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(** * Common Subexpression Elimination for PHOAS Syntax *)
Require Import Coq.ZArith.ZArith.
Require Import Coq.Sorting.Mergesort.
Require Import Coq.Structures.Orders.
Require Import Crypto.Compilers.Syntax.
Require Import Crypto.Compilers.Z.Syntax.
Require Import Crypto.Compilers.Z.Syntax.Util.
Require Import Crypto.Compilers.CommonSubexpressionElimination.
Require Import Crypto.Compilers.CommonSubexpressionEliminationProperties.
Local Set Decidable Equality Schemes.
Local Set Boolean Equality Schemes.
Inductive symbolic_op :=
| SOpConst (z : Z)
| SAdd
| SSub
| SMul
| SShl
| SShr
| SLand
| SLor
| SOpp
| SIdWithAlt
| SZselect
| SMulSplit (bitwidth : Z)
| SAddWithCarry
| SAddWithGetCarry (bitwidth : Z)
| SSubWithBorrow
| SSubWithGetBorrow (bitwidth : Z)
.
Definition symbolic_op_leb (x y : symbolic_op) : bool
:= match x, y with
| SOpConst z1, SOpConst z2 => Z.leb z1 z2
| SAddWithGetCarry bw1, SAddWithGetCarry bw2 => Z.leb bw1 bw2
| SSubWithGetBorrow bw1, SSubWithGetBorrow bw2 => Z.leb bw1 bw2
| SOpConst _, _ => true
| _, SOpConst _ => false
| SAdd, _ => true
| _, SAdd => false
| SSub, _ => true
| _, SSub => false
| SMul, _ => true
| _, SMul => false
| SShl, _ => true
| _, SShl => false
| SShr, _ => true
| _, SShr => false
| SLand, _ => true
| _, SLand => false
| SLor, _ => true
| _, SLor => false
| SOpp, _ => true
| _, SOpp => false
| SIdWithAlt, _ => true
| _, SIdWithAlt => false
| SZselect, _ => true
| _, SZselect => false
| SMulSplit _, _ => true
| _, SMulSplit _ => false
| SAddWithCarry, _ => true
| _, SAddWithCarry => false
| SAddWithGetCarry _, _ => true
| _, SAddWithGetCarry _ => false
| SSubWithBorrow, _ => true
| _, SSubWithBorrow => false
(*| SSubWithGetBorrow _, _ => true
| _, SSubWithGetBorrow _ => false*)
end.
Local Notation symbolic_expr := (@symbolic_expr base_type symbolic_op).
Local Notation symbolic_expr_beq := (@symbolic_expr_beq base_type symbolic_op base_type_beq symbolic_op_beq).
Local Notation symbolic_expr_leb := (@symbolic_expr_leb base_type symbolic_op base_type_beq symbolic_op_beq symbolic_op_leb base_type_leb).
Definition symbolize_op s d (opc : op s d) : symbolic_op
:= match opc with
| OpConst T z => SOpConst z
| Add T1 T2 Tout => SAdd
| Sub T1 T2 Tout => SSub
| Mul T1 T2 Tout => SMul
| Shl T1 T2 Tout => SShl
| Shr T1 T2 Tout => SShr
| Land T1 T2 Tout => SLand
| Lor T1 T2 Tout => SLor
| Opp T Tout => SOpp
| IdWithAlt T1 T2 Tout => SIdWithAlt
| Zselect T1 T2 T3 Tout => SZselect
| MulSplit bitwidth T1 T2 Tout1 Tout2 => SMulSplit bitwidth
| AddWithCarry T1 T2 T3 Tout => SAddWithCarry
| AddWithGetCarry bitwidth T1 T2 T3 Tout1 Tout2 => SAddWithGetCarry bitwidth
| SubWithBorrow T1 T2 T3 Tout => SSubWithBorrow
| SubWithGetBorrow bitwidth T1 T2 T3 Tout1 Tout2 => SSubWithGetBorrow bitwidth
end.
Definition denote_symbolic_op s d (opc : symbolic_op) : option (op s d)
:= match opc, s, d with
| SOpConst z, Unit, Tbase T => Some (OpConst z)
| SAdd, Prod (Tbase _) (Tbase _), Tbase _ => Some (Add _ _ _)
| SSub, Prod (Tbase _) (Tbase _), Tbase _ => Some (Sub _ _ _)
| SMul, Prod (Tbase _) (Tbase _), Tbase _ => Some (Mul _ _ _)
| SShl, Prod (Tbase _) (Tbase _), Tbase _ => Some (Shl _ _ _)
| SShr, Prod (Tbase _) (Tbase _), Tbase _ => Some (Shr _ _ _)
| SLand, Prod (Tbase _) (Tbase _), Tbase _ => Some (Land _ _ _)
| SLor, Prod (Tbase _) (Tbase _), Tbase _ => Some (Lor _ _ _)
| SOpp, Tbase _, Tbase _ => Some (Opp _ _)
| SIdWithAlt, Prod (Tbase _) (Tbase _), Tbase _ => Some (IdWithAlt _ _ _)
| SZselect, Prod (Prod (Tbase _) (Tbase _)) (Tbase _), Tbase _ => Some (Zselect _ _ _ _)
| SMulSplit bitwidth, Prod (Tbase _) (Tbase _), Prod (Tbase _) (Tbase _)
=> Some (MulSplit bitwidth _ _ _ _)
| SAddWithCarry, Prod (Prod (Tbase _) (Tbase _)) (Tbase _), Tbase _ => Some (AddWithCarry _ _ _ _)
| SAddWithGetCarry bitwidth, Prod (Prod (Tbase _) (Tbase _)) (Tbase _), Prod (Tbase _) (Tbase _)
=> Some (AddWithGetCarry bitwidth _ _ _ _ _)
| SSubWithBorrow, Prod (Prod (Tbase _) (Tbase _)) (Tbase _), Tbase _ => Some (SubWithBorrow _ _ _ _)
| SSubWithGetBorrow bitwidth, Prod (Prod (Tbase _) (Tbase _)) (Tbase _), Prod (Tbase _) (Tbase _)
=> Some (SubWithGetBorrow bitwidth _ _ _ _ _)
| SAdd, _, _
| SSub, _, _
| SMul, _, _
| SShl, _, _
| SShr, _, _
| SLand, _, _
| SLor, _, _
| SOpp, _, _
| SOpConst _, _, _
| SIdWithAlt, _, _
| SZselect, _, _
| SMulSplit _, _, _
| SAddWithCarry, _, _
| SAddWithGetCarry _, _, _
| SSubWithBorrow, _, _
| SSubWithGetBorrow _, _, _
=> None
end.
Lemma symbolic_op_leb_total
: forall a1 a2, symbolic_op_leb a1 a2 = true \/ symbolic_op_leb a2 a1 = true.
Proof.
induction a1, a2; simpl; auto;
rewrite !Z.leb_le; omega.
Qed.
Module SymbolicExprOrder <: TotalLeBool.
Definition t := (flat_type base_type * symbolic_expr)%type.
Definition leb (x y : t) : bool := symbolic_expr_leb (snd x) (snd y).
Theorem leb_total : forall a1 a2, leb a1 a2 = true \/ leb a2 a1 = true.
Proof.
intros; apply symbolic_expr_leb_total;
auto using internal_base_type_dec_bl, internal_base_type_dec_lb, internal_symbolic_op_dec_bl, internal_symbolic_op_dec_lb, base_type_leb_total, symbolic_op_leb_total.
Qed.
End SymbolicExprOrder.
Module Import SymbolicExprSort := Sort SymbolicExprOrder.
Fixpoint symbolic_op_args_to_list (t : flat_type base_type)
(opc : symbolic_op) (args : symbolic_expr)
: list (flat_type base_type * symbolic_expr)
:= match args, t with
| SOp argT opc' args', _
=> if symbolic_op_beq opc opc'
then symbolic_op_args_to_list argT opc args'
else (t, args)::nil
| SPair x y, Prod A B
=> symbolic_op_args_to_list A opc x ++ symbolic_op_args_to_list B opc y
| SPair x y, Unit
=> symbolic_op_args_to_list Unit opc x ++ symbolic_op_args_to_list Unit opc y
| STT, _
| SVar _, _
| SPair _ _, _
| SFst _ _ _, _
| SSnd _ _ _, _
| SInvalid, _
=> (t, args)::nil
end%list.
Fixpoint symbolic_op_list_to_args (args : list (flat_type base_type * symbolic_expr)) : symbolic_expr
:= match args with
| nil => SInvalid
| (t, arg)::nil => arg
| (t1, arg1)::(t2, arg2)::nil
=> SPair arg1 arg2
| (t1, arg1)::(((t2, arg2)::args'') as args')
=> SPair arg1 (SOp t2 SAdd (symbolic_op_list_to_args args'))
end%list.
Definition normalize_symbolic_expr_mod_c (opc : symbolic_op) (args : symbolic_expr) : symbolic_expr
:= match opc with
| SAdd
| SMul
| SLand
| SLor
=> let ls := symbolic_op_args_to_list Unit opc args in
let ls := sort ls in
symbolic_op_list_to_args ls
| SOpConst _
| SSub
| SShl
| SShr
| SOpp
| SIdWithAlt
| SZselect
| SMulSplit _
| SAddWithCarry
| SAddWithGetCarry _
| SSubWithBorrow
| SSubWithGetBorrow _
=> args
end.
Definition csef inline_symbolic_expr_in_lookup {var t} (v : exprf _ _ t) xs
:= @csef base_type symbolic_op base_type_beq symbolic_op_beq
internal_base_type_dec_bl op symbolize_op
normalize_symbolic_expr_mod_c
var inline_symbolic_expr_in_lookup t v xs.
Definition cse inline_symbolic_expr_in_lookup {var} (prefix : list _) {t} (v : expr _ _ t) xs
:= @cse base_type symbolic_op base_type_beq symbolic_op_beq
internal_base_type_dec_bl op symbolize_op
normalize_symbolic_expr_mod_c
inline_symbolic_expr_in_lookup var prefix t v xs.
Definition CSE_gen inline_symbolic_expr_in_lookup {t} (e : Expr t) (prefix : forall var, list { t : flat_type base_type & exprf _ _ t })
: Expr t
:= @CSE base_type symbolic_op base_type_beq symbolic_op_beq
internal_base_type_dec_bl op symbolize_op
normalize_symbolic_expr_mod_c
inline_symbolic_expr_in_lookup t e prefix.
Definition CSE inline_symbolic_expr_in_lookup {t} (e : Expr t)
: Expr t
:= @CSE_gen inline_symbolic_expr_in_lookup t e (fun _ => nil).
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