src/HOL/Lifting.thy
author haftmann
Thu, 14 Feb 2013 12:24:42 +0100
changeset 51112 da97167e03f7
parent 48891 c0eafbd55de3
child 51374 84d01fd733cf
permissions -rw-r--r--
abandoned theory Plain
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
     1
(*  Title:      HOL/Lifting.thy
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
     2
    Author:     Brian Huffman and Ondrej Kuncar
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
     3
    Author:     Cezary Kaliszyk and Christian Urban
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
     4
*)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
     5
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
     6
header {* Lifting package *}
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
     7
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
     8
theory Lifting
51112
da97167e03f7 abandoned theory Plain
haftmann
parents: 48891
diff changeset
     9
imports Equiv_Relations Transfer
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    10
keywords
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    11
  "print_quotmaps" "print_quotients" :: diag and
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    12
  "lift_definition" :: thy_goal and
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    13
  "setup_lifting" :: thy_decl
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    14
begin
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    15
47325
ec6187036495 new transfer proof method
huffman
parents: 47308
diff changeset
    16
subsection {* Function map *}
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    17
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    18
notation map_fun (infixr "--->" 55)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    19
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    20
lemma map_fun_id:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    21
  "(id ---> id) = id"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    22
  by (simp add: fun_eq_iff)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    23
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    24
subsection {* Quotient Predicate *}
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    25
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    26
definition
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    27
  "Quotient R Abs Rep T \<longleftrightarrow>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    28
     (\<forall>a. Abs (Rep a) = a) \<and> 
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    29
     (\<forall>a. R (Rep a) (Rep a)) \<and>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    30
     (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s) \<and>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    31
     T = (\<lambda>x y. R x x \<and> Abs x = y)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    32
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    33
lemma QuotientI:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    34
  assumes "\<And>a. Abs (Rep a) = a"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    35
    and "\<And>a. R (Rep a) (Rep a)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    36
    and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    37
    and "T = (\<lambda>x y. R x x \<and> Abs x = y)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    38
  shows "Quotient R Abs Rep T"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    39
  using assms unfolding Quotient_def by blast
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    40
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    41
context
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    42
  fixes R Abs Rep T
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    43
  assumes a: "Quotient R Abs Rep T"
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    44
begin
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    45
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    46
lemma Quotient_abs_rep: "Abs (Rep a) = a"
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    47
  using a unfolding Quotient_def
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    48
  by simp
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    49
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    50
lemma Quotient_rep_reflp: "R (Rep a) (Rep a)"
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    51
  using a unfolding Quotient_def
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    52
  by blast
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    53
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    54
lemma Quotient_rel:
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    55
  "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    56
  using a unfolding Quotient_def
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    57
  by blast
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    58
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    59
lemma Quotient_cr_rel: "T = (\<lambda>x y. R x x \<and> Abs x = y)"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    60
  using a unfolding Quotient_def
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    61
  by blast
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    62
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    63
lemma Quotient_refl1: "R r s \<Longrightarrow> R r r"
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    64
  using a unfolding Quotient_def
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    65
  by fast
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    66
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    67
lemma Quotient_refl2: "R r s \<Longrightarrow> R s s"
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    68
  using a unfolding Quotient_def
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    69
  by fast
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    70
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    71
lemma Quotient_rel_rep: "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    72
  using a unfolding Quotient_def
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    73
  by metis
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    74
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    75
lemma Quotient_rep_abs: "R r r \<Longrightarrow> R (Rep (Abs r)) r"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    76
  using a unfolding Quotient_def
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    77
  by blast
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    78
47937
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    79
lemma Quotient_rep_abs_fold_unmap: 
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    80
  assumes "x' \<equiv> Abs x" and "R x x" and "Rep x' \<equiv> Rep' x'" 
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    81
  shows "R (Rep' x') x"
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    82
proof -
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    83
  have "R (Rep x') x" using assms(1-2) Quotient_rep_abs by auto
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    84
  then show ?thesis using assms(3) by simp
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    85
qed
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    86
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    87
lemma Quotient_Rep_eq:
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    88
  assumes "x' \<equiv> Abs x" 
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    89
  shows "Rep x' \<equiv> Rep x'"
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    90
by simp
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    91
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    92
lemma Quotient_rel_abs: "R r s \<Longrightarrow> Abs r = Abs s"
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    93
  using a unfolding Quotient_def
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    94
  by blast
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    95
47937
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    96
lemma Quotient_rel_abs2:
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    97
  assumes "R (Rep x) y"
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    98
  shows "x = Abs y"
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
    99
proof -
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
   100
  from assms have "Abs (Rep x) = Abs y" by (auto intro: Quotient_rel_abs)
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
   101
  then show ?thesis using assms(1) by (simp add: Quotient_abs_rep)
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
   102
qed
70375fa2679d generate abs_eq, use it as a code equation for total quotients; no_abs_code renamed to no_code; added no_code for quotient_type command
kuncar
parents: 47936
diff changeset
   103
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
   104
lemma Quotient_symp: "symp R"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   105
  using a unfolding Quotient_def using sympI by (metis (full_types))
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   106
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
   107
lemma Quotient_transp: "transp R"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   108
  using a unfolding Quotient_def using transpI by (metis (full_types))
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   109
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
   110
lemma Quotient_part_equivp: "part_equivp R"
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
   111
by (metis Quotient_rep_reflp Quotient_symp Quotient_transp part_equivpI)
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
   112
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
   113
end
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   114
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   115
lemma identity_quotient: "Quotient (op =) id id (op =)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   116
unfolding Quotient_def by simp 
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   117
47652
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   118
text {* TODO: Use one of these alternatives as the real definition. *}
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   119
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   120
lemma Quotient_alt_def:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   121
  "Quotient R Abs Rep T \<longleftrightarrow>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   122
    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   123
    (\<forall>b. T (Rep b) b) \<and>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   124
    (\<forall>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   125
apply safe
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   126
apply (simp (no_asm_use) only: Quotient_def, fast)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   127
apply (simp (no_asm_use) only: Quotient_def, fast)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   128
apply (simp (no_asm_use) only: Quotient_def, fast)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   129
apply (simp (no_asm_use) only: Quotient_def, fast)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   130
apply (simp (no_asm_use) only: Quotient_def, fast)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   131
apply (simp (no_asm_use) only: Quotient_def, fast)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   132
apply (rule QuotientI)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   133
apply simp
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   134
apply metis
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   135
apply simp
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   136
apply (rule ext, rule ext, metis)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   137
done
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   138
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   139
lemma Quotient_alt_def2:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   140
  "Quotient R Abs Rep T \<longleftrightarrow>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   141
    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   142
    (\<forall>b. T (Rep b) b) \<and>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   143
    (\<forall>x y. R x y \<longleftrightarrow> T x (Abs y) \<and> T y (Abs x))"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   144
  unfolding Quotient_alt_def by (safe, metis+)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   145
47652
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   146
lemma Quotient_alt_def3:
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   147
  "Quotient R Abs Rep T \<longleftrightarrow>
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   148
    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and>
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   149
    (\<forall>x y. R x y \<longleftrightarrow> (\<exists>z. T x z \<and> T y z))"
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   150
  unfolding Quotient_alt_def2 by (safe, metis+)
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   151
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   152
lemma Quotient_alt_def4:
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   153
  "Quotient R Abs Rep T \<longleftrightarrow>
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   154
    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> R = T OO conversep T"
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   155
  unfolding Quotient_alt_def3 fun_eq_iff by auto
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   156
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   157
lemma fun_quotient:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   158
  assumes 1: "Quotient R1 abs1 rep1 T1"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   159
  assumes 2: "Quotient R2 abs2 rep2 T2"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   160
  shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2) (T1 ===> T2)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   161
  using assms unfolding Quotient_alt_def2
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   162
  unfolding fun_rel_def fun_eq_iff map_fun_apply
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   163
  by (safe, metis+)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   164
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   165
lemma apply_rsp:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   166
  fixes f g::"'a \<Rightarrow> 'c"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   167
  assumes q: "Quotient R1 Abs1 Rep1 T1"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   168
  and     a: "(R1 ===> R2) f g" "R1 x y"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   169
  shows "R2 (f x) (g y)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   170
  using a by (auto elim: fun_relE)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   171
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   172
lemma apply_rsp':
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   173
  assumes a: "(R1 ===> R2) f g" "R1 x y"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   174
  shows "R2 (f x) (g y)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   175
  using a by (auto elim: fun_relE)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   176
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   177
lemma apply_rsp'':
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   178
  assumes "Quotient R Abs Rep T"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   179
  and "(R ===> S) f f"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   180
  shows "S (f (Rep x)) (f (Rep x))"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   181
proof -
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   182
  from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   183
  then show ?thesis using assms(2) by (auto intro: apply_rsp')
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   184
qed
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   185
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   186
subsection {* Quotient composition *}
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   187
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   188
lemma Quotient_compose:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   189
  assumes 1: "Quotient R1 Abs1 Rep1 T1"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   190
  assumes 2: "Quotient R2 Abs2 Rep2 T2"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   191
  shows "Quotient (T1 OO R2 OO conversep T1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2) (T1 OO T2)"
47652
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   192
  using assms unfolding Quotient_alt_def4 by (auto intro!: ext)
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   193
47521
69f95ac85c3d tuned the setup of lifting; generate transfer rules for typedef and Quotient thms
kuncar
parents: 47501
diff changeset
   194
lemma equivp_reflp2:
69f95ac85c3d tuned the setup of lifting; generate transfer rules for typedef and Quotient thms
kuncar
parents: 47501
diff changeset
   195
  "equivp R \<Longrightarrow> reflp R"
69f95ac85c3d tuned the setup of lifting; generate transfer rules for typedef and Quotient thms
kuncar
parents: 47501
diff changeset
   196
  by (erule equivpE)
69f95ac85c3d tuned the setup of lifting; generate transfer rules for typedef and Quotient thms
kuncar
parents: 47501
diff changeset
   197
47544
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   198
subsection {* Respects predicate *}
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   199
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   200
definition Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   201
  where "Respects R = {x. R x x}"
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   202
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   203
lemma in_respects: "x \<in> Respects R \<longleftrightarrow> R x x"
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   204
  unfolding Respects_def by simp
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   205
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   206
subsection {* Invariant *}
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   207
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   208
definition invariant :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" 
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   209
  where "invariant R = (\<lambda>x y. R x \<and> x = y)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   210
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   211
lemma invariant_to_eq:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   212
  assumes "invariant P x y"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   213
  shows "x = y"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   214
using assms by (simp add: invariant_def)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   215
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   216
lemma fun_rel_eq_invariant:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   217
  shows "((invariant R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   218
by (auto simp add: invariant_def fun_rel_def)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   219
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   220
lemma invariant_same_args:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   221
  shows "invariant P x x \<equiv> P x"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   222
using assms by (auto simp add: invariant_def)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   223
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   224
lemma UNIV_typedef_to_Quotient:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   225
  assumes "type_definition Rep Abs UNIV"
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   226
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   227
  shows "Quotient (op =) Abs Rep T"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   228
proof -
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   229
  interpret type_definition Rep Abs UNIV by fact
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   230
  from Abs_inject Rep_inverse Abs_inverse T_def show ?thesis 
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   231
    by (fastforce intro!: QuotientI fun_eq_iff)
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   232
qed
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   233
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   234
lemma UNIV_typedef_to_equivp:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   235
  fixes Abs :: "'a \<Rightarrow> 'b"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   236
  and Rep :: "'b \<Rightarrow> 'a"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   237
  assumes "type_definition Rep Abs (UNIV::'a set)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   238
  shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   239
by (rule identity_equivp)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   240
47354
95846613e414 update keywords file
huffman
parents: 47351
diff changeset
   241
lemma typedef_to_Quotient:
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   242
  assumes "type_definition Rep Abs S"
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   243
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
47501
0b9294e093db leave Lifting prefix
kuncar
parents: 47436
diff changeset
   244
  shows "Quotient (invariant (\<lambda>x. x \<in> S)) Abs Rep T"
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   245
proof -
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   246
  interpret type_definition Rep Abs S by fact
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   247
  from Rep Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   248
    by (auto intro!: QuotientI simp: invariant_def fun_eq_iff)
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   249
qed
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   250
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   251
lemma typedef_to_part_equivp:
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   252
  assumes "type_definition Rep Abs S"
47501
0b9294e093db leave Lifting prefix
kuncar
parents: 47436
diff changeset
   253
  shows "part_equivp (invariant (\<lambda>x. x \<in> S))"
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   254
proof (intro part_equivpI)
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   255
  interpret type_definition Rep Abs S by fact
47501
0b9294e093db leave Lifting prefix
kuncar
parents: 47436
diff changeset
   256
  show "\<exists>x. invariant (\<lambda>x. x \<in> S) x x" using Rep by (auto simp: invariant_def)
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   257
next
47501
0b9294e093db leave Lifting prefix
kuncar
parents: 47436
diff changeset
   258
  show "symp (invariant (\<lambda>x. x \<in> S))" by (auto intro: sympI simp: invariant_def)
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   259
next
47501
0b9294e093db leave Lifting prefix
kuncar
parents: 47436
diff changeset
   260
  show "transp (invariant (\<lambda>x. x \<in> S))" by (auto intro: transpI simp: invariant_def)
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   261
qed
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   262
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   263
lemma open_typedef_to_Quotient:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   264
  assumes "type_definition Rep Abs {x. P x}"
47354
95846613e414 update keywords file
huffman
parents: 47351
diff changeset
   265
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   266
  shows "Quotient (invariant P) Abs Rep T"
47651
8e4f50afd21a tuned proofs
huffman
parents: 47575
diff changeset
   267
  using typedef_to_Quotient [OF assms] by simp
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   268
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   269
lemma open_typedef_to_part_equivp:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   270
  assumes "type_definition Rep Abs {x. P x}"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   271
  shows "part_equivp (invariant P)"
47651
8e4f50afd21a tuned proofs
huffman
parents: 47575
diff changeset
   272
  using typedef_to_part_equivp [OF assms] by simp
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   273
47376
776254f89a18 add transfer lemmas for quotients
huffman
parents: 47369
diff changeset
   274
text {* Generating transfer rules for quotients. *}
776254f89a18 add transfer lemmas for quotients
huffman
parents: 47369
diff changeset
   275
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   276
context
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   277
  fixes R Abs Rep T
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   278
  assumes 1: "Quotient R Abs Rep T"
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   279
begin
47376
776254f89a18 add transfer lemmas for quotients
huffman
parents: 47369
diff changeset
   280
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   281
lemma Quotient_right_unique: "right_unique T"
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   282
  using 1 unfolding Quotient_alt_def right_unique_def by metis
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   283
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   284
lemma Quotient_right_total: "right_total T"
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   285
  using 1 unfolding Quotient_alt_def right_total_def by metis
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   286
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   287
lemma Quotient_rel_eq_transfer: "(T ===> T ===> op =) R (op =)"
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   288
  using 1 unfolding Quotient_alt_def fun_rel_def by simp
47376
776254f89a18 add transfer lemmas for quotients
huffman
parents: 47369
diff changeset
   289
47538
1f0ec5b8135a add lemma Quotient_abs_induct
huffman
parents: 47537
diff changeset
   290
lemma Quotient_abs_induct:
1f0ec5b8135a add lemma Quotient_abs_induct
huffman
parents: 47537
diff changeset
   291
  assumes "\<And>y. R y y \<Longrightarrow> P (Abs y)" shows "P x"
1f0ec5b8135a add lemma Quotient_abs_induct
huffman
parents: 47537
diff changeset
   292
  using 1 assms unfolding Quotient_def by metis
1f0ec5b8135a add lemma Quotient_abs_induct
huffman
parents: 47537
diff changeset
   293
47544
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   294
lemma Quotient_All_transfer:
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   295
  "((T ===> op =) ===> op =) (Ball (Respects R)) All"
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   296
  unfolding fun_rel_def Respects_def Quotient_cr_rel [OF 1]
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   297
  by (auto, metis Quotient_abs_induct)
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   298
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   299
lemma Quotient_Ex_transfer:
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   300
  "((T ===> op =) ===> op =) (Bex (Respects R)) Ex"
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   301
  unfolding fun_rel_def Respects_def Quotient_cr_rel [OF 1]
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   302
  by (auto, metis Quotient_rep_reflp [OF 1] Quotient_abs_rep [OF 1])
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   303
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   304
lemma Quotient_forall_transfer:
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   305
  "((T ===> op =) ===> op =) (transfer_bforall (\<lambda>x. R x x)) transfer_forall"
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   306
  using Quotient_All_transfer
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   307
  unfolding transfer_forall_def transfer_bforall_def
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   308
    Ball_def [abs_def] in_respects .
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   309
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   310
end
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   311
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   312
text {* Generating transfer rules for total quotients. *}
47376
776254f89a18 add transfer lemmas for quotients
huffman
parents: 47369
diff changeset
   313
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   314
context
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   315
  fixes R Abs Rep T
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   316
  assumes 1: "Quotient R Abs Rep T" and 2: "reflp R"
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   317
begin
47376
776254f89a18 add transfer lemmas for quotients
huffman
parents: 47369
diff changeset
   318
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   319
lemma Quotient_bi_total: "bi_total T"
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   320
  using 1 2 unfolding Quotient_alt_def bi_total_def reflp_def by auto
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   321
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   322
lemma Quotient_id_abs_transfer: "(op = ===> T) (\<lambda>x. x) Abs"
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   323
  using 1 2 unfolding Quotient_alt_def reflp_def fun_rel_def by simp
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   324
47575
b90cd7016d4f generate abs_induct rules for quotient types
huffman
parents: 47545
diff changeset
   325
lemma Quotient_total_abs_induct: "(\<And>y. P (Abs y)) \<Longrightarrow> P x"
b90cd7016d4f generate abs_induct rules for quotient types
huffman
parents: 47545
diff changeset
   326
  using 1 2 assms unfolding Quotient_alt_def reflp_def by metis
b90cd7016d4f generate abs_induct rules for quotient types
huffman
parents: 47545
diff changeset
   327
47889
29212a4bb866 lifting package produces abs_eq_iff rules for total quotients
huffman
parents: 47777
diff changeset
   328
lemma Quotient_total_abs_eq_iff: "Abs x = Abs y \<longleftrightarrow> R x y"
29212a4bb866 lifting package produces abs_eq_iff rules for total quotients
huffman
parents: 47777
diff changeset
   329
  using Quotient_rel [OF 1] 2 unfolding reflp_def by simp
29212a4bb866 lifting package produces abs_eq_iff rules for total quotients
huffman
parents: 47777
diff changeset
   330
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   331
end
47376
776254f89a18 add transfer lemmas for quotients
huffman
parents: 47369
diff changeset
   332
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   333
text {* Generating transfer rules for a type defined with @{text "typedef"}. *}
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   334
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   335
context
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   336
  fixes Rep Abs A T
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   337
  assumes type: "type_definition Rep Abs A"
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   338
  assumes T_def: "T \<equiv> (\<lambda>(x::'a) (y::'b). x = Rep y)"
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   339
begin
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   340
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   341
lemma typedef_bi_unique: "bi_unique T"
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   342
  unfolding bi_unique_def T_def
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   343
  by (simp add: type_definition.Rep_inject [OF type])
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   344
47535
0f94b02fda1c lifting_setup generates transfer rule for rep of typedefs
huffman
parents: 47534
diff changeset
   345
lemma typedef_rep_transfer: "(T ===> op =) (\<lambda>x. x) Rep"
0f94b02fda1c lifting_setup generates transfer rule for rep of typedefs
huffman
parents: 47534
diff changeset
   346
  unfolding fun_rel_def T_def by simp
0f94b02fda1c lifting_setup generates transfer rule for rep of typedefs
huffman
parents: 47534
diff changeset
   347
47545
a2850a16e30f Lifting: generate more thms & note them & tuned
kuncar
parents: 47544
diff changeset
   348
lemma typedef_All_transfer: "((T ===> op =) ===> op =) (Ball A) All"
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   349
  unfolding T_def fun_rel_def
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   350
  by (metis type_definition.Rep [OF type]
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   351
    type_definition.Abs_inverse [OF type])
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   352
47545
a2850a16e30f Lifting: generate more thms & note them & tuned
kuncar
parents: 47544
diff changeset
   353
lemma typedef_Ex_transfer: "((T ===> op =) ===> op =) (Bex A) Ex"
a2850a16e30f Lifting: generate more thms & note them & tuned
kuncar
parents: 47544
diff changeset
   354
  unfolding T_def fun_rel_def
a2850a16e30f Lifting: generate more thms & note them & tuned
kuncar
parents: 47544
diff changeset
   355
  by (metis type_definition.Rep [OF type]
a2850a16e30f Lifting: generate more thms & note them & tuned
kuncar
parents: 47544
diff changeset
   356
    type_definition.Abs_inverse [OF type])
a2850a16e30f Lifting: generate more thms & note them & tuned
kuncar
parents: 47544
diff changeset
   357
a2850a16e30f Lifting: generate more thms & note them & tuned
kuncar
parents: 47544
diff changeset
   358
lemma typedef_forall_transfer:
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   359
  "((T ===> op =) ===> op =)
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   360
    (transfer_bforall (\<lambda>x. x \<in> A)) transfer_forall"
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   361
  unfolding transfer_bforall_def transfer_forall_def Ball_def [symmetric]
47545
a2850a16e30f Lifting: generate more thms & note them & tuned
kuncar
parents: 47544
diff changeset
   362
  by (rule typedef_All_transfer)
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   363
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   364
end
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   365
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   366
text {* Generating the correspondence rule for a constant defined with
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   367
  @{text "lift_definition"}. *}
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   368
47351
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   369
lemma Quotient_to_transfer:
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   370
  assumes "Quotient R Abs Rep T" and "R c c" and "c' \<equiv> Abs c"
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   371
  shows "T c c'"
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   372
  using assms by (auto dest: Quotient_cr_rel)
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   373
47982
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   374
text {* Proving reflexivity *}
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   375
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   376
definition left_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   377
  where "left_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y)"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   378
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   379
lemma left_totalI:
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   380
  "(\<And>x. \<exists>y. R x y) \<Longrightarrow> left_total R"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   381
unfolding left_total_def by blast
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   382
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   383
lemma left_totalE:
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   384
  assumes "left_total R"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   385
  obtains "(\<And>x. \<exists>y. R x y)"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   386
using assms unfolding left_total_def by blast
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   387
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   388
lemma Quotient_to_left_total:
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   389
  assumes q: "Quotient R Abs Rep T"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   390
  and r_R: "reflp R"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   391
  shows "left_total T"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   392
using r_R Quotient_cr_rel[OF q] unfolding left_total_def by (auto elim: reflpE)
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   393
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   394
lemma reflp_Quotient_composition:
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   395
  assumes lt_R1: "left_total R1"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   396
  and r_R2: "reflp R2"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   397
  shows "reflp (R1 OO R2 OO R1\<inverse>\<inverse>)"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   398
using assms
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   399
proof -
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   400
  {
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   401
    fix x
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   402
    from lt_R1 obtain y where "R1 x y" unfolding left_total_def by blast
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   403
    moreover then have "R1\<inverse>\<inverse> y x" by simp
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   404
    moreover have "R2 y y" using r_R2 by (auto elim: reflpE)
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   405
    ultimately have "(R1 OO R2 OO R1\<inverse>\<inverse>) x x" by auto
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   406
  }
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   407
  then show ?thesis by (auto intro: reflpI)
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   408
qed
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   409
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   410
lemma reflp_equality: "reflp (op =)"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   411
by (auto intro: reflpI)
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   412
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   413
subsection {* ML setup *}
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   414
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   415
ML_file "Tools/Lifting/lifting_util.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   416
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   417
ML_file "Tools/Lifting/lifting_info.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   418
setup Lifting_Info.setup
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   419
47777
f29e7dcd7c40 use a quot_map theorem attribute instead of the complicated map attribute
kuncar
parents: 47698
diff changeset
   420
declare fun_quotient[quot_map]
47982
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   421
lemmas [reflexivity_rule] = reflp_equality reflp_Quotient_composition
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   422
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   423
ML_file "Tools/Lifting/lifting_term.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   424
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   425
ML_file "Tools/Lifting/lifting_def.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   426
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   427
ML_file "Tools/Lifting/lifting_setup.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   428
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   429
hide_const (open) invariant
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   430
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   431
end