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fma_matrix_vec_mult.v
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From Coq Require Import ZArith Reals Psatz.
From Flocq Require Import Binary.
From mathcomp Require Import all_ssreflect ssralg ssrnat (*all_algebra*) seq matrix.
From mathcomp Require Import Rstruct.
Import List ListNotations.
From vcfloat Require Import FPStdLib.
Require Import floatlib fma_floating_point_model inf_norm_properties.
Require Import common fma_dot_acc float_acc_lems dotprod_model.
Set Bullet Behavior "Strict Subproofs".
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Require Import lemmas fma_is_finite.
Open Scope ring_scope.
Delimit Scope ring_scope with Ri.
Delimit Scope R_scope with Re.
Import Order.TTheory GRing.Theory (*Num.Def Num.Theory*).
Section WITHNANS.
Context {NANS: FPCore.Nans}.
Notation "A +f B" := (addmx_float A B) (at level 80).
Notation "-f A" := (opp_mat A) (at level 50).
Notation "A *f B" := (mulmx_float A B) (at level 70).
Notation "A -f B" := (sub_mat A B) (at level 80).
Definition e_i {n:nat} {ty} (i : 'I_n.+1)
(A: 'M[ftype ty]_n.+1) (v: 'cV[ftype ty]_n.+1) :=
let l1 := vec_to_list_float n.+1 (\row_(j < n.+1) A i j)^T in
let l2 := vec_to_list_float n.+1 v in
let L := combine l1 l2 in
let prods := map (uncurry Rmult) (map Rabsp (map FR2 L)) in
let rs:= sum_fold prods in
(g ty (length l1) * Rabs rs + g1 ty (length l1) (length l1 - 1))%Re.
Lemma e_i_pos {n ty} (i : 'I_n.+1) (A : 'M[ftype ty]_n.+1) (v : 'cV[ftype ty]_n.+1) :
0 <= e_i i A v.
Proof.
rewrite /e_i. apply Rplus_le_le_0_compat.
+ apply Rmult_le_pos; [apply g_pos | apply Rabs_pos].
+ apply g1_pos.
Qed.
Definition extract_elements {T} (idx : seq.seq nat) (l : list T) (default : T) :=
map (fun i => nth i l default) idx.
Lemma extract_elements_length {T} (idx : seq.seq nat) (l : list T) (default : T):
length (extract_elements idx l default) = length idx.
Proof.
induction idx as [|h idx'].
+ simpl. auto.
+ simpl. rewrite IHidx'. auto.
Qed.
Lemma extract_elements_succ {T} (idx : seq.seq nat) (x : T) (l : list T) (default : T) :
extract_elements (map S idx) (x :: l) default = extract_elements idx l default.
Proof.
induction idx as [|i0 idx'].
+ simpl. auto.
+ simpl. f_equal. auto.
Qed.
Fixpoint extract_nonzero_idx_aux {ty} (l : list (ftype ty)) (k : nat) : list nat :=
match l with
| [] => []
| h :: t => if (BCMP Eq false h (Zconst ty 0))
then k :: extract_nonzero_idx_aux t (S k)
else extract_nonzero_idx_aux t (S k)
end.
Definition extract_nonzero_idx {ty} (l : list (ftype ty)) :=
extract_nonzero_idx_aux l 0.
Definition a := map (Zconst Tsingle) ([0; 3; 0; 5; 7; 0; 9])%Z.
Definition b := extract_nonzero_idx a.
Compute b.
Fixpoint nat_extract_nonzero_idx_aux (l : list nat) (k : nat) : list nat :=
match l with
| [] => []
| h :: t => if (negb (Nat.eqb h 0))
then k :: nat_extract_nonzero_idx_aux t (S k)
else nat_extract_nonzero_idx_aux t (S k)
end.
Definition nat_extract_nonzero_idx (l : list nat) :=
nat_extract_nonzero_idx_aux l 0.
Definition extract_nonzero_elmt {ty} (l : list (ftype ty)) :=
extract_elements (extract_nonzero_idx l) l (Zconst ty 0).
Lemma extract_nonzero_idx_nil {ty} : @extract_nonzero_idx ty [] = [].
Proof.
unfold extract_nonzero_idx. simpl. auto.
Qed.
Lemma extract_nonzero_elmt_nil {ty} : @extract_nonzero_elmt ty [] = [].
Proof.
unfold extract_nonzero_elmt. rewrite extract_nonzero_idx_nil. simpl. auto.
Qed.
Lemma extract_nonzero_idx_aux_cons {ty} : forall x l k,
@extract_nonzero_idx_aux ty (x :: l) k =
if (BCMP Eq false x (Zconst ty 0)) then k%nat :: (extract_nonzero_idx_aux l k.+1) else (extract_nonzero_idx_aux l k.+1).
Proof.
intros. simpl. destruct (BCMP Eq false x (Zconst ty 0)) eqn:E; auto.
Qed.
Lemma extract_nonzero_idx_aux_succ {ty} : forall l k,
@extract_nonzero_idx_aux ty l (k.+1) = map S (extract_nonzero_idx_aux l k).
Proof.
intros. revert k. induction l as [|h l'].
+ simpl. reflexivity.
+ intros. repeat rewrite extract_nonzero_idx_aux_cons.
destruct (BCMP Eq false h (Zconst ty 0)) eqn:E.
- simpl. rewrite IHl'. reflexivity.
- simpl. rewrite IHl'. reflexivity.
Qed.
Lemma extract_nonzero_idx_cons {ty} : forall x l,
@extract_nonzero_idx ty (x :: l) =
if (BCMP Eq false x (Zconst ty 0)) then 0%nat :: map S (extract_nonzero_idx l) else map S (extract_nonzero_idx l).
Proof.
intros. destruct (BCMP Eq false x (Zconst ty 0)) eqn:E.
+ unfold extract_nonzero_idx in *. rewrite <- extract_nonzero_idx_aux_succ.
rewrite extract_nonzero_idx_aux_cons. rewrite E. auto.
+ unfold extract_nonzero_idx in *. rewrite <- extract_nonzero_idx_aux_succ.
rewrite extract_nonzero_idx_aux_cons. rewrite E. auto.
Qed.
Lemma extract_nonzero_elmt_cons {ty} : forall x l,
@extract_nonzero_elmt ty (x :: l) =
if (BCMP Eq false x (Zconst ty 0)) then x :: extract_nonzero_elmt l else extract_nonzero_elmt l.
Proof.
intros. unfold extract_nonzero_elmt.
destruct (BCMP Eq false x (Zconst ty 0)) eqn:E.
+ rewrite extract_nonzero_idx_cons. rewrite E. simpl.
f_equal. rewrite extract_elements_succ. auto.
+ rewrite extract_nonzero_idx_cons. rewrite E. simpl.
rewrite extract_elements_succ. auto.
Qed.
Definition sparsity_fac_aux {ty} (l : list (ftype ty)) :=
length (extract_nonzero_idx l).
Definition sparsity_fac {n : nat} {ty} (v : 'cV[ftype ty]_n.+1) :=
sparsity_fac_aux (vec_to_list_float n.+1 v).
Definition is_r_sparse_aux {ty} (l : list (ftype ty)) (r : nat) :=
le (sparsity_fac_aux l) r.
Definition is_r_sparse {n : nat} {ty} (v : 'cV[ftype ty]_n.+1) (r : nat) :=
is_r_sparse_aux (vec_to_list_float n.+1 v) r.
Lemma extract_nonzero_length {ty} (l : list (ftype ty)):
length (@extract_nonzero_elmt ty l) = length (@extract_nonzero_idx ty l).
Proof.
unfold extract_nonzero_elmt.
rewrite extract_elements_length.
reflexivity.
Qed.
Definition sparsity_fac_mat_row {n : nat} {ty} (A : 'M[ftype ty]_n.+1) (i : 'I_n.+1) :=
sparsity_fac (\row_(j < n.+1) A i j)^T.
Definition is_r_sparse_mat {n : nat} {ty} (A : 'M[ftype ty]_n.+1) (r : nat) :=
forall (i : 'I_n.+1), is_r_sparse (\row_j A (i) j)^T r.
Definition sparsity_fac_mat {n : nat} {ty} (A: 'M[ftype ty]_n.+1) :=
foldr maxn 0%nat [seq (sparsity_fac_mat_row A i) | i <- enum 'I_n.+1].
Definition e_i_sparse {n : nat} {ty} (i : 'I_n.+1)
(A : 'M[ftype ty]_n.+1) (v : 'cV[ftype ty]_n.+1)
(r : nat) (HA : is_r_sparse_mat A r) :=
let l1 := vec_to_list_float n.+1 (\row_(j < n.+1) A i j)^T in
let l2 := vec_to_list_float n.+1 v in
let L := combine l1 l2 in
let prods := map (uncurry Rmult) (map Rabsp (map FR2 L)) in
let rs:= sum_fold prods in
(g ty r * Rabs rs + g1 ty r (r - 1))%Re.
Definition mat_vec_mult_err_bnd {n : nat} {ty}
(A : 'M[ftype ty]_n.+1) (v : 'cV[ftype ty]_n.+1) :=
\big[Order.Def.max/0%Re]_(i <- enum 'I_n.+1) (e_i i A v).
(* bigmaxr deprecated *)
(* Definition mat_vec_mult_err_bnd {n:nat} {ty}
(A: 'M[ftype ty]_n.+1) (v: 'cV[ftype ty]_n.+1):=
bigmaxr 0%Re [seq (e_i i A v) | i <- enum 'I_n.+1]. *)
Definition mat_vec_mult_err_bnd_sparse {n : nat} {ty}
(A : 'M[ftype ty]_n.+1) (v : 'cV[ftype ty]_n.+1)
(r : nat) (HA : is_r_sparse_mat A r) :=
\big[Order.Def.max/0%Re]_(i <- enum 'I_n.+1) (@e_i_sparse n ty i A v r HA).
(* Definition mat_vec_mult_err_bnd_sparse {n : nat} {ty}
(A : 'M[ftype ty]_n.+1) (v : 'cV[ftype ty]_n.+1)
(r : nat) (HA : is_r_sparse_mat A r) :=
bigmaxr 0%Re [seq (@e_i_sparse n ty i A v r HA) | i <- enum 'I_n.+1]. *)
Lemma dotprod_cons {t: type} (v1 v2: list (ftype t)) (x y : ftype t):
length v1 = length v2 ->
dotprod_r (x :: v1) (y :: v2) =
BFMA x y (dotprod_r v1 v2).
Proof.
intros. by unfold dotprod_r.
Qed.
Lemma fma_dot_prod_rel_holds {n:nat} {ty} m i
(A: 'M[ftype ty]_n.+1) (v : 'cV[ftype ty]_n.+1):
fma_dot_prod_rel
(combine
(@vec_to_list_float _ n m (\row_j A (inord i) j)^T)
(@vec_to_list_float _ n m v))
(let l1 :=
@vec_to_list_float _ n m (\row_j A (inord i) j)^T
in
let l2 := @vec_to_list_float _ n m v in
dotprod_r l1 l2).
Proof.
induction m.
+ simpl. unfold dotprod_r. simpl. apply fma_dot_prod_rel_nil.
+ simpl. rewrite !mxE.
assert ( (dotprod_r
(A (inord i) (inord m)
:: vec_to_list_float m (\row_j A (inord i) j)^T)
(v (inord m) ord0 :: vec_to_list_float m v)) =
BFMA (A (inord i) (inord m)) (v (inord m) ord0)
(dotprod_r (vec_to_list_float m (\row_j A (inord i) j)^T)
(vec_to_list_float m v))).
{ apply dotprod_cons. by rewrite !length_veclist. }
rewrite H. by apply fma_dot_prod_rel_cons.
Qed.
Lemma R_dot_prod_rel_holds {n:nat} {ty} m i (le_n_m : (m <= n.+1)%nat)
(A: 'M[ftype ty]_n.+1) (v : 'cV[ftype ty]_n.+1):
R_dot_prod_rel
(combine
(map FT2R
(@vec_to_list_float _ n m
(\row_j A (inord i) j)^T))
(map FT2R (@vec_to_list_float _ n m v)))
(\sum_(j < m)
FT2R_mat A (inord i) (@widen_ord m n.+1 le_n_m j) *
FT2R_mat v (@widen_ord m n.+1 le_n_m j) ord0).
Proof.
induction m.
+ simpl. rewrite big_ord0 //=. apply R_dot_prod_rel_nil.
+ simpl. rewrite !mxE. rewrite big_ord_recr //=.
rewrite -RplusE -RmultE.
assert ((widen_ord le_n_m ord_max) = (inord m)).
{ unfold widen_ord.
apply val_inj. simpl. by rewrite inordK.
} rewrite H. rewrite Rplus_comm. rewrite !mxE.
apply R_dot_prod_rel_cons.
assert ((m <= n.+1)%nat). { by apply ltnW. }
specialize (IHm H0).
assert (\sum_(j < m)
FT2R_mat A (inord i)
(widen_ord H0 j) *
FT2R_mat v (widen_ord H0 j) ord0 =
\sum_(i0 < m)
FT2R_mat A (inord i)
(widen_ord le_n_m
(widen_ord (leqnSn m) i0)) *
FT2R_mat v
(widen_ord le_n_m
(widen_ord (leqnSn m) i0)) ord0).
{ apply eq_big. by []. intros.
assert ((widen_ord le_n_m
(widen_ord (leqnSn m) i0))=
(widen_ord H0 i0)).
{ unfold widen_ord.
apply val_inj. by simpl.
} by rewrite H2.
} rewrite -H1. apply IHm.
Qed.
Lemma R_dot_prod_rel_abs_holds {n:nat} {ty} m i
(A: 'M[ftype ty]_n.+1) (v : 'cV[ftype ty]_n.+1):
R_dot_prod_rel
(combine
(map Rabs
(map FT2R
(@vec_to_list_float _ n m
(\row_j A (inord i) j)^T)))
(map Rabs
(map FT2R (@vec_to_list_float _ n m v))))
(sum_fold
(map (uncurry Rmult)
(map Rabsp
(map FR2
(combine
(@vec_to_list_float _ n m
(\row_j A (inord i) j)^T)
(@vec_to_list_float _ n m v)))))).
Proof.
induction m.
+ simpl. apply R_dot_prod_rel_nil.
+ simpl. rewrite !mxE. by apply R_dot_prod_rel_cons.
Qed.
(** Write a lemma for matrix-vector multiplication **)
Lemma matrix_vec_mult_bound {n:nat} {ty}:
forall (A: 'M[ftype ty]_n.+1) (v : 'cV[ftype ty]_n.+1),
(forall (xy : ftype ty * ftype ty) (i : 'I_n.+1),
In xy
(combine
(vec_to_list_float n.+1
(\row_j A (inord i) j)^T)
(vec_to_list_float n.+1 v)) ->
finite xy.1 /\finite xy.2) ->
(forall (i : 'I_n.+1),
finite (let l1 := vec_to_list_float n.+1 (\row_j A (inord i) j)^T in
let l2 := vec_to_list_float n.+1 (\col_j v j ord0) in
dotprod_r l1 l2)) ->
vec_inf_norm (FT2R_mat (A *f v) - (FT2R_mat A) *m (FT2R_mat v)) <=
mat_vec_mult_err_bnd A v.
Proof.
intros. rewrite /vec_inf_norm /mat_vec_mult_err_bnd.
apply bigmax_le_0head.
{ rewrite size_map size_enum_ord; auto. }
2:{ intros. move /mapP in H1. destruct H1 as [x0 H1 H2].
rewrite H2. apply /RleP. apply Rabs_pos. }
intros. rewrite (@nth_map _ ord0).
2:{ rewrite size_enum_ord. rewrite size_map size_enum_ord in H1. auto. }
rewrite enum_inord.
2:{ rewrite size_map size_enum_ord in H1. auto. }
assert (H99: length (vec_to_list_float n.+1 (\row_j A (inord i) j)^T) =
length (vec_to_list_float n.+1 v))
by rewrite !length_veclist //= .
pose proof (fma_dotprod_forward_error _ ty (vec_to_list_float n.+1 (\row_j A (inord i) j)^T)
(vec_to_list_float n.+1 v) H99). clear H99.
apply Rle_trans with (e_i (inord i) A v).
2:{ remember ([seq e_i i0 A v | i0 <- enum 'I_n.+1]) as eis.
assert (\big[Order.Def.max/0%Re]_(i0 <- enum 'I_n.+1) e_i i0 A v =
\big[Order.Def.max/0%Re]_(i0 <- enum 'I_(size eis)) seq.nth 0%Re eis i0).
{ rewrite Heqeis //= size_map. rewrite size_enum_ord. apply eq_big; auto.
intros. rewrite (@nth_map 'I_n.+1 ord0) //=. rewrite nth_ord_enum //=.
rewrite size_enum_ord. apply leq_ord. }
rewrite {}H3. rewrite -bigmax_seq_enum_eq. apply bigmax_ler_0head.
+ rewrite Heqeis. apply (@map_f _ _ (fun i => (e_i i A v))). apply mem_enum.
+ rewrite Heqeis. intros. move /mapP in H3. destruct H3 as [i0 H3 H4]. rewrite H4.
apply /RleP. apply e_i_pos. }
rewrite mxE mxE mxE mxE //=. apply H2; clear H2.
+ pose proof (@fma_dot_prod_rel_holds n ty n.+1 i A v).
remember (vec_to_list_float n.+1 (\row_j A (inord i) j)^T) as l1.
remember (vec_to_list_float n.+1 v) as l2. simpl in H2.
assert (((\row_j A (inord i) j)^T (inord n) ord0 :: vec_to_list_float n (\row_j A (inord i) j)^T) = l1).
{ rewrite Heql1. rewrite /vec_to_list_float //=. } rewrite {}H3.
assert (\col_j v j ord0 = v).
{ apply matrixP. unfold eqrel. intros. rewrite !mxE /=.
assert (y = ord0). { apply ord1. } by rewrite H3. } rewrite {}H3.
assert ((v (inord n) ord0 :: vec_to_list_float n v) = l2).
{ rewrite Heql2. rewrite /vec_to_list_float //=. } rewrite {}H3 //=.
+ pose proof (@R_dot_prod_rel_holds n ty n.+1 i (leqnn n.+1) A v).
assert ((\sum_(j < n.+1) FT2R_mat A (inord i) (widen_ord (leqnn n.+1) j) * FT2R_mat v (widen_ord (leqnn n.+1) j) ord0) =
(FT2R_mat A *m FT2R_mat v) (inord i) ord0).
{ rewrite !mxE //=. apply eq_big; auto. intros.
assert (widen_ord (leqnn n.+1) i0 = i0).
{ unfold widen_ord. apply val_inj. by simpl. } by rewrite H4. }
by rewrite {}H3 in H2.
+ apply @R_dot_prod_rel_abs_holds.
+ intros. specialize (H0 (@inord n i)). rewrite inord_val in H0. apply H0.
Qed.
Lemma bcmp_zero {ty} (x : ftype ty) :
BCMP Eq false x (Zconst ty 0) = false ->
finite x ->
feq x (Zconst ty 0).
Proof.
intros. destruct x; auto; inversion H0.
unfold BCMP in H. unfold extend_comp in H. unfold Bcompare in H.
unfold BinarySingleNaN.Bcompare in H. simpl in H. destruct s; inversion H.
Qed.
Lemma bcmp_nonzero {ty} (x : ftype ty) :
BCMP Eq false x (Zconst ty 0) = true ->
finite x ->
~ feq x (Zconst ty 0).
Proof.
intros. destruct x; auto; inversion H0.
Qed.
Lemma bfma_bcmp_zero {ty} (x y z : ftype ty):
BCMP Eq false x (Zconst ty 0) = false ->
finite x ->
finite y ->
feq (BFMA x y z) z.
Proof.
intros. rewrite (bcmp_zero H); auto.
rewrite BFMA_zero1; auto.
Qed.
Definition list_finite {ty} (l : list (ftype ty)) :=
forall x, In x l -> finite x.
Lemma list_finite_nil {ty} : list_finite (@nil (ftype ty)).
Proof.
unfold list_finite. intros. inversion H.
Qed.
Lemma list_finite_cons {ty} : forall x l,
finite x ->
@list_finite ty l ->
list_finite (x :: l).
Proof.
unfold list_finite. intros. simpl in H1. destruct H1.
+ subst x0. auto.
+ apply H0. auto.
Qed.
Lemma list_finite_cons_inv {ty} : forall x l,
@list_finite ty (x :: l) ->
finite x /\ list_finite l.
Proof.
unfold list_finite. intros. split.
+ apply H. simpl. auto.
+ intros. apply H. simpl. auto.
Qed.
Lemma extract_elements_inclusion {ty} : forall (l : list (ftype ty)) idx d x,
In x (extract_elements idx l d) ->
In x l \/ x = d.
Proof.
intros. induction idx as [|i' idx'].
+ simpl in H. destruct H.
+ simpl in H. destruct H.
- pose proof (nth_in_or_default i' l d). subst x.
destruct H0; [left | right]; auto.
- apply (IHidx' H).
Qed.
Lemma list_finite_extract {ty} : forall (l : list (ftype ty)) idx d,
list_finite l ->
finite d ->
list_finite (extract_elements idx l d).
Proof.
unfold list_finite. intros.
pose proof (extract_elements_inclusion H1). destruct H2.
+ apply (H x H2).
+ subst x. apply H0.
Qed.
Definition list_list_finite {ty} (l : list (list (ftype ty))) :=
forall x, In x l -> list_finite x.
Lemma list_list_finite_nil {ty} : list_list_finite (@nil (list (ftype ty))).
Proof.
unfold list_list_finite. intros. inversion H.
Qed.
Lemma list_list_finite_cons {ty} : forall x l,
list_finite x ->
@list_list_finite ty l ->
list_list_finite (x :: l).
Proof.
unfold list_list_finite. intros. simpl in H1. destruct H1.
+ subst x0. auto.
+ apply H0. auto.
Qed.
Lemma reduce_sparse_vec_vec_mult {ty}:
forall (l1 l2 : seq.seq (ftype ty)),
let l1_nonzero := @extract_nonzero_elmt ty l1 in
let l2_nonzero := extract_elements (@extract_nonzero_idx ty l1) l2 (Zconst ty 0) in
length l1 = length l2 ->
list_finite l1 ->
list_finite l2 ->
feq (dotprod_r l1 l2) (dotprod_r l1_nonzero l2_nonzero).
Proof.
intros.
revert l2 H H1 l2_nonzero.
induction l1 as [| x1 l1']; intros.
+ simpl in H. destruct l2; auto.
+ destruct l2 as [| x2 l2']; [inversion H|].
simpl in *. inversion H. clear H.
pose proof (proj2 (list_finite_cons_inv H0)).
pose proof (proj2 (list_finite_cons_inv H1)).
specialize (IHl1' H l2' H3 H2).
rewrite dotprod_cons; auto.
destruct (BCMP Eq false x1 (Zconst ty 0)) eqn:E.
- subst l1_nonzero. rewrite extract_nonzero_elmt_cons. rewrite E.
subst l2_nonzero. rewrite extract_nonzero_idx_cons. rewrite E. simpl.
rewrite extract_elements_succ. rewrite dotprod_cons.
2:{ rewrite extract_elements_length. rewrite extract_nonzero_length. auto. }
rewrite IHl1'. auto.
- pose proof (proj1 (list_finite_cons_inv H0)).
pose proof (proj1 (list_finite_cons_inv H1)).
rewrite bfma_bcmp_zero; auto.
subst l1_nonzero. rewrite extract_nonzero_elmt_cons. rewrite E.
subst l2_nonzero. rewrite extract_nonzero_idx_cons. rewrite E. simpl.
rewrite extract_elements_succ. rewrite IHl1'. auto.
Qed.
Lemma op_defs_BFMA_zero1 {ty} y s:
strict_feq y y ->
feq (op_defs.BFMA (Zconst ty 0) y s) s.
Proof.
intros.
intros.
change (Zconst ty 0) with
(Binary.B754_zero (fprec ty) (femax ty) false).
unfold BFMA, BPLUS, BINOP in *.
destruct y, s; try discriminate; simpl; auto.
Qed.
Lemma fma_dot_prod_rel_holds_vec {ty} (l1 l2 : seq.seq (ftype ty)) :
length l1 = length l2 ->
fma_dot_prod_rel (combine l1 l2) (dotprod_r l1 l2).
Proof.
intros. revert l2 H. induction l1 as [|x1 l1']; intros.
+ destruct l2. 2:{ inversion H. }
unfold dotprod_r. simpl. apply fma_dot_prod_rel_nil.
+ destruct l2 as [|x2 l2']; inversion H.
specialize (IHl1' l2' H1).
apply fma_dot_prod_rel_cons. apply IHl1'.
Qed.
Lemma fma_dot_prod_rel_hold_vec_inv {ty} (l1 l2 : seq.seq (ftype ty)) :
length l1 = length l2 ->
forall p, fma_dot_prod_rel (combine l1 l2) p -> feq p (dotprod_r l1 l2).
Proof.
revert l2. induction l1 as [|x1 l1']; intros.
+ destruct l2; try inversion H.
inversion H0. auto.
+ destruct l2 as [|x2 l2']; inversion H. inversion H0.
specialize (IHl1' l2' H2 s H5).
rewrite dotprod_cons; auto. simpl. rewrite <- IHl1'. auto.
Qed.
Lemma feq_zero {ty} (x : ftype ty) :
feq x (Zconst ty 0) ->
x = B754_zero (fprec ty) (femax ty) true \/ x = B754_zero _ _ false.
Proof.
intros. destruct x; try inversion H.
destruct s; auto.
Qed.
Lemma bfma_zero_eq {ty} (x y z : ftype ty) :
feq x (Zconst ty 0) ->
finite y ->
BFMA x y z = z.
Proof.
intros. pose proof (feq_zero H). destruct H1 eqn:Ex; subst.
+ unfold BFMA. unfold Bfma. unfold BSN2B. unfold BinarySingleNaN.Bfma. simpl.
destruct y; try inversion H0; simpl.
- destruct z.
* simpl. unfold BinarySingleNaN.Bfma_szero. simpl.
destruct s eqn:E; destruct s0 eqn:E0; simpl; auto.
Abort.
Lemma fma_dot_prod_rel_holds_sparse {ty} :
forall (l1 l2 : seq.seq (ftype ty)),
let l1_nonzero := @extract_nonzero_elmt ty l1 in
let l2_nonzero := extract_elements (@extract_nonzero_idx ty l1) l2 (Zconst ty 0) in
length l1 = length l2 ->
list_finite l1 ->
list_finite l2 ->
forall fp,
fma_dot_prod_rel (combine l1 l2) fp ->
fma_dot_prod_rel (combine l1_nonzero l2_nonzero) fp.
Abort.
Lemma fma_dot_prod_rel_holds_sparse {ty} :
forall (l1 l2 : seq.seq (ftype ty)),
let l1_nonzero := @extract_nonzero_elmt ty l1 in
let l2_nonzero := extract_elements (@extract_nonzero_idx ty l1) l2 (Zconst ty 0) in
length l1 = length l2 ->
list_finite l1 ->
list_finite l2 ->
forall fp,
fma_dot_prod_rel (combine l1 l2) fp ->
exists fp', feq fp fp' /\ fma_dot_prod_rel (combine l1_nonzero l2_nonzero) fp'.
Proof.
intros. generalize dependent l2. revert fp.
induction l1 as [| x1 l1']; intros.
+ destruct l2; try inversion H. inversion H2.
exists fp. subst fp. split; auto.
+ destruct l2 as [| x2 l2']; inversion H.
simpl in *. clear H.
pose proof (proj2 (list_finite_cons_inv H0)).
pose proof (proj2 (list_finite_cons_inv H1)).
inversion H2. simpl in H6. rename s into fps. subst xy l. simpl in *.
specialize (IHl1' H fps l2' H4 H3 H8).
destruct IHl1' as [fps' [? ?]].
pose proof (proj1 (list_finite_cons_inv H0)).
destruct (BCMP Eq false x1 (Zconst ty 0)) eqn:E.
- subst l1_nonzero. rewrite extract_nonzero_elmt_cons. rewrite E.
subst l2_nonzero. rewrite extract_nonzero_idx_cons. rewrite E. simpl.
rewrite extract_elements_succ. exists (op_defs.BFMA x1 x2 fps'). split.
2:{ constructor. auto. }
rewrite H5. auto.
- subst l1_nonzero. rewrite extract_nonzero_elmt_cons. rewrite E.
subst l2_nonzero. rewrite extract_nonzero_idx_cons. rewrite E. simpl.
rewrite extract_elements_succ.
pose proof (bcmp_zero E (proj1 (list_finite_cons_inv H0))).
assert (feq (op_defs.BFMA x1 x2 fps) fps).
{ rewrite H10. rewrite op_defs_BFMA_zero1. auto.
pose proof (proj1 (list_finite_cons_inv H1)). auto. }
exists fps'. split; auto.
rewrite H11. auto.
Qed.
Lemma bcmp_zero_b754zero {ty} (x : ftype ty) :
BCMP Eq false x (Zconst ty 0) = false ->
finite x ->
exists b, x = B754_zero _ _ b.
Proof.
intros. pose proof (bcmp_zero H H0).
destruct x; inversion H1.
exists s; auto.
Qed.
Lemma R_dot_prod_rel_holds_sparse {ty} (v1 v2 : list (ftype ty)) (rp : R):
let v1_nonzero := @extract_nonzero_elmt ty v1 in
let v2_nonzero := extract_elements (@extract_nonzero_idx ty v1) v2 (Zconst ty 0) in
length v1 = length v2 ->
list_finite v1 ->
list_finite v2 ->
R_dot_prod_rel (combine (map FT2R v1) (map FT2R v2)) rp ->
R_dot_prod_rel (combine (map FT2R v1_nonzero) (map FT2R v2_nonzero)) rp.
Proof.
intros. generalize dependent v2. revert rp. induction v1 as [| x1 v1']; intros.
+ destruct v2; inversion H. simpl.
inversion H2. subst. constructor.
+ destruct v2 as [| x2 v2']; inversion H.
simpl in *. clear H.
pose proof (proj2 (list_finite_cons_inv H0)).
pose proof (proj2 (list_finite_cons_inv H1)).
inversion H2; subst. rename s into rp'; simpl in *.
specialize (IHv1' H rp' v2' H4 H3 H8).
destruct (BCMP Eq false x1 (Zconst ty 0)) eqn:E.
- subst v1_nonzero. rewrite extract_nonzero_elmt_cons. rewrite E.
subst v2_nonzero. rewrite extract_nonzero_idx_cons. rewrite E. simpl.
constructor. rewrite extract_elements_succ. auto.
- subst v1_nonzero. rewrite extract_nonzero_elmt_cons. rewrite E.
subst v2_nonzero. rewrite extract_nonzero_idx_cons. rewrite E. simpl.
rewrite extract_elements_succ.
pose proof (bcmp_zero_b754zero E (proj1 (list_finite_cons_inv H0))).
destruct H5 as [b ?]. subst x1. simpl.
rewrite Rmult_0_l. rewrite Rplus_0_l. auto.
Qed.
Lemma R_dot_prod_rel_abs_holds_sparse {ty} (v1 v2 : list (ftype ty)) (rp_abs : R):
let v1_nonzero := @extract_nonzero_elmt ty v1 in
let v2_nonzero := extract_elements (@extract_nonzero_idx ty v1) v2 (Zconst ty 0) in
length v1 = length v2 ->
list_finite v1 ->
list_finite v2 ->
R_dot_prod_rel (combine (map Rabs (map FT2R v1)) (map Rabs (map FT2R v2))) rp_abs ->
R_dot_prod_rel (combine (map Rabs (map FT2R v1_nonzero)) (map Rabs (map FT2R v2_nonzero))) rp_abs.
Proof.
intros. generalize dependent v2. revert rp_abs. induction v1 as [| x1 v1']; intros.
+ destruct v2; inversion H. simpl.
inversion H2. subst. constructor.
+ destruct v2 as [| x2 v2']; inversion H.
simpl in *. clear H.
pose proof (proj2 (list_finite_cons_inv H0)).
pose proof (proj2 (list_finite_cons_inv H1)).
inversion H2; subst. rename s into rp'; simpl in *.
specialize (IHv1' H rp' v2' H4 H3 H8).
destruct (BCMP Eq false x1 (Zconst ty 0)) eqn:E.
- subst v1_nonzero. rewrite extract_nonzero_elmt_cons. rewrite E.
subst v2_nonzero. rewrite extract_nonzero_idx_cons. rewrite E. simpl.
constructor. rewrite extract_elements_succ. auto.
- subst v1_nonzero. rewrite extract_nonzero_elmt_cons. rewrite E.
subst v2_nonzero. rewrite extract_nonzero_idx_cons. rewrite E. simpl.
rewrite extract_elements_succ.
pose proof (bcmp_zero_b754zero E (proj1 (list_finite_cons_inv H0))).
destruct H5 as [b ?]. subst x1. simpl.
rewrite Rabs_R0. rewrite Rmult_0_l. rewrite Rplus_0_l. auto.
Qed.
Lemma feq_finite {ty} (x y : ftype ty) :
feq x y ->
finite x ->
finite y.
Proof.
intros. destruct x; destruct y; inversion H; inversion H0; simpl; auto.
Qed.
Lemma feq_to_r {ty} (x y : ftype ty) :
feq x y ->
FT2R x = FT2R y.
Proof.
intros.
destruct x; destruct y; inversion H; simpl; auto.
destruct H1; subst s0 m0 e1. auto.
Qed.
Lemma g_increasing {ty} (n n' : nat) :
le n n' ->
Rle (g ty n) (g ty n').
Proof.
intros. unfold g.
pose proof (default_rel_ge_0 ty).
remember (default_rel ty) as p.
clear Heqp. induction H.
+ right. auto.
+ eapply Rle_trans; [apply IHle |].
unfold pow; fold pow.
apply Rplus_le_compat_r.
rewrite Rmult_comm.
rewrite <- (Rmult_1_r ((1+p)^m)) at 1.
apply Rmult_le_compat_l; try lra.
apply Rlt_le. apply pow_lt. lra.
Qed.
Lemma g1_increasing {ty} (n n' m m': nat) :
le n n' ->
le m m' ->
Rle (g1 ty n m) (g1 ty n' m').
Proof.
intros. unfold g1.
pose proof (default_abs_ge_0 ty).
eapply Rmult_le_compat.
+ apply Rmult_le_pos.
- apply pos_INR.
- apply H1.
+ pose proof (g_pos ty m). lra.
+ pose proof (le_INR n n' H).
pose proof (pos_INR n).
apply Rmult_le_compat_r; auto.
+ pose proof (@g_increasing ty m m' H0). lra.
Qed.
Lemma le_minus_1 {a b : nat} :
(a <= b)%coq_nat -> (a - 1 <= b - 1)%coq_nat.
Proof.
intros H.
destruct a; destruct b; try lia.
+ unfold subn. simpl. lia.
+ unfold subn. simpl. unfold subn. simpl. lia.
Qed.
Lemma fma_dotprod_forward_error_sparse {ty}:
forall (v1 v2 : seq.seq (ftype ty)) (r : nat) (HA : is_r_sparse_aux v1 r),
length v1 = length v2 ->
list_finite v1 ->
list_finite v2 ->
forall (fp : ftype ty) (rp rp_abs : R),
fma_dot_prod_rel (combine v1 v2) fp ->
R_dot_prod_rel (combine (map FT2R v1) (map FT2R v2)) rp ->
R_dot_prod_rel (combine (map Rabs (map FT2R v1)) (map Rabs (map FT2R v2))) rp_abs ->
finite fp ->
(Rabs (FT2R fp - rp) <= g ty r * Rabs rp_abs + g1 ty r (r-1)%nat)%Re.
Proof.
intros.
remember (@extract_nonzero_elmt ty v1) as v1_nonzero.
remember (extract_elements (@extract_nonzero_idx ty v1) v2 (Zconst ty 0)) as v2_nonzero.
assert (length v1_nonzero = length v2_nonzero) as Hl.
{ unfold extract_nonzero_elmt in Heqv1_nonzero.
subst. repeat rewrite extract_elements_length. auto. }
pose proof (fma_dotprod_forward_error _ _ v1_nonzero v2_nonzero Hl).
pose proof (fma_dot_prod_rel_holds_sparse H H0 H1 H2).
destruct H7 as [fp' [? ?]].
rewrite <- Heqv1_nonzero, <- Heqv2_nonzero in H8.
pose proof (R_dot_prod_rel_holds_sparse H H0 H1 H3).
pose proof (R_dot_prod_rel_abs_holds_sparse H H0 H1 H4).
pose proof (feq_finite H7 H5).
rewrite <- Heqv1_nonzero, <- Heqv2_nonzero in H9.
rewrite <- Heqv1_nonzero, <- Heqv2_nonzero in H10.
specialize (H6 fp' rp rp_abs H8 H9 H10 H11).
clear H8 H9 H10 H11.
rewrite (feq_to_r H7).
eapply Rle_trans; [apply H6 |].
assert (length v1_nonzero <= r)%coq_nat.
{ rewrite Heqv1_nonzero. unfold is_r_sparse_aux, sparsity_fac_aux in HA.
unfold extract_nonzero_elmt. rewrite extract_elements_length. auto. }
assert (length v1_nonzero - 1 <= r - 1)%coq_nat.
{ apply le_minus_1. auto. }
pose proof (@g_increasing ty (length v1_nonzero) r H8).
pose proof (@g1_increasing ty (length v1_nonzero) r (length v1_nonzero - 1) (r - 1) H8 H9).
apply Rplus_le_compat.
+ apply Rmult_le_compat_r; auto. apply Rabs_pos.
+ apply H11.
Qed.
Lemma in_combine_inv_l {A} (l1 l2 : list A) (x : A) :
In x l1 ->
length l1 = length l2 ->
exists y, In (x, y) (combine l1 l2).
Proof.
intros. generalize dependent l2.
induction l1 as [| x1 l1']; intros.
+ inversion H.
+ destruct l2 as [| x2 l2']; inversion H0.
simpl in *. destruct H.
- subst x1. exists x2. auto.
- destruct (IHl1' H l2' H2) as [y ?].
exists y. auto.
Qed.
Lemma in_combine_inv_r {A} (l1 l2 : list A) (y : A) :
In y l2 ->
length l1 = length l2 ->
exists x, In (x, y) (combine l1 l2).
Proof.
intros. generalize dependent l1.
induction l2 as [| x2 l2']; intros.
+ inversion H.
+ destruct l1 as [| x1 l1']; inversion H0.
simpl in *. destruct H.
- subst. exists x1. auto.
- destruct (IHl2' H l1' H2) as [x ?].
exists x. auto.
Qed.
Lemma combine_finite {ty} (l1 l2 : seq.seq (ftype ty)) :
length l1 = length l2 ->
(forall xy , In xy (combine l1 l2) -> finite xy.1 /\ finite xy.2) ->
list_finite l1 /\ list_finite l2.
Proof.
intros. split.
+ unfold list_finite. intros.
destruct (in_combine_inv_l H1 H) as [y ?].
specialize (H0 (x, y) H2). auto. destruct H0. auto.
+ unfold list_finite. intros.
destruct (in_combine_inv_r H1 H) as [y ?].
specialize (H0 (y, x) H2). auto. destruct H0. auto.
Qed.
(* Lemma matrix_vec_mult_bound_sparse {n : nat} {ty}:
forall (A: 'M[ftype ty]_n.+1) (v : 'cV[ftype ty]_n.+1)
{r : nat} {HA : is_r_sparse_mat A r},
(forall (xy : ftype ty * ftype ty) (i : nat),
In xy
(combine
(vec_to_list_float n.+1
(\row_j A (inord i) j)^T)
(vec_to_list_float n.+1 v)) ->
finite xy.1 /\finite xy.2) ->
(forall (i : 'I_n.+1),
finite (let l1 := vec_to_list_float n.+1 (\row_j A (inord i) j)^T in
let l2 := vec_to_list_float n.+1 (\col_j v j ord0) in
dotprod_r l1 l2)) ->
vec_inf_norm (FT2R_mat (A *f v) - (FT2R_mat A) *m (FT2R_mat v)) <=
@mat_vec_mult_err_bnd_sparse n ty A v r HA.
Proof.
intros. unfold vec_inf_norm, mat_vec_mult_err_bnd_sparse.
(*apply /RleP.*) apply lemmas.bigmax_le; first by rewrite size_map size_enum_ord.
intros. rewrite seq_equiv.
rewrite nth_mkseq; last by rewrite size_map size_enum_ord in H1.
remember (vec_to_list_float n.+1 (\row_j A (inord i) j)^T) as l1.
remember (vec_to_list_float n.+1 (\col_j v j ord0)) as l2.
pose proof (@fma_dotprod_forward_error_sparse ty l1 l2 r).
assert (is_r_sparse_aux l1 r).
{ unfold is_r_sparse_mat in HA. unfold is_r_sparse in HA.
rewrite Heql1. specialize (HA (inord i)). apply HA. }
assert (length l1 = length l2).
{ rewrite Heql1. rewrite Heql2. rewrite !length_veclist. auto. }
assert (list_finite l1).
{ pose proof (@combine_finite ty l1 l2 H4). apply H5.
intros. rewrite Heql1 Heql2 in H6.
assert (v = \col_j v j ord0).
{ apply matrixP. unfold eqrel. intros. rewrite !mxE /=.
assert ( y = ord0). { apply ord1. } by rewrite H7. }
rewrite -H7 in H6. clear H7.
apply (H xy (i)). apply H6.
}
assert (list_finite l2).
{ pose proof (@combine_finite ty l1 l2 H4). apply H6.
intros. rewrite Heql1 Heql2 in H7.
assert (v = \col_j v j ord0).
{ apply matrixP. unfold eqrel. intros. rewrite !mxE /=.
assert ( y = ord0). { apply ord1. } by rewrite H8. }
rewrite -H8 in H7. clear H8.
apply (H xy (i)). apply H7.
}
specialize (H2 H3 H4 H5 H6).
apply Rle_trans with (@e_i_sparse n ty (@inord n i) A v r HA).
+ unfold e_i_sparse. rewrite !mxE -RminusE.
apply H2.
- rewrite <- Heql1.
change GRing.zero with (@ord0 O).
rewrite <- Heql2. simpl. subst.
apply fma_dot_prod_rel_holds.
- pose proof (@R_dot_prod_rel_holds n ty n.+1 i (leqnn n.+1) A v).
subst.
assert (\sum_(j < n.+1)
FT2R_mat A (inord i) (widen_ord (leqnn n.+1) j) *
FT2R_mat v (widen_ord (leqnn n.+1) j) ord0 =
\sum_j
FT2R_mat A (inord i) j * FT2R_mat v j ord0).
{ apply eq_big. by []. intros.
assert (widen_ord (leqnn n.+1) i0 = i0).
{ unfold widen_ord. apply val_inj. by simpl. }
by rewrite H9. }
rewrite -H8.
assert (v = \col_j v j ord0).
{ apply matrixP. unfold eqrel. intros. rewrite !mxE /=.
assert ( y = ord0). { apply ord1. } by rewrite H9.
} rewrite -H9. auto.
- pose proof (@R_dot_prod_rel_abs_holds n ty n.+1 i A v).
rewrite Heql1 Heql2.
assert (v = \col_j v j ord0).
{ apply matrixP. unfold eqrel. intros. rewrite !mxE /=.
assert ( y = ord0). { apply ord1. } rewrite H8. auto. }
rewrite -H8. auto.
- intros. specialize (H0 (@inord n i)).
rewrite inord_val in H0.
rewrite -Heql1 -Heql2. simpl. subst l1 l2. apply H0.
+ assert (@e_i_sparse n ty (inord i) A v r HA = [seq @e_i_sparse n ty i0 A v r HA | i0 <- enum 'I_n.+1]`_i).
{ rewrite seq_equiv nth_mkseq. nra. by rewrite size_map size_enum_ord in H1. }
rewrite H7. apply /RleP.
pose proof (@le_bigmax _ _ _ 0%Re (fun i => @e_i_sparse n ty i A v r HA)) (inord i).
simpl in H8.
replace ([seq e_i_sparse i0 v HA | i0 <- enum 'I_n.+1]`_i) with (e_i_sparse (inord i) v HA) by auto.
rewrite big_enum //.
Qed. *)
Definition FT2R_abs {m n: nat} (A : 'M[R]_(m.+1, n.+1)) :=
\matrix_(i,j) Rabs (A i j).
Lemma sum_abs_eq {n:nat} (f: 'I_n.+1 -> R):
(forall i, (0 <= f i)%Re) ->
Rabs (\sum_j (f j)) = \sum_j (f j).
Proof.
intros.
induction n.
+ simpl. rewrite big_ord_recr /= big_ord0 /=.
rewrite add0r. rewrite Rabs_right. by []. apply Rle_ge. by apply H.
+ simpl. rewrite big_ord_recr /=. rewrite Rabs_right.
by []. rewrite -RplusE. apply Rle_ge.
apply Rplus_le_le_0_compat.
- rewrite -IHn. apply Rabs_pos.
intros. apply H.
- apply H.
Qed.
Lemma sum_fold_mathcomp_equiv {n:nat} {ty} m i (le_n_m : (m <= n.+1)%nat)
(A: 'M[ftype ty]_n.+1) (v : 'cV[ftype ty]_n.+1) :
\sum_(j < m) FT2R_abs (FT2R_mat A) (inord i) (@widen_ord m n.+1 le_n_m j)
* FT2R_abs (FT2R_mat v) (@widen_ord m n.+1 le_n_m j) ord0 =
sum_fold
(map (uncurry Rmult)
(map Rabsp
(map FR2
(combine
(@vec_to_list_float _ n m
(\row_j A (inord i) j)^T)
(@vec_to_list_float _ n m v))))).
Proof.
induction m.
+ simpl. by rewrite big_ord0 /=.
+ rewrite big_ord_recr /= !mxE.
assert ((widen_ord le_n_m ord_max) = (inord m)).
{ unfold widen_ord.
apply val_inj. simpl. by rewrite inordK.
} rewrite H. rewrite Rplus_comm.
assert ((m <= n.+1)%nat). { by apply ltnW. }
specialize (IHm H0).
assert (\sum_(j < m)
FT2R_abs (FT2R_mat A) (inord i)
(widen_ord H0 j) *
FT2R_abs (FT2R_mat v) (widen_ord H0 j) ord0 =
\sum_(i0 < m)
FT2R_abs (FT2R_mat A) (inord i)
(widen_ord le_n_m (widen_ord (leqnSn m) i0)) *
FT2R_abs (FT2R_mat v)
(widen_ord le_n_m (widen_ord (leqnSn m) i0)) ord0).
{ apply eq_big. by []. intros.
assert ((widen_ord le_n_m
(widen_ord (leqnSn m) i0))=
(widen_ord H0 i0)).
{ unfold widen_ord.
apply val_inj. by simpl.
} by rewrite H2.
} rewrite -H1. by rewrite IHm.
Qed.
Lemma e_i_eq_form {n : nat} {ty}
(A : 'M[ftype ty]_n.+1) (v : 'cV[ftype ty]_n.+1) (i : 'I_n.+1):
e_i i A v =
g ty n.+1 * (Rabs ((FT2R_abs (FT2R_mat A) *m FT2R_abs (FT2R_mat v)) i ord0) ) +
g1 ty n.+1 (n.+1 - 1).
Proof.