2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
7 ||A|| This file is distributed under the terms of the
8 \ / GNU General Public License Version 2
10 V_______________________________________________________________ *)
12 include "lambda/reduction.ma".
15 inductive T : Type[0] ≝
19 | Lambda: T → T → T (* type, body *)
20 | Prod: T → T → T (* type, body *)
25 inductive conv : T →T → Prop ≝
26 | cbeta: ∀P,M,N. conv (App (Lambda P M) N) (M[0 ≝ N])
27 | cdapp: ∀M,N. conv (App (D M) N) (D (App M N))
28 | cdlam: ∀M,N. conv (Lambda M (D N)) (D (Lambda M N))
29 | cappl: ∀M,M1,N. conv M M1 → conv (App M N) (App M1 N)
30 | cappr: ∀M,N,N1. conv N N1 → conv (App M N) (App M N1)
31 | claml: ∀M,M1,N. conv M M1 → conv (Lambda M N) (Lambda M1 N)
32 | clamr: ∀M,N,N1. conv N N1 → conv (Lambda M N) (Lambda M N1)
33 | cprodl: ∀M,M1,N. conv M M1 → conv (Prod M N) (Prod M1 N)
34 | cprodr: ∀M,N,N1. conv N N1 → conv (Prod M N) (Prod M N1)
35 | cd: ∀M,M1. conv (D M) (D M1).
37 definition CO ≝ star … conv.
39 lemma red_to_conv: ∀M,N. red M N → conv M N.
40 #M #N #redMN (elim redMN) /2/
43 inductive d_eq : T →T → Prop ≝
45 | ed: ∀M,M1. d_eq (D M) (D M1)
46 | eapp: ∀M1,M2,N1,N2. d_eq M1 M2 → d_eq N1 N2 →
47 d_eq (App M1 N1) (App M2 N2)
48 | elam: ∀M1,M2,N1,N2. d_eq M1 M2 → d_eq N1 N2 →
49 d_eq (Lambda M1 N1) (Lambda M2 N2)
50 | eprod: ∀M1,M2,N1,N2. d_eq M1 M2 → d_eq N1 N2 →
51 d_eq (Prod M1 N1) (Prod M2 N2).
53 lemma conv_to_deq: ∀M,N. conv M N → red M N ∨ d_eq M N.
54 #M #N #coMN (elim coMN)
58 |#P #M1 #N1 #coPM1 * [#redP %1 /2/ | #eqPM1 %2 /3/]
59 |#P #M1 #N1 #coPM1 * [#redP %1 /2/ | #eqPM1 %2 /3/]
60 |#P #M1 #N1 #coPM1 * [#redP %1 /2/ | #eqPM1 %2 /3/]
61 |#P #M1 #N1 #coPM1 * [#redP %1 /2/ | #eqPM1 %2 /3/]
62 |#P #M1 #N1 #coPM1 * [#redP %1 /2/ | #eqPM1 %2 /3/]
63 |#P #M1 #N1 #coPM1 * [#redP %1 /2/ | #eqPM1 %2 /3/]
67 (* FG: THIS IN NOT COMPLETE
68 theorem main1: ∀M,N. CO M N →
69 ∃P,Q. star … red M P ∧ star … red N Q ∧ d_eq P Q.
70 #M #N #coMN (elim coMN)
71 [#B #C #rMB #convBC * #P1 * #Q1 * * #redMP1 #redBQ1
72 #deqP1Q1 (cases (conv_to_deq … convBC))
74 |@(ex_intro … M) @(ex_intro … M) % // % //
77 lemma red_d : ∀M,P. red (D M) P → ∃N. P = D N ∧ red M N.
78 #M #P #redMP (inversion redMP)
79 [#P1 #M1 #N1 #eqH destruct
80 |#M1 #N1 #eqH destruct
81 |#M1 #N1 #eqH destruct
82 |4,5,6,7,8,9:#Q1 #Q2 #N1 #red1 #_ #eqH destruct
83 |#Q1 #M1 #red1 #_ #eqH destruct #eqP @(ex_intro … M1) /2/
87 lemma red_lambda : ∀M,N,P. red (Lambda M N) P →
88 (∃M1. P = (Lambda M1 N) ∧ red M M1) ∨
89 (∃N1. P = (Lambda M N1) ∧ red N N1) ∨
90 (∃Q. N = D Q ∧ P = D (Lambda M Q)).
91 #M #N #P #redMNP (inversion redMNP)
92 [#P1 #M1 #N1 #eqH destruct
93 |#M1 #N1 #eqH destruct
94 |#M1 #N1 #eqH destruct #eqP %2 (@(ex_intro … N1)) % //
95 |4,5,8,9:#Q1 #Q2 #N1 #red1 #_ #eqH destruct
96 |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %1 %1
97 (@(ex_intro … M1)) % //
98 |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %1 %2
99 (@(ex_intro … N1)) % //
100 |#Q1 #M1 #red1 #_ #eqH destruct
104 lemma red_prod : ∀M,N,P. red (Prod M N) P →
105 (∃M1. P = (Prod M1 N) ∧ red M M1) ∨
106 (∃N1. P = (Prod M N1) ∧ red N N1).
107 #M #N #P #redMNP (inversion redMNP)
108 [#P1 #M1 #N1 #eqH destruct
109 |2,3: #M1 #N1 #eqH destruct
110 |4,5,6,7:#Q1 #Q2 #N1 #red1 #_ #eqH destruct
111 |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %1
112 (@(ex_intro … M1)) % //
113 |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %2
114 (@(ex_intro … N1)) % //
115 |#Q1 #M1 #red1 #_ #eqH destruct
119 lemma red_app : ∀M,N,P. red (App M N) P →
120 (∃M1,N1. M = (Lambda M1 N1) ∧ P = N1[0:=N]) ∨
121 (∃M1. M = (D M1) ∧ P = D (App M1 N)) ∨
122 (∃M1. P = (App M1 N) ∧ red M M1) ∨
123 (∃N1. P = (App M N1) ∧ red N N1).
124 #M #N #P #redMNP (inversion redMNP)
125 [#P1 #M1 #N1 #eqH destruct #eqP %1 %1 %1
126 @(ex_intro … P1) @(ex_intro … M1) % //
127 |#M1 #N1 #eqH destruct #eqP %1 %1 %2 /3/
128 |#M1 #N1 #eqH destruct
129 |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %1 %2
130 (@(ex_intro … M1)) % //
131 |#Q1 #M1 #N1 #red1 #_ #eqH destruct #eqP %2
132 (@(ex_intro … N1)) % //
133 |6,7,8,9:#Q1 #Q2 #N1 #red1 #_ #eqH destruct
134 |#Q1 #M1 #red1 #_ #eqH destruct
138 definition reduct ≝ λn,m. red m n.
140 definition SN ≝ WF ? reduct.
142 definition NF ≝ λM. ∀N. ¬ (reduct N M).
144 theorem NF_to_SN: ∀M. NF M → SN M.
145 #M #nfM % #a #red @False_ind /2/
148 lemma NF_Sort: ∀i. NF (Sort i).
149 #i #N % #redN (inversion redN)
150 [1: #P #N #M #H destruct
151 |2,3 :#N #M #H destruct
152 |4,5,6,7,8,9: #N #M #P #_ #_ #H destruct
153 |#M #N #_ #_ #H destruct
157 lemma NF_Rel: ∀i. NF (Rel i).
158 #i #N % #redN (inversion redN)
159 [1: #P #N #M #H destruct
160 |2,3 :#N #M #H destruct
161 |4,5,6,7,8,9: #N #M #P #_ #_ #H destruct
162 |#M #N #_ #_ #H destruct
166 lemma red_subst : ∀N,M,M1,i. red M M1 → red M[i≝N] M1[i≝N].
167 #N @Telim_size #P (cases P)
168 [1,2:#j #Hind #M1 #i #r1 @False_ind /2/
169 |#P #Q #Hind #M1 #i #r1 (cases (red_app … r1))
172 [* #M2 * #N2 * #eqP #eqM1 >eqP normalize
173 >eqM1 >(plus_n_O i) >(subst_lemma N2) <(plus_n_O i)
174 (cut (i+1 =S i)) [//] #Hcut >Hcut @rbeta
175 |* #M2 * #eqP #eqM1 >eqM1 >eqP normalize @rdapp
177 |* #M2 * #eqM1 #rP >eqM1 normalize @rappl @Hind /2/
179 |* #N2 * #eqM1 #rQ >eqM1 normalize @rappr @Hind /2/
181 |#P #Q #Hind #M1 #i #r1 (cases (red_lambda …r1))
183 [* #P1 * #eqM1 #redP >eqM1 normalize @rlaml @Hind /2/
184 |* #Q1 * #eqM1 #redP >eqM1 normalize @rlamr @Hind /2/
186 |* #M2 * #eqQ #eqM1 >eqM1 >eqQ normalize @rdlam
188 |#P #Q #Hind #M1 #i #r1 (cases (red_prod …r1))
189 [* #P1 * #eqM1 #redP >eqM1 normalize @rprodl @Hind /2/
190 |* #P1 * #eqM1 #redP >eqM1 normalize @rprodr @Hind /2/
192 |#P #Hind #M1 #i #r1 (cases (red_d …r1))
193 #P1 * #eqM1 #redP >eqM1 normalize @d @Hind /2/
197 lemma red_lift: ∀N,N1,n. red N N1 → ∀k. red (lift N k n) (lift N1 k n).
198 #N #N1 #n #r1 (elim r1) normalize /2/
202 lemma star_appl: ∀M,M1,N. star … red M M1 →
203 star … red (App M N) (App M1 N).
204 #M #M1 #N #star1 (elim star1) //
205 #B #C #starMB #redBC #H @(inj … H) /2/
208 lemma star_appr: ∀M,N,N1. star … red N N1 →
209 star … red (App M N) (App M N1).
210 #M #N #N1 #star1 (elim star1) //
211 #B #C #starMB #redBC #H @(inj … H) /2/
214 lemma star_app: ∀M,M1,N,N1. star … red M M1 → star … red N N1 →
215 star … red (App M N) (App M1 N1).
216 #M #M1 #N #N1 #redM #redN @(trans_star ??? (App M1 N)) /2/
219 lemma star_laml: ∀M,M1,N. star … red M M1 →
220 star … red (Lambda M N) (Lambda M1 N).
221 #M #M1 #N #star1 (elim star1) //
222 #B #C #starMB #redBC #H @(inj … H) /2/
225 lemma star_lamr: ∀M,N,N1. star … red N N1 →
226 star … red (Lambda M N) (Lambda M N1).
227 #M #N #N1 #star1 (elim star1) //
228 #B #C #starMB #redBC #H @(inj … H) /2/
231 lemma star_lam: ∀M,M1,N,N1. star … red M M1 → star … red N N1 →
232 star … red (Lambda M N) (Lambda M1 N1).
233 #M #M1 #N #N1 #redM #redN @(trans_star ??? (Lambda M1 N)) /2/
236 lemma star_prodl: ∀M,M1,N. star … red M M1 →
237 star … red (Prod M N) (Prod M1 N).
238 #M #M1 #N #star1 (elim star1) //
239 #B #C #starMB #redBC #H @(inj … H) /2/
242 lemma star_prodr: ∀M,N,N1. star … red N N1 →
243 star … red (Prod M N) (Prod M N1).
244 #M #N #N1 #star1 (elim star1) //
245 #B #C #starMB #redBC #H @(inj … H) /2/
248 lemma star_prod: ∀M,M1,N,N1. star … red M M1 → star … red N N1 →
249 star … red (Prod M N) (Prod M1 N1).
250 #M #M1 #N #N1 #redM #redN @(trans_star ??? (Prod M1 N)) /2/
253 lemma star_d: ∀M,M1. star … red M M1 →
254 star … red (D M) (D M1).
255 #M #M1 #redM (elim redM) // #B #C #starMB #redBC #H @(inj … H) /2/
258 lemma red_subst1 : ∀M,N,N1,i. red N N1 →
259 (star … red) M[i≝N] M[i≝N1].
262 |#i #P #Q #n #r1 (cases (true_or_false (leb i n)))
263 [#lein (cases (le_to_or_lt_eq i n (leb_true_to_le … lein)))
264 [#ltin >(subst_rel1 … ltin) >(subst_rel1 … ltin) //
265 |#eqin >eqin >subst_rel2 >subst_rel2 @R_to_star /2/
267 |#lefalse (cut (n < i)) [@not_le_to_lt /2/] #ltni
268 >(subst_rel3 … ltni) >(subst_rel3 … ltni) //
270 |#P #Q #Hind1 #Hind2 #M1 #N1 #i #r1 normalize @star_app /2/
271 |#P #Q #Hind1 #Hind2 #M1 #N1 #i #r1 normalize @star_lam /2/
272 |#P #Q #Hind1 #Hind2 #M1 #N1 #i #r1 normalize @star_prod /2/
273 |#P #Hind #M #N #i #r1 normalize @star_d /2/
277 lemma SN_d : ∀M. SN M → SN (D M).
278 #M #snM (elim snM) #b #H #Hind % #a #redd (cases (red_d … redd))
279 #Q * #eqa #redbQ >eqa @Hind //
282 lemma SN_step: ∀N. SN N → ∀M. reduct M N → SN M.
283 #N * #b #H #M #red @H //.
286 lemma SN_star: ∀M,N. (star … red) N M → SN N → SN M.
287 #M #N #rstar (elim rstar) //
288 #Q #P #HbQ #redQP #snNQ #snN @(SN_step …redQP) /2/
291 lemma sub_red: ∀M,N.subterm N M → ∀N1.red N N1 →
292 ∃M1.subterm N1 M1 ∧ red M M1.
293 #M #N #subN (elim subN) /4/
294 (* trsansitive case *)
295 #P #Q #S #subPQ #subQS #H1 #H2 #A #redP (cases (H1 ? redP))
296 #B * #subA #redQ (cases (H2 ? redQ)) #C * #subBC #redSC
300 axiom sub_star_red: ∀M,N.(star … subterm) N M → ∀N1.red N N1 →
301 ∃M1.subterm N1 M1 ∧ red M M1.
303 lemma SN_subterm: ∀M. SN M → ∀N.subterm N M → SN N.
304 #M #snM (elim snM) #M #snM #HindM #N #subNM % #N1 #redN
305 (cases (sub_red … subNM ? redN)) #M1 *
306 #subN1M1 #redMM1 @(HindM … redMM1) //
309 lemma SN_subterm_star: ∀M. SN M → ∀N.(star … subterm N M) → SN N.
310 #M #snM #N #Hstar (cases (star_inv T subterm M N)) #_ #H
311 lapply (H Hstar) #Hstari (elim Hstari) //
312 #M #N #_ #subNM #snM @(SN_subterm …subNM) //
315 definition shrink ≝ λN,M. reduct N M ∨ (TC … subterm) N M.
317 definition SH ≝ WF ? shrink.
319 lemma SH_subterm: ∀M. SH M → ∀N.(star … subterm) N M → SH N.
320 #M #snM (elim snM) #M
321 #snM #HindM #N #subNM (cases (star_case ???? subNM))
324 [#redN (cases (sub_star_red … subNM ? redN)) #M1 *
325 #subN1M1 #redMM1 @(HindM M1) /2/
326 |#subN1 @(HindM N) /2/
331 theorem SN_to_SH: ∀N. SN N → SH N.
332 #N #snN (elim snN) (@Telim_size)
333 #b #Hsize #snb #Hind % #a * /2/ #subab @Hsize;
335 [#c #subac @size_subterm //
336 |#b #c #subab #subbc #sab @(transitive_lt … sab) @size_subterm //
338 |@SN_step @(SN_subterm_star b);
339 [% /2/ |@TC_to_star @subab] % @snb
340 |#a1 #reda1 cases(sub_star_red b a ?? reda1);
341 [#a2 * #suba1 #redba2 @(SH_subterm a2) /2/ |/2/ ]
345 lemma SH_to_SN: ∀N. SH N → SN N.
346 @WF_antimonotonic /2/ qed.
348 lemma SN_Lambda: ∀N.SN N → ∀M.SN M → SN (Lambda N M).
349 #N #snN (elim snN) #P #shP #HindP #M #snM
350 (* for M we proceed by induction on SH *)
351 (lapply (SN_to_SH ? snM)) #shM (elim shM)
352 #Q #shQ #HindQ % #a #redH (cases (red_lambda … redH))
354 [* #S * #eqa #redPS >eqa @(HindP S ? Q ?) //
356 |* #S * #eqa #redQS >eqa @(HindQ S) /2/
358 |* #S * #eqQ #eqa >eqa @SN_d @(HindQ S) /3/
363 lemma SH_Lambda: ∀N.SH N → ∀M.SH M → SN (Lambda N M).
364 #N #snN (elim snN) #P #snP #HindP #M #snM (elim snM)
365 #Q #snQ #HindQ % #a #redH (cases (red_lambda … redH))
367 [* #S * #eqa #redPS >eqa @(HindP S ? Q ?) /2/
369 |* #S * #eqa #redQS >eqa @(HindQ S) /2/
371 |* #S * #eqQ #eqa >eqa @SN_d @(HindQ S) /3/
375 lemma SN_Prod: ∀N.SN N → ∀M.SN M → SN (Prod N M).
376 #N #snN (elim snN) #P #shP #HindP #M #snM (elim snM)
377 #Q #snQ #HindQ % #a #redH (cases (red_prod … redH))
378 [* #S * #eqa #redPS >eqa @(HindP S ? Q ?) //
380 |* #S * #eqa #redQS >eqa @(HindQ S) /2/
384 lemma SN_subst: ∀i,N,M.SN M[i ≝ N] → SN M.
385 #i #N (cut (∀P.SN P → ∀M.P=M[i ≝ N] → SN M));
386 [#P #H (elim H) #Q #snQ #Hind #M #eqM % #M1 #redM
387 @(Hind M1[i:=N]) // >eqM /2/
388 |#Hcut #M #snM @(Hcut … snM) //
391 lemma SN_DAPP: ∀N,M. SN (App M N) → SN (App (D M) N).
392 cut (∀P. SN P → ∀M,N. P = App M N → SN (App (D M) N)); [|/2/]
393 #P #snP (elim snP) #Q #snQ #Hind
394 #M #N #eqQ % #A #rA (cases (red_app … rA))
397 [* #M1 * #N1 * #eqH destruct
398 |* #M1 * #eqH destruct #eqA >eqA @SN_d % @snQ
400 |* #M1 * #eqA #red1 (cases (red_d …red1))
401 #M2 * #eqM1 #r2 >eqA >eqM1 @(Hind (App M2 N)) /2/
403 |* #M2 * #eqA >eqA #r2 @(Hind (App M M2)) /2/
407 lemma SN_APP: ∀P.SN P → ∀N. SN N → ∀M.
408 SN M[0:=N] → SN (App (Lambda P M) N).
409 #P #snP (elim snP) #A #snA #HindA
410 #N #snN (elim snN) #B #snB #HindB
411 #M #snM1 (cut (SH M)) [@SN_to_SH @(SN_subst … snM1)] #shM
412 (generalize in match snM1) (elim shM)
413 #C #shC #HindC #snC1 % #Q #redQ (cases (red_app … redQ))
416 [* #M2 * #N2 * #eqlam destruct #eqQ //
417 |* #M2 * #eqlam destruct
419 |* #M2 * #eqQ #redlam >eqQ (cases (red_lambda …redlam))
421 [* #M3 * #eqM2 #r2 >eqM2 @HindA // % /2/
422 |* #M3 * #eqM2 #r2 >eqM2 @HindC;
423 [%1 // |@(SN_step … snC1) /2/]
425 |* #M3 * #eqC #eqM2 >eqM2 @SN_DAPP @HindC;
427 |@(SN_subterm … snC1) >eqC normalize //
431 |* #M2 * #eqQ #r2 >eqQ @HindB // @(SN_star … snC1)