(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
(*                                                                     *)
(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
(*                                                                     *)
(* *********************************************************************)
(** The RTL intermediate language: abstract syntax and semantics.  RTL stands for "Register Transfer Language". This is the first intermediate language after Cminor and CminorSel. *)
(** * Abstract syntax *)
(** RTL is organized as instructions, functions and programs. Instructions correspond roughly to elementary instructions of the target processor, but take their arguments and leave their results in pseudo-registers (also called temporaries in textbooks). Infinitely many pseudo-registers are available, and each function has its own set of pseudo-registers, unaffected by function calls.  Instructions are organized as a control-flow graph: a function is a finite map from ``nodes'' (abstract program points) to instructions, and each instruction lists explicitly the nodes of its successors. *)
Definition node := positive.
Inductive instruction: Type :=
| Inop: node -> instruction
(** No operation -- just branch to the successor. *)
| Iop: operation -> list reg -> reg -> node -> instruction
(** [Iop op args dest succ] performs the arithmetic operation [op] over the values of registers [args], stores the result in [dest], and branches to [succ]. *)
| Iload: memory_chunk -> addressing -> list reg -> reg -> node -> instruction
(** [Iload chunk addr args dest succ] loads a [chunk] quantity from the address determined by the addressing mode [addr] and the values of the [args] registers, stores the quantity just read into [dest], and branches to [succ]. *)
| Istore: memory_chunk -> addressing -> list reg -> reg -> node -> instruction
(** [Istore chunk addr args src succ] stores the value of register [src] in the [chunk] quantity at the the address determined by the addressing mode [addr] and the values of the [args] registers, then branches to [succ]. *)
| Icall: signature -> reg + ident -> list reg -> reg -> node -> instruction
(** [Icall sig fn args dest succ] invokes the function determined by [fn] (either a function pointer found in a register or a function name), giving it the values of registers [args] as arguments.  It stores the return value in [dest] and branches to [succ]. *)
| Itailcall: signature -> reg + ident -> list reg -> instruction
(** [Itailcall sig fn args] performs a function invocation in tail-call position.  *)
| Ibuiltin: external_function -> list (builtin_arg reg) -> builtin_res reg -> node -> instruction
(** [Ibuiltin ef args dest succ] calls the built-in function identified by [ef], giving it the values of [args] as arguments. It stores the return value in [dest] and branches to [succ]. *)
| Icond: condition -> list reg -> node -> node -> instruction
(** [Icond cond args ifso ifnot] evaluates the boolean condition [cond] over the values of registers [args].  If the condition is true, it transitions to [ifso].  If the condition is false, it transitions to [ifnot]. *)
| Ijumptable: reg -> list node -> instruction
(** [Ijumptable arg tbl] transitions to the node that is the [n]-th element of the list [tbl], where [n] is the signed integer value of register [arg]. *)
| Ireturn: option reg -> instruction.
(** [Ireturn] terminates the execution of the current function (it has no successor).  It returns the value of the given register, or [Vundef] if none is given. *)
Definition code: Type := PTree.t instruction.
Record function: Type := mkfunction {
fn_sig: signature;
fn_params: list reg;
fn_stacksize: Z;
fn_code: code;
fn_entrypoint: node
}.
(** A function description comprises a control-flow graph (CFG) [fn_code] (a partial finite mapping from nodes to instructions).  As in Cminor, [fn_sig] is the function signature and [fn_stacksize] the number of bytes for its stack-allocated activation record.  [fn_params] is the list of registers that are bound to the values of arguments at call time. [fn_entrypoint] is the node of the first instruction of the function in the CFG. *)
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 *)
Definition genv := Genv.t fundef unit.
Definition regset := Regmap.t val.
Fixpoint init_regs (vl: list val) (rl: list reg) {struct rl} : regset :=
match rl, vl with
| r1 :: rs, v1 :: vs => Regmap.set r1 v1 (init_regs vs rs)
| _, _ => Regmap.init Vundef
end.
(** The dynamic semantics of RTL is given in small-step style, as a set of transitions between states.  A state captures the current point in the execution.  Three kinds of states appear in the transitions:  - [State cs f sp pc rs m] describes an execution point within a function. [f] is the current function. [sp] is the pointer to the stack block for its current activation (as in Cminor). [pc] is the current program point (CFG node) within the code [c]. [rs] gives the current values for the pseudo-registers. [m] is the current memory state. - [Callstate cs f args m] is an intermediate state that appears during function calls. [f] is the function definition that we are calling. [args] (a list of values) are the arguments for this call. [m] is the current memory state. - [Returnstate cs v m] is an intermediate state that appears when a function terminates and returns to its caller. [v] is the return value and [m] the current memory state.  In all three kinds of states, the [cs] parameter represents the call stack. It is a list of frames [Stackframe res f sp pc rs].  Each frame represents a function call in progress. [res] is the pseudo-register that will receive the result of the call. [f] is the calling function. [sp] is its stack pointer. [pc] is the program point for the instruction that follows the call. [rs] is the state of registers in the calling function. *)
Inductive stackframe : Type :=
| Stackframe:
forall (res: reg)            (**r where to store the result *)
(f: function)         (**r calling function *)
(sp: val)             (**r stack pointer in calling function *)
(pc: node)            (**r program point in calling function *)
(rs: regset),         (**r register state in calling function *)
stackframe.
Inductive state : Type :=
| State:
forall (stack: list stackframe) (**r call stack *)
(f: function)            (**r current function *)
(sp: val)                (**r stack pointer *)
(pc: node)               (**r current program point in [c] *)
(rs: regset)             (**r register state *)
(m: mem),                (**r memory state *)
state
| Callstate:
forall (stack: list stackframe) (**r call stack *)
(f: fundef)              (**r function to call *)
(args: list val)         (**r arguments to the call *)
(m: mem),                (**r memory state *)
state
| Returnstate:
forall (stack: list stackframe) (**r call stack *)
(v: val)                 (**r return value for the call *)
(m: mem),                (**r memory state *)
state.
Section RELSEM.
Variable ge: genv.
Definition find_function
(ros: reg + ident) (rs: regset) : option fundef :=
match ros with
| inl r => Genv.find_funct ge rs#r
| inr symb =>
match Genv.find_symbol ge symb with
| None => None
| Some b => Genv.find_funct_ptr ge b
end
end.
(** The transitions are presented as an inductive predicate [step ge st1 t st2], where [ge] is the global environment, [st1] the initial state, [st2] the final state, and [t] the trace of system calls performed during this transition. *)
Inductive step: state -> trace -> state -> Prop :=
| exec_Inop:
forall s f sp pc rs m pc',
(fn_code f)!pc = Some(Inop pc') ->
step (State s f sp pc rs m)
E0 (State s f sp pc' rs m)
| exec_Iop:
forall s f sp pc rs m op args res pc' v,
(fn_code f)!pc = Some(Iop op args res pc') ->
eval_operation ge sp op rs##args m = Some v ->
step (State s f sp pc rs m)
E0 (State s f sp pc' (rs#res <- v) m)
| exec_Iload:
forall s f sp pc rs m chunk addr args dst pc' a v,
(fn_code f)!pc = Some(Iload chunk addr args dst pc') ->
eval_addressing ge sp addr rs##args = Some a ->
Mem.loadv chunk m a = Some v ->
step (State s f sp pc rs m)
E0 (State s f sp pc' (rs#dst <- v) m)
| exec_Istore:
forall s f sp pc rs m chunk addr args src pc' a m',
(fn_code f)!pc = Some(Istore chunk addr args src pc') ->
eval_addressing ge sp addr rs##args = Some a ->
Mem.storev chunk m a rs#src = Some m' ->
step (State s f sp pc rs m)
E0 (State s f sp pc' rs m')
| exec_Icall:
forall s f sp pc rs m sig ros args res pc' fd,
(fn_code f)!pc = Some(Icall sig ros args res pc') ->
find_function ros rs = Some fd ->
funsig fd = sig ->
step (State s f sp pc rs m)
E0 (Callstate (Stackframe res f sp pc' rs :: s) fd rs##args m)
| exec_Itailcall:
forall s f stk pc rs m sig ros args fd m',
(fn_code f)!pc = Some(Itailcall sig ros args) ->
find_function ros rs = Some fd ->
funsig fd = sig ->
Mem.free m stk 0 f.(fn_stacksize) = Some m' ->
step (State s f (Vptr stk Ptrofs.zero) pc rs m)
E0 (Callstate s fd rs##args m')
| exec_Ibuiltin:
forall s f sp pc rs m ef args res pc' vargs t vres m',
(fn_code f)!pc = Some(Ibuiltin ef args res pc') ->
eval_builtin_args ge (fun r => rs#r) sp m args vargs ->
external_call ef ge vargs m t vres m' ->
step (State s f sp pc rs m)
t (State s f sp pc' (regmap_setres res vres rs) m')
| exec_Icond:
forall s f sp pc rs m cond args ifso ifnot b pc',
(fn_code f)!pc = Some(Icond cond args ifso ifnot) ->
eval_condition cond rs##args m = Some b ->
pc' = (if b then ifso else ifnot) ->
step (State s f sp pc rs m)
E0 (State s f sp pc' rs m)
| exec_Ijumptable:
forall s f sp pc rs m arg tbl n pc',
(fn_code f)!pc = Some(Ijumptable arg tbl) ->
rs#arg = Vint n ->
list_nth_z tbl (Int.unsigned n) = Some pc' ->
step (State s f sp pc rs m)
E0 (State s f sp pc' rs m)
| exec_Ireturn:
forall s f stk pc rs m or m',
(fn_code f)!pc = Some(Ireturn or) ->
Mem.free m stk 0 f.(fn_stacksize) = Some m' ->
step (State s f (Vptr stk Ptrofs.zero) pc rs m)
E0 (Returnstate s (regmap_optget or Vundef rs) m')
| exec_function_internal:
forall s f args m m' stk,
Mem.alloc m 0 f.(fn_stacksize) = (m', stk) ->
step (Callstate s (Internal f) args m)
E0 (State s
f
(Vptr stk Ptrofs.zero)
f.(fn_entrypoint)
(init_regs args f.(fn_params))
m')
| exec_function_external:
forall s ef args res t m m',
external_call ef ge args m t res m' ->
step (Callstate s (External ef) args m)
t (Returnstate s res m')
| exec_return:
forall res f sp pc rs s vres m,
step (Returnstate (Stackframe res f sp pc rs :: s) vres m)
E0 (State s f sp pc (rs#res <- vres) m).

Lemma exec_Iop':
forall s f sp pc rs m op args res pc' rs' v,
(fn_code f)!pc = Some(Iop op args res pc') ->
eval_operation ge sp op rs##args m = Some v ->
rs' = (rs#res <- v) ->
step (State s f sp pc rs m)
E0 (State s f sp pc' rs' m).