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;
| S index' ⇒ iso_nat_nat_union s (minus m (s index)) index']].
alias symbol "eq" = "leibnitz's equality".
-naxiom plus_n_O: ∀n. plus n O = n.
+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 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_S: ∀a,b. S a - S b = a - 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) →
| napply le_S; nassumption ]##]##]##]
nqed.
-
-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 ≝
- 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 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 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_elim_CProp0: ∀n,m. ∀P: bool → CProp[0]. (n < m → P true) → (¬ (n < m) → P false) → P (ltb n m).
-
-nlemma partition_splits_card_output:
- ∀A. ∀P:partition A. ∀n,s.
- ∀f:isomorphism ?? (Nat_ (S n)) (indexes ? P).
- ∀fi:∀i. isomorphism ?? (Nat_ (s i)) (class ? P (iso_f ???? f i)).
- ∀x. x ∈ Nat_ (big_plus (S n) (λi.λ_.s i)) →
- ∃i1.∃i2. partition_splits_card_map A P (S n) s f fi x n = iso_f ???? (fi i1) i2.
- #A; #P; #n; #s; #f; #fi;
- nelim n in ⊢ (? → % → ??(λ_.??(λ_.???(????????%)?)))
- [ #x; nnormalize in ⊢ (% → ?); nrewrite > (plus_n_O (s O)); nchange in ⊢ (% → ?) with (x < s O);
- #H; napply (ex_intro … O); napply (ex_intro … x); nwhd in ⊢ (???%?);
- nrewrite > (ltb_t … H); nwhd in ⊢ (???%?); napply refl
- | #n'; #Hrec; #x; #Hx; nwhd in ⊢ (??(λ_.??(λ_.???%?))); nwhd in Hx; nwhd in Hx: (??%);
- nelim (ltb_cases x (s (S n'))); *; #K1; #K2; nrewrite > K2; nwhd in ⊢ (??(λ_.??(λ_.???%?)))
- [ napply (ex_intro … (S n')); napply (ex_intro … x); napply refl
- | napply (Hrec (x - s (S n')) ?); nwhd; nrewrite < (minus_S x (s (S n')) ?)
- [ napply ad_hoc9; nassumption | napply not_lt_to_le; nassumption ]##]
+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
+ [ nrewrite > (minus_S_S n i1 …); 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; #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: (? → ? → %);
+ *; #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)
+ [ 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;
*; #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 ⊢ (? → ? → %); #Hx; #Hx';
- nelim (partition_splits_card_output A P n s f fi x Hx); #i1x; *; #i2x; #Ex;
- nelim (partition_splits_card_output A P n s f fi x' Hx'); #i1x'; *; #i2x'; #Ex';
- ngeneralize in match (? :
- iso_f ???? fi i1x(* ≬ iso_f ???? (fi i1x'))*)) in ⊢ ?;
- #E; napply (f_inj ???? (fi i1x));
-
- nelim n in ⊢ (% → % → (???(????????%)(????????%)) → ?)
- [ nnormalize in ⊢ (% → % → ?); nrewrite > (plus_n_O (s O));
- nchange in ⊢ (% → ?) with (x < s O);
- nchange in ⊢ (? → % → ?) with (x' < s O);
- #H1; #H2; nwhd in ⊢ (???%% → ?);
- nrewrite > (ltb_t … H1); nrewrite > (ltb_t … H2); nwhd in ⊢ (???%% → ?);
- napply f_inj; nassumption
- | #n'; #Hrec; #Hx; #Hx'; nwhd in ⊢ (???%% → ?);
- ]
+ [ napply iso_nat_nat_union_pre [ napply le_S_S_to_le; nassumption | nassumption ]
+ ##| nwhd in ⊢ (???%%); napply (.= ?) [ nassumption|##skip]
+ ngeneralize in match (iso_nat_nat_union_char n s xxx ?) in ⊢ ?
+ [##2: napply iso_nat_nat_union_pre [ napply le_S_S_to_le; nassumption | nassumption]##]
+ *; *; #K1; #K2; #K3;
+ ngeneralize in match
+ (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)) ?????) in ⊢ ?
+ [ *; #E1; #E2; nrewrite > E1; nrewrite > E2; napply refl
+ | napply le_S_S_to_le; nassumption
+ |##*: nassumption]##]
+##| #x; #x'; nnormalize in ⊢ (? → ? → %); #Hx; #Hx'; #E;
+ ngeneralize in match (? : ∀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') in ⊢ ?
+ [##2: #i1; #i2; #i1'; #i2'; #Hi1; #Hi1'; #Hi2; #Hi2'; #E;
+ ngeneralize in match (disjoint ? P (iso_f ???? f i1) (iso_f ???? f i1') ???) in ⊢ ?
+ [##2,3: napply f_closed; nassumption
+ |##4: napply ex_intro [ napply (iso_f ???? (fi i1) i2) ] napply conj
+ [ napply f_closed; nassumption ##| napply (. ?‡#) [##2: nassumption | ##3: ##skip]
+ nwhd; napply f_closed; nassumption]##]
+ #E'; ngeneralize in match (? : i1=i1') in ⊢ ?
+ [##2: napply (f_inj … E'); nassumption
+ | #E''; nrewrite < E''; napply conj
+ [ napply refl | 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';
+ ngeneralize in match (K … E) in ⊢ ?
+ [##2,3: napply le_to_le_S_S; nassumption
+ |##4,5: 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 **********************)
projects / helm.git / commitdiff