24 lemma prod_quotient:

    24 lemma prod_quotient:

    25   assumes q1: "Quotient R1 Abs1 Rep1"

    25   assumes q1: "Quotient R1 Abs1 Rep1"

    26   assumes q2: "Quotient R2 Abs2 Rep2"

    26   assumes q2: "Quotient R2 Abs2 Rep2"

    27   shows "Quotient (prod_rel R1 R2) (prod_fun Abs1 Abs2) (prod_fun Rep1 Rep2)"

    27   shows "Quotient (prod_rel R1 R2) (prod_fun Abs1 Abs2) (prod_fun Rep1 Rep2)"

    28 unfolding Quotient_def

    28 unfolding Quotient_def

    30 using Quotient_rel[OF q1] Quotient_rel[OF q2] by blast

    31 using Quotient_rel[OF q1] Quotient_rel[OF q2] by blast

    31 

    32 

    32 lemma pair_rsp:

    33 lemma pair_rsp:

    33   assumes q1: "Quotient R1 Abs1 Rep1"

    34   assumes q1: "Quotient R1 Abs1 Rep1"

    34   assumes q2: "Quotient R2 Abs2 Rep2"

    35   assumes q2: "Quotient R2 Abs2 Rep2"

    38 lemma pair_prs_aux:

    39 lemma pair_prs_aux:

    39   assumes q1: "Quotient R1 Abs1 Rep1"

    40   assumes q1: "Quotient R1 Abs1 Rep1"

    40   assumes q2: "Quotient R2 Abs2 Rep2"

    41   assumes q2: "Quotient R2 Abs2 Rep2"

    41   shows "(prod_fun Abs1 Abs2) (Rep1 l, Rep2 r) \<equiv> (l, r)"

    42   shows "(prod_fun Abs1 Abs2) (Rep1 l, Rep2 r) \<equiv> (l, r)"

    42   by (simp add: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])

    43   by (simp add: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])

    45 

    44 

    46 lemma pair_prs[quot_preserve]:

    45 lemma pair_prs[quot_preserve]:

    47   assumes q1: "Quotient R1 Abs1 Rep1"

    46   assumes q1: "Quotient R1 Abs1 Rep1"

    48   assumes q2: "Quotient R2 Abs2 Rep2"

    47   assumes q2: "Quotient R2 Abs2 Rep2"

    49   shows "(Rep1 ---> Rep2 ---> (prod_fun Abs1 Abs2)) Pair = Pair"

    48   shows "(Rep1 ---> Rep2 ---> (prod_fun Abs1 Abs2)) Pair = Pair"