(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.(** * Abstract syntax *)

(** CminorSel programs share the general structure of Cminor programs:
  functions, statements and expressions.  However, CminorSel uses
  machine-dependent operations, addressing modes and conditions,
  as defined in module [Op] and used in lower-level intermediate
  languages ([RTL] and below).  Moreover, to express sharing of
  sub-computations, a "let" binding is provided (constructions
  [Elet] and [Eletvar]), using de Bruijn indices to refer to "let"-bound
  variables. *)

Inductive expr : Type :=
  | Evar : ident -> expr
  | Eop : operation -> exprlist -> expr
  | Eload : memory_chunk -> addressing -> exprlist -> expr
  | Econdition : condexpr -> expr -> expr -> expr
  | Elet : expr -> expr -> expr
  | Eletvar : nat -> expr
  | Ebuiltin : external_function -> exprlist -> expr
  | Eexternal : ident -> signature -> exprlist -> expr

with exprlist : Type :=
  | Enil: exprlist
  | Econs: expr -> exprlist -> exprlist

with condexpr : Type :=
  | CEcond : condition -> exprlist -> condexpr
  | CEcondition : condexpr -> condexpr -> condexpr -> condexpr
  | CElet: expr -> condexpr -> condexpr.Infix ":::" := Econs (at level 60, right associativity) : cminorsel_scope.(** Conditional expressions [condexpr] are expressions that are evaluated
  not for their exact value, but for their [true]/[false] Boolean value.
  Likewise, exit expressions [exitexpr] are expressions that evaluate
  to an exit number.  They are used to compile the [Sswitch] statement
  of Cminor. *)

Inductive exitexpr : Type :=
  | XEexit: nat -> exitexpr
  | XEjumptable: expr -> list nat -> exitexpr
  | XEcondition: condexpr -> exitexpr -> exitexpr -> exitexpr
  | XElet: expr -> exitexpr -> exitexpr.(** Conditional expressions [condexpr] are expressions that are evaluated
  not for their exact value, but for their [true]/[false] Boolean value.
  Likewise, exit expressions [exitexpr] are expressions that evaluate
  to an exit number.  They are used to compile the [Sswitch] statement
  of Cminor. *)

Inductive exitexpr : Type :=
  | XEexit: nat -> exitexpr
  | XEjumptable: expr -> list nat -> exitexpr
  | XEcondition: condexpr -> exitexpr -> exitexpr -> exitexpr
  | XElet: expr -> exitexpr -> exitexpr.(** Conditional expressions [condexpr] are expressions that are evaluated
  not for their exact value, but for their [true]/[false] Boolean value.
  Likewise, exit expressions [exitexpr] are expressions that evaluate
  to an exit number.  They are used to compile the [Sswitch] statement
  of Cminor. *)

Inductive exitexpr : Type :=
  | XEexit: nat -> exitexpr
  | XEjumptable: expr -> list nat -> exitexpr
  | XEcondition: condexpr -> exitexpr -> exitexpr -> exitexpr
  | XElet: expr -> exitexpr -> exitexpr.(** Conditional expressions [condexpr] are expressions that are evaluated
  not for their exact value, but for their [true]/[false] Boolean value.
  Likewise, exit expressions [exitexpr] are expressions that evaluate
  to an exit number.  They are used to compile the [Sswitch] statement
  of Cminor. *)

Inductive exitexpr : Type :=
  | XEexit: nat -> exitexpr
  | XEjumptable: expr -> list nat -> exitexpr
  | XEcondition: condexpr -> exitexpr -> exitexpr -> exitexpr
  | XElet: expr -> exitexpr -> exitexpr.(** Conditional expressions [condexpr] are expressions that are evaluated
  not for their exact value, but for their [true]/[false] Boolean value.
  Likewise, exit expressions [exitexpr] are expressions that evaluate
  to an exit number.  They are used to compile the [Sswitch] statement
  of Cminor. *)

Inductive exitexpr : Type :=
  | XEexit: nat -> exitexpr
  | XEjumptable: expr -> list nat -> exitexpr
  | XEcondition: condexpr -> exitexpr -> exitexpr -> exitexpr
  | XElet: expr -> exitexpr -> exitexpr.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.(** Statements are as in Cminor, except that the [Sifthenelse]
  construct uses a conditional expression, and the [Sstore] construct
  uses a machine-dependent addressing mode. *)

Inductive stmt : Type :=
  | Sskip: stmt
  | Sassign : ident -> expr -> stmt
  | Sstore : memory_chunk -> addressing -> exprlist -> expr -> stmt
  | Scall : option ident -> signature -> expr + ident -> exprlist -> stmt
  | Stailcall: signature -> expr + ident -> exprlist -> stmt
  | Sbuiltin : builtin_res ident -> external_function -> list (builtin_arg expr) -> stmt
  | Sseq: stmt -> stmt -> stmt
  | Sifthenelse: condexpr -> stmt -> stmt -> stmt
  | Sloop: stmt -> stmt
  | Sblock: stmt -> stmt
  | Sexit: nat -> stmt
  | Sswitch: exitexpr -> stmt
  | Sreturn: option expr -> stmt
  | Slabel: label -> stmt -> stmt
  | Sgoto: label -> stmt.Record function : Type := mkfunction {
  fn_sig: signature;
  fn_params: list ident;
  fn_vars: list ident;
  fn_stackspace: Z;
  fn_body: stmt
}.Record function : Type := mkfunction {
  fn_sig: signature;
  fn_params: list ident;
  fn_vars: list ident;
  fn_stackspace: Z;
  fn_body: stmt
}.Record function : Type := mkfunction {
  fn_sig: signature;
  fn_params: list ident;
  fn_vars: list ident;
  fn_stackspace: Z;
  fn_body: stmt
}.Record function : Type := mkfunction {
  fn_sig: signature;
  fn_params: list ident;
  fn_vars: list ident;
  fn_stackspace: Z;
  fn_body: stmt
}.Record function : Type := mkfunction {
  fn_sig: signature;
  fn_params: list ident;
  fn_vars: list ident;
  fn_stackspace: Z;
  fn_body: stmt
}.Record function : Type := mkfunction {
  fn_sig: signature;
  fn_params: list ident;
  fn_vars: list ident;
  fn_stackspace: Z;
  fn_body: stmt
}.Record function : Type := mkfunction {
  fn_sig: signature;
  fn_params: list ident;
  fn_vars: list ident;
  fn_stackspace: Z;
  fn_body: stmt
}.Definition fundef := AST.fundef function.Definition program := AST.program fundef unit.Definition funsig (fd: fundef) :=
  match fd with
  | Internal f => fn_sig f
  | External ef => ef_sig ef
  end.(** * Operational semantics *)

(** Three kinds of evaluation environments are involved:
- [genv]: global environments, define symbols and functions;
- [env]: local environments, map local variables to values;
- [lenv]: let environments, map de Bruijn indices to values.
*)

Definition genv := Genv.t fundef unit.Definition letenv := list val.(** Continuations *)

Inductive cont: Type :=
  | Kstop: cont                         (**r stop program execution *)
  | Kseq: stmt -> cont -> cont          (**r execute stmt, then cont *)
  | Kblock: cont -> cont                (**r exit a block, then do cont *)
  | Kcall: option ident -> function -> val -> env -> cont -> cont.(** Continuations *)

Inductive cont: Type :=
  | Kstop: cont                         (**r stop program execution *)
  | Kseq: stmt -> cont -> cont          (**r execute stmt, then cont *)
  | Kblock: cont -> cont                (**r exit a block, then do cont *)
  | Kcall: option ident -> function -> val -> env -> cont -> cont.(** Continuations *)

Inductive cont: Type :=
  | Kstop: cont                         (**r stop program execution *)
  | Kseq: stmt -> cont -> cont          (**r execute stmt, then cont *)
  | Kblock: cont -> cont                (**r exit a block, then do cont *)
  | Kcall: option ident -> function -> val -> env -> cont -> cont.(** Continuations *)

Inductive cont: Type :=
  | Kstop: cont                         (**r stop program execution *)
  | Kseq: stmt -> cont -> cont          (**r execute stmt, then cont *)
  | Kblock: cont -> cont                (**r exit a block, then do cont *)
  | Kcall: option ident -> function -> val -> env -> cont -> cont.(** Continuations *)

Inductive cont: Type :=
  | Kstop: cont                         (**r stop program execution *)
  | Kseq: stmt -> cont -> cont          (**r execute stmt, then cont *)
  | Kblock: cont -> cont                (**r exit a block, then do cont *)
  | Kcall: option ident -> function -> val -> env -> cont -> cont.(**r return to caller *)

(** States *)

Inductive state: Type :=
  | State:                              (**r execution within a function *)
      forall (f: function)              (**r currently executing function  *)
             (s: stmt)                  (**r statement under consideration *)
             (k: cont)                  (**r its continuation -- what to do next *)
             (sp: val)                  (**r current stack pointer *)
             (e: env)                   (**r current local environment *)
             (m: mem),                  (**r current memory state *)
      state
  | Callstate:                          (**r invocation of a fundef  *)
      forall (f: fundef)                (**r fundef to invoke *)
             (args: list val)           (**r arguments provided by caller *)
             (k: cont)                  (**r what to do next  *)
             (m: mem),                  (**r memory state *)
      state
  | Returnstate:
      forall (v: val)                   (**r return value *)
             (k: cont)                  (**r what to do next *)
             (m: mem),                  (**r memory state *)
      state.(**r return to caller *)

(** States *)

Inductive state: Type :=
  | State:                              (**r execution within a function *)
      forall (f: function)              (**r currently executing function  *)
             (s: stmt)                  (**r statement under consideration *)
             (k: cont)                  (**r its continuation -- what to do next *)
             (sp: val)                  (**r current stack pointer *)
             (e: env)                   (**r current local environment *)
             (m: mem),                  (**r current memory state *)
      state
  | Callstate:                          (**r invocation of a fundef  *)
      forall (f: fundef)                (**r fundef to invoke *)
             (args: list val)           (**r arguments provided by caller *)
             (k: cont)                  (**r what to do next  *)
             (m: mem),                  (**r memory state *)
      state
  | Returnstate:
      forall (v: val)                   (**r return value *)
             (k: cont)                  (**r what to do next *)
             (m: mem),                  (**r memory state *)
      state.(**r return to caller *)

(** States *)

Inductive state: Type :=
  | State:                              (**r execution within a function *)
      forall (f: function)              (**r currently executing function  *)
             (s: stmt)                  (**r statement under consideration *)
             (k: cont)                  (**r its continuation -- what to do next *)
             (sp: val)                  (**r current stack pointer *)
             (e: env)                   (**r current local environment *)
             (m: mem),                  (**r current memory state *)
      state
  | Callstate:                          (**r invocation of a fundef  *)
      forall (f: fundef)                (**r fundef to invoke *)
             (args: list val)           (**r arguments provided by caller *)
             (k: cont)                  (**r what to do next  *)
             (m: mem),                  (**r memory state *)
      state
  | Returnstate:
      forall (v: val)                   (**r return value *)
             (k: cont)                  (**r what to do next *)
             (m: mem),                  (**r memory state *)
      state.(**r return to caller *)

(** States *)

Inductive state: Type :=
  | State:                              (**r execution within a function *)
      forall (f: function)              (**r currently executing function  *)
             (s: stmt)                  (**r statement under consideration *)
             (k: cont)                  (**r its continuation -- what to do next *)
             (sp: val)                  (**r current stack pointer *)
             (e: env)                   (**r current local environment *)
             (m: mem),                  (**r current memory state *)
      state
  | Callstate:                          (**r invocation of a fundef  *)
      forall (f: fundef)                (**r fundef to invoke *)
             (args: list val)           (**r arguments provided by caller *)
             (k: cont)                  (**r what to do next  *)
             (m: mem),                  (**r memory state *)
      state
  | Returnstate:
      forall (v: val)                   (**r return value *)
             (k: cont)                  (**r what to do next *)
             (m: mem),                  (**r memory state *)
      state.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Inductive eval_expr: letenv -> expr -> val -> Prop :=
  | eval_Evar: forall le id v,
      PTree.get id e = Some v ->
      eval_expr le (Evar id) v
  | eval_Eop: forall le op al vl v,
      eval_exprlist le al vl ->
      eval_operation ge sp op vl m = Some v ->
      eval_expr le (Eop op al) v
  | eval_Eload: forall le chunk addr al vl vaddr v,
      eval_exprlist le al vl ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.loadv chunk m vaddr = Some v ->
      eval_expr le (Eload chunk addr al) v
  | eval_Econdition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_expr le (if va then b else c) v ->
      eval_expr le (Econdition a b c) v
  | eval_Elet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_expr (v1 :: le) b v2 ->
      eval_expr le (Elet a b) v2
  | eval_Eletvar: forall le n v,
      nth_error le n = Some v ->
      eval_expr le (Eletvar n) v
  | eval_Ebuiltin: forall le ef al vl v,
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Ebuiltin ef al) v
  | eval_Eexternal: forall le id sg al b ef vl v,
      Genv.find_symbol ge id = Some b ->
      Genv.find_funct_ptr ge b = Some (External ef) ->
      ef_sig ef = sg ->
      eval_exprlist le al vl ->
      external_call ef ge vl m E0 v m ->
      eval_expr le (Eexternal id sg al) v

with eval_exprlist: letenv -> exprlist -> list val -> Prop :=
  | eval_Enil: forall le,
      eval_exprlist le Enil nil
  | eval_Econs: forall le a1 al v1 vl,
      eval_expr le a1 v1 -> eval_exprlist le al vl ->
      eval_exprlist le (Econs a1 al) (v1 :: vl)

with eval_condexpr: letenv -> condexpr -> bool -> Prop :=
  | eval_CEcond: forall le cond al vl vb,
      eval_exprlist le al vl ->
      eval_condition cond vl m = Some vb ->
      eval_condexpr le (CEcond cond al) vb
  | eval_CEcondition: forall le a b c va v,
      eval_condexpr le a va ->
      eval_condexpr le (if va then b else c) v ->
      eval_condexpr le (CEcondition a b c) v
  | eval_CElet: forall le a b v1 v2,
      eval_expr le a v1 ->
      eval_condexpr (v1 :: le) b v2 ->
      eval_condexpr le (CElet a b) v2.Scheme eval_expr_ind3 := Minimality for eval_expr Sort Prop
  with eval_exprlist_ind3 := Minimality for eval_exprlist Sort Prop
  with eval_condexpr_ind3 := Minimality for eval_condexpr Sort Prop.Inductive eval_exitexpr: letenv -> exitexpr -> nat -> Prop :=
  | eval_XEexit: forall le x,
      eval_exitexpr le (XEexit x) x
  | eval_XEjumptable: forall le a tbl n x,
      eval_expr le a (Vint n) ->
      list_nth_z tbl (Int.unsigned n) = Some x ->
      eval_exitexpr le (XEjumptable a tbl) x
  | eval_XEcondition: forall le a b c va x,
      eval_condexpr le a va ->
      eval_exitexpr le (if va then b else c) x ->
      eval_exitexpr le (XEcondition a b c) x
  | eval_XElet: forall le a b v x,
      eval_expr le a v ->
      eval_exitexpr (v :: le) b x ->
      eval_exitexpr le (XElet a b) x.Inductive eval_exitexpr: letenv -> exitexpr -> nat -> Prop :=
  | eval_XEexit: forall le x,
      eval_exitexpr le (XEexit x) x
  | eval_XEjumptable: forall le a tbl n x,
      eval_expr le a (Vint n) ->
      list_nth_z tbl (Int.unsigned n) = Some x ->
      eval_exitexpr le (XEjumptable a tbl) x
  | eval_XEcondition: forall le a b c va x,
      eval_condexpr le a va ->
      eval_exitexpr le (if va then b else c) x ->
      eval_exitexpr le (XEcondition a b c) x
  | eval_XElet: forall le a b v x,
      eval_expr le a v ->
      eval_exitexpr (v :: le) b x ->
      eval_exitexpr le (XElet a b) x.Inductive eval_exitexpr: letenv -> exitexpr -> nat -> Prop :=
  | eval_XEexit: forall le x,
      eval_exitexpr le (XEexit x) x
  | eval_XEjumptable: forall le a tbl n x,
      eval_expr le a (Vint n) ->
      list_nth_z tbl (Int.unsigned n) = Some x ->
      eval_exitexpr le (XEjumptable a tbl) x
  | eval_XEcondition: forall le a b c va x,
      eval_condexpr le a va ->
      eval_exitexpr le (if va then b else c) x ->
      eval_exitexpr le (XEcondition a b c) x
  | eval_XElet: forall le a b v x,
      eval_expr le a v ->
      eval_exitexpr (v :: le) b x ->
      eval_exitexpr le (XElet a b) x.Inductive eval_exitexpr: letenv -> exitexpr -> nat -> Prop :=
  | eval_XEexit: forall le x,
      eval_exitexpr le (XEexit x) x
  | eval_XEjumptable: forall le a tbl n x,
      eval_expr le a (Vint n) ->
      list_nth_z tbl (Int.unsigned n) = Some x ->
      eval_exitexpr le (XEjumptable a tbl) x
  | eval_XEcondition: forall le a b c va x,
      eval_condexpr le a va ->
      eval_exitexpr le (if va then b else c) x ->
      eval_exitexpr le (XEcondition a b c) x
  | eval_XElet: forall le a b v x,
      eval_expr le a v ->
      eval_exitexpr (v :: le) b x ->
      eval_exitexpr le (XElet a b) x.Inductive eval_exitexpr: letenv -> exitexpr -> nat -> Prop :=
  | eval_XEexit: forall le x,
      eval_exitexpr le (XEexit x) x
  | eval_XEjumptable: forall le a tbl n x,
      eval_expr le a (Vint n) ->
      list_nth_z tbl (Int.unsigned n) = Some x ->
      eval_exitexpr le (XEjumptable a tbl) x
  | eval_XEcondition: forall le a b c va x,
      eval_condexpr le a va ->
      eval_exitexpr le (if va then b else c) x ->
      eval_exitexpr le (XEcondition a b c) x
  | eval_XElet: forall le a b v x,
      eval_expr le a v ->
      eval_exitexpr (v :: le) b x ->
      eval_exitexpr le (XElet a b) x.Inductive eval_expr_or_symbol: letenv -> expr + ident -> val -> Prop :=
  | eval_eos_e: forall le e v,
      eval_expr le e v ->
      eval_expr_or_symbol le (inl _ e) v
  | eval_eos_s: forall le id b,
      Genv.find_symbol ge id = Some b ->
      eval_expr_or_symbol le (inr _ id) (Vptr b Ptrofs.zero).Inductive eval_expr_or_symbol: letenv -> expr + ident -> val -> Prop :=
  | eval_eos_e: forall le e v,
      eval_expr le e v ->
      eval_expr_or_symbol le (inl _ e) v
  | eval_eos_s: forall le id b,
      Genv.find_symbol ge id = Some b ->
      eval_expr_or_symbol le (inr _ id) (Vptr b Ptrofs.zero).Inductive eval_expr_or_symbol: letenv -> expr + ident -> val -> Prop :=
  | eval_eos_e: forall le e v,
      eval_expr le e v ->
      eval_expr_or_symbol le (inl _ e) v
  | eval_eos_s: forall le id b,
      Genv.find_symbol ge id = Some b ->
      eval_expr_or_symbol le (inr _ id) (Vptr b Ptrofs.zero).Inductive eval_builtin_arg: builtin_arg expr -> val -> Prop :=
  | eval_BA: forall a v,
      eval_expr nil a v ->
      eval_builtin_arg (BA a) v
  | eval_BA_int: forall n,
      eval_builtin_arg (BA_int n) (Vint n)
  | eval_BA_long: forall n,
      eval_builtin_arg (BA_long n) (Vlong n)
  | eval_BA_float: forall n,
      eval_builtin_arg (BA_float n) (Vfloat n)
  | eval_BA_single: forall n,
      eval_builtin_arg (BA_single n) (Vsingle n)
  | eval_BA_loadstack: forall chunk ofs v,
      Mem.loadv chunk m (Val.offset_ptr sp ofs) = Some v ->
      eval_builtin_arg (BA_loadstack chunk ofs) v
  | eval_BA_addrstack: forall ofs,
      eval_builtin_arg (BA_addrstack ofs) (Val.offset_ptr sp ofs)
  | eval_BA_loadglobal: forall chunk id ofs v,
      Mem.loadv chunk m (Genv.symbol_address ge id ofs) = Some v ->
      eval_builtin_arg (BA_loadglobal chunk id ofs) v
  | eval_BA_addrglobal: forall id ofs,
      eval_builtin_arg (BA_addrglobal id ofs) (Genv.symbol_address ge id ofs)
  | eval_BA_splitlong: forall a1 a2 v1 v2,
      eval_expr nil a1 v1 -> eval_expr nil a2 v2 ->
      eval_builtin_arg (BA_splitlong (BA a1) (BA a2)) (Val.longofwords v1 v2)
  | eval_BA_addptr: forall a1 v1 a2 v2,
      eval_builtin_arg a1 v1 -> eval_builtin_arg a2 v2 ->
      eval_builtin_arg (BA_addptr a1 a2)
                       (if Archi.ptr64 then Val.addl v1 v2 else Val.add v1 v2).Inductive eval_builtin_arg: builtin_arg expr -> val -> Prop :=
  | eval_BA: forall a v,
      eval_expr nil a v ->
      eval_builtin_arg (BA a) v
  | eval_BA_int: forall n,
      eval_builtin_arg (BA_int n) (Vint n)
  | eval_BA_long: forall n,
      eval_builtin_arg (BA_long n) (Vlong n)
  | eval_BA_float: forall n,
      eval_builtin_arg (BA_float n) (Vfloat n)
  | eval_BA_single: forall n,
      eval_builtin_arg (BA_single n) (Vsingle n)
  | eval_BA_loadstack: forall chunk ofs v,
      Mem.loadv chunk m (Val.offset_ptr sp ofs) = Some v ->
      eval_builtin_arg (BA_loadstack chunk ofs) v
  | eval_BA_addrstack: forall ofs,
      eval_builtin_arg (BA_addrstack ofs) (Val.offset_ptr sp ofs)
  | eval_BA_loadglobal: forall chunk id ofs v,
      Mem.loadv chunk m (Genv.symbol_address ge id ofs) = Some v ->
      eval_builtin_arg (BA_loadglobal chunk id ofs) v
  | eval_BA_addrglobal: forall id ofs,
      eval_builtin_arg (BA_addrglobal id ofs) (Genv.symbol_address ge id ofs)
  | eval_BA_splitlong: forall a1 a2 v1 v2,
      eval_expr nil a1 v1 -> eval_expr nil a2 v2 ->
      eval_builtin_arg (BA_splitlong (BA a1) (BA a2)) (Val.longofwords v1 v2)
  | eval_BA_addptr: forall a1 v1 a2 v2,
      eval_builtin_arg a1 v1 -> eval_builtin_arg a2 v2 ->
      eval_builtin_arg (BA_addptr a1 a2)
                       (if Archi.ptr64 then Val.addl v1 v2 else Val.add v1 v2).Inductive eval_builtin_arg: builtin_arg expr -> val -> Prop :=
  | eval_BA: forall a v,
      eval_expr nil a v ->
      eval_builtin_arg (BA a) v
  | eval_BA_int: forall n,
      eval_builtin_arg (BA_int n) (Vint n)
  | eval_BA_long: forall n,
      eval_builtin_arg (BA_long n) (Vlong n)
  | eval_BA_float: forall n,
      eval_builtin_arg (BA_float n) (Vfloat n)
  | eval_BA_single: forall n,
      eval_builtin_arg (BA_single n) (Vsingle n)
  | eval_BA_loadstack: forall chunk ofs v,
      Mem.loadv chunk m (Val.offset_ptr sp ofs) = Some v ->
      eval_builtin_arg (BA_loadstack chunk ofs) v
  | eval_BA_addrstack: forall ofs,
      eval_builtin_arg (BA_addrstack ofs) (Val.offset_ptr sp ofs)
  | eval_BA_loadglobal: forall chunk id ofs v,
      Mem.loadv chunk m (Genv.symbol_address ge id ofs) = Some v ->
      eval_builtin_arg (BA_loadglobal chunk id ofs) v
  | eval_BA_addrglobal: forall id ofs,
      eval_builtin_arg (BA_addrglobal id ofs) (Genv.symbol_address ge id ofs)
  | eval_BA_splitlong: forall a1 a2 v1 v2,
      eval_expr nil a1 v1 -> eval_expr nil a2 v2 ->
      eval_builtin_arg (BA_splitlong (BA a1) (BA a2)) (Val.longofwords v1 v2)
  | eval_BA_addptr: forall a1 v1 a2 v2,
      eval_builtin_arg a1 v1 -> eval_builtin_arg a2 v2 ->
      eval_builtin_arg (BA_addptr a1 a2)
                       (if Archi.ptr64 then Val.addl v1 v2 else Val.add v1 v2).Inductive eval_builtin_arg: builtin_arg expr -> val -> Prop :=
  | eval_BA: forall a v,
      eval_expr nil a v ->
      eval_builtin_arg (BA a) v
  | eval_BA_int: forall n,
      eval_builtin_arg (BA_int n) (Vint n)
  | eval_BA_long: forall n,
      eval_builtin_arg (BA_long n) (Vlong n)
  | eval_BA_float: forall n,
      eval_builtin_arg (BA_float n) (Vfloat n)
  | eval_BA_single: forall n,
      eval_builtin_arg (BA_single n) (Vsingle n)
  | eval_BA_loadstack: forall chunk ofs v,
      Mem.loadv chunk m (Val.offset_ptr sp ofs) = Some v ->
      eval_builtin_arg (BA_loadstack chunk ofs) v
  | eval_BA_addrstack: forall ofs,
      eval_builtin_arg (BA_addrstack ofs) (Val.offset_ptr sp ofs)
  | eval_BA_loadglobal: forall chunk id ofs v,
      Mem.loadv chunk m (Genv.symbol_address ge id ofs) = Some v ->
      eval_builtin_arg (BA_loadglobal chunk id ofs) v
  | eval_BA_addrglobal: forall id ofs,
      eval_builtin_arg (BA_addrglobal id ofs) (Genv.symbol_address ge id ofs)
  | eval_BA_splitlong: forall a1 a2 v1 v2,
      eval_expr nil a1 v1 -> eval_expr nil a2 v2 ->
      eval_builtin_arg (BA_splitlong (BA a1) (BA a2)) (Val.longofwords v1 v2)
  | eval_BA_addptr: forall a1 v1 a2 v2,
      eval_builtin_arg a1 v1 -> eval_builtin_arg a2 v2 ->
      eval_builtin_arg (BA_addptr a1 a2)
                       (if Archi.ptr64 then Val.addl v1 v2 else Val.add v1 v2).Inductive eval_builtin_arg: builtin_arg expr -> val -> Prop :=
  | eval_BA: forall a v,
      eval_expr nil a v ->
      eval_builtin_arg (BA a) v
  | eval_BA_int: forall n,
      eval_builtin_arg (BA_int n) (Vint n)
  | eval_BA_long: forall n,
      eval_builtin_arg (BA_long n) (Vlong n)
  | eval_BA_float: forall n,
      eval_builtin_arg (BA_float n) (Vfloat n)
  | eval_BA_single: forall n,
      eval_builtin_arg (BA_single n) (Vsingle n)
  | eval_BA_loadstack: forall chunk ofs v,
      Mem.loadv chunk m (Val.offset_ptr sp ofs) = Some v ->
      eval_builtin_arg (BA_loadstack chunk ofs) v
  | eval_BA_addrstack: forall ofs,
      eval_builtin_arg (BA_addrstack ofs) (Val.offset_ptr sp ofs)
  | eval_BA_loadglobal: forall chunk id ofs v,
      Mem.loadv chunk m (Genv.symbol_address ge id ofs) = Some v ->
      eval_builtin_arg (BA_loadglobal chunk id ofs) v
  | eval_BA_addrglobal: forall id ofs,
      eval_builtin_arg (BA_addrglobal id ofs) (Genv.symbol_address ge id ofs)
  | eval_BA_splitlong: forall a1 a2 v1 v2,
      eval_expr nil a1 v1 -> eval_expr nil a2 v2 ->
      eval_builtin_arg (BA_splitlong (BA a1) (BA a2)) (Val.longofwords v1 v2)
  | eval_BA_addptr: forall a1 v1 a2 v2,
      eval_builtin_arg a1 v1 -> eval_builtin_arg a2 v2 ->
      eval_builtin_arg (BA_addptr a1 a2)
                       (if Archi.ptr64 then Val.addl v1 v2 else Val.add v1 v2).Inductive eval_builtin_arg: builtin_arg expr -> val -> Prop :=
  | eval_BA: forall a v,
      eval_expr nil a v ->
      eval_builtin_arg (BA a) v
  | eval_BA_int: forall n,
      eval_builtin_arg (BA_int n) (Vint n)
  | eval_BA_long: forall n,
      eval_builtin_arg (BA_long n) (Vlong n)
  | eval_BA_float: forall n,
      eval_builtin_arg (BA_float n) (Vfloat n)
  | eval_BA_single: forall n,
      eval_builtin_arg (BA_single n) (Vsingle n)
  | eval_BA_loadstack: forall chunk ofs v,
      Mem.loadv chunk m (Val.offset_ptr sp ofs) = Some v ->
      eval_builtin_arg (BA_loadstack chunk ofs) v
  | eval_BA_addrstack: forall ofs,
      eval_builtin_arg (BA_addrstack ofs) (Val.offset_ptr sp ofs)
  | eval_BA_loadglobal: forall chunk id ofs v,
      Mem.loadv chunk m (Genv.symbol_address ge id ofs) = Some v ->
      eval_builtin_arg (BA_loadglobal chunk id ofs) v
  | eval_BA_addrglobal: forall id ofs,
      eval_builtin_arg (BA_addrglobal id ofs) (Genv.symbol_address ge id ofs)
  | eval_BA_splitlong: forall a1 a2 v1 v2,
      eval_expr nil a1 v1 -> eval_expr nil a2 v2 ->
      eval_builtin_arg (BA_splitlong (BA a1) (BA a2)) (Val.longofwords v1 v2)
  | eval_BA_addptr: forall a1 v1 a2 v2,
      eval_builtin_arg a1 v1 -> eval_builtin_arg a2 v2 ->
      eval_builtin_arg (BA_addptr a1 a2)
                       (if Archi.ptr64 then Val.addl v1 v2 else Val.add v1 v2).Inductive eval_builtin_arg: builtin_arg expr -> val -> Prop :=
  | eval_BA: forall a v,
      eval_expr nil a v ->
      eval_builtin_arg (BA a) v
  | eval_BA_int: forall n,
      eval_builtin_arg (BA_int n) (Vint n)
  | eval_BA_long: forall n,
      eval_builtin_arg (BA_long n) (Vlong n)
  | eval_BA_float: forall n,
      eval_builtin_arg (BA_float n) (Vfloat n)
  | eval_BA_single: forall n,
      eval_builtin_arg (BA_single n) (Vsingle n)
  | eval_BA_loadstack: forall chunk ofs v,
      Mem.loadv chunk m (Val.offset_ptr sp ofs) = Some v ->
      eval_builtin_arg (BA_loadstack chunk ofs) v
  | eval_BA_addrstack: forall ofs,
      eval_builtin_arg (BA_addrstack ofs) (Val.offset_ptr sp ofs)
  | eval_BA_loadglobal: forall chunk id ofs v,
      Mem.loadv chunk m (Genv.symbol_address ge id ofs) = Some v ->
      eval_builtin_arg (BA_loadglobal chunk id ofs) v
  | eval_BA_addrglobal: forall id ofs,
      eval_builtin_arg (BA_addrglobal id ofs) (Genv.symbol_address ge id ofs)
  | eval_BA_splitlong: forall a1 a2 v1 v2,
      eval_expr nil a1 v1 -> eval_expr nil a2 v2 ->
      eval_builtin_arg (BA_splitlong (BA a1) (BA a2)) (Val.longofwords v1 v2)
  | eval_BA_addptr: forall a1 v1 a2 v2,
      eval_builtin_arg a1 v1 -> eval_builtin_arg a2 v2 ->
      eval_builtin_arg (BA_addptr a1 a2)
                       (if Archi.ptr64 then Val.addl v1 v2 else Val.add v1 v2).Inductive eval_builtin_arg: builtin_arg expr -> val -> Prop :=
  | eval_BA: forall a v,
      eval_expr nil a v ->
      eval_builtin_arg (BA a) v
  | eval_BA_int: forall n,
      eval_builtin_arg (BA_int n) (Vint n)
  | eval_BA_long: forall n,
      eval_builtin_arg (BA_long n) (Vlong n)
  | eval_BA_float: forall n,
      eval_builtin_arg (BA_float n) (Vfloat n)
  | eval_BA_single: forall n,
      eval_builtin_arg (BA_single n) (Vsingle n)
  | eval_BA_loadstack: forall chunk ofs v,
      Mem.loadv chunk m (Val.offset_ptr sp ofs) = Some v ->
      eval_builtin_arg (BA_loadstack chunk ofs) v
  | eval_BA_addrstack: forall ofs,
      eval_builtin_arg (BA_addrstack ofs) (Val.offset_ptr sp ofs)
  | eval_BA_loadglobal: forall chunk id ofs v,
      Mem.loadv chunk m (Genv.symbol_address ge id ofs) = Some v ->
      eval_builtin_arg (BA_loadglobal chunk id ofs) v
  | eval_BA_addrglobal: forall id ofs,
      eval_builtin_arg (BA_addrglobal id ofs) (Genv.symbol_address ge id ofs)
  | eval_BA_splitlong: forall a1 a2 v1 v2,
      eval_expr nil a1 v1 -> eval_expr nil a2 v2 ->
      eval_builtin_arg (BA_splitlong (BA a1) (BA a2)) (Val.longofwords v1 v2)
  | eval_BA_addptr: forall a1 v1 a2 v2,
      eval_builtin_arg a1 v1 -> eval_builtin_arg a2 v2 ->
      eval_builtin_arg (BA_addptr a1 a2)
                       (if Archi.ptr64 then Val.addl v1 v2 else Val.add v1 v2).Inductive eval_builtin_arg: builtin_arg expr -> val -> Prop :=
  | eval_BA: forall a v,
      eval_expr nil a v ->
      eval_builtin_arg (BA a) v
  | eval_BA_int: forall n,
      eval_builtin_arg (BA_int n) (Vint n)
  | eval_BA_long: forall n,
      eval_builtin_arg (BA_long n) (Vlong n)
  | eval_BA_float: forall n,
      eval_builtin_arg (BA_float n) (Vfloat n)
  | eval_BA_single: forall n,
      eval_builtin_arg (BA_single n) (Vsingle n)
  | eval_BA_loadstack: forall chunk ofs v,
      Mem.loadv chunk m (Val.offset_ptr sp ofs) = Some v ->
      eval_builtin_arg (BA_loadstack chunk ofs) v
  | eval_BA_addrstack: forall ofs,
      eval_builtin_arg (BA_addrstack ofs) (Val.offset_ptr sp ofs)
  | eval_BA_loadglobal: forall chunk id ofs v,
      Mem.loadv chunk m (Genv.symbol_address ge id ofs) = Some v ->
      eval_builtin_arg (BA_loadglobal chunk id ofs) v
  | eval_BA_addrglobal: forall id ofs,
      eval_builtin_arg (BA_addrglobal id ofs) (Genv.symbol_address ge id ofs)
  | eval_BA_splitlong: forall a1 a2 v1 v2,
      eval_expr nil a1 v1 -> eval_expr nil a2 v2 ->
      eval_builtin_arg (BA_splitlong (BA a1) (BA a2)) (Val.longofwords v1 v2)
  | eval_BA_addptr: forall a1 v1 a2 v2,
      eval_builtin_arg a1 v1 -> eval_builtin_arg a2 v2 ->
      eval_builtin_arg (BA_addptr a1 a2)
                       (if Archi.ptr64 then Val.addl v1 v2 else Val.add v1 v2).Inductive eval_builtin_arg: builtin_arg expr -> val -> Prop :=
  | eval_BA: forall a v,
      eval_expr nil a v ->
      eval_builtin_arg (BA a) v
  | eval_BA_int: forall n,
      eval_builtin_arg (BA_int n) (Vint n)
  | eval_BA_long: forall n,
      eval_builtin_arg (BA_long n) (Vlong n)
  | eval_BA_float: forall n,
      eval_builtin_arg (BA_float n) (Vfloat n)
  | eval_BA_single: forall n,
      eval_builtin_arg (BA_single n) (Vsingle n)
  | eval_BA_loadstack: forall chunk ofs v,
      Mem.loadv chunk m (Val.offset_ptr sp ofs) = Some v ->
      eval_builtin_arg (BA_loadstack chunk ofs) v
  | eval_BA_addrstack: forall ofs,
      eval_builtin_arg (BA_addrstack ofs) (Val.offset_ptr sp ofs)
  | eval_BA_loadglobal: forall chunk id ofs v,
      Mem.loadv chunk m (Genv.symbol_address ge id ofs) = Some v ->
      eval_builtin_arg (BA_loadglobal chunk id ofs) v
  | eval_BA_addrglobal: forall id ofs,
      eval_builtin_arg (BA_addrglobal id ofs) (Genv.symbol_address ge id ofs)
  | eval_BA_splitlong: forall a1 a2 v1 v2,
      eval_expr nil a1 v1 -> eval_expr nil a2 v2 ->
      eval_builtin_arg (BA_splitlong (BA a1) (BA a2)) (Val.longofwords v1 v2)
  | eval_BA_addptr: forall a1 v1 a2 v2,
      eval_builtin_arg a1 v1 -> eval_builtin_arg a2 v2 ->
      eval_builtin_arg (BA_addptr a1 a2)
                       (if Archi.ptr64 then Val.addl v1 v2 else Val.add v1 v2).Inductive eval_builtin_arg: builtin_arg expr -> val -> Prop :=
  | eval_BA: forall a v,
      eval_expr nil a v ->
      eval_builtin_arg (BA a) v
  | eval_BA_int: forall n,
      eval_builtin_arg (BA_int n) (Vint n)
  | eval_BA_long: forall n,
      eval_builtin_arg (BA_long n) (Vlong n)
  | eval_BA_float: forall n,
      eval_builtin_arg (BA_float n) (Vfloat n)
  | eval_BA_single: forall n,
      eval_builtin_arg (BA_single n) (Vsingle n)
  | eval_BA_loadstack: forall chunk ofs v,
      Mem.loadv chunk m (Val.offset_ptr sp ofs) = Some v ->
      eval_builtin_arg (BA_loadstack chunk ofs) v
  | eval_BA_addrstack: forall ofs,
      eval_builtin_arg (BA_addrstack ofs) (Val.offset_ptr sp ofs)
  | eval_BA_loadglobal: forall chunk id ofs v,
      Mem.loadv chunk m (Genv.symbol_address ge id ofs) = Some v ->
      eval_builtin_arg (BA_loadglobal chunk id ofs) v
  | eval_BA_addrglobal: forall id ofs,
      eval_builtin_arg (BA_addrglobal id ofs) (Genv.symbol_address ge id ofs)
  | eval_BA_splitlong: forall a1 a2 v1 v2,
      eval_expr nil a1 v1 -> eval_expr nil a2 v2 ->
      eval_builtin_arg (BA_splitlong (BA a1) (BA a2)) (Val.longofwords v1 v2)
  | eval_BA_addptr: forall a1 v1 a2 v2,
      eval_builtin_arg a1 v1 -> eval_builtin_arg a2 v2 ->
      eval_builtin_arg (BA_addptr a1 a2)
                       (if Archi.ptr64 then Val.addl v1 v2 else Val.add v1 v2).Inductive eval_builtin_arg: builtin_arg expr -> val -> Prop :=
  | eval_BA: forall a v,
      eval_expr nil a v ->
      eval_builtin_arg (BA a) v
  | eval_BA_int: forall n,
      eval_builtin_arg (BA_int n) (Vint n)
  | eval_BA_long: forall n,
      eval_builtin_arg (BA_long n) (Vlong n)
  | eval_BA_float: forall n,
      eval_builtin_arg (BA_float n) (Vfloat n)
  | eval_BA_single: forall n,
      eval_builtin_arg (BA_single n) (Vsingle n)
  | eval_BA_loadstack: forall chunk ofs v,
      Mem.loadv chunk m (Val.offset_ptr sp ofs) = Some v ->
      eval_builtin_arg (BA_loadstack chunk ofs) v
  | eval_BA_addrstack: forall ofs,
      eval_builtin_arg (BA_addrstack ofs) (Val.offset_ptr sp ofs)
  | eval_BA_loadglobal: forall chunk id ofs v,
      Mem.loadv chunk m (Genv.symbol_address ge id ofs) = Some v ->
      eval_builtin_arg (BA_loadglobal chunk id ofs) v
  | eval_BA_addrglobal: forall id ofs,
      eval_builtin_arg (BA_addrglobal id ofs) (Genv.symbol_address ge id ofs)
  | eval_BA_splitlong: forall a1 a2 v1 v2,
      eval_expr nil a1 v1 -> eval_expr nil a2 v2 ->
      eval_builtin_arg (BA_splitlong (BA a1) (BA a2)) (Val.longofwords v1 v2)
  | eval_BA_addptr: forall a1 v1 a2 v2,
      eval_builtin_arg a1 v1 -> eval_builtin_arg a2 v2 ->
      eval_builtin_arg (BA_addptr a1 a2)
                       (if Archi.ptr64 then Val.addl v1 v2 else Val.add v1 v2).(** Update local environment with the result of a builtin. *)

Definition set_builtin_res (res: builtin_res ident) (v: val) (e: env) : env :=
  match res with
  | BR id => PTree.set id v e
  | _ => e
  end.(** Pop continuation until a call or stop *)

Fixpoint call_cont (k: cont) : cont :=
  match k with
  | Kseq s k => call_cont k
  | Kblock k => call_cont k
  | _ => k
  end.Definition is_call_cont (k: cont) : Prop :=
  match k with
  | Kstop => True
  | Kcall _ _ _ _ _ => True
  | _ => False
  end.(** Find the statement and manufacture the continuation
  corresponding to a label *)

Fixpoint find_label (lbl: label) (s: stmt) (k: cont)
                    {struct s}: option (stmt * cont) :=
  match s with
  | Sseq s1 s2 =>
      match find_label lbl s1 (Kseq s2 k) with
      | Some sk => Some sk
      | None => find_label lbl s2 k
      end
  | Sifthenelse c s1 s2 =>
      match find_label lbl s1 k with
      | Some sk => Some sk
      | None => find_label lbl s2 k
      end
  | Sloop s1 =>
      find_label lbl s1 (Kseq (Sloop s1) k)
  | Sblock s1 =>
      find_label lbl s1 (Kblock k)
  | Slabel lbl' s' =>
      if ident_eq lbl lbl' then Some(s', k) else find_label lbl s' k
  | _ => None
  end.(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).(** One step of execution *)

Inductive step: state -> trace -> state -> Prop :=

  | step_skip_seq: forall f s k sp e m,
      step (State f Sskip (Kseq s k) sp e m)
        E0 (State f s k sp e m)
  | step_skip_block: forall f k sp e m,
      step (State f Sskip (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_skip_call: forall f k sp e m m',
      is_call_cont k ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f Sskip k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef k m')

  | step_assign: forall f id a k sp e m v,
      eval_expr sp e m nil a v ->
      step (State f (Sassign id a) k sp e m)
        E0 (State f Sskip k sp (PTree.set id v e) m)

  | step_store: forall f chunk addr al b k sp e m vl v vaddr m',
      eval_exprlist sp e m nil al vl ->
      eval_expr sp e m nil b v ->
      eval_addressing ge sp addr vl = Some vaddr ->
      Mem.storev chunk m vaddr v = Some m' ->
      step (State f (Sstore chunk addr al b) k sp e m)
        E0 (State f Sskip k sp e m')

  | step_call: forall f optid sig a bl k sp e m vf vargs fd,
      eval_expr_or_symbol sp e m nil a vf ->
      eval_exprlist sp e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      step (State f (Scall optid sig a bl) k sp e m)
        E0 (Callstate fd vargs (Kcall optid f sp e k) m)

  | step_tailcall: forall f sig a bl k sp e m vf vargs fd m',
      eval_expr_or_symbol (Vptr sp Ptrofs.zero) e m nil a vf ->
      eval_exprlist (Vptr sp Ptrofs.zero) e m nil bl vargs ->
      Genv.find_funct ge vf = Some fd ->
      funsig fd = sig ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Stailcall sig a bl) k (Vptr sp Ptrofs.zero) e m)
        E0 (Callstate fd vargs (call_cont k) m')

  | step_builtin: forall f res ef al k sp e m vl t v m',
      list_forall2 (eval_builtin_arg sp e m) al vl ->
      external_call ef ge vl m t v m' ->
      step (State f (Sbuiltin res ef al) k sp e m)
         t (State f Sskip k sp (set_builtin_res res v e) m')

  | step_seq: forall f s1 s2 k sp e m,
      step (State f (Sseq s1 s2) k sp e m)
        E0 (State f s1 (Kseq s2 k) sp e m)

  | step_ifthenelse: forall f c s1 s2 k sp e m b,
      eval_condexpr sp e m nil c b ->
      step (State f (Sifthenelse c s1 s2) k sp e m)
        E0 (State f (if b then s1 else s2) k sp e m)

  | step_loop: forall f s k sp e m,
      step (State f (Sloop s) k sp e m)
        E0 (State f s (Kseq (Sloop s) k) sp e m)

  | step_block: forall f s k sp e m,
      step (State f (Sblock s) k sp e m)
        E0 (State f s (Kblock k) sp e m)

  | step_exit_seq: forall f n s k sp e m,
      step (State f (Sexit n) (Kseq s k) sp e m)
        E0 (State f (Sexit n) k sp e m)
  | step_exit_block_0: forall f k sp e m,
      step (State f (Sexit O) (Kblock k) sp e m)
        E0 (State f Sskip k sp e m)
  | step_exit_block_S: forall f n k sp e m,
      step (State f (Sexit (S n)) (Kblock k) sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_switch: forall f a k sp e m n,
      eval_exitexpr sp e m nil a n ->
      step (State f (Sswitch a) k sp e m)
        E0 (State f (Sexit n) k sp e m)

  | step_return_0: forall f k sp e m m',
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn None) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate Vundef (call_cont k) m')
  | step_return_1: forall f a k sp e m v m',
      eval_expr (Vptr sp Ptrofs.zero) e m nil a v ->
      Mem.free m sp 0 f.(fn_stackspace) = Some m' ->
      step (State f (Sreturn (Some a)) k (Vptr sp Ptrofs.zero) e m)
        E0 (Returnstate v (call_cont k) m')

  | step_label: forall f lbl s k sp e m,
      step (State f (Slabel lbl s) k sp e m)
        E0 (State f s k sp e m)

  | step_goto: forall f lbl k sp e m s' k',
      find_label lbl f.(fn_body) (call_cont k) = Some(s', k') ->
      step (State f (Sgoto lbl) k sp e m)
        E0 (State f s' k' sp e m)

  | step_internal_function: forall f vargs k m m' sp e,
      Mem.alloc m 0 f.(fn_stackspace) = (m', sp) ->
      set_locals f.(fn_vars) (set_params vargs f.(fn_params)) = e ->
      step (Callstate (Internal f) vargs k m)
        E0 (State f f.(fn_body) k (Vptr sp Ptrofs.zero) e m')
  | step_external_function: forall ef vargs k m t vres m',
      external_call ef ge vargs m t vres m' ->
      step (Callstate (External ef) vargs k m)
         t (Returnstate vres k m')

  | step_return: forall v optid f sp e k m,
      step (Returnstate v (Kcall optid f sp e k) m)
        E0 (State f Sskip k sp (set_optvar optid v e) m).Inductive initial_state (p: program): state -> Prop :=
  | initial_state_intro: forall b f m0,
      let ge := Genv.globalenv p in
      Genv.init_mem p = Some m0 ->
      Genv.find_symbol ge p.(prog_main) = Some b ->
      Genv.find_funct_ptr ge b = Some f ->
      funsig f = signature_main ->
      initial_state p (Callstate f nil Kstop m0).Inductive initial_state (p: program): state -> Prop :=
  | initial_state_intro: forall b f m0,
      let ge := Genv.globalenv p in
      Genv.init_mem p = Some m0 ->
      Genv.find_symbol ge p.(prog_main) = Some b ->
      Genv.find_funct_ptr ge b = Some f ->
      funsig f = signature_main ->
      initial_state p (Callstate f nil Kstop m0).Inductive initial_state (p: program): state -> Prop :=
  | initial_state_intro: forall b f m0,
      let ge := Genv.globalenv p in
      Genv.init_mem p = Some m0 ->
      Genv.find_symbol ge p.(prog_main) = Some b ->
      Genv.find_funct_ptr ge b = Some f ->
      funsig f = signature_main ->
      initial_state p (Callstate f nil Kstop m0).Inductive final_state: state -> int -> Prop :=
  | final_state_intro: forall r m,
      final_state (Returnstate (Vint r) Kstop m) r.Inductive final_state: state -> int -> Prop :=
  | final_state_intro: forall r m,
      final_state (Returnstate (Vint r) Kstop m) r.Definition semantics (p: program) :=
  Semantics step (initial_state p) final_state (Genv.globalenv p).(** * Lifting of let-bound variables *)

(** Instruction selection sometimes generate [Elet] constructs to
  share the evaluation of a subexpression.  Owing to the use of de
  Bruijn indices for let-bound variables, we need to shift de Bruijn
  indices when an expression [b] is put in a [Elet a b] context. *)

Fixpoint lift_expr (p: nat) (a: expr) {struct a}: expr :=
  match a with
  | Evar id => Evar id
  | Eop op bl => Eop op (lift_exprlist p bl)
  | Eload chunk addr bl => Eload chunk addr (lift_exprlist p bl)
  | Econdition a b c =>
      Econdition (lift_condexpr p a) (lift_expr p b) (lift_expr p c)
  | Elet b c => Elet (lift_expr p b) (lift_expr (S p) c)
  | Eletvar n =>
      if le_gt_dec p n then Eletvar (S n) else Eletvar n
  | Ebuiltin ef bl => Ebuiltin ef (lift_exprlist p bl)
  | Eexternal id sg bl => Eexternal id sg (lift_exprlist p bl)
  end

with lift_exprlist (p: nat) (a: exprlist) {struct a}: exprlist :=
  match a with
  | Enil => Enil
  | Econs b cl => Econs (lift_expr p b) (lift_exprlist p cl)
  end

with lift_condexpr (p: nat) (a: condexpr) {struct a}: condexpr :=
  match a with
  | CEcond c al => CEcond c (lift_exprlist p al)
  | CEcondition a b c => CEcondition (lift_condexpr p a) (lift_condexpr p b) (lift_condexpr p c)
  | CElet a b => CElet (lift_expr p a) (lift_condexpr (S p) b)
  end.Definition lift (a: expr): expr := lift_expr O a.(** We now relate the evaluation of a lifted expression with that
    of the original expression. *)

Inductive insert_lenv: letenv -> nat -> val -> letenv -> Prop :=
  | insert_lenv_0:
      forall le v,
      insert_lenv le O v (v :: le)
  | insert_lenv_S:
      forall le p w le' v,
      insert_lenv le p w le' ->
      insert_lenv (v :: le) (S p) w (v :: le').(** We now relate the evaluation of a lifted expression with that
    of the original expression. *)

Inductive insert_lenv: letenv -> nat -> val -> letenv -> Prop :=
  | insert_lenv_0:
      forall le v,
      insert_lenv le O v (v :: le)
  | insert_lenv_S:
      forall le p w le' v,
      insert_lenv le p w le' ->
      insert_lenv (v :: le) (S p) w (v :: le').(** We now relate the evaluation of a lifted expression with that
    of the original expression. *)

Inductive insert_lenv: letenv -> nat -> val -> letenv -> Prop :=
  | insert_lenv_0:
      forall le v,
      insert_lenv le O v (v :: le)
  | insert_lenv_S:
      forall le p w le' v,
      insert_lenv le p w le' ->
      insert_lenv (v :: le) (S p) w (v :: le').

(* Specifications: *)
Lemma insert_lenv_lookup1: forall le p w le', insert_lenv le p w le' -> forall n v, nth_error le n = Some v -> (p > n)%nat -> nth_error le' n = Some v.
Lemma insert_lenv_lookup2: forall le p w le', insert_lenv le p w le' -> forall n v, nth_error le n = Some v -> (p <= n)%nat -> nth_error le' (S n) = Some v.
Lemma eval_lift_expr: forall ge sp e m w le a v, eval_expr ge sp e m le a v -> forall p le', insert_lenv le p w le' -> eval_expr ge sp e m le' (lift_expr p a) v.
Lemma eval_lift: forall ge sp e m le a v w, eval_expr ge sp e m le a v -> eval_expr ge sp e m (w::le) (lift a) v.