alias symbol "eq" = "setoid1 eq".
alias symbol "eq" = "setoid eq".
alias symbol "eq" = "setoid1 eq".
+alias symbol "eq" = "setoid eq".
+alias symbol "eq" = "setoid1 eq".
+alias symbol "eq" = "setoid eq".
nrecord partition (A: setoid) : Type[1] ≝
{ support: setoid;
indexes: qpowerclass support;
naxiom minus_O_n: ∀n. O = minus O n.
naxiom le_O_to_eq: ∀n. n ≤ O → n=O.
naxiom lt_to_minus_to_S: ∀n,m. m < n → ∃k. minus n m = S k.
-naxiom ltb_t: ∀n,m. n < m → ltb n m = true.
+naxiom ltb_t: ∀n,m. n < m → ltb n m = true.
+naxiom ltb_f: ∀n,m. ¬ (n < m) → ltb n m = false.
+naxiom plus_n_O: ∀n. plus n O = n.
+naxiom not_lt_plus: ∀n,m. ¬ (plus n m < n).
+naxiom lt_to_lt_plus: ∀n,m,l. n < m → n < m + l.
+naxiom S_plus: ∀n,m. S (n + m) = n + S m.
+naxiom big_plus_ext: ∀n,f,f'. (∀i,p. f i p = f' i p) → big_plus n f = big_plus n f'.
+naxiom ad_hoc1: ∀n,m,l. n + (m + l) = l + (n + m).
+naxiom assoc: ∀n,m,l. n + m + l = n + (m + l).
+naxiom lt_canc: ∀n,m,p. n < m → p + n < p + m.
+naxiom ad_hoc2: ∀a,b. a < b → b - a - (b - S a) = S O.
+naxiom ad_hoc3: ∀a,b. b < a → S (O + (a - S b) + b) = a.
+naxiom ad_hoc4: ∀a,b. a - S b ≤ a - b.
+naxiom ad_hoc5: ∀a. S a - a = S O.
+naxiom ad_hoc6: ∀a,b. b ≤ a → a - b + b = a.
+naxiom ad_hoc7: ∀a,b,c. a + (b + O) + c - b = a + c.
+naxiom ad_hoc8: ∀a,b,c. ¬ (a + (b + O) + c < b).
+
+
+naxiom split_big_plus:
+ ∀n,m,f. m ≤ n →
+ big_plus n f = big_plus m (λi,p.f i ?) + big_plus (n - m) (λi.λp.f (i + m) ?).
+ nelim daemon.
+nqed.
nlemma partition_splits_card:
∀A. ∀P:partition A. ∀n,s.
ngeneralize in match (f_sur ???? (fi nindex) y ?) in ⊢ ?
[##2: napply (. #‡(†?));##[##3: napply Hni2 |##2: ##skip | nassumption]##]
*; #nindex2; *; #Hni21; #Hni22;
- nletin xxx ≝ (plus match minus n nindex return λ_.nat with [ O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.s (S (plus i nindex)))] nindex2);
+ nletin xxx ≝ (plus (big_plus (minus n nindex) (λi.λ_.s (S (plus i nindex)))) nindex2);
napply (ex_intro … xxx); napply conj
- [ nwhd in Hni1; nwhd; nelim daemon
- | nwhd in ⊢ (???%?);
+ [ nwhd in Hni1; nwhd; nwhd in ⊢ (?(? %)%);
+ nchange with (? < plus (s n) (big_plus n ?));
+ nelim (le_to_lt_or_eq … (le_S_S_to_le … Hni1))
+ [##2: #E; nrewrite < E; nrewrite < (minus_canc nindex);
+ nwhd in ⊢ (?%?); nrewrite < E; napply lt_to_lt_plus; nassumption
+ | #L; nrewrite > (split_big_plus n (S nindex) (λm.λ_.s m) L);
+ nrewrite > (split_big_plus (n - nindex) (n - S nindex) (λi.λ_.s (S (i+nindex))) ?)
+ [ ngeneralize in match (big_plus_ext (n - S nindex)
+ (λi,p.s (S (i+nindex))) (λi,p.s (i + S nindex)) ?) in ⊢ ?
+ [ #E;
+ napply (eq_rect_CProp0_r ??
+ (λx:nat.λ_. x + big_plus (n - nindex - (n - S nindex))
+ (λi,p.s (S (i + (n - S nindex)+nindex))) + nindex2 <
+ s n + (big_plus (S nindex) (λi,p.s i) +
+ big_plus (n - S nindex) (λi,p. s (i + S nindex)))) ? ? E);
+ nrewrite > (ad_hoc1 (s n) (big_plus (S nindex) (λi,p.s i))
+ (big_plus (n - S nindex) (λi,p. s (i + S nindex))));
+ napply (eq_rect_CProp0_r
+ ?? (λx.λ_.x < ?) ?? (assoc
+ (big_plus (n - S nindex) (λi,p.s (i + S nindex)))
+ (big_plus (n - nindex - (n - S nindex))
+ (λi,p.s (S (i + (n - S nindex)+nindex))))
+ nindex2));
+ napply lt_canc;
+ nrewrite > (ad_hoc2 … L); nwhd in ⊢ (?(?%?)?);
+ nrewrite > (ad_hoc3 … L);
+ napply (eq_rect_CProp0_r ?? (λx.λ_.x < ?) ?? (assoc …));
+ napply lt_canc; nnormalize in ⊢ (?%?); nwhd in ⊢ (??%);
+ napply lt_to_lt_plus; nassumption
+ ##|##2: #i; #_; nrewrite > (S_plus i nindex); napply refl]
+ ##| napply ad_hoc4]##]
+ ##| nwhd in ⊢ (???%?);
nchange in Hni1 with (nindex < S n);
ngeneralize in match (le_S_S_to_le … Hni1) in ⊢ ?;
nwhd in ⊢ (? → ???(???????%?)?);
napply (nat_rect_CProp0
(λx. nindex ≤ x →
- partition_splits_card_map A P (S n) s f fi
+ eq_rel (carr A) (eq A)
+ (partition_splits_card_map A P (S n) s f fi
(plus
- match minus x nindex with
- [ O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.s (S (plus i nindex)))]
- nindex2) x = y) ?? n)
+ (big_plus (minus x nindex) (λi.λ_:i < minus x nindex.s (S (plus i nindex))))
+ nindex2) x) y) ?? n)
[ #K; nrewrite < (minus_O_n nindex); nwhd in ⊢ (???(???????%?)?);
nwhd in ⊢ (???%?); nchange in Hni21 with (nindex2 < s nindex);
ngeneralize in match (le_O_to_eq … K) in ⊢ ?; #K';
| #n'; #Hrec; #HH; nelim (le_to_lt_or_eq … HH)
[##2: #K; nrewrite < K; nrewrite < (minus_canc nindex);
nwhd in ⊢ (???(???????%?)?);
- (*???????*)
- ##| #K; nwhd in ⊢ (???%?);
- nrewrite > (minus_S n' nindex ?) [##2: napply le_S_S_to_le; nassumption]
- ngeneralize in match (? :
- match S (minus n' nindex) with [O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.s (S (plus i nindex)))]
- = big_plus (minus n' nindex) (λi.λ_.s (S (plus i nindex)))) in ⊢ ? [##2: napply refl]
- #He; napply (eq_rect_CProp0_r ??
- (λx.λ_.
- match ltb (plus x nindex2) (s (S n')) with
- [ true ⇒ iso_f ???? (fi (S n')) (plus x nindex2)
- | false ⇒ ?(*partition_splits_card_map A P (S n) s f fi
- (minus (plus x nindex2) (s (S n'))) n'*)
- ] = y)
- ?? He);
- ngeneralize in match (? :
- ltb (plus (big_plus (minus n' nindex) (λi.λ_.s (S (plus i nindex)))) nindex2)
- (s (S n')) = false) in ⊢ ?
- [ #Hc; nrewrite > Hc; nwhd in ⊢ (???%?);
- nelim (le_to_lt_or_eq … (le_S_S_to_le … K))
- [
- ##| #E; ngeneralize in match Hc in ⊢ ?;
- nrewrite < E; nrewrite < (minus_canc nindex);
- nwhd in ⊢ (??(?%?)? → ?);
- nrewrite > E in Hni21; #E'; nchange in E' with (nindex2 < s n');
- ngeneralize in match Hni21 in ⊢ ?;
-
-
- ngeneralize in match (? :
- minus (plus (big_plus (minus n' nindex) (λi.λ_.s (S (plus i nindex)))) nindex2)
- (s (S n'))
- =
- plus
- match minus n' nindex with
- [ O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.s (S (plus i nindex)))] nindex2)
- in ⊢ ?
- [ #F; nrewrite > F; napply Hrec; napply le_S_S_to_le; nassumption
- | nelim (le_to_lt_or_eq … (le_S_S_to_le … K))
- [
- ##| #E; nrewrite < E; nrewrite < (minus_canc nindex); nnormalize;
-
- nwhd in ⊢ (???%);
- ]
-
-
- nrewrite > He;
-
-
- nnormalize in ⊢ (???%?);
-
-
-
- nelim (le_to_lt_or_eq … K)
- [##2: #K'; nrewrite > K'; nrewrite < (minus_canc n); nnormalize;
- napply (eq_rect_CProp0 nat nindex (λx:nat.λ_.partition_splits_card_map A P (S n) s f fi nindex2 x = y) ? n K');
- nchange in Hni21 with (nindex2 < s nindex); ngeneralize in match Hni21 in ⊢ ?;
- ngeneralize in match Hni22 in ⊢ ?;
- nelim nindex
- [ #X1; #X2; nwhd in ⊢ (??? % ?);
- napply (lt_to_ltb_t ???? X2); #D; nwhd in ⊢ (??? % ?); nassumption
- | #n0; #_; #X1; #X2; nwhd in ⊢ (??? % ?);
- napply (lt_to_ltb_t ???? X2); #D; nwhd in ⊢ (??? % ?); nassumption]
- ##| #K'; ngeneralize in match (lt_to_minus … K') in ⊢ ?; #K2;
- napply (eq_rect_CProp0 ?? (λx.λ_.?) ? ? K2); (* uffa, ancora??? *)
- nwhd in ⊢ (??? (???????(?%?)?) ?);
- ngeneralize in match K' in ⊢ ?;
- napply (nat_rect_CProp0
- (λx. nindex < x →
- partition_splits_card_map A P (S n) s f fi
- (plus (big_op plus_magma_type (minus (minus x nindex) (S O))
- (λi.λ_.s (S (plus i nindex))) O) nindex2) x = y) ?? n)
- [ #A; nelim (not_lt_O … A)
- | #n'; #Hrec; #X; nwhd in ⊢ (???%?);
- ngeneralize in match
- (? : ¬ ((plus (big_op plus_magma_type (minus (minus (S n') nindex) (S O))
- (λi.λ_.s (S (plus i nindex))) O) nindex2) < s (S n'))) in ⊢ ?
- [ #B1; napply (lt_to_ltb_f ???? B1); #B1'; nwhd in ⊢ (???%?);
- nrewrite > (minus_S n' nindex …) [##2: napply le_S_S_to_le; nassumption]
- ngeneralize in match (le_S_S_to_le … X) in ⊢ ?; #X';
- nelim (le_to_lt_or_eq … X')
- [##2: #X'';
- nchange in Hni21 with (nindex2 < s nindex); ngeneralize in match Hni21 in ⊢ ?;
- nrewrite > X''; nrewrite < (minus_canc n');
- nrewrite < (minus_canc (S O)); nnormalize in ⊢ (? → %);
- nelim n'
- [ #Y; nwhd in ⊢ (??? % ?);
- ngeneralize in match (minus_lt_to_lt ? (s (S O)) ? Y) in ⊢ ?; #Y';
- napply (lt_to_ltb_t … Y'); #H; nwhd in ⊢ (???%?);
-
- nrewrite > (minus_S (minus n' nindex) (S O) …) [##2:
-
- XXX;
-
- nelim n in f K' ⊢ ?
- [ #A; nelim daemon;
-
- (* BEL POSTO DOVE FARE UN LEMMA *)
- (* invariante: Hni1; altre premesse: Hni1, Hni22 *)
- nelim n in ⊢ (% → ??? (????????%) ?)
- [ #A (* decompose *)
- | #index'; #Hrec; #K; nwhd in ⊢ (???%?);
- nelim (ltb xxx (s (S index')));
- #K1; nwhd in ⊢ (???%?)
- [
-
- nindex < S index' + 1
- +^{nindex} (s i) w < s (S index')
- S index' == nindex
-
- |
- ]
- ]
- ]
- | #x; #x'; nnormalize in ⊢ (? → ? → %);
+ nrewrite > K;
+ nwhd in ⊢ (???%?); nrewrite < K; nrewrite > (ltb_t … Hni21);
+ nwhd in ⊢ (???%?); nassumption
+ ##| #K; ngeneralize in match (le_S_S_to_le … K) in ⊢ ?; #K';
+ nwhd in ⊢ (???%?);
+ ngeneralize in match (?:
+ ¬ (big_plus (S n' - nindex) (λi,p.s (S (i+nindex))) + nindex2 < s (S n'))) in ⊢ ?
+ [ #N; nrewrite > (ltb_f … N); nwhd in ⊢ (???%?);
+ ngeneralize in match (Hrec K') in ⊢ ?; #Hrec';
+ napply (eq_rect_CProp0_r ??
+ (λx,p. eq_rel (carr A) (eq A) (partition_splits_card_map A P (S n) s f fi
+ (big_plus x ? + ? - ?) n') y) ?? (minus_S n' nindex K'));
+ nrewrite > (split_big_plus (S (n' - nindex)) (n' - nindex)
+ (λi,p.s (S (i+nindex))) (le_S ?? (le_n ?)));
+ nrewrite > (ad_hoc5 (n' - nindex));
+ nnormalize in ⊢ (???(???????(?(?(??%)?)?)?)?);
+ nrewrite > (ad_hoc6 … K');
+ nrewrite > (ad_hoc7 (big_plus (n' - nindex) (λi,p.s (S (i+nindex))))
+ (s (S n')) nindex2);
+ nassumption
+ | nrewrite > (minus_S … K');
+ nrewrite > (split_big_plus (S (n' - nindex)) (n' - nindex)
+ (λi,p.s (S (i+nindex))) (le_S ?? (le_n ?)));
+ nrewrite > (ad_hoc5 (n' - nindex));
+ nnormalize in ⊢ (?(?(?(??%)?)?));
+ nrewrite > (ad_hoc6 … K');
+ napply ad_hoc8]##]##]##]
+##| #x; #x'; nnormalize in ⊢ (? → ? → %);
nelim daemon
]
nqed.
#A; #R; napply mk_partition
[ napply (quotient ? R)
| napply Full_set
- | #a; napply mk_qpowerclass
- [ napply {x | R x a}
- | #x; #x'; #H; nnormalize; napply mk_iff; #K; nelim daemon]
+ | napply mk_unary_morphism1
+ [ #a; napply mk_qpowerclass
+ [ napply {x | R x a}
+ | #x; #x'; #H; nnormalize; napply mk_iff; #K; nelim daemon]
+ ##| #a; #a'; #H; napply conj; #x; nnormalize; #K [ nelim daemon | nelim daemon]##]
##| #x; #_; nnormalize; napply (ex_intro … x); napply conj; napply refl
| #x; #x'; #_; #_; nnormalize; *; #x''; *; #H1; #H2; napply (trans ?????? H2);
napply sym; nassumption
projects / helm.git / blobdiff