula_sound t (mkstruct F)

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 nat_abs_succ (n : Int) :
  n ≥ 0 → (n + 1).natAbs = n.natAbs + 1

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

lemma remove_not_in_r (h1 h2 : state) (p : loc) :
  p ∉ h2 →
  (h1 ∪ h2).erase p = h1.erase p ∪ h2

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

lemma xwhile_inv_lemma (I : Bool -> α -> hProp)
  (F1 F2 : formula) :
    WellFounded R ->
    (H ==> ∃ʰ b a, I b a) ->
    (∀ (S: formula) b X, structural S ->
        (∀ b a', R a' X -> (I b a') ==> S fun _ => ∃ʰ a, I false a) ->
        I b X ==> wpgen_if_trm F1 (wpgen_seq F2 S) (wpgen_val val_unit) fun _ => ∃ʰ a, I false a) ->
    H ==> wpgen_while F1 F2 (fun _ => ∃ʰ a, I false a)

@[simp]
lemma Heap.add_union_validInter (h₁ h₂ : Heap.heap val) {h : h₁ ⊥ʰ h₂} :
  h₁ +ʰ h₂ = h₁ ∪ h₂

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

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

lemma validInter_comm (h₁ h₂ : heap) :
  h₁ ⊥ʰ h₂ = h₂ ⊥ʰ h₁

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

lemma hexists_inv A (J : A → hProp) h :
  (hexists J) h → exists x, J x h

lemma hadd_single_gen (v v' : val) [PartialCommMonoid val] :
  PartialCommMonoid.valid v ->
  PartialCommMonoid.valid v' ->
  (p ~~> v) + (p ~~> v') = p ~~> (v + v')

lemma qstarE α (Q1 : α → hProp)  (H : hProp):
  Q1 ∗ H = fun x => Q1 x ∗ H

lemma wpgen_let_sound F1 F2of x t1 t2 :
  formula_sound t1 F1 →
  (forall v, formula_sound (subst x v t2) (F2of v)) →
  formula_sound (trm_let x t1 t2) (wpgen_let F1 F2of)

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

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_r H1 H2 H2' :
  H2 ==> H2' →
  (H1 ∗ H2) ==> (H1 ∗ H2')

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

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 kmerge_comm_perm (l₁ l₂ : List (Sigma (fun _ : loc => val))) (nd₁ : l₁.NodupKeys) (nd₂ : l₂.NodupKeys) :
  (kmerge l₁ l₂).Perm $ (kmerge l₂ l₁)

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

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 hstars_simp_keep_lemma :
  H1 ==> (H ∗ Ha) ∗ Ht →
  H1 ==> Ha ∗ H ∗ Ht

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

lemma qwand_equiv H A (Q1 Q2 : A → hProp) :
  H ==> (Q1 -∗ Q2) ↔ (Q1 ∗ H) ===> Q2

lemma union_diff_disjoint_r (h₁ h₂ h₃ : state) :
  h₂.Disjoint h₃ →
  (h₁ ∪ h₂) \ h₃ = (h₁ \ h₃) ∪ h₂

lemma hseg_cons v p j L :
  hseg (v :: L) p j = hcell v p j ∗ hseg L p (j + 1)

@[simp]
lemma qimpl_refl_resolve (Q : α -> hProp) : (Q ===> Q) = True

lemma hstar_pick_last_4 :
  h1 ∗ h2 ∗ h3 ∗ h4 = h4 ∗ h1 ∗ h2 ∗ h3

lemma validInter_of_disjoint (h₁ h₂ : heap) :
  h₁.Disjoint h₂ ->  h₁ ⊥ʰ h₂

lemma haffine_hforall A {_ : Inhabited A} (J : A → hProp) :
  (forall x, haffine (J x)) →
  haffine (hforall J)

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

lemma intersect_comm (s2 d : state) (a₁ : loc) (b₁ : val) (a₂ : loc) (b₂ : val) :
  (fun s x _ ↦ if x ∈ s2 then s else Finmap.erase x s)
      ((fun s x _ ↦ if x ∈ s2 then s else Finmap.erase x s) d a₁ b₁) a₂ b₂ =
    (fun s x _ ↦ if x ∈ s2 then s else Finmap.erase x s)
      ((fun s x _ ↦ if x ∈ s2 then s else Finmap.erase x s) d a₂ b₂) a₁ b₁

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

lemma haffine_hempty : haffine emp

lemma triple_for_raw (x:var) (n1 n2: Int) t3 H (Q:val->hProp) :
  triple (
    if (n1 < n2)
      then (trm_seq (subst x n1 t3) (trm_for x (val_int $ n1+1) n2 t3))
      else val_unit) H Q ->
  triple (trm_for x n1 n2 t3) H Q

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

lemma mkstruct_frame F H Q :
  (mkstruct F Q) ∗ H ==> mkstruct F (Q ∗ H)

lemma kmerge_mem2 (l₁ l₂ : List (Sigma (fun _ : loc => val))) (nd₁ : l₁.NodupKeys) (nd₂ : l₂.NodupKeys) -- necessary
    (a : Sigma (fun _ : loc => val)) : a ∈ (kmerge l₁ l₂) ↔
      if a.1 ∈ l₁.keys
      then (if a.1 ∈ l₂.keys then Option.merge (· + ·) (l₁.dlookup a.1) (l₂.dlookup a.1) = .some a.2 else a ∈ l₁)
      else a ∈ l₂

lemma hstar_pick_8 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ h8 ∗ h = h8 ∗ h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ h

lemma diff_disjoint_eq (s₁ s₂ s₃ : state) :
  s₁.Disjoint s₂ →
  s₂.keys = s₃.keys →
  (s₁ ∪ s₂) \ s₃  = s₁

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

lemma qwand_cancel A (Q1 Q2 : A → hProp) :
  Q1 ∗ (Q1 -∗ Q2) ===> Q2

lemma hind_hstar_hempty :
  ⊤⊤ ∗ emp ==> ⊤⊤

lemma trm_apps_app :
  trm_apps t1 (ts ++ ts') = trm_apps (trm_apps t1 ts) ts'

lemma Int.Ico_insert_succ_left (h : i < j) : insert i ⟦i+1, j⟧ = ⟦i, j⟧

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

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

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

lemma wp_frame t H Q :
  (wp t Q) ∗ H ==> wp t (Q ∗ H)

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

lemma eval_conseq s t Q1 Q2 :
  eval s t Q1 →
  Q1 ===> Q2 →
  eval s t Q2

lemma intersect_foldl_mem (s₂ : state) (l₁ l₂ : @AList loc fun _ ↦ val):
  p ∈ List.foldl (fun d s ↦ if s.fst ∈ s₂ then d else Finmap.erase s.fst d)
    l₁.toFinmap l₂.entries →
  p ∈ l₁

lemma hwand_curry_eq H1 H2 H3 :
  (H1 ∗ H2) -∗ H3 = H1 -∗ (H2 -∗ H3)

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 hstar_pick_10 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ h8 ∗ h9 ∗ h10 ∗ h = h10 ∗ h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ h8 ∗ h9 ∗ h

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

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

<local_ctx>
import Mathlib.Data.Finmap

import Mathlib.Data.Finset.Basic

import Mathlib.Data.Multiset.Nodup

import SPLean.Common.State

import SPLean.Common.Util

import SPLean.Theories.HProp

import SPLean.Theories.XSimp

open trm val prim

notation "funloc" p "↦" H =>
  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)

section evalProp
</local_ctx>

<target_theorem>
lemma evalExact_post :
  eval s t Q → evalExact s t Q' → Q' ===> Q :=
</target_theorem>

