(**************************************************************************)
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" = "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);
+ 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
- [ true ⇒ iso_f ???? (fi index) m
+ [ 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']].
-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 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 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 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; @
+ [ @ [ napply refl | napply le_n ] ##| nassumption ]
+##| #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'); napply refl
+ | nnormalize; napply le_n]
+ ##| nnormalize; nassumption ]
+ ##| nchange in H with (m < s (S n') + big_plus (S n') (λi.λ_.s i));
+ nlapply (Hrec (m - s (S n')) ?)
+ [ napply ad_hoc9; nassumption] *; *; #Hrec1; #Hrec2; #Hrec3; @
+ [##2: nassumption
+ |@
+ [nrewrite > (split_big_plus …); ##[##2:napply ad_hoc11;##|##3:##skip]
+ nrewrite > (ad_hoc12 …); ##[##2: nassumption]
+ nwhd in ⊢ (????(?(??%)?));
+ nrewrite > (ad_hoc13 …);##[##2: nassumption]
+ napply ad_hoc14 [ napply not_lt_to_le; nassumption ]
+ nwhd in ⊢ (???(?(??%)?));
+ nrewrite > (plus_n_O …);
+ nassumption;
+ ##| napply le_S; nassumption ]##]##]##]
+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).
+ #n; #s; #i1; #i2; #H1; #H2;
+ nrewrite > (split_big_plus (S n) (S i1) (λi.λ_.s i) ?)
+ [##2: napply le_to_le_S_S; nassumption]
+ napply ad_hoc15
+ [ nwhd in ⊢ (???(?%?));
+ napply big_plus_preserves_ext; #i; #_;
+ nrewrite > (plus_n_S i i1); napply refl
+ | nrewrite > (split_big_plus (S i1) i1 (λi.λ_.s i) ?) [##2: napply le_S; napply le_n]
+ napply ad_hoc16; nrewrite > (minus_S i1); nnormalize; nrewrite > (plus_n_O (s i1) …);
+ nassumption ]
+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; #Sn; ncases Sn
+#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: (? → ? → %);
+ 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 mk_isomorphism
- [ napply mk_unary_morphism
- [ napply (λm.partition_splits_card_map A P (S n) s f fi m n)
+ | #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;
- 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]##]
+ 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" = "refl".
+alias symbol "prop1" = "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);
- 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
- ]
+ @ 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)) ?????)
+ [##6: *; #E1; #E2; nrewrite > E1; nrewrite > E2; napply refl
+ |##5: napply le_S_S_to_le; nassumption
+ |##*: nassumption]##]
+##| #x; #x'; nnormalize in ⊢ (? → ? → %); #Hx; #Hx'; #E;
+ ncut(∀i1,i2,i1',i2'. i1 ∈ Nat_ (S n) → i1' ∈ Nat_ (S n) → i2 ∈ pc ? (Nat_ (s i1)) → i2' ∈ pc ? (Nat_ (s i1')) → eq_rel (carr A) (eq A) (iso_f ???? (fi i1) i2) (iso_f ???? (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; nassumption
+ |##1: @ (fi i1 i2); @;
+ ##[ napply f_closed; nassumption ##| alias symbol "refl" = "refl1".
+napply (. E‡#);
+ nwhd; napply f_closed; nassumption]##]
+ #E'; ncut(i1 = i1'); ##[ napply (f_inj … E'); nassumption; ##]
+ #E''; nrewrite < E''; @;
+ ##[ @;
+ ##| nrewrite < E'' in E; #E'''; napply (f_inj … E''')
+ [ nassumption | nrewrite > E''; nassumption ]##]##]
+ ##] #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 **********************)
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
+nqed.
projects / helm.git / blobdiff