Product Types

There is one library file in the directory lib_prod with the definition of (non-dependent) cross-product. As well as the basic definitions there are also theorems giving the $\eta$-rule for pairs and the fact that pairing is extensional.

 ** Module lib_prod Imports lib_eq_basics
 $prod : (Type)->(Type)->Type(prod)
 $Pair : {A|Type}{B|Type}A->B->prod A B
 $prod_elim :
    {A|Type}{B|Type}{C_prod:(prod A B)->Type}
    ({a:A}{b:B}C_prod (Pair a b))->{z:prod A B}C_prod z
 ** Label (!prod!) prod
 ** Label (!prod elim!) prod_elim
 ** Label (!prod Pair!) Pair
[[A|Type][B|Type][C_prod:(prod A B)->Type]
 [f_Pair:{a:A}{b:B}C_prod (Pair a b)][a:A][b:B]
    prod_elim C_prod f_Pair (Pair a b)  ==>  f_Pair a b]

   Gen (!prod is Pair!) as prod_is_Pair = ... :
    {A|Type}{B|Type}(prod A B)->Type
   Gen (!prod Pair injective!) as prod_Pair_injective = ... :
    {A|Type}{B|Type}{ix0,iy0|A}{ix1,iy1|B}
    (Eq (Pair ix0 ix1) (Pair iy0 iy1))->{P|Type}
    ((Eq ix0 iy0)->(Eq ix1 iy1)->P)->P
  pair1 = ... : {A:Type}{B:Type}A->B->prod A B
  prod_rec = ... : {s|Type}{t|Type}{u|Type}(s->t->u)->(prod s t)->u
  prod_ind = ... :
    {s|Type}{t|Type}{P:(prod s t)->Prop}({a:s}{b:t}P (Pair a b))->
    {p:prod s t}P p
  Fst = ... : {s|Type}{t|Type}(prod s t)->s
  Snd = ... : {s|Type}{t|Type}(prod s t)->t
  prod_eta = ... :
    {s|Type}{t|Type}{p:prod s t}Eq p (Pair (Fst p) (Snd p))
  prod_ext = ... :
    {s|Type}{t|Type}{p,q:prod s t}(Eq (Fst p) (Fst q))->
    (Eq (Snd p) (Snd q))->Eq p q