-- SPLean/Theories/WP1.lean

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

-- Repository definitions used:
content: notation:max "emp" => hempty

syntax "if " lang "then " lang "end " : lang

syntax " := " : bop

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

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

syntax "⟨" term "⟩" : lang

syntax "⟨" term ":" term "⟩" : lang

content: notation:max "⌜" P "⌝" => hpure P

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

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

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

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

content: 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)

content: 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
  )

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 hempty : hProp :=
  fun h => (h = ∅)

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 null : loc := 0

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

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

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

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

content: 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))

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

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

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

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

content: notation:max "emp" => hempty

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

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

content: notation:max "⌜" P "⌝" => hpure P

content: infixr:55 " -∗ " => HWand.hWand

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

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

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

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

content: 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

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

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

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

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

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

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

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

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

content: 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)

content: 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)

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

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

content: 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)

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

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

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

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

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

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

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

content: 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

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

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

content: lemma hstar_comm_assoc (H1 H2 H3 : hProp) :
  H1 ∗ H2 ∗ H3 = H2 ∗ H1 ∗ H3

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

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

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

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

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

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

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

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

content: 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')

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

content: lemma hstar_hempty_l H :
  emp ∗ H = H

content: lemma hstar_hempty_r H :
  H ∗ emp = H

content: lemma himpl_refl H : H ==> H

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

-- Local definitions used:
content: def wp (t : trm) (Q : val → hProp) : hProp :=
  fun s ↦ eval s t Q

-- Local lemmas used:
content: lemma mem_conseq :
  x ∈ conseq L p → p ≤ x

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

-- Target theorem and proof:
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
:=
by
  move=> ? h /hforall_inv hwp
  apply eval.eval_alloc=> // > *
  apply (eval_conseq _ _ (Q ∗ ⌜p ≠ null⌝ ∗ ∃ʰ L, ⌜↑L.length = n⌝ ∗ hrange L p))
  { move: (hwp p)=> /(hwand_inv sb)
    srw Finmap.Disjoint.symm_iff=> {}hwp
    apply hwp=> // ; subst sb
    apply hrange_of_conseq }
  move=> > s ![>] ? ![>] /hpure_inv [_ ->] /==
  move=> /hexists_inv [L] ![>] /hpure_inv [? ->] /== ? _ -> _ -> ? ->
  srw diff_disjoint_eq=> // ; subst sb
  sby apply hrange_eq_conseq