-- SPLean/Theories/SepLog.lean

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

-- Repository definitions used:
content: syntax "if " lang "then " lang "end " : lang

syntax " := " : bop

syntax "let" ident " := " lang " in" ppDedent(ppLine lang) : lang

syntax "fun" ident+ " => " lang : lang

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

syntax "⟨" term "⟩" : lang

syntax "⟨" term ":" term "⟩" : lang

content: macro_rules
  | `([lang| ()])                       => `(trm_val (val_unit))
  | `([lang| $n:num])                   => `(trm_val (val_int $n))
  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])
  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])
  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))
  | `([lang| let $x := $t1:lang in $t2:lang])     =>
    `(trm_let $(%x) [lang| $t1] [lang| $t2])
  | `([lang| ref $x := $t1:lang in $t2:lang])     =>
    `(trm_ref $(%x) [lang| $t1] [lang| $t2])
  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>
    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])
  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])
  | `([lang| fun_ $xs* => $t])             => do
    let xs <- xs.mapM fun x => `(term| $(%x))
    `(trm_funs [ $xs,* ] [lang| $t])
  | `([lang| fun $xs* => $t])             => do
    let xs <- xs.mapM fun x => `(term| $(%x))
    `(val_funs [ $xs,* ] [lang| $t])
  | `([lang| fix_ $f $xs* => $t])    => do
      let xs <- xs.mapM fun x => `(term| $(%x))
      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])
  | `([lang| fix $f $xs* => $t])    => do
      let xs <- xs.mapM fun x => `(term| $(%x))
      `(val_fixs $(%f) [ $xs,* ] [lang| $t])
  
  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])
  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])
  
  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])
  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])
  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])
  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])
  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])
  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])
  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])
  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])
  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])
  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])
  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])
  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])
  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])
  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])
  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])
  | `([lang| ($t)]) => `([lang| $t])
  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))
  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>
    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])
  | `([lang| while $c:lang { $t:lang } ]) =>
     `(trm_while [lang| $c] [lang| $t] )

content: macro_rules
  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])
  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])
  
  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])
  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])

content: macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)

  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

  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

content: abbrev var := String

content: abbrev loc := Nat

content: def hProp := heap -> Prop

content: abbrev heap := Heap.heap val

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

content: 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

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

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

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: 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))

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

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

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

content: abbrev state := Finmap (λ _ : loc ↦ val)

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

content: infixr:51 " ==> " => himpl

content: infixr:51 " ===> " => qimpl

content: infixr:60 " ~~> " => hsingle

content: infixr:55 " ∗ " => HStar.hStar

-- Repository lemmas used:
content: lemma disjoint_single (h : state) :
  p ∉ h →
  h.Disjoint (Finmap.singleton p v)

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

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

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

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

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

-- Local definitions used:
content: abbrev triple (t : trm) (H : hProp) (Q : val → hProp) : Prop :=
  forall s, H s → eval s t Q

-- Local lemmas used:
content: lemma eval_conseq s t Q1 Q2 :
  eval s t Q1 →
  Q1 ===> Q2 →
  eval s t Q2

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

-- Target theorem and proof:
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
:=
by
  move=> htriple h ?
  apply eval.eval_ref
  { sby apply (eval.eval_val h v (fun v' h' ↦ v' = v ∧ h' = h)) }
  move=> > [->->] > ?
  move: (htriple p)=> /triple_conseq {}htriple
  have eqn:(triple (subst x p t2) (H ∗ p ~~> v) fun v s ↦ Q v (s.erase p)) := by
    apply htriple=> //
    move=> > h /= ![>] ? /hexists_inv [v'] /hsingl_inv ->
    sby move=> /union_singleton_eq_erase /[apply] <-
  move=> {htriple}
  apply eqn
  exists h, Finmap.singleton p v
  move=> ⟨//|⟩ ⟨|⟩
  apply hsingle_intro=> ⟨|⟩
  apply disjoint_single=>//
  sby apply insert_eq_union_single=> //