include "nat/compare.ma".
include "nat/minus.ma".
-(* sbaglia a fare le proiezioni *)
-nrecord finite_partition (A: Type[0]) : Type[1] ≝
- { card: nat;
- class: ∀n. lt n card → Ω \sup A;
- inhabited: ∀i,p. class i p ≬ class i p(*;
- disjoint: ∀i,j,p,p'. class i p ≬ class j p' → i=j;
- covers: big_union ?? class = full_set A*)
- }.
-
-nrecord has_card (A: Type[0]) (S: Ω \sup A) (n: nat) : CProp[0] ≝
- { f: ∀m:nat. lt m n → A;
- in_S: ∀m.∀p:lt m n. f ? p ∈ S (*;
- f_inj: injective ?? f;
- f_sur: surjective ?? f*)
- }.
-
-(*
-nlemma subset_of_finite:
- ∀A. ∃n. has_card ? (full_subset A) n → ∀S. ∃m. has_card ? S m.
-nqed.
-*)
+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".
+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;
+ class: unary_morphism1 (setoid1_of_setoid support) (qpowerclass_setoid A);
+ inhabited: ∀i. i ∈ indexes → class i ≬ class i;
+ disjoint: ∀i,j. i ∈ indexes → j ∈ indexes → class i ≬ class j → i=j;
+ covers: big_union support ? ? (λx.class x) = full_set A
+ }. napply indexes; nqed.
naxiom daemon: False.
nlet rec partition_splits_card_map
- A (P: finite_partition A) (s: ∀i. lt i (card ? P) → nat)
- (H:∀i.∀p: lt i (card ? P). has_card ? (class ? P i p) (s i p))
- m index on index:
- le (S index) (card ? P) → lt m (big_plus (S index) (λi,p. s i ?)) → lt index (card ? P) → A ≝
- match index return λx. le (S x) (card ? P) → lt m (big_plus (S x) ?) → lt x (card ? P) → ? with
- [ O ⇒ λL,H1,p.f ??? (H O p) m ?
- | S index' ⇒ λL,H1,p.
- match ltb m (s (S index') p) with
- [ or_introl K ⇒ f ??? (H (S index') p) m K
- | or_intror _ ⇒ partition_splits_card_map A P s H (minus m (s (S index') p)) index' ??? ]].
-##[##3: napply lt_minus; nelim daemon (*nassumption*)
- |##4: napply lt_Sn_m; nassumption
- |##5: napply (lt_le_trans … p); nassumption
-##|##2: napply lt_to_le; nassumption
-##|##1: nnormalize in H1; nelim daemon ]
+ A (P:partition A) n s (f:isomorphism ?? (Nat_ n) (indexes ? P))
+ (fi: ∀i. isomorphism ?? (Nat_ (s i)) (class ? P (iso_f ???? f i))) m index
+ on index : A ≝
+ match ltb m (s index) with
+ [ true ⇒ iso_f ???? (fi index) m
+ | false ⇒
+ match index with
+ [ O ⇒ (* dummy value: it could be an elim False: *) iso_f ???? (fi O) O
+ | S index' ⇒
+ partition_splits_card_map A P n s f fi (minus m (s index)) index']].
+
+naxiom big_union_preserves_iso:
+ ∀A,A',B,T,T',f.
+ ∀g: isomorphism A' A T' T.
+ big_union A B T f = big_union A' B T' (λx.f (iso_f ???? g x)).
+
+naxiom le_to_lt_or_eq: ∀n,m. n ≤ m → n < m ∨ n = m.
+alias symbol "eq" = "leibnitz's equality".
+naxiom minus_canc: ∀n. O = minus n n.
+naxiom lt_to_ltb_t: ∀n,m. ∀P: bool → CProp[0]. P true → n < m → P (ltb n m).
+naxiom lt_to_ltb_f: ∀n,m. ∀P: bool → CProp[0]. P false → ¬ (n < m) → P (ltb n m).
+naxiom lt_to_minus: ∀n,m. n < m → S (minus (minus m n) (S O)) = minus m n.
+naxiom not_lt_O: ∀n. ¬ (n < O).
+naxiom minus_S: ∀n,m. m ≤ n → minus (S n) m = S (minus n m).
+naxiom minus_lt_to_lt: ∀n,m,p. n < p → minus n m < p.
+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_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: finite_partition A. ∀s: ∀i. lt i (card ? P) → nat.
- (∀i.∀p: lt i (card ? P). has_card ? (class ? P i p) (s i p)) →
- has_card A (full_set A) (big_plus (card ? P) s).
- #A; #P; #s; #H; ncases (card A P)
- [ nnormalize; napply mk_has_card
- [ #m; #H; nelim daemon
- | #m; #H; nelim daemon ]
-##| #c; napply mk_has_card
- [ #m; #H1; napply partition_splits_card_map A P s H m H1 (pred (card ? P))
- |
+ ∀A. ∀P:partition A. ∀n,s.
+ ∀f:isomorphism ?? (Nat_ n) (indexes ? P).
+ (∀i. isomorphism ?? (Nat_ (s i)) (class ? P (iso_f ???? f i))) →
+ (isomorphism ?? (Nat_ (big_plus n (λi.λ_.s i))) (Full_set A)).
+ #A; #P; #Sn; ncases Sn
+ [ #s; #f; #fi;
+ ngeneralize in match (covers ? P) in ⊢ ?; *; #_; #H;
+ ngeneralize in match
+ (big_union_preserves_iso ??? (indexes A P) (Nat_ O) (λx.class ? P x) f) in ⊢ ?;
+ *; #K; #_; nwhd in K: (? → ? → %);
+ nelim daemon (* impossibile *)
+ | #n; #s; #f; #fi; napply mk_isomorphism
+ [ napply mk_unary_morphism
+ [ napply (λm.partition_splits_card_map A P (S n) s f fi m n)
+ | #a; #a'; #H; nrewrite < H; napply refl ]
+##| #y; #_;
+ ngeneralize in match (covers ? P) in ⊢ ?; *; #_; #Hc;
+ ngeneralize in match (Hc y I) in ⊢ ?; *; #index; *; #Hi1; #Hi2;
+ ngeneralize in match (f_sur ???? f ? Hi1) in ⊢ ?; *; #nindex; *; #Hni1; #Hni2;
+ ngeneralize in match (f_sur ???? (fi nindex) y ?) in ⊢ ?
+ [##2: napply (. #‡(†?));##[##3: napply Hni2 |##2: ##skip | nassumption]##]
+ *; #nindex2; *; #Hni21; #Hni22;
+ 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; 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 →
+ eq_rel (carr A) (eq A)
+ (partition_splits_card_map A P (S n) s f fi
+ (plus
+ (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';
+ ngeneralize in match Hni21 in ⊢ ?;
+ ngeneralize in match Hni22 in ⊢ ?;
+ nrewrite > K' in ⊢ (% → % → ?); #K1; #K2;
+ nrewrite > (ltb_t … K2);
+ nwhd in ⊢ (???%?); nassumption
+ | #n'; #Hrec; #HH; nelim (le_to_lt_or_eq … HH)
+ [##2: #K; nrewrite < K; nrewrite < (minus_canc nindex);
+ nwhd 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.
-*)
\ No newline at end of file
+
+(************** equivalence relations vs partitions **********************)
+
+ndefinition partition_of_compatible_equivalence_relation:
+ ∀A:setoid. compatible_equivalence_relation A → partition A.
+ #A; #R; napply mk_partition
+ [ napply (quotient ? R)
+ | napply Full_set
+ | 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
+ | nnormalize; napply conj
+ [ #a; #_; napply I | #a; #_; napply (ex_intro … a); napply conj [ napply I | napply refl]##]
+nqed.
\ No newline at end of file
projects / helm.git / blobdiff