source: src/Clight/frontend_misc.ma @ 2500

Last change on this file since 2500 was 2500, checked in by garnier, 7 years ago

Continuing work on simulation in Clight/Cminor?

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1(* Various small homeless results. *)
2
3include "Clight/TypeComparison.ma".
4include "Clight/Csem.ma".
5include "common/Pointers.ma".
6include "basics/jmeq.ma".
7include "basics/star.ma". (* well-founded relations *)
8include "common/IOMonad.ma".
9include "common/IO.ma".
10include "basics/lists/listb.ma".
11include "basics/lists/list.ma".
12
13
14(* --------------------------------------------------------------------------- *)
15(* [cthulhu] plays the same role as daemon. It should be droppable. *)
16(* --------------------------------------------------------------------------- *)
17
18axiom cthulhu : ∀A:Prop. A. (* Because of the nightmares. *)
19
20(* --------------------------------------------------------------------------- *)
21(* Misc. *)
22(* --------------------------------------------------------------------------- *)
23
24lemma eq_intsize_identity : ∀id. eq_intsize id id = true.
25* normalize //
26qed.
27
28lemma sz_eq_identity : ∀s. sz_eq_dec s s = inl ?? (refl ? s).
29* normalize //
30qed.
31
32lemma type_eq_identity : ∀t. type_eq t t = true.
33#t normalize elim (type_eq_dec t t)
34[ 1: #Heq normalize //
35| 2: #H destruct elim H #Hcontr elim (Hcontr (refl ? t)) ] qed.
36
37lemma type_neq_not_identity : ∀t1, t2. t1 ≠ t2 → type_eq t1 t2 = false.
38#t1 #t2 #Hneq normalize elim (type_eq_dec t1 t2)
39[ 1: #Heq destruct elim Hneq #Hcontr elim (Hcontr (refl ? t2))
40| 2: #Hneq' normalize // ] qed.
41
42lemma le_S_O_absurd : ∀x:nat. S x ≤ 0 → False. /2 by absurd/ qed.
43
44lemma some_inj : ∀A : Type[0]. ∀a,b : A. Some ? a = Some ? b → a = b. #A #a #b #H destruct (H) @refl qed.
45
46lemma prod_eq_lproj : ∀A,B : Type[0]. ∀a : A. ∀b : B. ∀c : A × B. 〈a,b〉 = c → a = \fst c.
47#A #B #a #b * #a' #b' #H destruct @refl
48qed.
49
50lemma prod_eq_rproj : ∀A,B : Type[0]. ∀a : A. ∀b : B. ∀c : A × B. 〈a,b〉 = c → b = \snd c.
51#A #B #a #b * #a' #b' #H destruct @refl
52qed.
53
54lemma bindIO_Error : ∀err,f. bindIO io_out io_in (val×trace) (trace×state) (Error … err) f = Wrong io_out io_in (trace×state) err.
55// qed.
56
57lemma bindIO_Value : ∀v,f. bindIO io_out io_in (val×trace) (trace×state) (Value … v) f = (f v).
58// qed.
59
60lemma bindIO_elim :
61         ∀A.
62         ∀P : (IO io_out io_in A) → Prop.
63         ∀e : res A. (*IO io_out io_in A.*)
64         ∀f.
65         (∀v. (e:IO io_out io_in A) = OK … v →  P (f v)) →
66         (∀err. (e:IO io_out io_in A) = Error … err →  P (Wrong ??? err)) →
67         P (bindIO io_out io_in ? A (e:IO io_out io_in A) f).
68#A #P * try /2/ qed.
69
70lemma opt_to_io_Value :
71  ∀A,B,C,err,x,res. opt_to_io A B C err x = return res → x = Some ? res.
72#A #B #C #err #x cases x normalize
73[ 1: #res #Habsurd destruct
74| 2: #c #res #Heq destruct @refl ]
75qed. 
76
77lemma some_inversion :
78  ∀A,B:Type[0].
79  ∀e:option A.
80  ∀res:B.
81  ∀f:A → option B.
82 match e with
83 [ None ⇒ None ?
84 | Some x ⇒ f x ] = Some ? res →
85 ∃x. e = Some ? x ∧ f x = Some ? res.
86 #A #B #e #res #f cases e normalize nodelta
87[ 1: #Habsurd destruct (Habsurd)
88| 2: #a #Heq %{a} @conj >Heq @refl ]
89qed.
90
91lemma res_inversion :
92  ∀A,B:Type[0].
93  ∀e:option A.
94  ∀errmsg.
95  ∀result:B.
96  ∀f:A → res B.
97 match e with
98 [ None ⇒ Error ? errmsg
99 | Some x ⇒ f x ] = OK ? result →
100 ∃x. e = Some ? x ∧ f x = OK ? result.
101 #A #B #e #errmsg #result #f cases e normalize nodelta
102[ 1: #Habsurd destruct (Habsurd)
103| 2: #a #Heq %{a} @conj >Heq @refl ]
104qed.
105
106lemma cons_inversion :
107  ∀A,B:Type[0].
108  ∀e:list A.
109  ∀res:B.
110  ∀f:A → list A → option B.
111 match e with
112 [ nil ⇒ None ?
113 | cons hd tl ⇒ f hd tl ] = Some ? res →
114 ∃hd,tl. e = hd::tl ∧ f hd tl = Some ? res.
115#A #B #e #res #f cases e normalize nodelta
116[ 1: #Habsurd destruct (Habsurd)
117| 2: #hd #tl #Heq %{hd} %{tl} @conj >Heq @refl ]
118qed.
119
120lemma if_opt_inversion :
121  ∀A:Type[0].
122  ∀x.
123  ∀y:A.
124  ∀e:bool.
125 (if e then
126    x
127  else
128    None ?) = Some ? y →
129 e = true ∧ x = Some ? y.
130#A #x #y * normalize
131#H destruct @conj @refl
132qed.
133
134lemma opt_to_res_inversion :
135  ∀A:Type[0]. ∀errmsg. ∀opt. ∀val. opt_to_res A errmsg opt = OK ? val →
136  opt = Some ? val.
137#A #errmsg *
138[ 1: #val normalize #Habsurd destruct
139| 2: #res #val normalize #Heq destruct @refl ]
140qed.
141
142lemma andb_inversion :
143  ∀a,b : bool. andb a b = true → a = true ∧ b = true.
144* * normalize /2 by conj, refl/ qed. 
145
146lemma identifier_eq_i_i : ∀tag,i. ∃pf. identifier_eq tag i i = inl … pf.
147#tag #i cases (identifier_eq ? i i)
148[ 1: #H %{H} @refl
149| 2: * #Habsurd @(False_ind … (Habsurd … (refl ? i))) ]
150qed.
151
152(* --------------------------------------------------------------------------- *)
153(* Useful facts on various boolean operations. *)
154(* --------------------------------------------------------------------------- *)
155 
156lemma andb_lsimpl_true : ∀x. andb true x = x. // qed.
157lemma andb_lsimpl_false : ∀x. andb false x = false. normalize // qed.
158lemma andb_comm : ∀x,y. andb x y = andb y x. // qed.
159lemma notb_true : notb true = false. // qed.
160lemma notb_false : notb false = true. % #H destruct qed.
161lemma notb_fold : ∀x. if x then false else true = (¬x). // qed.
162
163(* --------------------------------------------------------------------------- *)
164(* Useful facts on Z. *)
165(* --------------------------------------------------------------------------- *)
166
167lemma Zltb_to_Zleb_true : ∀a,b. Zltb a b = true → Zleb a b = true.
168#a #b #Hltb lapply (Zltb_true_to_Zlt … Hltb) #Hlt @Zle_to_Zleb_true
169/3 by Zlt_to_Zle, transitive_Zle/ qed.
170
171lemma Zle_to_Zle_to_eq : ∀a,b. Zle a b → Zle b a → a = b.
172#a #b elim b
173[ 1: elim a // #a' normalize [ 1: @False_ind | 2: #_ @False_ind ] ]
174#b' elim a normalize
175[ 1: #_ @False_ind
176| 2: #a' #H1 #H2 >(le_to_le_to_eq … H1 H2) @refl
177| 3: #a' #_ @False_ind
178| 4: @False_ind
179| 5: #a' @False_ind
180| 6: #a' #H2 #H1 >(le_to_le_to_eq … H1 H2) @refl
181] qed.
182
183lemma Zleb_true_to_Zleb_true_to_eq : ∀a,b. (Zleb a b = true) → (Zleb b a = true) → a = b.
184#a #b #H1 #H2
185/3 by Zle_to_Zle_to_eq, Zleb_true_to_Zle/
186qed.
187
188lemma Zltb_dec : ∀a,b. (Zltb a b = true) ∨ (Zltb a b = false ∧ Zleb b a = true).
189#a #b
190lapply (Z_compare_to_Prop … a b)
191cases a
192[ 1: | 2,3: #a' ]
193cases b
194whd in match (Z_compare OZ OZ); normalize nodelta
195[ 2,3,5,6,8,9: #b' ]
196whd in match (Zleb ? ?);
197try /3 by or_introl, or_intror, conj, I, refl/
198whd in match (Zltb ??);
199whd in match (Zleb ??); #_
200[ 1: cases (decidable_le (succ a') b')
201     [ 1: #H lapply (le_to_leb_true … H) #H %1 assumption
202     | 2:  #Hnotle lapply (not_le_to_lt … Hnotle) #Hlt %2  @conj try @le_to_leb_true
203           /2 by monotonic_pred/ @(not_le_to_leb_false … Hnotle) ]
204| 2: cases (decidable_le (succ b') a')
205     [ 1: #H lapply (le_to_leb_true … H) #H %1 assumption
206     | 2:  #Hnotle lapply (not_le_to_lt … Hnotle) #Hlt %2  @conj try @le_to_leb_true
207           /2 by monotonic_pred/ @(not_le_to_leb_false … Hnotle) ]
208] qed.
209
210lemma Zleb_unsigned_OZ : ∀bv. Zleb OZ (Z_of_unsigned_bitvector 16 bv) = true.
211#bv elim bv try // #n' * #tl normalize /2/ qed.
212
213lemma Zltb_unsigned_OZ : ∀bv. Zltb (Z_of_unsigned_bitvector 16 bv) OZ = false.
214#bv elim bv try // #n' * #tl normalize /2/ qed.
215
216lemma Z_of_unsigned_not_neg : ∀bv.
217  (Z_of_unsigned_bitvector 16 bv = OZ) ∨ (∃p. Z_of_unsigned_bitvector 16 bv = pos p).
218#bv elim bv
219[ 1: normalize %1 @refl
220| 2: #n #hd #tl #Hind cases hd
221     [ 1: normalize %2 /2 by ex_intro/
222     | 2: cases Hind normalize [ 1: #H %1 @H | 2: * #p #H >H %2 %{p} @refl ]
223     ]
224] qed.
225
226lemma free_not_in_bounds : ∀x. if Zleb (pos one) x
227                                then Zltb x OZ 
228                                else false = false.
229#x lapply (Zltb_to_Zleb_true x OZ)
230elim (Zltb_dec … x OZ)
231[ 1: #Hlt >Hlt #H lapply (H (refl ??)) elim x
232     [ 2,3: #x' ] normalize in ⊢ (% → ?);
233     [ 1: #Habsurd destruct (Habsurd)
234     | 2,3: #_ @refl ]
235| 2: * #Hlt #Hle #_ >Hlt cases (Zleb (pos one) x) @refl ]
236qed.
237
238lemma free_not_valid : ∀x. if Zleb (pos one) x
239                            then Zltb x OZ 
240                            else false = false.
241#x
242cut (Zle (pos one) x ∧ Zlt x OZ → False)
243[ * #H1 #H2 lapply (transitive_Zle … (monotonic_Zle_Zsucc … H1) (Zlt_to_Zle … H2)) #H @H ] #Hguard
244cut (Zleb (pos one) x = true ∧ Zltb x OZ = true → False)
245[ * #H1 #H2 @Hguard @conj /2 by Zleb_true_to_Zle/ @Zltb_true_to_Zlt assumption ]
246cases (Zleb (pos one) x) cases (Zltb x OZ)
247/2 by refl/ #H @(False_ind … (H (conj … (refl ??) (refl ??))))
248qed.
249
250(* follows from (0 ≤ a < b → mod a b = a) *)
251axiom Z_of_unsigned_bitvector_of_small_Z :
252  ∀z. OZ ≤ z → z < Z_two_power_nat 16 → Z_of_unsigned_bitvector 16 (bitvector_of_Z 16 z) = z.
253
254theorem Zle_to_Zlt_to_Zlt: ∀n,m,p:Z. n ≤ m → m < p → n < p.
255#n #m #p #Hle #Hlt /4 by monotonic_Zle_Zplus_r, Zle_to_Zlt, Zlt_to_Zle, transitive_Zle/
256qed.
257
258(* --------------------------------------------------------------------------- *)
259(* Useful facts on blocks. *)
260(* --------------------------------------------------------------------------- *)
261
262lemma not_eq_block : ∀b1,b2. b1 ≠ b2 → eq_block b1 b2 = false.
263#b1 #b2 #Hneq
264@(eq_block_elim … b1 b2)
265[ 1: #Heq destruct elim Hneq #H @(False_ind … (H (refl ??)))
266| 2: #_ @refl ] qed.
267
268lemma not_eq_block_rev : ∀b1,b2. b2 ≠ b1 → eq_block b1 b2 = false.
269#b1 #b2 #Hneq
270@(eq_block_elim … b1 b2)
271[ 1: #Heq destruct elim Hneq #H @(False_ind … (H (refl ??)))
272| 2: #_ @refl ] qed.
273
274definition block_DeqSet : DeqSet ≝ mk_DeqSet block eq_block ?.
275* #r1 #id1 * #r2 #id2 @(eqZb_elim … id1 id2)
276[ 1: #Heq >Heq cases r1 cases r2 normalize
277     >eqZb_z_z normalize try // @conj #H destruct (H)
278     try @refl
279| 2: #Hneq cases r1 cases r2 normalize
280     >(eqZb_false … Hneq) normalize @conj
281     #H destruct (H) elim Hneq #H @(False_ind … (H (refl ??)))
282] qed.
283
284(* --------------------------------------------------------------------------- *)
285(* General results on lists. *)
286(* --------------------------------------------------------------------------- *)
287
288(* If mem succeeds in finding an element, then the list can be partitioned around this element. *)
289lemma list_mem_split : ∀A. ∀l : list A. ∀x : A. mem … x l → ∃l1,l2. l = l1 @ [x] @ l2.
290#A #l elim l
291[ 1: normalize #x @False_ind
292| 2: #hd #tl #Hind #x whd in ⊢ (% → ?); *
293     [ 1: #Heq %{(nil ?)} %{tl} destruct @refl
294     | 2: #Hmem lapply (Hind … Hmem) * #l1 * #l2 #Heq_tl >Heq_tl
295          %{(hd :: l1)} %{l2} @refl
296     ]
297] qed.
298
299lemma cons_to_append : ∀A. ∀hd : A. ∀l : list A. hd :: l = [hd] @ l. #A #hd #l @refl qed.
300
301lemma fold_append :
302  ∀A,B:Type[0]. ∀l1, l2 : list A. ∀f:A → B → B. ∀seed. foldr ?? f seed (l1 @ l2) = foldr ?? f (foldr ?? f seed l2) l1.
303#A #B #l1 elim l1 //
304#hd #tl #Hind #l2 #f #seed normalize >Hind @refl
305qed.
306
307lemma filter_append : ∀A:Type[0]. ∀l1,l2 : list A. ∀f. filter ? f (l1 @ l2) = (filter ? f l1) @ (filter ? f l2).
308#A #l1 elim l1 //
309#hd #tl #Hind #l2 #f
310>cons_to_append >associative_append
311normalize cases (f hd) normalize
312<Hind @refl
313qed.
314
315lemma filter_cons_unfold : ∀A:Type[0]. ∀f. ∀hd,tl.
316  filter ? f (hd :: tl) =
317  if f hd then
318    (hd :: filter A f tl)
319  else
320    (filter A f tl).
321#A #f #hd #tl elim tl // qed.
322
323
324lemma filter_elt_empty : ∀A:DeqSet. ∀elt,l. filter (carr A) (λx.¬(x==elt)) l = [ ] → All (carr A) (λx. x = elt) l.
325#A #elt #l elim l
326[ 1: //
327| 2: #hd #tl #Hind >filter_cons_unfold
328     lapply (eqb_true A hd elt)
329     cases (hd==elt) normalize nodelta
330     [ 2: #_ #Habsurd destruct
331     | 1: * #H1 #H2 #Heq lapply (Hind Heq) #Hall whd @conj //
332          @H1 @refl
333     ]
334] qed.
335
336lemma nil_append : ∀A. ∀(l : list A). [ ] @ l = l. // qed.
337
338alias id "meml" = "cic:/matita/basics/lists/list/mem.fix(0,2,1)".
339
340lemma mem_append : ∀A:Type[0]. ∀elt : A. ∀l1,l2. mem … elt (l1 @ l2) ↔ (mem … elt l1) ∨ (mem … elt l2).
341#A #elt #l1 elim l1
342[ 1: #l2 normalize @conj [ 1: #H %2 @H | 2: * [ 1: @False_ind | 2: #H @H ] ]
343| 2: #hd #tl #Hind #l2 @conj
344     [ 1: whd in match (meml ???); *
345          [ 1: #Heq >Heq %1 normalize %1 @refl
346          | 2: #H1 lapply (Hind l2) * #HA #HB normalize
347               elim (HA H1)
348               [ 1: #H %1 %2 assumption | 2: #H %2 assumption ]
349          ]
350     | 2: normalize *
351          [ 1: * /2 by or_introl, or_intror/
352               #H %2 elim (Hind l2) #_ #H' @H' %1 @H
353          | 2: #H %2 elim (Hind l2) #_ #H' @H' %2 @H ]
354     ]
355] qed.
356
357lemma mem_append_forward : ∀A:Type[0]. ∀elt : A. ∀l1,l2. mem … elt (l1 @ l2) → (mem … elt l1) ∨ (mem … elt l2).
358#A #elt #l1 #l2 #H elim (mem_append … elt l1 l2) #H' #_ @H' @H qed.
359
360lemma mem_append_backwards : ∀A:Type[0]. ∀elt : A. ∀l1,l2. (mem … elt l1) ∨ (mem … elt l2) → mem … elt (l1 @ l2) .
361#A #elt #l1 #l2 #H elim (mem_append … elt l1 l2) #_ #H' @H' @H qed.
362
363(* "Observational" equivalence on list implies concrete equivalence. Useful to
364 * prove equality from some reasoning on indexings. Needs a particular induction
365 * principle. *)
366 
367let rec double_list_ind
368  (A : Type[0])
369  (P : list A → list A → Prop)
370  (base_nil  : P [ ] [ ])
371  (base_l1   : ∀hd1,l1. P (hd1::l1) [ ])
372  (base_l2   : ∀hd2,l2. P [ ] (hd2::l2))
373  (ind  : ∀hd1,hd2,tl1,tl2. P tl1 tl2 → P (hd1 :: tl1) (hd2 :: tl2))
374  (l1, l2 : list A) on l1 ≝
375match l1 with
376[ nil ⇒
377  match l2 with
378  [ nil ⇒ base_nil
379  | cons hd2 tl2 ⇒ base_l2 hd2 tl2 ]
380| cons hd1 tl1 ⇒ 
381  match l2 with
382  [ nil ⇒ base_l1 hd1 tl1
383  | cons hd2 tl2 ⇒
384    ind hd1 hd2 tl1 tl2 (double_list_ind A P base_nil base_l1 base_l2 ind tl1 tl2)
385  ]
386]. 
387
388lemma nth_eq_tl :
389  ∀A:Type[0]. ∀l1,l2:list A. ∀hd1,hd2.
390  (∀i. nth_opt A i (hd1::l1) = nth_opt A i (hd2::l2)) →
391  (∀i. nth_opt A i l1 = nth_opt A i l2).
392#A #l1 #l2 @(double_list_ind … l1 l2)
393[ 1: #hd1 #hd2 #_ #i elim i try /2/
394| 2: #hd1' #l1' #hd1 #hd2 #H lapply (H 1) normalize #Habsurd destruct
395| 3: #hd2' #l2' #hd1 #hd2 #H lapply (H 1) normalize #Habsurd destruct
396| 4: #hd1' #hd2' #tl1' #tl2' #H #hd1 #hd2
397     #Hind
398     @(λi. Hind (S i))
399] qed.     
400
401
402lemma nth_eq_to_eq :
403  ∀A:Type[0]. ∀l1,l2:list A. (∀i. nth_opt A i l1 = nth_opt A i l2) → l1 = l2.
404#A #l1 elim l1
405[ 1: #l2 #H lapply (H 0) normalize
406     cases l2
407     [ 1: //
408     | 2: #hd2 #tl2 normalize #Habsurd destruct ]
409| 2: #hd1 #tl1 #Hind *
410     [ 1: #H lapply (H 0) normalize #Habsurd destruct
411     | 2: #hd2 #tl2 #H lapply (H 0) normalize #Heq destruct (Heq)
412          >(Hind tl2) try @refl @(nth_eq_tl … H)
413     ]
414] qed.
415
416(* --------------------------------------------------------------------------- *)
417(* General results on vectors. *)
418(* --------------------------------------------------------------------------- *)
419
420(* copied from AssemblyProof, TODO get rid of the redundant stuff. *)
421lemma Vector_O: ∀A. ∀v: Vector A 0. v ≃ VEmpty A.
422 #A #v generalize in match (refl … 0); cases v in ⊢ (??%? → ?%%??); //
423 #n #hd #tl #abs @⊥ destruct (abs)
424qed.
425
426lemma Vector_Sn: ∀A. ∀n.∀v:Vector A (S n).
427 ∃hd.∃tl.v ≃ VCons A n hd tl.
428 #A #n #v generalize in match (refl … (S n)); cases v in ⊢ (??%? → ??(λ_.??(λ_.?%%??)));
429 [ #abs @⊥ destruct (abs)
430 | #m #hd #tl #EQ <(injective_S … EQ) %[@hd] %[@tl] // ]
431qed.
432
433lemma vector_append_zero:
434  ∀A,m.
435  ∀v: Vector A m.
436  ∀q: Vector A 0.
437    v = q@@v.
438  #A #m #v #q
439  >(Vector_O A q) %
440qed.
441
442corollary prod_vector_zero_eq_left:
443  ∀A, n.
444  ∀q: Vector A O.
445  ∀r: Vector A n.
446    〈q, r〉 = 〈[[ ]], r〉.
447  #A #n #q #r
448  generalize in match (Vector_O A q …);
449  #hyp
450  >hyp in ⊢ (??%?);
451  %
452qed.
453 
454lemma vsplit_eq : ∀A. ∀m,n. ∀v : Vector A ((S m) + n).  ∃v1:Vector A (S m). ∃v2:Vector A n. v = v1 @@ v2.
455# A #m #n elim m
456[ 1: normalize #v
457  elim (Vector_Sn ?? v) #hd * #tl #Eq
458  @(ex_intro … (hd ::: [[]])) @(ex_intro … tl)
459  >Eq normalize //
460| 2: #n' #Hind #v
461  elim (Vector_Sn ?? v) #hd * #tl #Eq
462  elim (Hind tl)
463  #tl1 * #tl2 #Eq_tl
464  @(ex_intro … (hd ::: tl1))
465  @(ex_intro … tl2) 
466  destruct normalize //
467] qed.
468
469lemma vsplit_zero:
470  ∀A,m.
471  ∀v: Vector A m.
472    〈[[]], v〉 = vsplit A 0 m v.
473  #A #m #v
474  elim v
475  [ %
476  | #n #hd #tl #ih
477    normalize in ⊢ (???%); %
478  ]
479qed.
480
481lemma vsplit_zero2:
482  ∀A,m.
483  ∀v: Vector A m.
484    〈[[]], v〉 = vsplit' A 0 m v.
485  #A #m #v
486  elim v
487  [ %
488  | #n #hd #tl #ih
489    normalize in ⊢ (???%); %
490  ]
491qed.
492
493lemma vsplit_eq2 : ∀A. ∀m,n : nat. ∀v : Vector A (m + n).  ∃v1:Vector A m. ∃v2:Vector A n. v = v1 @@ v2.
494# A #m #n elim m
495[ 1: normalize #v @(ex_intro … (VEmpty ?)) @(ex_intro … v) normalize //
496| 2: #n' #Hind #v
497  elim (Vector_Sn ?? v) #hd * #tl #Eq
498  elim (Hind tl)
499  #tl1 * #tl2 #Eq_tl
500  @(ex_intro … (hd ::: tl1))
501  @(ex_intro … tl2) 
502  destruct normalize //
503] qed.
504
505(* This is not very nice. Note that this axiom was copied verbatim from ASM/AssemblyProof.ma.
506 * TODO sync with AssemblyProof.ma, in a better world we shouldn't have to copy all of this. *)
507axiom vsplit_succ:
508  ∀A, m, n.
509  ∀l: Vector A m.
510  ∀r: Vector A n.
511  ∀v: Vector A (m + n).
512  ∀hd.
513    v = l@@r → (〈l, r〉 = vsplit ? m n v → 〈hd:::l, r〉 = vsplit ? (S m) n (hd:::v)).
514
515axiom vsplit_succ2:
516  ∀A, m, n.
517  ∀l: Vector A m.
518  ∀r: Vector A n.
519  ∀v: Vector A (m + n).
520  ∀hd.
521    v = l@@r → (〈l, r〉 = vsplit' ? m n v → 〈hd:::l, r〉 = vsplit' ? (S m) n (hd:::v)).
522     
523lemma vsplit_prod2:
524  ∀A,m,n.
525  ∀p: Vector A (m + n).
526  ∀v: Vector A m.
527  ∀q: Vector A n.
528    p = v@@q → 〈v, q〉 = vsplit' A m n p.
529  #A #m
530  elim m
531  [ #n #p #v #q #hyp
532    >hyp <(vector_append_zero A n q v)
533    >(prod_vector_zero_eq_left A …)
534    @vsplit_zero2
535  | #r #ih #n #p #v #q #hyp
536    >hyp
537    cases (Vector_Sn A r v)
538    #hd #exists
539    cases exists
540    #tl #jmeq >jmeq
541    @vsplit_succ2 [1: % |2: @ih % ]
542  ]
543qed.
544
545lemma vsplit_prod:
546  ∀A,m,n.
547  ∀p: Vector A (m + n).
548  ∀v: Vector A m.
549  ∀q: Vector A n.
550    p = v@@q → 〈v, q〉 = vsplit A m n p.
551  #A #m
552  elim m
553  [ #n #p #v #q #hyp
554    >hyp <(vector_append_zero A n q v)
555    >(prod_vector_zero_eq_left A …)
556    @vsplit_zero
557  | #r #ih #n #p #v #q #hyp
558    >hyp
559    cases (Vector_Sn A r v)
560    #hd #exists
561    cases exists
562    #tl #jmeq >jmeq
563    @vsplit_succ [1: % |2: @ih % ]
564  ]
565qed.
566
567
568(* --------------------------------------------------------------------------- *)
569(* Generic properties of equivalence relations *)
570(* --------------------------------------------------------------------------- *)
571
572lemma triangle_diagram :
573  ∀acctype : Type[0].
574  ∀eqrel : acctype → acctype → Prop.
575  ∀refl_eqrel  : reflexive ? eqrel.
576  ∀trans_eqrel : transitive ? eqrel.
577  ∀sym_eqrel   : symmetric ? eqrel.
578  ∀a,b,c. eqrel a b → eqrel a c → eqrel b c.
579#acctype #eqrel #R #T #S #a #b #c
580#H1 #H2 @(T … (S … H1) H2)
581qed.
582
583lemma cotriangle_diagram :
584  ∀acctype : Type[0].
585  ∀eqrel : acctype → acctype → Prop.
586  ∀refl_eqrel  : reflexive ? eqrel.
587  ∀trans_eqrel : transitive ? eqrel.
588  ∀sym_eqrel   : symmetric ? eqrel.
589  ∀a,b,c. eqrel b a → eqrel c a → eqrel b c.
590#acctype #eqrel #R #T #S #a #b #c
591#H1 #H2 @S @(T … H2 (S … H1))
592qed.
593
594(* --------------------------------------------------------------------------- *)
595(* Quick and dirty implementation of finite sets relying on lists. The
596 * implementation is split in two: an abstract equivalence defined using inclusion
597 * of lists, and a concrete one where equivalence is defined as the closure of
598 * duplication, contraction and transposition of elements. We rely on the latter
599 * to prove stuff on folds over sets.  *)
600(* --------------------------------------------------------------------------- *)
601
602definition lset ≝ λA:Type[0]. list A.
603
604(* The empty set. *)
605definition empty_lset ≝ λA:Type[0]. nil A.
606
607(* Standard operations. *)
608definition lset_union ≝ λA:Type[0].λl1,l2 : lset A. l1 @ l2.
609
610definition lset_remove ≝ λA:DeqSet.λl:lset (carr A).λelt:carr A. (filter … (λx.¬x==elt) l).
611
612definition lset_difference ≝ λA:DeqSet.λl1,l2:lset (carr A). (filter … (λx.¬ (memb ? x l2)) l1).
613
614(* Standard predicates on sets *)
615definition lset_in ≝ λA:Type[0].λx : A. λl : lset A. mem … x l.
616
617definition lset_disjoint ≝ λA:Type[0].λl1, l2 : lset A.
618  ∀x,y. mem … x l1 → mem … y l2 → x ≠ y.
619 
620definition lset_inclusion ≝ λA:Type[0].λl1,l2.
621  All A (λx1. mem … x1 l2) l1.
622
623(* Definition of abstract set equivalence. *)
624definition lset_eq ≝ λA:Type[0].λl1,l2.
625  lset_inclusion A l1 l2 ∧
626  lset_inclusion A l2 l1.
627
628(* Properties of inclusion. *) 
629lemma reflexive_lset_inclusion : ∀A. ∀l. lset_inclusion A l l.
630#A #l elim l try //
631#hd #tl #Hind whd @conj
632[ 1: %1 @refl
633| 2: whd in Hind; @(All_mp … Hind)
634     #a #H whd %2 @H
635] qed.
636
637lemma transitive_lset_inclusion : ∀A. ∀l1,l2,l3. lset_inclusion A l1 l2 → lset_inclusion A l2 l3 → lset_inclusion A l1 l3 .
638#A #l1 #l2 #l3
639#Hincl1 #Hincl2 @(All_mp … Hincl1)
640whd in Hincl2;
641#a elim l2 in Hincl2;
642[ 1: normalize #_ @False_ind
643| 2: #hd #tl #Hind whd in ⊢ (% → ?);
644     * #Hmem #Hincl_tl whd in ⊢ (% → ?);
645     * [ 1: #Heq destruct @Hmem
646       | 2: #Hmem_tl @Hind assumption ]
647] qed.
648
649lemma cons_preserves_inclusion : ∀A. ∀l1,l2. lset_inclusion A l1 l2 → ∀x. lset_inclusion A l1 (x::l2).
650#A #l1 #l2 #Hincl #x @(All_mp … Hincl) #a #Hmem whd %2 @Hmem qed.
651
652lemma cons_monotonic_inclusion : ∀A. ∀l1,l2. lset_inclusion A l1 l2 → ∀x. lset_inclusion A (x::l1) (x::l2).
653#A #l1 #l2 #Hincl #x whd @conj
654[ 1: /2 by or_introl/
655| 2: @(All_mp … Hincl) #a #Hmem whd %2 @Hmem ] qed.
656
657lemma lset_inclusion_concat : ∀A. ∀l1,l2. lset_inclusion A l1 l2 → ∀l3. lset_inclusion A l1 (l3 @ l2).
658#A #l1 #l2 #Hincl #l3 elim l3
659try /2 by cons_preserves_inclusion/
660qed.
661
662lemma lset_inclusion_concat_monotonic : ∀A. ∀l1,l2. lset_inclusion A l1 l2 → ∀l3. lset_inclusion A (l3 @ l1) (l3 @ l2).
663#A #l1 #l2 #Hincl #l3 elim l3
664try @Hincl #hd #tl #Hind @cons_monotonic_inclusion @Hind
665qed.
666 
667(* lset_eq is an equivalence relation. *)
668lemma reflexive_lset_eq : ∀A. ∀l. lset_eq A l l. /2 by conj/ qed.
669
670lemma transitive_lset_eq : ∀A. ∀l1,l2,l3. lset_eq A l1 l2 → lset_eq A l2 l3 → lset_eq A l1 l3.
671#A #l1 #l2 #l3 * #Hincl1 #Hincl2 * #Hincl3 #Hincl4
672@conj @(transitive_lset_inclusion ??l2) assumption
673qed.
674
675lemma symmetric_lset_eq : ∀A. ∀l1,l2. lset_eq A l1 l2 → lset_eq A l2 l1.
676#A #l1 #l2 * #Hincl1 #Hincl2 @conj assumption
677qed.
678
679(* Properties of inclusion vs union and equality. *)
680lemma lset_union_inclusion : ∀A:Type[0]. ∀a,b,c. 
681  lset_inclusion A a c → lset_inclusion ? b c → lset_inclusion ? (lset_union ? a b) c.
682#A #a #b #c #H1 #H2 whd whd in match (lset_union ???);
683@All_append assumption qed.
684
685lemma lset_inclusion_union : ∀A:Type[0]. ∀a,b,c. 
686  lset_inclusion A a b ∨ lset_inclusion A a c → lset_inclusion ? a (lset_union ? b c).
687#A #a #b #c *
688[ 1: @All_mp #elt #Hmem @mem_append_backwards %1 @Hmem
689| 2: @All_mp #elt #Hmem @mem_append_backwards %2 @Hmem
690] qed.
691
692lemma lset_inclusion_eq : ∀A : Type[0]. ∀a,b,c : lset A.
693  lset_eq ? a b → lset_inclusion ? b c → lset_inclusion ? a c.
694#A #a #b #c * #H1 #H2 #H3 @(transitive_lset_inclusion … H1 H3)
695qed.
696
697lemma lset_inclusion_eq2 : ∀A : Type[0]. ∀a,b,c : lset A.
698  lset_eq ? b c → lset_inclusion ? a b → lset_inclusion ? a c.
699#A #a #b #c * #H1 #H2 #H3 @(transitive_lset_inclusion … H3 H1)
700qed.
701
702(* Properties of lset_eq and mem *)
703lemma lset_eq_mem :
704  ∀A:Type[0].
705  ∀s1,s2 : lset A.
706  lset_eq ? s1 s2 →
707  ∀b.mem ? b s1 → mem ? b s2.
708#A #s1 #s2 * #Hincl12 #_ #b
709whd in Hincl12; elim s1 in Hincl12;
710[ 1: normalize #_ *
711| 2: #hd #tl #Hind whd in ⊢ (% → % → ?); * #Hmem' #Hall * #Heq
712     [ 1: destruct (Heq) assumption
713     | 2: @Hind assumption ]
714] qed.
715
716lemma lset_eq_memb :
717  ∀A : DeqSet.
718  ∀s1,s2 : lset (carr A).
719  lset_eq ? s1 s2 →
720  ∀b.memb ? b s1 = true → memb ? b s2 = true.
721#A #s1 #s2 #Heq #b
722lapply (memb_to_mem A s1 b) #H1 #H2
723lapply (H1 H2) #Hmem lapply (lset_eq_mem … Heq ? Hmem) @mem_to_memb
724qed.
725
726lemma lset_eq_memb_eq :
727  ∀A : DeqSet.
728  ∀s1,s2 : lset (carr A).
729  lset_eq ? s1 s2 →
730  ∀b.memb ? b s1 = memb ? b s2.
731#A #s1 #s2 #Hlset_eq #b
732lapply (lset_eq_memb … Hlset_eq b)
733lapply (lset_eq_memb … (symmetric_lset_eq … Hlset_eq) b) 
734cases (b∈s1)
735[ 1: #_ #H lapply (H (refl ??)) #Hmem >H @refl
736| 2: cases (b∈s2) // #H lapply (H (refl ??)) #Habsurd destruct
737] qed.
738
739lemma lset_eq_filter_eq :
740  ∀A : DeqSet.
741  ∀s1,s2 : lset (carr A).
742  lset_eq ? s1 s2 → 
743  ∀l.
744     (filter ? (λwb.¬wb∈s1) l) =
745     (filter ? (λwb.¬wb∈s2) l).
746#A #s1 #s2 #Heq #l elim l
747[ 1: @refl
748| 2: #hd #tl #Hind >filter_cons_unfold >filter_cons_unfold
749      >(lset_eq_memb_eq … Heq) cases (hd∈s2)
750      normalize in match (notb ?); normalize nodelta
751      try @Hind >Hind @refl
752] qed.
753
754lemma cons_monotonic_eq : ∀A. ∀l1,l2 : lset A. lset_eq A l1 l2 → ∀x. lset_eq A (x::l1) (x::l2).
755#A #l1 #l2 #Heq #x cases Heq #Hincl1 #Hincl2
756@conj
757[ 1: @cons_monotonic_inclusion
758| 2: @cons_monotonic_inclusion ]
759assumption
760qed.
761
762(* Properties of difference and remove *)
763lemma lset_difference_unfold :
764  ∀A : DeqSet.
765  ∀s1, s2 : lset (carr A).
766  ∀hd. lset_difference A (hd :: s1) s2 =
767    if hd∈s2 then
768      lset_difference A s1 s2
769    else
770      hd :: (lset_difference A s1 s2).
771#A #s1 #s2 #hd normalize
772cases (hd∈s2) @refl
773qed.
774
775lemma lset_difference_unfold2 :
776  ∀A : DeqSet.
777  ∀s1, s2 : lset (carr A).
778  ∀hd. lset_difference A s1 (hd :: s2) =
779       lset_difference A (lset_remove ? s1 hd) s2.
780#A #s1
781elim s1
782[ 1: //
783| 2: #hd1 #tl1 #Hind #s2 #hd
784     whd in match (lset_remove ???);
785     whd in match (lset_difference A ??);
786     whd in match (memb ???);
787     lapply (eqb_true … hd1 hd)
788     cases (hd1==hd) normalize nodelta
789     [ 1: * #H #_ lapply (H (refl ??)) -H #H
790           @Hind
791     | 2: * #_ #Hguard >lset_difference_unfold
792          cases (hd1∈s2) normalize in match (notb ?); normalize nodelta
793          <Hind @refl ]
794] qed.         
795
796lemma lset_difference_disjoint :
797 ∀A : DeqSet.
798 ∀s1,s2 : lset (carr A).
799  lset_disjoint A s1 (lset_difference A s2 s1).
800#A #s1 elim s1
801[ 1: #s2 normalize #x #y *
802| 2: #hd1 #tl1 #Hind #s2 >lset_difference_unfold2 #x #y
803     whd in ⊢ (% → ?); *
804     [ 2: @Hind
805     | 1: #Heq >Heq elim s2
806          [ 1: normalize *
807          | 2: #hd2 #tl2 #Hind2 whd in match (lset_remove ???);
808               lapply (eqb_true … hd2 hd1)
809               cases (hd2 == hd1) normalize in match (notb ?); normalize nodelta * #H1 #H2
810               [ 1: @Hind2
811               | 2: >lset_difference_unfold cases (hd2 ∈ tl1) normalize nodelta try @Hind2
812                     whd in ⊢ (% → ?); *
813                     [ 1: #Hyhd2 destruct % #Heq lapply (H2 … (sym_eq … Heq)) #Habsurd destruct
814                     | 2: @Hind2 ]
815               ]
816          ]
817    ]
818] qed.
819
820
821lemma lset_remove_split : ∀A : DeqSet. ∀l1,l2 : lset A. ∀elt. lset_remove A (l1 @ l2) elt = (lset_remove … l1 elt) @ (lset_remove … l2 elt).
822#A #l1 #l2 #elt /2 by filter_append/ qed.
823
824lemma lset_inclusion_remove :
825  ∀A : DeqSet.
826  ∀s1, s2 : lset A.
827  lset_inclusion ? s1 s2 →
828  ∀elt. lset_inclusion ? (lset_remove ? s1 elt) (lset_remove ? s2 elt).
829#A #s1 elim s1
830[ 1: normalize //
831| 2: #hd1 #tl1 #Hind1 #s2 * #Hmem #Hincl
832     elim (list_mem_split ??? Hmem) #s2A * #s2B #Hs2_eq destruct #elt
833     whd in match (lset_remove ???);
834     @(match (hd1 == elt)
835       return λx. (hd1 == elt = x) → ?
836       with
837       [ true ⇒ λH. ?
838       | false ⇒ λH. ? ] (refl ? (hd1 == elt))) normalize in match (notb ?);
839     normalize nodelta
840     [ 1:  @Hind1 @Hincl
841     | 2: whd @conj
842          [ 2: @(Hind1 … Hincl)
843          | 1: >lset_remove_split >lset_remove_split
844               normalize in match (lset_remove A [hd1] elt);
845               >H normalize nodelta @mem_append_backwards %2
846               @mem_append_backwards %1 normalize %1 @refl ]
847     ]
848] qed.
849
850lemma lset_difference_lset_eq :
851  ∀A : DeqSet. ∀a,b,c.
852   lset_eq A b c →
853   lset_eq A (lset_difference A a b) (lset_difference A a c).
854#A #a #b #c #Heq
855whd in match (lset_difference ???) in ⊢ (??%%);   
856elim a
857[ 1: normalize @conj @I
858| 2: #hda #tla #Hind whd in match (foldr ?????) in ⊢ (??%%);
859     >(lset_eq_memb_eq … Heq hda) cases (hda∈c)
860     normalize in match (notb ?); normalize nodelta
861     try @Hind @cons_monotonic_eq @Hind
862] qed.
863
864lemma lset_difference_lset_remove_commute :
865  ∀A:DeqSet.
866  ∀elt,s1,s2.
867  (lset_difference A (lset_remove ? s1 elt) s2) =
868  (lset_remove A (lset_difference ? s1 s2) elt).
869#A #elt #s1 #s2
870elim s1 try //
871#hd #tl #Hind
872>lset_difference_unfold
873whd in match (lset_remove ???);
874@(match (hd==elt) return λx. (hd==elt) = x → ?
875  with
876  [ true ⇒ λHhd. ?
877  | false ⇒ λHhd. ?
878  ] (refl ? (hd==elt)))
879@(match (hd∈s2) return λx. (hd∈s2) = x → ?
880  with
881  [ true ⇒ λHmem. ?
882  | false ⇒ λHmem. ?
883  ] (refl ? (hd∈s2)))
884>notb_true >notb_false normalize nodelta try //
885try @Hind
886[ 1:  whd in match (lset_remove ???) in ⊢ (???%); >Hhd
887     elim (eqb_true ? elt elt) #_ #H >(H (refl ??))
888     normalize in match (notb ?); normalize nodelta @Hind
889| 2: >lset_difference_unfold >Hmem @Hind
890| 3: whd in match (lset_remove ???) in ⊢ (???%);
891     >lset_difference_unfold >Hhd >Hmem
892     normalize in match (notb ?);
893     normalize nodelta >Hind @refl
894] qed.
895
896(* Inversion lemma on emptyness *)
897lemma lset_eq_empty_inv : ∀A. ∀l. lset_eq A l [ ] → l = [ ].
898#A #l elim l //
899#hd' #tl' normalize #Hind * * @False_ind
900qed.
901
902(* Inversion lemma on singletons *)
903lemma lset_eq_singleton_inv : ∀A,hd,l. lset_eq A [hd] (hd::l) → All ? (λx.x=hd) l.
904#A #hd #l
905* #Hincl1 whd in ⊢ (% → ?); * #_ @All_mp
906normalize #a * [ 1: #H @H | 2: @False_ind ]
907qed.
908
909(* Permutation of two elements on top of the list is ok. *)
910lemma lset_eq_permute : ∀A,l,x1,x2. lset_eq A (x1 :: x2 :: l) (x2 :: x1 :: l).
911#A #l #x1 #x2 @conj normalize
912[ 1: /5 by cons_preserves_inclusion, All_mp, or_introl, or_intror, conj/
913| 2: /5 by cons_preserves_inclusion, All_mp, or_introl, or_intror, conj/
914] qed.
915
916(* "contraction" of an element. *)
917lemma lset_eq_contract : ∀A,l,x. lset_eq A (x :: x :: l) (x :: l).
918#A #l #x @conj
919[ 1: /5 by or_introl, conj, transitive_lset_inclusion/
920| 2: /5 by cons_preserves_inclusion, cons_monotonic_inclusion/ ]
921qed.
922
923(* We don't need more than one instance of each element. *)
924lemma lset_eq_filter : ∀A:DeqSet.∀tl.∀hd.
925  lset_eq A (hd :: (filter ? (λy.notb (eqb A y hd)) tl)) (hd :: tl).
926#A #tl elim tl
927[ 1: #hd normalize /4 by or_introl, conj, I/
928| 2: #hd' #tl' #Hind #hd >filter_cons_unfold
929     lapply (eqb_true A hd' hd) cases (hd'==hd)
930     [ 2: #_ normalize in match (notb ?); normalize nodelta
931          lapply (cons_monotonic_eq … (Hind hd) hd') #H
932          lapply (lset_eq_permute ? tl' hd' hd) #H'
933          @(transitive_lset_eq ? ? (hd'::hd::tl') ? … H')
934          @(transitive_lset_eq ? ?? (hd'::hd::tl') … H)
935          @lset_eq_permute
936     | 1: * #Heq #_ >(Heq (refl ??)) normalize in match (notb ?); normalize nodelta
937          lapply (Hind hd) #H
938          @(transitive_lset_eq ? ? (hd::tl') (hd::hd::tl') H)
939          @conj
940          [ 1: whd @conj /2 by or_introl/ @cons_preserves_inclusion @cons_preserves_inclusion
941               @reflexive_lset_inclusion
942          | 2: whd @conj /2 by or_introl/ ]
943     ]
944] qed.
945
946lemma lset_inclusion_filter_self :
947  ∀A:DeqSet.∀l,pred.
948    lset_inclusion A (filter ? pred l) l.
949#A #l #pred elim l
950[ 1: normalize @I
951| 2: #hd #tl #Hind whd in match (filter ???);
952     cases (pred hd) normalize nodelta
953     [ 1: @cons_monotonic_inclusion @Hind
954     | 2: @cons_preserves_inclusion @Hind ]
955] qed.   
956
957lemma lset_inclusion_filter_monotonic :
958  ∀A:DeqSet.∀l1,l2. lset_inclusion (carr A) l1 l2 →
959  ∀elt. lset_inclusion A (filter ? (λx.¬(x==elt)) l1) (filter ? (λx.¬(x==elt)) l2).
960#A #l1 elim l1
961[ 1: #l2 normalize //
962| 2: #hd1 #tl1 #Hind #l2 whd in ⊢ (% → ?); * #Hmem1 #Htl1_incl #elt
963     whd >filter_cons_unfold
964     lapply (eqb_true A hd1 elt) cases (hd1==elt)
965     [ 1: * #H1 #_ lapply (H1 (refl ??)) #H1_eq >H1_eq in Hmem1; #Hmem
966          normalize in match (notb ?); normalize nodelta @Hind assumption
967     | 2: * #_ #Hneq normalize in match (notb ?); normalize nodelta
968          whd @conj
969          [ 1: elim l2 in Hmem1; try //
970               #hd2 #tl2 #Hincl whd in ⊢ (% → ?); *
971               [ 1: #Heq >Heq in Hneq; normalize
972                    lapply (eqb_true A hd2 elt) cases (hd2==elt)
973                    [ 1: * #H #_ #H2 lapply (H2 (H (refl ??))) #Habsurd destruct (Habsurd)
974                    | 2: #_ normalize nodelta #_ /2 by or_introl/ ]
975               | 2: #H lapply (Hincl H) #Hok
976                    normalize cases (hd2==elt) normalize nodelta
977                    [ 1: @Hok
978                    | 2: %2 @Hok ] ]
979          | 2: @Hind assumption ] ] ]
980qed.
981
982(* removing an element of two equivalent sets conserves equivalence. *)
983lemma lset_eq_filter_monotonic :
984  ∀A:DeqSet.∀l1,l2. lset_eq (carr A) l1 l2 →
985  ∀elt. lset_eq A (filter ? (λx.¬(x==elt)) l1) (filter ? (λx.¬(x==elt)) l2).
986#A #l1 #l2 * #Hincl1 #Hincl2 #elt @conj
987/2 by lset_inclusion_filter_monotonic/
988qed.
989
990(* ---------------- Concrete implementation of sets --------------------- *)
991
992(* We can give an explicit characterization of equivalent sets: they are permutations with repetitions, i,e.
993   a composition of transpositions and duplications. We restrict ourselves to DeqSets. *)
994inductive iso_step_lset (A : DeqSet) : lset A → lset A → Prop ≝
995| lset_transpose : ∀a,x,b,y,c. iso_step_lset A (a @ [x] @ b @ [y] @ c) (a @ [y] @ b @ [x] @ c)
996| lset_weaken : ∀a,x,b. iso_step_lset A (a @ [x] @ b) (a @ [x] @ [x] @ b)
997| lset_contract : ∀a,x,b. iso_step_lset A (a @ [x] @ [x] @ b) (a @ [x] @ b).
998
999(* The equivalence is the reflexive, transitive and symmetric closure of iso_step_lset *)
1000inductive lset_eq_concrete (A : DeqSet) : lset A → lset A → Prop ≝
1001| lset_trans : ∀a,b,c. iso_step_lset A a b → lset_eq_concrete A b c → lset_eq_concrete A a c
1002| lset_refl  : ∀a. lset_eq_concrete A a a.
1003
1004(* lset_eq_concrete is symmetric and transitive *)
1005lemma transitive_lset_eq_concrete : ∀A,l1,l2,l3. lset_eq_concrete A l1 l2 → lset_eq_concrete A l2 l3 → lset_eq_concrete A l1 l3.
1006#A #l1 #l2 #l3 #Hequiv
1007elim Hequiv //
1008#a #b #c #Hstep #Hequiv #Hind #Hcl3 lapply (Hind Hcl3) #Hbl3
1009@(lset_trans ???? Hstep Hbl3)
1010qed.
1011
1012lemma symmetric_step : ∀A,l1,l2. iso_step_lset A l1 l2 → iso_step_lset A l2 l1.
1013#A #l1 #l2 * /2/ qed.
1014
1015lemma symmetric_lset_eq_concrete : ∀A,l1,l2. lset_eq_concrete A l1 l2 → lset_eq_concrete A l2 l1.
1016#A #l1 #l2 #H elim H //
1017#a #b #c #Hab #Hbc #Hcb
1018@(transitive_lset_eq_concrete ???? Hcb ?)
1019@(lset_trans … (symmetric_step ??? Hab) (lset_refl …))
1020qed.
1021 
1022(* lset_eq_concrete is conserved by cons. *)
1023lemma equivalent_step_cons : ∀A,l1,l2. iso_step_lset A l1 l2 → ∀x. iso_step_lset A (x :: l1) (x :: l2).
1024#A #l1 #l2 * // qed. (* That // was impressive. *)
1025
1026lemma lset_eq_concrete_cons : ∀A,l1,l2. lset_eq_concrete A l1 l2 → ∀x. lset_eq_concrete A (x :: l1) (x :: l2).
1027#A #l1 #l2 #Hequiv elim Hequiv try //
1028#a #b #c #Hab #Hbc #Hind #x %1{(equivalent_step_cons ??? Hab x) (Hind x)}
1029qed.
1030
1031lemma absurd_list_eq_append : ∀A.∀x.∀l1,l2:list A. [ ] = l1 @ [x] @ l2 → False.
1032#A #x #l1 #l2 elim l1 normalize
1033[ 1: #Habsurd destruct
1034| 2: #hd #tl #_ #Habsurd destruct
1035] qed.
1036
1037(* Inversion lemma for emptyness, step case *)
1038lemma equivalent_iso_step_empty_inv : ∀A,l. iso_step_lset A l [] → l = [ ].
1039#A #l elim l //
1040#hd #tl #Hind #H inversion H
1041[ 1: #a #x #b #y #c #_ #Habsurd
1042      @(False_ind … (absurd_list_eq_append ? y … Habsurd))
1043| 2: #a #x #b #_ #Habsurd
1044      @(False_ind … (absurd_list_eq_append ? x … Habsurd))
1045| 3: #a #x #b #_ #Habsurd
1046      @(False_ind … (absurd_list_eq_append ? x … Habsurd))
1047] qed.
1048
1049(* Same thing for non-emptyness *)
1050lemma equivalent_iso_step_cons_inv : ∀A,l1,l2. l1 ≠ [ ] → iso_step_lset A l1 l2 → l2 ≠ [ ].
1051#A #l1 elim l1
1052[ 1: #l2 * #H @(False_ind … (H (refl ??)))
1053| 2: #hd #tl #H #l2 #_ #Hstep % #Hl2 >Hl2 in Hstep; #Hstep
1054     lapply (equivalent_iso_step_empty_inv … Hstep) #Habsurd destruct
1055] qed.
1056
1057lemma lset_eq_concrete_cons_inv : ∀A,l1,l2. l1 ≠ [ ] → lset_eq_concrete A l1 l2 → l2 ≠ [ ].
1058#A #l1 #l2 #Hl1 #H lapply Hl1 elim H
1059[ 2: #a #H @H
1060| 1: #a #b #c #Hab #Hbc #H #Ha lapply (equivalent_iso_step_cons_inv … Ha Hab) #Hb @H @Hb
1061] qed.
1062
1063lemma lset_eq_concrete_empty_inv : ∀A,l1,l2. l1 = [ ] → lset_eq_concrete A l1 l2 → l2 = [ ].
1064#A #l1 #l2 #Hl1 #H lapply Hl1 elim H //
1065#a #b #c #Hab #Hbc #Hbc_eq #Ha >Ha in Hab; #H_b lapply (equivalent_iso_step_empty_inv … ?? (symmetric_step … H_b))
1066#Hb @Hbc_eq @Hb
1067qed.
1068
1069(* Square equivalence diagram *)
1070lemma square_lset_eq_concrete :
1071  ∀A. ∀a,b,a',b'. lset_eq_concrete A a b → lset_eq_concrete A a a' → lset_eq_concrete A b b' → lset_eq_concrete A a' b'.
1072#A #a #b #a' #b' #H1 #H2 #H3
1073@(transitive_lset_eq_concrete ???? (symmetric_lset_eq_concrete … H2))
1074@(transitive_lset_eq_concrete ???? H1)
1075@H3
1076qed.
1077
1078(* Make the transposition of elements visible at top-level *)
1079lemma transpose_lset_eq_concrete :
1080  ∀A. ∀x,y,a,b,c,a',b',c'.
1081  lset_eq_concrete A (a @ [x] @ b @ [y] @ c) (a' @ [x] @ b' @ [y] @ c') →
1082  lset_eq_concrete A (a @ [y] @ b @ [x] @ c) (a' @ [y] @ b' @ [x] @ c').
1083#A #x #y #a #b #c #a' #b' #c
1084#H @(square_lset_eq_concrete ????? H) /2 by lset_trans, lset_refl, lset_transpose/
1085qed.
1086
1087lemma switch_lset_eq_concrete :
1088  ∀A. ∀a,b,c. lset_eq_concrete A (a@[b]@c) ([b]@a@c).
1089#A #a elim a //
1090#hda #tla #Hind #b #c lapply (Hind hda c) #H
1091lapply (lset_eq_concrete_cons … H b)
1092#H' normalize in H' ⊢ %; @symmetric_lset_eq_concrete
1093/3 by lset_transpose, lset_trans, symmetric_lset_eq_concrete/
1094qed.
1095
1096(* Folding a commutative and idempotent function on equivalent sets yields the same result. *)
1097lemma lset_eq_concrete_fold :
1098  ∀A : DeqSet.
1099  ∀acctype : Type[0].
1100  ∀l1,l2 : list (carr A).
1101  lset_eq_concrete A l1 l2 →
1102  ∀f:carr A → acctype → acctype.
1103  (∀x1,x2. ∀acc. f x1 (f x2 acc) = f x2 (f x1 acc)) →
1104  (∀x.∀acc. f x (f x acc) = f x acc) →
1105  ∀acc. foldr ?? f acc l1 = foldr ?? f acc l2.
1106#A #acctype #l1 #l2 #Heq #f #Hcomm #Hidem
1107elim Heq
1108try //
1109#a' #b' #c' #Hstep #Hbc #H #acc <H -H
1110elim Hstep
1111[ 1: #a #x #b #y #c
1112     >fold_append >fold_append >fold_append >fold_append
1113     >fold_append >fold_append >fold_append >fold_append
1114     normalize
1115     cut (f x (foldr A acctype f (f y (foldr A acctype f acc c)) b) =
1116          f y (foldr A acctype f (f x (foldr A acctype f acc c)) b)) [   
1117     elim c
1118     [ 1: normalize elim b
1119          [ 1: normalize >(Hcomm x y) @refl
1120          | 2: #hdb #tlb #Hind normalize
1121               >(Hcomm x hdb) >(Hcomm y hdb) >Hind @refl ]
1122     | 2: #hdc #tlc #Hind normalize elim b
1123          [ 1: normalize >(Hcomm x y) @refl
1124          | 2: #hdb #tlb #Hind normalize
1125               >(Hcomm x hdb) >(Hcomm y hdb) >Hind @refl ] ]
1126     ]
1127     #Hind >Hind @refl
1128| 2: #a #x #b
1129     >fold_append >fold_append >fold_append >(fold_append ?? ([x]@[x]))
1130     normalize >Hidem @refl
1131| 3: #a #x #b
1132     >fold_append  >(fold_append ?? ([x]@[x])) >fold_append >fold_append
1133     normalize >Hidem @refl
1134] qed.
1135
1136(* Folding over equivalent sets yields equivalent results, for any equivalence. *)
1137lemma inj_to_fold_inj :
1138  ∀A,acctype : Type[0].
1139  ∀eqrel : acctype → acctype → Prop.
1140  ∀refl_eqrel  : reflexive ? eqrel.
1141  ∀trans_eqrel : transitive ? eqrel.
1142  ∀sym_eqrel   : symmetric ? eqrel.
1143  ∀f           : A → acctype → acctype.
1144  (∀x:A.∀acc1:acctype.∀acc2:acctype.eqrel acc1 acc2→eqrel (f x acc1) (f x acc2)) →
1145  ∀l : list A. ∀acc1, acc2 : acctype. eqrel acc1 acc2 → eqrel (foldr … f acc1 l) (foldr … f acc2 l).
1146#A #acctype #eqrel #R #T #S #f #Hinj #l elim l
1147//
1148#hd #tl #Hind #acc1 #acc2 #Heq normalize @Hinj @Hind @Heq
1149qed.
1150
1151(* We need to extend the above proof to arbitrary equivalence relation instead of
1152   just standard equality. *)
1153lemma lset_eq_concrete_fold_ext :
1154  ∀A : DeqSet.
1155  ∀acctype : Type[0].
1156  ∀eqrel : acctype → acctype → Prop.
1157  ∀refl_eqrel  : reflexive ? eqrel.
1158  ∀trans_eqrel : transitive ? eqrel.
1159  ∀sym_eqrel   : symmetric ? eqrel.
1160  ∀f:carr A → acctype → acctype.
1161  (∀x,acc1,acc2. eqrel acc1 acc2 → eqrel (f x acc1) (f x acc2)) →
1162  (∀x1,x2. ∀acc. eqrel (f x1 (f x2 acc)) (f x2 (f x1 acc))) →
1163  (∀x.∀acc. eqrel (f x (f x acc)) (f x acc)) →
1164  ∀l1,l2 : list (carr A).
1165  lset_eq_concrete A l1 l2 → 
1166  ∀acc. eqrel (foldr ?? f acc l1) (foldr ?? f acc l2).
1167#A #acctype #eqrel #R #T #S #f #Hinj #Hcomm #Hidem #l1 #l2 #Heq
1168elim Heq
1169try //
1170#a' #b' #c' #Hstep #Hbc #H inversion Hstep
1171[ 1: #a #x #b #y #c #HlA #HlB #_ #acc
1172     >HlB in H; #H @(T … ? (H acc))
1173     >fold_append >fold_append >fold_append >fold_append
1174     >fold_append >fold_append >fold_append >fold_append
1175     normalize
1176     cut (eqrel (f x (foldr ? acctype f (f y (foldr ? acctype f acc c)) b))
1177                (f y (foldr ? acctype f (f x (foldr ? acctype f acc c)) b)))
1178     [ 1:
1179     elim c
1180     [ 1: normalize elim b
1181          [ 1: normalize @(Hcomm x y)
1182          | 2: #hdb #tlb #Hind normalize
1183               lapply (Hinj hdb ?? Hind) #Hind'
1184               lapply (T … Hind' (Hcomm ???)) #Hind''
1185               @S @(triangle_diagram ? eqrel R T S … Hind'') @Hcomm ]
1186     | 2: #hdc #tlc #Hind normalize elim b
1187          [ 1: normalize @(Hcomm x y)
1188          | 2: #hdb #tlb #Hind normalize
1189               lapply (Hinj hdb ?? Hind) #Hind'
1190               lapply (T … Hind' (Hcomm ???)) #Hind''
1191               @S @(triangle_diagram ? eqrel R T S … Hind'') @Hcomm ]
1192     ] ]
1193     #Hind @(inj_to_fold_inj … eqrel R T S ? Hinj … Hind)
1194| 2: #a #x #b #HeqA #HeqB #_ #acc >HeqB in H; #H @(T … (H acc))
1195     >fold_append >fold_append >fold_append >(fold_append ?? ([x]@[x]))
1196     normalize @(inj_to_fold_inj … eqrel R T S ? Hinj) @S @Hidem
1197| 3: #a #x #b #HeqA #HeqB #_ #acc >HeqB in H; #H @(T … (H acc))
1198     >fold_append >(fold_append ?? ([x]@[x])) >fold_append >fold_append
1199     normalize @(inj_to_fold_inj … eqrel R T S ? Hinj) @Hidem
1200] qed.
1201
1202(* Prepare some well-founded induction principles on lists. The idea is to perform
1203   an induction on the sequence of filterees of a list : taking the first element,
1204   filtering it out of the tail, etc. We give such principles for pairs of lists
1205   and isolated lists.  *)
1206
1207(* The two lists [l1,l2] share at least the head of l1. *)
1208definition head_shared ≝ λA. λl1,l2 : list A.
1209match l1 with
1210[ nil ⇒ l2 = (nil ?)
1211| cons hd _ ⇒  mem … hd l2
1212].
1213
1214(* Relation on pairs of lists, as described above. *)
1215definition filtered_lists : ∀A:DeqSet. relation (list (carr A)×(list (carr A))) ≝
1216λA:DeqSet. λll1,ll2.
1217let 〈la1,lb1〉 ≝ ll1 in
1218let 〈la2,lb2〉 ≝ ll2 in
1219match la2 with
1220[ nil ⇒ False
1221| cons hda2 tla2 ⇒
1222    head_shared ? la2 lb2 ∧
1223    la1 = filter … (λx.¬(x==hda2)) tla2 ∧
1224    lb1 = filter … (λx.¬(x==hda2)) lb2
1225].
1226
1227(* Notice the absence of plural : this relation works on a simple list, not a pair. *)
1228definition filtered_list : ∀A:DeqSet. relation (list (carr A)) ≝
1229λA:DeqSet. λl1,l2.
1230match l2 with
1231[ nil ⇒ False
1232| cons hd2 tl2 ⇒
1233    l1 = filter … (λx.¬(x==hd2)) l2
1234].
1235
1236(* Relation on lists based on their lengths. We know this one is well-founded. *)
1237definition length_lt : ∀A:DeqSet. relation (list (carr A)) ≝
1238λA:DeqSet. λl1,l2. lt (length ? l1) (length ? l2).
1239
1240(* length_lt can be extended on pairs by just measuring the first component *)
1241definition length_fst_lt : ∀A:DeqSet. relation (list (carr A) × (list (carr A))) ≝
1242λA:DeqSet. λll1,ll2. lt (length ? (\fst ll1)) (length ? (\fst ll2)).
1243
1244lemma filter_length : ∀A. ∀l. ∀f. |filter A f l| ≤ |l|.
1245#A #l #f elim l //
1246#hd #tl #Hind whd in match (filter ???); cases (f hd) normalize nodelta
1247[ 1: /2 by le_S_S/
1248| 2: @le_S @Hind
1249] qed.
1250
1251(* The order on lists defined by their length is wf *)
1252lemma length_lt_wf : ∀A. ∀l. WF (list (carr A)) (length_lt A) l.
1253#A #l % elim l
1254[ 1: #a normalize in ⊢ (% → ?); #H lapply (le_S_O_absurd ? H) @False_ind
1255| 2: #hd #tl #Hind #a #Hlt % #a' #Hlt' @Hind whd in Hlt Hlt'; whd
1256@(transitive_le … Hlt') @(monotonic_pred … Hlt)
1257qed.
1258
1259(* Order on pairs of list by measuring the first proj *)
1260lemma length_fst_lt_wf : ∀A. ∀ll. WF ? (length_fst_lt A) ll.
1261#A * #l1 #l2 % elim l1
1262[ 1: * #a1 #a2 normalize in ⊢ (% → ?); #H lapply (le_S_O_absurd ? H) @False_ind
1263| 2: #hd1 #tl1 #Hind #a #Hlt % #a' #Hlt' @Hind whd in Hlt Hlt'; whd
1264@(transitive_le … Hlt') @(monotonic_pred … Hlt)
1265qed.
1266
1267lemma length_to_filtered_lists : ∀A. subR ? (filtered_lists A) (length_fst_lt A).
1268#A whd * #a1 #a2 * #b1 #b2 elim b1
1269[ 1: @False_ind
1270| 2: #hd1 #tl1 #Hind whd in ⊢ (% → ?); * * #Hmem #Ha1 #Ha2 whd
1271     >Ha1 whd in match (length ??) in ⊢ (??%); @le_S_S @filter_length
1272] qed.
1273
1274lemma length_to_filtered_list : ∀A. subR ? (filtered_list A) (length_lt A).
1275#A whd #a #b elim b
1276[ 1: @False_ind
1277| 2: #hd #tl #Hind whd in ⊢ (% → ?); whd in match (filter ???);
1278     lapply (eqb_true ? hd hd) * #_ #H >(H (refl ??)) normalize in match (notb ?);
1279     normalize nodelta #Ha whd @le_S_S >Ha @filter_length ]
1280qed.
1281
1282(* Prove well-foundedness by embedding in lt *)
1283lemma filtered_lists_wf : ∀A. ∀ll. WF ? (filtered_lists A) ll.
1284#A #ll @(WF_antimonotonic … (length_to_filtered_lists A)) @length_fst_lt_wf
1285qed.
1286
1287lemma filtered_list_wf : ∀A. ∀l. WF ? (filtered_list A) l.
1288#A #l @(WF_antimonotonic … (length_to_filtered_list A)) @length_lt_wf
1289qed.
1290
1291definition Acc_elim : ∀A,R,x. WF A R x → (∀a. R a x → WF A R a) ≝
1292λA,R,x,acc.
1293match acc with
1294[ wf _ a0 ⇒ a0 ].
1295
1296(* Stolen from Coq. Warped to avoid prop-to-type restriction. *)
1297let rec WF_rect
1298  (A : Type[0])
1299  (R : A → A → Prop)
1300  (P : A → Type[0])
1301  (f : ∀ x : A.
1302       (∀ y : A. R y x → WF ? R y) →
1303       (∀ y : A. R y x → P y) → P x)
1304  (x : A)
1305  (a : WF A R x) on a : P x ≝
1306f x (Acc_elim … a) (λy:A. λRel:R y x. WF_rect A R P f y ((Acc_elim … a) y Rel)).
1307
1308lemma lset_eq_concrete_filter : ∀A:DeqSet.∀tl.∀hd.
1309  lset_eq_concrete A (hd :: (filter ? (λy.notb (eqb A y hd)) tl)) (hd :: tl).
1310#A #tl elim tl
1311[ 1: #hd //
1312| 2: #hd' #tl' #Hind #hd >filter_cons_unfold
1313     lapply (eqb_true A hd' hd)
1314     cases (hd'==hd)
1315     [ 2: #_ normalize in match (notb false); normalize nodelta
1316          >cons_to_append >(cons_to_append … hd')
1317          >cons_to_append in ⊢ (???%); >(cons_to_append … hd') in ⊢ (???%);
1318          @(transpose_lset_eq_concrete ? hd' hd [ ] [ ] (filter A (λy:A.¬y==hd) tl') [ ] [ ] tl')
1319          >nil_append >nil_append >nil_append >nil_append
1320          @lset_eq_concrete_cons >nil_append >nil_append
1321          @Hind
1322     | 1: * #H1 #_ lapply (H1 (refl ??)) #Heq normalize in match (notb ?); normalize nodelta
1323          >Heq >cons_to_append >cons_to_append in ⊢ (???%); >cons_to_append in ⊢ (???(???%));
1324          @(square_lset_eq_concrete A ([hd]@filter A (λy:A.¬y==hd) tl') ([hd]@tl'))
1325          [ 1: @Hind
1326          | 2: %2
1327          | 3: %1{???? ? (lset_refl ??)} /2 by lset_weaken/ ]
1328     ]
1329] qed.
1330
1331
1332(* The "abstract", observational definition of set equivalence implies the concrete one. *)
1333
1334lemma lset_eq_to_lset_eq_concrete_aux :
1335  ∀A,ll.
1336    head_shared … (\fst ll) (\snd ll) →
1337    lset_eq (carr A) (\fst ll) (\snd ll) →
1338    lset_eq_concrete A (\fst ll) (\snd ll).
1339#A #ll @(WF_ind ????? (filtered_lists_wf A ll))
1340* *
1341[ 1: #b2 #Hwf #Hind_wf whd in ⊢ (% → ?); #Hb2 >Hb2 #_ %2
1342| 2: #hd1 #tl1 #b2 #Hwf #Hind_wf whd in ⊢ (% → ?); #Hmem
1343     lapply (list_mem_split ??? Hmem) * #l2A * #l2B #Hb2 #Heq
1344     destruct
1345     lapply (Hind_wf 〈filter … (λx.¬x==hd1) tl1,filter … (λx.¬x==hd1) (l2A@l2B)〉)
1346     cut (filtered_lists A 〈filter A (λx:A.¬x==hd1) tl1,filter A (λx:A.¬x==hd1) (l2A@l2B)〉 〈hd1::tl1,l2A@[hd1]@l2B〉)
1347     [ @conj try @conj try @refl whd
1348       [ 1: /2 by /
1349       | 2: >filter_append in ⊢ (???%); >filter_append in ⊢ (???%);
1350            whd in match (filter ?? [hd1]);
1351            elim (eqb_true A hd1 hd1) #_ #H >(H (refl ??)) normalize in match (notb ?);
1352            normalize nodelta <filter_append @refl ] ]
1353     #Hfiltered #Hind_aux lapply (Hind_aux Hfiltered) -Hind_aux
1354     cut (lset_eq A (filter A (λx:A.¬x==hd1) tl1) (filter A (λx:A.¬x==hd1) (l2A@l2B)))
1355     [ 1: lapply (lset_eq_filter_monotonic … Heq hd1)
1356          >filter_cons_unfold >filter_append >(filter_append … [hd1])
1357          whd in match (filter ?? [hd1]);
1358          elim (eqb_true A hd1 hd1) #_ #H >(H (refl ??)) normalize in match (notb ?);
1359          normalize nodelta <filter_append #Hsol @Hsol ]
1360     #Hset_eq
1361     cut (head_shared A (filter A (λx:A.¬x==hd1) tl1) (filter A (λx:A.¬x==hd1) (l2A@l2B)))
1362     [ 1: lapply Hset_eq cases (filter A (λx:A.¬x==hd1) tl1)
1363          [ 1: whd in ⊢ (% → ?); * #_ elim (filter A (λx:A.¬x==hd1) (l2A@l2B)) //
1364               #hd' #tl' normalize #Hind * @False_ind
1365          | 2: #hd' #tl' * #Hincl1 #Hincl2 whd elim Hincl1 #Hsol #_ @Hsol ] ]
1366     #Hshared #Hind_aux lapply (Hind_aux Hshared Hset_eq)
1367     #Hconcrete_set_eq
1368     >cons_to_append
1369     @(transitive_lset_eq_concrete ? ([hd1]@tl1) ([hd1]@l2A@l2B) (l2A@[hd1]@l2B))
1370     [ 2: @symmetric_lset_eq_concrete @switch_lset_eq_concrete ]
1371     lapply (lset_eq_concrete_cons … Hconcrete_set_eq hd1) #Hconcrete_cons_eq
1372     @(square_lset_eq_concrete … Hconcrete_cons_eq)
1373     [ 1: @(lset_eq_concrete_filter ? tl1 hd1)
1374     | 2: @(lset_eq_concrete_filter ? (l2A@l2B) hd1) ]
1375] qed.
1376
1377lemma lset_eq_to_lset_eq_concrete : ∀A,l1,l2. lset_eq (carr A) l1 l2 → lset_eq_concrete A l1 l2.
1378#A *
1379[ 1: #l2 #Heq >(lset_eq_empty_inv … (symmetric_lset_eq … Heq)) //
1380| 2: #hd1 #tl1 #l2 #Hincl lapply Hincl lapply (lset_eq_to_lset_eq_concrete_aux ? 〈hd1::tl1,l2〉) #H @H
1381     whd elim Hincl * //
1382] qed.
1383
1384
1385(* The concrete one implies the abstract one. *)
1386lemma lset_eq_concrete_to_lset_eq : ∀A,l1,l2. lset_eq_concrete A l1 l2 → lset_eq A l1 l2.
1387#A #l1 #l2 #Hconcrete
1388elim Hconcrete try //
1389#a #b #c #Hstep #Heq_bc_concrete #Heq_bc
1390cut (lset_eq A a b)
1391[ 1: elim Hstep
1392     [ 1: #a' elim a'
1393          [ 2: #hda #tla #Hind #x #b' #y #c' >cons_to_append
1394               >(associative_append ? [hda] tla ?)
1395               >(associative_append ? [hda] tla ?)
1396               @cons_monotonic_eq >nil_append >nil_append @Hind
1397          | 1: #x #b' #y #c' >nil_append >nil_append
1398               elim b' try //
1399               #hdb #tlc #Hind >append_cons >append_cons in ⊢ (???%);
1400               >associative_append >associative_append
1401               lapply (cons_monotonic_eq … Hind hdb) #Hind'               
1402               @(transitive_lset_eq ? ([x]@[hdb]@tlc@[y]@c') ([hdb]@[x]@tlc@[y]@c'))
1403               /2 by transitive_lset_eq/ ]
1404     | 2: #a' elim a'
1405          [ 2: #hda #tla #Hind #x #b' >cons_to_append
1406               >(associative_append ? [hda] tla ?)
1407               >(associative_append ? [hda] tla ?)
1408               @cons_monotonic_eq >nil_append >nil_append @Hind
1409          | 1: #x #b' >nil_append >nil_append @conj normalize
1410               [ 1: @conj [ 1: %1 @refl ] elim b'
1411                    [ 1: @I
1412                    | 2: #hdb #tlb #Hind normalize @conj
1413                         [ 1: %2 %2 %1 @refl
1414                         | 2: @(All_mp … Hind) #a0 *
1415                              [ 1: #Heq %1 @Heq
1416                              | 2: * /2 by or_introl, or_intror/ ] ] ]
1417                    #H %2 %2 %2 @H
1418               | 2: @conj try @conj try /2 by or_introl, or_intror/ elim b'
1419                    [ 1: @I
1420                    | 2: #hdb #tlb #Hind normalize @conj
1421                         [ 1: %2 %1 @refl
1422                         | 2: @(All_mp … Hind) #a0 *
1423                              [ 1: #Heq %1 @Heq
1424                              | 2: #H %2 %2 @H ] ] ] ] ]
1425     | 3: #a #x #b elim a try @lset_eq_contract
1426          #hda #tla #Hind @cons_monotonic_eq @Hind ] ]
1427#Heq_ab @(transitive_lset_eq … Heq_ab Heq_bc)
1428qed.
1429
1430lemma lset_eq_fold :
1431  ∀A : DeqSet.
1432  ∀acctype : Type[0].
1433  ∀eqrel : acctype → acctype → Prop.
1434  ∀refl_eqrel  : reflexive ? eqrel.
1435  ∀trans_eqrel : transitive ? eqrel.
1436  ∀sym_eqrel   : symmetric ? eqrel.
1437  ∀f:carr A → acctype → acctype.
1438  (∀x,acc1,acc2. eqrel acc1 acc2 → eqrel (f x acc1) (f x acc2)) →
1439  (∀x1,x2. ∀acc. eqrel (f x1 (f x2 acc)) (f x2 (f x1 acc))) →
1440  (∀x.∀acc. eqrel (f x (f x acc)) (f x acc)) →
1441  ∀l1,l2 : list (carr A).
1442  lset_eq A l1 l2 → 
1443  ∀acc. eqrel (foldr ?? f acc l1) (foldr ?? f acc l2).
1444#A #acctype #eqrel #refl_eqrel #trans_eqrel #sym_eqrel #f #Hinj #Hsym #Hcontract #l1 #l2 #Heq #acc
1445lapply (lset_eq_to_lset_eq_concrete … Heq) #Heq_concrete
1446@(lset_eq_concrete_fold_ext A acctype eqrel refl_eqrel trans_eqrel sym_eqrel f Hinj Hsym Hcontract l1 l2 Heq_concrete acc)
1447qed.
1448
1449(* Additional lemmas on lsets *)
1450
1451lemma lset_difference_empty :
1452  ∀A : DeqSet.
1453  ∀s1. lset_difference A s1 [ ] = s1.
1454#A #s1 elim s1 try //
1455#hd #tl #Hind >lset_difference_unfold >Hind @refl
1456qed.
1457
1458lemma lset_not_mem_difference :
1459  ∀A : DeqSet. ∀s1,s2,s3. lset_inclusion (carr A) s1 (lset_difference ? s2 s3) → ∀x. mem ? x s1 → ¬(mem ? x s3).
1460#A #s1 #s2 #s3 #Hincl #x #Hmem
1461lapply (lset_difference_disjoint ? s3 s2) whd in ⊢ (% → ?); #Hdisjoint % #Hmem_s3
1462elim s1 in Hincl Hmem;
1463[ 1: #_ *
1464| 2: #hd #tl #Hind whd in ⊢ (% → %); * #Hmem_hd #Hall *
1465     [ 2: #Hmem_x_tl @Hind assumption
1466     | 1: #Heq lapply (Hdisjoint … Hmem_s3 Hmem_hd) * #H @H @Heq ]
1467] qed.
1468
1469lemma lset_mem_inclusion_mem :
1470  ∀A,s1,s2,elt.
1471  mem A elt s1 → lset_inclusion ? s1 s2 → mem ? elt s2.
1472#A #s1 elim s1
1473[ 1: #s2 #elt *
1474| 2: #hd #tl #Hind #s2 #elt *
1475     [ 1: #Heq destruct * //
1476     | 2: #Hmem_tl * #Hmem #Hall elim tl in Hall Hmem_tl;
1477          [ 1: #_ *
1478          | 2: #hd' #tl' #Hind * #Hmem' #Hall *
1479               [ 1: #Heq destruct @Hmem'
1480               | 2: #Hmem'' @Hind assumption ] ] ] ]
1481qed.
1482
1483lemma lset_remove_inclusion :
1484  ∀A : DeqSet. ∀s,elt.
1485    lset_inclusion A (lset_remove ? s elt) s.
1486#A #s elim s try // qed.
1487
1488lemma lset_difference_remove_inclusion :
1489  ∀A : DeqSet. ∀s1,s2,elt.
1490    lset_inclusion A
1491      (lset_difference ? (lset_remove ? s1 elt) s2) 
1492      (lset_difference ? s1 s2).
1493#A #s elim s try // qed.
1494
1495lemma lset_difference_permute :
1496  ∀A : DeqSet. ∀s1,s2,s3.
1497    lset_difference A s1 (s2 @ s3) =
1498    lset_difference A s1 (s3 @ s2).
1499#A #s1 #s2 elim s2 try //
1500#hd #tl #Hind #s3 >lset_difference_unfold2 >lset_difference_lset_remove_commute
1501>Hind elim s3 try //
1502#hd' #tl' #Hind' >cons_to_append >associative_append
1503>associative_append >(cons_to_append … hd tl)
1504>lset_difference_unfold2 >lset_difference_lset_remove_commute >nil_append
1505>lset_difference_unfold2 >lset_difference_lset_remove_commute >nil_append
1506<Hind' generalize in match (lset_difference ???); #foo
1507whd in match (lset_remove ???); whd in match (lset_remove ???) in ⊢ (??(?????%)?);
1508whd in match (lset_remove ???) in ⊢ (???%); whd in match (lset_remove ???) in ⊢ (???(?????%));
1509elim foo
1510[ 1: normalize @refl
1511| 2: #hd'' #tl'' #Hind normalize
1512      @(match (hd''==hd') return λx. ((hd''==hd') = x) → ? with
1513        [ true ⇒ λH. ?
1514        | false ⇒ λH. ?
1515        ] (refl ? (hd''==hd')))
1516      @(match (hd''==hd) return λx. ((hd''==hd) = x) → ? with
1517        [ true ⇒ λH'. ?
1518        | false ⇒ λH'. ?
1519        ] (refl ? (hd''==hd)))
1520      normalize nodelta
1521      try @Hind
1522[ 1: normalize >H normalize nodelta @Hind
1523| 2: normalize >H' normalize nodelta @Hind
1524| 3: normalize >H >H' normalize nodelta >Hind @refl
1525] qed.
1526
1527
1528
1529lemma lset_disjoint_dec :
1530  ∀A : DeqSet.
1531  ∀s1,elt,s2.
1532  mem ? elt s1 ∨ mem ? elt (lset_difference A (elt :: s2) s1).
1533#A #s1 #elt #s2
1534@(match elt ∈ s1 return λx. ((elt ∈ s1) = x) → ?
1535  with
1536  [ false ⇒ λHA. ?
1537  | true ⇒ λHA. ? ] (refl ? (elt ∈ s1)))
1538[ 1: lapply (memb_to_mem … HA) #Hmem
1539     %1 @Hmem
1540| 2: %2 elim s1 in HA;
1541     [ 1: #_ whd %1 @refl
1542     | 2: #hd1 #tl1 #Hind normalize in ⊢ (% → ?);
1543          >lset_difference_unfold
1544          >lset_difference_unfold2
1545          lapply (eqb_true ? elt hd1) whd in match (memb ???) in ⊢ (? → ? → %);
1546          cases (elt==hd1) normalize nodelta
1547          [ 1: #_ #Habsurd destruct
1548          | 2: #HA #HB >HB normalize nodelta %1 @refl ] ] ]
1549qed.
1550
1551lemma mem_filter : ∀A : DeqSet. ∀l,elt1,elt2.
1552  mem A elt1 (filter A (λx:A.¬x==elt2) l) → mem A elt1 l.
1553#A #l elim l try // #hd #tl #Hind #elt1 #elt2 /2 by lset_mem_inclusion_mem/
1554qed.
1555
1556lemma lset_inclusion_difference_aux :
1557  ∀A : DeqSet. ∀s1,s2.
1558  lset_inclusion A s1 s2 →
1559  (lset_eq A s2 (s1@lset_difference A s2 s1)).
1560#A #s1
1561@(WF_ind ????? (filtered_list_wf A s1))
1562*
1563[ 1: #_ #_ #s2 #_ >nil_append >lset_difference_empty @reflexive_lset_eq
1564| 2: #hd1 #tl1 #Hwf #Hind #s2 * #Hmem #Hincl
1565     lapply (Hind (filter ? (λx.¬x==hd1) tl1) ?)
1566     [ 1: whd normalize
1567          >(proj2 … (eqb_true ? hd1 hd1) (refl ??)) normalize nodelta @refl ]
1568     #Hind_wf     
1569     elim (list_mem_split ??? Hmem) #s2A * #s2B #Heq >Heq
1570     >cons_to_append in ⊢ (???%); >associative_append
1571     >lset_difference_unfold2
1572     >nil_append
1573     >lset_remove_split >lset_remove_split
1574     normalize in match (lset_remove ? [hd1] hd1);
1575     >(proj2 … (eqb_true ? hd1 hd1) (refl ??)) normalize nodelta
1576     >nil_append <lset_remove_split >lset_difference_lset_remove_commute
1577     lapply (Hind_wf (lset_remove A (s2A@s2B) hd1) ?)
1578     [ 1: lapply (lset_inclusion_remove … Hincl hd1)
1579          >Heq @lset_inclusion_eq2
1580          >lset_remove_split >lset_remove_split >lset_remove_split
1581          normalize in match (lset_remove ? [hd1] hd1);
1582          >(proj2 … (eqb_true ? hd1 hd1) (refl ??)) normalize nodelta
1583          >nil_append @reflexive_lset_eq ]
1584     #Hind >lset_difference_lset_remove_commute in Hind; <lset_remove_split #Hind
1585     @lset_eq_concrete_to_lset_eq
1586     lapply (lset_eq_to_lset_eq_concrete … (cons_monotonic_eq … Hind hd1)) #Hindc
1587     @(square_lset_eq_concrete ????? Hindc) -Hindc -Hind
1588     [ 1: @(transitive_lset_eq_concrete ?? ([hd1]@s2A@s2B) (s2A@[hd1]@s2B))
1589          [ 2: @symmetric_lset_eq_concrete @switch_lset_eq_concrete
1590          | 1: @lset_eq_to_lset_eq_concrete @lset_eq_filter ]
1591     | 2: @lset_eq_to_lset_eq_concrete @(transitive_lset_eq A … (lset_eq_filter ? ? hd1 …))
1592          elim (s2A@s2B)
1593          [ 1: normalize in match (lset_difference ???); @reflexive_lset_eq
1594          | 2: #hd2 #tl2 #Hind >lset_difference_unfold >lset_difference_unfold
1595               @(match (hd2∈filter A (λx:A.¬x==hd1) tl1)
1596                 return λx. ((hd2∈filter A (λx:A.¬x==hd1) tl1) = x) → ?
1597                 with
1598                 [ false ⇒ λH. ?
1599                 | true ⇒ λH. ?
1600                 ] (refl ? (hd2∈filter A (λx:A.¬x==hd1) tl1))) normalize nodelta
1601               [ 1: lapply (memb_to_mem … H) #Hfilter >(mem_to_memb … (mem_filter … Hfilter))
1602                    normalize nodelta @Hind
1603               | 2: @(match (hd2∈tl1)
1604                      return λx. ((hd2∈tl1) = x) → ?
1605                      with
1606                      [ false ⇒ λH'. ?
1607                      | true ⇒ λH'. ?
1608                      ] (refl ? (hd2∈tl1))) normalize nodelta
1609                      [ 1: (* We have hd2 = hd1 *)
1610                            cut (hd2 = hd1)
1611                            [ elim tl1 in H H';
1612                              [ 1: normalize #_ #Habsurd destruct (Habsurd)
1613                              | 2: #hdtl1 #tltl1 #Hind normalize in ⊢ (% → % → ?);
1614                                    lapply (eqb_true ? hdtl1 hd1)
1615                                    cases (hdtl1==hd1) normalize nodelta
1616                                    [ 1: * #H >(H (refl ??)) #_
1617                                         lapply (eqb_true ? hd2 hd1)
1618                                         cases (hd2==hd1) normalize nodelta *
1619                                         [ 1: #H' >(H' (refl ??)) #_ #_ #_ @refl
1620                                         | 2: #_ #_ @Hind ]
1621                                    | 2: #_ normalize lapply (eqb_true ? hd2 hdtl1)
1622                                         cases (hd2 == hdtl1) normalize nodelta *
1623                                         [ 1: #_ #_ #Habsurd destruct (Habsurd)
1624                                         | 2: #_ #_ @Hind ] ] ] ]
1625                           #Heq_hd2hd1 destruct (Heq_hd2hd1)
1626                           @lset_eq_concrete_to_lset_eq lapply (lset_eq_to_lset_eq_concrete … Hind)
1627                           #Hind' @(square_lset_eq_concrete … Hind')
1628                           [ 2: @lset_refl
1629                           | 1: >cons_to_append >cons_to_append in ⊢ (???%);
1630                                @(transitive_lset_eq_concrete … ([hd1]@[hd1]@tl1@lset_difference A tl2 (filter A (λx:A.¬x==hd1) tl1)))
1631                                [ 1: @lset_eq_to_lset_eq_concrete @symmetric_lset_eq @lset_eq_contract
1632                                | 2: >(cons_to_append … hd1 (lset_difference ???))
1633                                     @lset_eq_concrete_cons >nil_append >nil_append
1634                                     @symmetric_lset_eq_concrete @switch_lset_eq_concrete ] ]
1635                        | 2: @(match hd2 == hd1
1636                               return λx. ((hd2 == hd1) = x) → ?
1637                               with
1638                               [ true ⇒ λH''. ?
1639                               | false ⇒ λH''. ?
1640                               ] (refl ? (hd2 == hd1)))
1641                             [ 1: whd in match (lset_remove ???) in ⊢ (???%);
1642                                  >H'' normalize nodelta >((proj1 … (eqb_true …)) H'')
1643                                  @(transitive_lset_eq … Hind)
1644                                  @(transitive_lset_eq … (hd1::hd1::tl1@lset_difference A tl2 (filter A (λx:A.¬x==hd1) tl1)))
1645                                  [ 2: @lset_eq_contract ]                                                                   
1646                                  @lset_eq_concrete_to_lset_eq @lset_eq_concrete_cons                                 
1647                                  @switch_lset_eq_concrete
1648                             | 2: whd in match (lset_remove ???) in ⊢ (???%);
1649                                  >H'' >notb_false normalize nodelta
1650                                  @lset_eq_concrete_to_lset_eq
1651                                  lapply (lset_eq_to_lset_eq_concrete … Hind) #Hindc
1652                                  lapply (lset_eq_concrete_cons … Hindc hd2) #Hindc' -Hindc
1653                                  @(square_lset_eq_concrete … Hindc')
1654                                  [ 1: @symmetric_lset_eq_concrete
1655                                       >cons_to_append >cons_to_append in ⊢ (???%);
1656                                       >(cons_to_append … hd2) >(cons_to_append … hd1) in ⊢ (???%);
1657                                       @(switch_lset_eq_concrete ? ([hd1]@tl1) hd2 ?)
1658                                  | 2: @symmetric_lset_eq_concrete @(switch_lset_eq_concrete ? ([hd1]@tl1) hd2 ?)
1659                                  ]
1660                              ]
1661                        ]
1662                    ]
1663             ]
1664      ]
1665] qed.             
1666                                                       
1667lemma lset_inclusion_difference :
1668  ∀A : DeqSet.
1669  ∀s1,s2 : lset (carr A).
1670    lset_inclusion ? s1 s2 →
1671    ∃s2'. lset_eq ? s2 (s1 @ s2') ∧
1672          lset_disjoint ? s1 s2' ∧
1673          lset_eq ? s2' (lset_difference ? s2 s1).
1674#A #s1 #s2 #Hincl %{(lset_difference A s2 s1)} @conj try @conj
1675[ 1: @lset_inclusion_difference_aux @Hincl
1676| 2: /2 by lset_difference_disjoint/
1677| 3,4: @reflexive_lset_inclusion ]
1678qed.
1679
1680(* --------------------------------------------------------------------------- *)
1681(* Stuff on bitvectors, previously in memoryInjections.ma *)
1682(* --------------------------------------------------------------------------- *)
1683(* --------------------------------------------------------------------------- *)   
1684(* Some general lemmas on bitvectors (offsets /are/ bitvectors) *)
1685(* --------------------------------------------------------------------------- *)
1686 
1687lemma add_with_carries_n_O : ∀n,bv. add_with_carries n bv (zero n) false = 〈bv,zero n〉.
1688#n #bv whd in match (add_with_carries ????); elim bv //
1689#n #hd #tl #Hind whd in match (fold_right2_i ????????);
1690>Hind normalize
1691cases n in tl;
1692[ 1: #tl cases hd normalize @refl
1693| 2: #n' #tl cases hd normalize @refl ]
1694qed.
1695
1696lemma addition_n_0 : ∀n,bv. addition_n n bv (zero n) = bv.
1697#n #bv whd in match (addition_n ???);
1698>add_with_carries_n_O //
1699qed.
1700
1701lemma replicate_Sn : ∀A,sz,elt.
1702  replicate A (S sz) elt = elt ::: (replicate A sz elt).
1703// qed.
1704
1705lemma zero_Sn : ∀n. zero (S n) = false ::: (zero n). // qed.
1706
1707lemma negation_bv_Sn : ∀n. ∀xa. ∀a : BitVector n. negation_bv … (xa ::: a) = (notb xa) ::: (negation_bv … a).
1708#n #xa #a normalize @refl qed.
1709
1710(* useful facts on carry_of *)
1711lemma carry_of_TT : ∀x. carry_of true true x = true. // qed.
1712lemma carry_of_TF : ∀x. carry_of true false x = x. // qed.
1713lemma carry_of_FF : ∀x. carry_of false false x = false. // qed.
1714lemma carry_of_lcomm : ∀x,y,z. carry_of x y z = carry_of y x z. * * * // qed.
1715lemma carry_of_rcomm : ∀x,y,z. carry_of x y z = carry_of x z y. * * * // qed.
1716
1717
1718
1719definition one_bv ≝ λn. (\fst (add_with_carries … (zero n) (zero n) true)).
1720
1721lemma one_bv_Sn_aux : ∀n. ∀bits,flags : BitVector (S n).
1722    add_with_carries … (zero (S n)) (zero (S n)) true = 〈bits, flags〉 →
1723    add_with_carries … (zero (S (S n))) (zero (S (S n))) true = 〈false ::: bits, false ::: flags〉.
1724#n elim n
1725[ 1: #bits #flags elim (BitVector_Sn … bits) #hd_bits * #tl_bits #Heq_bits
1726     elim (BitVector_Sn … flags) #hd_flags * #tl_flags #Heq_flags
1727     >(BitVector_O … tl_flags) >(BitVector_O … tl_bits)
1728     normalize #Heq destruct (Heq) @refl
1729| 2: #n' #Hind #bits #flags elim (BitVector_Sn … bits) #hd_bits * #tl_bits #Heq_bits
1730     destruct #Hind >add_with_carries_Sn >replicate_Sn
1731     whd in match (zero ?) in Hind; lapply Hind
1732     elim (add_with_carries (S (S n'))
1733            (false:::replicate bool (S n') false)
1734            (false:::replicate bool (S n') false) true) #bits #flags #Heq destruct
1735            normalize >add_with_carries_Sn in Hind;
1736     elim (add_with_carries (S n') (replicate bool (S n') false)
1737                    (replicate bool (S n') false) true) #flags' #bits'
1738     normalize
1739     cases (match bits' in Vector return λsz:ℕ.(λfoo:Vector bool sz.bool) with 
1740            [VEmpty⇒true|VCons (sz:ℕ)   (cy:bool)   (tl:(Vector bool sz))⇒cy])
1741     normalize #Heq destruct @refl
1742] qed.     
1743
1744lemma one_bv_Sn : ∀n. one_bv (S (S n)) = false ::: (one_bv (S n)).
1745#n lapply (one_bv_Sn_aux n)
1746whd in match (one_bv ?) in ⊢ (? → (??%%));
1747elim (add_with_carries (S n) (zero (S n)) (zero (S n)) true) #bits #flags
1748#H lapply (H bits flags (refl ??)) #H2 >H2 @refl
1749qed.
1750
1751lemma increment_to_addition_n_aux : ∀n. ∀a : BitVector n.
1752    add_with_carries ? a (zero n) true = add_with_carries ? a (one_bv n) false.
1753#n   
1754elim n
1755[ 1: #a >(BitVector_O … a) normalize @refl
1756| 2: #n' cases n'
1757     [ 1: #Hind #a elim (BitVector_Sn ? a) #xa * #tl #Heq destruct
1758          >(BitVector_O … tl) normalize cases xa @refl
1759     | 2: #n'' #Hind #a elim (BitVector_Sn ? a) #xa * #tl #Heq destruct
1760          >one_bv_Sn >zero_Sn
1761          lapply (Hind tl)
1762          >add_with_carries_Sn >add_with_carries_Sn
1763          #Hind >Hind elim (add_with_carries (S n'') tl (one_bv (S n'')) false) #bits #flags
1764          normalize nodelta elim (BitVector_Sn … flags) #flags_hd * #flags_tl #Hflags_eq >Hflags_eq
1765          normalize nodelta @refl
1766] qed.         
1767
1768(* In order to use associativity on increment, we hide it under addition_n. *)
1769lemma increment_to_addition_n : ∀n. ∀a : BitVector n. increment ? a = addition_n ? a (one_bv n).
1770#n
1771whd in match (increment ??) in ⊢ (∀_.??%?);
1772whd in match (addition_n ???) in ⊢ (∀_.???%);
1773#a lapply (increment_to_addition_n_aux n a)
1774#Heq >Heq cases (add_with_carries n a (one_bv n) false) #bits #flags @refl
1775qed.
1776
1777(* Explicit formulation of addition *)
1778
1779(* Explicit formulation of the last carry bit *)
1780let rec ith_carry (n : nat) (a,b : BitVector n) (init : bool) on n : bool ≝
1781match n return λx. BitVector x → BitVector x → bool with
1782[ O ⇒ λ_,_. init
1783| S x ⇒ λa',b'.
1784  let hd_a ≝ head' … a' in
1785  let hd_b ≝ head' … b' in
1786  let tl_a ≝ tail … a' in
1787  let tl_b ≝ tail … b' in
1788  carry_of hd_a hd_b (ith_carry x tl_a tl_b init)
1789] a b.
1790
1791lemma ith_carry_unfold : ∀n. ∀init. ∀a,b : BitVector (S n).
1792  ith_carry ? a b init = (carry_of (head' … a) (head' … b) (ith_carry ? (tail … a) (tail … b) init)).
1793#n #init #a #b @refl qed.
1794
1795lemma ith_carry_Sn : ∀n. ∀init. ∀xa,xb. ∀a,b : BitVector n.
1796  ith_carry ? (xa ::: a) (xb ::: b) init = (carry_of xa xb (ith_carry ? a b init)). // qed.
1797
1798(* correction of [ith_carry] *)
1799lemma ith_carry_ok : ∀n. ∀init. ∀a,b,res_ab,flags_ab : BitVector (S n).
1800  〈res_ab,flags_ab〉 = add_with_carries ? a b init →
1801  head' … flags_ab = ith_carry ? a b init.
1802#n elim n
1803[ 1: #init #a #b #res_ab #flags_ab
1804     elim (BitVector_Sn … a) #hd_a * #tl_a #Heq_a
1805     elim (BitVector_Sn … b) #hd_b * #tl_b #Heq_b
1806     elim (BitVector_Sn … res_ab) #hd_res * #tl_res #Heq_res
1807     elim (BitVector_Sn … flags_ab) #hd_flags * #tl_flags #Heq_flags
1808     destruct
1809     >(BitVector_O … tl_a) >(BitVector_O … tl_b)
1810     cases hd_a cases hd_b cases init normalize #Heq destruct (Heq)
1811     @refl
1812| 2: #n' #Hind #init #a #b #res_ab #flags_ab
1813     elim (BitVector_Sn … a) #hd_a * #tl_a #Heq_a
1814     elim (BitVector_Sn … b) #hd_b * #tl_b #Heq_b
1815     elim (BitVector_Sn … res_ab) #hd_res * #tl_res #Heq_res
1816     elim (BitVector_Sn … flags_ab) #hd_flags * #tl_flags #Heq_flags
1817     destruct
1818     lapply (Hind … init tl_a tl_b tl_res tl_flags)
1819     >add_with_carries_Sn >(ith_carry_Sn (S n'))
1820     elim (add_with_carries (S n') tl_a tl_b init) #res_ab #flags_ab
1821     elim (BitVector_Sn … flags_ab) #hd_flags_ab * #tl_flags_ab #Heq_flags_ab >Heq_flags_ab
1822     normalize nodelta cases hd_flags_ab normalize nodelta
1823     whd in match (head' ? (S n') ?); #H1 #H2
1824     destruct (H2) lapply (H1 (refl ??)) whd in match (head' ???); #Heq <Heq @refl
1825] qed.
1826
1827(* Explicit formulation of ith bit of an addition, with explicit initial carry bit. *)
1828definition ith_bit ≝ λ(n : nat).λ(a,b : BitVector n).λinit.
1829match n return λx. BitVector x → BitVector x → bool with
1830[ O ⇒ λ_,_. init
1831| S x ⇒ λa',b'.
1832  let hd_a ≝ head' … a' in
1833  let hd_b ≝ head' … b' in
1834  let tl_a ≝ tail … a' in
1835  let tl_b ≝ tail … b' in
1836  xorb (xorb hd_a hd_b) (ith_carry x tl_a tl_b init)
1837] a b.
1838
1839lemma ith_bit_unfold : ∀n. ∀init. ∀a,b : BitVector (S n).
1840  ith_bit ? a b init =  xorb (xorb (head' … a) (head' … b)) (ith_carry ? (tail … a) (tail … b) init).
1841#n #a #b // qed.
1842
1843lemma ith_bit_Sn : ∀n. ∀init. ∀xa,xb. ∀a,b : BitVector n.
1844  ith_bit ? (xa ::: a) (xb ::: b) init =  xorb (xorb xa xb) (ith_carry ? a b init). // qed.
1845
1846(* correction of ith_bit *)
1847lemma ith_bit_ok : ∀n. ∀init. ∀a,b,res_ab,flags_ab : BitVector (S n).
1848  〈res_ab,flags_ab〉 = add_with_carries ? a b init →
1849  head' … res_ab = ith_bit ? a b init.
1850#n
1851cases n
1852[ 1: #init #a #b #res_ab #flags_ab
1853     elim (BitVector_Sn … a) #hd_a * #tl_a #Heq_a
1854     elim (BitVector_Sn … b) #hd_b * #tl_b #Heq_b
1855     elim (BitVector_Sn … res_ab) #hd_res * #tl_res #Heq_res
1856     elim (BitVector_Sn … flags_ab) #hd_flags * #tl_flags #Heq_flags
1857     destruct
1858     >(BitVector_O … tl_a) >(BitVector_O … tl_b)
1859     >(BitVector_O … tl_flags) >(BitVector_O … tl_res)
1860     normalize cases init cases hd_a cases hd_b normalize #Heq destruct @refl
1861| 2: #n' #init #a #b #res_ab #flags_ab
1862     elim (BitVector_Sn … a) #hd_a * #tl_a #Heq_a
1863     elim (BitVector_Sn … b) #hd_b * #tl_b #Heq_b
1864     elim (BitVector_Sn … res_ab) #hd_res * #tl_res #Heq_res
1865     elim (BitVector_Sn … flags_ab) #hd_flags * #tl_flags #Heq_flags
1866     destruct
1867     lapply (ith_carry_ok … init tl_a tl_b tl_res tl_flags)
1868     #Hcarry >add_with_carries_Sn elim (add_with_carries ? tl_a tl_b init) in Hcarry;
1869     #res #flags normalize nodelta elim (BitVector_Sn … flags) #hd_flags' * #tl_flags' #Heq_flags'
1870     >Heq_flags' normalize nodelta cases hd_flags' normalize nodelta #H1 #H2 destruct (H2)
1871     cases hd_a cases hd_b >ith_bit_Sn whd in match (head' ???) in H1 ⊢ %;
1872     <(H1 (refl ??)) @refl
1873] qed.
1874
1875(* Transform a function from bit-vectors to bits into a vector by folding *)
1876let rec bitvector_fold (n : nat) (v : BitVector n) (f : ∀sz. BitVector sz → bool) on v : BitVector n ≝
1877match v with
1878[ VEmpty ⇒ VEmpty ?
1879| VCons sz elt tl ⇒
1880  let bit ≝ f ? v in
1881  bit ::: (bitvector_fold ? tl f)
1882].
1883
1884(* Two-arguments version *)
1885let rec bitvector_fold2 (n : nat) (v1, v2 : BitVector n) (f : ∀sz. BitVector sz → BitVector sz → bool) on v1 : BitVector n ≝
1886match v1  with
1887[ VEmpty ⇒ λ_. VEmpty ?
1888| VCons sz elt tl ⇒ λv2'.
1889  let bit ≝ f ? v1 v2 in
1890  bit ::: (bitvector_fold2 ? tl (tail … v2') f)
1891] v2.
1892
1893lemma bitvector_fold2_Sn : ∀n,x1,x2,v1,v2,f.
1894  bitvector_fold2 (S n) (x1 ::: v1) (x2 ::: v2) f = (f ? (x1 ::: v1) (x2 ::: v2)) ::: (bitvector_fold2 … v1 v2 f). // qed.
1895
1896(* These functions pack all the relevant information (including carries) directly. *)
1897definition addition_n_direct ≝ λn,v1,v2,init. bitvector_fold2 n v1 v2 (λn,v1,v2. ith_bit n v1 v2 init).
1898
1899lemma addition_n_direct_Sn : ∀n,x1,x2,v1,v2,init.
1900  addition_n_direct (S n) (x1 ::: v1) (x2 ::: v2) init = (ith_bit ? (x1 ::: v1) (x2 ::: v2) init) ::: (addition_n_direct … v1 v2 init). // qed.
1901 
1902lemma tail_Sn : ∀n. ∀x. ∀a : BitVector n. tail … (x ::: a) = a. // qed.
1903
1904(* Prove the equivalence of addition_n_direct with add_with_carries *)
1905lemma addition_n_direct_ok : ∀n,carry,v1,v2.
1906  (\fst (add_with_carries n v1 v2 carry)) = addition_n_direct n v1 v2 carry.
1907#n elim n
1908[ 1: #carry #v1 #v2 >(BitVector_O … v1) >(BitVector_O … v2) normalize @refl
1909| 2: #n' #Hind #carry #v1 #v2
1910     elim (BitVector_Sn … v1) #hd1 * #tl1 #Heq1
1911     elim (BitVector_Sn … v2) #hd2 * #tl2 #Heq2
1912     lapply (Hind carry tl1 tl2)
1913     lapply (ith_bit_ok ? carry v1 v2)
1914     lapply (ith_carry_ok ? carry v1 v2)
1915     destruct
1916     #Hind >addition_n_direct_Sn
1917     >ith_bit_Sn >add_with_carries_Sn
1918     elim (add_with_carries n' tl1 tl2 carry) #bits #flags normalize nodelta
1919     cases (match flags in Vector return λsz:ℕ.(λfoo:Vector bool sz.bool) with 
1920            [VEmpty⇒carry|VCons (sz:ℕ)   (cy:bool)   (tl:(Vector bool sz))⇒cy])
1921     normalize nodelta #Hcarry' lapply (Hcarry' ?? (refl ??))
1922     whd in match head'; normalize nodelta
1923     #H1 #H2 >H1 >H2 @refl
1924] qed.
1925
1926lemma addition_n_direct_ok2 : ∀n,carry,v1,v2.
1927  (let 〈a,b〉 ≝ add_with_carries n v1 v2 carry in a) = addition_n_direct n v1 v2 carry.
1928#n #carry #v1 #v2 <addition_n_direct_ok
1929cases (add_with_carries ????) //
1930qed.
1931 
1932(* trivially lift associativity to our new setting *)     
1933lemma associative_addition_n_direct : ∀n. ∀carry1,carry2. ∀v1,v2,v3 : BitVector n.
1934  addition_n_direct ? (addition_n_direct ? v1 v2 carry1) v3 carry2 =
1935  addition_n_direct ? v1 (addition_n_direct ? v2 v3 carry1) carry2.
1936#n #carry1 #carry2 #v1 #v2 #v3
1937<addition_n_direct_ok <addition_n_direct_ok
1938<addition_n_direct_ok <addition_n_direct_ok
1939lapply (associative_add_with_carries … carry1 carry2 v1 v2 v3)
1940elim (add_with_carries n v2 v3 carry1) #bits #carries normalize nodelta
1941elim (add_with_carries n v1 v2 carry1) #bits' #carries' normalize nodelta
1942#H @(sym_eq … H)
1943qed.
1944
1945lemma commutative_addition_n_direct : ∀n. ∀v1,v2 : BitVector n.
1946  addition_n_direct ? v1 v2 false = addition_n_direct ? v2 v1 false.
1947#n #v1 #v2 /by associative_addition_n, addition_n_direct_ok/
1948qed.
1949
1950definition increment_direct ≝ λn,v. addition_n_direct n v (one_bv ?) false.
1951definition twocomp_neg_direct ≝ λn,v. increment_direct n (negation_bv n v).
1952
1953
1954(* fold andb on a bitvector. *)
1955let rec andb_fold (n : nat) (b : BitVector n) on b : bool ≝
1956match b with
1957[ VEmpty ⇒ true
1958| VCons sz elt tl ⇒
1959  andb elt (andb_fold ? tl)
1960].
1961
1962lemma andb_fold_Sn : ∀n. ∀x. ∀b : BitVector n. andb_fold (S n) (x ::: b) = andb x (andb_fold … n b). // qed.
1963
1964lemma andb_fold_inversion : ∀n. ∀elt,x. andb_fold (S n) (elt ::: x) = true → elt = true ∧ andb_fold n x = true.
1965#n #elt #x cases elt normalize #H @conj destruct (H) try assumption @refl
1966qed.
1967
1968lemma ith_increment_carry : ∀n. ∀a : BitVector (S n).
1969  ith_carry … a (one_bv ?) false = andb_fold … a.
1970#n elim n
1971[ 1: #a elim (BitVector_Sn … a) #hd * #tl #Heq >Heq >(BitVector_O … tl)
1972     cases hd normalize @refl
1973| 2: #n' #Hind #a
1974     elim (BitVector_Sn … a) #hd * #tl #Heq >Heq
1975     lapply (Hind … tl) #Hind >one_bv_Sn
1976     >ith_carry_Sn whd in match (andb_fold ??);
1977     cases hd >Hind @refl
1978] qed.
1979
1980lemma ith_increment_bit : ∀n. ∀a : BitVector (S n).
1981  ith_bit … a (one_bv ?) false = xorb (head' … a) (andb_fold … (tail … a)).
1982#n #a
1983elim (BitVector_Sn … a) #hd * #tl #Heq >Heq
1984whd in match (head' ???);
1985-a cases n in tl;
1986[ 1: #tl >(BitVector_O … tl) cases hd normalize try //
1987| 2: #n' #tl >one_bv_Sn >ith_bit_Sn
1988     >ith_increment_carry >tail_Sn
1989     cases hd try //
1990] qed.
1991
1992(* Lemma used to prove involutivity of two-complement negation *)
1993lemma twocomp_neg_involutive_aux : ∀n. ∀v : BitVector (S n).
1994   (andb_fold (S n) (negation_bv (S n) v) =
1995    andb_fold (S n) (negation_bv (S n) (addition_n_direct (S n) (negation_bv (S n) v) (one_bv (S n)) false))).
1996#n elim n
1997[ 1: #v elim (BitVector_Sn … v) #hd * #tl #Heq >Heq >(BitVector_O … tl) cases hd @refl
1998| 2: #n' #Hind #v elim (BitVector_Sn … v) #hd * #tl #Heq >Heq
1999     lapply (Hind tl) -Hind #Hind >negation_bv_Sn >one_bv_Sn >addition_n_direct_Sn
2000     >andb_fold_Sn >ith_bit_Sn >negation_bv_Sn >andb_fold_Sn <Hind
2001     cases hd normalize nodelta
2002     [ 1: >xorb_false >(xorb_comm false ?) >xorb_false
2003     | 2: >xorb_false >(xorb_comm true ?) >xorb_true ]
2004     >ith_increment_carry
2005     cases (andb_fold (S n') (negation_bv (S n') tl)) @refl
2006] qed.
2007   
2008(* Test of the 'direct' approach: proof of the involutivity of two-complement negation. *)
2009lemma twocomp_neg_involutive : ∀n. ∀v : BitVector n. twocomp_neg_direct ? (twocomp_neg_direct ? v) = v.
2010#n elim n
2011[ 1: #v >(BitVector_O v) @refl
2012| 2: #n' cases n'
2013     [ 1: #Hind #v elim (BitVector_Sn … v) #hd * #tl #Heq >Heq
2014          >(BitVector_O … tl) normalize cases hd @refl
2015     | 2: #n'' #Hind #v elim (BitVector_Sn … v) #hd * #tl #Heq >Heq
2016          lapply (Hind tl) -Hind #Hind <Hind in ⊢ (???%);
2017          whd in match twocomp_neg_direct; normalize nodelta
2018          whd in match increment_direct; normalize nodelta
2019          >(negation_bv_Sn ? hd tl) >one_bv_Sn >(addition_n_direct_Sn ? (¬hd) false ??)
2020          >ith_bit_Sn >negation_bv_Sn >addition_n_direct_Sn >ith_bit_Sn
2021          generalize in match (addition_n_direct (S n'')
2022                                                   (negation_bv (S n'')
2023                                                   (addition_n_direct (S n'') (negation_bv (S n'') tl) (one_bv (S n'')) false))
2024                                                   (one_bv (S n'')) false); #tail
2025          >ith_increment_carry >ith_increment_carry
2026          cases hd normalize nodelta
2027          [ 1: normalize in match (xorb false false); >(xorb_comm false ?) >xorb_false >xorb_false
2028          | 2: normalize in match (xorb true false); >(xorb_comm true ?) >xorb_true >xorb_false ]
2029          <twocomp_neg_involutive_aux
2030          cases (andb_fold (S n'') (negation_bv (S n'') tl)) @refl
2031      ]
2032] qed.
2033
2034lemma bitvector_cons_inj_inv : ∀n. ∀a,b. ∀va,vb : BitVector n. a ::: va = b ::: vb → a =b ∧ va = vb.
2035#n #a #b #va #vb #H destruct (H) @conj @refl qed.
2036
2037lemma bitvector_cons_eq : ∀n. ∀a,b. ∀v : BitVector n. a = b → a ::: v = b ::: v. // qed.
2038
2039(* Injectivity of increment *)
2040lemma increment_inj : ∀n. ∀a,b : BitVector n.
2041  increment_direct ? a = increment_direct ? b →
2042  a = b ∧ (ith_carry n a (one_bv n) false = ith_carry n b (one_bv n) false).
2043#n whd in match increment_direct; normalize nodelta elim n
2044[ 1: #a #b >(BitVector_O … a) >(BitVector_O … b) normalize #_ @conj //
2045| 2: #n' cases n'
2046   [ 1: #_ #a #b
2047        elim (BitVector_Sn … a) #hd_a * #tl_a #Heq_a >Heq_a
2048        elim (BitVector_Sn … b) #hd_b * #tl_b #Heq_b >Heq_b
2049        >(BitVector_O … tl_a) >(BitVector_O … tl_b) cases hd_a cases hd_b
2050        normalize #H @conj try //
2051   | 2: #n'' #Hind #a #b
2052        elim (BitVector_Sn … a) #hd_a * #tl_a #Heq_a >Heq_a
2053        elim (BitVector_Sn … b) #hd_b * #tl_b #Heq_b >Heq_b
2054        lapply (Hind … tl_a tl_b) -Hind #Hind
2055        >one_bv_Sn >addition_n_direct_Sn >ith_bit_Sn
2056        >addition_n_direct_Sn >ith_bit_Sn >xorb_false >xorb_false
2057        #H elim (bitvector_cons_inj_inv … H) #Heq1 #Heq2
2058        lapply (Hind Heq2) * #Heq3 #Heq4
2059        cut (hd_a = hd_b)
2060        [ 1: >Heq4 in Heq1; #Heq5 lapply (xorb_inj (ith_carry ? tl_b (one_bv ?) false) hd_a hd_b)
2061             * #Heq6 #_ >xorb_comm in Heq6; >(xorb_comm  ? hd_b) #Heq6 >(Heq6 Heq5)
2062             @refl ]
2063        #Heq5 @conj [ 1: >Heq3 >Heq5 @refl ]
2064        >ith_carry_Sn >ith_carry_Sn >Heq4 >Heq5 @refl
2065] qed.
2066
2067(* Inverse of injecivity of increment, does not lose information (cf increment_inj) *)
2068lemma increment_inj_inv : ∀n. ∀a,b : BitVector n.
2069  a = b → increment_direct ? a = increment_direct ? b. // qed.
2070
2071(* A more general result. *)
2072lemma addition_n_direct_inj : ∀n. ∀x,y,delta: BitVector n.
2073  addition_n_direct ? x delta false = addition_n_direct ? y delta false →
2074  x = y ∧ (ith_carry n x delta false = ith_carry n y delta false).
2075#n elim n
2076[ 1: #x #y #delta >(BitVector_O … x) >(BitVector_O … y) >(BitVector_O … delta) * @conj @refl
2077| 2: #n' #Hind #x #y #delta
2078     elim (BitVector_Sn … x) #hdx * #tlx #Heqx >Heqx
2079     elim (BitVector_Sn … y) #hdy * #tly #Heqy >Heqy
2080     elim (BitVector_Sn … delta) #hdd * #tld #Heqd >Heqd
2081     >addition_n_direct_Sn >ith_bit_Sn
2082     >addition_n_direct_Sn >ith_bit_Sn
2083     >ith_carry_Sn >ith_carry_Sn
2084     lapply (Hind … tlx tly tld) -Hind #Hind #Heq
2085     elim (bitvector_cons_inj_inv … Heq) #Heq_hd #Heq_tl
2086     lapply (Hind Heq_tl) -Hind * #HindA #HindB
2087     >HindA >HindB >HindB in Heq_hd; #Heq_hd
2088     cut (hdx = hdy)
2089     [ 1: cases hdd in Heq_hd; cases (ith_carry n' tly tld false)
2090          cases hdx cases hdy normalize #H try @H try @refl
2091          >H try @refl ]
2092     #Heq_hd >Heq_hd @conj @refl
2093] qed.
2094
2095(* We also need it the other way around. *)
2096lemma addition_n_direct_inj_inv : ∀n. ∀x,y,delta: BitVector n.
2097  x ≠ y → (* ∧ (ith_carry n x delta false = ith_carry n y delta false). *)
2098   addition_n_direct ? x delta false ≠ addition_n_direct ? y delta false.
2099#n elim n
2100[ 1: #x #y #delta >(BitVector_O … x) >(BitVector_O … y) >(BitVector_O … delta) * #H @(False_ind … (H (refl ??)))
2101| 2: #n' #Hind #x #y #delta
2102     elim (BitVector_Sn … x) #hdx * #tlx #Heqx >Heqx
2103     elim (BitVector_Sn … y) #hdy * #tly #Heqy >Heqy
2104     elim (BitVector_Sn … delta) #hdd * #tld #Heqd >Heqd
2105     #Hneq
2106     cut (hdx ≠ hdy ∨ tlx ≠ tly)
2107     [ @(eq_bv_elim … tlx tly)
2108       [ #Heq_tl >Heq_tl >Heq_tl in Hneq;
2109         #Hneq cut (hdx ≠ hdy) [ % #Heq_hd >Heq_hd in Hneq; *
2110                                 #H @H @refl ]
2111         #H %1 @H
2112       | #H %2 @H ] ]
2113     -Hneq #Hneq
2114     >addition_n_direct_Sn >addition_n_direct_Sn
2115     >ith_bit_Sn >ith_bit_Sn cases Hneq
2116     [ 1: #Hneq_hd
2117          lapply (addition_n_direct_inj … tlx tly tld)         
2118          @(eq_bv_elim … (addition_n_direct ? tlx tld false) (addition_n_direct ? tly tld false))
2119          [ 1: #Heq -Hind #Hind elim (Hind Heq) #Heq_tl >Heq_tl #Heq_carry >Heq_carry
2120               % #Habsurd elim (bitvector_cons_inj_inv … Habsurd) -Habsurd
2121               lapply Hneq_hd
2122               cases hdx cases hdd cases hdy cases (ith_carry ? tly tld false)
2123               normalize in ⊢ (? → % → ?); #Hneq_hd #Heq_hd #_
2124               try @(absurd … Heq_hd Hneq_hd)
2125               elim Hneq_hd -Hneq_hd #Hneq_hd @Hneq_hd
2126               try @refl try assumption try @(sym_eq … Heq_hd)
2127          | 2: #Htl_not_eq #_ % #Habsurd elim (bitvector_cons_inj_inv … Habsurd) #_
2128               elim Htl_not_eq -Htl_not_eq #HA #HB @HA @HB ]
2129     | 2: #Htl_not_eq lapply (Hind tlx tly tld Htl_not_eq) -Hind #Hind
2130          % #Habsurd elim (bitvector_cons_inj_inv … Habsurd) #_
2131          elim Hind -Hind #HA #HB @HA @HB ]
2132] qed.
2133
2134lemma carry_notb : ∀a,b,c. notb (carry_of a b c) = carry_of (notb a) (notb b) (notb c). * * * @refl qed.
2135
2136lemma increment_to_carry_aux : ∀n. ∀a : BitVector (S n).
2137   ith_carry (S n) a (one_bv (S n)) false
2138   = ith_carry (S n) a (zero (S n)) true.
2139#n elim n
2140[ 1: #a elim (BitVector_Sn ? a) #hd_a * #tl_a #Heq >Heq >(BitVector_O … tl_a) @refl
2141| 2: #n' #Hind #a elim (BitVector_Sn ? a) #hd_a * #tl_a #Heq >Heq
2142     lapply (Hind tl_a) #Hind
2143     >one_bv_Sn >zero_Sn >ith_carry_Sn >ith_carry_Sn >Hind @refl
2144] qed.
2145
2146lemma neutral_addition_n_direct_aux : ∀n. ∀v. ith_carry n v (zero n) false = false.
2147#n elim n //
2148#n' #Hind #v elim (BitVector_Sn … v) #hd * #tl #Heq >Heq >zero_Sn
2149>ith_carry_Sn >(Hind tl) cases hd @refl.
2150qed.
2151
2152lemma neutral_addition_n_direct : ∀n. ∀v : BitVector n.
2153  addition_n_direct ? v (zero ?) false = v.
2154#n elim n
2155[ 1: #v >(BitVector_O … v) normalize @refl
2156| 2: #n' #Hind #v elim (BitVector_Sn … v) #hd * #tl #Heq >Heq
2157     lapply (Hind … tl) #H >zero_Sn >addition_n_direct_Sn
2158     >ith_bit_Sn >H >xorb_false >neutral_addition_n_direct_aux
2159     >xorb_false @refl
2160] qed.
2161
2162lemma increment_to_carry_zero : ∀n. ∀a : BitVector n. addition_n_direct ? a (one_bv ?) false = addition_n_direct ? a (zero ?) true.
2163#n elim n
2164[ 1: #a >(BitVector_O … a) normalize @refl
2165| 2: #n' cases n'
2166     [ 1: #_ #a elim (BitVector_Sn … a) #hd_a * #tl_a #Heq >Heq >(BitVector_O … tl_a) cases hd_a @refl
2167     | 2: #n'' #Hind #a
2168          elim (BitVector_Sn … a) #hd_a * #tl_a #Heq >Heq
2169          lapply (Hind tl_a) -Hind #Hind
2170          >one_bv_Sn >zero_Sn >addition_n_direct_Sn >ith_bit_Sn
2171          >addition_n_direct_Sn >ith_bit_Sn
2172          >xorb_false >Hind @bitvector_cons_eq
2173          >increment_to_carry_aux @refl
2174     ]
2175] qed.
2176
2177lemma increment_to_carry : ∀n. ∀a,b : BitVector n.
2178  addition_n_direct ? a (addition_n_direct ? b (one_bv ?) false) false = addition_n_direct ? a b true.
2179#n #a #b >increment_to_carry_zero <associative_addition_n_direct
2180>neutral_addition_n_direct @refl
2181qed.
2182
2183lemma increment_direct_ok : ∀n,v. increment_direct n v = increment n v.
2184#n #v whd in match (increment ??);
2185>addition_n_direct_ok <increment_to_carry_zero @refl
2186qed.
2187
2188(* Prove -(a + b) = -a + -b *)
2189lemma twocomp_neg_plus : ∀n. ∀a,b : BitVector n.
2190  twocomp_neg_direct ? (addition_n_direct ? a b false) = addition_n_direct ? (twocomp_neg_direct … a) (twocomp_neg_direct … b) false.
2191whd in match twocomp_neg_direct; normalize nodelta
2192lapply increment_inj_inv
2193whd in match increment_direct; normalize nodelta
2194#H #n #a #b
2195<associative_addition_n_direct @H
2196>associative_addition_n_direct >(commutative_addition_n_direct ? (one_bv n))
2197>increment_to_carry
2198-H lapply b lapply a -b -a
2199cases n
2200[ 1: #a #b >(BitVector_O … a) >(BitVector_O … b) @refl
2201| 2: #n' #a #b
2202     cut (negation_bv ? (addition_n_direct ? a b false)
2203           = addition_n_direct ? (negation_bv ? a) (negation_bv ? b) true ∧
2204          notb (ith_carry ? a b false) = (ith_carry ? (negation_bv ? a) (negation_bv ? b) true))
2205     [ -n lapply b lapply a elim n'
2206     [ 1: #a #b elim (BitVector_Sn … a) #hd_a * #tl_a #Heqa >Heqa >(BitVector_O … tl_a)
2207          elim (BitVector_Sn … b) #hd_b * #tl_b #Heqb >Heqb >(BitVector_O … tl_b)
2208          cases hd_a cases hd_b normalize @conj @refl
2209     | 2: #n #Hind #a #b
2210          elim (BitVector_Sn … a) #hd_a * #tl_a #Heqa >Heqa
2211          elim (BitVector_Sn … b) #hd_b * #tl_b #Heqb >Heqb
2212          lapply (Hind tl_a tl_b) * #H1 #H2
2213          @conj
2214          [ 2: >ith_carry_Sn >negation_bv_Sn >negation_bv_Sn >ith_carry_Sn
2215               >carry_notb >H2 @refl
2216          | 1: >addition_n_direct_Sn >ith_bit_Sn >negation_bv_Sn
2217               >negation_bv_Sn >negation_bv_Sn
2218               >addition_n_direct_Sn >ith_bit_Sn >H1 @bitvector_cons_eq
2219               >xorb_lneg >xorb_rneg >notb_notb
2220               <xorb_rneg >H2 @refl
2221          ]
2222      ] ]
2223      * #H1 #H2 @H1
2224] qed.
2225
2226lemma addition_n_direct_neg : ∀n. ∀a.
2227 (addition_n_direct n a (negation_bv n a) false) = replicate ?? true
2228 ∧ (ith_carry n a (negation_bv n a) false = false).
2229#n elim n
2230[ 1: #a >(BitVector_O … a) @conj @refl
2231| 2: #n' #Hind #a elim (BitVector_Sn … a) #hd * #tl #Heq >Heq
2232     lapply (Hind … tl) -Hind * #HA #HB
2233     @conj
2234     [ 2: >negation_bv_Sn >ith_carry_Sn >HB cases hd @refl
2235     | 1: >negation_bv_Sn >addition_n_direct_Sn
2236          >ith_bit_Sn >HB >xorb_false >HA
2237          @bitvector_cons_eq elim hd @refl
2238     ]
2239] qed.
2240
2241(* -a + a = 0 *)
2242lemma bitvector_opp_direct : ∀n. ∀a : BitVector n. addition_n_direct ? a (twocomp_neg_direct ? a) false = (zero ?).
2243whd in match twocomp_neg_direct;
2244whd in match increment_direct;
2245normalize nodelta
2246#n #a <associative_addition_n_direct
2247elim (addition_n_direct_neg … a) #H #_ >H
2248-H -a
2249cases n try //
2250#n'
2251cut ((addition_n_direct (S n') (replicate bool ? true) (one_bv ?) false = (zero (S n')))
2252       ∧ (ith_carry ? (replicate bool (S n') true) (one_bv (S n')) false = true))
2253[ elim n'
2254     [ 1: @conj @refl
2255     | 2: #n' * #HA #HB @conj
2256          [ 1: >replicate_Sn >one_bv_Sn  >addition_n_direct_Sn
2257               >ith_bit_Sn >HA >zero_Sn @bitvector_cons_eq >HB @refl
2258          | 2: >replicate_Sn >one_bv_Sn >ith_carry_Sn >HB @refl ]
2259     ]
2260] * #H1 #H2 @H1
2261qed.
2262
2263(* Lift back the previous result to standard operations. *)
2264lemma twocomp_neg_direct_ok : ∀n. ∀v. twocomp_neg_direct ? v = two_complement_negation n v.
2265#n #v whd in match twocomp_neg_direct; normalize nodelta
2266whd in match increment_direct; normalize nodelta
2267whd in match two_complement_negation; normalize nodelta
2268>increment_to_addition_n <addition_n_direct_ok
2269whd in match addition_n; normalize nodelta
2270elim (add_with_carries ????) #a #b @refl
2271qed.
2272
2273lemma two_complement_negation_plus : ∀n. ∀a,b : BitVector n.
2274  two_complement_negation ? (addition_n ? a b) = addition_n ? (two_complement_negation ? a) (two_complement_negation ? b).
2275#n #a #b
2276lapply (twocomp_neg_plus ? a b)
2277>twocomp_neg_direct_ok >twocomp_neg_direct_ok >twocomp_neg_direct_ok
2278<addition_n_direct_ok <addition_n_direct_ok
2279whd in match addition_n; normalize nodelta
2280elim (add_with_carries n a b false) #bits #flags normalize nodelta
2281elim (add_with_carries n (two_complement_negation n a) (two_complement_negation n b) false) #bits' #flags'
2282normalize nodelta #H @H
2283qed.
2284
2285lemma bitvector_opp_addition_n : ∀n. ∀a : BitVector n. addition_n ? a (two_complement_negation ? a) = (zero ?).
2286#n #a lapply (bitvector_opp_direct ? a)
2287>twocomp_neg_direct_ok <addition_n_direct_ok
2288whd in match (addition_n ???);
2289elim (add_with_carries n a (two_complement_negation n a) false) #bits #flags #H @H
2290qed.
2291
2292lemma neutral_addition_n : ∀n. ∀a : BitVector n. addition_n ? a (zero ?) = a.
2293#n #a
2294lapply (neutral_addition_n_direct n a)
2295<addition_n_direct_ok
2296whd in match (addition_n ???);
2297elim (add_with_carries n a (zero n) false) #bits #flags #H @H
2298qed.
2299
2300lemma injective_addition_n : ∀n. ∀x,y,delta : BitVector n.
2301  addition_n ? x delta = addition_n ? y delta → x = y. 
2302#n #x #y #delta 
2303lapply (addition_n_direct_inj … x y delta)
2304<addition_n_direct_ok <addition_n_direct_ok
2305whd in match addition_n; normalize nodelta
2306elim (add_with_carries n x delta false) #bitsx #flagsx
2307elim (add_with_carries n y delta false) #bitsy #flagsy
2308normalize #H1 #H2 elim (H1 H2) #Heq #_ @Heq
2309qed.
2310
2311lemma injective_inv_addition_n : ∀n. ∀x,y,delta : BitVector n.
2312  x ≠ y → addition_n ? x delta ≠ addition_n ? y delta. 
2313#n #x #y #delta 
2314lapply (addition_n_direct_inj_inv … x y delta)
2315<addition_n_direct_ok <addition_n_direct_ok
2316whd in match addition_n; normalize nodelta
2317elim (add_with_carries n x delta false) #bitsx #flagsx
2318elim (add_with_carries n y delta false) #bitsy #flagsy
2319normalize #H1 #H2 @(H1 H2)
2320qed.
2321
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