1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "sets/sets.ma".
16 include "nat/plus.ma".
17 include "nat/compare.ma".
18 include "nat/minus.ma".
20 alias symbol "eq" = "setoid eq".
21 alias symbol "eq" = "setoid1 eq".
22 nrecord partition (A: setoid) : Type[1] ≝
24 indexes: qpowerclass support;
25 class: support → qpowerclass A;
26 inhabited: ∀i. i ∈ indexes → class i ≬ class i;
27 disjoint: ∀i,j. i ∈ indexes → j ∈ indexes → class i ≬ class j → i=j;
28 covers: big_union support ? ? (λx.class x) = full_set A
29 }. napply indexes; nqed.
33 nlet rec partition_splits_card_map
34 A (P:partition A) n s (f:isomorphism ?? (Nat_ n) (indexes ? P))
35 (fi: ∀i. isomorphism ?? (Nat_ (s i)) (class ? P (iso_f ???? f i))) m index
37 match ltb m (s index) with
38 [ or_introl _ ⇒ iso_f ???? (fi index) m
41 [ O ⇒ (* dummy value: it could be an elim False: *) iso_f ???? (fi O) O
43 partition_splits_card_map A P n s f fi (minus m (s index)) index']].
45 nlemma partition_splits_card:
46 ∀A. ∀P:partition A. ∀n,s.
47 ∀f:isomorphism ?? (Nat_ n) (indexes ? P).
48 (∀i. isomorphism ?? (Nat_ (s i)) (class ? P (iso_f ???? f i))) →
49 (isomorphism ?? (Nat_ (big_plus n (λi.λ_.s i))) (Full_set A)).
50 #A; #P; #n; #s; #f; #fi; napply mk_isomorphism
51 [ napply mk_unary_morphism
52 [ napply (λm.partition_splits_card_map A P n s f fi m n)
53 | #a; #a'; #H; nrewrite < H; napply refl ]
55 ngeneralize in match (covers ? P) in ⊢ ?; *; #_; #Hc;
56 ngeneralize in match (Hc y I) in ⊢ ?; *; #index; *; #Hi1; #Hi2;
58 | #x; #x'; nnormalize in ⊢ (? → ? → %);