Commit a13cdb39 authored by Heiko Becker's avatar Heiko Becker
Browse files

general lemma about constants

parent 9b427ee4
Formalization of the base expression language for the daisy framework
Require Import Coq.Reals.Reals.
Require Import Coq.Reals.Reals Interval.Interval_tactic.
Require Import Daisy.realConversion.
Set Implicit Arguments.
......@@ -94,6 +94,59 @@ Proof.
rewrite (IHe1 v0 v4); auto.
rewrite (IHe2 v3 v5); auto.
Lemma perturb_0_val (v:R) (delta:R):
(Rabs delta <= 0)%R ->
perturb v delta = v.
intros abs_0; apply Rabs_0_impl_eq in abs_0; subst.
unfold perturb.
rewrite Rmult_plus_distr_l.
rewrite Rmult_0_r.
rewrite Rmult_1_r.
rewrite Rplus_0_r; auto.
Lemma Rsub_eq_Ropp_Rplus (a:R) (b:R) :
(a - b = a + (- b))%R.
field_simplify; reflexivity.
Lemma const_abs_err_bounded (n:R) (nR:R) (nF:R):
forall cenv:nat -> R,
eval_exp 0%R cenv (Const n) nR ->
eval_exp machineEpsilon cenv (Const n) nF ->
(Rabs (nR - nF) <= Rabs (n * machineEpsilon))%R.
intros cenv eval_real eval_float.
inversion eval_real; subst.
rewrite perturb_0_val; auto.
inversion eval_float; subst.
unfold perturb; simpl.
rewrite Rmult_plus_distr_l.
rewrite Rmult_1_r.
rewrite Rsub_eq_Ropp_Rplus.
rewrite Ropp_plus_distr.
assert (-n + (- (n * delta0)) = ( -(n * delta0) + - n))%R by (rewrite Rplus_comm; auto).
rewrite H.
rewrite <- Rsub_eq_Ropp_Rplus.
rewrite Rplus_minus.
rewrite Rabs_Ropp.
repeat rewrite Rabs_mult.
apply Rmult_le_compat_l.
- apply Rabs_pos.
- assert (Rabs machineEpsilon = machineEpsilon).
+ unfold machineEpsilon.
unfold realFromNum, negPow.
unfold Rabs.
destruct Rcase_abs; auto.
apply (Rlt_not_ge _ _ r).
+ rewrite H2; auto.
Using the parametric expressions, define boolean expressions for conditionals
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