\ / GNU General Public License Version 2
V_______________________________________________________________ *)
-include "basics/lists/listb.ma".
+include "basics/deqlist.ma".
(****** DeqSet: a set with a decidable equality ******)
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.
-definition FinBool ≝ mk_FinSet DeqBool [true;false] bool_enum_unique.
+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
definition DeqNat ≝ mk_DeqSet nat eqb eqbnat_true.
-let rec enumn n ≝
- match n with [ O ⇒ [ ] | S p ⇒ p::enumn p ].
+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 memb_enumn: ∀m,n. n ≤ m → (¬ (memb DeqNat m (enumn n))) = true.
-#m #n elim n // #n1 #Hind #ltm @sym_eq @noteq_to_eqnot @sym_not_eq
+(* 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)) @sym_not_eq /2/
- |<(notb_notb (memb …)) >Hind normalize /2/
+ [whd in ⊢ (??%?→?); #H1 @(absurd … ltni) @le_to_not_lt
+ >(eqb_true_to_eq … H1) @le_n
+ |<(notb_notb (memb …)) >Hind normalize /2 by lt_to_le, absurd/
]
+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 DeqNat (enumn n) = true.
-#n elim n // #m #Hind @true_to_andb_true /2/
+
+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 DeqNat (enumn n) (enumn_unique n).
+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.
-example tipa: ∀n.∃x: initN (S n). x = n.
-#n @(ex_intro … 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.
-example inject : ∃f: initN 2 → initN 4. injective ?? f.
-@(ex_intro … S) //
+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.
-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,
definition enum_prod ≝ λA,B:DeqSet.λl1.λl2.
compose ??? (mk_Prod A B) l1 l2.
-
-axiom enum_prod_unique: ∀A,B,l1,l2.
+
+lemma enum_prod_unique: ∀A,B,l1,l2.
uniqueb A l1 = true → uniqueb B l2 = true →
uniqueb ? (enum_prod A B l1 l2) = true.
+#A #B #l1 elim l1 //
+ #a #tl #Hind #l2 #H1 #H2 @uniqueb_append
+ [@unique_map_inj [#x #y #Heq @(eq_f … \snd … Heq) | //]
+ |@Hind // @(andb_true_r … H1)
+ |#p #H3 cases (memb_map_to_exists … H3) #b *
+ #Hmemb #eqp <eqp @(not_to_not ? (memb ? a tl = true))
+ [2: @sym_not_eq @eqnot_to_noteq @sym_eq @(andb_true_l … H1)
+ |elim tl
+ [normalize //
+ |#a1 #tl1 #Hind2 #H4 cases (memb_append … H4) -H4 #H4
+ [cases (memb_map_to_exists … H4) #b1 * #memb1 #H destruct (H)
+ normalize >(\b (refl ? a)) //
+ |@orb_true_r2 @Hind2 @H4
+ ]
+ ]
+ ]
+ ]
+qed.
+
+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.
definition FinProd ≝
λA,B:FinSet.mk_FinSet (DeqProd A B)
(enum_prod A B (enum A) (enum B))
- (enum_prod_unique A B (enum A) (enum B) (enum_unique A) (enum_unique B) ).
+ (enum_prod_unique A B … (enum_unique A) (enum_unique B))
+ (enum_prod_complete A B … (enum_complete A) (enum_complete B)).
unification hint 0 ≔ C1,C2;
T1 ≟ FinSetcarr C1,
(* ---------------------------------------- *) ⊢
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 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.
+
+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.
+
+(* FinFun *)
+
+definition enum_fun_raw: ∀A,B:DeqSet.list A → list B → list (list (DeqProd A B)) ≝
+ λA,B,lA,lB.foldr A (list (list (DeqProd A B)))
+ (λa.compose ??? (λb. cons ? 〈a,b〉) lB) [[]] lA.
+
+lemma enum_fun_raw_cons: ∀A,B,a,lA,lB.
+ enum_fun_raw A B (a::lA) lB =
+ compose ??? (λb. cons ? 〈a,b〉) lB (enum_fun_raw A B lA lB).
+//
+qed.
+
+definition is_functional ≝ λA,B:DeqSet.λlA:list A.λl: list (DeqProd A B).
+ map ?? (fst A B) l = lA.
+
+definition carr_fun ≝ λA,B:FinSet.
+ DeqSig (DeqList (DeqProd A B)) (is_functional A B (enum A)).
+
+definition carr_fun_l ≝ λA,B:DeqSet.λl.
+ DeqSig (DeqList (DeqProd A B)) (is_functional A B l).
+
+lemma compose_spec1 : ∀A,B,C:DeqSet.∀f:A→B→C.∀a:A.∀b:B.∀lA:list A.∀lB:list B.
+ a ∈ lA = true → b ∈ lB = true → ((f a b) ∈ (compose A B C f lA lB)) = true.
+#A #B #C #f #a #b #lA elim lA
+ [normalize #lB #H destruct
+ |#a1 #tl #Hind #lB #Ha #Hb cases (orb_true_l ?? Ha) #Hcase
+ [>(\P Hcase) normalize @memb_append_l1 @memb_map //
+ |@memb_append_l2 @Hind //
+ ]
+ ]
+qed.
+
+lemma compose_cons: ∀A,B,C.∀f:A→B→C.∀l1,l2,a.
+ compose A B C f (a::l1) l2 =
+ (map ?? (f a) l2)@(compose A B C f l1 l2).
+// qed.
+
+lemma compose_spec2 : ∀A,B,C:DeqSet.∀f:A→B→C.∀c:C.∀lA:list A.∀lB:list B.
+ c ∈ (compose A B C f lA lB) = true →
+ ∃a,b.a ∈ lA = true ∧ b ∈ lB = true ∧ c = f a b.
+#A #B #C #f #c #lA elim lA
+ [normalize #lB #H destruct
+ |#a1 #tl #Hind #lB >compose_cons #Hc cases (memb_append … Hc) #Hcase
+ [lapply(memb_map_to_exists … Hcase) * #b * #Hb #Hf
+ %{a1} %{b} /3/
+ |lapply(Hind ? Hcase) * #a2 * #b * * #Ha #Hb #Hf %{a2} %{b} % // % //
+ @orb_true_r2 //
+ ]
+ ]
+qed.
+
+definition compose2 ≝
+ λA,B:DeqSet.λa:A.λl. compose B (carr_fun_l A B l) (carr_fun_l A B (a::l))
+ (λb,tl. mk_Sig ?? (〈a,b〉::(pi1 … tl)) ?).
+normalize @eq_f @(pi2 … tl)
+qed.
+
+let rec Dfoldr (A:Type[0]) (B:list A → Type[0])
+ (f:∀a:A.∀l.B l → B (a::l)) (b:B [ ]) (l:list A) on l : B l ≝
+ match l with [ nil ⇒ b | cons a l ⇒ f a l (Dfoldr A B f b l)].
+
+definition empty_graph: ∀A,B:DeqSet. carr_fun_l A B [].
+#A #B %{[]} // qed.
+
+definition enum_fun: ∀A,B:DeqSet.∀lA:list A.list B → list (carr_fun_l A B lA) ≝
+ λA,B,lA,lB.Dfoldr A (λl.list (carr_fun_l A B l))
+ (λa,l.compose2 ?? a l lB) [empty_graph A B] lA.
+
+lemma mem_enum_fun: ∀A,B:DeqSet.∀lA,lB.∀x:carr_fun_l A B lA.
+ pi1 … x ∈ map ?? (pi1 … ) (enum_fun A B lA lB) = true →
+ x ∈ enum_fun A B lA lB = true .
+#A #B #lA #lB #x @(memb_map_inj
+ (DeqSig (DeqList (DeqProd A B))
+ (λx0:DeqList (DeqProd A B).is_functional A B lA x0))
+ (DeqList (DeqProd A B)) (pi1 …))
+* #l1 #H1 * #l2 #H2 #Heq lapply H1 lapply H2 >Heq //
+qed.
+
+lemma enum_fun_cons: ∀A,B,a,lA,lB.
+ enum_fun A B (a::lA) lB =
+ compose ??? (λb,tl. mk_Sig ?? (〈a,b〉::(pi1 … tl)) ?) lB (enum_fun A B lA lB).
+//
+qed.
+
+lemma map_map: ∀A,B,C.∀f:A→B.∀g:B→C.∀l.
+ map ?? g (map ?? f l) = map ?? (g ∘ f) l.
+#A #B #C #f #g #l elim l [//]
+#a #tl #Hind normalize @eq_f @Hind
+qed.
+
+lemma map_compose: ∀A,B,C,D.∀f:A→B→C.∀g:C→D.∀l1,l2.
+ map ?? g (compose A B C f l1 l2) = compose A B D (λa,b. g (f a b)) l1 l2.
+#A #B #C #D #f #g #l1 elim l1 [//]
+#a #tl #Hind #l2 >compose_cons >compose_cons <map_append @eq_f2
+ [@map_map |@Hind]
+qed.
+
+definition enum_fun_graphs: ∀A,B,lA,lB.
+ map ?? (pi1 … ) (enum_fun A B lA lB) = enum_fun_raw A B lA lB.
+#A #B #lA elim lA [normalize //]
+#a #tl #Hind #lB >(enum_fun_cons A B a tl lB) >enum_fun_raw_cons >map_compose
+cut (∀lB2. compose B (Σx:DeqList (DeqProd A B).is_functional A B tl x)
+ (DeqList (DeqProd A B))
+ (λa0:B
+ .λb:Σx:DeqList (DeqProd A B).is_functional A B tl x
+ .〈a,a0〉
+ ::pi1 (list (A×B)) (λx:DeqList (DeqProd A B).is_functional A B tl x) b) lB
+ (enum_fun A B tl lB2)
+ =compose B (list (A×B)) (list (A×B)) (λb:B.cons (A×B) 〈a,b〉) lB
+ (enum_fun_raw A B tl lB2))
+ [#lB2 elim lB
+ [normalize //
+ |#b #tlb #Hindb >compose_cons in ⊢ (???%); >compose_cons
+ @eq_f2 [<Hind >map_map // |@Hindb]]]
+#Hcut @Hcut
+qed.
+
+lemma uniqueb_compose: ∀A,B,C:DeqSet.∀f,l1,l2.
+ (∀a1,a2,b1,b2. f a1 b1 = f a2 b2 → a1 = a2 ∧ b1 = b2) →
+ uniqueb ? l1 = true → uniqueb ? l2 = true →
+ uniqueb ? (compose A B C f l1 l2) = true.
+#A #B #C #f #l1 #l2 #Hinj elim l1 //
+#a #tl #Hind #HuA #HuB >compose_cons @uniqueb_append
+ [@(unique_map_inj … HuB) #b1 #b2 #Hb1b2 @(proj2 … (Hinj … Hb1b2))
+ |@Hind // @(andb_true_r … HuA)
+ |#c #Hc lapply(memb_map_to_exists … Hc) * #b * #Hb2 #Hfab % #Hc
+ lapply(compose_spec2 … Hc) * #a1 * #b1 * * #Ha1 #Hb1 <Hfab #H
+ @(absurd (a=a1))
+ [@(proj1 … (Hinj … H))
+ |% #eqa @(absurd … Ha1) % <eqa #H lapply(andb_true_l … HuA) >H
+ normalize #H1 destruct (H1)
+ ]
+ ]
+qed.
+
+lemma enum_fun_unique: ∀A,B:DeqSet.∀lA,lB.
+ uniqueb ? lA = true → uniqueb ? lB = true →
+ uniqueb ? (enum_fun A B lA lB) = true.
+#A #B #lA elim lA
+ [#lB #_ #ulB //
+ |#a #tlA #Hind #lB #uA #uB lapply (enum_fun_cons A B a tlA lB) #H >H
+ @(uniqueb_compose B (carr_fun_l A B tlA) (carr_fun_l A B (a::tlA)))
+ [#b1 #b2 * #l1 #funl1 * #l2 #funl2 #H1 destruct (H1) /2/
+ |//
+ |@(Hind … uB) @(andb_true_r … uA)
+ ]
+ ]
+qed.
+
+lemma enum_fun_complete: ∀A,B:FinSet.∀l1,l2.
+ (∀x:A. memb A x l1 = true) →
+ (∀x:B. memb B x l2 = true) →
+ ∀x:carr_fun_l A B l1. memb ? x (enum_fun A B l1 l2) = true.
+#A #B #l1 #l2 #H1 #H2 * #g #H @mem_enum_fun >enum_fun_graphs
+lapply H -H lapply g -g elim l1
+ [* // #p #tlg normalize #H destruct (H)
+ |#a #tl #Hind #g cases g
+ [normalize in ⊢ (%→?); #H destruct (H)
+ |* #a1 #b #tl1 normalize in ⊢ (%→?); #H
+ cut (is_functional A B tl tl1) [destruct (H) //] #Hfun
+ >(cons_injective_l ????? H)
+ >(enum_fun_raw_cons … ) @(compose_spec1 … (λb. cons ? 〈a,b〉))
+ [@H2 |@Hind @Hfun]
+ ]
+ ]
+qed.
+
+definition FinFun ≝
+λA,B:FinSet.mk_FinSet (carr_fun A B)
+ (enum_fun A B (enum A) (enum B))
+ (enum_fun_unique A B … (enum_unique A) (enum_unique B))
+ (enum_fun_complete A B … (enum_complete A) (enum_complete B)).
+
+(*
+unification hint 0 ≔ C1,C2;
+ T1 ≟ FinSetcarr C1,
+ T2 ≟ FinSetcarr C2,
+ X ≟ FinProd C1 C2
+(* ---------------------------------------- *) ⊢
+ T1×T2 ≡ FinSetcarr X. *)
\ No newline at end of file
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