1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "sets/sets.ma".
16 include "nat/plus.ma".
17 include "nat/compare.ma".
18 include "nat/minus.ma".
19 include "datatypes/pairs.ma".
21 alias symbol "eq" = "setoid eq".
22 alias symbol "eq" = "setoid1 eq".
23 alias symbol "eq" = "setoid eq".
24 alias symbol "eq" = "setoid1 eq".
25 alias symbol "eq" = "setoid eq".
26 alias symbol "eq" = "setoid1 eq".
27 alias symbol "eq" = "setoid eq".
28 alias symbol "eq" = "setoid1 eq".
29 alias symbol "eq" = "setoid eq".
30 alias symbol "eq" = "setoid1 eq".
31 alias symbol "eq" = "setoid eq".
32 alias symbol "eq" = "setoid1 eq".
33 alias symbol "eq" = "setoid eq".
34 nrecord partition (A: setoid) : Type[1] ≝
36 indexes: qpowerclass support;
37 class: unary_morphism1 (setoid1_of_setoid support) (qpowerclass_setoid A);
38 inhabited: ∀i. i ∈ indexes → class i ≬ class i;
39 disjoint: ∀i,j. i ∈ indexes → j ∈ indexes → class i ≬ class j → i=j;
40 covers: big_union support ? ? (λx.class x) = full_set A
41 }. napply indexes; nqed.
45 nlet rec iso_nat_nat_union (s: nat → nat) m index on index : pair nat nat ≝
46 match ltb m (s index) with
47 [ true ⇒ mk_pair … index m
50 [ O ⇒ (* dummy value: it could be an elim False: *) mk_pair … O O
51 | S index' ⇒ iso_nat_nat_union s (minus m (s index)) index']].
53 alias symbol "eq" = "leibnitz's equality".
54 naxiom plus_n_O: ∀n. plus n O = n.
55 naxiom ltb_t: ∀n,m. n < m → ltb n m = true.
56 naxiom ltb_f: ∀n,m. ¬ (n < m) → ltb n m = false.
57 naxiom ltb_cases: ∀n,m. (n < m ∧ ltb n m = true) ∨ (¬ (n < m) ∧ ltb n m = false).
58 naxiom minus_canc: ∀n. minus n n = O.
59 naxiom ad_hoc9: ∀a,b,c. a < b + c → a - b < c.
60 naxiom ad_hoc10: ∀a,b,c. a - b = c → a = b + c.
61 naxiom ad_hoc11: ∀a,b. a - b ≤ S a - b.
62 naxiom ad_hoc12: ∀a,b. b ≤ a → S a - b - (a - b) = S O.
63 naxiom ad_hoc13: ∀a,b. b ≤ a → (O + (a - b)) + b = a.
64 naxiom ad_hoc14: ∀a,b,c,d,e. c ≤ a → a - c = b + d + e → a = b + (c + d) + e.
65 naxiom not_lt_to_le: ∀a,b. ¬ (a < b) → b ≤ a.
66 naxiom split_big_plus:
68 big_plus n f = big_plus m (λi,p.f i ?) + big_plus (n - m) (λi.λp.f (i + m) ?).
72 ntheorem iso_nat_nat_union_char:
73 ∀n:nat. ∀s: nat → nat. ∀m:nat. m < big_plus (S n) (λi.λ_.s i) →
74 let p ≝ iso_nat_nat_union s m n in
75 m = big_plus (n - fst … p) (λi.λ_.s (S (i + fst … p))) + snd … p ∧
76 fst … p ≤ n ∧ snd … p < s (fst … p).
78 [ #m; nwhd in ⊢ (??% → let p ≝ % in ?); nwhd in ⊢ (??(??%) → ?);
79 nrewrite > (plus_n_O (s O)); #H; nrewrite > (ltb_t … H); nnormalize;
80 napply conj [ napply conj [ napply refl | napply le_n ] ##| nassumption ]
81 ##| #n'; #Hrec; #m; nwhd in ⊢ (??% → let p ≝ % in ?); #H;
82 ncases (ltb_cases m (s (S n'))); *; #H1; #H2; nrewrite > H2;
83 nwhd in ⊢ (let p ≝ % in ?); nwhd
84 [ napply conj [napply conj
85 [ nwhd in ⊢ (????(?(?%(λ_.λ_:(??%).?))%)); nrewrite > (minus_canc n'); napply refl
86 | nnormalize; napply le_n]
87 ##| nnormalize; nassumption ]
88 ##| nchange in H with (m < s (S n') + big_plus (S n') (λi.λ_.s i));
89 ngeneralize in match (Hrec (m - s (S n')) ?) in ⊢ ?
90 [##2: napply ad_hoc9; nassumption] *; *; #Hrec1; #Hrec2; #Hrec3; napply conj
93 [napply (eq_rect_CProp0_r ?? (λx.λ_. m = x + snd … (iso_nat_nat_union s (m - s (S n')) n')) ??
95 (S n' - fst … (iso_nat_nat_union s (m - s (S n')) n'))
96 (n' - fst … (iso_nat_nat_union s (m - s (S n')) n'))
97 (λi.λ_.s (S (i + fst … (iso_nat_nat_union s (m - s (S n')) n'))))?))
98 [##2: napply ad_hoc11]
99 napply (eq_rect_CProp0_r ?? (λx.λ_. ? = ? + big_plus x (λ_.λ_:? < x.?) + ?)
100 ?? (ad_hoc12 n' (fst … (iso_nat_nat_union s (m - s (S n')) n')) ?))
102 nwhd in ⊢ (???(?(??%)?));
103 nrewrite > (ad_hoc13 n' (fst … (iso_nat_nat_union s (m - s (S n')) n')) ?)
105 napply ad_hoc14 [ napply not_lt_to_le; nassumption ]
106 nwhd in ⊢ (???(?(??%)?));
107 napply (eq_rect_CProp0_r ?? (λx.λ_. ? = x + ?) ??
108 (plus_n_O (big_plus (n' - fst … (iso_nat_nat_union s (m - s (S n')) n'))
109 (λi.λ_.s (S (i + fst … (iso_nat_nat_union s (m - s (S n')) n')))))));
111 | napply le_S; nassumption ]##]##]##]
115 nlet rec partition_splits_card_map
116 A (P:partition A) n s (f:isomorphism ?? (Nat_ n) (indexes ? P))
117 (fi: ∀i. isomorphism ?? (Nat_ (s i)) (class ? P (iso_f ???? f i))) m index
119 match ltb m (s index) with
120 [ true ⇒ iso_f ???? (fi index) m
123 [ O ⇒ (* dummy value: it could be an elim False: *) iso_f ???? (fi O) O
125 partition_splits_card_map A P n s f fi (minus m (s index)) index']].
127 naxiom big_union_preserves_iso:
129 ∀g: isomorphism A' A T' T.
130 big_union A B T f = big_union A' B T' (λx.f (iso_f ???? g x)).
132 naxiom le_to_lt_or_eq: ∀n,m. n ≤ m → n < m ∨ n = m.
133 alias symbol "eq" = "leibnitz's equality".
134 naxiom lt_to_ltb_t: ∀n,m. ∀P: bool → CProp[0]. P true → n < m → P (ltb n m).
135 naxiom lt_to_ltb_f: ∀n,m. ∀P: bool → CProp[0]. P false → ¬ (n < m) → P (ltb n m).
136 naxiom lt_to_minus: ∀n,m. n < m → S (minus (minus m n) (S O)) = minus m n.
137 naxiom not_lt_O: ∀n. ¬ (n < O).
138 naxiom minus_S: ∀n,m. m ≤ n → minus (S n) m = S (minus n m).
139 naxiom minus_lt_to_lt: ∀n,m,p. n < p → minus n m < p.
140 naxiom minus_O_n: ∀n. O = minus O n.
141 naxiom le_O_to_eq: ∀n. n ≤ O → n=O.
142 naxiom lt_to_minus_to_S: ∀n,m. m < n → ∃k. minus n m = S k.
143 naxiom not_lt_plus: ∀n,m. ¬ (plus n m < n).
144 naxiom lt_to_lt_plus: ∀n,m,l. n < m → n < m + l.
145 naxiom S_plus: ∀n,m. S (n + m) = n + S m.
146 naxiom big_plus_ext: ∀n,f,f'. (∀i,p. f i p = f' i p) → big_plus n f = big_plus n f'.
147 naxiom ad_hoc1: ∀n,m,l. n + (m + l) = l + (n + m).
148 naxiom assoc: ∀n,m,l. n + m + l = n + (m + l).
149 naxiom lt_canc: ∀n,m,p. n < m → p + n < p + m.
150 naxiom ad_hoc2: ∀a,b. a < b → b - a - (b - S a) = S O.
151 naxiom ad_hoc3: ∀a,b. b < a → S (O + (a - S b) + b) = a.
152 naxiom ad_hoc4: ∀a,b. a - S b ≤ a - b.
153 naxiom ad_hoc5: ∀a. S a - a = S O.
154 naxiom ad_hoc6: ∀a,b. b ≤ a → a - b + b = a.
155 naxiom ad_hoc7: ∀a,b,c. a + (b + O) + c - b = a + c.
156 naxiom ad_hoc8: ∀a,b,c. ¬ (a + (b + O) + c < b).
157 naxiom ltb_elim_CProp0: ∀n,m. ∀P: bool → CProp[0]. (n < m → P true) → (¬ (n < m) → P false) → P (ltb n m).
159 nlemma partition_splits_card_output:
160 ∀A. ∀P:partition A. ∀n,s.
161 ∀f:isomorphism ?? (Nat_ (S n)) (indexes ? P).
162 ∀fi:∀i. isomorphism ?? (Nat_ (s i)) (class ? P (iso_f ???? f i)).
163 ∀x. x ∈ Nat_ (big_plus (S n) (λi.λ_.s i)) →
164 ∃i1.∃i2. partition_splits_card_map A P (S n) s f fi x n = iso_f ???? (fi i1) i2.
165 #A; #P; #n; #s; #f; #fi;
166 nelim n in ⊢ (? → % → ??(λ_.??(λ_.???(????????%)?)))
167 [ #x; nnormalize in ⊢ (% → ?); nrewrite > (plus_n_O (s O)); nchange in ⊢ (% → ?) with (x < s O);
168 #H; napply (ex_intro … O); napply (ex_intro … x); nwhd in ⊢ (???%?);
169 nrewrite > (ltb_t … H); nwhd in ⊢ (???%?); napply refl
170 | #n'; #Hrec; #x; #Hx; nwhd in ⊢ (??(λ_.??(λ_.???%?))); nwhd in Hx; nwhd in Hx: (??%);
171 nelim (ltb_cases x (s (S n'))); *; #K1; #K2; nrewrite > K2; nwhd in ⊢ (??(λ_.??(λ_.???%?)))
172 [ napply (ex_intro … (S n')); napply (ex_intro … x); napply refl
173 | napply (Hrec (x - s (S n')) ?); nwhd; nrewrite < (minus_S x (s (S n')) ?)
174 [ napply ad_hoc9; nassumption | napply not_lt_to_le; nassumption ]##]
177 nlemma partition_splits_card:
178 ∀A. ∀P:partition A. ∀n,s.
179 ∀f:isomorphism ?? (Nat_ n) (indexes ? P).
180 (∀i. isomorphism ?? (Nat_ (s i)) (class ? P (iso_f ???? f i))) →
181 (isomorphism ?? (Nat_ (big_plus n (λi.λ_.s i))) (Full_set A)).
182 #A; #P; #Sn; ncases Sn
184 ngeneralize in match (covers ? P) in ⊢ ?; *; #_; #H;
186 (big_union_preserves_iso ??? (indexes A P) (Nat_ O) (λx.class ? P x) f) in ⊢ ?;
187 *; #K; #_; nwhd in K: (? → ? → %);
188 nelim daemon (* impossibile *)
189 | #n; #s; #f; #fi; napply mk_isomorphism
190 [ napply mk_unary_morphism
191 [ napply (λm.partition_splits_card_map A P (S n) s f fi m n)
192 | #a; #a'; #H; nrewrite < H; napply refl ]
194 ngeneralize in match (covers ? P) in ⊢ ?; *; #_; #Hc;
195 ngeneralize in match (Hc y I) in ⊢ ?; *; #index; *; #Hi1; #Hi2;
196 ngeneralize in match (f_sur ???? f ? Hi1) in ⊢ ?; *; #nindex; *; #Hni1; #Hni2;
197 ngeneralize in match (f_sur ???? (fi nindex) y ?) in ⊢ ?
198 [##2: napply (. #‡(†?));##[##3: napply Hni2 |##2: ##skip | nassumption]##]
199 *; #nindex2; *; #Hni21; #Hni22;
200 nletin xxx ≝ (plus (big_plus (minus n nindex) (λi.λ_.s (S (plus i nindex)))) nindex2);
201 napply (ex_intro … xxx); napply conj
202 [ nwhd in Hni1; nwhd; nwhd in ⊢ (?(? %)%);
203 nchange with (? < plus (s n) (big_plus n ?));
204 nelim (le_to_lt_or_eq … (le_S_S_to_le … Hni1))
205 [##2: #E; nrewrite < E; nrewrite < (minus_canc nindex);
206 nwhd in ⊢ (?%?); nrewrite < E; napply lt_to_lt_plus; nassumption
207 | #L; nrewrite > (split_big_plus n (S nindex) (λm.λ_.s m) L);
208 nrewrite > (split_big_plus (n - nindex) (n - S nindex) (λi.λ_.s (S (i+nindex))) ?)
209 [ ngeneralize in match (big_plus_ext (n - S nindex)
210 (λi,p.s (S (i+nindex))) (λi,p.s (i + S nindex)) ?) in ⊢ ?
212 napply (eq_rect_CProp0_r ??
213 (λx:nat.λ_. x + big_plus (n - nindex - (n - S nindex))
214 (λi,p.s (S (i + (n - S nindex)+nindex))) + nindex2 <
215 s n + (big_plus (S nindex) (λi,p.s i) +
216 big_plus (n - S nindex) (λi,p. s (i + S nindex)))) ? ? E);
217 nrewrite > (ad_hoc1 (s n) (big_plus (S nindex) (λi,p.s i))
218 (big_plus (n - S nindex) (λi,p. s (i + S nindex))));
219 napply (eq_rect_CProp0_r
220 ?? (λx.λ_.x < ?) ?? (assoc
221 (big_plus (n - S nindex) (λi,p.s (i + S nindex)))
222 (big_plus (n - nindex - (n - S nindex))
223 (λi,p.s (S (i + (n - S nindex)+nindex))))
226 nrewrite > (ad_hoc2 … L); nwhd in ⊢ (?(?%?)?);
227 nrewrite > (ad_hoc3 … L);
228 napply (eq_rect_CProp0_r ?? (λx.λ_.x < ?) ?? (assoc …));
229 napply lt_canc; nnormalize in ⊢ (?%?); nwhd in ⊢ (??%);
230 napply lt_to_lt_plus; nassumption
231 ##|##2: #i; #_; nrewrite > (S_plus i nindex); napply refl]
232 ##| napply ad_hoc4]##]
233 ##| nwhd in ⊢ (???%?);
234 nchange in Hni1 with (nindex < S n);
235 ngeneralize in match (le_S_S_to_le … Hni1) in ⊢ ?;
236 nwhd in ⊢ (? → ???(???????%?)?);
237 napply (nat_rect_CProp0
239 eq_rel (carr A) (eq A)
240 (partition_splits_card_map A P (S n) s f fi
242 (big_plus (minus x nindex) (λi.λ_:i < minus x nindex.s (S (plus i nindex))))
244 [ #K; nrewrite < (minus_O_n nindex); nwhd in ⊢ (???(???????%?)?);
245 nwhd in ⊢ (???%?); nchange in Hni21 with (nindex2 < s nindex);
246 ngeneralize in match (le_O_to_eq … K) in ⊢ ?; #K';
247 ngeneralize in match Hni21 in ⊢ ?;
248 ngeneralize in match Hni22 in ⊢ ?;
249 nrewrite > K' in ⊢ (% → % → ?); #K1; #K2;
250 nrewrite > (ltb_t … K2);
251 nwhd in ⊢ (???%?); nassumption
252 | #n'; #Hrec; #HH; nelim (le_to_lt_or_eq … HH)
253 [##2: #K; nrewrite < K; nrewrite < (minus_canc nindex);
254 nwhd in ⊢ (???(???????%?)?);
256 nwhd in ⊢ (???%?); nrewrite < K; nrewrite > (ltb_t … Hni21);
257 nwhd in ⊢ (???%?); nassumption
258 ##| #K; ngeneralize in match (le_S_S_to_le … K) in ⊢ ?; #K';
260 ngeneralize in match (?:
261 ¬ (big_plus (S n' - nindex) (λi,p.s (S (i+nindex))) + nindex2 < s (S n'))) in ⊢ ?
262 [ #N; nrewrite > (ltb_f … N); nwhd in ⊢ (???%?);
263 ngeneralize in match (Hrec K') in ⊢ ?; #Hrec';
264 napply (eq_rect_CProp0_r ??
265 (λx,p. eq_rel (carr A) (eq A) (partition_splits_card_map A P (S n) s f fi
266 (big_plus x ? + ? - ?) n') y) ?? (minus_S n' nindex K'));
267 nrewrite > (split_big_plus (S (n' - nindex)) (n' - nindex)
268 (λi,p.s (S (i+nindex))) (le_S ?? (le_n ?)));
269 nrewrite > (ad_hoc5 (n' - nindex));
270 nnormalize in ⊢ (???(???????(?(?(??%)?)?)?)?);
271 nrewrite > (ad_hoc6 … K');
272 nrewrite > (ad_hoc7 (big_plus (n' - nindex) (λi,p.s (S (i+nindex))))
275 | nrewrite > (minus_S … K');
276 nrewrite > (split_big_plus (S (n' - nindex)) (n' - nindex)
277 (λi,p.s (S (i+nindex))) (le_S ?? (le_n ?)));
278 nrewrite > (ad_hoc5 (n' - nindex));
279 nnormalize in ⊢ (?(?(?(??%)?)?));
280 nrewrite > (ad_hoc6 … K');
281 napply ad_hoc8]##]##]##]
282 ##| #x; #x'; nnormalize in ⊢ (? → ? → %); #Hx; #Hx';
283 nelim (partition_splits_card_output A P n s f fi x Hx); #i1x; *; #i2x; #Ex;
284 nelim (partition_splits_card_output A P n s f fi x' Hx'); #i1x'; *; #i2x'; #Ex';
285 ngeneralize in match (? :
286 iso_f ???? fi i1x(* ≬ iso_f ???? (fi i1x'))*)) in ⊢ ?;
287 #E; napply (f_inj ???? (fi i1x));
289 nelim n in ⊢ (% → % → (???(????????%)(????????%)) → ?)
290 [ nnormalize in ⊢ (% → % → ?); nrewrite > (plus_n_O (s O));
291 nchange in ⊢ (% → ?) with (x < s O);
292 nchange in ⊢ (? → % → ?) with (x' < s O);
293 #H1; #H2; nwhd in ⊢ (???%% → ?);
294 nrewrite > (ltb_t … H1); nrewrite > (ltb_t … H2); nwhd in ⊢ (???%% → ?);
295 napply f_inj; nassumption
296 | #n'; #Hrec; #Hx; #Hx'; nwhd in ⊢ (???%% → ?);
300 (************** equivalence relations vs partitions **********************)
302 ndefinition partition_of_compatible_equivalence_relation:
303 ∀A:setoid. compatible_equivalence_relation A → partition A.
304 #A; #R; napply mk_partition
305 [ napply (quotient ? R)
307 | napply mk_unary_morphism1
308 [ #a; napply mk_qpowerclass
310 | #x; #x'; #H; nnormalize; napply mk_iff; #K; nelim daemon]
311 ##| #a; #a'; #H; napply conj; #x; nnormalize; #K [ nelim daemon | nelim daemon]##]
312 ##| #x; #_; nnormalize; napply (ex_intro … x); napply conj; napply refl
313 | #x; #x'; #_; #_; nnormalize; *; #x''; *; #H1; #H2; napply (trans ?????? H2);
314 napply sym; nassumption
315 | nnormalize; napply conj
316 [ #a; #_; napply I | #a; #_; napply (ex_intro … a); napply conj [ napply I | napply refl]##]