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*)
- }.
+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".
+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.
-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.
-*)
+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 (H1: lt m (big_plus ? s)) index (p: lt index (card ? P)) on index : A ≝
- match index return λx. lt x (card ? P) → ? with
- [ O ⇒ λp'.f ??? (H O p') m ?
- | S index' ⇒ λp'.
- match ltb m (s index p) with
- [ or_introl K ⇒ f ??? (H index p) m K
- | or_intror _ ⇒ partition_splits_card_map A P s H (minus m (s index p)) ? index' ? ]] p.
-##[##2: napply lt_minus; nassumption
- |##3: napply lt_Sn_m; nassumption
- |
-nqed.
+ 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.
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; 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 match minus n nindex return λ_.nat with [ O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.s (S (plus i nindex)))] nindex2);
+ napply (ex_intro … xxx); napply conj
+ [ nwhd in Hni1; nwhd; nelim daemon
+ | 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
+ (plus
+ match minus x nindex with
+ [ O ⇒ O | S nn ⇒ big_plus nn (λi.λ_.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 ⊢ (???(???????%?)?);
+ (*???????*)
+ ##| #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 ⊢ (? → ? → %);
+ nelim daemon
]
+nqed.
+
+(************** 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
+ | #a; napply mk_qpowerclass
+ [ napply {x | R x a}
+ | #x; #x'; #H; nnormalize; napply mk_iff; #K; 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