diff options
Diffstat (limited to 'theories/Ints/num/GenLift.v')
| -rw-r--r-- | theories/Ints/num/GenLift.v | 62 |
1 files changed, 62 insertions, 0 deletions
diff --git a/theories/Ints/num/GenLift.v b/theories/Ints/num/GenLift.v index c1283aefb9..11455b8044 100644 --- a/theories/Ints/num/GenLift.v +++ b/theories/Ints/num/GenLift.v @@ -117,8 +117,10 @@ Section GenLift. | Gt => [[x]] > [[y]] end. Variable spec_ww_digits : ww_Digits = xO w_digits. + Variable spec_w_head00 : forall x, [|x|] = 0 -> [|w_head0 x|] = Zpos w_digits. Variable spec_w_head0 : forall x, 0 < [|x|] -> wB/ 2 <= 2 ^ ([|w_head0 x|]) * [|x|] < wB. + Variable spec_w_tail00 : forall x, [|x|] = 0 -> [|w_tail0 x|] = Zpos w_digits. Variable spec_w_tail0 : forall x, 0 < [|x|] -> exists y, 0 <= y /\ [|x|] = (2* y + 1) * (2 ^ [|w_tail0 x|]). Variable spec_w_add_mul_div : forall x y p, @@ -134,13 +136,42 @@ Section GenLift. Variable spec_zdigits : [| w_zdigits |] = Zpos w_digits. Variable spec_low: forall x, [| low x|] = [[x]] mod wB. + Variable spec_ww_zdigits : [[ww_zdigits]] = Zpos ww_Digits. Hint Resolve div_le_0 div_lt w_to_Z_wwB: lift. Ltac zarith := auto with zarith lift. + + Lemma spec_ww_head00 : forall x, [[x]] = 0 -> [[ww_head0 x]] = Zpos ww_Digits. + Proof. + intros x; case x; unfold ww_head0. + intros HH; rewrite spec_ww_zdigits; auto. + intros xh xl; simpl; intros Hx. + case (spec_to_Z xh); intros Hx1 Hx2. + case (spec_to_Z xl); intros Hy1 Hy2. + assert (F1: [|xh|] = 0). + case (Zle_lt_or_eq _ _ Hy1); auto; intros Hy3. + absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. + apply Zlt_le_trans with (1 := Hy3); auto with zarith. + pattern [|xl|] at 1; rewrite <- (Zplus_0_l [|xl|]). + apply Zplus_le_compat_r; auto with zarith. + case (Zle_lt_or_eq _ _ Hx1); auto; intros Hx3. + absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. + rewrite <- Hy3; rewrite Zplus_0_r; auto with zarith. + apply Zmult_lt_0_compat; auto with zarith. + generalize (spec_compare w_0 xh); case w_compare. + intros H; simpl. + rewrite spec_w_add; rewrite spec_w_head00. + rewrite spec_zdigits; rewrite spec_ww_digits. + rewrite Zpos_xO; auto with zarith. + rewrite F1 in Hx; auto with zarith. + rewrite spec_w_0; auto with zarith. + rewrite spec_w_0; auto with zarith. + Qed. Lemma spec_ww_head0 : forall x, 0 < [[x]] -> wwB/ 2 <= 2 ^ [[ww_head0 x]] * [[x]] < wwB. Proof. + clear spec_ww_zdigits. rewrite wwB_div_2;rewrite Zmult_comm;rewrite wwB_wBwB. assert (U:= lt_0_wB w_digits); destruct x as [ |xh xl];simpl ww_to_Z;intros H. unfold Zlt in H;discriminate H. @@ -182,9 +213,38 @@ Section GenLift. assert (H1 := spec_to_Z xh);zarith. Qed. + Lemma spec_ww_tail00 : forall x, [[x]] = 0 -> [[ww_tail0 x]] = Zpos ww_Digits. + Proof. + intros x; case x; unfold ww_tail0. + intros HH; rewrite spec_ww_zdigits; auto. + intros xh xl; simpl; intros Hx. + case (spec_to_Z xh); intros Hx1 Hx2. + case (spec_to_Z xl); intros Hy1 Hy2. + assert (F1: [|xh|] = 0). + case (Zle_lt_or_eq _ _ Hy1); auto; intros Hy3. + absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. + apply Zlt_le_trans with (1 := Hy3); auto with zarith. + pattern [|xl|] at 1; rewrite <- (Zplus_0_l [|xl|]). + apply Zplus_le_compat_r; auto with zarith. + case (Zle_lt_or_eq _ _ Hx1); auto; intros Hx3. + absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. + rewrite <- Hy3; rewrite Zplus_0_r; auto with zarith. + apply Zmult_lt_0_compat; auto with zarith. + assert (F2: [|xl|] = 0). + rewrite F1 in Hx; auto with zarith. + generalize (spec_compare w_0 xl); case w_compare. + intros H; simpl. + rewrite spec_w_add; rewrite spec_w_tail00; auto. + rewrite spec_zdigits; rewrite spec_ww_digits. + rewrite Zpos_xO; auto with zarith. + rewrite spec_w_0; auto with zarith. + rewrite spec_w_0; auto with zarith. + Qed. + Lemma spec_ww_tail0 : forall x, 0 < [[x]] -> exists y, 0 <= y /\ [[x]] = (2 * y + 1) * 2 ^ [[ww_tail0 x]]. Proof. + clear spec_ww_zdigits. destruct x as [ |xh xl];simpl ww_to_Z;intros H. unfold Zlt in H;discriminate H. assert (H0 := spec_compare w_0 xl);rewrite spec_w_0 in H0. @@ -251,6 +311,7 @@ Section GenLift. ([[WW xh xl]] * (2^[[p]]) + [[WW yh yl]] / (2^(Zpos (xO w_digits) - [[p]]))) mod wwB. Proof. + clear spec_ww_zdigits. intros xh xl yh yl p zdigits;assert (HwwB := wwB_pos w_digits). case (spec_to_w_Z p); intros Hv1 Hv2. replace (Zpos (xO w_digits)) with (Zpos w_digits + Zpos w_digits). @@ -374,6 +435,7 @@ Section GenLift. ([[x]] * (2^[[p]]) + [[y]] / (2^(Zpos (xO w_digits) - [[p]]))) mod wwB. Proof. + clear spec_ww_zdigits. intros x y p H. destruct x as [ |xh xl]; [assert (H1 := @spec_ww_add_mul_div_aux w_0 w_0) |