# DecimalFacts : some facts about Decimal numbers

Require Import Decimal.

Lemma uint_dec (d d' : uint) : { d = d' } + { d <> d' }.

Lemma rev_revapp d d' :
rev (revapp d d') = revapp d' d.

Lemma rev_rev d : rev (rev d) = d.

Normalization on little-endian numbers

Fixpoint nztail d :=
match d with
| Nil => Nil
| D0 d => match nztail d with Nil => Nil | d' => D0 d' end
| D1 d => D1 (nztail d)
| D2 d => D2 (nztail d)
| D3 d => D3 (nztail d)
| D4 d => D4 (nztail d)
| D5 d => D5 (nztail d)
| D6 d => D6 (nztail d)
| D7 d => D7 (nztail d)
| D8 d => D8 (nztail d)
| D9 d => D9 (nztail d)
end.

Definition lnorm d :=
match nztail d with
| Nil => zero
| d => d
end.

Lemma nzhead_revapp_0 d d' : nztail d = Nil ->
nzhead (revapp d d') = nzhead d'.

Lemma nzhead_revapp d d' : nztail d <> Nil ->
nzhead (revapp d d') = revapp (nztail d) d'.

Lemma nzhead_rev d : nztail d <> Nil ->
nzhead (rev d) = rev (nztail d).

Lemma rev_nztail_rev d :
rev (nztail (rev d)) = nzhead d.

Lemma revapp_nil_inv d d' : revapp d d' = Nil -> d = Nil /\ d' = Nil.

Lemma rev_nil_inv d : rev d = Nil -> d = Nil.

Lemma rev_lnorm_rev d :
rev (lnorm (rev d)) = unorm d.

Lemma nzhead_nonzero d d' : nzhead d <> D0 d'.

Lemma unorm_0 d : unorm d = zero <-> nzhead d = Nil.

Lemma unorm_nonnil d : unorm d <> Nil.