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lecture2.v
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(******************************************************************************)
(* *)
(* LECTURE : Floating-point numbers and formal proof *)
(* Laurent.Thery@inria.fr 12/05/2016 *)
(* *)
(******************************************************************************)
(*
How to define floating point numbers ?
We are going to use the flocq library at http://flocq.gforge.inria.fr/
*)
Require Import Psatz Reals.
From Flocq Require Import Core.
Section Lecture2.
Open Scope R_scope.
Axiom todo : forall P, P.
Ltac todo := apply todo.
(* We start very abstractly *)
Variable F : R -> Prop. (* The predicate "to be a float" *)
Variables vx vy : R. (* Some values for our examples *)
Check F vx. (* vx is a floating-point number *)
Variable P : R -> R -> Prop. (* The relation
"to be a rounded value" *)
Check P vx vy. (* vy is a rounded value of vx *)
(******************************************************************************)
(* ROUNDING RELATION *)
(******************************************************************************)
(* What do we require for our relation "to be a rounded value"? *)
Lemma round_pred_P : round_pred P.
Proof.
split; red.
todo. (* - to be total *)
todo. (* - to be monotone *)
Qed.
Definition rnd := let (f, _) := round_fun_of_pred P round_pred_P in f.
Check rnd. (* the function associated with the
relation *)
Lemma rndP : forall x, P x (rnd x). (* its associated relation *)
Proof. unfold rnd; destruct (round_fun_of_pred P round_pred_P); auto. Qed.
(******************************************************************************)
(* ROUNDING DOWN, UP, TO ZERO *)
(******************************************************************************)
(* DOWN *)
Check Rnd_DN_pt F vx vy.
Eval lazy beta delta [Rnd_DN_pt] in Rnd_DN_pt F vx vy.
Check Rnd_DN F rnd.
Eval lazy beta delta [Rnd_DN] in Rnd_DN F rnd.
(* UP *)
Check Rnd_UP_pt F vx vy.
Eval lazy beta delta [Rnd_UP_pt] in Rnd_UP_pt F vx vy.
Check Rnd_UP F rnd.
Eval lazy beta delta [Rnd_UP] in Rnd_UP F rnd.
(* ZERO *)
Check Rnd_ZR_pt F vx vy.
Eval lazy beta delta [Rnd_ZR_pt] in Rnd_ZR_pt F vx vy.
Check Rnd_ZR F rnd.
Eval lazy beta delta [Rnd_ZR] in Rnd_ZR F rnd.
(******************************************************************************)
(* IDEMPOTENCE AND MONOTONY *)
(******************************************************************************)
Fact ex1 : forall x, F x -> Rnd_DN_pt F x x. (* Rnd_Dn_pt idempotent on F *)
Proof.
intros x Fx; repeat split.
- assumption.
- lra.
- intros g Fg gLx; assumption.
Qed.
Fact ex2 : round_pred_monotone (Rnd_DN_pt F). (* Rnd_Dn_pt monotone *)
Proof.
intros x y f g Rxf Rxg xLy.
destruct Rxf as [Ff [fLx fdown]].
destruct Rxg as [Fg [gLy gdown]].
apply gdown; try lra; assumption.
Qed.
(*
Prove that UP is also idempotent and monotone
Fact ex3 : forall x, F x -> Rnd_UP_pt F x x.
Proof.
...
Qed.
Fact ex4 : round_pred_monotone (Rnd_UP_pt F).
Proof.
...
Qed.
Prove that ZR is idempotent but only monotone if 0 is a float
Fact ex5 : forall x, F x -> Rnd_ZR_pt F x x.
Proof.
...
Qed.
Fact ex6 : F 0 -> round_pred_monotone (Rnd_ZR_pt F).
Proof.
...
Qed.
Hint:
In order to perform a case analysis on the fact that x is smaller to y or not
one can use the tactic "destruct (Rle_lt_dec x y) as [xLy | yLx]"
*)
(******************************************************************************)
(* TOTALITY *)
(******************************************************************************)
Fact ex7 : (forall x, F x -> F (-x)) -> (* the format is symmetric *)
round_pred_total (Rnd_DN_pt F) -> round_pred_total (Rnd_UP_pt F).
Proof.
intros sym tDN.
intros x.
destruct (tDN (-x)) as [y [Fy [yLx yM]]].
exists (-y); repeat split.
- apply sym; auto.
- lra.
- intros g Fg xLg.
assert (HH := yM _ (sym _ Fg)).
lra.
Qed.
Print satisfies_any.
Hypothesis SAF : satisfies_any F.
(*
Prove that DN, UP, ZR are rounding predicates
Fact ex8 : round_pred (Rnd_DN_pt F).
Proof.
...
Qed.
Fact ex9 : round_pred (Rnd_UP_pt F).
Proof.
...
Qed.
Fact ex10 : round_pred (Rnd_ZR_pt F).
Proof.
...
Qed.
*)
(******************************************************************************)
(* ROUNDING TO THE CLOSEST *)
(******************************************************************************)
Check Rnd_N_pt F vx vy.
Eval lazy beta delta [Rnd_N_pt] in Rnd_N_pt F vx vy.
Check Rnd_N.
Eval lazy beta delta [Rnd_N] in Rnd_N F rnd.
(*
Prove that N is idempotent, that is either UP or DOWN and that it is strictly
monotone
Fact ex11 : forall x, F x -> Rnd_N_pt F x x.
Proof.
...
Qed.
Fact ex12 : forall x f, Rnd_N_pt F x f -> Rnd_DN_pt F x f \/ Rnd_UP_pt F x f.
Proof.
...
Qed.
Fact ex13 : forall x y f g,
Rnd_N_pt F x f -> Rnd_N_pt F y g -> x < y -> f <= g.
Proof.
...
Qed.
Hints : some theorems about absolute values
Check Rabs_R0.
Check Rabs_pos.
Check Rabs_right.
Check Rabs_left.
*)
Variable T : R -> R -> Prop. (* Tie-break rule *)
Check Rnd_NG_pt F T vx vy.
Eval lazy beta delta [Rnd_NG_pt] in Rnd_NG_pt F T vx vy.
Check NG_existence_prop F T. (* condition to ensure existance *)
Eval lazy beta delta [NG_existence_prop] in NG_existence_prop F T.
Search NG_existence_prop.
Check Rnd_NG_pt_unique_prop F T.
Eval lazy beta delta [Rnd_NG_pt_unique_prop] in Rnd_NG_pt_unique_prop F T.
Check Rnd_NG_pt_monotone F T.
Search Rnd_NG_pt_unique_prop.
Definition Taway x f := Rabs f >= Rabs x.
(*
Prove that Taway verifies the two properties to build a rounding mode
Fact ex14 : NG_existence_prop F Taway.
Proof.
...
Qed.
Fact ex15 : F 0 -> Rnd_NG_pt_unicity_prop F Taway.
Proof.
...
Qed.
*)
(*
Resume
- "generic" version of format and roundind modes
- 4 rounding functions : up, down, to zero, to the nearest.
*)
End Lecture2.