Definition val_has_type_opt' (v: option val) (ty: typ) :=
 match v with
 | None => True
 | Some v' => Val.has_type v' ty
 end.Definition val_has_type_opt (v: option val) (sig: signature) :=
  val_has_type_opt' v (proj_sig_res sig).(** A "core semantics represents" a fairly traditional, sequential,
   small step semantics of computation.  They are designed to
   cooperate with "extensions" which give semantics to primtive
   constructs not defined by the extensible semantics (e.g., external
   function calls).

   The [G] type parameter is the type of global environments, the type
   [C] is the type of core states, and the type [E] is the type of
   extension requests.  The [at_external] function gives a way to
   determine when the sequential execution is blocked on an extension
   call, and to extract the data necessary to execute the call.
   [after_external] give a way to inject the extension call results
   back into the sequential state so execution can continue.

(* Previsouly, we had additionally

   a) a type parameter [D] that stood for the type of initialization
       data, eg list (ident * globvar V).

   b) a clause [initial_mem] that characterized memories
      suitable at module entry.

But we eliminated these, since these constructs refer to PROGRAMS,
not CORES.

*)

   [initial_core] that produces the core state
      corresponding to an entry point of a module.
      The arguments are the genv, a pointer to the
      function to run, and the arguments for that function.

   The [halted] predicate indicates when a program state
   has reached a halted state, and what it's exit code/return value is
   when it has reached such a state.

   [corestep] is the fundamental small-step relation for
   the sequential semantics.

   The remaining properties give basic sanity properties which constrain
   the behavior of programs.
    1) a state cannot be both blocked on an extension call
        and also step,
    2) a state cannot both step and be halted
    3) a state cannot both be halted and blocked on an external call
    4) after external calls, cores are back in a "runnable" state
       (NOTE: this axiom may be removed at some point) *)

Record CoreSemantics {G C M (*D only needed in initial_mem*):Type}: Type :=
  { (*Removed: is a propert of programs, not of cores
      initial_mem: G -> M -> D -> Prop;*)
    initial_core : G -> val -> list val -> option C;
    at_external : C -> option (external_function * signature * list val);
    after_external : option val -> C -> option C;
    halted : C -> option val;
    corestep : G -> C -> M -> C -> M -> Prop;

    corestep_not_at_external: forall ge m q m' q',
      corestep ge q m q' m' -> at_external q = None;
    corestep_not_halted: forall ge m q m' q',
      corestep ge q m q' m' -> halted q = None;
    at_external_halted_excl: forall q,
      at_external q = None \/ halted q = None;
    after_at_external_excl : forall retv q q',
      after_external retv q = Some q' -> at_external q' = None
  }.(** A "core semantics represents" a fairly traditional, sequential,
   small step semantics of computation.  They are designed to
   cooperate with "extensions" which give semantics to primtive
   constructs not defined by the extensible semantics (e.g., external
   function calls).

   The [G] type parameter is the type of global environments, the type
   [C] is the type of core states, and the type [E] is the type of
   extension requests.  The [at_external] function gives a way to
   determine when the sequential execution is blocked on an extension
   call, and to extract the data necessary to execute the call.
   [after_external] give a way to inject the extension call results
   back into the sequential state so execution can continue.

(* Previsouly, we had additionally

   a) a type parameter [D] that stood for the type of initialization
       data, eg list (ident * globvar V).

   b) a clause [initial_mem] that characterized memories
      suitable at module entry.

But we eliminated these, since these constructs refer to PROGRAMS,
not CORES.

*)

   [initial_core] that produces the core state
      corresponding to an entry point of a module.
      The arguments are the genv, a pointer to the
      function to run, and the arguments for that function.

   The [halted] predicate indicates when a program state
   has reached a halted state, and what it's exit code/return value is
   when it has reached such a state.

   [corestep] is the fundamental small-step relation for
   the sequential semantics.

   The remaining properties give basic sanity properties which constrain
   the behavior of programs.
    1) a state cannot be both blocked on an extension call
        and also step,
    2) a state cannot both step and be halted
    3) a state cannot both be halted and blocked on an external call
    4) after external calls, cores are back in a "runnable" state
       (NOTE: this axiom may be removed at some point) *)

Record CoreSemantics {G C M (*D only needed in initial_mem*):Type}: Type :=
  { (*Removed: is a propert of programs, not of cores
      initial_mem: G -> M -> D -> Prop;*)
    initial_core : G -> val -> list val -> option C;
    at_external : C -> option (external_function * signature * list val);
    after_external : option val -> C -> option C;
    halted : C -> option val;
    corestep : G -> C -> M -> C -> M -> Prop;

    corestep_not_at_external: forall ge m q m' q',
      corestep ge q m q' m' -> at_external q = None;
    corestep_not_halted: forall ge m q m' q',
      corestep ge q m q' m' -> halted q = None;
    at_external_halted_excl: forall q,
      at_external q = None \/ halted q = None;
    after_at_external_excl : forall retv q q',
      after_external retv q = Some q' -> at_external q' = None
  }.(** A "core semantics represents" a fairly traditional, sequential,
   small step semantics of computation.  They are designed to
   cooperate with "extensions" which give semantics to primtive
   constructs not defined by the extensible semantics (e.g., external
   function calls).

   The [G] type parameter is the type of global environments, the type
   [C] is the type of core states, and the type [E] is the type of
   extension requests.  The [at_external] function gives a way to
   determine when the sequential execution is blocked on an extension
   call, and to extract the data necessary to execute the call.
   [after_external] give a way to inject the extension call results
   back into the sequential state so execution can continue.

(* Previsouly, we had additionally

   a) a type parameter [D] that stood for the type of initialization
       data, eg list (ident * globvar V).

   b) a clause [initial_mem] that characterized memories
      suitable at module entry.

But we eliminated these, since these constructs refer to PROGRAMS,
not CORES.

*)

   [initial_core] that produces the core state
      corresponding to an entry point of a module.
      The arguments are the genv, a pointer to the
      function to run, and the arguments for that function.

   The [halted] predicate indicates when a program state
   has reached a halted state, and what it's exit code/return value is
   when it has reached such a state.

   [corestep] is the fundamental small-step relation for
   the sequential semantics.

   The remaining properties give basic sanity properties which constrain
   the behavior of programs.
    1) a state cannot be both blocked on an extension call
        and also step,
    2) a state cannot both step and be halted
    3) a state cannot both be halted and blocked on an external call
    4) after external calls, cores are back in a "runnable" state
       (NOTE: this axiom may be removed at some point) *)

Record CoreSemantics {G C M (*D only needed in initial_mem*):Type}: Type :=
  { (*Removed: is a propert of programs, not of cores
      initial_mem: G -> M -> D -> Prop;*)
    initial_core : G -> val -> list val -> option C;
    at_external : C -> option (external_function * signature * list val);
    after_external : option val -> C -> option C;
    halted : C -> option val;
    corestep : G -> C -> M -> C -> M -> Prop;

    corestep_not_at_external: forall ge m q m' q',
      corestep ge q m q' m' -> at_external q = None;
    corestep_not_halted: forall ge m q m' q',
      corestep ge q m q' m' -> halted q = None;
    at_external_halted_excl: forall q,
      at_external q = None \/ halted q = None;
    after_at_external_excl : forall retv q q',
      after_external retv q = Some q' -> at_external q' = None
  }.(** A "core semantics represents" a fairly traditional, sequential,
   small step semantics of computation.  They are designed to
   cooperate with "extensions" which give semantics to primtive
   constructs not defined by the extensible semantics (e.g., external
   function calls).

   The [G] type parameter is the type of global environments, the type
   [C] is the type of core states, and the type [E] is the type of
   extension requests.  The [at_external] function gives a way to
   determine when the sequential execution is blocked on an extension
   call, and to extract the data necessary to execute the call.
   [after_external] give a way to inject the extension call results
   back into the sequential state so execution can continue.

(* Previsouly, we had additionally

   a) a type parameter [D] that stood for the type of initialization
       data, eg list (ident * globvar V).

   b) a clause [initial_mem] that characterized memories
      suitable at module entry.

But we eliminated these, since these constructs refer to PROGRAMS,
not CORES.

*)

   [initial_core] that produces the core state
      corresponding to an entry point of a module.
      The arguments are the genv, a pointer to the
      function to run, and the arguments for that function.

   The [halted] predicate indicates when a program state
   has reached a halted state, and what it's exit code/return value is
   when it has reached such a state.

   [corestep] is the fundamental small-step relation for
   the sequential semantics.

   The remaining properties give basic sanity properties which constrain
   the behavior of programs.
    1) a state cannot be both blocked on an extension call
        and also step,
    2) a state cannot both step and be halted
    3) a state cannot both be halted and blocked on an external call
    4) after external calls, cores are back in a "runnable" state
       (NOTE: this axiom may be removed at some point) *)

Record CoreSemantics {G C M (*D only needed in initial_mem*):Type}: Type :=
  { (*Removed: is a propert of programs, not of cores
      initial_mem: G -> M -> D -> Prop;*)
    initial_core : G -> val -> list val -> option C;
    at_external : C -> option (external_function * signature * list val);
    after_external : option val -> C -> option C;
    halted : C -> option val;
    corestep : G -> C -> M -> C -> M -> Prop;

    corestep_not_at_external: forall ge m q m' q',
      corestep ge q m q' m' -> at_external q = None;
    corestep_not_halted: forall ge m q m' q',
      corestep ge q m q' m' -> halted q = None;
    at_external_halted_excl: forall q,
      at_external q = None \/ halted q = None;
    after_at_external_excl : forall retv q q',
      after_external retv q = Some q' -> at_external q' = None
  }.(** A "core semantics represents" a fairly traditional, sequential,
   small step semantics of computation.  They are designed to
   cooperate with "extensions" which give semantics to primtive
   constructs not defined by the extensible semantics (e.g., external
   function calls).

   The [G] type parameter is the type of global environments, the type
   [C] is the type of core states, and the type [E] is the type of
   extension requests.  The [at_external] function gives a way to
   determine when the sequential execution is blocked on an extension
   call, and to extract the data necessary to execute the call.
   [after_external] give a way to inject the extension call results
   back into the sequential state so execution can continue.

(* Previsouly, we had additionally

   a) a type parameter [D] that stood for the type of initialization
       data, eg list (ident * globvar V).

   b) a clause [initial_mem] that characterized memories
      suitable at module entry.

But we eliminated these, since these constructs refer to PROGRAMS,
not CORES.

*)

   [initial_core] that produces the core state
      corresponding to an entry point of a module.
      The arguments are the genv, a pointer to the
      function to run, and the arguments for that function.

   The [halted] predicate indicates when a program state
   has reached a halted state, and what it's exit code/return value is
   when it has reached such a state.

   [corestep] is the fundamental small-step relation for
   the sequential semantics.

   The remaining properties give basic sanity properties which constrain
   the behavior of programs.
    1) a state cannot be both blocked on an extension call
        and also step,
    2) a state cannot both step and be halted
    3) a state cannot both be halted and blocked on an external call
    4) after external calls, cores are back in a "runnable" state
       (NOTE: this axiom may be removed at some point) *)

Record CoreSemantics {G C M (*D only needed in initial_mem*):Type}: Type :=
  { (*Removed: is a propert of programs, not of cores
      initial_mem: G -> M -> D -> Prop;*)
    initial_core : G -> val -> list val -> option C;
    at_external : C -> option (external_function * signature * list val);
    after_external : option val -> C -> option C;
    halted : C -> option val;
    corestep : G -> C -> M -> C -> M -> Prop;

    corestep_not_at_external: forall ge m q m' q',
      corestep ge q m q' m' -> at_external q = None;
    corestep_not_halted: forall ge m q m' q',
      corestep ge q m q' m' -> halted q = None;
    at_external_halted_excl: forall q,
      at_external q = None \/ halted q = None;
    after_at_external_excl : forall retv q q',
      after_external retv q = Some q' -> at_external q' = None
  }.(** A "core semantics represents" a fairly traditional, sequential,
   small step semantics of computation.  They are designed to
   cooperate with "extensions" which give semantics to primtive
   constructs not defined by the extensible semantics (e.g., external
   function calls).

   The [G] type parameter is the type of global environments, the type
   [C] is the type of core states, and the type [E] is the type of
   extension requests.  The [at_external] function gives a way to
   determine when the sequential execution is blocked on an extension
   call, and to extract the data necessary to execute the call.
   [after_external] give a way to inject the extension call results
   back into the sequential state so execution can continue.

(* Previsouly, we had additionally

   a) a type parameter [D] that stood for the type of initialization
       data, eg list (ident * globvar V).

   b) a clause [initial_mem] that characterized memories
      suitable at module entry.

But we eliminated these, since these constructs refer to PROGRAMS,
not CORES.

*)

   [initial_core] that produces the core state
      corresponding to an entry point of a module.
      The arguments are the genv, a pointer to the
      function to run, and the arguments for that function.

   The [halted] predicate indicates when a program state
   has reached a halted state, and what it's exit code/return value is
   when it has reached such a state.

   [corestep] is the fundamental small-step relation for
   the sequential semantics.

   The remaining properties give basic sanity properties which constrain
   the behavior of programs.
    1) a state cannot be both blocked on an extension call
        and also step,
    2) a state cannot both step and be halted
    3) a state cannot both be halted and blocked on an external call
    4) after external calls, cores are back in a "runnable" state
       (NOTE: this axiom may be removed at some point) *)

Record CoreSemantics {G C M (*D only needed in initial_mem*):Type}: Type :=
  { (*Removed: is a propert of programs, not of cores
      initial_mem: G -> M -> D -> Prop;*)
    initial_core : G -> val -> list val -> option C;
    at_external : C -> option (external_function * signature * list val);
    after_external : option val -> C -> option C;
    halted : C -> option val;
    corestep : G -> C -> M -> C -> M -> Prop;

    corestep_not_at_external: forall ge m q m' q',
      corestep ge q m q' m' -> at_external q = None;
    corestep_not_halted: forall ge m q m' q',
      corestep ge q m q' m' -> halted q = None;
    at_external_halted_excl: forall q,
      at_external q = None \/ halted q = None;
    after_at_external_excl : forall retv q q',
      after_external retv q = Some q' -> at_external q' = None
  }.(** A "core semantics represents" a fairly traditional, sequential,
   small step semantics of computation.  They are designed to
   cooperate with "extensions" which give semantics to primtive
   constructs not defined by the extensible semantics (e.g., external
   function calls).

   The [G] type parameter is the type of global environments, the type
   [C] is the type of core states, and the type [E] is the type of
   extension requests.  The [at_external] function gives a way to
   determine when the sequential execution is blocked on an extension
   call, and to extract the data necessary to execute the call.
   [after_external] give a way to inject the extension call results
   back into the sequential state so execution can continue.

(* Previsouly, we had additionally

   a) a type parameter [D] that stood for the type of initialization
       data, eg list (ident * globvar V).

   b) a clause [initial_mem] that characterized memories
      suitable at module entry.

But we eliminated these, since these constructs refer to PROGRAMS,
not CORES.

*)

   [initial_core] that produces the core state
      corresponding to an entry point of a module.
      The arguments are the genv, a pointer to the
      function to run, and the arguments for that function.

   The [halted] predicate indicates when a program state
   has reached a halted state, and what it's exit code/return value is
   when it has reached such a state.

   [corestep] is the fundamental small-step relation for
   the sequential semantics.

   The remaining properties give basic sanity properties which constrain
   the behavior of programs.
    1) a state cannot be both blocked on an extension call
        and also step,
    2) a state cannot both step and be halted
    3) a state cannot both be halted and blocked on an external call
    4) after external calls, cores are back in a "runnable" state
       (NOTE: this axiom may be removed at some point) *)

Record CoreSemantics {G C M (*D only needed in initial_mem*):Type}: Type :=
  { (*Removed: is a propert of programs, not of cores
      initial_mem: G -> M -> D -> Prop;*)
    initial_core : G -> val -> list val -> option C;
    at_external : C -> option (external_function * signature * list val);
    after_external : option val -> C -> option C;
    halted : C -> option val;
    corestep : G -> C -> M -> C -> M -> Prop;

    corestep_not_at_external: forall ge m q m' q',
      corestep ge q m q' m' -> at_external q = None;
    corestep_not_halted: forall ge m q m' q',
      corestep ge q m q' m' -> halted q = None;
    at_external_halted_excl: forall q,
      at_external q = None \/ halted q = None;
    after_at_external_excl : forall retv q q',
      after_external retv q = Some q' -> at_external q' = None
  }.(** A "core semantics represents" a fairly traditional, sequential,
   small step semantics of computation.  They are designed to
   cooperate with "extensions" which give semantics to primtive
   constructs not defined by the extensible semantics (e.g., external
   function calls).

   The [G] type parameter is the type of global environments, the type
   [C] is the type of core states, and the type [E] is the type of
   extension requests.  The [at_external] function gives a way to
   determine when the sequential execution is blocked on an extension
   call, and to extract the data necessary to execute the call.
   [after_external] give a way to inject the extension call results
   back into the sequential state so execution can continue.

(* Previsouly, we had additionally

   a) a type parameter [D] that stood for the type of initialization
       data, eg list (ident * globvar V).

   b) a clause [initial_mem] that characterized memories
      suitable at module entry.

But we eliminated these, since these constructs refer to PROGRAMS,
not CORES.

*)

   [initial_core] that produces the core state
      corresponding to an entry point of a module.
      The arguments are the genv, a pointer to the
      function to run, and the arguments for that function.

   The [halted] predicate indicates when a program state
   has reached a halted state, and what it's exit code/return value is
   when it has reached such a state.

   [corestep] is the fundamental small-step relation for
   the sequential semantics.

   The remaining properties give basic sanity properties which constrain
   the behavior of programs.
    1) a state cannot be both blocked on an extension call
        and also step,
    2) a state cannot both step and be halted
    3) a state cannot both be halted and blocked on an external call
    4) after external calls, cores are back in a "runnable" state
       (NOTE: this axiom may be removed at some point) *)

Record CoreSemantics {G C M (*D only needed in initial_mem*):Type}: Type :=
  { (*Removed: is a propert of programs, not of cores
      initial_mem: G -> M -> D -> Prop;*)
    initial_core : G -> val -> list val -> option C;
    at_external : C -> option (external_function * signature * list val);
    after_external : option val -> C -> option C;
    halted : C -> option val;
    corestep : G -> C -> M -> C -> M -> Prop;

    corestep_not_at_external: forall ge m q m' q',
      corestep ge q m q' m' -> at_external q = None;
    corestep_not_halted: forall ge m q m' q',
      corestep ge q m q' m' -> halted q = None;
    at_external_halted_excl: forall q,
      at_external q = None \/ halted q = None;
    after_at_external_excl : forall retv q q',
      after_external retv q = Some q' -> at_external q' = None
  }.(** A "core semantics represents" a fairly traditional, sequential,
   small step semantics of computation.  They are designed to
   cooperate with "extensions" which give semantics to primtive
   constructs not defined by the extensible semantics (e.g., external
   function calls).

   The [G] type parameter is the type of global environments, the type
   [C] is the type of core states, and the type [E] is the type of
   extension requests.  The [at_external] function gives a way to
   determine when the sequential execution is blocked on an extension
   call, and to extract the data necessary to execute the call.
   [after_external] give a way to inject the extension call results
   back into the sequential state so execution can continue.

(* Previsouly, we had additionally

   a) a type parameter [D] that stood for the type of initialization
       data, eg list (ident * globvar V).

   b) a clause [initial_mem] that characterized memories
      suitable at module entry.

But we eliminated these, since these constructs refer to PROGRAMS,
not CORES.

*)

   [initial_core] that produces the core state
      corresponding to an entry point of a module.
      The arguments are the genv, a pointer to the
      function to run, and the arguments for that function.

   The [halted] predicate indicates when a program state
   has reached a halted state, and what it's exit code/return value is
   when it has reached such a state.

   [corestep] is the fundamental small-step relation for
   the sequential semantics.

   The remaining properties give basic sanity properties which constrain
   the behavior of programs.
    1) a state cannot be both blocked on an extension call
        and also step,
    2) a state cannot both step and be halted
    3) a state cannot both be halted and blocked on an external call
    4) after external calls, cores are back in a "runnable" state
       (NOTE: this axiom may be removed at some point) *)

Record CoreSemantics {G C M (*D only needed in initial_mem*):Type}: Type :=
  { (*Removed: is a propert of programs, not of cores
      initial_mem: G -> M -> D -> Prop;*)
    initial_core : G -> val -> list val -> option C;
    at_external : C -> option (external_function * signature * list val);
    after_external : option val -> C -> option C;
    halted : C -> option val;
    corestep : G -> C -> M -> C -> M -> Prop;

    corestep_not_at_external: forall ge m q m' q',
      corestep ge q m q' m' -> at_external q = None;
    corestep_not_halted: forall ge m q m' q',
      corestep ge q m q' m' -> halted q = None;
    at_external_halted_excl: forall q,
      at_external q = None \/ halted q = None;
    after_at_external_excl : forall retv q q',
      after_external retv q = Some q' -> at_external q' = None
  }.(** A "core semantics represents" a fairly traditional, sequential,
   small step semantics of computation.  They are designed to
   cooperate with "extensions" which give semantics to primtive
   constructs not defined by the extensible semantics (e.g., external
   function calls).

   The [G] type parameter is the type of global environments, the type
   [C] is the type of core states, and the type [E] is the type of
   extension requests.  The [at_external] function gives a way to
   determine when the sequential execution is blocked on an extension
   call, and to extract the data necessary to execute the call.
   [after_external] give a way to inject the extension call results
   back into the sequential state so execution can continue.

(* Previsouly, we had additionally

   a) a type parameter [D] that stood for the type of initialization
       data, eg list (ident * globvar V).

   b) a clause [initial_mem] that characterized memories
      suitable at module entry.

But we eliminated these, since these constructs refer to PROGRAMS,
not CORES.

*)

   [initial_core] that produces the core state
      corresponding to an entry point of a module.
      The arguments are the genv, a pointer to the
      function to run, and the arguments for that function.

   The [halted] predicate indicates when a program state
   has reached a halted state, and what it's exit code/return value is
   when it has reached such a state.

   [corestep] is the fundamental small-step relation for
   the sequential semantics.

   The remaining properties give basic sanity properties which constrain
   the behavior of programs.
    1) a state cannot be both blocked on an extension call
        and also step,
    2) a state cannot both step and be halted
    3) a state cannot both be halted and blocked on an external call
    4) after external calls, cores are back in a "runnable" state
       (NOTE: this axiom may be removed at some point) *)

Record CoreSemantics {G C M (*D only needed in initial_mem*):Type}: Type :=
  { (*Removed: is a propert of programs, not of cores
      initial_mem: G -> M -> D -> Prop;*)
    initial_core : G -> val -> list val -> option C;
    at_external : C -> option (external_function * signature * list val);
    after_external : option val -> C -> option C;
    halted : C -> option val;
    corestep : G -> C -> M -> C -> M -> Prop;

    corestep_not_at_external: forall ge m q m' q',
      corestep ge q m q' m' -> at_external q = None;
    corestep_not_halted: forall ge m q m' q',
      corestep ge q m q' m' -> halted q = None;
    at_external_halted_excl: forall q,
      at_external q = None \/ halted q = None;
    after_at_external_excl : forall retv q q',
      after_external retv q = Some q' -> at_external q' = None
  }.Fixpoint corestepN (n:nat) : C -> M -> C -> M -> Prop :=
    match n with
      | O => fun c m c' m' => (c,m) = (c',m')
      | S k => fun c1 m1 c3 m3 => exists c2, exists m2,
        corestep Sem ge c1 m1 c2 m2 /\
        corestepN k c2 m2 c3 m3
    end.

(* Specifications: *)
Lemma corestepN_add : forall n m c1 m1 c3 m3, corestepN (n+m) c1 m1 c3 m3 <-> exists c2, exists m2, corestepN n c1 m1 c2 m2 /\ corestepN m c2 m2 c3 m3.