diff options
Diffstat (limited to 'src/BaseSystemProofs.v')
| -rw-r--r-- | src/BaseSystemProofs.v | 144 |
1 files changed, 82 insertions, 62 deletions
diff --git a/src/BaseSystemProofs.v b/src/BaseSystemProofs.v index 1c2fe0fbe..409d8b7db 100644 --- a/src/BaseSystemProofs.v +++ b/src/BaseSystemProofs.v @@ -15,14 +15,14 @@ Section BaseSystemProofs. Context `(base_vector : BaseVector). Lemma decode'_truncate : forall bs us, decode' bs us = decode' bs (firstn (length bs) us). - Proof. + Proof using Type. unfold decode'; intros; f_equal; apply combine_truncate_l. Qed. Lemma decode'_splice : forall xs ys bs, decode' bs (xs ++ ys) = decode' (firstn (length xs) bs) xs + decode' (skipn (length xs) bs) ys. - Proof. + Proof using Type. unfold decode'. induction xs; destruct ys, bs; boring. + rewrite combine_truncate_r. @@ -32,17 +32,19 @@ Section BaseSystemProofs. Qed. Lemma add_rep : forall bs us vs, decode' bs (add us vs) = decode' bs us + decode' bs vs. - Proof. + Proof using Type. unfold decode', accumulate; induction bs; destruct us, vs; boring; ring. Qed. Lemma decode_nil : forall bs, decode' bs nil = 0. + Proof using Type. + auto. Qed. Hint Rewrite decode_nil. Lemma decode_base_nil : forall us, decode' nil us = 0. - Proof. + Proof using Type. intros; rewrite decode'_truncate; auto. Qed. @@ -50,26 +52,26 @@ Section BaseSystemProofs. Lemma mul_each_rep : forall bs u vs, decode' bs (mul_each u vs) = u * decode' bs vs. - Proof. + Proof using Type. unfold decode', accumulate; induction bs; destruct vs; boring; ring. Qed. Lemma base_eq_1cons: base = 1 :: skipn 1 base. - Proof. + Proof using Type*. pose proof (b0_1 0) as H. destruct base; compute in H; try discriminate; boring. Qed. Lemma decode'_cons : forall x1 x2 xs1 xs2, decode' (x1 :: xs1) (x2 :: xs2) = x1 * x2 + decode' xs1 xs2. - Proof. + Proof using Type. unfold decode', accumulate; boring; ring. Qed. Hint Rewrite decode'_cons. Lemma decode_cons : forall x us, decode base (x :: us) = x + decode base (0 :: us). - Proof. + Proof using Type*. unfold decode; intros. rewrite base_eq_1cons. autorewrite with core; ring_simplify; auto. @@ -78,7 +80,7 @@ Section BaseSystemProofs. Lemma decode'_map_mul : forall v xs bs, decode' (map (Z.mul v) bs) xs = Z.mul v (decode' bs xs). - Proof. + Proof using Type. unfold decode'. induction xs; destruct bs; boring. unfold accumulate; simpl; nia. @@ -87,18 +89,18 @@ Section BaseSystemProofs. Lemma decode_map_mul : forall v xs, decode (map (Z.mul v) base) xs = Z.mul v (decode base xs). - Proof. + Proof using Type. unfold decode; intros; apply decode'_map_mul. Qed. Lemma sub_rep : forall bs us vs, decode' bs (sub us vs) = decode' bs us - decode' bs vs. - Proof. + Proof using Type. induction bs; destruct us; destruct vs; boring; ring. Qed. Lemma nth_default_base_nonzero : forall d, d <> 0 -> forall i, nth_default d base i <> 0. - Proof. + Proof using Type*. intros. rewrite nth_default_eq. destruct (nth_in_or_default i base d). @@ -108,7 +110,7 @@ Section BaseSystemProofs. Lemma nth_default_base_pos : forall d, 0 < d -> forall i, 0 < nth_default d base i. - Proof. + Proof using Type*. intros. rewrite nth_default_eq. destruct (nth_in_or_default i base d). @@ -118,7 +120,7 @@ Section BaseSystemProofs. Lemma mul_each_base : forall us bs c, decode' bs (mul_each c us) = decode' (mul_each c bs) us. - Proof. + Proof using Type. induction us; destruct bs; boring; ring. Qed. @@ -128,14 +130,14 @@ Section BaseSystemProofs. Lemma base_app : forall us low high, decode' (low ++ high) us = decode' low (firstn (length low) us) + decode' high (skipn (length low) us). - Proof. + Proof using Type. induction us; destruct low; boring. Qed. Lemma base_mul_app : forall low c us, decode' (low ++ mul_each c low) us = decode' low (firstn (length low) us) + c * decode' low (skipn (length low) us). - Proof. + Proof using Type. intros. rewrite base_app; f_equal. rewrite <- mul_each_rep. @@ -144,27 +146,31 @@ Section BaseSystemProofs. Qed. Lemma zeros_rep : forall bs n, decode' bs (zeros n) = 0. + Proof using Type. + induction bs; destruct n; boring. Qed. Lemma length_zeros : forall n, length (zeros n) = n. + Proof using Type. + induction n; boring. Qed. Hint Rewrite length_zeros. Lemma app_zeros_zeros : forall n m, zeros n ++ zeros m = zeros (n + m)%nat. - Proof. + Proof using Type. induction n; boring. Qed. Hint Rewrite app_zeros_zeros. Lemma zeros_app0 : forall m, zeros m ++ 0 :: nil = zeros (S m). - Proof. + Proof using Type. induction m; boring. Qed. Hint Rewrite zeros_app0. Lemma nth_default_zeros : forall n i, nth_default 0 (BaseSystem.zeros n) i = 0. - Proof. + Proof using Type. induction n; intros; [ cbv [BaseSystem.zeros]; apply nth_default_nil | ]. rewrite <-zeros_app0, nth_default_app. rewrite length_zeros. @@ -176,7 +182,7 @@ Section BaseSystemProofs. Qed. Lemma rev_zeros : forall n, rev (zeros n) = zeros n. - Proof. + Proof using Type. induction n; boring. Qed. Hint Rewrite rev_zeros. @@ -185,19 +191,19 @@ Section BaseSystemProofs. Lemma decode_single : forall n bs x, decode' bs (zeros n ++ x :: nil) = nth_default 0 bs n * x. - Proof. + Proof using Type. induction n; destruct bs; boring. Qed. Hint Rewrite decode_single. Lemma peel_decode : forall xs ys x y, decode' (x::xs) (y::ys) = x*y + decode' xs ys. - Proof. + Proof using Type. boring. Qed. Hint Rewrite zeros_rep peel_decode. Lemma decode_Proper : Proper (Logic.eq ==> (Forall2 Logic.eq) ==> Logic.eq) decode'. - Proof. + Proof using Type. repeat intro; subst. revert y y0 H0; induction x0; intros. + inversion H0. rewrite !decode_nil. @@ -210,18 +216,20 @@ Section BaseSystemProofs. Qed. Lemma decode_highzeros : forall xs bs n, decode' bs (xs ++ zeros n) = decode' bs xs. - Proof. + Proof using Type. induction xs; destruct bs; boring. Qed. Lemma mul_bi'_zeros : forall n m, mul_bi' base n (zeros m) = zeros m. + Proof using Type. + induction m; boring. Qed. Hint Rewrite mul_bi'_zeros. Lemma nth_error_base_nonzero : forall n x, nth_error base n = Some x -> x <> 0. - Proof. + Proof using Type*. eauto using (@nth_error_value_In Z), Z.gt0_neq0, base_positive. Qed. @@ -230,7 +238,7 @@ Section BaseSystemProofs. Lemma mul_bi_single : forall m n x, (n + m < length base)%nat -> decode base (mul_bi base n (zeros m ++ x :: nil)) = nth_default 0 base m * x * nth_default 0 base n. - Proof. + Proof using Type*. unfold mul_bi, decode. destruct m; simpl; simpl_list; simpl; intros. { pose proof nth_error_base_nonzero as nth_nonzero. @@ -268,12 +276,14 @@ Section BaseSystemProofs. Qed. Lemma set_higher' : forall vs x, vs++x::nil = vs .+ (zeros (length vs) ++ x :: nil). + Proof using Type. + induction vs; boring; f_equal; ring. Qed. Lemma set_higher : forall bs vs x, decode' bs (vs++x::nil) = decode' bs vs + nth_default 0 bs (length vs) * x. - Proof. + Proof using Type. intros. rewrite set_higher'. rewrite add_rep. @@ -282,41 +292,49 @@ Section BaseSystemProofs. Qed. Lemma zeros_plus_zeros : forall n, zeros n = zeros n .+ zeros n. + Proof using Type. + induction n; auto. simpl; f_equal; auto. Qed. Lemma mul_bi'_n_nil : forall n, mul_bi' base n nil = nil. - Proof. + Proof using Type. unfold mul_bi; auto. Qed. Hint Rewrite mul_bi'_n_nil. Lemma add_nil_l : forall us, nil .+ us = us. + Proof using Type. + induction us; auto. Qed. Hint Rewrite add_nil_l. Lemma add_nil_r : forall us, us .+ nil = us. + Proof using Type. + induction us; auto. Qed. Hint Rewrite add_nil_r. Lemma add_first_terms : forall us vs a b, (a :: us) .+ (b :: vs) = (a + b) :: (us .+ vs). + Proof using Type. + auto. Qed. Hint Rewrite add_first_terms. Lemma mul_bi'_cons : forall n x us, mul_bi' base n (x :: us) = x * crosscoef base n (length us) :: mul_bi' base n us. - Proof. + Proof using Type. unfold mul_bi'; auto. Qed. Lemma add_same_length : forall us vs l, (length us = l) -> (length vs = l) -> length (us .+ vs) = l. - Proof. + Proof using Type. induction us, vs; boring. erewrite (IHus vs (pred l)); boring. Qed. @@ -327,7 +345,7 @@ Section BaseSystemProofs. Lemma add_snoc_same_length : forall l us vs a b, (length us = l) -> (length vs = l) -> (us ++ a :: nil) .+ (vs ++ b :: nil) = (us .+ vs) ++ (a + b) :: nil. - Proof. + Proof using Type. induction l, us, vs; boring; discriminate. Qed. @@ -336,7 +354,7 @@ Section BaseSystemProofs. (Hlvs: length vs = l), mul_bi' base n (rev (us .+ vs)) = mul_bi' base n (rev us) .+ mul_bi' base n (rev vs). - Proof. + Proof using Type. (* TODO(adamc): please help prettify this *) induction us using rev_ind; try solve [destruct vs; boring; congruence]. @@ -360,18 +378,20 @@ Section BaseSystemProofs. Qed. Lemma zeros_cons0 : forall n, 0 :: zeros n = zeros (S n). + Proof using Type. + auto. Qed. Lemma add_leading_zeros : forall n us vs, (zeros n ++ us) .+ (zeros n ++ vs) = zeros n ++ (us .+ vs). - Proof. + Proof using Type. induction n; boring. Qed. Lemma rev_add_rev : forall us vs l, (length us = l) -> (length vs = l) -> (rev us) .+ (rev vs) = rev (us .+ vs). - Proof. + Proof using Type. induction us, vs; boring; try solve [subst; discriminate]. rewrite (add_snoc_same_length (pred l) _ _ _ _) by (subst; simpl_list; omega). rewrite (IHus vs (pred l)) by omega; auto. @@ -379,13 +399,13 @@ Section BaseSystemProofs. Hint Rewrite rev_add_rev. Lemma mul_bi'_length : forall us n, length (mul_bi' base n us) = length us. - Proof. + Proof using Type. induction us, n; boring. Qed. Hint Rewrite mul_bi'_length. Lemma add_comm : forall us vs, us .+ vs = vs .+ us. - Proof. + Proof using Type. induction us, vs; boring; f_equal; auto. Qed. @@ -394,7 +414,7 @@ Section BaseSystemProofs. Lemma mul_bi_add_same_length : forall n us vs l, (length us = l) -> (length vs = l) -> mul_bi base n (us .+ vs) = mul_bi base n us .+ mul_bi base n vs. - Proof. + Proof using Type. unfold mul_bi; boring. rewrite add_leading_zeros. erewrite mul_bi'_add; boring. @@ -402,7 +422,7 @@ Section BaseSystemProofs. Qed. Lemma add_zeros_same_length : forall us, us .+ (zeros (length us)) = us. - Proof. + Proof using Type. induction us; boring; f_equal; omega. Qed. @@ -411,13 +431,13 @@ Section BaseSystemProofs. Lemma add_trailing_zeros : forall us vs, (length us >= length vs)%nat -> us .+ vs = us .+ (vs ++ (zeros (length us - length vs)%nat)). - Proof. + Proof using Type. induction us, vs; boring; f_equal; boring. Qed. Lemma length_add_ge : forall us vs, (length us >= length vs)%nat -> (length (us .+ vs) <= length us)%nat. - Proof. + Proof using Type. intros. rewrite add_trailing_zeros by trivial. erewrite add_same_length by (pose proof app_length; boring); omega. @@ -425,25 +445,25 @@ Section BaseSystemProofs. Lemma add_length_le_max : forall us vs, (length (us .+ vs) <= max (length us) (length vs))%nat. - Proof. + Proof using Type. intros; case_max; (rewrite add_comm; apply length_add_ge; omega) || (apply length_add_ge; omega) . Qed. Lemma sub_nil_length: forall us : digits, length (sub nil us) = length us. - Proof. + Proof using Type. induction us; boring. Qed. Lemma sub_length : forall us vs, (length (sub us vs) = max (length us) (length vs))%nat. - Proof. + Proof using Type. induction us, vs; boring. rewrite sub_nil_length; auto. Qed. Lemma mul_bi_length : forall us n, length (mul_bi base n us) = (length us + n)%nat. - Proof. + Proof using Type. pose proof mul_bi'_length; unfold mul_bi. destruct us; repeat progress (simpl_list; boring). Qed. @@ -451,7 +471,7 @@ Section BaseSystemProofs. Lemma mul_bi_trailing_zeros : forall m n us, mul_bi base n us ++ zeros m = mul_bi base n (us ++ zeros m). - Proof. + Proof using Type. unfold mul_bi. induction m; intros; try solve [boring]. rewrite <- zeros_app0. @@ -462,7 +482,7 @@ Section BaseSystemProofs. Lemma mul_bi_add_longer : forall n us vs, (length us >= length vs)%nat -> mul_bi base n (us .+ vs) = mul_bi base n us .+ mul_bi base n vs. - Proof. + Proof using Type. boring. rewrite add_trailing_zeros by auto. rewrite (add_trailing_zeros (mul_bi base n us) (mul_bi base n vs)) @@ -475,7 +495,7 @@ Section BaseSystemProofs. Lemma mul_bi_add : forall n us vs, mul_bi base n (us .+ vs) = (mul_bi base n us) .+ (mul_bi base n vs). - Proof. + Proof using Type. intros; pose proof mul_bi_add_longer. destruct (le_ge_dec (length us) (length vs)). { rewrite add_comm. @@ -489,7 +509,7 @@ Section BaseSystemProofs. Lemma mul_bi_rep : forall i vs, (i + length vs < length base)%nat -> decode base (mul_bi base i vs) = decode base vs * nth_default 0 base i. - Proof. + Proof using Type*. unfold decode. induction vs using rev_ind; intros; try solve [unfold mul_bi; boring]. assert (i + length vs < length base)%nat by @@ -511,7 +531,7 @@ Section BaseSystemProofs. Lemma mul'_rep : forall us vs, (length us + length vs <= length base)%nat -> decode base (mul' (rev us) vs) = decode base us * decode base vs. - Proof. + Proof using Type*. unfold decode. induction us using rev_ind; boring. @@ -530,13 +550,13 @@ Section BaseSystemProofs. Lemma mul_rep : forall us vs, (length us + length vs <= length base)%nat -> decode base (mul us vs) = decode base us * decode base vs. - Proof. + Proof using Type*. exact mul'_rep. Qed. Lemma mul'_length: forall us vs, (length (mul' us vs) <= length us + length vs)%nat. - Proof. + Proof using Type. pose proof add_length_le_max. induction us; boring. unfold mul_each. @@ -545,7 +565,7 @@ Section BaseSystemProofs. Lemma mul_length: forall us vs, (length (mul us vs) <= length us + length vs)%nat. - Proof. + Proof using Type. intros; unfold BaseSystem.mul. rewrite mul'_length. rewrite rev_length; omega. @@ -553,7 +573,7 @@ Section BaseSystemProofs. Lemma add_length_exact : forall us vs, length (us .+ vs) = max (length us) (length vs). - Proof. + Proof using Type. induction us; destruct vs; boring. Qed. @@ -564,7 +584,7 @@ Section BaseSystemProofs. | 0 => 0%nat | _ => pred (length us + length vs) end)%nat. - Proof. + Proof using Type. induction us; intros; try solve [boring]. unfold BaseSystem.mul'; fold mul'. unfold mul_each. @@ -583,7 +603,7 @@ Section BaseSystemProofs. Lemma mul'_length_exact: forall us vs, (length us <= length vs)%nat -> us <> nil -> (length (mul' us vs) = pred (length us + length vs))%nat. - Proof. + Proof using Type. intros; rewrite mul'_length_exact_full; destruct us; simpl; congruence. Qed. @@ -592,7 +612,7 @@ Section BaseSystemProofs. | 0 => 0 | _ => pred (length us + length vs) end)%nat. - Proof. + Proof using Type. intros; unfold BaseSystem.mul; autorewrite with distr_length; reflexivity. Qed. @@ -603,7 +623,7 @@ Section BaseSystemProofs. Lemma mul_length_exact: forall us vs, (length us <= length vs)%nat -> us <> nil -> (length (mul us vs) = pred (length us + length vs))%nat. - Proof. + Proof using Type. intros; unfold BaseSystem.mul. rewrite mul'_length_exact; rewrite ?rev_length; try omega. intro rev_nil. @@ -618,7 +638,7 @@ Section BaseSystemProofs. Hint Resolve encode'_zero encode'_succ. Lemma encode'_length : forall z max i, length (encode' base z max i) = i. - Proof. + Proof using Type. induction i; auto. rewrite encode'_succ, app_length, IHi. cbv [length]. @@ -634,7 +654,7 @@ Section BaseSystemProofs. Lemma encode'_spec : forall z max, 0 < max -> base_max_succ_divide max -> forall i, (i <= length base)%nat -> decode' base (encode' base z max i) = z mod (nth_default max base i). - Proof. + Proof using Type*. induction i; intros. + rewrite encode'_zero, b0_1, Z.mod_1_r. apply decode_nil. @@ -653,7 +673,7 @@ Section BaseSystemProofs. Lemma encode_rep : forall z max, 0 <= z < max -> base_max_succ_divide max -> decode base (encode base z max) = z. - Proof. + Proof using Type*. unfold encode; intros. rewrite encode'_spec, nth_default_out_of_bounds by (omega || auto). apply Z.mod_small; omega. @@ -669,7 +689,7 @@ Section MultiBaseSystemProofs. Lemma decode_short_initial : forall (us : digits), (firstn (length us) base0 = firstn (length us) base1) -> decode base0 us = decode base1 us. - Proof. + Proof using Type. intros us H. unfold decode, decode'. rewrite (combine_truncate_r us base0), (combine_truncate_r us base1), H. @@ -681,7 +701,7 @@ Section MultiBaseSystemProofs. -> firstn (length us) base0 = firstn (length us) base1 -> firstn (length vs) base0 = firstn (length vs) base1 -> (decode base0 us) * (decode base0 vs) = decode base1 (mul base1 us vs). - Proof. + Proof using base_vector1. intros. rewrite mul_rep by trivial. apply f_equal2; apply decode_short_initial; assumption. |