Codebase list coq-float / HEAD Closest2Plus.v
HEAD

Tree @HEAD (Download .tar.gz)

Closest2Plus.v @HEAD — raw · history · blame 

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
(****************************************************************************
                                                                             
          IEEE754  :  Closest2Plus                                                     
                                                                             
          Laurent Thery                                                      
                                                                             
  ******************************************************************************)
Require Export ClosestPlus.
Require Export Closest2Prop.
Section F2.
Variable b : Fbound.
Variable precision : nat.
 
Let radix := 2%Z.
 
Let FtoRradix := FtoR radix.
Local Coercion FtoRradix : float >-> R.
 
Theorem TwoMoreThanOne : (1 < radix)%Z.
red in |- *; simpl in |- *; auto.
Qed.
Hint Resolve TwoMoreThanOne.
Hypothesis precisionNotZero : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.
 
Theorem plusUpperBound :
 forall P,
 RoundedModeP b 2%nat P ->
 forall p q pq : float,
 P (p + q)%R pq ->
 Fbounded b p ->
 Fbounded b q -> (Rabs pq <= radix * Rmax (Rabs p) (Rabs q))%R.
intros P H' p q pq H'0 H'1 H'2.
rewrite <- (Rabs_right radix); auto with real zarith.
rewrite <- RmaxRmult; auto with real.
repeat rewrite <- Rabs_mult.
case (Rle_or_lt p q); intros Rltp.
apply RmaxAbs; auto.
apply (RoundedModeMultLess b radix) with (P := P) (r := (p + q)%R); auto.
replace (radix * FtoR radix p)%R with (p + p)%R;
 [ auto with real | unfold radix at 1; fold FtoRradix; ring].
unfold FtoRradix in |- *;
 apply (RoundedModeMult b radix) with (P := P) (r := (p + q)%R); 
 auto.
replace (radix * FtoR radix q)%R with (q + q)%R;
 [ auto with real | unfold radix at 1; fold FtoRradix; ring ].
rewrite RmaxSym.
apply RmaxAbs; auto.
apply (RoundedModeMultLess b radix) with (P := P) (r := (p + q)%R); auto.
replace (radix * FtoR radix q)%R with (q + q)%R;
 [ auto with real | unfold radix at 1; fold FtoRradix; ring ].
unfold FtoRradix in |- *;
 apply (RoundedModeMult b radix) with (P := P) (r := (p + q)%R); 
 auto.
replace (radix * FtoR radix p)%R with (p + p)%R;
 [ auto with real zarith | unfold radix at 1; fold FtoRradix; ring].
Qed.
 
Theorem plusErrorBound2 :
 forall p q r : float,
 Fbounded b p ->
 Fbounded b q ->
 Closest b radix (p + q) r ->
 ~ is_Fzero r ->
 (Rabs (r - (p + q)) < radix * / pPred (vNum b) * Rmax (Rabs p) (Rabs q))%R.
intros p q r H' H'0 H'1 H'2.
apply
 Rlt_le_trans
  with (Rabs (FtoR radix r) * / radix * (radix * / pPred (vNum b)))%R; 
 auto.
unfold FtoRradix in |- *; apply plusErrorBound1 with (precision := precision);
 auto with arith.
replace (Rabs (FtoR radix r) * / radix * (radix * / pPred (vNum b)))%R with
 (radix * / pPred (vNum b) * (Rabs r * / radix))%R; 
 [ idtac | fold FtoRradix; ring ].
apply Rmult_le_compat_l; auto.
replace 0%R with (radix * 0)%R; [ apply Rmult_le_compat_l | ring ].
cut (0 <= radix)%Z; auto with real zarith.
apply Rlt_le; apply Rinv_0_lt_compat; cut (0 < pPred (vNum b))%Z;
 auto with real zarith.
unfold pPred in |- *; apply Zlt_succ_pred; simpl in |- *.
apply vNumbMoreThanOne with (radix := radix) (precision := precision);
 auto with real arith.
apply Rmult_le_reg_l with (r := IZR radix); auto with real.
rewrite (Rmult_comm (Rabs r)); rewrite <- Rmult_assoc; rewrite Rinv_r;
 auto with real zarith; rewrite Rmult_1_l.
apply plusUpperBound with (P := Closest b radix); auto.
apply ClosestRoundedModeP with (precision := precision); auto.
Qed.
 
Theorem plusClosestLowerBoundAux1 :
 forall p q pq : float,
 (Rabs q <= p)%R ->
 Closest b radix (p + q) pq ->
 Fbounded b p -> Fbounded b q -> pq <> (p + q)%R :>R -> (/ radix * p <= pq)%R.
intros p q pq H' H'0 H'1 H'2 H'3.
cut (0 <= p)%R;
 [ intros Rl0; Casec Rl0; intros H0
 | apply Rle_trans with (2 := H'); auto with real ].
apply (FmultRadixInv b radix precision) with (5 := H'0); auto.
case (Rle_or_lt 0 q); intros Rl0.
apply Rlt_le_trans with (FtoRradix p); auto.
apply Rlt_RinvDouble; auto.
pattern (FtoRradix p) at 1 in |- *; replace (FtoRradix p) with (p + 0)%R;
 auto with real.
apply Rmult_lt_reg_l with (r := IZR radix); auto with real.
rewrite <- Rmult_assoc; rewrite Rinv_r.
rewrite Rmult_plus_distr_l.
rewrite Rmult_1_l.
apply Rplus_lt_reg_l with (r := (- (radix * p))%R).
replace (- (radix * p) + FtoR radix p)%R with (- p)%R;
 [ idtac | unfold radix at 1; unfold FtoRradix; ring].
replace (- (radix * p) + (radix * p + radix * q))%R with (radix * q)%R;
 [ idtac | simpl in |- *; ring ].
rewrite <- (Ropp_involutive (radix * q)); apply Ropp_lt_contravar.
replace (- (radix * q))%R with (radix * - q)%R; [ idtac | ring ].
case (Rle_or_lt (FtoRradix p) (radix * - q)); auto.
intros H'4; Contradict H'3.
rewrite <- (Fplus_correct radix); auto.
unfold FtoRradix in |- *; apply sym_eq; apply ClosestIdem with (b := b); auto.
replace (Fplus radix p q) with (Fminus radix p (Fopp q)).
rewrite <- Fopp_Fminus.
apply oppBounded; auto.
apply Sterbenz; auto.
apply oppBounded; auto.
apply Rmult_le_reg_l with (r := IZR radix); auto with real.
rewrite <- Rmult_assoc; rewrite Rinv_r.
rewrite Rmult_1_l; rewrite Fopp_correct; auto.
replace 0%R with (INR 0); auto with real arith.
apply Rle_trans with (FtoR 2%nat p); auto with real.
rewrite Fopp_correct; auto.
rewrite <- Faux.Rabsolu_left1; auto.
apply Rlt_le; auto.
unfold Fminus in |- *; rewrite Fopp_Fopp; auto.
apply
 (ClosestCompatible b radix (p + q)%R (FtoR radix (Fplus radix p q)) pq pq);
 auto.
apply sym_eq; unfold FtoRradix in |- *; apply Fplus_correct; auto.
apply
 RoundedModeBounded
  with (radix := radix) (P := Closest b radix) (r := (p + q)%R); 
 auto.
apply ClosestRoundedModeP with (precision := precision); auto.
replace 0%R with (INR 0); auto with real arith.
rewrite <- H0; rewrite Rmult_0_r.
replace (FtoRradix pq) with (FtoRradix p); auto.
rewrite <- H0; auto with real.
unfold FtoRradix in |- *; apply ClosestIdem with (b := b); auto.
apply (ClosestCompatible b radix (p + q)%R (FtoR radix p) pq pq); auto.
replace (FtoR 2%nat p) with (FtoRradix p); auto.
fold FtoRradix; rewrite <- H0; replace (FtoRradix q) with 0%R; try ring.
generalize H'; unfold Rabs in |- *; case (Rcase_abs q); auto.
intros H'4 H'5; Contradict H'5; rewrite <- H0; auto with real.
apply Rlt_not_le; auto with real.
intros H'4 H'5; apply Rle_antisym; auto with real.
rewrite H0; auto.
apply
 RoundedModeBounded
  with (radix := radix) (P := Closest b radix) (r := (p + q)%R); 
 auto.
apply ClosestRoundedModeP with (precision := precision); auto.
Qed.
 
Theorem plusClosestLowerBoundAux2 :
 forall p q pq : float,
 Closest b radix (p + q) pq ->
 Fbounded b p ->
 Fbounded b q ->
 pq <> (p + q)%R :>R ->
 (Rabs p <= Rabs q)%R -> (/ radix * Rabs q <= Rabs pq)%R.
intros p q pq H' H'0 H'1 H'2 H'3.
cut (Fbounded b pq);
 [ intros Fb0
 | apply
    RoundedModeBounded
     with (radix := radix) (P := Closest b radix) (r := (p + q)%R); 
    auto; apply ClosestRoundedModeP with (precision := precision) ]; 
 auto.
case (Rle_or_lt 0 q); intros Rl2;
 [ idtac | cut (q <= 0)%R; [ intros Rl2' | apply Rlt_le; auto ] ].
repeat rewrite Rabs_right; auto with real.
apply plusClosestLowerBoundAux1 with (q := p); auto.
rewrite <- (Rabs_right q); auto with real.
apply (ClosestCompatible b radix (p + q)%R (q + p)%R pq pq); auto; try ring.
rewrite Rplus_comm; auto with real.
apply Rle_ge; unfold FtoRradix in |- *;
 apply
  RleRoundedR0
   with
     (b := b)
     (precision := precision)
     (P := Closest b radix)
     (r := (p + q)%R); auto.
apply ClosestRoundedModeP with (precision := precision); auto.
case (Rle_or_lt 0 p); intros Rl3;
 [ idtac | cut (p <= 0)%R; [ intros Rl3' | apply Rlt_le; auto ] ];
 auto with real.
replace 0%R with (0 + 0)%R; auto with real.
apply Rplus_le_reg_l with (r := (- p)%R).
replace (- p + 0)%R with (- p)%R; [ idtac | ring ].
replace (- p + (p + q))%R with (FtoRradix q); [ idtac | ring ].
rewrite <- (Faux.Rabsolu_left1 (FtoRradix p)); auto with real.
rewrite <- (Rabs_right (FtoRradix q)); auto with real.
repeat rewrite Faux.Rabsolu_left1; auto with real.
unfold FtoRradix in |- *; repeat rewrite <- (Fopp_correct 2%nat); auto.
apply plusClosestLowerBoundAux1 with (q := Fopp p); auto.
unfold FtoRradix in |- *; repeat rewrite Fopp_correct; rewrite Rabs_Ropp;
 rewrite <- Faux.Rabsolu_left1; auto with real.
apply
 (ClosestCompatible b radix (- (p + q))%R (Fopp q + Fopp p)%R (
    Fopp pq) (Fopp pq)); auto.
apply ClosestOpp; auto.
unfold FtoRradix in |- *; repeat rewrite Fopp_correct; ring.
apply oppBounded; auto.
apply oppBounded; auto.
apply oppBounded; auto.
unfold FtoRradix in |- *; repeat rewrite Fopp_correct; Contradict H'2.
unfold FtoRradix in |- *; rewrite <- (Ropp_involutive (FtoR radix pq));
 rewrite H'2; ring.
unfold FtoRradix in |- *;
 apply
  RleRoundedLessR0
   with
     (b := b)
     (precision := precision)
     (P := Closest b radix)
     (r := (p + q)%R); auto.
apply ClosestRoundedModeP with (precision := precision); auto.
case (Rle_or_lt 0 p); intros Rl3;
 [ idtac | cut (p <= 0)%R; [ intros Rl3' | apply Rlt_le; auto ] ];
 auto with real.
apply Rplus_le_reg_l with (r := (- q)%R).
replace (- q + 0)%R with (- q)%R; [ idtac | ring ].
replace (- q + (p + q))%R with (FtoRradix p); [ idtac | ring ].
rewrite <- (Rabs_right (FtoRradix p)); auto with real.
rewrite <- (Rabs_left (FtoRradix q)); auto with real.
replace 0%R with (0 + 0)%R; auto with real.
Qed.
 
Theorem plusClosestLowerBound :
 forall p q pq : float,
 Closest b radix (p + q) pq ->
 Fbounded b p ->
 Fbounded b q ->
 pq <> (p + q)%R :>R -> (/ radix * Rmax (Rabs p) (Rabs q) <= Rabs pq)%R.
intros p q pq H' H'0 H'1 H'2.
cut (Fbounded b pq);
 [ intros Fb0
 | apply
    RoundedModeBounded
     with (radix := radix) (P := Closest b radix) (r := (p + q)%R); 
    auto; apply ClosestRoundedModeP with (precision := precision) ]; 
 auto.
unfold Rmax in |- *.
case (Rle_dec (Rabs p) (Rabs q)); intros Rl1.
apply plusClosestLowerBoundAux2 with (p := p); auto.
apply plusClosestLowerBoundAux2 with (p := q); auto.
apply (ClosestCompatible b radix (p + q)%R (q + p)%R pq pq); auto; try ring.
rewrite Rplus_comm; auto.
case (Rle_or_lt (Rabs q) (Rabs p)); auto; intros H'3; Contradict Rl1;
 apply Rlt_le; auto.
Qed.
End F2.