theory QuotList

imports QuotScript List

begin

fun

  list_rel

where

  "list_rel R [] [] = True"

| "list_rel R (x#xs) [] = False"

| "list_rel R [] (x#xs) = False"

| "list_rel R (x#xs) (y#ys) = (R x y \<and> list_rel R xs ys)"

  assumes a: "equivp R"

  shows "equivp (list_rel R)"

  assumes q: "Quotient R Abs Rep"

  shows "list_rel R r s = (list_rel R r r \<and> list_rel R s s \<and> (map Abs r = map Abs s))"

lemma list_quotient:

  assumes q: "Quotient R Abs Rep"

  shows "Quotient (list_rel R) (map Abs) (map Rep)"

lemma CONS_PRS:

  assumes q: "Quotient R Abs Rep"

  shows "(h#t) = (map Abs) ((Rep h)#(map Rep t))"

by (induct t) (simp_all add: Quotient_ABS_REP[OF q])

lemma CONS_RSP:

  assumes q: "Quotient R Abs Rep"

  and     a: "R h1 h2" "list_rel R t1 t2"

  shows "list_rel R (h1#t1) (h2#t2)"

using a by (auto)

lemma NIL_PRS:

  assumes q: "Quotient R Abs Rep"

  shows "[] = (map Abs [])"

by (simp)

lemma NIL_RSP:

  assumes q: "Quotient R Abs Rep"

  shows "list_rel R [] []"

by simp

lemma MAP_PRS:

  assumes q1: "Quotient R1 Abs1 Rep1"

  and     q2: "Quotient R2 Abs2 Rep2"

  shows "map f l = (map Abs2) (map ((Abs1 ---> Rep2) f) (map Rep1 l))"

by (induct l)

   (simp_all add: Quotient_ABS_REP[OF q1] Quotient_ABS_REP[OF q2])

lemma MAP_RSP:

  assumes q1: "Quotient R1 Abs1 Rep1"

  and     q2: "Quotient R2 Abs2 Rep2"

  and     a: "(R1 ===> R2) f1 f2"

  and     b: "list_rel R1 l1 l2"

  shows "list_rel R2 (map f1 l1) (map f2 l2)"

using b a

by (induct l1 l2 rule: list_induct2')

   (simp_all)