+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. *)
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