1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
|
Require Import Coq.Lists.List Coq.Lists.SetoidList. Import ListNotations.
Require Import Crypto.Util.ListUtil.
Require Import Algebra. Import Monoid ScalarMult.
Require Import VerdiTactics.
Require Import Crypto.Util.Option.
Section AddChainExp.
Function fold_chain {T} (id:T) (op:T->T->T) (is:list (nat*nat)) (acc:list T) {struct is} : T :=
match is with
| nil =>
match acc with
| nil => id
| ret::_ => ret
end
| (i,j)::is' =>
let ijx := op (nth_default id acc i) (nth_default id acc j) in
fold_chain id op is' (ijx::acc)
end.
Example wikipedia_addition_chain : fold_chain 0 plus [
(0, 0); (* 2 = 1 + 1 *) (* the indices say how far back the chain to look *)
(0, 1); (* 3 = 2 + 1 *)
(0, 0); (* 6 = 3 + 3 *)
(0, 0); (* 12 = 6 + 6 *)
(0, 0); (* 24 = 12 + 12 *)
(0, 2); (* 30 = 24 + 6 *)
(0, 6)] (* 31 = 30 + 1 *)
[1] = 31. reflexivity. Qed.
Context {G eq op id} {monoid:@Algebra.monoid G eq op id}.
Context {scalarmult} {is_scalarmult:@ScalarMult.is_scalarmult G eq op id scalarmult}.
Local Infix "=" := eq : type_scope.
Lemma fold_chain_exp' : forall (x:G) is acc ref
(H:forall i, nth_default id acc i = scalarmult (nth_default 0 ref i) x)
(Hl:Logic.eq (length acc) (length ref)),
fold_chain id op is acc = scalarmult (fold_chain 0 plus is ref) x.
Proof.
induction is; intros; simpl @fold_chain.
{ repeat break_match; specialize (H 0); rewrite ?nth_default_cons, ?nth_default_cons_S in H;
solve [ simpl length in *; discriminate | apply H | rewrite scalarmult_0_l; reflexivity ]. }
{ repeat break_match. eapply IHis; intros.
{ match goal with
[H:context [nth_default _ ?l _] |- context[nth_default _ (?h::?l) ?i] ]
=> destruct i; rewrite ?nth_default_cons, ?nth_default_cons_S;
solve [ apply H | rewrite !H, <-scalarmult_add_l; reflexivity ]
end. }
{ auto with distr_length. } }
Qed.
Lemma fold_chain_exp x is: fold_chain id op is [x] = scalarmult (fold_chain 0 plus is [1]) x.
Proof.
eapply fold_chain_exp'; intros; trivial.
destruct i; try destruct i; rewrite ?nth_default_cons_S, ?nth_default_cons, ?nth_default_nil;
rewrite ?scalarmult_1_l, ?scalarmult_0_l; reflexivity.
Qed.
End AddChainExp.
|
