1 files changed, 3 insertions, 467 deletions

diff --git a/src/BaseSystem.v b/src/BaseSystem.v
index e9e4bc9d4..f3a1a0fdb 100644
--- a/src/BaseSystem.v
+++ b/src/BaseSystem.v
@@ -16,7 +16,7 @@ Class BaseVector (base : list Z):= {
 }.
 
 Section BaseSystem.
-  Context (base : list Z) (base_vector : BaseVector base).
+  Context (base : list Z).
   (** [BaseSystem] implements an constrained positional number system.
       A wide variety of bases are supported: the base coefficients are not
       required to be powers of 2, and it is NOT necessarily the case that
@@ -32,7 +32,6 @@ Section BaseSystem.
   Definition accumulate p acc := fst p * snd p + acc.
   Definition decode' bs u  := fold_right accumulate 0 (combine u bs).
   Definition decode := decode' base.
-  Hint Unfold accumulate.
   (* Does not carry; z becomes the lowest and only digit. *)
   Definition encode (z : Z) := z :: nil.
 
@@ -51,50 +50,7 @@ Section BaseSystem.
 
   Hint Extern 1 (@eq Z _ _) => ring.
 
-  Lemma add_rep : forall bs us vs, decode' bs (add us vs) = decode' bs us + decode' bs vs.
-  Proof.
-    unfold decode, decode'; induction bs; destruct us; destruct vs; boring.
-  Qed.
-
-  Lemma decode_nil : forall bs, decode' bs nil = 0.
-    auto.
-  Qed.
-  Hint Rewrite decode_nil.
-
-  Lemma decode_base_nil : forall us, decode' nil us = 0.
-  Proof.
-    intros; rewrite decode'_truncate; auto.
-  Qed.
-  Hint Rewrite decode_base_nil.
-
   Definition mul_each u := map (Z.mul u).
-  Lemma mul_each_rep : forall bs u vs,
-    decode' bs (mul_each u vs) = u * decode' bs vs.
-  Proof.
-    unfold decode'; induction bs; destruct vs; boring.
-  Qed.
-
-  Lemma base_eq_1cons: base = 1 :: skipn 1 base.
-  Proof.
-    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.
-    unfold decode'; boring.
-  Qed.
-  Hint Rewrite decode'_cons.
-
-  Lemma decode_cons : forall x us,
-    decode (x :: us) = x + decode (0 :: us).
-  Proof.
-    unfold decode; intros.
-    rewrite base_eq_1cons.
-    autorewrite with core; ring_simplify; auto.
-  Qed.
-
   Fixpoint sub (us vs:digits) : digits :=
     match us,vs with
       | u::us', v::vs' => u-v :: sub us' vs'
@@ -102,76 +58,13 @@ Section BaseSystem.
       | nil, v::vs' => (0-v)::sub nil vs'
     end.
 
-  Lemma sub_rep : forall bs us vs, decode' bs (sub us vs) = decode' bs us - decode' bs vs.
-  Proof.
-    induction bs; destruct us; destruct vs; boring.
-  Qed.
-
-  Lemma encode_rep : forall z, decode (encode z) = z.
-  Proof.
-    pose proof base_eq_1cons.
-    unfold decode, encode; destruct z; boring.
-  Qed.
-
-  Lemma mul_each_base : forall us bs c, 
-      decode' bs (mul_each c us) = decode' (mul_each c bs) us.
-  Proof.
-    induction us; destruct bs; boring.
-  Qed.
-
-  Hint Rewrite (@nth_default_nil Z).
-  Hint Rewrite (@firstn_nil Z).
-  Hint Rewrite (@skipn_nil Z).
-
-  Lemma base_app : forall us low high,
-      decode' (low ++ high) us = decode' low (firstn (length low) us) + decode' high (skipn (length low) us).
-  Proof.
-    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.
-    intros.
-    rewrite base_app; f_equal.
-    rewrite <- mul_each_rep.
-    rewrite mul_each_base.
-    reflexivity.
-  Qed.
-
   Definition crosscoef i j : Z := 
     let b := nth_default 0 base in
     (b(i) * b(j)) / b(i+j)%nat.
   Hint Unfold crosscoef.
 
   Fixpoint zeros n := match n with O => nil | S n' => 0::zeros n' end.
-  Lemma zeros_rep : forall bs n, decode' bs (zeros n) = 0.
-    induction bs; destruct n; boring.
-  Qed.
-  Lemma length_zeros : forall n, length (zeros n) = n.
-    induction n; boring.
-  Qed.
-  Hint Rewrite length_zeros.
-
-  Lemma app_zeros_zeros : forall n m, zeros n ++ zeros m = zeros (n + m).
-  Proof.
-    induction n; boring.
-  Qed.
-  Hint Rewrite app_zeros_zeros.
-
-  Lemma zeros_app0 : forall m, zeros m ++ 0 :: nil = zeros (S m).
-  Proof.
-    induction m; boring.
-  Qed.
-  Hint Rewrite zeros_app0.
-
-  Lemma rev_zeros : forall n, rev (zeros n) = zeros n.
-  Proof.
-    induction n; boring.
-  Qed.
-  Hint Rewrite rev_zeros.
-
+  
   (* mul' is multiplication with the SECOND ARGUMENT REVERSED and OUTPUT REVERSED *)
   Fixpoint mul_bi' (i:nat) (vsr:digits) := 
     match vsr with
@@ -180,317 +73,7 @@ Section BaseSystem.
     end.
   Definition mul_bi (i:nat) (vs:digits) : digits :=
     zeros i ++ rev (mul_bi' i (rev vs)).
-
-  Hint Unfold nth_default.
-
-  Lemma decode_single : forall n bs x,
-    decode' bs (zeros n ++ x :: nil) = nth_default 0 bs n * x.
-  Proof.
-    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.
-    boring.
-  Qed.
-  Hint Rewrite zeros_rep peel_decode.
-
-  Lemma decode_highzeros : forall xs bs n, decode' bs (xs ++ zeros n) = decode' bs xs.
-  Proof.
-    induction xs; destruct bs; boring.
-  Qed.
-
-  Lemma mul_bi'_zeros : forall n m, mul_bi' n (zeros m) = zeros m.
-    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.
-    eauto using (@nth_error_value_In Z), Zgt0_neq0, base_positive.
-  Qed.
-
-  Hint Rewrite plus_0_r.
-
-  Lemma mul_bi_single : forall m n x,
-    (n + m < length base)%nat ->
-    decode (mul_bi n (zeros m ++ x :: nil)) = nth_default 0 base m * x * nth_default 0 base n.
-  Proof.
-    unfold mul_bi, decode.
-    destruct m; simpl; simpl_list; simpl; intros. {
-      pose proof nth_error_base_nonzero as nth_nonzero.
-      case_eq base; [intros; boring | intros z l base_eq].
-      specialize (b0_1 0); intro b0_1'.
-      rewrite base_eq in *.
-      rewrite nth_default_cons in b0_1'.
-      rewrite b0_1' in *.
-      boring; nth_tac.
-      rewrite Z_div_mul'; eauto.
-      destruct x; ring.
-    } {
-      ssimpl_list.
-      autorewrite with core.
-      rewrite app_assoc.
-      autorewrite with core.
-      unfold crosscoef; simpl; ring_simplify.
-      rewrite Nat.add_1_r.
-      rewrite base_good by auto.
-      rewrite Z_div_mult by (apply base_positive; rewrite nth_default_eq; apply nth_In; auto).
-      rewrite <- Z.mul_assoc.
-      rewrite <- Z.mul_comm.
-      rewrite <- Z.mul_assoc.
-      rewrite <- Z.mul_assoc.
-      destruct (Z.eq_dec x 0); subst; try ring.
-      rewrite Z.mul_cancel_l by auto.
-      rewrite <- base_good by auto.
-      ring.
-      }
-  Qed.
-
-  Lemma set_higher' : forall vs x, vs++x::nil = vs .+ (zeros (length vs) ++ x :: nil).
-    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.
-    intros.
-    rewrite set_higher'.
-    rewrite add_rep.
-    f_equal.
-    apply decode_single.
-  Qed.
-
-  Lemma zeros_plus_zeros : forall n, zeros n = zeros n .+ zeros n.
-    induction n; auto.
-    simpl; f_equal; auto. 
-  Qed.
-
-  Lemma mul_bi'_n_nil : forall n, mul_bi' n nil = nil.
-  Proof.
-    unfold mul_bi; auto.
-  Qed.
-  Hint Rewrite mul_bi'_n_nil.
-
-  Lemma add_nil_l : forall us, nil .+ us = us.
-    induction us; auto.
-  Qed.
-  Hint Rewrite add_nil_l.
-
-  Lemma add_nil_r : forall us, us .+ nil = us.
-    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).
-    auto.
-  Qed.
-  Hint Rewrite add_first_terms.
- 
-  Lemma mul_bi'_cons : forall n x us,
-    mul_bi' n (x :: us) = x * crosscoef n (length us) :: mul_bi' n us.
-  Proof.
-    unfold mul_bi'; auto.
-  Qed.
-
-  Lemma add_same_length : forall us vs l, (length us = l) -> (length vs = l) ->
-    length (us .+ vs) = l.
-  Proof.
-    induction us, vs; boring.
-    erewrite (IHus vs (pred l)); boring.
-  Qed.
-
-  Hint Rewrite app_nil_l.
-  Hint Rewrite app_nil_r.
-
-  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.
-    induction l, us, vs; boring; discriminate.
-  Qed.
-
-  Lemma mul_bi'_add : forall us n vs l
-    (Hlus: length us = l)
-    (Hlvs: length vs = l),
-    mul_bi' n (rev (us .+ vs)) = 
-    mul_bi' n (rev us) .+ mul_bi' n (rev vs).
-  Proof.
-    (* TODO(adamc): please help prettify this *)
-    induction us using rev_ind;
-      try solve [destruct vs; boring; congruence].
-    destruct vs using rev_ind; boring; clear IHvs; simpl_list.
-    erewrite (add_snoc_same_length (pred l) us vs _ _); simpl_list.
-    repeat rewrite mul_bi'_cons; rewrite add_first_terms; simpl_list.
-    rewrite (IHus n vs (pred l)).
-    replace (length us) with (pred l).
-    replace (length vs) with (pred l).
-    rewrite (add_same_length us vs (pred l)).
-    f_equal; ring.
-
-    erewrite length_snoc; eauto.
-    erewrite length_snoc; eauto.
-    erewrite length_snoc; eauto.
-    erewrite length_snoc; eauto.
-    erewrite length_snoc; eauto.
-    erewrite length_snoc; eauto.
-    erewrite length_snoc; eauto.
-    erewrite length_snoc; eauto.
-   Qed.
-
-  Lemma zeros_cons0 : forall n, 0 :: zeros n = zeros (S n).
-    auto.
-  Qed.
-
-  Lemma add_leading_zeros : forall n us vs,
-    (zeros n ++ us) .+ (zeros n ++ vs) = zeros n ++ (us .+ vs).
-  Proof.
-    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.
-    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.
-  Qed.
-  Hint Rewrite rev_add_rev.
-
-  Lemma mul_bi'_length : forall us n, length (mul_bi' n us) = length us.
-  Proof.
-    induction us, n; boring.
-  Qed.
-  Hint Rewrite mul_bi'_length.
-
-  Lemma add_comm : forall us vs, us .+ vs = vs .+ us.
-  Proof.
-    induction us, vs; boring; f_equal; auto.
-  Qed.
-
-  Hint Rewrite rev_length.
-
-  Lemma mul_bi_add_same_length : forall n us vs l,
-    (length us = l) -> (length vs = l) ->
-    mul_bi n (us .+ vs) = mul_bi n us .+ mul_bi n vs.
-  Proof.
-    unfold mul_bi; boring.
-    rewrite add_leading_zeros.
-    erewrite mul_bi'_add; boring.
-    erewrite rev_add_rev; boring.
-  Qed.
-
-  Lemma add_zeros_same_length : forall us, us .+ (zeros (length us)) = us.
-  Proof.
-    induction us; boring; f_equal; omega.
-  Qed.
-
-  Hint Rewrite add_zeros_same_length.
-  Hint Rewrite minus_diag.
-
-  Lemma add_trailing_zeros : forall us vs, (length us >= length vs)%nat ->
-    us .+ vs = us .+ (vs ++ (zeros (length us - length vs))).
-  Proof.
-    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.
-    intros.
-    rewrite add_trailing_zeros by trivial.
-    erewrite add_same_length by (pose proof app_length; boring); omega.
-  Qed.
-
-  Lemma add_length_le_max : forall us vs,
-      (length (us .+ vs) <= max (length us) (length vs))%nat.
-  Proof.
-    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.
-     induction us; boring.
-  Qed.
-
-  Lemma sub_length_le_max : forall us vs,
-      (length (sub us vs) <= max (length us) (length vs))%nat.
-  Proof.
-    induction us, vs; boring.
-    rewrite sub_nil_length; auto.
-  Qed.
-
-  Lemma mul_bi_length : forall us n, length (mul_bi n us) = (length us + n)%nat.
-  Proof.
-    pose proof mul_bi'_length; unfold mul_bi.
-    destruct us; repeat progress (simpl_list; boring).
-  Qed.
-  Hint Rewrite mul_bi_length.
-
-  Lemma mul_bi_trailing_zeros : forall m n us,
-    mul_bi n us ++ zeros m = mul_bi n (us ++ zeros m).
-  Proof.
-    unfold mul_bi.
-    induction m; intros; try solve [boring].
-    rewrite <- zeros_app0.
-    rewrite app_assoc.
-    repeat progress (boring; rewrite rev_app_distr).
-  Qed.
-
-  Lemma mul_bi_add_longer : forall n us vs,
-    (length us >= length vs)%nat ->
-    mul_bi n (us .+ vs) = mul_bi n us .+ mul_bi n vs.
-  Proof.
-    boring.
-    rewrite add_trailing_zeros by auto.
-    rewrite (add_trailing_zeros (mul_bi n us) (mul_bi n vs))
-      by (repeat (rewrite mul_bi_length); omega).
-    erewrite mul_bi_add_same_length by
-      (eauto; simpl_list; rewrite length_zeros; omega).
-    rewrite mul_bi_trailing_zeros.
-    repeat (f_equal; boring).
-  Qed.
-
-  Lemma mul_bi_add : forall n us vs,
-    mul_bi n (us .+ vs) = (mul_bi n  us) .+ (mul_bi n vs).
-  Proof.
-    intros; pose proof mul_bi_add_longer.
-    destruct (le_ge_dec (length us) (length vs)). {
-      replace (mul_bi n us .+ mul_bi n vs)
-         with (mul_bi n vs .+ mul_bi n us)
-           by (apply add_comm).
-      replace (us .+ vs)
-         with (vs .+ us)
-           by (apply add_comm).
-      boring.
-    } {
-      boring.
-    }
-  Qed.
-
-  Lemma mul_bi_rep : forall i vs,
-    (i + length vs < length base)%nat ->
-    decode (mul_bi i vs) = decode vs * nth_default 0 base i.
-  Proof.
-    unfold decode.
-    induction vs using rev_ind; intros; try solve [unfold mul_bi; boring].
-    assert (i + length vs < length base)%nat by
-      (rewrite app_length in *; boring).
-
-    rewrite set_higher.
-    ring_simplify.
-    rewrite <- IHvs by auto; clear IHvs.
-    rewrite <- mul_bi_single by auto.
-    rewrite <- add_rep.
-    rewrite <- mul_bi_add.
-    rewrite set_higher'.
-    auto.
-  Qed.
-
   (* mul' is multiplication with the FIRST ARGUMENT REVERSED *)
   Fixpoint mul' (usr vs:digits) : digits :=
     match usr with
@@ -500,51 +83,6 @@ Section BaseSystem.
     end.
   Definition mul us := mul' (rev us).
 
-  Lemma mul'_rep : forall us vs,
-    (length us + length vs <= length base)%nat ->
-    decode (mul' (rev us) vs) = decode us * decode vs.
-  Proof.
-    unfold decode.
-    induction us using rev_ind; boring.
-
-    assert (length us + length vs < length base)%nat by
-      (rewrite app_length in *; boring).
-
-    ssimpl_list.
-    rewrite add_rep.
-    boring.
-    rewrite set_higher.
-    rewrite mul_each_rep.
-    rewrite mul_bi_rep by auto.
-    unfold decode; ring.
-  Qed.
-
-  Lemma mul_rep : forall us vs,
-    (length us + length vs <= length base)%nat ->
-    decode (mul us vs) = decode us * decode vs.
-  Proof.
-    exact mul'_rep.
-  Qed.
-
-  Lemma mul'_length: forall us vs,
-      (length (mul' us vs) <= length us + length vs)%nat.
-  Proof.
-    pose proof add_length_le_max.
-    induction us; boring.
-    unfold mul_each.
-    simpl_list; case_max; boring; omega.
-  Qed.
-
-  Lemma mul_length: forall us vs,
-      (length (mul us vs) <= length us + length vs)%nat.
-  Proof.
-    intros; unfold mul.
-    rewrite mul'_length.
-    rewrite rev_length; omega.
-  Qed.
-
-(* Print Assumptions mul_rep. *)
-
 End BaseSystem.
 
 Section PolynomialBaseCoefs.
@@ -615,7 +153,6 @@ Section PolynomialBaseCoefs.
     split; apply Z.pow_nonzero; try apply Zle_0_nat; try solve [intro H; inversion H].
   Qed.
 
-  Print BaseVector.
   Instance PolyBaseVector : BaseVector poly_base := {
     base_positive := poly_base_positive;
     b0_1 := poly_b0_1;
@@ -633,8 +170,7 @@ Section BaseSystemExample.
   Qed.
   Definition base2 := poly_base 2 baseLength.
 
-  About mul.
-  Example three_times_two : @mul base2 [1;1;0] [0;1;0] = [0;1;1;0;0].
+  Example three_times_two : mul base2 [1;1;0] [0;1;0] = [0;1;1;0;0].
   Proof.
     reflexivity.
   Qed.