slicing madness

Author: mbbarbosa-lectures

proof/correctness/avx2/MLKEM_InnerPKE_avx2.ec

@@ -1819,7 +1819,7 @@ op sliceget256_16_256 (arr: W16.t Array256.t) (offset: int) : W256.t =  W256.bit
 
 lemma flatten_take_drop_16  (l : W16.t list) (csize offset bit : int) :
   0 <= offset =>
-  offset + csize < 16 * size l =>
+  offset + csize <= 16 * size l =>
   0 <= bit < csize =>
   nth false (take csize (drop offset (flatten (map W16.w2bits l)))) bit =
             (nth witness l ((offset + bit) %/ 16)).[(offset + bit) %% 16].
@@ -1833,17 +1833,21 @@ rewrite -get_w2bits;congr.
 by rewrite (nth_map witness) 1:/#. 
 qed.
 
+lemma size_flatten_W16_w2bits (a : W16.t list) :
+  (size (flatten (map W16.w2bits (a))))  =  16 * size a.
+proof.
+  rewrite size_flatten -map_comp /(\o) /=.
+  rewrite StdBigop.Bigint.sumzE /= StdBigop.Bigint.BIA.big_mapT /(\o) /=. 
+  rewrite StdBigop.Bigint.big_constz count_predT /#.
+qed.
+
 lemma aligned_get256_16_256 arr offset :
-0 <= offset < 16*256 - 256 =>
+0 <= offset <= 16*256 - 256 =>
 256 %| offset  => 
 sliceget256_16_256 arr offset =
 WArray512.get256 (WArray512.init16 (fun (i_0 : int) => arr.[i_0])) (offset %/ 256).
 move => Ho1 Ho2; rewrite /sliceget256_16_256.
-have sz : size (take 256 (drop offset (flatten (map W16.w2bits (to_list arr))))) = 256.
-+ rewrite size_take 1:/# size_drop 1:/# /max /=.
-  rewrite size_flatten -map_comp /(\o) /=.
-  rewrite StdBigop.Bigint.sumzE /= StdBigop.Bigint.BIA.big_mapT /(\o) /=. 
-  rewrite StdBigop.Bigint.big_constz count_predT;smt(Array256.size_to_list).
+have sz : size (take 256 (drop offset (flatten (map W16.w2bits (to_list arr))))) = 256 by  rewrite size_take 1:/# size_drop 1:/# /max /=;smt(Array256.size_to_list size_flatten_W16_w2bits).
 rewrite wordP => i ib; rewrite get_bits2w //.
 rewrite flatten_take_drop_16;1..3:smt(Array256.size_to_list).
 rewrite nth_mkseq 1:/# /=.
@@ -1857,9 +1861,7 @@ bind op [W16.t & W256.t & Array256.t] sliceget256_16_256 "asliceget".
 realize bvaslicegetP.
 move => *; rewrite /sliceget256_16_256 bits2wK // size_take //= size_drop //=.
 admit. (* bounds are incomplete! 0 <= offset <= 16 * 256 - 256 *)
-rewrite size_flatten -map_comp /(\o) /=.
-rewrite StdBigop.Bigint.sumzE /= StdBigop.Bigint.BIA.big_mapT /(\o) /=. 
-by rewrite StdBigop.Bigint.big_constz count_predT;smt(Array256.size_to_list).
+by smt(Array256.size_to_list size_flatten_W16_w2bits).
 qed.
 
 import BitEncoding BS2Int BitChunking.
@@ -1869,29 +1871,83 @@ op sliceset256_16_256 (arr: W16.t Array256.t) (offset: int) (bv: W256.t) : W16.t
 
 
 lemma aligned_set256_16_256 arr offset bv :
-0 <= offset < 16*256 - 256 =>
+0 <= offset <= 16*256 - 256 =>
 256 %| offset  => 
 sliceset256_16_256 arr offset bv =
 Array256.init (fun (i3 : int) => get16 (set256 ((init16 (fun (i_0 : int) => arr.[i_0])))%WArray512 (offset %/ 256) bv) i3).
 rewrite /sliceset256_16_256 tP /= => ?? i ib.
 rewrite !initiE 1,2:/# /=.
-rewrite get16_set256E 1,2:/# /= (nth_map witness).
-+ admit. 
-admitted.
+rewrite get16_set256E 1,2:/# /= (nth_map []).
++ rewrite size_chunk // !size_cat  !size_take 1:/# !size_drop 1:/# /max /=.
+   by smt(Array256.size_to_list size_flatten_W16_w2bits). 
+rewrite JWordList.nth_chunk //= 1:/#.
+rewrite !size_cat  !size_take 1:/# !size_drop 1:/# /max /=.
+   by smt(Array256.size_to_list size_flatten_W16_w2bits). 
+case (32 * (offset %/ 256) <= 2 * i);last first.
++ move => ? /=.  have ? : 16*i < offset.  smt().
+  rewrite get16_init16 1:/# -catA drop_cat ifT;1: by smt(size_take W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0). 
+  rewrite take_cat_le ifT;1: by smt(size_take size_drop  W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0). 
+  have -> : offset = 16 * (offset %/ 16) by smt().
+  rewrite take_flatten_ctt; 1:   by smt(mapP W16.size_w2bits).
+  rewrite -map_take.
+  rewrite -(W16.w2bitsK arr.[i]);congr.
+  apply (eq_from_nth false).  
+  + rewrite size_w2bits size_take // size_drop 1:/# /= /max /=;smt(size_take size_drop  W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0). 
+  move => k kb; rewrite  flatten_take_drop_16 1:/#. 
+   + rewrite size_take 1:/# size_to_list //= 1:/#.
+     by smt(size_take size_drop  W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0). 
+   rewrite nth_take 1:/#. smt(size_take size_drop  W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0). 
+   rewrite get_w2bits;congr; rewrite ?get_to_list;smt(size_take size_drop  W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0).
+case (2 * i < 32 * (offset %/ 256 + 1));last first.
++ move => ? /=.  have ? : offset + 256 <= 16*i .  smt().
+  rewrite get16_init16 1:/# -catA drop_cat ifF;1: by smt(size_take W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0). 
+  rewrite drop_cat ifF;1: by smt(size_take W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0). 
+  rewrite size_take 1:/# size_flatten_W16_w2bits size_to_list /= ifT 1:/#. 
+  have -> : offset + 256 = 16 * ((offset + 256) %/ 16) by smt().
+  rewrite drop_flatten_ctt; 1:   by smt(mapP W16.size_w2bits).
+  have -> : 16 * i - offset - 256 = 16 * (i - offset %/ 16 - 16)  by smt().
+  rewrite drop_flatten_ctt; 1:   by smt(mapP W16.size_w2bits mem_drop).
+  rewrite drop_drop 1,2:/# /= => ?.
+  rewrite -(W16.w2bitsK arr.[i]);congr.
+  apply (eq_from_nth false).  
+  + rewrite  -map_drop size_take // size_flatten_W16_w2bits size_drop 1:/#; smt(Array256.size_to_list W16.size_w2bits). 
+  move => k kb. 
+   have -> : i - offset %/ 16 - 16 + (offset + 256) %/ 16 = i by smt().  
+   rewrite -(drop_flatten_ctt 16); 1: smt(mapP W16.size_w2bits).
+   rewrite flatten_take_drop_16; 1..3: by smt(size_take size_drop  W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0). 
+   rewrite get_w2bits;congr; rewrite ?get_to_list;smt(size_take size_drop  W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0).
+
++ move => ?? /=. have ? : offset <= 16*i < offset + 256.  smt().
+  rewrite -!catA drop_cat ifF;1: by smt(size_take W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0). 
+  rewrite !drop_cat ifT;1: by smt(size_take W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0). 
+  rewrite size_take 1:/# size_flatten_W16_w2bits size_to_list /= ifT 1:/#. 
+  rewrite take_cat_le ifT;1: by rewrite size_drop 1:/# size_w2bits /= /max ifT /#.
+     rewrite -(W16.w2bitsK ((bv \bits16 i - 16 * (offset %/ 256))));congr.
+  apply (eq_from_nth false).  
+  + rewrite  size_take // size_drop 1:/#; smt(Array256.size_to_list W16.size_w2bits). 
+  move => k kb. 
+   rewrite nth_take; 1,2: by smt(size_take size_drop  W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0). 
+   rewrite nth_drop; 1,2: by smt(size_take size_drop  W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0). 
+   rewrite !get_w2bits get_bits16;by smt(size_take size_drop  W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0). 
+qed.
+
+
 
 bind op [W16.t & W256.t & Array256.t] sliceset256_16_256 "asliceset".
-realize bvaslicesetP. 
-move => arr offset bv *.
+realize bvaslicesetP.  (* bounds are incomplete! 0 <= offset <= 16 * 256 - 256 *)
+move => arr offset bv *. have ? : 0 <= offset by admit.
 rewrite /sliceset256_16_256 of_listK.
-+ admit.
++ rewrite size_map size_chunk // !size_cat size_take 1:/#.
+  by smt(size_take size_drop  W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0). 
 rewrite -(map_comp W16.w2bits W16.bits2w) /(\o). 
 have := eq_in_map ((fun (x : bool list) => w2bits ((bits2w x))%W16)) idfun (chunk 16
         (take offset (flatten (map W16.w2bits (to_list arr))) ++ w2bits bv ++
          drop (offset + 256) (flatten (map W16.w2bits (to_list arr))))).
-rewrite iffE => [#] -> *.
-+ admit.
+rewrite iffE => [#] -> * /=. 
++ by smt(in_chunk_size W16.bits2wK).
 rewrite map_id /= chunkK //.
-+ admit.
++ rewrite !size_cat size_take 1:/#.
+  by smt(size_take size_drop  W16.size_w2bits size_cat Array256.size_to_list size_flatten_W16_w2bits size_ge0). 
 qed.
 
 op sliceget32_8_256 (arr: W8.t Array32.t) (i: int) : W256.t = get256 (WArray32.init8 (fun (i_0 : int) => pvc_shufbidx_s.[i_0])) (i%/256).