include "nat/minus.ma".
include "datatypes/pairs.ma".
-alias symbol "eq" (instance 2) = "leibnitz's equality".
-alias symbol "eq" (instance 1) = "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] ≝
indexes: qpowerclass support;
class: unary_morphism1 (setoid1_of_setoid support) (qpowerclass_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 ≝
(isomorphism ?? (Nat_ (big_plus n (λi.λ_.s i))) (Full_set A)).
#A; #P; #Sn; ncases Sn
[ #s; #f; #fi;
- ngeneralize in match (covers ? P) in ⊢ ?; *; #_; #H;
+ nlapply (covers ? P); *; #_; #H;
(*
nlapply
(big_union_preserves_iso ??? (indexes A P) (Nat_ O) (λx.class ? P x) f);
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 (. #‡(†?));##[##2: napply Hni2 |##1: ##skip | nassumption]##]
+ [##2: 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