src/HOL/Lifting.thy
author wenzelm
Mon, 07 Dec 2015 10:38:04 +0100
changeset 61799 4cf66f21b764
parent 61630 608520e0e8e2
child 63092 a949b2a5f51d
permissions -rw-r--r--
isabelle update_cartouches -c -t;
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
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60229
diff changeset
     6
section \<open>Lifting package\<close>
47308
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
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
    11
  "parametric" and
53219
ca237b9e4542 use only one data slot; rename print_quotmaps to print_quot_maps; tuned
kuncar
parents: 53151
diff changeset
    12
  "print_quot_maps" "print_quotients" :: diag and
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    13
  "lift_definition" :: thy_goal and
53651
ee90c67502c9 restoring Transfer/Lifting context
kuncar
parents: 53219
diff changeset
    14
  "setup_lifting" "lifting_forget" "lifting_update" :: thy_decl
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    15
begin
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    16
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60229
diff changeset
    17
subsection \<open>Function map\<close>
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    18
53011
aeee0a4be6cf introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents: 52307
diff changeset
    19
context
aeee0a4be6cf introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents: 52307
diff changeset
    20
begin
aeee0a4be6cf introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents: 52307
diff changeset
    21
interpretation lifting_syntax .
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    22
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    23
lemma map_fun_id:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    24
  "(id ---> id) = id"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    25
  by (simp add: fun_eq_iff)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    26
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60229
diff changeset
    27
subsection \<open>Quotient Predicate\<close>
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    28
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    29
definition
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    30
  "Quotient R Abs Rep T \<longleftrightarrow>
58186
a6c3962ea907 named interpretations
blanchet
parents: 58177
diff changeset
    31
     (\<forall>a. Abs (Rep a) = a) \<and>
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    32
     (\<forall>a. R (Rep a) (Rep a)) \<and>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    33
     (\<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
    34
     T = (\<lambda>x y. R x x \<and> Abs x = y)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    35
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    36
lemma QuotientI:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    37
  assumes "\<And>a. Abs (Rep a) = a"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    38
    and "\<And>a. R (Rep a) (Rep a)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    39
    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
    40
    and "T = (\<lambda>x y. R x x \<and> Abs x = y)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    41
  shows "Quotient R Abs Rep T"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    42
  using assms unfolding Quotient_def by blast
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    43
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    44
context
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    45
  fixes R Abs Rep T
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    46
  assumes a: "Quotient R Abs Rep T"
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    47
begin
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    48
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    49
lemma Quotient_abs_rep: "Abs (Rep a) = a"
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    50
  using a unfolding Quotient_def
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    51
  by simp
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    52
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    53
lemma Quotient_rep_reflp: "R (Rep a) (Rep a)"
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    54
  using a unfolding Quotient_def
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    55
  by blast
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    56
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    57
lemma Quotient_rel:
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61630
diff changeset
    58
  "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" \<comment> \<open>orientation does not loop on rewriting\<close>
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    59
  using a unfolding Quotient_def
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    60
  by blast
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    61
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    62
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
    63
  using a unfolding Quotient_def
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    64
  by blast
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    65
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    66
lemma Quotient_refl1: "R r s \<Longrightarrow> R r r"
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    67
  using a unfolding Quotient_def
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    68
  by fast
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    69
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    70
lemma Quotient_refl2: "R r s \<Longrightarrow> R s s"
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    71
  using a unfolding Quotient_def
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    72
  by fast
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    73
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    74
lemma Quotient_rel_rep: "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    75
  using a unfolding Quotient_def
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    76
  by metis
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    77
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    78
lemma Quotient_rep_abs: "R r r \<Longrightarrow> R (Rep (Abs r)) r"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    79
  using a unfolding Quotient_def
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    80
  by blast
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
    81
55610
9066b603dff6 refactoring; generate rep_eq always, not only when it would be accepted by the code generator
kuncar
parents: 55604
diff changeset
    82
lemma Quotient_rep_abs_eq: "R t t \<Longrightarrow> R \<le> op= \<Longrightarrow> Rep (Abs t) = t"
9066b603dff6 refactoring; generate rep_eq always, not only when it would be accepted by the code generator
kuncar
parents: 55604
diff changeset
    83
  using a unfolding Quotient_def
9066b603dff6 refactoring; generate rep_eq always, not only when it would be accepted by the code generator
kuncar
parents: 55604
diff changeset
    84
  by blast
9066b603dff6 refactoring; generate rep_eq always, not only when it would be accepted by the code generator
kuncar
parents: 55604
diff changeset
    85
58186
a6c3962ea907 named interpretations
blanchet
parents: 58177
diff changeset
    86
lemma Quotient_rep_abs_fold_unmap:
a6c3962ea907 named interpretations
blanchet
parents: 58177
diff changeset
    87
  assumes "x' \<equiv> Abs x" and "R x x" and "Rep x' \<equiv> Rep' x'"
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
    88
  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
    89
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
    90
  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
    91
  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
    92
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
    93
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
    94
lemma Quotient_Rep_eq:
58186
a6c3962ea907 named interpretations
blanchet
parents: 58177
diff changeset
    95
  assumes "x' \<equiv> Abs x"
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
  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
    97
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
    98
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
    99
lemma Quotient_rel_abs: "R r s \<Longrightarrow> Abs r = Abs s"
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
   100
  using a unfolding Quotient_def
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
   101
  by blast
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
   102
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
   103
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
   104
  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
   105
  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
   106
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
   107
  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
   108
  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
   109
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
   110
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
   111
lemma Quotient_symp: "symp R"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   112
  using a unfolding Quotient_def using sympI by (metis (full_types))
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   113
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
   114
lemma Quotient_transp: "transp R"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   115
  using a unfolding Quotient_def using transpI by (metis (full_types))
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   116
47536
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
   117
lemma Quotient_part_equivp: "part_equivp R"
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
   118
by (metis Quotient_rep_reflp Quotient_symp Quotient_transp part_equivpI)
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
   119
8474a865a4e5 use context block
huffman
parents: 47535
diff changeset
   120
end
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   121
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   122
lemma identity_quotient: "Quotient (op =) id id (op =)"
58186
a6c3962ea907 named interpretations
blanchet
parents: 58177
diff changeset
   123
unfolding Quotient_def by simp
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   124
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60229
diff changeset
   125
text \<open>TODO: Use one of these alternatives as the real definition.\<close>
47652
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   126
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   127
lemma Quotient_alt_def:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   128
  "Quotient R Abs Rep T \<longleftrightarrow>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   129
    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   130
    (\<forall>b. T (Rep b) b) \<and>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   131
    (\<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
   132
apply safe
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   133
apply (simp (no_asm_use) only: Quotient_def, fast)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   134
apply (simp (no_asm_use) only: Quotient_def, fast)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   135
apply (simp (no_asm_use) only: Quotient_def, fast)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   136
apply (simp (no_asm_use) only: Quotient_def, fast)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   137
apply (simp (no_asm_use) only: Quotient_def, fast)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   138
apply (simp (no_asm_use) only: Quotient_def, fast)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   139
apply (rule QuotientI)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   140
apply simp
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   141
apply metis
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   142
apply simp
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   143
apply (rule ext, rule ext, metis)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   144
done
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   145
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   146
lemma Quotient_alt_def2:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   147
  "Quotient R Abs Rep T \<longleftrightarrow>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   148
    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   149
    (\<forall>b. T (Rep b) b) \<and>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   150
    (\<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
   151
  unfolding Quotient_alt_def by (safe, metis+)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   152
47652
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   153
lemma Quotient_alt_def3:
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   154
  "Quotient R Abs Rep T \<longleftrightarrow>
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   155
    (\<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
   156
    (\<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
   157
  unfolding Quotient_alt_def2 by (safe, metis+)
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   158
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   159
lemma Quotient_alt_def4:
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   160
  "Quotient R Abs Rep T \<longleftrightarrow>
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   161
    (\<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
   162
  unfolding Quotient_alt_def3 fun_eq_iff by auto
1b722b100301 move alternative definition lemmas into Lifting.thy;
huffman
parents: 47651
diff changeset
   163
56524
f4ba736040fa setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents: 56519
diff changeset
   164
lemma Quotient_alt_def5:
f4ba736040fa setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents: 56519
diff changeset
   165
  "Quotient R Abs Rep T \<longleftrightarrow>
57398
882091eb1e9a merged two small theory files
blanchet
parents: 56524
diff changeset
   166
    T \<le> BNF_Def.Grp UNIV Abs \<and> BNF_Def.Grp UNIV Rep \<le> T\<inverse>\<inverse> \<and> R = T OO T\<inverse>\<inverse>"
56524
f4ba736040fa setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents: 56519
diff changeset
   167
  unfolding Quotient_alt_def4 Grp_def by blast
f4ba736040fa setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents: 56519
diff changeset
   168
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   169
lemma fun_quotient:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   170
  assumes 1: "Quotient R1 abs1 rep1 T1"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   171
  assumes 2: "Quotient R2 abs2 rep2 T2"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   172
  shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2) (T1 ===> T2)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   173
  using assms unfolding Quotient_alt_def2
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   174
  unfolding rel_fun_def fun_eq_iff map_fun_apply
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   175
  by (safe, metis+)
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
  fixes f g::"'a \<Rightarrow> 'c"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   179
  assumes q: "Quotient R1 Abs1 Rep1 T1"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   180
  and     a: "(R1 ===> R2) f g" "R1 x y"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   181
  shows "R2 (f x) (g y)"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   182
  using a by (auto elim: rel_funE)
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   183
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   184
lemma apply_rsp':
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   185
  assumes a: "(R1 ===> R2) f g" "R1 x y"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   186
  shows "R2 (f x) (g y)"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   187
  using a by (auto elim: rel_funE)
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   188
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   189
lemma apply_rsp'':
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   190
  assumes "Quotient R Abs Rep T"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   191
  and "(R ===> S) f f"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   192
  shows "S (f (Rep x)) (f (Rep x))"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   193
proof -
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   194
  from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   195
  then show ?thesis using assms(2) by (auto intro: apply_rsp')
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   196
qed
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   197
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60229
diff changeset
   198
subsection \<open>Quotient composition\<close>
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   199
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   200
lemma Quotient_compose:
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   201
  assumes 1: "Quotient R1 Abs1 Rep1 T1"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   202
  assumes 2: "Quotient R2 Abs2 Rep2 T2"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   203
  shows "Quotient (T1 OO R2 OO conversep T1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2) (T1 OO T2)"
51994
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   204
  using assms unfolding Quotient_alt_def4 by fastforce
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   205
47521
69f95ac85c3d tuned the setup of lifting; generate transfer rules for typedef and Quotient thms
kuncar
parents: 47501
diff changeset
   206
lemma equivp_reflp2:
69f95ac85c3d tuned the setup of lifting; generate transfer rules for typedef and Quotient thms
kuncar
parents: 47501
diff changeset
   207
  "equivp R \<Longrightarrow> reflp R"
69f95ac85c3d tuned the setup of lifting; generate transfer rules for typedef and Quotient thms
kuncar
parents: 47501
diff changeset
   208
  by (erule equivpE)
69f95ac85c3d tuned the setup of lifting; generate transfer rules for typedef and Quotient thms
kuncar
parents: 47501
diff changeset
   209
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60229
diff changeset
   210
subsection \<open>Respects predicate\<close>
47544
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   211
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   212
definition Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   213
  where "Respects R = {x. R x x}"
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   214
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   215
lemma in_respects: "x \<in> Respects R \<longleftrightarrow> R x x"
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   216
  unfolding Respects_def by simp
e455cdaac479 move constant 'Respects' into Lifting.thy;
huffman
parents: 47538
diff changeset
   217
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   218
lemma UNIV_typedef_to_Quotient:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   219
  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
   220
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   221
  shows "Quotient (op =) Abs Rep T"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   222
proof -
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   223
  interpret type_definition Rep Abs UNIV by fact
58186
a6c3962ea907 named interpretations
blanchet
parents: 58177
diff changeset
   224
  from Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   225
    by (fastforce intro!: QuotientI fun_eq_iff)
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   226
qed
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   227
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   228
lemma UNIV_typedef_to_equivp:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   229
  fixes Abs :: "'a \<Rightarrow> 'b"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   230
  and Rep :: "'b \<Rightarrow> 'a"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   231
  assumes "type_definition Rep Abs (UNIV::'a set)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   232
  shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   233
by (rule identity_equivp)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   234
47354
95846613e414 update keywords file
huffman
parents: 47351
diff changeset
   235
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
   236
  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
   237
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   238
  shows "Quotient (eq_onp (\<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
   239
proof -
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   240
  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
   241
  from Rep Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   242
    by (auto intro!: QuotientI simp: eq_onp_def fun_eq_iff)
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   243
qed
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   244
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   245
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
   246
  assumes "type_definition Rep Abs S"
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   247
  shows "part_equivp (eq_onp (\<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
   248
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
   249
  interpret type_definition Rep Abs S by fact
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   250
  show "\<exists>x. eq_onp (\<lambda>x. x \<in> S) x x" using Rep by (auto simp: eq_onp_def)
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   251
next
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   252
  show "symp (eq_onp (\<lambda>x. x \<in> S))" by (auto intro: sympI simp: eq_onp_def)
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   253
next
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   254
  show "transp (eq_onp (\<lambda>x. x \<in> S))" by (auto intro: transpI simp: eq_onp_def)
47361
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   255
qed
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   256
87c0eaf04bad support non-open typedefs; define cr_rel in terms of a rep function for typedefs
kuncar
parents: 47354
diff changeset
   257
lemma open_typedef_to_Quotient:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   258
  assumes "type_definition Rep Abs {x. P x}"
47354
95846613e414 update keywords file
huffman
parents: 47351
diff changeset
   259
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   260
  shows "Quotient (eq_onp P) Abs Rep T"
47651
8e4f50afd21a tuned proofs
huffman
parents: 47575
diff changeset
   261
  using typedef_to_Quotient [OF assms] by simp
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   262
47361
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_part_equivp:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   264
  assumes "type_definition Rep Abs {x. P x}"
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   265
  shows "part_equivp (eq_onp P)"
47651
8e4f50afd21a tuned proofs
huffman
parents: 47575
diff changeset
   266
  using typedef_to_part_equivp [OF assms] by simp
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   267
60229
4cd6462c1fda Workaround that allows us to execute lifted constants that have as a return type a datatype containing a subtype
kuncar
parents: 59726
diff changeset
   268
lemma type_definition_Quotient_not_empty: "Quotient (eq_onp P) Abs Rep T \<Longrightarrow> \<exists>x. P x"
4cd6462c1fda Workaround that allows us to execute lifted constants that have as a return type a datatype containing a subtype
kuncar
parents: 59726
diff changeset
   269
unfolding eq_onp_def by (drule Quotient_rep_reflp) blast
4cd6462c1fda Workaround that allows us to execute lifted constants that have as a return type a datatype containing a subtype
kuncar
parents: 59726
diff changeset
   270
4cd6462c1fda Workaround that allows us to execute lifted constants that have as a return type a datatype containing a subtype
kuncar
parents: 59726
diff changeset
   271
lemma type_definition_Quotient_not_empty_witness: "Quotient (eq_onp P) Abs Rep T \<Longrightarrow> P (Rep undefined)"
4cd6462c1fda Workaround that allows us to execute lifted constants that have as a return type a datatype containing a subtype
kuncar
parents: 59726
diff changeset
   272
unfolding eq_onp_def by (drule Quotient_rep_reflp) blast
4cd6462c1fda Workaround that allows us to execute lifted constants that have as a return type a datatype containing a subtype
kuncar
parents: 59726
diff changeset
   273
4cd6462c1fda Workaround that allows us to execute lifted constants that have as a return type a datatype containing a subtype
kuncar
parents: 59726
diff changeset
   274
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60229
diff changeset
   275
text \<open>Generating transfer rules for quotients.\<close>
47376
776254f89a18 add transfer lemmas for quotients
huffman
parents: 47369
diff changeset
   276
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   277
context
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   278
  fixes R Abs Rep T
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   279
  assumes 1: "Quotient R Abs Rep T"
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   280
begin
47376
776254f89a18 add transfer lemmas for quotients
huffman
parents: 47369
diff changeset
   281
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   282
lemma Quotient_right_unique: "right_unique T"
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   283
  using 1 unfolding Quotient_alt_def right_unique_def by metis
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   284
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   285
lemma Quotient_right_total: "right_total T"
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   286
  using 1 unfolding Quotient_alt_def right_total_def by metis
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   287
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   288
lemma Quotient_rel_eq_transfer: "(T ===> T ===> op =) R (op =)"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   289
  using 1 unfolding Quotient_alt_def rel_fun_def by simp
47376
776254f89a18 add transfer lemmas for quotients
huffman
parents: 47369
diff changeset
   290
47538
1f0ec5b8135a add lemma Quotient_abs_induct
huffman
parents: 47537
diff changeset
   291
lemma Quotient_abs_induct:
1f0ec5b8135a add lemma Quotient_abs_induct
huffman
parents: 47537
diff changeset
   292
  assumes "\<And>y. R y y \<Longrightarrow> P (Abs y)" shows "P x"
1f0ec5b8135a add lemma Quotient_abs_induct
huffman
parents: 47537
diff changeset
   293
  using 1 assms unfolding Quotient_def by metis
1f0ec5b8135a add lemma Quotient_abs_induct
huffman
parents: 47537
diff changeset
   294
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   295
end
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   296
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60229
diff changeset
   297
text \<open>Generating transfer rules for total quotients.\<close>
47376
776254f89a18 add transfer lemmas for quotients
huffman
parents: 47369
diff changeset
   298
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   299
context
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   300
  fixes R Abs Rep T
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   301
  assumes 1: "Quotient R Abs Rep T" and 2: "reflp R"
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   302
begin
47376
776254f89a18 add transfer lemmas for quotients
huffman
parents: 47369
diff changeset
   303
56518
beb3b6851665 left_total and left_unique rules are now transfer rules (cleaner solution, reflexvity_rule attribute not needed anymore)
kuncar
parents: 56517
diff changeset
   304
lemma Quotient_left_total: "left_total T"
beb3b6851665 left_total and left_unique rules are now transfer rules (cleaner solution, reflexvity_rule attribute not needed anymore)
kuncar
parents: 56517
diff changeset
   305
  using 1 2 unfolding Quotient_alt_def left_total_def reflp_def by auto
beb3b6851665 left_total and left_unique rules are now transfer rules (cleaner solution, reflexvity_rule attribute not needed anymore)
kuncar
parents: 56517
diff changeset
   306
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   307
lemma Quotient_bi_total: "bi_total T"
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   308
  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
   309
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   310
lemma Quotient_id_abs_transfer: "(op = ===> T) (\<lambda>x. x) Abs"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   311
  using 1 2 unfolding Quotient_alt_def reflp_def rel_fun_def by simp
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   312
47575
b90cd7016d4f generate abs_induct rules for quotient types
huffman
parents: 47545
diff changeset
   313
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
   314
  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
   315
47889
29212a4bb866 lifting package produces abs_eq_iff rules for total quotients
huffman
parents: 47777
diff changeset
   316
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
   317
  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
   318
47537
b06be48923a4 more usage of context blocks
huffman
parents: 47536
diff changeset
   319
end
47376
776254f89a18 add transfer lemmas for quotients
huffman
parents: 47369
diff changeset
   320
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61630
diff changeset
   321
text \<open>Generating transfer rules for a type defined with \<open>typedef\<close>.\<close>
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   322
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   323
context
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   324
  fixes Rep Abs A T
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   325
  assumes type: "type_definition Rep Abs A"
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   326
  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
   327
begin
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   328
51994
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   329
lemma typedef_left_unique: "left_unique T"
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   330
  unfolding left_unique_def T_def
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   331
  by (simp add: type_definition.Rep_inject [OF type])
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   332
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   333
lemma typedef_bi_unique: "bi_unique T"
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   334
  unfolding bi_unique_def T_def
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   335
  by (simp add: type_definition.Rep_inject [OF type])
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   336
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   337
(* the following two theorems are here only for convinience *)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   338
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   339
lemma typedef_right_unique: "right_unique T"
58186
a6c3962ea907 named interpretations
blanchet
parents: 58177
diff changeset
   340
  using T_def type Quotient_right_unique typedef_to_Quotient
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   341
  by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   342
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   343
lemma typedef_right_total: "right_total T"
58186
a6c3962ea907 named interpretations
blanchet
parents: 58177
diff changeset
   344
  using T_def type Quotient_right_total typedef_to_Quotient
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   345
  by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   346
47535
0f94b02fda1c lifting_setup generates transfer rule for rep of typedefs
huffman
parents: 47534
diff changeset
   347
lemma typedef_rep_transfer: "(T ===> op =) (\<lambda>x. x) Rep"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   348
  unfolding rel_fun_def T_def by simp
47535
0f94b02fda1c lifting_setup generates transfer rule for rep of typedefs
huffman
parents: 47534
diff changeset
   349
47534
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   350
end
06cc372a80ed use context block to organize typedef lifting theorems
huffman
parents: 47521
diff changeset
   351
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60229
diff changeset
   352
text \<open>Generating the correspondence rule for a constant defined with
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61630
diff changeset
   353
  \<open>lift_definition\<close>.\<close>
47368
4c522dff1f4d add lemmas for generating transfer rules for typedefs
huffman
parents: 47354
diff changeset
   354
47351
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   355
lemma Quotient_to_transfer:
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   356
  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
   357
  shows "T c c'"
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   358
  using assms by (auto dest: Quotient_cr_rel)
0193e663a19e lift_definition command generates transfer rule
huffman
parents: 47325
diff changeset
   359
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60229
diff changeset
   360
text \<open>Proving reflexivity\<close>
47982
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   361
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   362
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
   363
  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
   364
  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
   365
  shows "left_total T"
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   366
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
   367
55563
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   368
lemma Quotient_composition_ge_eq:
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   369
  assumes "left_total T"
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   370
  assumes "R \<ge> op="
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   371
  shows "(T OO R OO T\<inverse>\<inverse>) \<ge> op="
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   372
using assms unfolding left_total_def by fast
51994
82cc2aeb7d13 stronger reflexivity prover
kuncar
parents: 51956
diff changeset
   373
55563
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   374
lemma Quotient_composition_le_eq:
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   375
  assumes "left_unique T"
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   376
  assumes "R \<le> op="
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   377
  shows "(T OO R OO T\<inverse>\<inverse>) \<le> op="
55604
42e4e8c2e8dc less flex-flex pairs
noschinl
parents: 55563
diff changeset
   378
using assms unfolding left_unique_def by blast
47982
7aa35601ff65 prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents: 47937
diff changeset
   379
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   380
lemma eq_onp_le_eq:
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   381
  "eq_onp P \<le> op=" unfolding eq_onp_def by blast
55563
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   382
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   383
lemma reflp_ge_eq:
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   384
  "reflp R \<Longrightarrow> R \<ge> op=" unfolding reflp_def by blast
a64d49f49ca3 implement the reflexivity prover as a monotonicity prover that proves R >= op=; derive "reflexivity" rules for relators from mono rules and eq rules
kuncar
parents: 55083
diff changeset
   385
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60229
diff changeset
   386
text \<open>Proving a parametrized correspondence relation\<close>
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   387
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   388
definition POS :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   389
"POS A B \<equiv> A \<le> B"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   390
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   391
definition  NEG :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   392
"NEG A B \<equiv> B \<le> A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   393
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   394
lemma pos_OO_eq:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   395
  shows "POS (A OO op=) A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   396
unfolding POS_def OO_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   397
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   398
lemma pos_eq_OO:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   399
  shows "POS (op= OO A) A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   400
unfolding POS_def OO_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   401
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   402
lemma neg_OO_eq:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   403
  shows "NEG (A OO op=) A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   404
unfolding NEG_def OO_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   405
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   406
lemma neg_eq_OO:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   407
  shows "NEG (op= OO A) A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   408
unfolding NEG_def OO_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   409
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   410
lemma POS_trans:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   411
  assumes "POS A B"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   412
  assumes "POS B C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   413
  shows "POS A C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   414
using assms unfolding POS_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   415
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   416
lemma NEG_trans:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   417
  assumes "NEG A B"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   418
  assumes "NEG B C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   419
  shows "NEG A C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   420
using assms unfolding NEG_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   421
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   422
lemma POS_NEG:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   423
  "POS A B \<equiv> NEG B A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   424
  unfolding POS_def NEG_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   425
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   426
lemma NEG_POS:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   427
  "NEG A B \<equiv> POS B A"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   428
  unfolding POS_def NEG_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   429
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   430
lemma POS_pcr_rule:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   431
  assumes "POS (A OO B) C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   432
  shows "POS (A OO B OO X) (C OO X)"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   433
using assms unfolding POS_def OO_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   434
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   435
lemma NEG_pcr_rule:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   436
  assumes "NEG (A OO B) C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   437
  shows "NEG (A OO B OO X) (C OO X)"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   438
using assms unfolding NEG_def OO_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   439
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   440
lemma POS_apply:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   441
  assumes "POS R R'"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   442
  assumes "R f g"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   443
  shows "R' f g"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   444
using assms unfolding POS_def by auto
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   445
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60229
diff changeset
   446
text \<open>Proving a parametrized correspondence relation\<close>
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   447
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   448
lemma fun_mono:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   449
  assumes "A \<ge> C"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   450
  assumes "B \<le> D"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   451
  shows   "(A ===> B) \<le> (C ===> D)"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   452
using assms unfolding rel_fun_def by blast
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   453
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   454
lemma pos_fun_distr: "((R ===> S) OO (R' ===> S')) \<le> ((R OO R') ===> (S OO S'))"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   455
unfolding OO_def rel_fun_def by blast
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   456
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   457
lemma functional_relation: "right_unique R \<Longrightarrow> left_total R \<Longrightarrow> \<forall>x. \<exists>!y. R x y"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   458
unfolding right_unique_def left_total_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   459
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   460
lemma functional_converse_relation: "left_unique R \<Longrightarrow> right_total R \<Longrightarrow> \<forall>y. \<exists>!x. R x y"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   461
unfolding left_unique_def right_total_def by blast
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   462
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   463
lemma neg_fun_distr1:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   464
assumes 1: "left_unique R" "right_total R"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   465
assumes 2: "right_unique R'" "left_total R'"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   466
shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S')) "
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   467
  using functional_relation[OF 2] functional_converse_relation[OF 1]
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   468
  unfolding rel_fun_def OO_def
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   469
  apply clarify
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   470
  apply (subst all_comm)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   471
  apply (subst all_conj_distrib[symmetric])
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   472
  apply (intro choice)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   473
  by metis
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   474
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   475
lemma neg_fun_distr2:
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   476
assumes 1: "right_unique R'" "left_total R'"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   477
assumes 2: "left_unique S'" "right_total S'"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   478
shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S'))"
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   479
  using functional_converse_relation[OF 2] functional_relation[OF 1]
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   480
  unfolding rel_fun_def OO_def
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   481
  apply clarify
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   482
  apply (subst all_comm)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   483
  apply (subst all_conj_distrib[symmetric])
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   484
  apply (intro choice)
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   485
  by metis
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   486
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60229
diff changeset
   487
subsection \<open>Domains\<close>
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   488
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   489
lemma composed_equiv_rel_eq_onp:
55731
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   490
  assumes "left_unique R"
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   491
  assumes "(R ===> op=) P P'"
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   492
  assumes "Domainp R = P''"
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   493
  shows "(R OO eq_onp P' OO R\<inverse>\<inverse>) = eq_onp (inf P'' P)"
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   494
using assms unfolding OO_def conversep_iff Domainp_iff[abs_def] left_unique_def rel_fun_def eq_onp_def
55731
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   495
fun_eq_iff by blast
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   496
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   497
lemma composed_equiv_rel_eq_eq_onp:
55731
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   498
  assumes "left_unique R"
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   499
  assumes "Domainp R = P"
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   500
  shows "(R OO op= OO R\<inverse>\<inverse>) = eq_onp P"
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   501
using assms unfolding OO_def conversep_iff Domainp_iff[abs_def] left_unique_def eq_onp_def
55731
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   502
fun_eq_iff is_equality_def by metis
66df76dd2640 rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents: 55610
diff changeset
   503
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   504
lemma pcr_Domainp_par_left_total:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   505
  assumes "Domainp B = P"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   506
  assumes "left_total A"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   507
  assumes "(A ===> op=) P' P"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   508
  shows "Domainp (A OO B) = P'"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   509
using assms
58186
a6c3962ea907 named interpretations
blanchet
parents: 58177
diff changeset
   510
unfolding Domainp_iff[abs_def] OO_def bi_unique_def left_total_def rel_fun_def
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   511
by (fast intro: fun_eq_iff)
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   512
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   513
lemma pcr_Domainp_par:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   514
assumes "Domainp B = P2"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   515
assumes "Domainp A = P1"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   516
assumes "(A ===> op=) P2' P2"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   517
shows "Domainp (A OO B) = (inf P1 P2')"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55737
diff changeset
   518
using assms unfolding rel_fun_def Domainp_iff[abs_def] OO_def
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   519
by (fast intro: fun_eq_iff)
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   520
53151
fbf4d50dec91 remove OP
kuncar
parents: 53011
diff changeset
   521
definition rel_pred_comp :: "('a => 'b => bool) => ('b => bool) => 'a => bool"
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   522
where "rel_pred_comp R P \<equiv> \<lambda>x. \<exists>y. R x y \<and> P y"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   523
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   524
lemma pcr_Domainp:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   525
assumes "Domainp B = P"
53151
fbf4d50dec91 remove OP
kuncar
parents: 53011
diff changeset
   526
shows "Domainp (A OO B) = (\<lambda>x. \<exists>y. A x y \<and> P y)"
fbf4d50dec91 remove OP
kuncar
parents: 53011
diff changeset
   527
using assms by blast
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   528
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   529
lemma pcr_Domainp_total:
56518
beb3b6851665 left_total and left_unique rules are now transfer rules (cleaner solution, reflexvity_rule attribute not needed anymore)
kuncar
parents: 56517
diff changeset
   530
  assumes "left_total B"
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   531
  assumes "Domainp A = P"
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   532
  shows "Domainp (A OO B) = P"
58186
a6c3962ea907 named interpretations
blanchet
parents: 58177
diff changeset
   533
using assms unfolding left_total_def
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   534
by fast
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   535
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   536
lemma Quotient_to_Domainp:
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   537
  assumes "Quotient R Abs Rep T"
58186
a6c3962ea907 named interpretations
blanchet
parents: 58177
diff changeset
   538
  shows "Domainp T = (\<lambda>x. R x x)"
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   539
by (simp add: Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   540
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   541
lemma eq_onp_to_Domainp:
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   542
  assumes "Quotient (eq_onp P) Abs Rep T"
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   543
  shows "Domainp T = P"
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   544
by (simp add: eq_onp_def Domainp_iff[abs_def] Quotient_cr_rel[OF assms])
51956
a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents: 51374
diff changeset
   545
53011
aeee0a4be6cf introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents: 52307
diff changeset
   546
end
aeee0a4be6cf introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents: 52307
diff changeset
   547
60229
4cd6462c1fda Workaround that allows us to execute lifted constants that have as a return type a datatype containing a subtype
kuncar
parents: 59726
diff changeset
   548
(* needed for lifting_def_code_dt.ML (moved from Lifting_Set) *)
4cd6462c1fda Workaround that allows us to execute lifted constants that have as a return type a datatype containing a subtype
kuncar
parents: 59726
diff changeset
   549
lemma right_total_UNIV_transfer: 
4cd6462c1fda Workaround that allows us to execute lifted constants that have as a return type a datatype containing a subtype
kuncar
parents: 59726
diff changeset
   550
  assumes "right_total A"
4cd6462c1fda Workaround that allows us to execute lifted constants that have as a return type a datatype containing a subtype
kuncar
parents: 59726
diff changeset
   551
  shows "(rel_set A) (Collect (Domainp A)) UNIV"
4cd6462c1fda Workaround that allows us to execute lifted constants that have as a return type a datatype containing a subtype
kuncar
parents: 59726
diff changeset
   552
  using assms unfolding right_total_def rel_set_def Domainp_iff by blast
4cd6462c1fda Workaround that allows us to execute lifted constants that have as a return type a datatype containing a subtype
kuncar
parents: 59726
diff changeset
   553
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60229
diff changeset
   554
subsection \<open>ML setup\<close>
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   555
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   556
ML_file "Tools/Lifting/lifting_util.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   557
57961
10b2f60b70f0 updated to named_theorems;
wenzelm
parents: 57398
diff changeset
   558
named_theorems relator_eq_onp
10b2f60b70f0 updated to named_theorems;
wenzelm
parents: 57398
diff changeset
   559
  "theorems that a relator of an eq_onp is an eq_onp of the corresponding predicate"
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   560
ML_file "Tools/Lifting/lifting_info.ML"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   561
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   562
(* setup for the function type *)
47777
f29e7dcd7c40 use a quot_map theorem attribute instead of the complicated map attribute
kuncar
parents: 47698
diff changeset
   563
declare fun_quotient[quot_map]
51374
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   564
declare fun_mono[relator_mono]
84d01fd733cf lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents: 51112
diff changeset
   565
lemmas [relator_distr] = pos_fun_distr neg_fun_distr1 neg_fun_distr2
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   566
56524
f4ba736040fa setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents: 56519
diff changeset
   567
ML_file "Tools/Lifting/lifting_bnf.ML"
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   568
ML_file "Tools/Lifting/lifting_term.ML"
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   569
ML_file "Tools/Lifting/lifting_def.ML"
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47982
diff changeset
   570
ML_file "Tools/Lifting/lifting_setup.ML"
60229
4cd6462c1fda Workaround that allows us to execute lifted constants that have as a return type a datatype containing a subtype
kuncar
parents: 59726
diff changeset
   571
ML_file "Tools/Lifting/lifting_def_code_dt.ML"
58186
a6c3962ea907 named interpretations
blanchet
parents: 58177
diff changeset
   572
61630
608520e0e8e2 add various lemmas
Andreas Lochbihler
parents: 60758
diff changeset
   573
lemma pred_prod_beta: "pred_prod P Q xy \<longleftrightarrow> P (fst xy) \<and> Q (snd xy)"
608520e0e8e2 add various lemmas
Andreas Lochbihler
parents: 60758
diff changeset
   574
by(cases xy) simp
608520e0e8e2 add various lemmas
Andreas Lochbihler
parents: 60758
diff changeset
   575
608520e0e8e2 add various lemmas
Andreas Lochbihler
parents: 60758
diff changeset
   576
lemma pred_prod_split: "P (pred_prod Q R xy) \<longleftrightarrow> (\<forall>x y. xy = (x, y) \<longrightarrow> P (Q x \<and> R y))"
608520e0e8e2 add various lemmas
Andreas Lochbihler
parents: 60758
diff changeset
   577
by(cases xy) simp
608520e0e8e2 add various lemmas
Andreas Lochbihler
parents: 60758
diff changeset
   578
56519
c1048f5bbb45 more appropriate name (Lifting.invariant -> eq_onp)
kuncar
parents: 56518
diff changeset
   579
hide_const (open) POS NEG
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   580
9caab698dbe4 new package Lifting - initial commit
kuncar
parents:
diff changeset
   581
end