(**************************************************************************)
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" = "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);
+ 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 iso_nat_nat_union (s: nat → nat) m index on index : pair nat nat ≝
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:
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 conj [ napply conj [ napply refl | napply le_n ] ##| nassumption ]
+ 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
| nnormalize; napply le_n]
##| nnormalize; nassumption ]
##| nchange in H with (m < s (S n') + big_plus (S n') (λi.λ_.s i));
- ngeneralize in match (Hrec (m - s (S n')) ?) in ⊢ ?
- [##2: napply ad_hoc9; nassumption] *; *; #Hrec1; #Hrec2; #Hrec3; napply conj
+ nlapply (Hrec (m - s (S n')) ?)
+ [ napply ad_hoc9; nassumption] *; *; #Hrec1; #Hrec2; #Hrec3; @
[##2: nassumption
- |napply conj
+ |@
[nrewrite > (split_big_plus …); ##[##2:napply ad_hoc11;##|##3:##skip]
nrewrite > (ad_hoc12 …); ##[##2: nassumption]
nwhd in ⊢ (????(?(??%)?));
[##2: napply le_to_le_S_S; nassumption]
napply ad_hoc15
[ nwhd in ⊢ (???(?%?));
-
- napply (eq_rect_CProp0_r nat (n - i1) (λx.λy.?) ?? (minus_S_S …)); STOP
-
- nrewrite > (minus_S_S n i1); napply big_plus_preserves_ext; #i; #_;
+ 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) …);
∀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)).
- STOP #A; #P; #Sn; ncases Sn
+#A; #P; #Sn; ncases Sn
[ #s; #f; #fi;
- ngeneralize in match (covers ? P) in ⊢ ?; *; #_; #H;
+ nlapply (covers ? P); *; #_; #H;
(*
- ngeneralize in match
- (big_union_preserves_iso ??? (indexes A P) (Nat_ O) (λx.class ? P x) f) in ⊢ ?;
+ 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.let p ≝ iso_nat_nat_union s m n in iso_f ???? (fi (fst … p)) (snd … p))
+ | #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
+ @ xxx; @
[ 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]##]
+ ##| 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;
- ngeneralize in match
+ 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)) ?????) in ⊢ ?
- [ *; #E1; #E2; nrewrite > E1; nrewrite > E2; napply refl
- | napply le_S_S_to_le; nassumption
+ 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;
- 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 ⊢ ?
+ 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
- |##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''')
+ |##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';
- ngeneralize in match (K … E) in ⊢ ?
- [##2,3: napply le_to_le_S_S; nassumption
- |##4,5: nassumption]
+ 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);
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