1 theory QuotList

     2 imports QuotScript List

     3 begin

     4 

     5 fun

     6   list_rel

     7 where

     8   "list_rel R [] [] = True"

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

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

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

    12 

    13 lemma list_equivp:

    14   assumes a: "equivp R"

    15   shows "equivp (list_rel R)"

    16   unfolding equivp_def

    17   apply(rule allI)+

    18   apply(induct_tac x y rule: list_induct2')

    19   apply(simp_all add: expand_fun_eq)

    20   apply(metis list_rel.simps(1) list_rel.simps(2))

    21   apply(metis list_rel.simps(1) list_rel.simps(2))

    22   apply(rule iffI)

    23   apply(rule allI)

    24   apply(case_tac x)

    25   apply(simp_all)

    26   using a

    27   apply(unfold equivp_def)

    28   apply(auto)[1]

    29   apply(metis list_rel.simps(4))

    30   done

    31 

    32 lemma list_rel_rel:

    33   assumes q: "Quotient R Abs Rep"

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

    35   apply(induct r s rule: list_induct2')

    36   apply(simp_all)

    37   using Quotient_rel[OF q]

    38   apply(metis)

    39   done

    40 

    41 lemma list_quotient:

    42   assumes q: "Quotient R Abs Rep"

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

    44   unfolding Quotient_def

    45   apply(rule conjI)

    46   apply(rule allI)

    47   apply(induct_tac a)

    48   apply(simp)

    49   apply(simp add: Quotient_abs_rep[OF q])

    50   apply(rule conjI)

    51   apply(rule allI)

    52   apply(induct_tac a)

    53   apply(simp)

    54   apply(simp)

    55   apply(simp add: Quotient_rep_reflp[OF q])

    56   apply(rule allI)+

    57   apply(rule list_rel_rel[OF q])

    58   done

    59 

    60 

    61 lemma cons_prs:

    62   assumes q: "Quotient R Abs Rep"

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

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

    65 

    66 lemma cons_rsp:

    67   assumes q: "Quotient R Abs Rep"

    68   shows "(R ===> list_rel R ===> list_rel R) op # op #"

    69 by (auto)

    70 

    71 lemma nil_prs:

    72   assumes q: "Quotient R Abs Rep"

    73   shows "map Abs [] \<equiv> []"

    74 by (simp)

    75 

    76 lemma nil_rsp:

    77   assumes q: "Quotient R Abs Rep"

    78   shows "list_rel R [] []"

    79 by simp

    80 

    81 lemma map_prs:

    82   assumes a: "Quotient R1 abs1 rep1"

    83   and     b: "Quotient R2 abs2 rep2"

    84   shows "(map abs2) (map ((abs1 ---> rep2) f) (map rep1 l)) = map f l"

    85 by (induct l) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])

    86 

    87 lemma map_rsp:

    88   assumes q1: "Quotient R1 Abs1 Rep1"

    89   and     q2: "Quotient R2 Abs2 Rep2"

    90   shows "((R1 ===> R2) ===> (list_rel R1) ===> list_rel R2) map map"

    91 apply(simp)

    92 apply(rule allI)+

    93 apply(rule impI)

    94 apply(rule allI)+

    95 apply (induct_tac xa ya rule: list_induct2')

    96 apply simp_all

    97 done

    98 

    99 (* TODO: if the above is correct, we can remove this one *)

   100 lemma map_rsp_lo:

   101   assumes q1: "Quotient R1 Abs1 Rep1"

   102   and     q2: "Quotient R2 Abs2 Rep2"

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

   104   and     b: "list_rel R1 l1 l2"

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

   106 using b a

   107 by (induct l1 l2 rule: list_induct2') (simp_all)

   108 

   109 lemma foldr_prs:

   110   assumes a: "Quotient R1 abs1 rep1"

   111   and     b: "Quotient R2 abs2 rep2"

   112   shows "abs2 (foldr ((abs1 ---> abs2 ---> rep2) f) (map rep1 l) (rep2 e)) = foldr f l e"

   113 by (induct l) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])

   114 

   115 lemma foldl_prs:

   116   assumes a: "Quotient R1 abs1 rep1"

   117   and     b: "Quotient R2 abs2 rep2"

   118   shows "abs1 (foldl ((abs1 ---> abs2 ---> rep1) f) (rep1 e) (map rep2 l)) = foldl f e l"

   119 by (induct l arbitrary:e) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])

   120 

   121 lemma list_rel_empty: "list_rel R [] b \<Longrightarrow> length b = 0"

   122 by (induct b) (simp_all)

   123 

   124 lemma list_rel_len: "list_rel R a b \<Longrightarrow> length a = length b"

   125 apply (induct a arbitrary: b)

   126 apply (simp add: list_rel_empty)

   127 apply (case_tac b)

   128 apply simp_all

   129 done

   130 

   131 (* TODO: induct_tac doesn't accept 'arbitrary'.

   132          induct     doesn't accept 'rule'.

   133    that's why the proof uses manual generalisation and needs assumptions

   134    both in conclusion for induction and in assumptions. *)

   135 lemma foldl_rsp:

   136   assumes q1: "Quotient R1 Abs1 Rep1"

   137   and     q2: "Quotient R2 Abs2 Rep2"

   138   shows "((R1 ===> R2 ===> R1) ===> R1 ===> list_rel R2 ===> R1) foldl foldl"

   139 apply auto

   140 apply (subgoal_tac "R1 xa ya \<longrightarrow> list_rel R2 xb yb \<longrightarrow> R1 (foldl x xa xb) (foldl y ya yb)")

   141 apply simp

   142 apply (rule_tac x="xa" in spec)

   143 apply (rule_tac x="ya" in spec)

   144 apply (rule_tac xs="xb" and ys="yb" in list_induct2)

   145 apply (rule list_rel_len)

   146 apply (simp_all)

   147 done

   148 

   149 (* TODO: foldr_rsp should be similar *)

   150 

   151 

   152 

   153 

   154 (* TODO: Rest are unused *)

   155 

   156 lemma list_map_id:

   157   shows "map (\<lambda>x. x) = (\<lambda>x. x)"

   158   by simp

   159 

   160 lemma list_rel_eq:

   161   shows "list_rel (op =) \<equiv> (op =)"

   162 apply(rule eq_reflection)

   163 unfolding expand_fun_eq

   164 apply(rule allI)+

   165 apply(induct_tac x xa rule: list_induct2')

   166 apply(simp_all)

   167 done

   168 

   169 lemma list_rel_refl:

   170   assumes a: "\<And>x y. R x y = (R x = R y)"

   171   shows "list_rel R x x"

   172 by (induct x) (auto simp add: a)

   173 

   174 end