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author | 2016-11-03 14:23:41 -0400 | |
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committer | 2016-11-03 14:24:03 -0400 | |
commit | e094a3f049444dd4db93b0862b2b9ff847677aa8 (patch) | |
tree | edf6e5660bd393bdbd0d1cc760b87422f84f4cec /src/ModularArithmetic/ModularBaseSystemProofs.v | |
parent | 690d35a9f418fd3cf03ec7c746e201ebaa2fc1a0 (diff) |
Make [freeze] proofs consider machine integer width and hard input bounds separately
Diffstat (limited to 'src/ModularArithmetic/ModularBaseSystemProofs.v')
-rw-r--r-- | src/ModularArithmetic/ModularBaseSystemProofs.v | 32 |
1 files changed, 20 insertions, 12 deletions
diff --git a/src/ModularArithmetic/ModularBaseSystemProofs.v b/src/ModularArithmetic/ModularBaseSystemProofs.v index 59a6f33db..197a0dca6 100644 --- a/src/ModularArithmetic/ModularBaseSystemProofs.v +++ b/src/ModularArithmetic/ModularBaseSystemProofs.v @@ -521,16 +521,18 @@ End CarryProofs. Hint Rewrite @length_carry_and_reduce @length_carry : distr_length. -Class FreezePreconditions `{prm : PseudoMersenneBaseParams} B := +Class FreezePreconditions `{prm : PseudoMersenneBaseParams} B int_width := { lt_1_length_limb_widths : (1 < length limb_widths)%nat; + int_width_pos : 0 < int_width; + B_le_int_width : B <= int_width; B_pos : 0 < B; B_compat : forall w, In w limb_widths -> w < B; (* on the first reduce step, we add at most one bit of width to the first digit *) c_reduce1 : c * ((2 ^ B) >> nth_default 0 limb_widths (pred (length limb_widths))) <= 2 ^ (nth_default 0 limb_widths 0); (* on the second reduce step, we add at most one bit of width to the first digit, and leave room to carry c one more time after the highest bit is carried *) - c_reduce2 : c <= 2 ^ (nth_default 0 limb_widths 0) - c + c_reduce2 : c <= 2 ^ (nth_default 0 limb_widths 0) - c; }. Section CanonicalizationProofs. @@ -940,8 +942,14 @@ Section CanonicalizationProofs. congruence. Qed. + Lemma int_width_compat : forall x, In x limb_widths -> x < int_width. + Proof. + intros. apply B_compat in H. + eapply Z.lt_le_trans; eauto using B_le_int_width. + Qed. + Lemma minimal_rep_freeze : forall u, initial_bounds u -> - minimal_rep (freeze B u). + minimal_rep (freeze int_width u). Proof. repeat match goal with | |- _ => progress (cbv [freeze ModularBaseSystemList.freeze]) @@ -952,12 +960,12 @@ Section CanonicalizationProofs. | |- _ => apply conditional_subtract_lt_modulus | |- _ => apply conditional_subtract_modulus_preserves_bounded | |- bounded _ (carry_full _) => apply bounded_iff - | |- _ => solve [auto using Z.lt_le_incl, B_pos, B_compat, lt_1_length_limb_widths, length_carry_full, length_to_list] + | |- _ => solve [auto using Z.lt_le_incl, int_width_pos, int_width_compat, lt_1_length_limb_widths, length_carry_full, length_to_list] end. Qed. Lemma freeze_decode : forall u, - BaseSystem.decode base (to_list _ (freeze B u)) mod modulus = + BaseSystem.decode base (to_list _ (freeze int_width u)) mod modulus = BaseSystem.decode base (to_list _ u) mod modulus. Proof. repeat match goal with @@ -967,7 +975,7 @@ Section CanonicalizationProofs. | |- _ => rewrite Z.mod_add by (pose proof prime_modulus; prime_bound) | |- _ => rewrite to_list_from_list | |- _ => rewrite conditional_subtract_modulus_spec by - auto using Z.lt_le_incl, B_pos, B_compat, lt_1_length_limb_widths, length_carry_full, length_to_list, ge_modulus_01 + (auto using Z.lt_le_incl, int_width_pos, int_width_compat, lt_1_length_limb_widths, length_carry_full, length_to_list, ge_modulus_01) end. rewrite !decode_mod_Fdecode by auto using length_carry_full, length_to_list. cbv [carry_full]. @@ -986,7 +994,7 @@ Section CanonicalizationProofs. rewrite from_list_to_list; reflexivity. Qed. - Lemma freeze_rep : forall u x, rep u x -> rep (freeze B u) x. + Lemma freeze_rep : forall u x, rep u x -> rep (freeze int_width u) x. Proof. cbv [rep]; intros. apply F.eq_to_Z_iff. @@ -997,7 +1005,7 @@ Section CanonicalizationProofs. Lemma freeze_canonical : forall u v x y, rep u x -> rep v y -> initial_bounds u -> initial_bounds v -> - (x = y <-> fieldwise Logic.eq (freeze B u) (freeze B v)). + (x = y <-> fieldwise Logic.eq (freeze int_width u) (freeze int_width v)). Proof. intros; apply bounded_canonical; auto using freeze_rep, minimal_rep_freeze. Qed. @@ -1022,7 +1030,7 @@ Section SquareRootProofs. Lemma eqb_true_iff : forall u v x y, bounded_by u freeze_input_bounds -> bounded_by v freeze_input_bounds -> - u ~= x -> v ~= y -> (x = y <-> eqb B u v = true). + u ~= x -> v ~= y -> (x = y <-> eqb int_width u v = true). Proof. cbv [eqb freeze_input_bounds]. intros. rewrite fieldwiseb_fieldwise by (apply Z.eqb_eq). @@ -1031,10 +1039,10 @@ Section SquareRootProofs. Lemma eqb_false_iff : forall u v x y, bounded_by u freeze_input_bounds -> bounded_by v freeze_input_bounds -> - u ~= x -> v ~= y -> (x <> y <-> eqb B u v = false). + u ~= x -> v ~= y -> (x <> y <-> eqb int_width u v = false). Proof. intros. - case_eq (eqb B u v). + case_eq (eqb int_width u v). + rewrite <-eqb_true_iff by eassumption; split; intros; congruence || contradiction. + split; intros; auto. @@ -1069,7 +1077,7 @@ Section SquareRootProofs. bounded_by powx_squared freeze_input_bounds -> ModularBaseSystem.eq powx (pow u chain) -> ModularBaseSystem.eq powx_squared (mul powx powx) -> - (sqrt_5mod8 B powx powx_squared chain chain_correct sqrt_m1 u) ~= F.sqrt_5mod8 (decode sqrt_m1) x. + (sqrt_5mod8 int_width powx powx_squared chain chain_correct sqrt_m1 u) ~= F.sqrt_5mod8 (decode sqrt_m1) x. Proof. cbv [sqrt_5mod8 F.sqrt_5mod8]. intros. |