-- SPLean/Common/Heap.lean

import Lean

import Mathlib.Data.Finmap

import Mathlib.Algebra.Group.Basic

import Mathlib.Algebra.BigOperators.Group.Finset

import Mathlib.Algebra.BigOperators.Intervals

import Mathlib.Data.Int.Interval

import Mathlib.Order.Interval.Finset.Basic

import Batteries.Data.List.Perm

import Ssreflect.Lang

open Classical

abbrev loc := Nat

abbrev var := String

section Option.merge

variable {α : Type u} (f : α → α → α)

end Option.merge

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

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

open PartialCommMonoid (valid)

section

variable {val : Type} [PartialCommMonoid val] -- [Inhabited val]

local notation "heap" => Heap.heap val

private def kmerge1 (l : loc) (v : val) (l₂ : List (Sigma (fun _ : loc => val))) : val :=
  match l₂.dlookup l with
  | .some v' => v + v'
  | _ => 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)

noncomputable def AList.merge (h₁ h₂ : AList (fun _ : loc => val)) :  AList (fun _ : loc => val) :=
  ⟨kmerge h₁.entries h₂.entries, by admit /- proof elided -/
    ⟩

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

infixr:55 " +ʰ " => Heap.add

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

infixr:55 " ⊥ʰ " => validInter

-- Repository definitions used:
content: abbrev heap := Heap.heap val

  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

content: 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 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

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

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

content: 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

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

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

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

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

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

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

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

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

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

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

-- Repository lemmas used:
content: theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\n(l₁ ++ l₂)[i] = l₁[i]

-- Local definitions used:
content: abbrev loc := Nat

content: abbrev var := String

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

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

content: private def kmerge1 (l : loc) (v : val) (l₂ : List (Sigma (fun _ : loc => val))) : val :=
  match l₂.dlookup l with
  | .some v' => v + v'
  | _ => v

content: @[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)

content: noncomputable def AList.merge (h₁ h₂ : AList (fun _ : loc => val)) :  AList (fun _ : loc => val) :=
  ⟨kmerge h₁.entries h₂.entries, by admit /- proof elided -/
    ⟩

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

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

-- Local lemmas used:
content: lemma Option.merge_none_r (a : Option α) : Option.merge f a none = a

content: 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)

-- Target theorem and proof:
lemma validInter_assoc_l (h₁ h₂ h₃ : heap) :
  h₁ ⊥ʰ h₂ -> (h₁ +ʰ h₂) ⊥ʰ h₃ -> h₁ ⊥ʰ (h₂ +ʰ h₃)
:= by
  simp [validInter]
  move=> h1 h2 l hin1 /[tac (specialize h1 _ hin1 ; specialize h2 _ (Or.intro_left _ hin1))] [ hin2 | hin3 ]
  { rcases h : Finmap.lookup l h₃
    { rw [Option.merge_none_r] ; aesop }
    { srw h at h2 ; rw [← Option.merge_assoc, h2] ; apply Finmap.mem_of_lookup_eq_some at h=> //
      apply add_assoc } }
  { rw [← Option.merge_assoc, h2]=> //
    apply add_assoc }