source: src/ASM/PolicyFront.ma @ 2222

Last change on this file since 2222 was 2222, checked in by sacerdot, 7 years ago

More robust to possible future changes to the "in match" semantics
(that would be weakened).

File size: 36.3 KB
Line 
1include "ASM/ASM.ma".
2include "ASM/Arithmetic.ma".
3include "ASM/Fetch.ma".
4include "ASM/Status.ma".
5include "utilities/extralib.ma".
6include "ASM/Assembly.ma".
7
8(* Internal types *)
9
10(* ppc_pc_map: program length × (pseudo program counter ↦ 〈pc, jump_length〉) *)
11definition ppc_pc_map ≝ ℕ × (BitVectorTrie (ℕ × jump_length) 16).
12
13(* The different properties that we want/need to prove at some point *)
14(* During our iteration, everything not yet seen is None, and vice versa *)
15definition out_of_program_none ≝
16  λprefix:list labelled_instruction.λsigma:ppc_pc_map.
17  ∀i.i < 2^16 → (i > |prefix| ↔ bvt_lookup_opt … (bitvector_of_nat ? i) (\snd sigma) = None ?).
18
19(* If instruction i is a jump, then there will be something in the policy at
20 * position i *)
21definition is_jump' ≝
22  λx:preinstruction Identifier.
23  match x with
24  [ JC _ ⇒ True
25  | JNC _ ⇒ True
26  | JZ _ ⇒ True
27  | JNZ _ ⇒ True
28  | JB _ _ ⇒ True
29  | JNB _ _ ⇒ True
30  | JBC _ _ ⇒ True
31  | CJNE _ _ ⇒ True
32  | DJNZ _ _ ⇒ True
33  | _ ⇒ False
34  ].
35
36definition is_relative_jump ≝
37  λinstr:pseudo_instruction.
38  match instr with
39  [ Instruction i ⇒ is_jump' i
40  | _             ⇒ False
41  ].
42   
43definition is_jump ≝
44  λinstr:pseudo_instruction.
45  match instr with
46  [ Instruction i   ⇒ is_jump' i
47  | Call _ ⇒ True
48  | Jmp _ ⇒ True
49  | _ ⇒ False
50  ].
51
52definition is_call ≝
53  λinstr:pseudo_instruction.
54  match instr with
55  [ Call _ ⇒ True
56  | _ ⇒ False
57  ].
58 
59definition is_jump_to ≝
60  λx:pseudo_instruction.λd:Identifier.
61  match x with
62  [ Instruction i ⇒ match i with
63    [ JC j ⇒ d = j
64    | JNC j ⇒ d = j
65    | JZ j ⇒ d = j
66    | JNZ j ⇒ d = j
67    | JB _ j ⇒ d = j
68    | JNB _ j ⇒ d = j
69    | JBC _ j ⇒ d = j
70    | CJNE _ j ⇒ d = j
71    | DJNZ _ j ⇒ d = j
72    | _ ⇒ False
73    ]
74  | Call c ⇒ d = c
75  | Jmp j ⇒ d = j
76  | _ ⇒ False
77  ].
78 
79definition not_jump_default ≝
80  λprefix:list labelled_instruction.λsigma:ppc_pc_map.
81  ∀i:ℕ.i < |prefix| →
82  ¬is_jump (\snd (nth i ? prefix 〈None ?, Comment []〉)) →
83  \snd (bvt_lookup … (bitvector_of_nat ? i) (\snd sigma) 〈0,short_jump〉) = short_jump.
84 
85(* Between two policies, jumps cannot decrease *)
86definition jmpeqb: jump_length → jump_length → bool ≝
87  λj1.λj2.
88  match j1 with
89  [ short_jump ⇒ match j2 with [ short_jump ⇒ true | _ ⇒ false ]
90  | absolute_jump ⇒ match j2 with [ absolute_jump ⇒ true | _ ⇒ false ]
91  | long_jump ⇒ match j2 with [ long_jump ⇒ true | _ ⇒ false ]
92  ].
93
94lemma jmpeqb_to_eq: ∀j1,j2.jmpeqb j1 j2 → j1 = j2.
95 #j1 #j2 cases j1 cases j2
96 [1,5,9: / by /]
97 #H cases H
98qed.
99
100definition jmple: jump_length → jump_length → Prop ≝
101  λj1.λj2.
102  match j1 with
103  [ short_jump  ⇒
104    match j2 with
105    [ short_jump ⇒ False
106    | _          ⇒ True
107    ]
108  | absolute_jump ⇒
109    match j2 with
110    [ long_jump ⇒ True
111    | _         ⇒ False
112    ]
113  | long_jump   ⇒ False
114  ].
115
116definition jmpleq: jump_length → jump_length → Prop ≝
117  λj1.λj2.jmple j1 j2 ∨ j1 = j2.
118 
119definition jump_increase ≝
120 λprefix:list labelled_instruction.λop:ppc_pc_map.λp:ppc_pc_map.
121 ∀i.i ≤ |prefix| →
122   let 〈opc,oj〉 ≝ bvt_lookup … (bitvector_of_nat ? i) (\snd op) 〈0,short_jump〉 in
123   let 〈pc,j〉 ≝ bvt_lookup … (bitvector_of_nat ? i) (\snd p) 〈0,short_jump〉 in
124     jmpleq oj j.
125     
126(* this is the instruction size as determined by the jump length given *)
127definition expand_relative_jump_internal_unsafe:
128  jump_length → ([[relative]] → preinstruction [[relative]]) → list instruction ≝
129  λjmp_len:jump_length.λi.
130  match jmp_len with
131  [ short_jump ⇒ [ RealInstruction (i (RELATIVE (zero 8))) ]
132  | absolute_jump ⇒ [ ] (* this should not happen *)
133  | long_jump ⇒
134    [ RealInstruction (i (RELATIVE (bitvector_of_nat ? 2)));
135      SJMP (RELATIVE (bitvector_of_nat ? 3)); (* LJMP size? *)
136      LJMP (ADDR16 (zero 16))
137    ]
138  ].
139 @I
140qed.
141
142definition expand_relative_jump_unsafe:
143  jump_length → preinstruction Identifier → list instruction ≝
144  λjmp_len:jump_length.λi.
145  match i with
146  [ JC jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (JC ?)
147  | JNC jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (JNC ?)
148  | JB baddr jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (JB ? baddr)
149  | JZ jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (JZ ?)
150  | JNZ jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (JNZ ?)
151  | JBC baddr jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (JBC ? baddr)
152  | JNB baddr jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (JNB ? baddr)
153  | CJNE addr jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (CJNE ? addr)
154  | DJNZ addr jmp ⇒ expand_relative_jump_internal_unsafe jmp_len (DJNZ ? addr)
155  | ADD arg1 arg2 ⇒ [ ADD ? arg1 arg2 ]
156  | ADDC arg1 arg2 ⇒ [ ADDC ? arg1 arg2 ]
157  | SUBB arg1 arg2 ⇒ [ SUBB ? arg1 arg2 ]
158  | INC arg ⇒ [ INC ? arg ]
159  | DEC arg ⇒ [ DEC ? arg ]
160  | MUL arg1 arg2 ⇒ [ MUL ? arg1 arg2 ]
161  | DIV arg1 arg2 ⇒ [ DIV ? arg1 arg2 ]
162  | DA arg ⇒ [ DA ? arg ]
163  | ANL arg ⇒ [ ANL ? arg ]
164  | ORL arg ⇒ [ ORL ? arg ]
165  | XRL arg ⇒ [ XRL ? arg ]
166  | CLR arg ⇒ [ CLR ? arg ]
167  | CPL arg ⇒ [ CPL ? arg ]
168  | RL arg ⇒ [ RL ? arg ]
169  | RR arg ⇒ [ RR ? arg ]
170  | RLC arg ⇒ [ RLC ? arg ]
171  | RRC arg ⇒ [ RRC ? arg ]
172  | SWAP arg ⇒ [ SWAP ? arg ]
173  | MOV arg ⇒ [ MOV ? arg ]
174  | MOVX arg ⇒ [ MOVX ? arg ]
175  | SETB arg ⇒ [ SETB ? arg ]
176  | PUSH arg ⇒ [ PUSH ? arg ]
177  | POP arg ⇒ [ POP ? arg ]
178  | XCH arg1 arg2 ⇒ [ XCH ? arg1 arg2 ]
179  | XCHD arg1 arg2 ⇒ [ XCHD ? arg1 arg2 ]
180  | RET ⇒ [ RET ? ]
181  | RETI ⇒ [ RETI ? ]
182  | NOP ⇒ [ RealInstruction (NOP ?) ]
183  ].
184
185definition instruction_size_jmplen:
186 jump_length → pseudo_instruction → ℕ ≝
187  λjmp_len.
188  λi.
189  let pseudos ≝ match i with
190  [ Cost cost ⇒ [ ]
191  | Comment comment ⇒ [ ]
192  | Call call ⇒
193    match jmp_len with
194    [ short_jump ⇒ [ ] (* this should not happen *)
195    | absolute_jump ⇒ [ ACALL (ADDR11 (zero 11)) ]
196    | long_jump ⇒ [ LCALL (ADDR16 (zero 16)) ]
197    ]
198  | Mov d trgt ⇒
199     [ RealInstruction (MOV ? (inl ? ? (inl ? ? (inr ? ? 〈DPTR, DATA16 (zero 16)〉))))]
200  | Instruction instr ⇒ expand_relative_jump_unsafe jmp_len instr
201  | Jmp jmp ⇒
202    match jmp_len with
203    [ short_jump ⇒ [ SJMP (RELATIVE (zero 8)) ]
204    | absolute_jump ⇒ [ AJMP (ADDR11 (zero 11)) ]
205    | long_jump ⇒ [ LJMP (ADDR16 (zero 16)) ]
206    ]
207  ] in
208  let mapped ≝ map ? ? assembly1 pseudos in
209  let flattened ≝ flatten ? mapped in
210  let pc_len ≝ length ? flattened in
211    pc_len.
212 @I.
213qed.
214
215definition sigma_compact_unsafe ≝
216 λprefix:list labelled_instruction.λlabels:label_map.λsigma:ppc_pc_map.
217 ∀n.n < |prefix| →
218  match bvt_lookup_opt … (bitvector_of_nat ? n) (\snd sigma) with
219  [ None ⇒ False
220  | Some x ⇒ let 〈pc,j〉 ≝ x in
221    match bvt_lookup_opt … (bitvector_of_nat ? (S n)) (\snd sigma) with
222    [ None ⇒ False
223    | Some x1 ⇒ let 〈pc1,j1〉 ≝ x1 in
224       pc1 = pc + instruction_size_jmplen j (\snd (nth n ? prefix 〈None ?, Comment []〉))
225    ]
226  ].
227   
228(* new safety condition: sigma corresponds to program and resulting program is compact *)
229definition sigma_compact ≝
230 λprogram:list labelled_instruction.λlabels:label_map.λsigma:ppc_pc_map.
231 ∀n.n < |program| →
232  match bvt_lookup_opt … (bitvector_of_nat ? n) (\snd sigma) with
233  [ None ⇒ False
234  | Some x ⇒ let 〈pc,j〉 ≝ x in
235    match bvt_lookup_opt … (bitvector_of_nat ? (S n)) (\snd sigma) with
236    [ None ⇒ False
237    | Some x1 ⇒ let 〈pc1,j1〉 ≝ x1 in
238       pc1 = pc + instruction_size (λid.bitvector_of_nat ? (lookup_def ?? labels id 0))
239         (λppc.bitvector_of_nat ? (\fst (bvt_lookup ?? ppc (\snd sigma) 〈0,short_jump〉)))
240         (λppc.jmpeqb long_jump (\snd (bvt_lookup ?? ppc (\snd sigma) 〈0,short_jump〉)))
241         (bitvector_of_nat ? n) (\snd (nth n ? program 〈None ?, Comment []〉))
242    ]
243  ].
244
245(* jumps are of the proper size *)
246definition sigma_safe ≝
247 λprefix:list labelled_instruction.λlabels:label_map.λadded:ℕ.
248 λold_sigma:ppc_pc_map.λsigma:ppc_pc_map.
249 ∀i.i < |prefix| →
250 let 〈pc,j〉 ≝ bvt_lookup … (bitvector_of_nat ? i) (\snd sigma) 〈0,short_jump〉 in
251 let pc_plus_jmp_length ≝ bitvector_of_nat ? (\fst (bvt_lookup … (bitvector_of_nat ? (S i)) (\snd sigma) 〈0,short_jump〉)) in
252 let 〈label,instr〉 ≝ nth i ? prefix 〈None ?, Comment [ ]〉 in
253 ∀dest.is_jump_to instr dest →
254   let paddr ≝ lookup_def … labels dest 0 in
255   let addr ≝ bitvector_of_nat ? (if leb paddr (|prefix|) (* jump to address already known *)
256   then \fst (bvt_lookup … (bitvector_of_nat ? paddr) (\snd sigma) 〈0,short_jump〉)
257   else \fst (bvt_lookup … (bitvector_of_nat ? paddr) (\snd old_sigma) 〈0,short_jump〉)+added) in
258   match j with
259   [ short_jump ⇒ \fst (short_jump_cond pc_plus_jmp_length addr) = true ∧
260      ¬is_call instr
261   | absolute_jump ⇒  \fst (absolute_jump_cond pc_plus_jmp_length addr) = true ∧
262       \fst (short_jump_cond pc_plus_jmp_length addr) = false ∧
263       ¬is_relative_jump instr
264   | long_jump   ⇒ \fst (short_jump_cond pc_plus_jmp_length addr) = false ∧
265       \fst (absolute_jump_cond pc_plus_jmp_length addr) = false
266   ].
267 
268(* Definitions and theorems for the jump_length type (itself defined in Assembly) *)
269definition max_length: jump_length → jump_length → jump_length ≝
270  λj1.λj2.
271  match j1 with
272  [ long_jump   ⇒ long_jump
273  | absolute_jump ⇒
274    match j2 with
275    [ absolute_jump ⇒ absolute_jump
276    | _           ⇒ long_jump
277    ]
278  | short_jump  ⇒
279    match j2 with
280    [ short_jump ⇒ short_jump
281    | _          ⇒ long_jump
282    ]
283  ].
284
285lemma dec_jmple: ∀x,y:jump_length.Sum (jmple x y) (¬(jmple x y)).
286 #x #y cases x cases y /3 by inl, inr, nmk, I/
287qed.
288 
289lemma jmpleq_max_length: ∀ol,nl.
290  jmpleq ol (max_length ol nl).
291 #ol #nl cases ol cases nl
292 /2 by or_introl, or_intror, I/
293qed.
294
295lemma dec_eq_jump_length: ∀a,b:jump_length.Sum (a = b) (a ≠ b).
296  #a #b cases a cases b /2/
297  %2 @nmk #H destruct (H)
298qed.
299 
300(* The function that creates the label-to-address map *)
301definition create_label_map: ∀program:list labelled_instruction.
302  (Σlabels:label_map.
303    ∀l.occurs_exactly_once ?? l program →
304    And (bitvector_of_nat ? (lookup_def ?? labels l 0) =
305     address_of_word_labels_code_mem program l)
306    (lookup_def ?? labels l 0 < |program|)
307  ) ≝
308 λprogram.
309   \fst (create_label_cost_map program).
310 #l #Hl lapply (pi2 ?? (create_label_cost_map0 program)) @pair_elim
311 #labels #costs #EQ normalize nodelta #H whd in match create_label_cost_map;
312 normalize nodelta >EQ @(H l Hl)
313qed.
314
315(* General note on jump length selection: the jump displacement is added/replaced
316 * AFTER the fetch (and attendant PC increase), but we calculate before the
317 * fetch, which means that in the case of a short and medium jump we are 2
318 * bytes off and have to compensate.
319 * For the long jump we don't care, because the PC gets replaced integrally anyway. *)
320definition select_reljump_length: label_map → ppc_pc_map → ppc_pc_map → ℕ → ℕ →
321  Identifier → jump_length ≝
322  λlabels.λold_sigma.λinc_sigma.λadded.λppc.λlbl.
323  let pc ≝ \fst inc_sigma in
324  let pc_plus_jmp_length ≝ bitvector_of_nat ? (pc+2) in
325  let paddr ≝ lookup_def … labels lbl 0 in
326  let addr ≝ bitvector_of_nat ? (if leb paddr ppc (* jump to address already known *)
327  then \fst (bvt_lookup … (bitvector_of_nat 16 paddr) (\snd inc_sigma) 〈0,short_jump〉)
328  else \fst (bvt_lookup … (bitvector_of_nat 16 paddr) (\snd old_sigma) 〈0,short_jump〉)+added) in
329  let 〈sj_possible, disp〉 ≝ short_jump_cond pc_plus_jmp_length addr in
330  if sj_possible
331  then short_jump
332  else long_jump.
333
334definition select_call_length: label_map → ppc_pc_map → ppc_pc_map → ℕ → ℕ →
335  Identifier → jump_length ≝
336  λlabels.λold_sigma.λinc_sigma.λadded.λppc.λlbl.
337  let pc ≝ \fst inc_sigma in
338  let pc_plus_jmp_length ≝ bitvector_of_nat ? (pc+2) in
339  let paddr ≝ lookup_def ? ? labels lbl 0 in
340  let addr ≝ bitvector_of_nat ?
341    (if leb paddr ppc (* jump to address already known *)
342    then \fst (bvt_lookup … (bitvector_of_nat ? paddr) (\snd inc_sigma) 〈0,short_jump〉)
343    else \fst (bvt_lookup … (bitvector_of_nat ? paddr) (\snd old_sigma) 〈0,short_jump〉)+added) in
344  let 〈aj_possible, disp〉 ≝ absolute_jump_cond pc_plus_jmp_length addr in   
345  if aj_possible
346  then absolute_jump
347  else long_jump.
348 
349definition select_jump_length: label_map → ppc_pc_map → ppc_pc_map → ℕ → ℕ →
350  Identifier → jump_length ≝
351  λlabels.λold_sigma.λinc_sigma.λadded.λppc.λlbl.
352  let pc ≝ \fst inc_sigma in
353  let pc_plus_jmp_length ≝ bitvector_of_nat ? (pc+2) in
354  let paddr ≝ lookup_def … labels lbl 0 in
355  let addr ≝ bitvector_of_nat ? (if leb paddr ppc (* jump to address already known *)
356  then \fst (bvt_lookup … (bitvector_of_nat 16 paddr) (\snd inc_sigma) 〈0,short_jump〉)
357  else \fst (bvt_lookup … (bitvector_of_nat 16 paddr) (\snd old_sigma) 〈0,short_jump〉)+added) in
358  let 〈sj_possible, disp〉 ≝ short_jump_cond pc_plus_jmp_length addr in
359  if sj_possible
360  then short_jump
361  else select_call_length labels old_sigma inc_sigma added ppc lbl.
362 
363definition jump_expansion_step_instruction: label_map → ppc_pc_map → ppc_pc_map →
364  ℕ → ℕ → preinstruction Identifier → option jump_length ≝
365  λlabels.λold_sigma.λinc_sigma.λadded.λppc.λi.
366  match i with
367  [ JC j     ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j)
368  | JNC j    ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j)
369  | JZ j     ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j)
370  | JNZ j    ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j)
371  | JB _ j   ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j)
372  | JBC _ j  ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j)
373  | JNB _ j  ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j)
374  | CJNE _ j ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j)
375  | DJNZ _ j ⇒ Some ? (select_reljump_length labels old_sigma inc_sigma added ppc j)
376  | _        ⇒ None ?
377  ].
378
379lemma dec_is_jump: ∀x.Sum (is_jump x) (¬is_jump x).
380#i cases i
381[#id cases id
382 [1,2,3,6,7,33,34:
383  #x #y %2 whd in match (is_jump ?); /2 by nmk/
384 |4,5,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32:
385  #x %2 whd in match (is_jump ?); /2 by nmk/
386 |35,36,37: %2 whd in match (is_jump ?); /2 by nmk/
387 |9,10,14,15: #x %1 / by I/
388 |11,12,13,16,17: #x #y %1 / by I/
389 ]
390|2,3: #x %2 /2 by nmk/
391|4,5: #x %1 / by I/
392|6: #x #y %2 /2 by nmk/
393]
394qed.
395
396(* The first step of the jump expansion: everything to short. *)
397definition jump_expansion_start:
398  ∀program:(Σl:list labelled_instruction.S (|l|) < 2^16 ∧ is_well_labelled_p l).
399  ∀labels:label_map.
400  Σpolicy:option ppc_pc_map.
401    match policy with
402    [ None ⇒ True
403    | Some p ⇒ And (And (And (And (And
404       (not_jump_default (pi1 ?? program) p)
405       (\fst (bvt_lookup … (bitvector_of_nat ? 0) (\snd p) 〈0,short_jump〉) = 0))
406       (\fst p = \fst (bvt_lookup … (bitvector_of_nat ? (|program|)) (\snd p) 〈0,short_jump〉)))
407       (sigma_compact_unsafe program labels p))
408       (∀i.i ≤ |program| → ∃pc.
409         bvt_lookup_opt … (bitvector_of_nat ? i) (\snd p) = Some ? 〈pc,short_jump〉))
410       (\fst p ≤ 2^16)         
411    ] ≝
412  λprogram.λlabels.
413  let final_policy ≝ foldl_strong (option Identifier × pseudo_instruction)
414  (λprefix.Σpolicy:ppc_pc_map.And (And (And (And
415    (not_jump_default prefix policy)
416    (\fst (bvt_lookup … (bitvector_of_nat ? 0) (\snd policy) 〈0,short_jump〉) = 0))
417    (\fst policy = \fst (bvt_lookup … (bitvector_of_nat ? (|prefix|)) (\snd policy) 〈0,short_jump〉)))
418    (sigma_compact_unsafe prefix labels policy))
419    (∀i.i ≤ |prefix| → ∃pc.
420      bvt_lookup_opt … (bitvector_of_nat ? i) (\snd policy) = Some ? 〈pc,short_jump〉))
421  program
422  (λprefix.λx.λtl.λprf.λp.
423   let 〈pc,sigma〉 ≝ pi1 ?? p in
424   let 〈label,instr〉 ≝ x in
425   let isize ≝ instruction_size_jmplen short_jump instr in
426   〈pc + isize, bvt_insert … (bitvector_of_nat 16 (S (|prefix|))) 〈pc+isize,short_jump〉 sigma〉
427  ) 〈0, bvt_insert ?? (bitvector_of_nat 16 0) 〈0,short_jump〉 (Stub ??)〉 in
428  if gtb (\fst (pi1 ?? final_policy)) 2^16 then
429    None ?
430  else
431    Some ? (pi1 ?? final_policy).
432[ / by I/
433| lapply p -p cases final_policy -final_policy #p #Hp #hg
434  @conj [ @Hp | @not_lt_to_le @ltb_false_to_not_lt @hg ]
435| @conj [ @conj [ @conj [ @conj
436  [ (* not_jump_default *) cases p -p #p cases p -p #pc #sigma #Hp
437    cases x in prf; #lbl #ins #prf #i >append_length <commutative_plus #Hi
438    normalize in Hi; normalize nodelta cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
439    [ >lookup_insert_miss
440      [ (* USE[pass]: not_jump_default *)
441        lapply (proj1 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hp))) i Hi)
442        >nth_append_first
443        [ #H #H2 @H @H2
444        | @Hi
445        ]
446      | @bitvector_of_nat_abs
447        [ @(transitive_lt ??? Hi) @le_S_to_le]
448        [1,2: @(transitive_lt … (proj1 ?? (pi2 ?? program))) @le_S_S >prf >append_length
449          <plus_n_Sm @le_S_S @le_plus_n_r
450        | @lt_to_not_eq @le_S @Hi
451        ]
452      ]
453    | >Hi >lookup_insert_miss
454      [ #_ (* USE: everything is short *)
455        elim ((proj2 ?? Hp) (|prefix|) (le_n (|prefix|))) #pc #Hl
456        >(lookup_opt_lookup_hit … Hl 〈0,short_jump〉) @refl
457      | @bitvector_of_nat_abs
458        [ @(transitive_lt … (proj1 ?? (pi2 ?? program))) >prf @le_S_S >append_length @le_plus_n_r
459        | @(transitive_lt … (proj1 ?? (pi2 ?? program))) >prf @le_S_S >append_length <plus_n_Sm @le_S_S
460          @le_plus_n_r
461        | @lt_to_not_eq @le_n
462        ]
463      ]
464    ]
465  | (* 0 ↦ 0 *)
466    cases p -p #p cases p -p #pc #sigma #Hp cases x #lbl #instr normalize nodelta
467    >lookup_insert_miss
468    [ (* USE[pass]: 0 ↦ 0 *)
469      @(proj2 ?? (proj1 ?? (proj1 ?? (proj1 ?? Hp))))
470    | @bitvector_of_nat_abs
471      [ / by /
472      | @(transitive_lt … (proj1 ?? (pi2 ?? program))) >prf >append_length @le_S_S <plus_n_Sm
473        @le_S_S @le_plus_n_r
474      | @lt_to_not_eq / by /
475      ]
476    ]
477  ]
478  | (* fst p = pc *)
479    cases p -p #p cases p -p #pc #sigma #Hp cases x #lbl #instr normalize nodelta
480    >append_length >(commutative_plus (|prefix|)) >lookup_insert_hit @refl
481  ]
482  | (* policy_compact_unsafe *) #i >append_length <commutative_plus #Hi normalize in Hi;
483    cases p -p #p cases p -p #fpc #sigma #Hp cases x #lbl #instr normalize nodelta
484    cases (le_to_or_lt_eq … (le_S_S_to_le … Hi)) -Hi #Hi
485    [ >lookup_opt_insert_miss
486      [ >lookup_opt_insert_miss
487        [ (* USE[pass]: policy_compact_unsafe *)
488          lapply (proj2 ?? (proj1 ?? Hp) i Hi)
489          lapply (refl ? (bvt_lookup_opt … (bitvector_of_nat ? i) sigma))
490          cases (bvt_lookup_opt … (bitvector_of_nat ? i) sigma) in ⊢ (???% → %);
491          [ #_ normalize nodelta / by /
492          | #x cases x -x #pci #ji #EQi
493            lapply (refl ? (bvt_lookup_opt … (bitvector_of_nat ? (S i)) sigma))
494            cases (bvt_lookup_opt … (bitvector_of_nat ? (S i)) sigma) in ⊢ (???% → %);
495            [ #_ normalize nodelta / by /
496            | #x cases x -x #pcSi #jSi #EQSi normalize nodelta >nth_append_first
497              [ / by /
498              | @Hi
499              ]
500            ]
501          ]
502        ]
503      ]
504      [2: lapply (le_S_to_le … Hi) -Hi #Hi]
505      @bitvector_of_nat_abs
506      [1,4: @(transitive_lt … (proj1 ?? (pi2 ?? program))) >prf @le_S_S >append_length <commutative_plus
507        @le_plus_a @Hi
508      |2,5: @(transitive_lt … (proj1 ?? (pi2 ?? program))) >prf @le_S_S >append_length <plus_n_Sm
509        @le_S_S @le_plus_n_r
510      |3,6: @lt_to_not_eq @le_S_S @Hi
511      ]
512    | >lookup_opt_insert_miss
513      [ >Hi >lookup_opt_insert_hit normalize nodelta
514        (* USE: everything is short, fst p = pc *)
515        elim ((proj2 ?? Hp) (|prefix|) (le_n ?)) #pc #Hl
516        lapply (proj2 ?? (proj1 ?? (proj1 ?? Hp))) >Hl
517        >(lookup_opt_lookup_hit … Hl 〈0,short_jump〉) #EQ normalize nodelta >nth_append_second
518        [ <minus_n_n whd in match (nth ????); >EQ @refl
519        | @le_n
520        ]
521      | @bitvector_of_nat_abs
522        [ @(transitive_lt … (proj1 ?? (pi2 ?? program))) >Hi >prf @le_S_S >append_length <commutative_plus
523          @le_plus_a @le_n
524        | @(transitive_lt … (proj1 ?? (pi2 ?? program))) >prf @le_S_S >append_length <plus_n_Sm
525          @le_S_S @le_plus_n_r
526        | @lt_to_not_eq @le_S_S >Hi @le_n
527        ]
528      ]
529    ]
530  ]
531  | (* everything is short *) #i >append_length <commutative_plus #Hi normalize in Hi;
532    cases p -p #p cases p -p #pc #sigma #Hp cases x #lbl #instr normalize nodelta
533    cases (le_to_or_lt_eq … Hi) -Hi #Hi
534    [ >lookup_opt_insert_miss
535      [ (* USE[pass]: everything is short *)
536        @((proj2 ?? Hp) i (le_S_S_to_le … Hi))
537      | @bitvector_of_nat_abs
538        [ @(transitive_lt … (proj1 ?? (pi2 ?? program))) >prf >append_length @le_S_S
539          >commutative_plus @le_plus_a @le_S_S_to_le @Hi
540        | @(transitive_lt … (proj1 ?? (pi2 ?? program))) >prf >append_length <plus_n_Sm
541          @le_S_S @le_S_S @le_plus_n_r
542        | @lt_to_not_eq @Hi
543        ]
544      ]
545    | >Hi >lookup_opt_insert_hit @(ex_intro ?? (pc+instruction_size_jmplen short_jump instr))
546      @refl
547    ]
548  ]
549| @conj [ @conj [ @conj [ @conj
550  [ #i cases i
551    [ #Hi @⊥ @(absurd … Hi) @not_le_Sn_O
552    | -i #i #Hi #Hj @⊥ @(absurd … Hi) @not_le_Sn_O
553    ]
554  ] ]
555  >lookup_insert_hit @refl
556  | #i cases i
557    [ #Hi @⊥ @(absurd … Hi) @le_to_not_lt @le_n
558    | -i #i #Hi @⊥ @(absurd … Hi) @not_le_Sn_O
559    ]
560  ]
561  | #i cases i
562    [ #Hi >lookup_opt_insert_hit @(ex_intro ?? 0) @refl
563    | -i #i #Hi @⊥ @(absurd … Hi) @not_le_Sn_O
564    ]
565  ]
566]
567qed.
568
569(* NOTE: we only compare the first elements here because otherwise the
570 * added = 0 → policy_equal property of jump_expansion_step doesn't hold:
571 * if we have not added anything to the pc, we only know the PC hasn't changed,
572 * there might still have been a short/medium jump change *)
573definition sigma_pc_equal ≝
574  λprogram:list labelled_instruction.λp1,p2:ppc_pc_map.
575  (∀n.n ≤ |program| →
576    \fst (bvt_lookup … (bitvector_of_nat 16 n) (\snd p1) 〈0,short_jump〉) =
577    \fst (bvt_lookup … (bitvector_of_nat 16 n) (\snd p2) 〈0,short_jump〉)).
578
579definition sigma_jump_equal ≝
580  λprogram:list labelled_instruction.λp1,p2:ppc_pc_map.
581  (∀n.n < |program| →
582    \snd (bvt_lookup … (bitvector_of_nat 16 n) (\snd p1) 〈0,short_jump〉) =
583    \snd (bvt_lookup … (bitvector_of_nat 16 n) (\snd p2) 〈0,short_jump〉)).
584   
585definition nec_plus_ultra ≝
586  λprogram:list labelled_instruction.λp:ppc_pc_map.
587  ¬(∀i.i < |program| → is_jump (\snd (nth i ? program 〈None ?, Comment []〉)) →
588  \snd (bvt_lookup … (bitvector_of_nat 16 i) (\snd p) 〈0,short_jump〉) = long_jump).
589 
590(*include alias "common/Identifiers.ma".*)
591include alias "ASM/BitVector.ma".
592include alias "basics/lists/list.ma".
593include alias "arithmetics/nat.ma".
594include alias "basics/logic.ma".
595
596lemma jump_length_equal_max: ∀a,b,i.
597  is_jump i → instruction_size_jmplen (max_length a b) i = instruction_size_jmplen a i →
598  (max_length a b) = a.
599 #a #b #i cases i
600 [1: #pi cases pi
601   [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37:
602     try (#x #y #H #_) try (#x #H #_) try (#H #_) cases H
603   |9,10,11,12,13,14,15,16,17: #x [3,4,5,8,9: #y] #_ try (#_ %)
604     try (#abs normalize in abs; destruct (abs) @I)
605     cases a; cases b; try (#_ %) try (#abs normalize in abs; destruct(abs) @I)
606     try (@(subaddressing_mode_elim … x) #w #abs normalize in abs; destruct (abs) @I)
607     cases x * #a1 #a2 @(subaddressing_mode_elim … a2) #w
608     try ( #abs normalize in abs; destruct (abs) @I)
609     @(subaddressing_mode_elim … a1) #w2 #abs normalize in abs; destruct (abs)
610   ]
611  |2,3,6: #x [3: #y] #H cases H
612  |4,5: #id #_ cases a cases b
613    [2,3,4,6,11,12,13,15: normalize #H destruct (H)
614    |1,5,7,8,9,10,14,16,17,18: #H / by refl/
615    ]
616  ]
617qed.
618
619lemma jump_length_le_max: ∀a,b,i.is_jump i →
620  instruction_size_jmplen a i ≤ instruction_size_jmplen (max_length a b) i.
621 #a #b #i cases i
622 [2,3,6: #x [3: #y] #H cases H
623 |4,5: #id #_ cases a cases b / by le_n/
624 |1: #pi cases pi
625   [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37:
626     try (#x #y #H) try (#x #H) try (#H) cases H
627   |9,10,11,12,13,14,15,16,17: #x [3,4,5,8,9: #y]
628     #_ cases a cases b
629     [1,4,5,6,7,8,9: / by le_n/
630     |10,13,14,15,16,17,18: / by le_n/
631     |19,22,23,24,25,26,27: / by le_n/
632     |28,31,32,33,34,35,36: / by le_n/
633     |37,40,41,42,43,44,45: / by le_n/
634     |46,47,48,49,50,51,52,53,54: / by le_n/
635     |55,56,57,58,59,60,61,62,63: / by le_n/
636     |64,65,66,67,68,69,70,71,72: / by le_n/
637     |73,74,75,76,77,78,79,80,81: / by le_n/
638     ]
639     try (@(subaddressing_mode_elim … x) #w normalize @le_S @le_S @le_S @le_S @le_S @le_n)
640     cases x * #ad   
641     try (@(subaddressing_mode_elim … ad) #w #z normalize @le_S @le_S @le_S @le_S @le_S @le_n)
642     #z @(subaddressing_mode_elim … z) #w normalize / by /
643   ]
644 ]
645qed.
646
647lemma equal_compact_unsafe_compact: ∀program:(Σl.(S (|l|)) < 2^16 ∧ is_well_labelled_p l).
648  ∀old_sigma.∀sigma.
649  sigma_pc_equal program old_sigma sigma →
650  sigma_safe program (create_label_map program) 0 old_sigma sigma →
651  sigma_compact_unsafe program (create_label_map program) sigma →
652  sigma_compact program (create_label_map program) sigma.
653  #program cases program -program #program #Hprogram #old_sigma #sigma #Hequal
654  #Hsafe #Hcp_unsafe #i #Hi
655  lapply (Hcp_unsafe i Hi) lapply (Hsafe i Hi)
656  lapply (refl ? (lookup_opt … (bitvector_of_nat ? i) (\snd sigma)))
657  cases (lookup_opt … (bitvector_of_nat ? i) (\snd sigma)) in ⊢ (???% → %);
658  [ / by /
659  | #x cases x -x #x1 #x2 #EQ
660    cases (lookup_opt … (bitvector_of_nat ? (S i)) (\snd sigma))
661    [ / by /
662    | #y cases y -y #y1 #y2 normalize nodelta #H #H2
663      cut (instruction_size_jmplen x2
664       (\snd (nth i ? program 〈None ?, Comment []〉)) =
665       instruction_size … (bitvector_of_nat ? i)
666       (\snd (nth i ? program 〈None ?, Comment []〉)))
667      [5: #H3 <H3 @H2
668      |4: whd in match (instruction_size_jmplen ??);
669          whd in match (instruction_size …);
670          whd in match (assembly_1_pseudoinstruction …);
671          whd in match (expand_pseudo_instruction …);
672          normalize nodelta whd in match (append …) in H;
673          lapply (refl ? (nth i ? program 〈None ?, Comment []〉)) lapply H
674          cases (nth i ? program 〈None ?,Comment []〉) in ⊢ (% → ???% → %);
675          #lbl #instr cases instr
676          [2,3,6: #x [3: #y] normalize nodelta #H #_ %
677          |4,5: #x >(lookup_opt_lookup_hit … EQ 〈0,short_jump〉) #Hj #Heq
678            lapply (Hj x (refl ? x)) -Hj normalize nodelta
679            >add_bitvector_of_nat_plus <(plus_n_Sm i 0) <plus_n_O
680            cases x2 normalize nodelta
681            [1,4: whd in match short_jump_cond; normalize nodelta
682              cut (lookup_def ?? (create_label_map program) x 0 ≤ (|program|))
683              [1,3: cases (create_label_map program) #clm #Hclm
684                @le_S_to_le @(proj2 ?? (Hclm x ?))
685                [1: @(proj2 ?? Hprogram x (bitvector_of_nat ? i) ? (Jmp x) ??)
686                |2: @(proj2 ?? Hprogram x (bitvector_of_nat ? i) ? (Call x) ??)]
687                [1,4: >nat_of_bitvector_bitvector_of_nat_inverse
688                  [2,4: @(transitive_lt … (proj1 ?? Hprogram)) @le_S] @Hi
689                |2,5: whd in match fetch_pseudo_instruction; normalize nodelta
690                  >nth_safe_nth
691                  [1,3: >nat_of_bitvector_bitvector_of_nat_inverse
692                    [1,3: >Heq / by refl/
693                    |2,4: @(transitive_lt … (proj1 ?? Hprogram)) @le_S @Hi
694                    ]
695                  ]
696                |3,6: / by /
697                ]
698              |2,4: #H >(le_to_leb_true … H) normalize nodelta <plus_n_O
699              cases (sub_16_with_carry (bitvector_of_nat ??) (bitvector_of_nat ??) false)
700              #result #flags normalize nodelta
701              cases (vsplit bool 9 7 result) #upper #lower normalize nodelta
702              cases (get_index' bool 2 0 flags) normalize nodelta
703              [3,4: #H @⊥ @(absurd ?? (proj2 ?? H)) / by I/
704              |1,2: cases (eq_bv 9 upper ?)
705                [2,4: #H lapply (proj1 ?? H) #H3 destruct (H3)
706                |1,3: #_ normalize nodelta @refl
707                ]
708              ]
709              ]
710            |2,5: whd in match short_jump_cond; whd in match absolute_jump_cond;
711              cut (lookup_def ?? (create_label_map program) x 0 ≤ (|program|))
712              [1,3: cases (create_label_map program) #clm #Hclm
713                @le_S_to_le @(proj2 ?? (Hclm x ?))
714                [1: @(proj2 ?? Hprogram x (bitvector_of_nat ? i) ? (Jmp x) ??)
715                |2: @(proj2 ?? Hprogram x (bitvector_of_nat ? i) ? (Call x) ??)]
716                [1,4: >nat_of_bitvector_bitvector_of_nat_inverse
717                  [2,4: @(transitive_lt … (proj1 ?? Hprogram)) @le_S] @Hi
718                |2,5: whd in match fetch_pseudo_instruction; normalize nodelta
719                  >nth_safe_nth
720                  [1,3: >nat_of_bitvector_bitvector_of_nat_inverse
721                    [1,3: >Heq / by refl/
722                    |2,4: @(transitive_lt … (proj1 ?? Hprogram)) @le_S @Hi
723                    ]
724                  ]
725                |3,6: / by /
726                ]
727              |2,4: #H >(le_to_leb_true … H) normalize nodelta <plus_n_O
728              normalize nodelta cases (vsplit bool 5 11 ?) #addr1 #addr2
729              cases (vsplit bool 5 11 ?) #pc1 #pc2 normalize nodelta
730              cases (sub_16_with_carry (bitvector_of_nat ??) (bitvector_of_nat ??) false)
731              #result #flags normalize nodelta
732              cases (vsplit bool 9 7 result) #upper #lower normalize nodelta
733              cases (get_index' bool 2 0 flags) normalize nodelta
734              #H >(proj2 ?? (proj1 ?? H)) >(proj1 ?? (proj1 ?? H)) normalize nodelta @refl
735              ]
736            |3,6: whd in match short_jump_cond; whd in match absolute_jump_cond;
737              cut (lookup_def ?? (create_label_map program) x 0 ≤ (|program|))
738              [1,3: cases (create_label_map program) #clm #Hclm
739                @le_S_to_le @(proj2 ?? (Hclm x ?))
740                [1: @(proj2 ?? Hprogram x (bitvector_of_nat ? i) ? (Jmp x) ??)
741                |2: @(proj2 ?? Hprogram x (bitvector_of_nat ? i) ? (Call x) ??)]
742                [1,4: >nat_of_bitvector_bitvector_of_nat_inverse
743                  [2,4: @(transitive_lt … (proj1 ?? Hprogram)) @le_S] @Hi
744                |2,5: whd in match fetch_pseudo_instruction; normalize nodelta
745                  >nth_safe_nth
746                  [1,3: >nat_of_bitvector_bitvector_of_nat_inverse
747                    [1,3: >Heq / by refl/
748                    |2,4: @(transitive_lt … (proj1 ?? Hprogram)) @le_S @Hi
749                    ]
750                  ]
751                |3,6: / by /
752                ]
753              |2,4: #H >(le_to_leb_true … H) normalize nodelta <plus_n_O
754              cases (vsplit bool 5 11 ?) #addr1 #addr2
755              cases (vsplit bool 5 11 ?) #pc1 #pc2 normalize nodelta
756              cases (sub_16_with_carry (bitvector_of_nat ??) (bitvector_of_nat ??) false)
757              #result #flags normalize nodelta
758              cases (vsplit bool 9 7 result) #upper #lower normalize nodelta
759              cases (get_index' bool 2 0 flags) normalize nodelta
760              #H >(proj1 ?? H) >(proj2 ?? H) normalize nodelta @refl
761              ]
762            ]
763          |1: #pi cases pi
764            [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37:
765              [1,2,3,6,7,24,25: #x #y
766              |4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23: #x]
767              normalize nodelta #H #Heq @refl
768            |9,10,11,12,13,14,15,16,17: [1,2,6,7: #x |3,4,5,8,9: #y #x]
769              normalize nodelta >(lookup_opt_lookup_hit … EQ 〈0,short_jump〉)
770              #Hj #Heq lapply (Hj x (refl ? x)) -Hj
771              whd in match expand_relative_jump; normalize nodelta
772              whd in match expand_relative_jump_internal; normalize nodelta
773              whd in match expand_relative_jump_unsafe; normalize nodelta
774              whd in match expand_relative_jump_internal_unsafe;
775              normalize nodelta >(add_bitvector_of_nat_plus ? i 1)
776              <(plus_n_Sm i 0) <plus_n_O <plus_n_O cases x2 normalize nodelta
777              [1,4,7,10,13,16,19,22,25:
778                cut (∀A,B,ab.fst A B ab = (let 〈a,b〉 ≝ ab in a))
779                [1,3,5,7,9,11,13,15,17: #A #B * / by refl/ ]
780                #fst_foo >fst_foo @pair_elim #sj_possible #disp #H #H2
781                @(pair_replace ?????????? (eq_to_jmeq … H))
782                [1,3,5,7,9,11,13,15,17:
783                  cut (lookup_def ?? (create_label_map program) x 0 ≤ (|program|))
784                  [1,3,5,7,9,11,13,15,17: cases (create_label_map program) #clm #Hclm
785                    @le_S_to_le @(proj2 ?? (Hclm x ?))
786                    @(proj2 ?? Hprogram x (bitvector_of_nat ? i) ? (\snd (nth i ? program 〈None ?, Comment []〉)) ??)
787                    [1,4,7,10,13,16,19,22,25: >nat_of_bitvector_bitvector_of_nat_inverse
788                      [2,4,6,8,10,12,14,16,18: @(transitive_lt … (proj1 ?? Hprogram)) @le_S] @Hi
789                    |2,5,8,11,14,17,20,23,26: whd in match fetch_pseudo_instruction; normalize nodelta
790                      >nth_safe_nth
791                      [1,3,5,7,9,11,13,15,17: >nat_of_bitvector_bitvector_of_nat_inverse
792                        [1,3,5,7,9,11,13,15,17: >Heq %]
793                        @(transitive_lt … (proj1 ?? Hprogram)) @le_S @Hi
794                      ]
795                    ]
796                    >Heq / by /
797                  ]
798                  #X >(le_to_leb_true … X) @refl
799                ]
800                >(proj1 ?? H2) try (@refl) normalize nodelta
801                [1,2,3,5: @(subaddressing_mode_elim … y) #w %
802                | cases y * #sth #sth2 @(subaddressing_mode_elim … sth)
803                  @(subaddressing_mode_elim … sth2) #x [3,4: #x2] %
804                ]
805              |2,5,8,11,14,17,20,23,26: ** #_ #_ #abs cases abs #abs2 @⊥ @abs2 / by I/
806              ]
807              cut (∀A,B,ab.fst A B ab = (let 〈a,b〉 ≝ ab in a))
808              [1,3,5,7,9,11,13,15,17: #A #B * / by refl/ ]
809              #fst_foo * #H #_ >fst_foo in H; @pair_elim #sj_possible #disp #H
810              @(pair_replace ?????????? (eq_to_jmeq … H))
811                [1,3,5,7,9,11,13,15,17:
812                  cut (lookup_def ?? (create_label_map program) x 0 ≤ (|program|))
813                  [1,3,5,7,9,11,13,15,17: cases (create_label_map program) #clm #Hclm
814                    @le_S_to_le @(proj2 ?? (Hclm x ?))
815                    @(proj2 ?? Hprogram x (bitvector_of_nat ? i) ? (\snd (nth i ? program 〈None ?, Comment []〉)) ??)
816                    [1,4,7,10,13,16,19,22,25: >nat_of_bitvector_bitvector_of_nat_inverse
817                      [2,4,6,8,10,12,14,16,18: @(transitive_lt … (proj1 ?? Hprogram)) @le_S] @Hi
818                    |2,5,8,11,14,17,20,23,26: whd in match fetch_pseudo_instruction; normalize nodelta
819                      >nth_safe_nth
820                      [1,3,5,7,9,11,13,15,17: >nat_of_bitvector_bitvector_of_nat_inverse
821                        [1,3,5,7,9,11,13,15,17: >Heq %]
822                        @(transitive_lt … (proj1 ?? Hprogram)) @le_S @Hi
823                      ]
824                    ]
825                    >Heq / by /
826                  ]
827                  #X >(le_to_leb_true … X) @refl
828                ]
829                #H2 >H2 try (@refl) normalize nodelta
830                [1,2,3,5: @(subaddressing_mode_elim … y) #w %
831                | cases y * #sth #sth2 @(subaddressing_mode_elim … sth2) #w
832                  [1,2: %] whd in match (map ????); whd in match (flatten ??);
833                  whd in match (map ????) in ⊢ (???%); whd in match (flatten ??) in ⊢ (???%);
834                  >length_append >length_append @eq_f2 %
835                ]
836              ]
837            ]
838          ]
839        ]
840      ]
841qed.
842
843lemma instruction_size_irrelevant: ∀i.
844  ¬is_jump i → ∀j1,j2.instruction_size_jmplen j1 i = instruction_size_jmplen j2 i.
845 #i cases i
846 [2,3,6: #x [3: #y] #Hj #j1 #j2 %
847 |4,5: #x #Hi cases Hi #abs @⊥ @abs @I
848 |1: #pi cases pi
849   [1,2,3,4,5,6,7,8,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37:
850     [1,2,3,6,7,24,25: #x #y
851     |4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23: #x]
852     #Hj #j1 #j2 %
853   |9,10,11,12,13,14,15,16,17: [1,2,6,7: #x |3,4,5,8,9: #y #x]
854     #Hi cases Hi #abs @⊥ @abs @I
855   ]
856 ]
857qed.
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