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 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

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

abbrev loc := Nat

abbrev var := String

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

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

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

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

abbrev 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

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

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

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

AList.insert in Mathlib.Data.List.AList

Lean.Meta.unhygienic in Batteries.Lean.Meta.Basic

| 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

def delabNoNotations :=
  (withOptions (fun _ =>
    (((KVMap.empty.insert
      `pp.all true).insert
      `pp.universes false).insert
      `pp.deepTerms true).insert
      `pp.fullNames true) $ PrettyPrinter.delab ·)

instance AddHProp [PartialCommMonoid val] : Add hProp := ⟨hadd⟩

abbrev ofhProp (h : val -> hProp) : val -> heap -> Prop := h

@[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 ( )

Lean.Syntax in Init.Prelude

Bool in Init.Prelude

List.keys in Mathlib.Data.List.Sigma

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

Lean.Name.cmp in Lean.Data.Name

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

@[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 ( )

scase_if in Ssreflect.Elim

instHAdd in Init.Prelude

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

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

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

def validInter (h₁ h₂ : heap) : Prop :=
  ∀ l ∈ h₁, l ∈ h₂ -> ((h₁ +ʰ h₂).lookup l).any (valid (α := val))

Iff in Init.Core

Lean.unbracketedExplicitBinders in Init.NotationExtra

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

Int.toNat in Init.Data.Int.Basic

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

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

Nonempty in Init.Prelude

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

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

Lean.Elab.Tactic.getMainTarget in Lean.Elab.Tactic.Basic

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

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

Function.update in Mathlib.Logic.Function.Basic

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

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
  )

Finmap.Disjoint in Mathlib.Data.Finmap

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

def pickTripleLemma : TacticM Name := do
  let n <- getTripleFun (<- getMainTarget)
  let .some thm := (xappExtension.getState (← getEnv)).find? n | throwError "no triple lemma found"
  return thm

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

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

@[simp]
def kmerge : List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val))
  | [], l₂ => l₂
  | s :: l₁, l₂ =>
    (if s.1 ∈ l₂.keys then
      ⟨s.1, kmerge1 s.1 s.2 l₂⟩ :: kmerge l₁ (l₂.kerase s.1)
    else s :: kmerge l₁ l₂)

List.length in Init.Prelude

Zero in Init.Prelude

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

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

abbrev default_get' := default_get (val_int 0)

Lean.explicitBinders in Init.NotationExtra

Inhabited in Init.Prelude

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

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

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

Lean.Meta.isDefEq in Lean.Meta.Basic

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?

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

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

Disjoint in Mathlib.Order.Disjoint

Or in Init.Prelude

OfNat in Init.Prelude

Finset.Ico in Mathlib.Order.Interval.Finset.Defs

shave in Ssreflect.Have

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

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

Prod.mk in Init.Prelude

AList.lookup in Mathlib.Data.List.AList

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

Unit in Init.Prelude

@[simp]
def trms_to_vals (ts:List trm) : Option (List val) := do
  match ts with
  | [] => return []
  | (trm_val v) :: ts' => v :: (<- trms_to_vals ts')
  | _ => failure

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))))

scase in Ssreflect.Elim

skip in Talk.DemoLeanSSR

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

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

def wpgen_fail : formula :=
  fun _ ↦ ⌜false⌝

instance: GetElem (List α) Int α (fun L i => i.natAbs < L.length) :=
  ⟨fun xs i _ => xs[i.natAbs]⟩

def hheader (n : Int) (p : loc) : hProp :=
  p ~~> (val_int n)

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"

Finmap.singleton in Mathlib.Data.Finmap

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

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

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))

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

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

example (L : List val) (i : Int) (_ : i.natAbs < L.length) :
  L[i.natAbs]! = L[i]! :=

tryGoal in Ssreflect.Utils

HAdd.hAdd in Init.Prelude

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

L in Archive.Hairer

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 ]

Lean.Name.mkSimple in Init.Prelude

Lean.Expr.equal in Lean.Expr

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 ]

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

List.Perm in Init.Data.List.Basic

def _root_.List.mkAlist [DecidableEq α] (xs : List α) (vs : List β) :=
  ((xs.zip vs).map fun (x, y) => ⟨x, y⟩).toAList

elab "#hint_xapp" thm:ident : command => do
  Command.runTermElabM fun _ => do
    let thm := (<- Term.elabTerm thm none)
    let .some thmName := thm.getAppFn.constName? | throwError "invalid theorem"
    let thm <- Meta.inferType thm
    let thmFun <- Meta.forallTelescope thm fun _ thm => do
      getTripleFun thm
    xappExtension.add (thmFun, thmName)

Add in Init.Prelude

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

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

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

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

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

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

Finmap.liftOn₂ in Mathlib.Data.Finmap

Set in Mathlib.Data.Set.Defs

Lean.Meta.inferType in Lean.Meta.Basic
</all_available_defs>

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

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

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

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

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

lemma union_singleton_eq_erase (h h' : state) :
  h.Disjoint (Finmap.singleton p v) →
  h' = h ∪ Finmap.singleton p v →
  h = h'.erase p

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

lemma triple_val_minimal v :
  triple (trm_val v) emp (fun r ↦ ⌜r = v⌝)

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

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

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

lemma eval_val_minimal s v :
  eval s (trm_val v) (fun v' s' => (v' = v) /\ (s' = s))

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 evalExact_sat :
  evalExact s t Q → ∃ v s, Q v s

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

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

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

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

lemma wpgen_if_sound F1 F2 t0 t1 t2 :
  formula_sound t1 F1 →
  formula_sound t2 F2 →
  formula_sound (trm_if t0 t1 t2) (wpgen_if t0 F1 F2)

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

lemma wpgen_seq_sound F1 F2 t1 t2 :
  formula_sound t1 F1 →
  formula_sound t2 F2 →
  formula_sound (trm_seq t1 t2) (wpgen_seq F1 F2)

lemma xsimp_lr_hwand_hfalse :
  XSimp (Hla, emp, emp) ((⌜False⌝ -∗ H1) ∗ emp, emp, emp)

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

lemma xlet_lemma H F1 F2of Q :
  H ==> F1 (fun v => F2of v Q) →
  H ==> wpgen_let F1 F2of Q

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

lemma hseg_one v p j :
  hseg (v :: []) p j = hcell v p j

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

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

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 triple_fun x t1 H Q :
  H ==> Q (val_fun x t1) →
  triple (trm_fun x t1) H Q

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

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

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

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

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

lemma hstar_hgc_hgc : /GC ∗ /GC = /GC

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

lemma wp_conseq t Q1 Q2 :
  Q1 ===> Q2 →
  wp t Q1 ==> wp t Q2

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

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

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

lemma of_nat_nat n :
  OfNat.ofNat n = n

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

lemma AList_erase_entries (l : @AList loc (fun _ ↦ val)) :
  (l.erase p).entries = (l.entries).kerase p

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

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

lemma hprop_op_comm (op : hProp → hProp → hProp) :
  (forall H1 H2, op H1 H2 ==> op H2 H1) →
  (forall H1 H2, op H1 H2 = op H2 H1)

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

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

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

lemma mkstruct_wp t :
  mkstruct (wp t) = wp t

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

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

lemma wp_seq t1 t2 Q :
  wp t1 (fun _ ↦ wp t2 Q) ==> wp (trm_seq t1 t2) Q

lemma hseg_nil p j :
  hseg [] p j = emp

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

lemma xval_lemma v H Q :
  H ==> Q v →
  H ==> wpgen_val v Q

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

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

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

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

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

lemma hstar_pick_3 :
  h1 ∗ h2 ∗ h3 ∗ h = h3 ∗ h1 ∗ h2 ∗ h

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

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

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

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)

lemma wp_let x t1 t2 Q :
  wp t1 (fun v ↦ wp (subst x v t2) Q) ==> wp (trm_let x t1 t2) Q

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

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

lemma hstar_pick_1 :
  h1 ∗ h = h1 ∗ h

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

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

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

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

lemma triple_named_heap t H Q :
  (forall h, H h → triple t (fun h' ↦ h' = h) Q) →
  triple t H Q

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

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

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

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

@[simp]
lemma himpl_refl_resolve  H : (H ==> H) = True

lemma structural_imp : structural F ->
  Q ===> Q' -> F Q ==> F Q'

lemma eval_frame (h1 h2 : state) t (Q : val -> hProp) :
  eval h1 t (ofhProp Q) →
  Finmap.Disjoint h1 h2 →
  eval (h1 ∪ h2) t (Q ∗ (tohProp (fun h ↦ h = h2)))

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

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

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

lemma hstar_pick_last_1 :
  h1 = h1

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

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

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

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

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

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

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

lemma xwhile_inv_basic_lemma (I : Bool -> α -> hProp) R
  -- (F1 F2 : formula)
  :
  WellFounded R ->
  -- structural F1 ->
  -- structural F2 ->
  (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', ⌜R X' X⌝ ∗ I b X')) ->
  H ==> wp (trm_while F1 F2) (fun _ => H' ∗ ∃ʰ a, I false a)

@[simp]
lemma validE : PartialCommMonoid.valid (α := val) = fun _ => False

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

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

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

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

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

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

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

lemma wp_val v Q :
  Q v ==> wp (trm_val v) Q

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

lemma triple_let_val x v1 t2 H Q :
  triple (subst x v1 t2) H Q →
  triple (trm_let x v1 t2) H Q

lemma xfor_lemma (z n : ℤ) (x : var) (I : ℤ -> hProp) :
  z <= n ->
  (H ==> H' ∗ I z) ->
  (∀ i, z <= i -> i < n -> I i ==> wp (subst x i F1) (fun _ => I (i + 1))) ->
  ((fun _ => I n ∗ H') ===> Q) ->
  H ==> wp (trm_for x z n F1) Q

lemma hstar_pick_2  :
  h1 ∗ h2 ∗ h = h2 ∗ h1 ∗ h

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

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

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

lemma make_list_len (n : Int) (v : val) :
  (make_list n.natAbs v).length = n.natAbs

lemma hstar_hexists A (J : A → hProp) H :
  (hexists J) ∗ H = hexists (fun x => (J x) ∗ H)

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

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

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

lemma frame_eq_rw :
  s.Disjoint h2 →
  (fun v' s' ↦ v' = v ∧ s' = s ∪ h2) =
  (qstar (fun v' s' ↦ v' = v ∧ s' = s) (tohProp (fun h ↦ h = h2)))

lemma haffine_hany : forall H,
  haffine H

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

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

lemma triple_seq t1 t2 H Q H1 :
  triple t1 H (fun _ ↦ H1) →
  triple t2 H1 Q →
  triple (trm_seq t1 t2) H Q

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

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

lemma haffine_hempty : haffine emp

lemma mkstruct_conseq F Q1 Q2 :
  Q1 ===> Q2 →
  mkstruct F Q1 ==> mkstruct F Q2

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

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

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

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

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

lemma hstar_hempty_r H :
  H ∗ emp = H

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

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

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

lemma 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 wp_fun x t Q :
  Q (val_fun x t) ==> wp (trm_fun x t) Q

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

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

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

lemma hstar_flip_9 :
  h1 ∗ h2 ∗ h3 ∗ h4 ∗ h5 ∗ h6 ∗ h7 ∗ h8 ∗ h9 ∗ emp =
  h9 ∗ h8 ∗ h7 ∗ h6 ∗ h5 ∗ h4 ∗ h3 ∗ h2 ∗ h1 ∗ emp

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

@[simp]
lemma Heap.add_single (v v' : val) [PartialCommMonoid val] :
  (Finmap.singleton p v) +ʰ (Finmap.singleton p v') = (Finmap.singleton p (v + v'))

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

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

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

lemma AList.erase_insert_swap {α : Type u} {β : α → Type v} [DecidableEq α] (a a' : α) (b : β a) (l : AList β) :
  a ≠ a' → (AList.erase a' (AList.insert a b l)).entries.Perm (AList.insert a b (AList.erase a' l)).entries

lemma xif_lemma b H F1 F2 Q :
  (b = true -> H ==> F1 Q) →
  (b = false -> H ==> F2 Q) →
  H ==> wpgen_if b F1 F2 Q

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

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

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

section evalProp

end evalProp
</local_ctx>

<target_theorem>
lemma triple_ref (v : val) :
  (forall (p : loc), triple (subst x p t2) (H ∗ (p ~~> v)) (Q ∗ ∃ʰ v, p ~~> v)) →
  triple (trm_ref x (trm_val v) t2) H Q :=
</target_theorem>

