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".
alias symbol "eq" = "setoid eq".
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 ≝
| #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 (. #‡(†?));##[##2: napply Hni2 |##1: ##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 (.= ?) [##3: 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]##]
+ 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
+ nindex2 (snd … (iso_nat_nat_union s xxx n)) ?????)
+ [##2: *; #E1; #E2; nrewrite > E1; nrewrite > E2; napply refl
| napply le_S_S_to_le; nassumption
|##*: nassumption]##]
##| #x; #x'; nnormalize in ⊢ (? → ? → %); #Hx; #Hx'; #E;