record FinSet : Type[1] ≝
{ FinSetcarr:> DeqSet;
enum: list FinSetcarr;
- enum_unique: uniqueb FinSetcarr enum = true
+ enum_unique: uniqueb FinSetcarr enum = true;
+ enum_complete : ∀x:FinSetcarr. memb ? x enum = true
}.
+notation < "𝐅" non associative with precedence 90
+ for @{'bigF}.
+interpretation "FinSet" 'bigF = (mk_FinSet ???).
+
+(* bool *)
+lemma bool_enum_unique: uniqueb ? [true;false] = true.
+// qed.
+
+lemma bool_enum_complete: ∀x:bool. memb ? x [true;false] = true.
+* // qed.
+
+definition FinBool ≝
+ mk_FinSet DeqBool [true;false] bool_enum_unique bool_enum_complete.
+
+unification hint 0 ≔ ;
+ X ≟ FinBool
+(* ---------------------------------------- *) ⊢
+ bool ≡ FinSetcarr X.
+
+(* nat segment *)
+
+lemma eqbnat_true : ∀n,m. eqb n m = true ↔ n = m.
+#n #m % [@eqb_true_to_eq | @eq_to_eqb_true]
+qed.
+
+definition DeqNat ≝ mk_DeqSet nat eqb eqbnat_true.
+
+lemma lt_to_le : ∀n,m. n < m → n ≤ m.
+/2/ qed-.
+
+let rec enumnaux n m ≝
+ match n return (λn.n ≤ m → list (Σx.x < m)) with
+ [ O ⇒ λh.[ ] | S p ⇒ λh:p < m.(mk_Sig ?? p h)::enumnaux p m (lt_to_le p m h)].
+
+definition enumn ≝ λn.enumnaux n n (le_n n).
+
+definition Nat_to ≝ λn. DeqSig DeqNat (λi.i<n).
+
+(* lemma prova : ∀n. carr (Nat_to n) = (Σx.x<n). // *)
+
+lemma memb_enumn: ∀m,n,i:DeqNat. ∀h:n ≤ m. ∀k: i < m. n ≤ i →
+ (¬ (memb (Nat_to m) (mk_Sig ?? i k) (enumnaux n m h))) = true.
+#m #n elim n -n // #n #Hind #i #ltm #k #ltni @sym_eq @noteq_to_eqnot @sym_not_eq
+% #H cases (orb_true_l … H)
+ [#H1 @(absurd … (\P H1)) % #Hfalse
+ cut (∀A,P,a,a1,h,h1.mk_Sig A P a h = mk_Sig A P a1 h1 → a = a1)
+ [#A #P #a #a1 #h #h1 #H destruct (H) %] #Hcut
+ lapply (Hcut nat (λi.i<m) i n ? ? Hfalse) #Hfalse @(absurd … ltni)
+ @le_to_not_lt >Hfalse @le_n
+ |<(notb_notb (memb …)) >Hind normalize /2/
+ ]
+qed.
+
+
+lemma enumn_unique_aux: ∀n,m. ∀h:n ≤ m. uniqueb (Nat_to m) (enumnaux n m h) = true.
+#n elim n -n // #n #Hind #m #h @true_to_andb_true // @memb_enumn //
+qed.
+
+lemma enumn_unique: ∀n.uniqueb (Nat_to n) (enumn n) = true.
+#n @enumn_unique_aux
+qed.
+
+(* definition ltb ≝ λn,m.leb (S n) m. *)
+lemma enumn_complete_aux: ∀n,m,i.∀h:n ≤m.∀k:i<m.i<n →
+ memb (Nat_to m) (mk_Sig ?? i k) (enumnaux n m h) = true.
+#n elim n -n
+ [normalize #n #i #_ #_ #Hfalse @False_ind /2/
+ |#n #Hind #m #i #h #k #lein whd in ⊢ (??%?);
+ cases (le_to_or_lt_eq … (le_S_S_to_le … lein))
+ [#ltin cut (eqb (Nat_to m) (mk_Sig ?? i k) (mk_Sig ?? n h) = false)
+ [normalize @not_eq_to_eqb_false @lt_to_not_eq @ltin]
+ #Hcut >Hcut @Hind //
+ |#eqin cut (eqb (Nat_to m) (mk_Sig ?? i k) (mk_Sig ?? n h) = true)
+ [normalize @eq_to_eqb_true //
+ |#Hcut >Hcut %
+ ]
+ ]
+qed.
+
+lemma enumn_complete: ∀n.∀i:Nat_to n. memb ? i (enumn n) = true.
+#n whd in ⊢ (%→?); * #i #ltin @enumn_complete_aux //
+qed.
+
+definition initN ≝ λn.
+ mk_FinSet (Nat_to n) (enumn n) (enumn_unique n) (enumn_complete n).
+
+example tipa: ∀n.∃x: initN (S n). pi1 … x = n.
+#n @ex_intro [whd @mk_Sig [@n | @le_n] | //] qed.
+
+(* option *)
+definition enum_option ≝ λA:DeqSet.λl.
+ None A::(map ?? (Some A) l).
+
+lemma enum_option_def : ∀A:FinSet.∀l.
+ enum_option A l = None A :: (map ?? (Some A) l).
+// qed.
+
+lemma enum_option_unique: ∀A:DeqSet.∀l.
+ uniqueb A l = true →
+ uniqueb ? (enum_option A l) = true.
+#A #l #uA @true_to_andb_true
+ [generalize in match uA; -uA #_ elim l [%]
+ #a #tl #Hind @sym_eq @noteq_to_eqnot % #H
+ cases (orb_true_l … (sym_eq … H))
+ [#H1 @(absurd (None A = Some A a)) [@(\P H1) | % #H2 destruct]
+ |-H #H >H in Hind; normalize /2/
+ ]
+ |@unique_map_inj // #a #a1 #H destruct %
+ ]
+qed.
+
+lemma enum_option_complete: ∀A:DeqSet.∀l.
+ (∀x:A. memb A x l = true) →
+ ∀x:DeqOption A. memb ? x (enum_option A l) = true.
+#A #l #Hl * // #a @memb_cons @memb_map @Hl
+qed.
+
+definition FinOption ≝ λA:FinSet.
+ mk_FinSet (DeqOption A)
+ (enum_option A (enum A))
+ (enum_option_unique … (enum_unique A))
+ (enum_option_complete … (enum_complete A)).
+
+unification hint 0 ≔ C;
+ T ≟ FinSetcarr C,
+ X ≟ FinOption C
+(* ---------------------------------------- *) ⊢
+ option T ≡ FinSetcarr X.
+
+(* sum *)
definition enum_sum ≝ λA,B:DeqSet.λl1.λl2.
(map ?? (inl A B) l1) @ (map ?? (inr A B) l2).
-lemma enumAB_def : ∀A,B:FinSet.∀l1,l2. enum_sum A B l1 l2 =
+lemma enum_sum_def : ∀A,B:FinSet.∀l1,l2. enum_sum A B l1 l2 =
(map ?? (inl A B) l1) @ (map ?? (inr A B) l2).
// qed.
-axiom unique_map_inj: ∀A,B:DeqSet.∀f:A→B.∀l. injective A B f →
- uniqueb A l = true → uniqueb B (map ?? f l) = true .
-
-axiom memb_map_inj: ∀A,B:DeqSet.∀f:A→B.∀l,a. injective A B f →
- memb ? (f a) (map ?? f l) = true → memb ? a l = true.
-
-lemma enumAB_unique: ∀A,B:DeqSet.∀l1,l2.
+lemma enum_sum_unique: ∀A,B:DeqSet.∀l1,l2.
uniqueb A l1 = true → uniqueb B l2 = true →
uniqueb ? (enum_sum A B l1 l2) = true.
#A #B #l1 #l2 elim l1
]
qed.
+lemma enum_sum_complete: ∀A,B:DeqSet.∀l1,l2.
+ (∀x:A. memb A x l1 = true) →
+ (∀x:B. memb B x l2 = true) →
+ ∀x:DeqSum A B. memb ? x (enum_sum A B l1 l2) = true.
+#A #B #l1 #l2 #Hl1 #Hl2 *
+ [#a @memb_append_l1 @memb_map @Hl1
+ |#b @memb_append_l2 @memb_map @Hl2
+ ]
+qed.
+
definition FinSum ≝ λA,B:FinSet.
mk_FinSet (DeqSum A B)
(enum_sum A B (enum A) (enum B))
- (enumAB_unique … (enum_unique A) (enum_unique B)).
+ (enum_sum_unique … (enum_unique A) (enum_unique B))
+ (enum_sum_complete … (enum_complete A) (enum_complete B)).
+
+include alias "basics/types.ma".
unification hint 0 ≔ C1,C2;
T1 ≟ FinSetcarr C1,
(* ---------------------------------------- *) ⊢
T1+T2 ≡ FinSetcarr X.
+(* prod *)
-(*
-unification hint 0 ≔ ;
- X ≟ mk_DeqSet bool beqb beqb_true
-(* ---------------------------------------- *) ⊢
- bool ≡ carr X.
-
-unification hint 0 ≔ b1,b2:bool;
- X ≟ mk_DeqSet bool beqb beqb_true
-(* ---------------------------------------- *) ⊢
- beqb b1 b2 ≡ eqb X b1 b2.
-
-example exhint: ∀b:bool. (b == false) = true → b = false.
-#b #H @(\P H).
+definition enum_prod ≝ λA,B:DeqSet.λl1.λl2.
+ compose ??? (mk_Prod A B) l1 l2.
+
+axiom enum_prod_unique: ∀A,B,l1,l2.
+ uniqueb A l1 = true → uniqueb B l2 = true →
+ uniqueb ? (enum_prod A B l1 l2) = true.
+
+lemma enum_prod_complete:∀A,B:DeqSet.∀l1,l2.
+ (∀a. memb A a l1 = true) → (∀b.memb B b l2 = true) →
+ ∀p. memb ? p (enum_prod A B l1 l2) = true.
+#A #B #l1 #l2 #Hl1 #Hl2 * #a #b @memb_compose //
qed.
-(* pairs *)
-definition eq_pairs ≝
- λA,B:DeqSet.λp1,p2:A×B.(\fst p1 == \fst p2) ∧ (\snd p1 == \snd p2).
+definition FinProd ≝
+λA,B:FinSet.mk_FinSet (DeqProd A B)
+ (enum_prod A B (enum A) (enum B))
+ (enum_prod_unique A B … (enum_unique A) (enum_unique B))
+ (enum_prod_complete A B … (enum_complete A) (enum_complete B)).
-lemma eq_pairs_true: ∀A,B:DeqSet.∀p1,p2:A×B.
- eq_pairs A B p1 p2 = true ↔ p1 = p2.
-#A #B * #a1 #b1 * #a2 #b2 %
- [#H cases (andb_true …H) #eqa #eqb >(\P eqa) >(\P eqb) //
- |#H destruct normalize >(\b (refl … a2)) >(\b (refl … b2)) //
- ]
-qed.
+unification hint 0 ≔ C1,C2;
+ T1 ≟ FinSetcarr C1,
+ T2 ≟ FinSetcarr C2,
+ X ≟ FinProd C1 C2
+(* ---------------------------------------- *) ⊢
+ T1×T2 ≡ FinSetcarr X.
+
+(* graph of a function *)
+
+definition graph_of ≝ λA,B.λf:A→B.
+ Σp:A×B.f (\fst p) = \snd p.
-definition DeqProd ≝ λA,B:DeqSet.
- mk_DeqSet (A×B) (eq_pairs A B) (eq_pairs_true A B).
+definition graph_enum ≝ λA,B:FinSet.λf:A→B.
+ filter ? (λp.f (\fst p) == \snd p) (enum (FinProd A B)).
+lemma graph_enum_unique : ∀A,B,f.
+ uniqueb ? (graph_enum A B f) = true.
+#A #B #f @uniqueb_filter @(enum_unique (FinProd A B))
+qed.
+
+lemma graph_enum_correct: ∀A,B:FinSet.∀f:A→B. ∀a,b.
+memb ? 〈a,b〉 (graph_enum A B f) = true → f a = b.
+#A #B #f #a #b #membp @(\P ?) @(filter_true … membp)
+qed.
-example hint2: ∀b1,b2.
- 〈b1,true〉==〈false,b2〉=true → 〈b1,true〉=〈false,b2〉.
-#b1 #b2 #H @(\P H).
-*)
\ No newline at end of file
+lemma graph_enum_complete: ∀A,B:FinSet.∀f:A→B. ∀a,b.
+f a = b → memb ? 〈a,b〉 (graph_enum A B f) = true.
+#A #B #f #a #b #eqf @memb_filter_l [@(\b eqf)]
+@enum_prod_complete //
+qed.
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