Section FMinMax.
Variable b : Fbound.
Variable radix : Z.
Variable precision : nat.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ radixMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionNotZero : precision <> 0.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.
(* a function that returns a boundd greater than a given nat *)
Definition boundNat (n : nat) := Float 1%nat (digit radix n).
intros n; unfold FtoRradix, FtoR, boundNat in |- *; simpl in |- *.
rewrite Rmult_1_l.
rewrite <- Zpower_nat_Z_powerRZ; auto with real zarith.
rewrite INR_IZR_INZ; auto with real zarith.
apply Rle_lt_trans with (Z.abs n); [rewrite (Z.abs_eq (Z_of_nat n))|idtac];auto with real zarith.
Qed.
intros n; repeat split; unfold boundNat in |- *; simpl in |- *;
auto with zarith.
apply vNumbMoreThanOne with (radix := radix) (precision := precision);
auto with zarith.
apply Z.le_trans with 0%Z;[case (dExp b)|idtac]; auto with zarith.
Qed.
(* A function that returns a bounded greater than a given r *)
Definition boundR (r : R) := boundNat (Z.abs_nat (up (Rabs r))).
intros r; case (Rle_or_lt r 0); intros H'.
apply Rle_lt_trans with (1 := H').
unfold boundR, boundNat, FtoRradix, FtoR in |- *; simpl in |- *;
auto with real.
rewrite Rmult_1_l; auto with real zarith.
apply Rlt_trans with (2 := boundNatCorrect (Z.abs_nat (up (Rabs r)))).
replace (Rabs r) with r; auto with real.
apply Rlt_le_trans with (r2 := IZR (up r)); auto with real zarith.
case (archimed r); auto.
rewrite INR_IZR_INZ; auto with real zarith.
unfold Rabs in |- *; case (Rcase_abs r); auto with real.
intros H'0; Contradict H'0; auto with real.
Qed.
intros R; unfold boundR in |- *.
rewrite Rabs_Ropp; auto.
Qed.
intros r; case (Rle_or_lt r 0); intros H'.
rewrite boundRrOpp.
pattern r at 2 in |- *; rewrite <- (Ropp_involutive r).
unfold FtoRradix in |- *; rewrite Fopp_correct.
apply Ropp_lt_contravar; apply boundRCorrect1; auto.
apply Rle_lt_trans with 0%R; auto.
replace 0%R with (-0)%R; auto with real.
unfold FtoRradix in |- *; rewrite Fopp_correct.
apply Ropp_le_contravar.
unfold boundR, boundNat, FtoRradix, FtoR in |- *; simpl in |- *;
auto with real zarith.
rewrite Rmult_1_l; apply Rlt_le; auto with real zarith arith.
Qed.
(* A function that returns a list containing all the bounded smaller than a given real *)
Definition mBFloat (p : R) :=
map (fun p : Z * Z => Float (fst p) (snd p))
(mProd Z Z (Z * Z)
(mZlist (- pPred (vNum b)) (pPred (vNum b)))
(mZlist (- dExp b) (Fexp (boundR p)))).
intros r q.
case (Zle_or_lt (Fexp (boundR r)) (Fexp q)); intros H'.
intros H'0 H'1 H'2 H'3; case H'0.
apply is_Fzero_rep2 with (radix := radix); auto.
rewrite <-
FshiftCorrect with (n := Z.abs_nat (Fexp q - Fexp (boundR r))) (x := q);
auto with arith.
apply is_Fzero_rep1 with (radix := radix).
unfold is_Fzero in |- *.
cut (forall p : Z, (- 1%nat < p)%Z -> (p < 1%nat)%Z -> p = 0%Z);
[ intros tmp; apply tmp | idtac ].
replace (- 1%nat)%Z with (Fnum (Fopp (boundR r))).
apply Rlt_Fexp_eq_Zlt with (radix := radix); auto with real zarith.
rewrite FshiftCorrect; auto.
unfold Fshift in |- *; simpl in |- *.
rewrite (fun x y => inj_abs (x - y)); auto with zarith.
simpl in |- *; auto.
replace (Z_of_nat 1) with (Fnum (boundR r)).
apply Rlt_Fexp_eq_Zlt with (radix := radix); auto with zarith.
rewrite FshiftCorrect; auto.
unfold Fshift in |- *; simpl in |- *.
rewrite inj_abs; auto with zarith.
generalize H'; simpl in |- *; auto with zarith.
simpl in |- *; auto.
intros p0; case p0; simpl in |- *; auto with zarith.
intros H'0 H'1 H'2 H'3; unfold mBFloat in |- *.
replace q with
((fun p : Z * Z => Float (fst p) (snd p)) (Fnum q, Fexp q)).
apply in_map with (f := fun p : Z * Z => Float (fst p) (snd p));
auto.
apply mProd_correct; auto.
apply mZlist_correct; auto with float.
apply Zle_Zabs_inv1; auto with float.
unfold pPred in |- *; apply Zle_Zpred; auto with float.
apply Zle_Zabs_inv2; auto with float.
unfold pPred in |- *; apply Zle_Zpred; auto with float.
apply mZlist_correct; auto with float.
auto with zarith.
case q; simpl in |- *; auto with zarith.
Qed.
intros r; unfold mBFloat in |- *.
replace (boundR r) with
((fun p : Z * Z => Float (fst p) (snd p))
(Fnum (boundR r), Fexp (boundR r))).
apply in_map with (f := fun p : Z * Z => Float (fst p) (snd p));
auto.
apply mProd_correct; auto.
apply mZlist_correct; auto.
unfold boundR, boundNat in |- *; simpl in |- *; auto with zarith.
apply Z.le_trans with (- (0))%Z; auto with zarith.
apply Zle_Zopp; unfold pPred in |- *; apply Zle_Zpred; simpl in |- *.
apply Z.lt_trans with 1%Z; auto with zarith.
apply vNumbMoreThanOne with (3 := pGivesBound); auto.
unfold boundR, boundNat in |- *; simpl in |- *; auto with zarith.
unfold pPred in |- *; apply Zle_Zpred; simpl in |- *.
unfold boundR, boundNat in |- *; simpl in |- *; auto with zarith.
apply vNumbMoreThanOne with (3 := pGivesBound); auto.
apply mZlist_correct; auto.
unfold boundR, boundNat in |- *; simpl in |- *; auto with zarith.
apply Z.le_trans with 0%Z; auto with zarith arith.
case (dExp b); auto with zarith.
case (boundR r); simpl in |- *; auto with zarith.
case (boundR r); simpl in |- *; auto with zarith.
Qed.
intros r; unfold mBFloat in |- *.
replace (Fopp (boundR r)) with
((fun p : Z * Z => Float (fst p) (snd p))
(Fnum (Fopp (boundR r)), Fexp (Fopp (boundR r)))).
apply in_map with (f := fun p : Z * Z => Float (fst p) (snd p));
auto.
apply mProd_correct; auto.
apply mZlist_correct; auto.
unfold boundR, boundNat in |- *; simpl in |- *; auto with zarith.
replace (-1)%Z with (- Z_of_nat 1)%Z; auto with zarith.
apply Zle_Zopp.
unfold pPred in |- *; apply Zle_Zpred; simpl in |- *.
apply (vNumbMoreThanOne radix) with (precision := precision);
auto with zarith.
unfold pPred in |- *; apply Zle_Zpred; simpl in |- *.
red in |- *; simpl in |- *; auto.
apply mZlist_correct; auto.
unfold boundR, boundNat in |- *; simpl in |- *; auto with zarith.
apply Z.le_trans with 0%Z; auto with zarith.
case (dExp b); auto with zarith.
case (boundR r); simpl in |- *; auto with zarith.
case (boundR r); simpl in |- *; auto with zarith.
Qed.
intros p; unfold mBFloat in |- *.
replace (Float 0%nat (- dExp b)) with
((fun p : Z * Z => Float (fst p) (snd p))
(Fnum (Float 0%nat (- dExp b)), Fexp (Float 0%nat (- dExp b)))).
apply in_map with (f := fun p : Z * Z => Float (fst p) (snd p));
auto.
apply mProd_correct; auto.
apply mZlist_correct; auto.
simpl in |- *; auto with zarith.
replace 0%Z with (- (0))%Z; [ idtac | simpl in |- *; auto ].
apply Zle_Zopp; unfold pPred in |- *; apply Zle_Zpred.
red in |- *; simpl in |- *; auto with zarith.
simpl in |- *; auto with zarith.
unfold pPred in |- *; apply Zle_Zpred.
red in |- *; simpl in |- *; auto with zarith.
apply mZlist_correct; auto.
simpl in |- *; auto with zarith.
unfold boundR, boundNat in |- *; simpl in |- *; auto with zarith.
apply Z.le_trans with 0%Z; auto with zarith.
case (dExp b); auto with zarith.
simpl in |- *; auto with zarith.
Qed.
intros p r H'; red in |- *; repeat (split; auto).
apply Zpred_Zle_Zabs_intro.
apply mZlist_correct_rev1 with (q := Z.pred (Zpos (vNum b)));
auto with real.
apply
mProd_correct_rev1
with
(l2 := mZlist (- dExp b) (Fexp (boundR r)))
(C := (Z * Z)%type)
(b := Fexp p); auto.
apply
in_map_inv with (f := fun p : Z * Z => Float (fst p) (snd p));
auto.
intros a1 b1; case a1; case b1; simpl in |- *.
intros z z0 z1 z2 H'0; inversion H'0; auto.
generalize H'; case p; auto.
apply mZlist_correct_rev2 with (p := (- Z.pred (Zpos (vNum b)))%Z);
auto.
apply
mProd_correct_rev1
with
(l2 := mZlist (- dExp b) (Fexp (boundR r)))
(C := (Z * Z)%type)
(b := Fexp p); auto.
apply
in_map_inv with (f := fun p : Z * Z => Float (fst p) (snd p));
auto.
intros a1 b1; case a1; case b1; simpl in |- *.
intros z z0 z1 z2 H'0; inversion H'0; auto.
generalize H'; case p; auto.
apply mZlist_correct_rev1 with (q := Fexp (boundR r)); auto.
apply
mProd_correct_rev2
with
(l1 := mZlist (- pPred (vNum b)) (pPred (vNum b)))
(C := (Z * Z)%type)
(a := Fnum p); auto.
apply
in_map_inv with (f := fun p : Z * Z => Float (fst p) (snd p));
auto.
intros a1 b1; case a1; case b1; simpl in |- *.
intros z z0 z1 z2 H'0; inversion H'0; auto.
generalize H'; case p; auto.
Qed.
(* Some general properties of rounded predicate : -Projector A bounded is rounded to something equal to itself - Monotone : the rounded predicate is monotone *)
Definition ProjectorP (P : R -> float -> Prop) :=
forall p q : float, Fbounded b p -> P p q -> p = q :>R.
Definition MonotoneP (P : R -> float -> Prop) :=
forall (p q : R) (p' q' : float),
(p < q)%R -> P p p' -> P q q' -> (p' <= q')%R.
(* What it is to be a minimum*)
Definition isMin (r : R) (min : float) :=
Fbounded b min /\
(min <= r)%R /\
(forall f : float, Fbounded b f -> (f <= r)%R -> (f <= min)%R).
(* Min is a projector *)
intros p r H; case H; intros H1 H2; case H2; auto.
Qed.
red in |- *.
intros p q H' H'0; apply Rle_antisym.
elim H'0; intros H'1 H'2; elim H'2; intros H'3 H'4; apply H'4; clear H'2;
auto with real.
apply isMin_inv1 with (1 := H'0); auto.
Qed.
(* It is monotone *)
red in |- *.
intros p q p' q' H' H'0 H'1.
elim H'1; intros H'2 H'3; elim H'3; intros H'4 H'5; apply H'5; clear H'3 H'1;
auto.
case H'0; auto.
apply Rle_trans with p; auto.
apply isMin_inv1 with (1 := H'0); auto.
apply Rlt_le; auto.
Qed.
(* What it is to be a maximum *)
Definition isMax (r : R) (max : float) :=
Fbounded b max /\
(r <= max)%R /\
(forall f : float, Fbounded b f -> (r <= f)%R -> (max <= f)%R).
(* It is a projector *)
intros p r H; case H; intros H1 H2; case H2; auto.
Qed.
red in |- *.
intros p q H' H'0; apply Rle_antisym.
apply isMax_inv1 with (1 := H'0); auto.
elim H'0; intros H'1 H'2; elim H'2; intros H'3 H'4; apply H'4; clear H'2;
auto with real.
Qed.
(* It is monotone *)
red in |- *.
intros p q p' q' H' H'0 H'1.
elim H'0; intros H'2 H'3; elim H'3; intros H'4 H'5; apply H'5; clear H'3 H'0.
case H'1; auto.
apply Rle_trans with q; auto.
apply Rlt_le; auto.
apply isMax_inv1 with (1 := H'1); auto.
Qed.
(* Minimun is defined upto equality *)
intros p q r H' H'0; apply Rle_antisym.
elim H'0; intros H'1 H'2; elim H'2; intros H'3 H'4; apply H'4; clear H'2 H'0;
auto.
case H'; auto.
apply isMin_inv1 with (1 := H'); auto.
elim H'; intros H'1 H'2; elim H'2; intros H'3 H'4; apply H'4; clear H'2 H';
auto.
case H'0; auto.
apply isMin_inv1 with (1 := H'0); auto.
Qed.
(* Maximum is defined upto equality *)
intros p q r H' H'0; apply Rle_antisym.
elim H'; intros H'1 H'2; elim H'2; intros H'3 H'4; apply H'4; clear H'2 H';
auto.
case H'0; auto.
apply isMax_inv1 with (1 := H'0); auto.
elim H'0; intros H'1 H'2; elim H'2; intros H'3 H'4; apply H'4; clear H'2 H'0;
auto.
case H'; auto.
apply isMax_inv1 with (1 := H'); auto.
Qed.
(* Min and Max are related *)
intros p r H'; split.
apply oppBounded; case H'; auto.
split.
unfold FtoRradix in |- *; rewrite Fopp_correct.
apply Ropp_le_contravar; apply isMin_inv1 with (1 := H'); auto.
intros f H'0 H'1.
rewrite <- (Fopp_Fopp f).
unfold FtoRradix in |- *; rewrite Fopp_correct; rewrite Fopp_correct.
apply Ropp_le_contravar.
elim H'.
intros H'2 H'3; elim H'3; intros H'4 H'5; apply H'5; clear H'3.
apply oppBounded; case H'; auto.
rewrite <- (Ropp_involutive r).
unfold FtoRradix in |- *; rewrite Fopp_correct; auto with real.
Qed.
(* Max and Min are related *)
intros p r H'; split.
apply oppBounded; case H'; auto.
split.
unfold FtoRradix in |- *; rewrite Fopp_correct.
apply Ropp_le_contravar; apply isMax_inv1 with (1 := H'); auto.
intros f H'0 H'1.
rewrite <- (Fopp_Fopp f).
unfold FtoRradix in |- *; repeat rewrite Fopp_correct.
apply Ropp_le_contravar.
rewrite <- (Fopp_correct radix f).
elim H'.
intros H'2 H'3; elim H'3; intros H'4 H'5; apply H'5; clear H'3.
apply oppBounded; auto.
rewrite <- (Ropp_involutive r).
unfold FtoRradix in |- *; rewrite Fopp_correct; auto with real.
Qed.
(* If I have a strict min I can get a max using FNSucc *)
intros p r H' H'0.
split.
apply FcanonicBound with (radix := radix); auto with float.
apply FNSuccCanonic; auto.
inversion H'; auto.
split.
case (Rle_or_lt (FNSucc b radix precision p) r); intros H'2; auto.
absurd (FNSucc b radix precision p <= p)%R.
apply Rlt_not_le.
unfold FtoRradix in |- *; apply FNSuccLt; auto.
inversion H'; auto.
elim H0; intros H'1 H'3; apply H'3; auto.
apply FcanonicBound with (radix := radix); auto with float.
apply Rlt_le; auto.
intros f H'2 H'3.
replace (FtoRradix f) with (FtoRradix (Fnormalize radix b precision f)).
unfold FtoRradix in |- *; apply FNSuccProp; auto.
inversion H'; auto.
apply FcanonicBound with (radix := radix); auto with float.
apply Rlt_le_trans with r; auto.
case (Rle_or_lt r p); auto.
intros H'4; Contradict H'0.
apply Rle_antisym; auto; apply isMin_inv1 with (1 := H'); auto.
rewrite FnormalizeCorrect; auto.
unfold FtoRradix in |- *; apply FnormalizeCorrect; auto.
Qed.
(* Find a minimun in a given list if it exists *)
intros r L; elim L; simpl in |- *; auto.
left; intros f H'; elim H'.
intros a l H'.
elim H';
[ intros H'0; clear H'
| intros H'0; elim H'0; intros min E; elim E; intros H'1 H'2; elim H'2;
intros H'3 H'4; try exact H'4; clear H'2 E H'0 H' ].
case (Rle_or_lt a r); intros H'1.
right; exists a; repeat split; auto.
intros f H'; elim H';
[ intros H'2; rewrite <- H'2; clear H' | intros H'2; clear H' ];
auto with real.
intros H'; Contradict H'; auto with real.
apply Rlt_not_le; auto with real.
left; intros f H'; elim H';
[ intros H'2; rewrite <- H'2; clear H' | intros H'2; clear H' ];
auto.
case (Rle_or_lt a min); intros H'5.
right; exists min; repeat split; auto.
intros f H'; elim H';
[ intros H'0; rewrite <- H'0; clear H' | intros H'0; clear H' ];
auto.
case (Rle_or_lt a r); intros H'6.
right; exists a; repeat split; auto.
intros f H'; elim H';
[ intros H'0; rewrite <- H'0; clear H' | intros H'0; clear H' ];
auto with real.
intros H'; apply Rle_trans with (FtoRradix min); auto with real.
right; exists min; split; auto; split; auto.
intros f H'; elim H';
[ intros H'0; elim H'0; clear H' | intros H'0; clear H' ];
auto.
intros H'; Contradict H'6; auto with real.
apply Rle_not_lt; auto.
Qed.
intros r.
case (MinExList r (mBFloat r)).
intros H'0; absurd (Fopp (boundR r) <= r)%R; auto.
apply Rlt_not_le.
apply H'0.
apply mBFadic_correct3; auto.
(* A minimum always exists *)
apply Rlt_le.
apply boundRCorrect2; auto.
intros H'0; elim H'0; intros min E; elim E; intros H'1 H'2; elim H'2;
intros H'3 H'4; clear H'2 E H'0.
exists min; split; auto.
apply mBPadic_Fbounded with (r := r); auto.
split; auto.
intros f H'0 H'2.
case (Req_dec f 0); intros H'6.
replace (FtoRradix f) with (FtoRradix (Float 0%nat (- dExp b))).
apply H'4; auto.
apply mBFadic_correct4; auto.
replace (FtoRradix (Float 0%nat (- dExp b))) with (FtoRradix f); auto.
rewrite H'6.
unfold FtoRradix, FtoR in |- *; simpl in |- *; auto with real.
rewrite H'6.
unfold FtoRradix, FtoR in |- *; simpl in |- *; auto with real.
case (Rle_or_lt f (Fopp (boundR r))); intros H'5.
apply Rle_trans with (FtoRradix (Fopp (boundR r))); auto.
apply H'4; auto.
apply mBFadic_correct3; auto.
apply Rlt_le.
apply boundRCorrect2; auto.
case (Rle_or_lt (boundR r) f); intros H'7.
Contradict H'2; apply Rlt_not_le.
apply Rlt_le_trans with (FtoRradix (boundR r)); auto.
apply boundRCorrect1; auto.
apply H'4; auto.
apply mBFadic_correct1; auto.
Contradict H'6; unfold FtoRradix in |- *; apply is_Fzero_rep1; auto.
Qed.
intros r; case (MinEx r).
intros x H'.
case (Req_dec x r); intros H'1.
exists x.
rewrite <- H'1.
red in |- *; split; [ case H' | split ]; auto with real.
(* A maximum always exists *)
exists (FNSucc b radix precision x).
apply MinMax; auto.
Qed.
intros r p H' H'0 H'1.
split; auto.
split; auto.
intros f H'2 H'3.
case (Rle_or_lt f p); auto; intros H'5.
Contradict H'3.
(* If we are between a bound and its successor, it is our minimum *)
apply Rlt_not_le.
apply Rlt_le_trans with (1 := H'1); auto with real.
replace (FtoRradix f) with (FtoRradix (Fnormalize radix b precision f)).
unfold FtoRradix in |- *; apply FNSuccProp; auto; try apply FnormalizeCanonic;
auto.
unfold FtoRradix in |- *; repeat rewrite FnormalizeCorrect; auto with real.
apply FcanonicBound with (radix := radix); auto.
apply FnormalizeCanonic; auto.
unfold FtoRradix in |- *; rewrite FnormalizeCorrect; auto with real.
unfold FtoRradix in |- *; rewrite FnormalizeCorrect; auto with real.
Qed.

Theorem FminRep :
forall p q : float,
isMin p q -> exists m : Z, q = Float m (Fexp p) :>R.