(* Stratified Semantics -- the "product" of two generic semantics. *)
(* We use S' instead of S to avoid problems with Datatypes.S. *)
Class StratifiedSemantics {W V G S' U F}
                          `{WVG : GenericSemantics W V G}
                           (* The abstract semantics *)
                          `{S'UF : GenericSemantics S' U F}
                           (* The concrete semantics *)
                           (world_erase : W -> S')
                           (* An erasure function mapping worlds to states *)
                           (VU : relation W -> relation S' -> Prop)
                           (* Acceptable primitive command pairs *)
                           (GF : pfunc W val -> pfunc S' val -> Prop)
                           (* Acceptable primitive expression pairs *)
                           : Type := mkStratifiedSemantics {

  (* These axioms just enforce that VU and GF are subsets of the products
     V x U and G x F. *)
  ss_VU_wellformed : forall v u,
    VU v u
    -> V v /\ U u;

  ss_GF_wellformed : forall g f,
    GF g f
    -> G g /\ F f;

  (* Erasure and safety of primitive commands *)
  ss_primcom_erasure : forall v u w w' s s',
    VU v u
    -> world_erase w = s
    -> v w w'
    -> u s s'
    -> world_erase w' = s';

  ss_primcom_safety : forall v u w w' s,
    VU v u
    -> world_erase w = s
    -> v w w'
    -> exists s', u s s';

  (* Ditto for primitive expressions *)
  ss_primexpr_erasure : forall g f b1 b2 w s (Hgf : GF g f),
    world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exprEval F (EPrim f) s b2
    -> b1=b2;

  ss_primexpr_saftey : forall g f b1 w s,
    GF g f
    -> world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exists b2, exprEval F (EPrim f) s b2
}.(* Stratified Semantics -- the "product" of two generic semantics. *)
(* We use S' instead of S to avoid problems with Datatypes.S. *)
Class StratifiedSemantics {W V G S' U F}
                          `{WVG : GenericSemantics W V G}
                           (* The abstract semantics *)
                          `{S'UF : GenericSemantics S' U F}
                           (* The concrete semantics *)
                           (world_erase : W -> S')
                           (* An erasure function mapping worlds to states *)
                           (VU : relation W -> relation S' -> Prop)
                           (* Acceptable primitive command pairs *)
                           (GF : pfunc W val -> pfunc S' val -> Prop)
                           (* Acceptable primitive expression pairs *)
                           : Type := mkStratifiedSemantics {

  (* These axioms just enforce that VU and GF are subsets of the products
     V x U and G x F. *)
  ss_VU_wellformed : forall v u,
    VU v u
    -> V v /\ U u;

  ss_GF_wellformed : forall g f,
    GF g f
    -> G g /\ F f;

  (* Erasure and safety of primitive commands *)
  ss_primcom_erasure : forall v u w w' s s',
    VU v u
    -> world_erase w = s
    -> v w w'
    -> u s s'
    -> world_erase w' = s';

  ss_primcom_safety : forall v u w w' s,
    VU v u
    -> world_erase w = s
    -> v w w'
    -> exists s', u s s';

  (* Ditto for primitive expressions *)
  ss_primexpr_erasure : forall g f b1 b2 w s (Hgf : GF g f),
    world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exprEval F (EPrim f) s b2
    -> b1=b2;

  ss_primexpr_saftey : forall g f b1 w s,
    GF g f
    -> world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exists b2, exprEval F (EPrim f) s b2
}.(* Stratified Semantics -- the "product" of two generic semantics. *)
(* We use S' instead of S to avoid problems with Datatypes.S. *)
Class StratifiedSemantics {W V G S' U F}
                          `{WVG : GenericSemantics W V G}
                           (* The abstract semantics *)
                          `{S'UF : GenericSemantics S' U F}
                           (* The concrete semantics *)
                           (world_erase : W -> S')
                           (* An erasure function mapping worlds to states *)
                           (VU : relation W -> relation S' -> Prop)
                           (* Acceptable primitive command pairs *)
                           (GF : pfunc W val -> pfunc S' val -> Prop)
                           (* Acceptable primitive expression pairs *)
                           : Type := mkStratifiedSemantics {

  (* These axioms just enforce that VU and GF are subsets of the products
     V x U and G x F. *)
  ss_VU_wellformed : forall v u,
    VU v u
    -> V v /\ U u;

  ss_GF_wellformed : forall g f,
    GF g f
    -> G g /\ F f;

  (* Erasure and safety of primitive commands *)
  ss_primcom_erasure : forall v u w w' s s',
    VU v u
    -> world_erase w = s
    -> v w w'
    -> u s s'
    -> world_erase w' = s';

  ss_primcom_safety : forall v u w w' s,
    VU v u
    -> world_erase w = s
    -> v w w'
    -> exists s', u s s';

  (* Ditto for primitive expressions *)
  ss_primexpr_erasure : forall g f b1 b2 w s (Hgf : GF g f),
    world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exprEval F (EPrim f) s b2
    -> b1=b2;

  ss_primexpr_saftey : forall g f b1 w s,
    GF g f
    -> world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exists b2, exprEval F (EPrim f) s b2
}.(* Stratified Semantics -- the "product" of two generic semantics. *)
(* We use S' instead of S to avoid problems with Datatypes.S. *)
Class StratifiedSemantics {W V G S' U F}
                          `{WVG : GenericSemantics W V G}
                           (* The abstract semantics *)
                          `{S'UF : GenericSemantics S' U F}
                           (* The concrete semantics *)
                           (world_erase : W -> S')
                           (* An erasure function mapping worlds to states *)
                           (VU : relation W -> relation S' -> Prop)
                           (* Acceptable primitive command pairs *)
                           (GF : pfunc W val -> pfunc S' val -> Prop)
                           (* Acceptable primitive expression pairs *)
                           : Type := mkStratifiedSemantics {

  (* These axioms just enforce that VU and GF are subsets of the products
     V x U and G x F. *)
  ss_VU_wellformed : forall v u,
    VU v u
    -> V v /\ U u;

  ss_GF_wellformed : forall g f,
    GF g f
    -> G g /\ F f;

  (* Erasure and safety of primitive commands *)
  ss_primcom_erasure : forall v u w w' s s',
    VU v u
    -> world_erase w = s
    -> v w w'
    -> u s s'
    -> world_erase w' = s';

  ss_primcom_safety : forall v u w w' s,
    VU v u
    -> world_erase w = s
    -> v w w'
    -> exists s', u s s';

  (* Ditto for primitive expressions *)
  ss_primexpr_erasure : forall g f b1 b2 w s (Hgf : GF g f),
    world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exprEval F (EPrim f) s b2
    -> b1=b2;

  ss_primexpr_saftey : forall g f b1 w s,
    GF g f
    -> world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exists b2, exprEval F (EPrim f) s b2
}.(* Stratified Semantics -- the "product" of two generic semantics. *)
(* We use S' instead of S to avoid problems with Datatypes.S. *)
Class StratifiedSemantics {W V G S' U F}
                          `{WVG : GenericSemantics W V G}
                           (* The abstract semantics *)
                          `{S'UF : GenericSemantics S' U F}
                           (* The concrete semantics *)
                           (world_erase : W -> S')
                           (* An erasure function mapping worlds to states *)
                           (VU : relation W -> relation S' -> Prop)
                           (* Acceptable primitive command pairs *)
                           (GF : pfunc W val -> pfunc S' val -> Prop)
                           (* Acceptable primitive expression pairs *)
                           : Type := mkStratifiedSemantics {

  (* These axioms just enforce that VU and GF are subsets of the products
     V x U and G x F. *)
  ss_VU_wellformed : forall v u,
    VU v u
    -> V v /\ U u;

  ss_GF_wellformed : forall g f,
    GF g f
    -> G g /\ F f;

  (* Erasure and safety of primitive commands *)
  ss_primcom_erasure : forall v u w w' s s',
    VU v u
    -> world_erase w = s
    -> v w w'
    -> u s s'
    -> world_erase w' = s';

  ss_primcom_safety : forall v u w w' s,
    VU v u
    -> world_erase w = s
    -> v w w'
    -> exists s', u s s';

  (* Ditto for primitive expressions *)
  ss_primexpr_erasure : forall g f b1 b2 w s (Hgf : GF g f),
    world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exprEval F (EPrim f) s b2
    -> b1=b2;

  ss_primexpr_saftey : forall g f b1 w s,
    GF g f
    -> world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exists b2, exprEval F (EPrim f) s b2
}.(* Stratified Semantics -- the "product" of two generic semantics. *)
(* We use S' instead of S to avoid problems with Datatypes.S. *)
Class StratifiedSemantics {W V G S' U F}
                          `{WVG : GenericSemantics W V G}
                           (* The abstract semantics *)
                          `{S'UF : GenericSemantics S' U F}
                           (* The concrete semantics *)
                           (world_erase : W -> S')
                           (* An erasure function mapping worlds to states *)
                           (VU : relation W -> relation S' -> Prop)
                           (* Acceptable primitive command pairs *)
                           (GF : pfunc W val -> pfunc S' val -> Prop)
                           (* Acceptable primitive expression pairs *)
                           : Type := mkStratifiedSemantics {

  (* These axioms just enforce that VU and GF are subsets of the products
     V x U and G x F. *)
  ss_VU_wellformed : forall v u,
    VU v u
    -> V v /\ U u;

  ss_GF_wellformed : forall g f,
    GF g f
    -> G g /\ F f;

  (* Erasure and safety of primitive commands *)
  ss_primcom_erasure : forall v u w w' s s',
    VU v u
    -> world_erase w = s
    -> v w w'
    -> u s s'
    -> world_erase w' = s';

  ss_primcom_safety : forall v u w w' s,
    VU v u
    -> world_erase w = s
    -> v w w'
    -> exists s', u s s';

  (* Ditto for primitive expressions *)
  ss_primexpr_erasure : forall g f b1 b2 w s (Hgf : GF g f),
    world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exprEval F (EPrim f) s b2
    -> b1=b2;

  ss_primexpr_saftey : forall g f b1 w s,
    GF g f
    -> world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exists b2, exprEval F (EPrim f) s b2
}.(* Stratified Semantics -- the "product" of two generic semantics. *)
(* We use S' instead of S to avoid problems with Datatypes.S. *)
Class StratifiedSemantics {W V G S' U F}
                          `{WVG : GenericSemantics W V G}
                           (* The abstract semantics *)
                          `{S'UF : GenericSemantics S' U F}
                           (* The concrete semantics *)
                           (world_erase : W -> S')
                           (* An erasure function mapping worlds to states *)
                           (VU : relation W -> relation S' -> Prop)
                           (* Acceptable primitive command pairs *)
                           (GF : pfunc W val -> pfunc S' val -> Prop)
                           (* Acceptable primitive expression pairs *)
                           : Type := mkStratifiedSemantics {

  (* These axioms just enforce that VU and GF are subsets of the products
     V x U and G x F. *)
  ss_VU_wellformed : forall v u,
    VU v u
    -> V v /\ U u;

  ss_GF_wellformed : forall g f,
    GF g f
    -> G g /\ F f;

  (* Erasure and safety of primitive commands *)
  ss_primcom_erasure : forall v u w w' s s',
    VU v u
    -> world_erase w = s
    -> v w w'
    -> u s s'
    -> world_erase w' = s';

  ss_primcom_safety : forall v u w w' s,
    VU v u
    -> world_erase w = s
    -> v w w'
    -> exists s', u s s';

  (* Ditto for primitive expressions *)
  ss_primexpr_erasure : forall g f b1 b2 w s (Hgf : GF g f),
    world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exprEval F (EPrim f) s b2
    -> b1=b2;

  ss_primexpr_saftey : forall g f b1 w s,
    GF g f
    -> world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exists b2, exprEval F (EPrim f) s b2
}.(* Stratified Semantics -- the "product" of two generic semantics. *)
(* We use S' instead of S to avoid problems with Datatypes.S. *)
Class StratifiedSemantics {W V G S' U F}
                          `{WVG : GenericSemantics W V G}
                           (* The abstract semantics *)
                          `{S'UF : GenericSemantics S' U F}
                           (* The concrete semantics *)
                           (world_erase : W -> S')
                           (* An erasure function mapping worlds to states *)
                           (VU : relation W -> relation S' -> Prop)
                           (* Acceptable primitive command pairs *)
                           (GF : pfunc W val -> pfunc S' val -> Prop)
                           (* Acceptable primitive expression pairs *)
                           : Type := mkStratifiedSemantics {

  (* These axioms just enforce that VU and GF are subsets of the products
     V x U and G x F. *)
  ss_VU_wellformed : forall v u,
    VU v u
    -> V v /\ U u;

  ss_GF_wellformed : forall g f,
    GF g f
    -> G g /\ F f;

  (* Erasure and safety of primitive commands *)
  ss_primcom_erasure : forall v u w w' s s',
    VU v u
    -> world_erase w = s
    -> v w w'
    -> u s s'
    -> world_erase w' = s';

  ss_primcom_safety : forall v u w w' s,
    VU v u
    -> world_erase w = s
    -> v w w'
    -> exists s', u s s';

  (* Ditto for primitive expressions *)
  ss_primexpr_erasure : forall g f b1 b2 w s (Hgf : GF g f),
    world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exprEval F (EPrim f) s b2
    -> b1=b2;

  ss_primexpr_saftey : forall g f b1 w s,
    GF g f
    -> world_erase w = s
    -> exprEval G (EPrim g) w b1
    -> exists b2, exprEval F (EPrim f) s b2
}.

Lemma primcom_safety : forall v u w w' s, VU v u -> world_erase w = s -> v w w' -> exists s', u s s'.