(*|
..
   Vericert: Verified high-level synthesis.
   Copyright (C) 2020-2022 Yann Herklotz <yann@yannherklotz.com>

   This program is free software: you can redistribute it and/or modify
   it under the terms of the GNU General Public License as published by
   the Free Software Foundation, either version 3 of the License, or
   (at your option) any later version.

   This program is distributed in the hope that it will be useful,
   but WITHOUT ANY WARRANTY; without even the implied warranty of
   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
   GNU General Public License for more details.

   You should have received a copy of the GNU General Public License
   along with this program.  If not, see <https://www.gnu.org/licenses/>.

================
RTLBlockgenproof
================

.. coq:: none
|*)

Require compcert.backend.RTL.
Require Import compcert.common.AST.
Require Import compcert.common.Errors.
Require Import compcert.common.Globalenvs.
Require Import compcert.lib.Maps.
Require Import compcert.backend.Registers.
Require compcert.common.Smallstep.
Require Import compcert.common.Events.

Require Import vericert.common.Vericertlib.
Require Import vericert.hls.RTLBlockInstr.
Require Import vericert.hls.RTLBlock.
Require Import vericert.hls.RTLBlockgen.

#[local] Open Scope positive.

(*|
Defining a find block specification
===================================

Basically, it should be able to find the location of the block without using the
``find_block`` function, so that this is more useful for the proofs.  There are
various different types of options that could come up though:

1. The instruction is a standard instruction present inside of a basic block.
2. The instruction is a standard instruction which ends with a ``goto``.
3. The instruction is a control-flow instruction.

For case number 1, there should exist a value in the list of instructions, such
that the instructions match exactly, and the indices match as well.  In the
original code, this instruction must have been going from the current node to
the node - 1.

For case number 2, there should be an instruction at the right index again,
however, this time there will also be a ``goto`` instruction in the control-flow
part of the basic block.

For case number 3, there should be a ``nop`` instruction in the basic block, and
then the equivalent control-flow instruction ending the basic block.

In the end though, it seems like two cases are actually enough, as the last two
cases are similar enough that they can be merged into one.
|*)

Definition all_max {A} (c: PTree.t A) p: Prop :=
  Forall (fun x => x <= p) (map fst (PTree.elements c)).

Definition find_block_spec (c: code) (n1 n2: node): Prop :=
  forall x, all_max c x -> find_block x (map fst (PTree.elements c)) n1 = n2.

Definition offset (pc pc': positive): nat := Pos.to_nat pc' - Pos.to_nat pc.

Definition find_instr_spec (c: code) (n: node) (i: RTL.instruction)
           (n': node) (i': instr) :=
  find_block_spec c n n' /\
  exists il, c ! n' = Some il
        /\ nth_error il.(bb_body) (offset n n') = Some i'.

(*|
.. index::
   pair: semantics; transl_code_spec

Translation Specification
-------------------------

This specification should describe the translation that the unverified
transformation performs.  This should be proven from the validation of the
transformation.

This also specifies ``x'`` relative to x given the code.
|*)

Variant transl_code_spec (c: RTL.code) (tc: code) (x x': node) (i: RTL.instruction) (i': instr): Prop :=
| transl_code_standard_bb :
    c ! x = Some i ->
    Is_true (check_instr x i i') ->
    transl_code_spec c tc x x' i i'
| transl_code_standard_cf :
  forall il,
    c ! x = Some i ->
    tc ! x' = Some il ->
    Is_true (check_cf_instr_body i i') ->
    Is_true (check_cf_instr i il.(bb_exit)) ->
    transl_code_spec c tc x x' i i'
.

Section CORRECTNESS.

  Context (prog : RTL.program).
  Context (tprog : RTLBlock.program).

  Let ge : RTL.genv := Globalenvs.Genv.globalenv prog.
  Let tge : genv := Globalenvs.Genv.globalenv tprog.

(*|
Matches the basic block that should be present in the state.  This simulates the
small step execution of the basic block from the big step semantics that are
currently present.

Why does it not need to find the pc' value using ``find_block``?

It doesn't have to find the value because it's given as an input, and the
finding is actually done at that higher level already.
|*)

(*  Variant match_bblock (tc: code) (pc pc': node): list instr -> Prop :=
    | match_bblock_intro :
      forall bb cf,
        tc ! pc' = Some (mk_bblock bb cf) ->
        match_bblock tc pc pc' (list_drop (offset pc pc') bb).*)

  Definition imm_succ (pc pc': node) : Prop := pc' = Pos.succ pc.

  Inductive match_block (c: RTL.code) (pc: node): bb -> cf_instr -> Prop :=
  (*
   * Basic Block Instructions
   *)
  | match_block_nop b cf:
    c ! pc = Some (RTL.Inop (Pos.pred pc)) ->
    match_block c (Pos.pred pc) b cf ->
    match_block c pc (RBnop :: b) cf
  | match_block_op cf b op args dst:
    c ! pc = Some (RTL.Iop op args dst (Pos.pred pc)) ->
    match_block c (Pos.pred pc) b cf ->
    match_block c pc (RBop None op args dst :: b) cf
  | match_block_load cf b chunk addr args dst:
    c ! pc = Some (RTL.Iload chunk addr args dst (Pos.pred pc)) ->
    match_block c (Pos.pred pc) b cf ->
    match_block c pc (RBload None chunk addr args dst :: b) cf
  | match_block_store cf b chunk addr args src:
    c ! pc = Some (RTL.Istore chunk addr args src (Pos.pred pc)) ->
    match_block c (Pos.pred pc) b cf ->
    match_block c pc (RBstore None chunk addr args src :: b) cf
  (*
   * Control flow instructions using goto
   *)
  | match_block_goto pc':
    pc' <> Pos.pred pc ->
    c ! pc = Some (RTL.Inop pc') ->
    match_block c pc (RBnop :: nil) (RBgoto pc')
  | match_block_op_cf pc' op args dst:
    pc' <> Pos.pred pc ->
    c ! pc = Some (RTL.Iop op args dst pc') ->
    match_block c pc (RBop None op args dst :: nil) (RBgoto pc')
  | match_block_load_cf pc' chunk addr args dst:
    pc' <> Pos.pred pc ->
    c ! pc = Some (RTL.Iload chunk addr args dst pc') ->
    match_block c pc (RBload None chunk addr args dst :: nil) (RBgoto pc')
  | match_block_store_cf pc' chunk addr args src:
    pc' <> Pos.pred pc ->
    c ! pc = Some (RTL.Istore chunk addr args src pc') ->
    match_block c pc (RBstore None chunk addr args src :: nil) (RBgoto pc')
  (*
   * Standard control flow instructions
   *)
  | match_block_call sig ident args dst pc' :
    c ! pc = Some (RTL.Icall sig ident args dst pc') ->
    match_block c pc (RBnop :: nil) (RBcall sig ident args dst pc')
  | match_block_tailcall sig ident args :
    c ! pc = Some (RTL.Itailcall sig ident args) ->
    match_block c pc (RBnop :: nil) (RBtailcall sig ident args)
  | match_block_builtin ident args dst pc' :
    c ! pc = Some (RTL.Ibuiltin ident args dst pc') ->
    match_block c pc (RBnop :: nil) (RBbuiltin ident args dst pc')
  | match_block_cond cond args pct pcf :
    c ! pc = Some (RTL.Icond cond args pct pcf) ->
    match_block c pc (RBnop :: nil) (RBcond cond args pct pcf)
  | match_block_return r :
    c ! pc = Some (RTL.Ireturn r) ->
    match_block c pc (RBnop :: nil) (RBreturn r)
  .

(*|
Match the code
~~~~~~~~~~~~~~

The ``match_code`` predicate asserts that for any node in the original
control-flow graph, there is now a basic block in the new control- and data-flow
graph that contains the same instruction, but also that the whole basic block
matches some sequence of instructions starting at the node that corresponds to
the basic block.
|*)

  Definition match_code (c: RTL.code) (tc: code) (pc: node) :=
    forall n1 i,
      c ! n1 = Some i ->
      exists b,
        find_block_spec tc n1 pc /\ tc ! pc = Some b
        /\ match_block c pc b.(bb_body) b.(bb_exit).

  Definition match_code' (c: RTL.code) (tc: code) : Prop :=
    forall i pc pc',
      c ! pc = Some i ->
      In pc' (RTL.successors_instr i) ->
      ~ imm_succ pc pc' ->
      exists b, tc ! pc' = Some b /\ match_block c pc b.(bb_body) b.(bb_exit).


  Variant match_stackframe : RTL.stackframe -> stackframe -> Prop :=
    | match_stackframe_init :
      forall res f tf sp pc rs (TF: transl_function f = OK tf),
        match_stackframe (RTL.Stackframe res f sp pc rs)
                         (Stackframe res tf sp pc rs (PMap.init false)).

  Definition sem_extrap f pc sp in_s in_s' block :=
    forall out_s block2,
      step_instr_list tge sp in_s block out_s ->
      f.(fn_code) ! pc = Some block2 ->
      step_instr_list tge sp in_s' block2.(bb_body) out_s.

(*|
Matching states
~~~~~~~~~~~~~~~

Initially, the idea was to define the ``match_states`` predicate normally to
defines how to find the correct ``bb`` that should be executed, as well as the
value of ``pc``.  However, this does not quite work when proving the equivalence
of the translation from ``RTL`` to ``RTLBlock``, because one cannot match one
transition to a transition in RTLBlock.  The alternative to this is to include a
proof inside of the ``match_states`` that shows that the execution from the
``pc`` of the start of the basic block to the current point is the same as the
whole execution (in one big step) of the basic block.
|*)

  Variant match_states : option bb -> RTL.state -> RTLBlock.state -> Prop :=
    | match_state :
      forall stk stk' f tf sp pc rs m pc0 b rs0 m0
        (TF: transl_function f = OK tf)
        (CODE: match_code' f.(RTL.fn_code) tf.(fn_code))
        (BLOCK: forall i b',
            f.(RTL.fn_code) ! pc = Some i ->
            tf.(fn_code) ! pc0 = Some b' ->
            match_block f.(RTL.fn_code) pc b b'.(bb_exit))
        (STK: Forall2 match_stackframe stk stk')
        (STARSIMU: Smallstep.star RTL.step ge (RTL.State stk f sp pc0 rs0 m0)
                                  E0 (RTL.State stk f sp pc rs m))
        (BB: sem_extrap tf pc0 sp (mk_instr_state rs (PMap.init false) m)
                        (mk_instr_state rs0 (PMap.init false) m0) b),
        match_states (Some b) (RTL.State stk f sp pc rs m)
                     (State stk' tf sp pc0 rs0 (PMap.init false) m0)
    | match_callstate :
      forall cs cs' f tf args m
        (TF: transl_fundef f = OK tf)
        (STK: Forall2 match_stackframe cs cs'),
        match_states None (RTL.Callstate cs f args m) (Callstate cs' tf args m)
    | match_returnstate :
      forall cs cs' v m
        (STK: Forall2 match_stackframe cs cs'),
        match_states None (RTL.Returnstate cs v m) (Returnstate cs' v m)
  .

  Definition match_prog (p: RTL.program) (tp: RTLBlock.program) :=
    Linking.match_program (fun cu f tf => transl_fundef f = Errors.OK tf) eq p tp.

  Context (TRANSL : match_prog prog tprog).

  Lemma symbols_preserved:
    forall (s: AST.ident), Genv.find_symbol tge s = Genv.find_symbol ge s.
  Proof using TRANSL. intros. eapply (Genv.find_symbol_match TRANSL). Qed.

  Lemma senv_preserved:
    Senv.equiv (Genv.to_senv ge) (Genv.to_senv tge).
  Proof using TRANSL. intros; eapply (Genv.senv_transf_partial TRANSL). Qed.
  #[local] Hint Resolve senv_preserved : rtlgp.

  Lemma function_ptr_translated:
    forall b f,
      Genv.find_funct_ptr ge b = Some f ->
      exists tf, Genv.find_funct_ptr tge b = Some tf /\ transl_fundef f = OK tf.
  Proof
    (Genv.find_funct_ptr_transf_partial TRANSL).

  Lemma sig_transl_function:
    forall (f: RTL.fundef) (tf: RTLBlock.fundef),
      transl_fundef f = OK tf ->
      RTLBlock.funsig tf = RTL.funsig f.
  Proof using.
    unfold transl_fundef. unfold transf_partial_fundef.
    intros. destruct_match. unfold bind in *. destruct_match; try discriminate.
    inv H. unfold transl_function in Heqr.
    repeat (destruct_match; try discriminate). inv Heqr. auto.
    inv H; auto.
  Qed.

  Lemma transl_initial_states :
    forall s1, RTL.initial_state prog s1 ->
      exists b s2, RTLBlock.initial_state tprog s2 /\ match_states b s1 s2.
  Proof using TRANSL.
    induction 1.
    exploit function_ptr_translated; eauto. intros [tf [A B]].
    do 2 econstructor. simplify. econstructor.
    apply (Genv.init_mem_transf_partial TRANSL); eauto.
    replace (prog_main tprog) with (prog_main prog). rewrite symbols_preserved; eauto.
    symmetry; eapply Linking.match_program_main; eauto. eauto.
    erewrite sig_transl_function; eauto. constructor. auto. auto.
  Qed.

  Lemma transl_final_states :
    forall s1 s2 r b,
      match_states b s1 s2 ->
      RTL.final_state s1 r ->
      RTLBlock.final_state s2 r.
  Proof using.
    inversion 2; crush. subst. inv H. inv STK. econstructor.
  Qed.

  Compute (hd_error (list_drop 3 (1::2::3::4::5::nil))).
  Compute (nth_error (1::2::3::4::5::nil) 3).

  Lemma hd_nth_equiv:
    forall A n (l: list A), hd_error (list_drop n l) = nth_error l n.
  Proof. induction n; destruct l; crush. Qed.

  Lemma hd_error_Some_exists:
    forall A (l: list A) n, hd_error l = Some n -> l = n :: tl l.
  Proof. induction l; crush. Qed.

  Lemma transl_Inop_correct_nj:
    forall s f sp pc rs m stk' tf pc1 rs1 m1 b x,
      (RTL.fn_code f) ! pc = Some (RTL.Inop (Pos.pred pc)) ->
      match_states (Some (RBnop :: b)) (RTL.State s f sp pc rs m)
                   (State stk' tf sp pc1 rs1 (PMap.init false) m1) ->
      (RTL.fn_code f) ! pc = Some (RTL.Inop (Pos.pred pc)) ->
      match_block (RTL.fn_code f) (Pos.pred pc) b x ->
      exists b' s2',
        Smallstep.star step tge (State stk' tf sp pc1 rs1 (PMap.init false) m1) E0 s2' /\
          match_states b' (RTL.State s f sp (Pos.pred pc) rs m) s2'.
  Proof.
    intros s f sp pc rs m H stk' tf pc1 rs1 m1 b H0 x H1 H3. Admitted.

  Lemma transl_Inop_correct_j:
    forall s f sp pc rs m pc' stk' tf pc1 rs1 m1 x,
      (RTL.fn_code f) ! pc = Some (RTL.Inop pc') ->
      match_states (Some (RBnop :: nil)) (RTL.State s f sp pc rs m)
                   (State stk' tf sp pc1 rs1 (PMap.init false) m1) ->
      find_block_spec (fn_code tf) pc pc1 ->
      (fn_code tf) ! pc1 = Some x ->
      match_block (RTL.fn_code f) pc1 (bb_body x) (bb_exit x) ->
      RBgoto pc' = bb_exit x ->
      (RTL.fn_code f) ! pc = Some (RTL.Inop pc') ->
      pc' <> Pos.pred pc ->
      exists b' s2',
        Smallstep.star step tge (State stk' tf sp pc1 rs1 (PMap.init false) m1) E0 s2' /\
          match_states b' (RTL.State s f sp pc' rs m) s2'.
  Proof.
    intros * H H0 H2 H1 H4 H5 H8 H6.
    inv H0.
    do 3 econstructor. apply Smallstep.star_one. econstructor.
    eassumption. eapply BB; [econstructor; constructor | eassumption].
    setoid_rewrite <- H5. econstructor.

    econstructor; try eassumption. Admitted.
  (*   apply Smallstep.star_refl. admit. *)
  (* Admitted. *)

  Lemma transl_Inop_correct:
    forall s f sp pc rs m pc',
      (RTL.fn_code f) ! pc = Some (RTL.Inop pc') ->
      forall b s2, match_states b (RTL.State s f sp pc rs m) s2 ->
        exists b' s2', Smallstep.star step tge s2 Events.E0 s2'
               /\ match_states b' (RTL.State s f sp pc' rs m) s2'.
  Proof.
    intros s f sp pc rs m pc' H.
    inversion 1; subst; simplify.
    unfold match_code' in *.
    pose proof (CODE _ _ H) as X. simplify.
    pose proof (BLOCK _ _ H H1); simplify. inv H3; crush;
    eauto using transl_Inop_correct_nj, transl_Inop_correct_j.
  Qed.

  Lemma transl_Iop_correct:
    forall s f sp pc rs m op args res pc' v,
      (RTL.fn_code f) ! pc = Some (RTL.Iop op args res pc') ->
      Op.eval_operation ge sp op rs##args m = Some v ->
      forall s2, match_states (RTL.State s f sp pc rs m) s2 ->
        exists s2', Smallstep.plus step tge s2 Events.E0 s2'
               /\ match_states (RTL.State s f sp pc' (Registers.Regmap.set res v rs) m) s2'.
  Proof.
    intros s f sp pc rs m op args res pc' v H H0.
  Admitted.

  Lemma transl_Iload_correct:
    forall s f sp pc rs m chunk addr args dst pc' a v,
      (RTL.fn_code f) ! pc = Some (RTL.Iload chunk addr args dst pc') ->
      Op.eval_addressing ge sp addr rs##args = Some a ->
      Memory.Mem.loadv chunk m a = Some v ->
      forall s2, match_states (RTL.State s f sp pc rs m) s2 ->
        exists s2', Smallstep.plus step tge s2 Events.E0 s2'
               /\ match_states (RTL.State s f sp pc' (Registers.Regmap.set dst v rs) m) s2'.
  Proof.
    intros s f sp pc rs m chunk addr args dst pc' a v H H0 H1.
  Admitted.

  Lemma transl_correct:
    forall s1 t s1',
      RTL.step ge s1 t s1' ->
      forall s2, match_states s1 s2 ->
        exists s2', Smallstep.plus step tge s2 t s2' /\ match_states s1' s2'.
  Proof.
    induction 1.
    - eauto using transl_Inop_correct.
    - eauto using transl_Iop_correct.
    - eauto using transl_Iload_correct.
  Admitted.

  Theorem transf_program_correct:
    Smallstep.forward_simulation (RTL.semantics prog)
                                 (RTLBlock.semantics tprog).
  Proof using TRANSL.
    eapply Smallstep.forward_simulation_plus.
    apply senv_preserved.
    eauto using transl_initial_states.
    eapply transl_final_states.
    eauto using transl_correct.
  Qed.

End CORRECTNESS.