(*Unital*)
Instance discHeap_unital':
  @UnitalSeparationAlgebra Heap' discHeap_R' Heap_Join'.(*Residual*)
Instance discHeap_residual':
  @ResidualSeparationAlgebra Heap' discHeap_R' Heap_Join'.(** * Monotonic heap*)
Instance monHeap_R': Relation Heap' :=
  @pointwise_relation _ _ (@option01_relation _ eq).Instance po_monHeap_R': PreOrder Krelation :=
  @pointwise_preorder _ _ _ (@option01_preorder _ _ (eq_preorder _)).(* Upwards-closed*)
Instance monHeap_uSA':
  @UpwardsClosedSeparationAlgebra Heap' monHeap_R' Heap_Join'.(*Downwards-closed*)
Definition monHeap_dSA':
  @DownwardsClosedSeparationAlgebra Heap' monHeap_R' Heap_Join'.(* Increasing *)
Instance monHeap_increasing':
  @IncreasingSeparationAlgebra Heap' monHeap_R' Heap_Join'.(* Residual *)
Instance monHeap_residual':
  @ResidualSeparationAlgebra Heap' monHeap_R' Heap_Join'.(* Unital *)
Instance monHeap_unital':
  @UnitalSeparationAlgebra Heap' monHeap_R' Heap_Join'.Inductive htype:=
  |char
  |short1
  |short2.Inductive htype:=
  |char
  |short1
  |short2.Inductive htype:=
  |char
  |short1
  |short2.Inductive htype:=
  |char
  |short1
  |short2.Inductive ht_ord: htype -> htype -> Prop :=
    |htype_refl x: ht_ord x x
    |htype_sht1 : ht_ord char short1
    |htype_sht2 : ht_ord char short2.Inductive ht_ord: htype -> htype -> Prop :=
    |htype_refl x: ht_ord x x
    |htype_sht1 : ht_ord char short1
    |htype_sht2 : ht_ord char short2.Inductive ht_ord: htype -> htype -> Prop :=
    |htype_refl x: ht_ord x x
    |htype_sht1 : ht_ord char short1
    |htype_sht2 : ht_ord char short2.Inductive ht_ord: htype -> htype -> Prop :=
    |htype_refl x: ht_ord x x
    |htype_sht1 : ht_ord char short1
    |htype_sht2 : ht_ord char short2.Instance ht_preorder: PreOrder ht_ord.Notation THeap':= (nat -> option htype).Definition THvalid (TH:THeap'):=
     forall n, TH n = Some short1 <->
                   TH (S n) = Some short2.Record THeap: Type :=
    { theap:> nat -> option htype;
      th_wf1: THvalid theap}.Record THeap: Type :=
    { theap:> nat -> option htype;
      th_wf1: THvalid theap}.Record THeap: Type :=
    { theap:> nat -> option htype;
      th_wf1: THvalid theap}.Instance THeap_Join': Join THeap' :=
    @fun_Join _ _ (@option_Join _ (trivial_Join)).Inductive THeap_Join: Join THeap :=
  | th_join (h1 h2 h3: THeap): THeap_Join' h1 h2 h3 ->
                               THeap_Join h1 h2 h3.Inductive THeap_Join: Join THeap :=
  | th_join (h1 h2 h3: THeap): THeap_Join' h1 h2 h3 ->
                               THeap_Join h1 h2 h3.Instance THeap_SA': @SeparationAlgebra _ THeap_Join':=
  @fun_SA _ _ _ (@option_SA _ _ (trivial_SA)).

Lemma THeap_Join_valid: forall {h1 h2 h3}, THeap_Join' h1 h2 h3 -> THvalid h1 -> THvalid h2 -> THvalid h3.