Definition upd_compact_sum {A} {F} (l: list A) (v: compact_sum (map F l)) (a: A) (v0: F a) (H: forall a b: A, {a = b} + {a <> b}) : compact_sum (map F l).Definition proj_compact_prod {A: Type} {F: A -> Type} (a: A) (l: list A) (v: compact_prod (map F l)) (default: F a) (H: forall a b: A, {a = b} + {a <> b}) : F a.Definition proj_compact_sum {A: Type} {F: A -> Type} (a: A) (l: list A) (v: compact_sum (map F l)) (default: F a) (H: forall a b: A, {a = b} + {a <> b}) : F a.Definition compact_sum_inj {A: Type} {F: A -> Type} {l: list A} (v: compact_sum (map F l)) (a: A) (H: forall a b: A, {a = b} + {a <> b}): Prop.

Lemma compact_sum_proj_gen: forall {A: Type} {F: A -> Type} {l: list A} a df (filter: A -> bool) (gen: forall a, F a) (H: forall a b : A, {a = b} + {a <> b}), compact_sum_inj (compact_sum_gen filter gen l) a H -> proj_compact_sum a l (compact_sum_gen filter gen l) df H = gen a.