(* *********************************************************************)
(*                                                                     *)
(*              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.
*)

Require Import Coqlib Maps.
Require Import AST Integers Values Events Memory Globalenvs Smallstep.
Require Import Op Registers.

(** * 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.

(* Goal: Ple r (@fold_left positive positive Pos.max (fn_params f) xH) *)
(* Goal: Ple r (max_reg_function f) *)
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.

(* Goal: Ple r (@fold_left positive positive Pos.max (fn_params f) xH) *)
(* Goal: Ple r (max_reg_function f) *)
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).
Proof.
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
  intros. subst rs'. eapply exec_Iop; eauto.
Qed.

Lemma exec_Iload':
  forall s f sp pc rs m chunk addr args dst pc' rs' 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 ->
  rs' = (rs#dst <- v) ->
  step (State s f sp pc rs m)
    E0 (State s f sp pc' rs' m).
Proof.
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
  intros. subst rs'. eapply exec_Iload; eauto.
Qed.

End RELSEM.

(** Execution of whole programs are described as sequences of transitions
  from an initial state to a final state.  An initial state is a [Callstate]
  corresponding to the invocation of the ``main'' function of the program
  without arguments and with an empty call stack. *)

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 nil f nil m0).

(** A final state is a [Returnstate] with an empty call stack. *)

Inductive final_state: state -> int -> Prop :=
  | final_state_intro: forall r m,
      final_state (Returnstate nil (Vint r) m) r.

(** The small-step semantics for a program. *)

Definition semantics (p: program) :=
  Semantics step (initial_state p) final_state (Genv.globalenv p).

(** This semantics is receptive to changes in events. *)

Lemma semantics_receptive:
  forall (p: program), receptive (semantics p).
Proof.
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
  intros. constructor; simpl; intros.
(* receptiveness *)
(* Goal: @ex state (fun s2 : state => step (@Genv.globalenv fundef unit p) s t2 s2) *)
(* Goal: single_events (semantics p) *)
  assert (t1 = E0 -> exists s2, step (Genv.globalenv p) s t2 s2).
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
    intros. subst. inv H0. exists s1; auto.
(* Goal: @ex state (fun s2 : state => step (@Genv.globalenv fundef unit p) s t2 s2) *)
(* Goal: single_events (semantics p) *)
  inversion H; subst; auto.
(* Goal: @ex state (fun s2 : state => step (@Genv.globalenv fundef unit p) (Callstate s0 (@External function ef) args m) t2 s2) *)
(* Goal: single_events (semantics p) *)
  exploit external_call_receptive; eauto. intros [vres2 [m2 EC2]].
(* Goal: @ex state (fun s2 : state => step (@Genv.globalenv fundef unit p) (State s0 f sp pc rs m) t2 s2) *)
(* Goal: @ex state (fun s2 : state => step (@Genv.globalenv fundef unit p) (Callstate s0 (@External function ef) args m) t2 s2) *)
(* Goal: single_events (semantics p) *)
  exists (State s0 f sp pc' (regmap_setres res vres2 rs) m2). eapply exec_Ibuiltin; eauto.
(* Goal: @ex state (fun s2 : state => step (@Genv.globalenv fundef unit p) (Callstate s0 (@External function ef) args m) t2 s2) *)
(* Goal: single_events (semantics p) *)
  exploit external_call_receptive; eauto. intros [vres2 [m2 EC2]].
(* Goal: @ex state (fun s2 : state => step (@Genv.globalenv fundef unit p) (Callstate s0 (@External function ef) args m) t2 s2) *)
(* Goal: single_events (semantics p) *)
  exists (Returnstate s0 vres2 m2). econstructor; eauto.
(* trace length *)
(* Goal: single_events (semantics p) *)
  red; intros; inv H; simpl; try omega.
(* Goal: le (@Datatypes.length event t) (S O) *)
  eapply external_call_trace_length; eauto.
(* Goal: le (@Datatypes.length event t) (S O) *)
  eapply external_call_trace_length; eauto.
Qed.

(** * Operations on RTL abstract syntax *)

(** Transformation of a RTL function instruction by instruction.
  This applies a given transformation function to all instructions
  of a function and constructs a transformed function from that. *)

Section TRANSF.

Variable transf: node -> instruction -> instruction.

Definition transf_function (f: function) : function :=
  mkfunction
    f.(fn_sig)
    f.(fn_params)
    f.(fn_stacksize)
    (PTree.map transf f.(fn_code))
    f.(fn_entrypoint).

End TRANSF.

(** Computation of the possible successors of an instruction.
  This is used in particular for dataflow analyses. *)

Definition successors_instr (i: instruction) : list node :=
  match i with
  | Inop s => s :: nil
  | Iop op args res s => s :: nil
  | Iload chunk addr args dst s => s :: nil
  | Istore chunk addr args src s => s :: nil
  | Icall sig ros args res s => s :: nil
  | Itailcall sig ros args => nil
  | Ibuiltin ef args res s => s :: nil
  | Icond cond args ifso ifnot => ifso :: ifnot :: nil
  | Ijumptable arg tbl => tbl
  | Ireturn optarg => nil
  end.

Definition successors_map (f: function) : PTree.t (list node) :=
  PTree.map1 successors_instr f.(fn_code).

(** The registers used by an instruction *)

Definition instr_uses (i: instruction) : list reg :=
  match i with
  | Inop s => nil
  | Iop op args res s => args
  | Iload chunk addr args dst s => args
  | Istore chunk addr args src s => src :: args
  | Icall sig (inl r) args res s => r :: args
  | Icall sig (inr id) args res s => args
  | Itailcall sig (inl r) args => r :: args
  | Itailcall sig (inr id) args => args
  | Ibuiltin ef args res s => params_of_builtin_args args
  | Icond cond args ifso ifnot => args
  | Ijumptable arg tbl => arg :: nil
  | Ireturn None => nil
  | Ireturn (Some arg) => arg :: nil
  end.

(** The register defined by an instruction, if any *)

Definition instr_defs (i: instruction) : option reg :=
  match i with
  | Inop s => None
  | Iop op args res s => Some res
  | Iload chunk addr args dst s => Some dst
  | Istore chunk addr args src s => None
  | Icall sig ros args res s => Some res
  | Itailcall sig ros args => None
  | Ibuiltin ef args res s =>
      match res with BR r => Some r | _ => None end
  | Icond cond args ifso ifnot => None
  | Ijumptable arg tbl => None
  | Ireturn optarg => None
  end.

(** Maximum PC (node number) in the CFG of a function.  All nodes of
  the CFG of [f] are between 1 and [max_pc_function f] (inclusive). *)

Definition max_pc_function (f: function) :=
  PTree.fold (fun m pc i => Pos.max m pc) f.(fn_code) 1%positive.

Lemma max_pc_function_sound:
  forall f pc i, f.(fn_code)!pc = Some i -> Ple pc (max_pc_function f).
Proof.
(* Goal: forall (f : function) (pc : positive) (i : instruction) (_ : @eq (option instruction) (@PTree.get instruction pc (fn_code f)) (@Some instruction i)), Ple pc (max_pc_function f) *)
  intros until i. unfold max_pc_function.
(* Goal: forall _ : @eq (option instruction) (@PTree.get instruction pc (fn_code f)) (@Some instruction i), Ple pc (@PTree.fold instruction positive (fun (m pc : positive) (_ : instruction) => Pos.max m pc) (fn_code f) xH) *)
  apply PTree_Properties.fold_rec with (P := fun c m => c!pc = Some i -> Ple pc m).
  (* extensionality *)
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
  intros. apply H0. rewrite H; auto.
  (* base case *)
(* Goal: forall _ : @eq (option instruction) (@PTree.get instruction pc (PTree.empty instruction)) (@Some instruction i), Ple pc xH *)
(* Goal: forall (m : PTree.t instruction) (a : positive) (k : PTree.elt) (v : instruction) (_ : @eq (option instruction) (@PTree.get instruction k m) (@None instruction)) (_ : @eq (option instruction) (@PTree.get instruction k (fn_code f)) (@Some instruction v)) (_ : forall _ : @eq (option instruction) (@PTree.get instruction pc m) (@Some instruction i), Ple pc a) (_ : @eq (option instruction) (@PTree.get instruction pc (@PTree.set instruction k v m)) (@Some instruction i)), Ple pc (Pos.max a k) *)
  rewrite PTree.gempty. congruence.
  (* inductive case *)
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
  intros. rewrite PTree.gsspec in H2. destruct (peq pc k).
(* Goal: Ple pc (Pos.max a k) *)
(* Goal: Ple pc (Pos.max a k) *)
  inv H2. xomega.
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
  apply Ple_trans with a. auto. xomega.
Qed.

(** Maximum pseudo-register mentioned in a function.  All results or arguments
  of an instruction of [f], as well as all parameters of [f], are between
  1 and [max_reg_function] (inclusive). *)

Definition max_reg_instr (m: positive) (pc: node) (i: instruction) :=
  match i with
  | Inop s => m
  | Iop op args res s => fold_left Pos.max args (Pos.max res m)
  | Iload chunk addr args dst s => fold_left Pos.max args (Pos.max dst m)
  | Istore chunk addr args src s => fold_left Pos.max args (Pos.max src m)
  | Icall sig (inl r) args res s => fold_left Pos.max args (Pos.max r (Pos.max res m))
  | Icall sig (inr id) args res s => fold_left Pos.max args (Pos.max res m)
  | Itailcall sig (inl r) args => fold_left Pos.max args (Pos.max r m)
  | Itailcall sig (inr id) args => fold_left Pos.max args m
  | Ibuiltin ef args res s =>
      fold_left Pos.max (params_of_builtin_args args)
        (fold_left Pos.max (params_of_builtin_res res) m)
  | Icond cond args ifso ifnot => fold_left Pos.max args m
  | Ijumptable arg tbl => Pos.max arg m
  | Ireturn None => m
  | Ireturn (Some arg) => Pos.max arg m
  end.

Definition max_reg_function (f: function) :=
  Pos.max
    (PTree.fold max_reg_instr f.(fn_code) 1%positive)
    (fold_left Pos.max f.(fn_params) 1%positive).

Remark max_reg_instr_ge:
  forall m pc i, Ple m (max_reg_instr m pc i).
Proof.
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
  intros.
(* Goal: Ple m (max_reg_instr m pc i) *)
  assert (X: forall l n, Ple m n -> Ple m (fold_left Pos.max l n)).
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
  { induction l; simpl; intros.
(* Goal: or (@In reg r l) (Ple r (Pos.max n a)) *)
(* Goal: or (@In reg r l) (Ple r (Pos.max n a)) *)
    auto.
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
    apply IHl. xomega. }
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
  destruct i; simpl; try (destruct s0); repeat (apply X); try xomega.
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
  destruct o; xomega.
Qed.

Remark max_reg_instr_def:
  forall m pc i r, instr_defs i = Some r -> Ple r (max_reg_instr m pc i).
Proof.
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
  intros.
(* Goal: Ple r (max_reg_instr m pc i) *)
  assert (X: forall l n, Ple r n -> Ple r (fold_left Pos.max l n)).
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
  { induction l; simpl; intros. xomega. apply IHl. xomega. }
(* Goal: Ple r (max_reg_instr m pc i) *)
  destruct i; simpl in *; inv H.
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
- apply X. xomega.
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
- apply X. xomega.
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
- destruct s0; apply X; xomega.
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
- destruct b; inv H1. apply X. simpl. xomega.
Qed.

Remark max_reg_instr_uses:
  forall m pc i r, In r (instr_uses i) -> Ple r (max_reg_instr m pc i).
Proof.
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
  intros.
  assert (X: forall l n, In r l \/ Ple r n -> Ple r (fold_left Pos.max l n)).
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
  { induction l; simpl; intros.
(* Goal: or (@In reg r l) (Ple r (Pos.max n a)) *)
(* Goal: or (@In reg r l) (Ple r (Pos.max n a)) *)
    tauto.
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
    apply IHl. destruct H0 as [[A|A]|A]. right; subst; xomega. auto. right; xomega. }
(* Goal: Ple r (max_reg_instr m pc i) *)
  destruct i; simpl in *; try (destruct s0); try (apply X; auto).
- contradiction.
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
- destruct H. right; subst; xomega. auto.
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
- destruct H. right; subst; xomega. auto.
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
- destruct H. right; subst; xomega. auto.
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
- intuition. subst; xomega.
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
- destruct o; simpl in H; intuition. subst; xomega.
Qed.

Lemma max_reg_function_def:
  forall f pc i r,
  f.(fn_code)!pc = Some i -> instr_defs i = Some r -> Ple r (max_reg_function f).
Proof.
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
  intros.
(* Goal: Ple r (max_reg_function f) *)
  assert (Ple r (PTree.fold max_reg_instr f.(fn_code) 1%positive)).
(* Goal: Ple r (@fold_left positive positive Pos.max (fn_params f) xH) *)
(* Goal: Ple r (max_reg_function f) *)
  {  revert H.
     apply PTree_Properties.fold_rec with
       (P := fun c m => c!pc = Some i -> Ple r m).
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
   - intros. rewrite H in H1; auto.
(* Goal: forall _ : @eq (option instruction) (@None instruction) (@Some instruction i), Ple pc xH *)
(* Goal: forall (m : PTree.t instruction) (a : positive) (k : PTree.elt) (v : instruction) (_ : @eq (option instruction) (@PTree.get instruction k m) (@None instruction)) (_ : @eq (option instruction) (@PTree.get instruction k (fn_code f)) (@Some instruction v)) (_ : forall _ : @eq (option instruction) (@PTree.get instruction pc m) (@Some instruction i), Ple pc a) (_ : @eq (option instruction) (@PTree.get instruction pc (@PTree.set instruction k v m)) (@Some instruction i)), Ple pc (Pos.max a k) *)
   - rewrite PTree.gempty; congruence.
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
   - intros. rewrite PTree.gsspec in H3. destruct (peq pc k).
(* Goal: or (@In reg r l) (Ple r (Pos.max n a)) *)
(* Goal: or (@In reg r l) (Ple r (Pos.max n a)) *)
     + inv H3. eapply max_reg_instr_def; eauto.
(* Goal: Ple r (max_reg_instr a k v) *)
     + apply Ple_trans with a. auto. apply max_reg_instr_ge.
  }
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
  unfold max_reg_function. xomega.
Qed.

Lemma max_reg_function_use:
  forall f pc i r,
  f.(fn_code)!pc = Some i -> In r (instr_uses i) -> Ple r (max_reg_function f).
Proof.
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
  intros.
(* Goal: Ple r (max_reg_function f) *)
  assert (Ple r (PTree.fold max_reg_instr f.(fn_code) 1%positive)).
(* Goal: Ple r (@fold_left positive positive Pos.max (fn_params f) xH) *)
(* Goal: Ple r (max_reg_function f) *)
  {  revert H.
     apply PTree_Properties.fold_rec with
       (P := fun c m => c!pc = Some i -> Ple r m).
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
   - intros. rewrite H in H1; auto.
(* Goal: forall _ : @eq (option instruction) (@None instruction) (@Some instruction i), Ple pc xH *)
(* Goal: forall (m : PTree.t instruction) (a : positive) (k : PTree.elt) (v : instruction) (_ : @eq (option instruction) (@PTree.get instruction k m) (@None instruction)) (_ : @eq (option instruction) (@PTree.get instruction k (fn_code f)) (@Some instruction v)) (_ : forall _ : @eq (option instruction) (@PTree.get instruction pc m) (@Some instruction i), Ple pc a) (_ : @eq (option instruction) (@PTree.get instruction pc (@PTree.set instruction k v m)) (@Some instruction i)), Ple pc (Pos.max a k) *)
   - rewrite PTree.gempty; congruence.
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
   - intros. rewrite PTree.gsspec in H3. destruct (peq pc k).
(* Goal: or (@In reg r l) (Ple r (Pos.max n a)) *)
(* Goal: or (@In reg r l) (Ple r (Pos.max n a)) *)
     + inv H3. eapply max_reg_instr_uses; eauto.
(* Goal: Ple r (max_reg_instr a k v) *)
     + apply Ple_trans with a. auto. apply max_reg_instr_ge.
  }
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
  unfold max_reg_function. xomega.
Qed.

Lemma max_reg_function_params:
  forall f r, In r f.(fn_params) -> Ple r (max_reg_function f).
Proof.
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
  intros.
  assert (X: forall l n, In r l \/ Ple r n -> Ple r (fold_left Pos.max l n)).
(* Goal: forall (f : function) (r : reg) (_ : @In reg r (fn_params f)), Ple r (max_reg_function f) *)
  { induction l; simpl; intros.
(* Goal: or (@In reg r l) (Ple r (Pos.max n a)) *)
(* Goal: or (@In reg r l) (Ple r (Pos.max n a)) *)
    tauto.
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
    apply IHl. destruct H0 as [[A|A]|A]. right; subst; xomega. auto. right; xomega. }
(* Goal: Ple r (max_reg_function f) *)
  assert (Y: Ple r (fold_left Pos.max f.(fn_params) 1%positive)).
(* Goal: or (@In reg r l) (Ple r (Pos.max n a)) *)
(* Goal: or (@In reg r l) (Ple r (Pos.max n a)) *)
  { apply X; auto. }
(* Goal: Ple r (Pos.max (@PTree.fold instruction positive max_reg_instr (fn_code f) xH) (@fold_left positive positive Pos.max (fn_params f) xH)) *)
  unfold max_reg_function. xomega.
Qed.
