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(*** Interface for bounded arithmetic *)
Require Import Coq.ZArith.ZArith.
Require Import Crypto.Util.ZUtil.
Require Import Crypto.Util.Notations.
Local Open Scope Z_scope.
Local Open Scope type_scope.
Create HintDb push_decode discriminated.
Create HintDb pull_decode discriminated.
Hint Extern 1 => progress autorewrite with push_decode in * : push_decode.
Hint Extern 1 => progress autorewrite with pull_decode in * : pull_decode.
(* TODO(from jgross): Try dropping the record wrappers. See
https://github.com/mit-plv/fiat-crypto/pull/52#discussion_r74627992
and
https://github.com/mit-plv/fiat-crypto/pull/52#discussion_r74658417
and
https://github.com/mit-plv/fiat-crypto/pull/52#issuecomment-239536847.
The wrappers are here to make [autorewrite] databases feasable and
fast, based on design patterns learned from past experience. There
might be better ways. *)
Class decoder (n : Z) W :=
{ decode : W -> Z }.
Coercion decode : decoder >-> Funclass.
Global Arguments decode {n W _} _.
Class is_decode {n W} (decode : decoder n W) :=
decode_range : forall x, 0 <= decode x < 2^n.
Section InstructionGallery.
Context (n : Z) (* bit-width of width of [W] *)
{W : Type} (* bounded type, [W] for word *)
(Wdecoder : decoder n W).
Local Notation imm := Z (only parsing). (* immediate (compile-time) argument *)
Record load_immediate := { ldi :> imm -> W }.
Class is_load_immediate {ldi : load_immediate} :=
decode_load_immediate : forall x, 0 <= x < 2^n -> decode (ldi x) = x.
Record shift_right_doubleword_immediate := { shrd :> W -> W -> imm -> W }.
Class is_shift_right_doubleword_immediate (shrd : shift_right_doubleword_immediate) :=
decode_shift_right_doubleword :
forall high low count,
0 <= count < n
-> decode (shrd high low count) = (((decode high << n) + decode low) >> count) mod 2^n.
Record shift_left_immediate := { shl :> W -> imm -> W }.
Class is_shift_left_immediate (shl : shift_left_immediate) :=
decode_shift_left_immediate :
forall r count, 0 <= count < n -> decode (shl r count) = (decode r << count) mod 2^n.
Record spread_left_immediate := { sprl :> W -> imm -> W * W (* [(high, low)] *) }.
Class is_spread_left_immediate (sprl : spread_left_immediate) :=
{
decode_fst_spread_left_immediate : forall r count,
0 <= count < n
-> decode (fst (sprl r count)) = (decode r << count) >> n;
decode_snd_spread_left_immediate : forall r count,
0 <= count < n
-> decode (snd (sprl r count)) = (decode r << count) mod 2^n
}.
Record mask_keep_low := { mkl :> W -> imm -> W }.
Class is_mask_keep_low (mkl : mask_keep_low) :=
decode_mask_keep_low : forall r count,
0 <= count < n -> decode (mkl r count) = decode r mod 2^count.
Local Notation bit b := (if b then 1 else 0).
Record add_with_carry := { adc :> W -> W -> bool -> bool * W }.
Class is_add_with_carry (adc : add_with_carry) :=
{
bit_fst_add_with_carry : forall x y c, bit (fst (adc x y c)) = (decode x + decode y + bit c) >> n;
decode_snd_add_with_carry : forall x y c, decode (snd (adc x y c)) = (decode x + decode y + bit c) mod (2^n)
}.
Record sub_with_carry := { subc :> W -> W -> bool -> bool * W }.
Class is_sub_with_carry (subc:W->W->bool->bool*W) :=
{
fst_sub_with_carry : forall x y c, fst (subc x y c) = ((decode x - decode y - bit c) <? 0);
decode_snd_sub_with_carry : forall x y c, decode (snd (subc x y c)) = (decode x - decode y - bit c) mod 2^n
}.
Record multiply := { mul :> W -> W -> W }.
Class is_mul (mul : multiply) :=
decode_mul : forall x y, decode (mul x y) = (decode x * decode y) mod 2^n.
Record multiply_low_low := { mulhwll :> W -> W -> W }.
Record multiply_high_low := { mulhwhl :> W -> W -> W }.
Record multiply_high_high := { mulhwhh :> W -> W -> W }.
Class is_mul_low_low (w:Z) (mulhwll : multiply_low_low) :=
decode_mul_low_low :
forall x y, decode (mulhwll x y) = ((decode x mod 2^w) * (decode y mod 2^w)) mod 2^n.
Class is_mul_high_low (w:Z) (mulhwhl : multiply_high_low) :=
decode_mul_high_low :
forall x y, decode (mulhwhl x y) = ((decode x >> w) * (decode y mod 2^w)) mod 2^n.
Class is_mul_high_high (w:Z) (mulhwhh : multiply_high_high) :=
decode_mul_high_high :
forall x y, decode (mulhwhh x y) = ((decode x >> w) * (decode y >> w)) mod 2^n.
Record select_conditional := { selc :> bool -> W -> W -> W }.
Class is_select_conditional (selc : select_conditional) :=
decode_select_conditional : forall b x y,
decode (selc b x y) = if b then decode x else decode y.
Record add_modulo := { addm :> W -> W -> W (* modulus *) -> W }.
Class is_add_modulo (addm : add_modulo) :=
decode_add_modulo : forall x y modulus,
decode (addm x y modulus) = (decode x + decode y) mod (decode modulus).
End InstructionGallery.
Global Arguments load_immediate : clear implicits.
Global Arguments shift_right_doubleword_immediate : clear implicits.
Global Arguments shift_left_immediate : clear implicits.
Global Arguments spread_left_immediate : clear implicits.
Global Arguments mask_keep_low : clear implicits.
Global Arguments add_with_carry : clear implicits.
Global Arguments sub_with_carry : clear implicits.
Global Arguments multiply : clear implicits.
Global Arguments multiply_low_low : clear implicits.
Global Arguments multiply_high_low : clear implicits.
Global Arguments multiply_high_high : clear implicits.
Global Arguments select_conditional : clear implicits.
Global Arguments add_modulo : clear implicits.
Global Arguments ldi {_ _} _.
Global Arguments shrd {_ _} _ _ _.
Global Arguments shl {_ _} _ _.
Global Arguments sprl {_ _} _ _.
Global Arguments mkl {_ _} _ _.
Global Arguments adc {_ _} _ _ _.
Global Arguments subc {_ _} _ _ _.
Global Arguments mul {_ _} _ _.
Global Arguments mulhwll {_ _} _ _.
Global Arguments mulhwhl {_ _} _ _.
Global Arguments mulhwhh {_ _} _ _.
Global Arguments selc {_ _} _ _ _.
Global Arguments addm {_ _} _ _ _.
Existing Class load_immediate.
Existing Class shift_right_doubleword_immediate.
Existing Class shift_left_immediate.
Existing Class spread_left_immediate.
Existing Class mask_keep_low.
Existing Class add_with_carry.
Existing Class sub_with_carry.
Existing Class multiply.
Existing Class multiply_low_low.
Existing Class multiply_high_low.
Existing Class multiply_high_high.
Existing Class select_conditional.
Existing Class add_modulo.
Global Arguments is_decode {_ _} _.
Global Arguments is_load_immediate {_ _ _} _.
Global Arguments is_shift_right_doubleword_immediate {_ _ _} _.
Global Arguments is_shift_left_immediate {_ _ _} _.
Global Arguments is_spread_left_immediate {_ _ _} _.
Global Arguments is_mask_keep_low {_ _ _} _.
Global Arguments is_add_with_carry {_ _ _} _.
Global Arguments is_sub_with_carry {_ _ _} _.
Global Arguments is_mul {_ _ _} _.
Global Arguments is_mul_low_low {_ _ _} _ _.
Global Arguments is_mul_high_low {_ _ _} _ _.
Global Arguments is_mul_high_high {_ _ _} _ _.
Global Arguments is_select_conditional {_ _ _} _.
Global Arguments is_add_modulo {_ _ _} _.
Ltac bounded_sovlver_tac :=
solve [ eassumption | typeclasses eauto | omega | auto 6 using decode_range with typeclass_instances omega ].
Hint Rewrite @decode_load_immediate @decode_shift_right_doubleword @decode_shift_left_immediate @decode_fst_spread_left_immediate @decode_snd_spread_left_immediate @decode_mask_keep_low @bit_fst_add_with_carry @decode_snd_add_with_carry @fst_sub_with_carry @decode_snd_sub_with_carry @decode_mul @decode_mul_low_low @decode_mul_high_low @decode_mul_high_high @decode_select_conditional @decode_add_modulo using bounded_sovlver_tac : push_decode.
Ltac push_decode :=
repeat first [ erewrite !decode_load_immediate by bounded_sovlver_tac
| erewrite !decode_shift_right_doubleword by bounded_sovlver_tac
| erewrite !decode_shift_left_immediate by bounded_sovlver_tac
| erewrite !decode_fst_spread_left_immediate by bounded_sovlver_tac
| erewrite !decode_snd_spread_left_immediate by bounded_sovlver_tac
| erewrite !decode_mask_keep_low by bounded_sovlver_tac
| erewrite !bit_fst_add_with_carry by bounded_sovlver_tac
| erewrite !decode_snd_add_with_carry by bounded_sovlver_tac
| erewrite !fst_sub_with_carry by bounded_sovlver_tac
| erewrite !decode_snd_sub_with_carry by bounded_sovlver_tac
| erewrite !decode_mul by bounded_sovlver_tac
| erewrite !decode_mul_low_low by bounded_sovlver_tac
| erewrite !decode_mul_high_low by bounded_sovlver_tac
| erewrite !decode_mul_high_high by bounded_sovlver_tac
| erewrite !decode_select_conditional by bounded_sovlver_tac
| erewrite !decode_add_modulo by bounded_sovlver_tac ].
Ltac pull_decode :=
repeat first [ erewrite <- !decode_load_immediate by bounded_sovlver_tac
| erewrite <- !decode_shift_right_doubleword by bounded_sovlver_tac
| erewrite <- !decode_shift_left_immediate by bounded_sovlver_tac
| erewrite <- !decode_fst_spread_left_immediate by bounded_sovlver_tac
| erewrite <- !decode_snd_spread_left_immediate by bounded_sovlver_tac
| erewrite <- !decode_mask_keep_low by bounded_sovlver_tac
| erewrite <- !bit_fst_add_with_carry by bounded_sovlver_tac
| erewrite <- !decode_snd_add_with_carry by bounded_sovlver_tac
| erewrite <- !fst_sub_with_carry by bounded_sovlver_tac
| erewrite <- !decode_snd_sub_with_carry by bounded_sovlver_tac
| erewrite <- !decode_mul by bounded_sovlver_tac
| erewrite <- !decode_mul_low_low by bounded_sovlver_tac
| erewrite <- !decode_mul_high_low by bounded_sovlver_tac
| erewrite <- !decode_mul_high_high by bounded_sovlver_tac
| erewrite <- !decode_select_conditional by bounded_sovlver_tac
| erewrite <- !decode_add_modulo by bounded_sovlver_tac ].
Module fancy_machine.
Local Notation imm := Z (only parsing).
Class instructions (n : Z) :=
{
W : Type (* [n]-bit word *);
decode :> decoder n W;
ldi :> load_immediate W;
shrd :> shift_right_doubleword_immediate W;
shl :> shift_left_immediate W;
mkl :> mask_keep_low W;
adc :> add_with_carry W;
subc :> sub_with_carry W;
mulhwll :> multiply_low_low W;
mulhwhl :> multiply_high_low W;
mulhwhh :> multiply_high_high W;
selc :> select_conditional W;
addm :> add_modulo W
}.
Class arithmetic {n_over_two} (ops:instructions (2 * n_over_two)) :=
{
decode_range :> is_decode decode;
load_immediate :> is_load_immediate ldi;
shift_right_doubleword_immediate :> is_shift_right_doubleword_immediate shrd;
shift_left_immediate :> is_shift_left_immediate shl;
mask_keep_low :> is_mask_keep_low mkl;
add_with_carry :> is_add_with_carry adc;
sub_with_carry :> is_sub_with_carry subc;
multiply_low_low :> is_mul_low_low n_over_two mulhwll;
multiply_high_low :> is_mul_high_low n_over_two mulhwhl;
multiply_high_high :> is_mul_high_high n_over_two mulhwhh;
select_conditional :> is_select_conditional selc;
add_modulo :> is_add_modulo addm
}.
End fancy_machine.
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