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" = "setoid eq".
-alias symbol "eq" = "setoid1 eq".
-alias symbol "eq" = "setoid eq".
-alias symbol "eq" = "setoid1 eq".
+alias symbol "eq" (instance 7) = "setoid1 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 ? indexes (λx.class x) = full_set A
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 ]
+ 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'); 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 ]##]##]##]
+ [ 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).
- #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.
+/2/. nqed.
ntheorem iso_nat_nat_union_uniq:
∀n:nat. ∀s: nat → nat.
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".
+ [ 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);
*; *; #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]##]
+ 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 ∈ 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');
+ 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) (eq 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; nassumption
+ [##2,3: napply f_closed; //
|##1: @ (fi i1 i2); @;
- ##[ napply f_closed; nassumption ##| 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 ]##]##]
+ ##[ 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';
*; #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 ]
+ nrewrite > K1; nrewrite > K2; napply refl.
nqed.
(************** equivalence relations vs partitions **********************)
[ napply (quotient ? R)
| napply Full_set
| napply mk_unary_morphism1
- [ #a; napply mk_qpowerclass
+ [ #a; napply mk_ext_powerclass
[ 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.
+##| #x; #_; nnormalize; /3/
+ | #x; #x'; #_; #_; nnormalize; *; #x''; *; /3/
+ | nnormalize; napply conj; /4/ ]
+nqed.
\ No newline at end of file
projects / helm.git / blobdiff