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 hsingle (p : loc) (v : val) : hProp :=
  fun h => (h = Finmap.singleton p v)

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

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

abbrev triple (t : trm) (H : hProp) (Q : val → hProp) : Prop :=
  forall s, H s → eval s t Q

def hrange (L : List val) (p : loc) : hProp :=
  match L with
  | []      => emp
  | x :: L' => (p ~~> x) ∗ (hrange L' (p + 1))

def wpgen_alloc (x : var) (t1 t2 : trm) : formula :=
  fun Q ↦ ∃ʰ n : ℤ,
    ⌜n ≥ 0 ∧ t1 = trm_val n⌝ ∗
    h∀ p,
      (hrange (make_list n.natAbs val_uninit) p) -∗
      protect wp (subst x p t2) (Q ∗ ⌜p ≠ null⌝ ∗ ∃ʰ L, ⌜L.length = n⌝ ∗ hrange L p)

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
  )

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

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))

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

@[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 ( )

def hgc := htop

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

def hcell (v : val) (p : loc) (i : Int) : hProp :=
  ((p + 1 + (Int.natAbs i)) ~~> v) ∗ ⌜i >= 0⌝

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

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

Lean.Meta.forallTelescope in Lean.Meta.Basic

@[app_unexpander trm_fixs] def unexpandTFixs : 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 ( )

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

noncomputable def sP' (h : heap) (t : trm) : (val → hProp) :=
  if ex : ∃ Q, evalExact h t Q then choose ex else fun _ _ ↦ 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)

@[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 ( )

Lean.PrettyPrinter.Unexpander in Init.Prelude

fresh in Ssreflect.Utils

Lean.Meta.MetaM in Lean.Meta.Basic

instance : Coe loc val where
  coe l := val.val_loc l

def hstars_simp_start : TacticM Unit := withMainContext do
  let goal <- getGoalStxNN
  match goal with
  | `(himpl $_ $_) =>
      {| apply hstars_simp_start_lemma ; try srw ?hstar_assoc |}
  | _          => throwError "hstars_simp_start failure"

List.cons in Init.Prelude

outParam in Init.Prelude

Lean.Name.mkSimple in Init.Prelude

def _root_.Lean.TSyntax.isMVarStx (x : Term) : Bool :=
  match x with
  | `(?$_) => true
  | _ => false

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 ]

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 }

Lean.Meta.inferType in Lean.Meta.Basic

instance [Inhabited α] : Inhabited (Labeled α) := ⟨⟨0, default⟩⟩

Finmap.erase in Mathlib.Data.Finmap

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

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?

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 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
        () ]

@[app_unexpander val_loc] def unexpandLoc : Lean.PrettyPrinter.Unexpander
  | `($(_) $n:ident) => `($n:ident)
  | `($(_) $n:term) => `(lang|⟨$n:term⟩)
  | _ => throw ( )

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 ]

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

Inhabited in Init.Prelude

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

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"

AList.erase in Mathlib.Data.List.AList

List.zip in Init.Data.List.Basic

Lean.binderIdent in Init.Notation

Lean.trace in Lean.Util.Trace

Option.any in Init.Data.Option.Basic

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

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

List.ofFn in Init.Data.List.OfFn

instance : Coe var trm where
  coe x := trm.trm_var x

sdone in Ssreflect.Done

Finset.Icc in Mathlib.Order.Interval.Finset.Defs

@[heapSimp]
def protect (x : α) := x

abbrev add : val -> val -> val := fun _ _ => default

@[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 ( )

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

@[app_unexpander trm_let] def unexpandLet : Lean.PrettyPrinter.Unexpander
  | `($(_) $x:str [lang| $t1] [lang| $t2]) =>
    let str := x.getString
    let name := Lean.mkIdent $ Lean.Name.mkSimple str
    `([lang| let $name := $t1 in $t2])
  | _ => throw ( )

Sigma in Init.Core

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

skip in Talk.DemoLeanSSR

Lean.PrettyPrinter.delab in Lean.PrettyPrinter.Delaborator.Basic

def getGoalStxAll : Lean.Elab.Tactic.TacticM Syntax := do
  delabAll $ <- getMainTarget

Int.natAbs in Init.Data.Int.Basic

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

class AddCommMonoidWRT (α : Type) (add' : semiOutParam $ α -> α -> α) extends AddCommMonoid α where
  addE : (· + ·) = add'

@[default_instance]
instance : HWand hProp hProp hProp where
  hWand := hwand

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

Lean.Expr in Lean.Expr

def formula_sound (t : trm) (F : formula) : Prop :=
  forall Q, F Q ==> wp t Q

Lean.pp.all in Lean.PrettyPrinter.Delaborator.Options

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

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

Lean.Meta.subst in Lean.Meta.Tactic.Subst

Lean.Meta.mkAppOptM in Lean.Meta.AppBuilder

instance : Coe prim val where
  coe x := val.val_prim x

structure Labeled (α : Type*) where
  lab : LabType
  val : α

def wpgen_if (t : trm) (F1 F2 : formula) : formula :=
  fun Q ↦ ∃ʰ (b : Bool),
    ⌜t = trm_val (val_bool b)⌝ ∗ (if b then F1 Q else F2 Q)

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

instance [PartialCommMonoid val] : AddCommMonoid hProp where
  zero := emp
  add  := hadd
  nsmul := nsmulRec
  add_comm  := by admit /- proof elided -/

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

List.length in Init.Prelude

instance : Coe Int val where
  coe z := val.val_int z

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

AList.lookup in Mathlib.Data.List.AList

@[app_unexpander val_int] def unexpandInt : Lean.PrettyPrinter.Unexpander
  | `($(_) $n:num) => `($n:num)
  | `($(_) $n:ident) => `($n:ident)
  | `($(_) $n:term) => `(lang|⟨$n:term⟩)
  | _ => throw ( )

Function.update in Mathlib.Logic.Function.Basic

scase in Ssreflect.Elim

@[app_unexpander val_bool] def unexpandBool : Lean.PrettyPrinter.Unexpander
  | `($(_) $n:ident) => `($n:ident)
  | `($(_) $n:term) => `(lang|⟨$n:term⟩)
  | _ => throw ( )

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"

| step_seq_ctx : forall s1 s2 t1 t1' t2,
      step s1 t1 s2 t1' →
      step s1 (trm_seq t1 t2) s2 (trm_seq t1' t2)
  | step_let_ctx : forall s1 s2 x t1 t1' t2,
      step s1 t1 s2 t1' →
      step s1 (trm_let x t1 t2) s2 (trm_let x t1' t2)
  | step_app_arg_1 : forall s1 s2 t1 t1' t2,
      step s1 t1 s2 t1' →
      step s1 (trm_app t1 t2) s2 (trm_app t1' t2)
  | step_app_arg2 : forall s1 s2 v1 t2 t2',
      step s1 t2 s2 t2' →
      step s1 (trm_app v1 t2) s2 (trm_app v1 t2')

| 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

HAdd in Init.Prelude

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

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

Lean.Meta.withAssignableSyntheticOpaque in Lean.Meta.Basic

Unit in Init.Prelude

def wpgen_if_trm (F0 F1 F2 : formula) : formula :=
  wpgen_let F0 (fun v ↦ mkstruct (wpgen_if v F1 F2))

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

instance : OfNat val n where
  ofNat := val.val_int n

@[app_unexpander trm_ref] def unexpandRef : Lean.PrettyPrinter.Unexpander
  | `($(_) $x:str [lang| $t1] [lang| $t2]) =>
    let str := x.getString
    let name := Lean.mkIdent $ Lean.Name.mkSimple str
    `([lang| ref $name := $t1 in $t2])
  | _ => throw ( )

Finmap.lookup in Mathlib.Data.Finmap

@[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 ( )

Not in Init.Prelude

def st2 : state := ((Finmap.singleton 0 2).insert 2 2).insert 1 2

seq in Talk.DemoLeanSSR

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

List.zipWith in Init.Data.List.Basic

@[app_unexpander val_real] def unexpandReal : Lean.PrettyPrinter.Unexpander
  | `($(_) $n:num) => `($n:num)
  | `($(_) $n:ident) => `($n:ident)
  | `($(_) $n:term) => `(lang|⟨$n:term⟩)
  | _ => throw ( )

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))))
</all_available_defs>

<all_available_lemmas>
lemma hstar_comm_assoc (H1 H2 H3 : hProp) :
  H1 ∗ H2 ∗ H3 = H2 ∗ H1 ∗ H3

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

lemma himpl_refl H : H ==> H

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

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

lemma hstar_inv (H1 H2 : hProp) h:
  (H1 ∗ H2) h →
  exists h1 h2, H1 h1 ∧ H2 h2 ∧ Finmap.Disjoint h1 h2 ∧ h = h1 ∪ h2

lemma hstar_comm H1 H2 :
  H1 ∗ H2 = H2 ∗ H1

lemma hstar_assoc H1 H2 H3 :
  (H1 ∗ H2) ∗ H3 = H1 ∗ (H2 ∗ H3)

lemma hstar_hempty_l H :
  emp ∗ H = H

lemma hstar_hempty_r H :
  H ∗ emp = H

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

lemma hpure_inv P h :
  ⌜P⌝ h →
  P ∧ h = ∅

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

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

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

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

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

lemma himpl_hforall_r A (J : A → hProp) H :
  (forall x, H ==> J x) →
  H ==> (hforall J)

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

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

lemma hwand_equiv H0 H1 H2 :
  (H0 ==> H1 -∗ H2) ↔ (H1 ∗ H0 ==> H2)

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

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

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

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

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

lemma nonneg_eq_abs (n : Int) :
  0 ≤ n → n.natAbs = n

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

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

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

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

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

lemma harray_focus_read i L p :
  0 <= i ∧ i < L.length →
  harray L p ==>
  hcell L[Int.natAbs i]! p i ∗ (hcell L[Int.natAbs i]! p i -∗ harray L p)

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

lemma natabs_eq (a : Nat) :
  a = (Int.ofNat a).natAbs

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

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

lemma triple_conseq t H' Q' H Q :
  triple t H' Q' →
  H ==> H'→
  Q' ===> Q →
  triple t H Q

lemma triple_frame t H (Q : val -> hProp) H' :
  triple t H Q →
  triple t (H ∗ H') (Q ∗ H')

lemma triple_hpure t P H Q :
  (P → triple t H Q) →
  triple t (⌜P⌝ ∗ H) Q

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

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

lemma int_eq_sub (l m n : ℤ) :
  l + m = n → l = n - m

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

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 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

theorem diff_non_mem (h₁ h₂ : state) :
  p ∈ h₂ → p ∉ h₁ \ h₂

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

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

lemma lookup_diff_none (h₁ h₂ : state) :
  p ∈ h₂ →
  (h₁ \ h₂).lookup p = none

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

lemma conseq_nil B (l : Nat) :
  conseq ([] : List B) 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 disjoint_single_conseq B l l' L (v : B) :
  (l < l') ∨ (l ≥ l' + L.length) →
  Finmap.Disjoint (Finmap.singleton l v) (conseq L l')

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

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

lemma hstar_flip_5 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ emp = h5 ∗ h4 ∗ h3 ∗ h2 ∗ h1 ∗ emp

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

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

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

lemma xsimp_r_hexists α (x : α) hl hra hrg hrt j :
  XSimp hl (hra, hrg, j x ∗ hrt) ->
  XSimp hl (hra, hrg, (hexists j) ∗ hrt)

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

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

lemma hpure_inv_hempty P h :
  ⌜P⌝ h →
  P ∧ emp h

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

lemma non_mem_diff_helper2 (h₁ : state) (l : @AList loc fun _ ↦ val) :
  p ∈ List.foldl (fun d s ↦ Finmap.erase s.fst d) (Finmap.erase a h₁) l.entries →
  p ∈ List.foldl (fun d s ↦ Finmap.erase s.fst d) h₁ l.entries

lemma triple_app_fix_direct v2 f x t1 H Q :
  f ≠ x →
  triple (subst x v2 (subst f (val_fix f x t1) t1)) H Q →
  triple (trm_app (val_fix f x t1) v2) H Q

lemma Option.merge_assoc (h : Associative f) (a b c : Option α) :
  Option.merge f (Option.merge f a b) c = Option.merge f a (Option.merge f b c)

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

lemma sP'_strongest :
  eval h t Q → sP' h t ===> Q

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

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 xsimp_lr_cancel_same :
  XSimp (hla, hlw, hlt) (hra, hrg, hrt) ->
  XSimp (h ∗ hla, hlw, hlt) (hra, hrg, h ∗ hrt)

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

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

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 triple_lt n1 n2 :
  triple (trm_app val_lt (val_int n1) (val_int n2))
    emp
    (fun r ↦ ⌜r = val_bool (n1 < n2)⌝)

lemma hempty_inv h :
  emp h → h = ∅

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

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

lemma triple_array_length_hheader (n : Int) (p : loc) :
  triple (trm_app val_array_length p)
    (hheader n p)
    (fun r ↦ ⌜r = n⌝ ∗ hheader n p)

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

lemma hstar_pick_1 :
  h1 ∗ h = h1 ∗ h

lemma xsimpl_lr_hforall :
  (forall x, XSimp (emp, emp, Hla) (emp, emp, J x ∗ emp)) ->
  XSimp (Hla, emp, emp) ((hforall J) ∗ emp, emp, emp)

lemma triple_neg (b1 : Bool) :
  triple (trm_app val_neg b1)
    emp
    (fun r ↦ ⌜r = val_bool (¬b1)⌝)

lemma steps_trans s1 s2 s3 t1 t2 t3 :
  steps s1 t1 s2 t2 →
  steps s2 t2 s3 t3 →
  steps s1 t1 s3 t3

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

lemma htop_intro h :
  ⊤ h

lemma findIdx_spec' (arr : loc) (f : Int -> ℝ)
  (z n : ℤ) (_ : z <= n) (_ : 0 <= z) (N : ℕ) (_ : n <= N) :
  Set.InjOn f ⟦z, n⟧ ->
  i ∈ ⟦z, n⟧ ->
  { arr(arr, x in N => f x) }
  [ findIdx arr ⟨f i⟩ z n ]
  { v, ⌜ v = i ⌝ ∗ arr(arr, x in N => f x) }

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

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

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

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

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

lemma triple_fix f x t1 H Q :
  H ==> Q (val_fix f x t1) →
  triple (trm_fix f x t1) H Q

lemma triple_val v H Q :
  H ==> Q v →
  triple (trm_val v) H Q

lemma sum_single (v : β -> Int) (fs : Finset β) :
  (p ~~> 0) + ∑ i in fs, (p ~~> v i) = p ~~> val.val_int (∑ i in fs, v i)

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

lemma insert_mem_keys (s : state) :
  p ∈ s →
  (s.insert p v).keys = s.keys

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

lemma htop_eq :
  ⊤ = ∃ʰ H, H

lemma remove_disjoint_union_l (h1 h2 : state) (x : loc) :
  x ∈ h1 → Finmap.Disjoint h1 h2 →
  Finmap.erase x (h1 ∪ h2) = Finmap.erase x h1 ∪ h2

lemma union_same_keys (h₁ h₂ h₃ : state) :
  h₁.Disjoint h₃ → h₂.Disjoint h₃ →
  (h₁ ∪ h₃).keys = (h₂ ∪ h₃).keys →
  h₁.keys = h₂.keys

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

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

@[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) }

@[simp]
lemma labSet_mem :
  x ∈ ⟪l, s⟫ ↔ x.lab = l ∧ x.val ∈ s

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

@[simp]
lemma Heap.add_empty_l (h : heap) : ∅ +ʰ h = h

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 union_monotone_r (s₃ s₁ s₂ : state) :
  s₁ = s₂ →
  s₁ ∪ s₃ = s₂ ∪ s₃

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

lemma triple_add n1 n2 :
  triple (trm_app val_add (val_int n1) (val_int n2))
    emp
    (fun r ↦ ⌜r = val_int (n1 + n2)⌝)

lemma findIdx_spec (arr : loc) (f : ℤ -> ℝ) (target : ℝ)
  (z n : ℤ) (_ : z <= n) (_ : 0 <= z) (N : ℕ) (_ : n <= N) :
  Set.InjOn f ⟦z, n⟧ ->
  target ∈ f '' ⟦z, n⟧ ->
  { arr(arr, i in N => f i) }
  [ findIdx arr target z n ]
  { v, ⌜ v = f.invFunOn ⟦z, n⟧ target ⌝ ∗ arr(arr, i in N => f i) }

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

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

lemma list_middle_inv (A : Type) (n : Nat) (l : List A) :
  n < l.length →
  exists (x : A) (l1 l2 : List A),
    l = l1 ++ x :: l2 ∧ l1.length = n

lemma diff_insert_intersect_id (s₁ s₂ : state) :
  (s₁ \ s₂) ∪ (intersect s₁ s₂) = s₁

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

lemma himpl_hstar_trans_r H1 H2 H3 H4 :
  H1 ==> H2 →
  H3 ∗ H2 ==> H4 →
  H3 ∗ H1 ==> H4

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

lemma eval_div s n1 n2 Q :
  n2 ≠ 0 ->
  Q (val_int (n1 / n2)) s ->
  eval s (trm_app (trm_app val_div (val_int n1)) (val_int n2)) Q

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

lemma xsimpl_r_hgc_or_htop :
  XSimp HL (Hra, (H ∗ Hrg), Hrt) ->
  XSimp HL (Hra, Hrg, (H ∗ Hrt))

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 hpure_intro_hempty P h :
  emp h → P → ⌜P⌝ h

@[ext]
lemma hProp.ext (H1 H2 : hProp) :
  (∀ h, H1 h ↔ H2 h) → H1 = H2

lemma triple_harrayFun_set (f : ℤ -> val) (p : loc) (i : Int) (v : val) :
   0 <= i ∧ i < (n : ℕ) →
   triple (trm_app val_array_set p i v)
    (harrayFun f n p)
    (fun _ => harrayFun (Function.update f i v) n p)

lemma getElem!_nil_intint (n : Int) :
  ([] : List Int)[n]! = default

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

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

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

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

lemma add_pointer_spec (p q : loc) (n m : Int) (_ : m >= 0) :
  { p ~~> n ∗ q ~~> m }
  [ add_pointer p q ]
  { v, ⌜v = n+m⌝ ∗ ⊤  }

lemma xpull_protect (h : H1 ==> protect H2) : H1 ==> H2

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

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

lemma kmerge_assoc_perm (l₁ l₂ l₃ : List (Sigma (fun _ : loc => val))) (nd₁ : l₁.NodupKeys) (nd₂ : l₂.NodupKeys) (nd₃ : l₃.NodupKeys) :
  (kmerge (kmerge l₁ l₂) l₃).Perm $ (kmerge l₁ (kmerge l₂ l₃))

lemma hstar_pick_last_8 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ h8 = h8 ∗ h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7

lemma hProp.disjoint_sum (H : hProp) (fs : Finset β) :
  (∀ i ∈ fs, H.Disjoint (H' i)) ->
    H.Disjoint (fs.sum H')

lemma evalExact_frame_unop_binop :
  s.Disjoint h2 →
  evalExact (s ∪ h2) t (purepost (s ∪ h2) P) →
  evalExact (s ∪ h2) t (qstar (purepost s P) (tohProp fun h ↦ h = h2))

theorem mem_diff_rev (h₁ h₂ : state) :
  p ∈ h₁ ∧ p ∉ h₂ → p ∈ h₁ \ h₂

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

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

lemma Heap.add_assoc (h₁ h₂ h₃ : heap) : (h₁ +ʰ h₂) +ʰ h₃ = h₁ +ʰ h₂ +ʰ h₃

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

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 insert_same (h1 h2 : state) :
  p ∉ h1 → p ∉ h2 →
  (h1.insert p v).keys = (h2.insert p v').keys →
  h1.keys = h2.keys

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

lemma xsimp_l_hwand_reorder :
  h1 = h1' ->
  XSimp (hla, ((h1' -∗ h2) ∗ hlw), hlt) hr ->
  XSimp (hla, (h1 -∗ h2) ∗ hlw, hlt) hr

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

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

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

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

lemma triple_array_get_hcell (p : loc) (i : Int) (v : val) :
  triple [lang| val_array_get p i]
    (hcell v p i)
    (fun r ↦ ⌜r = v⌝ ∗ hcell v p i)

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

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

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

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

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

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

lemma hseg_app L1 L2 p j :
  hseg (L1 ++ L2) p j =
  hseg L1 p j ∗ hseg L2 p (j + L1.length)
</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

def wp (t : trm) (Q : val → hProp) : hProp :=
  fun s ↦ eval s t Q
</local_ctx>

<target_theorem>
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 :=
</target_theorem>

