src/Demo.v: a 200-line introduction to BaseSystem ideas

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+(* Following http://adam.chlipala.net/theses/andreser.pdf chapter 3 *)
+Require Import Coq.ZArith.ZArith Coq.micromega.Lia Crypto.Algebra.Nsatz.
+Require Import Crypto.Util.Tactics Crypto.Util.Decidable.
+Require Import Crypto.Util.Tuple Crypto.Util.Prod Crypto.Util.LetIn.
+Require Import Crypto.Util.ListUtil Coq.Lists.List Crypto.Util.NatUtil.
+Require Import QArith.QArith_base QArith.Qround Crypto.Util.QUtil.
+Require Import Crypto.Algebra.Ring Crypto.Util.Decidable.Bool2Prop.
+Import ListNotations. Local Open Scope Z_scope.
+    
+Definition runtime_mul := Z.mul.
+Definition runtime_add := Z.add.
+Delimit Scope runtime_scope with RT.
+Infix "*" := runtime_mul : runtime_scope.
+Infix "+" := runtime_add : runtime_scope. 
+
+Module Associational.
+  Definition eval (p:list (Z*Z)) : Z :=
+    fold_right Z.add 0%Z (map (fun t => fst t * snd t) p).
+  
+  Lemma eval_nil : eval nil = 0.
+  Proof. trivial.                                             Qed.
+  Lemma eval_cons p q : eval (p::q) = fst p * snd p + eval q.
+  Proof. trivial.                                             Qed.
+  Lemma eval_app p q: eval (p++q) = eval p + eval q.
+  Proof. induction p; rewrite <-?List.app_comm_cons;
+           rewrite ?eval_nil, ?eval_cons; nsatz.              Qed.
+  
+  Hint Rewrite eval_nil eval_cons eval_app : push_eval.
+  Local Ltac push := autorewrite with
+      push_eval push_map push_partition push_flat_map
+      push_fold_right push_nth_default cancel_pair.
+
+  Lemma eval_map_mul (a x:Z) (p:list (Z*Z))
+  : eval (List.map (fun t => (a*fst t, x*snd t)) p) = a*x*eval p.
+  Proof. induction p; push; nsatz.                            Qed.
+  Hint Rewrite eval_map_mul : push_eval.
+
+  Definition mul (p q:list (Z*Z)) : list (Z*Z) :=
+    flat_map (fun t =>
+      map (fun t' =>
+        (fst t * fst t', (snd t * snd t')%RT))
+    q) p.
+  Lemma eval_mul p q : eval (mul p q) = eval p * eval q.
+  Proof. induction p; cbv [mul]; push; nsatz.                 Qed.
+  Hint Rewrite eval_mul : push_eval.
+
+  Example base10_2digit_mul (a0:Z) (a1:Z) (b0:Z) (b1:Z) :
+    {ab| eval ab = eval [(10,a1);(1,a0)] * eval [(10,b1);(1,b0)]}.
+    eexists ?[ab].
+    (* Goal: eval ?ab = eval [(10,a1);(1,a0)] * eval [(10,b1);(1,b0)] *)
+    rewrite <-eval_mul.
+    (* Goal: eval ?ab = eval (mul [(10,a1);(1,a0)] [(10,b1);(1,b0)]) *)
+    cbv -[runtime_mul eval].
+    (* Goal: eval ?ab = eval [(100,(a1*b1));(10,a1*b0);(10,a0*b1);(1,a0*b0)]%RT *)
+    trivial.                                              Defined.
+
+  Definition split (s:Z) (p:list (Z*Z)) : list (Z*Z) * list (Z*Z)
+    := let hi_lo := partition (fun t => fst t mod s =? 0) p in
+       (snd hi_lo, map (fun t => (fst t / s, snd t)) (fst hi_lo)).
+  Lemma eval_split s p (s_nz:s<>0) :
+    eval (fst (split s p)) + s * eval (snd (split s p)) = eval p.
+  Proof. cbv [split]; induction p;
+    repeat match goal with
+    | |- context[?a/?b] =>
+      unique pose proof (Z_div_exact_full_2 a b ltac:(trivial) ltac:(trivial))
+    | _ => progress push
+    | _ => progress break_match
+    | _ => progress nsatz                                end. Qed.
+
+  Lemma reduction_rule a b s c (modulus_nz:s-c<>0) :
+    (a + s * b) mod (s - c) = (a + c * b) mod (s - c).
+  Proof. replace (a + s * b) with ((a + c*b) + b*(s-c)) by nsatz.
+    rewrite Z.add_mod,Z_mod_mult,Z.add_0_r,Z.mod_mod;trivial. Qed.
+
+  Definition reduce (s:Z) (c:list _) (p:list _) : list (Z*Z) :=
+    let lo_hi := split s p in fst lo_hi ++ mul c (snd lo_hi).
+
+  Lemma eval_reduce s c p (s_nz:s<>0) (modulus_nz:s-eval c<>0) :
+    eval (reduce s c p) mod (s - eval c) = eval p mod (s - eval c).
+  Proof. cbv [reduce]; push.
+         rewrite <-reduction_rule, eval_split; trivial.      Qed.
+  Hint Rewrite eval_reduce : push_eval.
+End Associational.
+
+Module Positional. Section Positional.
+  Context (weight : nat -> Z)
+          (weight_0 : weight 0%nat = 1)
+          (weight_nz : forall i, weight i <> 0).
+
+  Definition to_associational {n:nat} (xs:tuple Z n) : list (Z*Z)
+    := combine (map weight (List.seq 0 n)) (Tuple.to_list n xs).
+  Definition eval {n} x := Associational.eval (@to_associational n x).
+  Lemma eval_to_associational {n} x :
+    Associational.eval (@to_associational n x) = eval x.
+  Proof. trivial.                                             Qed.
+
+  (* SKIP over this: zeros, add_to_nth *)
+  Local Ltac push := autorewrite with push_eval push_map distr_length
+    push_flat_map push_fold_right push_nth_default cancel_pair natsimplify.
+  Program Definition zeros n : tuple Z n
+    := Tuple.from_list n (List.map (fun _ => 0) (List.seq 0 n)) _.
+  Next Obligation. push; reflexivity. Qed.
+  Lemma eval_zeros n : eval (zeros n) = 0.
+  Proof.
+    cbv [eval Associational.eval to_associational zeros];
+      rewrite Tuple.to_list_from_list.
+    generalize dependent (List.seq 0 n); intro xs.
+    induction xs; simpl; nsatz.                               Qed.
+  Program Definition add_to_nth {n} i x : tuple Z n -> tuple Z n
+    := Tuple.on_tuple (ListUtil.update_nth i (runtime_add x)) _.
+  Next Obligation. apply ListUtil.length_update_nth. Defined.
+  Lemma eval_add_to_nth {n} (i:nat) (H:(i<n)%nat) (x:Z) (xs:tuple Z n) :
+    eval (add_to_nth i x xs) = weight i * x + eval xs.
+  Proof.
+    cbv [eval to_associational add_to_nth Tuple.on_tuple runtime_add].
+    rewrite !Tuple.to_list_from_list.
+    rewrite ListUtil.combine_update_nth_r at 1.
+    rewrite <-(update_nth_id i (List.combine _ _)) at 2.
+    rewrite <-!(ListUtil.splice_nth_equiv_update_nth_update _ _
+      (weight 0, 0)) by (push; lia); cbv [ListUtil.splice_nth id].
+    repeat match goal with
+           | _ => progress push
+           | _ => progress break_match
+           | _ => progress (apply Zminus_eq; ring_simplify)
+           | _ => rewrite <-ListUtil.map_nth_default_always
+           end; lia.                                          Qed.
+  Hint Rewrite @eval_add_to_nth eval_zeros : push_eval.
+
+  Fixpoint place (t:Z*Z) (i:nat) : nat * Z :=
+    if dec (fst t mod weight i = 0)
+    then (i, let c := fst t / weight i in (c * snd t)%RT)
+    else match i with S i' => place t i' | O => (O, fst t * snd t)%RT end.
+  Lemma place_in_range (t:Z*Z) (n:nat) : (fst (place t n) < S n)%nat.
+  Proof. induction n; cbv [place] in *; break_match; autorewrite with cancel_pair; try omega. Qed.
+  Lemma weight_place t i : weight (fst (place t i)) * snd (place t i) = fst t * snd t.
+  Proof. induction i; cbv [place] in *; break_match; push;
+    repeat match goal with |- context[?a/?b] =>
+      unique pose proof (Z_div_exact_full_2 a b ltac:(auto) ltac:(auto))
+           end; nsatz.                                        Qed.
+  Hint Rewrite weight_place : push_eval.
+
+  Definition from_associational n (p:list (Z*Z)) :=
+    List.fold_right (fun t =>
+      let p := place t (pred n) in
+      add_to_nth (fst p) (snd p)    ) (zeros n) p.
+  Lemma eval_from_associational {n} p (n_nz:n<>O) :
+    eval (from_associational n p) = Associational.eval p.
+  Proof. induction p; cbv [from_associational] in *; push; try
+  pose proof place_in_range a (pred n); try omega; try nsatz. Qed.
+  Hint Rewrite @eval_from_associational : push_eval.
+
+  Section mulmod.
+    Context (m:Z) (m_nz:m <> 0) (s:Z) (s_nz:s <> 0)
+            (c:list (Z*Z)) (Hm:m = s - Associational.eval c).
+    Definition mulmod {n} (a b:tuple Z n) : tuple Z n
+      := let a_a := to_associational a in
+         let b_a := to_associational b in
+         let ab_a := Associational.mul a_a b_a in
+         let abm_a := Associational.reduce s c ab_a in
+         from_associational n abm_a.
+    Lemma eval_mulmod {n} (H:(n<>0)%nat) (f g:tuple Z n) :
+      eval (mulmod f g) mod m = (eval f * eval g) mod m.
+    Proof. cbv [mulmod]; rewrite Hm in *; push; trivial.      Qed.
+  End mulmod.
+End Positional. End Positional.
+
+Import Associational Positional.
+Local Coercion Z.of_nat : nat >-> Z.
+Local Coercion QArith_base.inject_Z : Z >-> Q.
+
+Definition w (i:nat) : Z := 2^Qceiling((25+1/2)*i).
+
+Example base_25_5_mul (f g:tuple Z 10) :
+  { fg : tuple Z 10 | (eval w fg) mod (2^255-19)
+                    = (eval w f * eval w g) mod (2^255-19) }.
+  (* manually assign names to limbs for pretty-printing *)
+  destruct f as [[[[[[[[[f9 f8]f7]f6]f5]f4]f3]f2]f1]f0].
+  destruct g as [[[[[[[[[g9 g8]g7]g6]g5]g4]g3]g2]g1]g0].
+  eexists ?[fg].
+  erewrite <-eval_mulmod with (s:=2^255) (c:=[(1,19)])
+      by (try eapply pow_ceil_mul_nat_nonzero; vm_decide).
+(*   eval w ?fg mod (2 ^ 255 - 19) = *)
+(*   eval w *)
+(*     (mulmod w (2^255) [(1, 19)] (f9,f8,f7,f6,f5,f4,f3,f2,f1,f0) *)
+(*        (g9,g8,g7,g6,g5,g4,g3,g2,g1,g0)) mod (2^255 - 19) *)
+  eapply f_equal2; [|trivial]. eapply f_equal.
+(*   ?fg = *)
+(*   mulmod w (2 ^ 255) [(1, 19)] (f9, f8, f7, f6, f5, f4, f3, f2, f1, f0) *)
+(*     (g9, g8, g7, g6, g5, g4, g3, g2, g1, g0) *)
+  cbv -[runtime_mul runtime_add]; cbv [runtime_mul runtime_add].
+  ring_simplify_subterms.
+(* ?fg =
+ (f0*g9+ f1*g8+    f2*g7+    f3*g6+    f4*g5+    f5*g4+    f6*g3+    f7*g2+    f8*g1+    f9*g0,   
+  f0*g8+ 2*f1*g7+  f2*g6+    2*f3*g5+  f4*g4+    2*f5*g3+  f6*g2+    2*f7*g1+  f8*g0+    38*f9*g9,
+  f0*g7+ f1*g6+    f2*g5+    f3*g4+    f4*g3+    f5*g2+    f6*g1+    f7*g0+    19*f8*g9+ 19*f9*g8,
+  f0*g6+ 2*f1*g5+  f2*g4+    2*f3*g3+  f4*g2+    2*f5*g1+  f6*g0+    38*f7*g9+ 19*f8*g8+ 38*f9*g7,
+  f0*g5+ f1*g4+    f2*g3+    f3*g2+    f4*g1+    f5*g0+    19*f6*g9+ 19*f7*g8+ 19*f8*g7+ 19*f9*g6,
+  f0*g4+ 2*f1*g3+  f2*g2+    2*f3*g1+  f4*g0+    38*f5*g9+ 19*f6*g8+ 38*f7*g7+ 19*f8*g6+ 38*f9*g5,
+  f0*g3+ f1*g2+    f2*g1+    f3*g0+    19*f4*g9+ 19*f5*g8+ 19*f6*g7+ 19*f7*g6+ 19*f8*g5+ 19*f9*g4,
+  f0*g2+ 2*f1*g1+  f2*g0+    38*f3*g9+ 19*f4*g8+ 38*f5*g7+ 19*f6*g6+ 38*f7*g5+ 19*f8*g4+ 38*f9*g3,
+  f0*g1+ f1*g0+    19*f2*g9+ 19*f3*g8+ 19*f4*g7+ 19*f5*g6+ 19*f6*g5+ 19*f7*g4+ 19*f8*g3+ 19*f9*g2,
+  f0*g0+ 38*f1*g9+ 19*f2*g8+ 38*f3*g7+ 19*f4*g6+ 38*f5*g5+ 19*f6*g4+ 38*f7*g3+ 19*f8*g2+ 38*f9*g1) *)
+  trivial.
+Defined.
+
+(* Eval cbv on this one would produce an ugly term due to the use of [destruct] *)