(**************************************************************************)
include "sets/sets.ma".
-include "nat/plus.ma".
+include "nat/plus.ma".
include "nat/compare.ma".
include "nat/minus.ma".
+include "datatypes/pairs.ma".
-alias symbol "eq" = "setoid eq".
-alias symbol "eq" = "setoid1 eq".
-alias symbol "eq" = "setoid eq".
+alias symbol "eq" (instance 7) = "setoid1 eq".
nrecord partition (A: setoid) : Type[1] ≝
{ support: setoid;
- indexes: qpowerclass support;
- class: support → qpowerclass A;
+ indexes: ext_powerclass support;
+ class: unary_morphism1 (setoid1_of_setoid support) (ext_powerclass_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.
-
+ disjoint: ∀i,j. i ∈ indexes → j ∈ indexes → class i ≬ class j → i = j;
+ covers: big_union support ? indexes (λx.class x) = full_set A
+ }.
+
naxiom daemon: False.
-nlet rec partition_splits_card_map
- 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 ≝
+nlet rec iso_nat_nat_union (s: nat → nat) m index on index : pair nat nat ≝
match ltb m (s index) with
- [ or_introl _ ⇒ iso_f ???? (fi index) m
- | or_intror _ ⇒
+ [ true ⇒ mk_pair … 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']].
+ [ O ⇒ (* dummy value: it could be an elim False: *) mk_pair … O O
+ | S index' ⇒ iso_nat_nat_union s (minus m (s index)) index']].
+
+alias symbol "eq" = "leibnitz's equality".
+naxiom plus_n_O: ∀n. n + O = n.
+naxiom plus_n_S: ∀n,m. n + S m = S (n + m).
+naxiom ltb_t: ∀n,m. n < m → ltb n m = true.
+naxiom ltb_f: ∀n,m. ¬ (n < m) → ltb n m = false.
+naxiom ltb_cases: ∀n,m. (n < m ∧ ltb n m = true) ∨ (¬ (n < m) ∧ ltb n m = false).
+naxiom minus_canc: ∀n. minus n n = O.
+naxiom ad_hoc9: ∀a,b,c. a < b + c → a - b < c.
+naxiom ad_hoc10: ∀a,b,c. a - b = c → a = b + c.
+naxiom ad_hoc11: ∀a,b. a - b ≤ S a - b.
+naxiom ad_hoc12: ∀a,b. b ≤ a → S a - b - (a - b) = S O.
+naxiom ad_hoc13: ∀a,b. b ≤ a → (O + (a - b)) + b = a.
+naxiom ad_hoc14: ∀a,b,c,d,e. c ≤ a → a - c = b + d + e → a = b + (c + d) + e.
+naxiom ad_hoc15: ∀a,a',b,c. a=a' → b < c → a + b < c + a'.
+naxiom ad_hoc16: ∀a,b,c. a < c → a < b + c.
+naxiom not_lt_to_le: ∀a,b. ¬ (a < b) → b ≤ a.
+naxiom le_to_le_S_S: ∀a,b. a ≤ b → S a ≤ S b.
+naxiom minus_S: ∀n. S n - n = S O.
+naxiom ad_hoc17: ∀a,b,c,d,d'. a+c+d=b+c+d' → a+d=b+d'.
+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.
+naxiom big_plus_preserves_ext:
+ ∀n,f,f'. (∀i,p. f i p = f' i p) → big_plus n f = big_plus n f'.
+
+ntheorem iso_nat_nat_union_char:
+ ∀n:nat. ∀s: nat → nat. ∀m:nat. m < big_plus (S n) (λi.λ_.s i) →
+ let p ≝ iso_nat_nat_union s m n in
+ m = big_plus (n - fst … p) (λi.λ_.s (S (i + fst … p))) + snd … p ∧
+ fst … p ≤ n ∧ snd … p < s (fst … p).
+ #n; #s; nelim n
+ [ #m; nwhd in ⊢ (??% → let p ≝ % in ?); nwhd in ⊢ (??(??%) → ?);
+ nrewrite > (plus_n_O (s O)); #H; nrewrite > (ltb_t … H); nnormalize; @; /2/
+##| #n'; #Hrec; #m; nwhd in ⊢ (??% → let p ≝ % in ?); #H;
+ ncases (ltb_cases m (s (S n'))); *; #H1; #H2; nrewrite > H2;
+ nwhd in ⊢ (let p ≝ % in ?); nwhd
+ [ napply conj [napply conj; //;
+ nwhd in ⊢ (???(?(?%(λ_.λ_:(??%).?))%)); nrewrite > (minus_canc n'); //
+ ##| nnormalize; // ]
+##| nchange in H with (m < s (S n') + big_plus (S n') (λi.λ_.s i));
+ nlapply (Hrec (m - s (S n')) ?); /2/; *; *; #Hrec1; #Hrec2; #Hrec3; @; //; @; /2/;
+ nrewrite > (split_big_plus …); ##[##2:napply ad_hoc11;##|##3:##skip]
+ nrewrite > (ad_hoc12 …); //;
+ nwhd in ⊢ (???(?(??%)?));
+ nrewrite > (ad_hoc13 …); //;
+ napply ad_hoc14; /2/;
+ nwhd in ⊢ (???(?(??%)?));
+ nrewrite > (plus_n_O …); // ##]##]
+nqed.
+
+ntheorem iso_nat_nat_union_pre:
+ ∀n:nat. ∀s: nat → nat.
+ ∀i1,i2. i1 ≤ n → i2 < s i1 →
+ big_plus (n - i1) (λi.λ_.s (S (i + i1))) + i2 < big_plus (S n) (λi.λ_.s i).
+/2/. nqed.
+
+ntheorem iso_nat_nat_union_uniq:
+ ∀n:nat. ∀s: nat → nat.
+ ∀i1,i1',i2,i2'. i1 ≤ n → i1' ≤ n → i2 < s i1 → i2' < s i1' →
+ big_plus (n - i1) (λi.λ_.s (S (i + i1))) + i2 = big_plus (n - i1') (λi.λ_.s (S (i + i1'))) + i2' →
+ i1 = i1' ∧ i2 = i2'.
+ #n; #s; #i1; #i1'; #i2; #i2'; #H1; #H1'; #H2; #H2'; #E;
+ nelim daemon.
+nqed.
nlemma partition_splits_card:
∀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; #n; #s; #f; #fi; napply mk_isomorphism
- [ napply mk_unary_morphism
- [ napply (λm.partition_splits_card_map A P n s f fi m n)
+#A; #P; #Sn; ncases Sn
+ [ #s; #f; #fi;
+ nlapply (covers ? P); *; #_; #H;
+ (*
+ nlapply
+ (big_union_preserves_iso ??? (indexes A P) (Nat_ O) (λx.class ? P x) f);
+ *; #K; #_; nwhd in K: (? → ? → %);*)
+ nelim daemon (* impossibile *)
+ | #n; #s; #f; #fi; @
+ [ @
+ [ napply (λm.let p ≝ (iso_nat_nat_union s m n) in iso_f ???? (fi (fst … p)) (snd … p))
| #a; #a'; #H; nrewrite < H; napply refl ]
+##| #x; #Hx; nwhd; napply I
##| #y; #_;
- ngeneralize in match (covers ? P) in ⊢ ?; *; #_; #Hc;
- ngeneralize in match (Hc y I) in ⊢ ?; *; #index; *; #Hi1; #Hi2;
- nelim daemon
- | #x; #x'; nnormalize in ⊢ (? → ? → %);
- nelim daemon
- ]
+ nlapply (covers ? P); *; #_; #Hc;
+ nlapply (Hc y I); *; #index; *; #Hi1; #Hi2;
+ nlapply (f_sur ???? f ? Hi1); *; #nindex; *; #Hni1; #Hni2;
+ nlapply (f_sur ???? (fi nindex) y ?)
+ [ alias symbol "refl" (instance 3) = "refl".
+alias symbol "prop2" (instance 2) = "prop21".
+alias symbol "prop1" (instance 4) = "prop11".
+napply (. #‡(†?));##[##2: napply Hni2 |##1: ##skip | nassumption]##]
+ *; #nindex2; *; #Hni21; #Hni22;
+ nletin xxx ≝ (plus (big_plus (minus n nindex) (λi.λ_.s (S (plus i nindex)))) nindex2);
+ @ xxx; @
+ [ napply iso_nat_nat_union_pre [ napply le_S_S_to_le; nassumption | nassumption ]
+ ##| nwhd in ⊢ (???%%); napply (.= ?) [##3: nassumption|##skip]
+ nlapply (iso_nat_nat_union_char n s xxx ?)
+ [napply iso_nat_nat_union_pre [ napply le_S_S_to_le; nassumption | nassumption]##]
+ *; *; #K1; #K2; #K3;
+ nlapply
+ (iso_nat_nat_union_uniq n s nindex (fst … (iso_nat_nat_union s xxx n))
+ nindex2 (snd … (iso_nat_nat_union s xxx n)) ?????); /2/
+ [##2: *; #E1; #E2; nrewrite > E1; nrewrite > E2; //
+ | nassumption ]##]
+##| #x; #x'; nnormalize in ⊢ (? → ? → %); #Hx; #Hx'; #E;
+ ncut(∀i1,i2,i1',i2'. i1 ∈ Nat_ (S n) → i1' ∈ Nat_ (S n) → i2 ∈ Nat_ (s i1) → i2' ∈ Nat_ (s i1') → eq_rel (carr A) (eq0 A) (fi i1 i2) (fi i1' i2') → i1=i1' ∧ i2=i2');
+ ##[ #i1; #i2; #i1'; #i2'; #Hi1; #Hi1'; #Hi2; #Hi2'; #E;
+ nlapply(disjoint … P (f i1) (f i1') ???)
+ [##2,3: napply f_closed; //
+ |##1: @ (fi i1 i2); @;
+ ##[ napply f_closed; // ##| alias symbol "refl" = "refl1".
+napply (. E‡#);
+ nwhd; napply f_closed; //]##]
+ #E'; ncut(i1 = i1'); ##[ napply (f_inj … E'); // ##]
+ #E''; nrewrite < E''; @; //;
+ nrewrite < E'' in E; #E'''; napply (f_inj … E'''); //;
+ nrewrite > E''; // ]##]
+ ##] #K;
+ nelim (iso_nat_nat_union_char n s x Hx); *; #i1x; #i2x; #i3x;
+ nelim (iso_nat_nat_union_char n s x' Hx'); *; #i1x'; #i2x'; #i3x';
+ nlapply (K … E)
+ [##1,2: nassumption;
+ ##|##3,4:napply le_to_le_S_S; nassumption; ##]
+ *; #K1; #K2;
+ napply (eq_rect_CProp0_r ?? (λX.λ_.? = X) ?? i1x');
+ napply (eq_rect_CProp0_r ?? (λX.λ_.X = ?) ?? i1x);
+ nrewrite > K1; nrewrite > K2; napply refl.
nqed.
(************** equivalence relations vs partitions **********************)
#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]##]
+ | napply mk_unary_morphism1
+ [ #a; napply mk_ext_powerclass
+ [ napply {x | rel ? 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; /3/
+ | #x; #x'; #_; #_; nnormalize; *; #x''; *; /3/
+ | nnormalize; napply conj; /4/ ]
nqed.
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