src/HOL/Relation.thy
author blanchet
Mon, 20 Jan 2014 20:21:12 +0100
changeset 55083 0a689157e3ce
parent 54611 31afce809794
child 55096 916b2ac758f4
permissions -rw-r--r--
move BNF_LFP up the dependency chain
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
10358
ef2a753cda2a converse: syntax \<inverse>;
wenzelm
parents: 10212
diff changeset
     1
(*  Title:      HOL/Relation.thy
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
     2
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory; Stefan Berghofer, TU Muenchen
1128
64b30e3cc6d4 Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff changeset
     3
*)
64b30e3cc6d4 Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff changeset
     4
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
     5
header {* Relations – as sets of pairs, and binary predicates *}
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
     6
15131
c69542757a4d New theory header syntax.
nipkow
parents: 13830
diff changeset
     7
theory Relation
54555
e8c5e95d338b rationalize imports
blanchet
parents: 54410
diff changeset
     8
imports Finite_Set
15131
c69542757a4d New theory header syntax.
nipkow
parents: 13830
diff changeset
     9
begin
5978
fa2c2dd74f8c moved diag (diagonal relation) from Univ to Relation
paulson
parents: 5608
diff changeset
    10
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    11
text {* A preliminary: classical rules for reasoning on predicates *}
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    12
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46833
diff changeset
    13
declare predicate1I [Pure.intro!, intro!]
6242b4bc05bc tuned simpset
noschinl
parents: 46833
diff changeset
    14
declare predicate1D [Pure.dest, dest]
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    15
declare predicate2I [Pure.intro!, intro!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    16
declare predicate2D [Pure.dest, dest]
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
    17
declare bot1E [elim!] 
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    18
declare bot2E [elim!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    19
declare top1I [intro!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    20
declare top2I [intro!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    21
declare inf1I [intro!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    22
declare inf2I [intro!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    23
declare inf1E [elim!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    24
declare inf2E [elim!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    25
declare sup1I1 [intro?]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    26
declare sup2I1 [intro?]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    27
declare sup1I2 [intro?]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    28
declare sup2I2 [intro?]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    29
declare sup1E [elim!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    30
declare sup2E [elim!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    31
declare sup1CI [intro!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    32
declare sup2CI [intro!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    33
declare INF1_I [intro!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    34
declare INF2_I [intro!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    35
declare INF1_D [elim]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    36
declare INF2_D [elim]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    37
declare INF1_E [elim]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    38
declare INF2_E [elim]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    39
declare SUP1_I [intro]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    40
declare SUP2_I [intro]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    41
declare SUP1_E [elim!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    42
declare SUP2_E [elim!]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    43
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    44
subsection {* Fundamental *}
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    45
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    46
subsubsection {* Relations as sets of pairs *}
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    47
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    48
type_synonym 'a rel = "('a * 'a) set"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    49
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    50
lemma subrelI: -- {* Version of @{thm [source] subsetI} for binary relations *}
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    51
  "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s) \<Longrightarrow> r \<subseteq> s"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    52
  by auto
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    53
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    54
lemma lfp_induct2: -- {* Version of @{thm [source] lfp_induct} for binary relations *}
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    55
  "(a, b) \<in> lfp f \<Longrightarrow> mono f \<Longrightarrow>
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    56
    (\<And>a b. (a, b) \<in> f (lfp f \<inter> {(x, y). P x y}) \<Longrightarrow> P a b) \<Longrightarrow> P a b"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    57
  using lfp_induct_set [of "(a, b)" f "prod_case P"] by auto
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    58
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    59
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
    60
subsubsection {* Conversions between set and predicate relations *}
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    61
46833
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
    62
lemma pred_equals_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S) \<longleftrightarrow> R = S"
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    63
  by (simp add: set_eq_iff fun_eq_iff)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    64
46833
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
    65
lemma pred_equals_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S) \<longleftrightarrow> R = S"
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    66
  by (simp add: set_eq_iff fun_eq_iff)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    67
46833
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
    68
lemma pred_subset_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S) \<longleftrightarrow> R \<subseteq> S"
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    69
  by (simp add: subset_iff le_fun_def)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    70
46833
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
    71
lemma pred_subset_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S) \<longleftrightarrow> R \<subseteq> S"
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    72
  by (simp add: subset_iff le_fun_def)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    73
46883
eec472dae593 tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents: 46882
diff changeset
    74
lemma bot_empty_eq [pred_set_conv]: "\<bottom> = (\<lambda>x. x \<in> {})"
46689
f559866a7aa2 marked candidates for rule declarations
haftmann
parents: 46664
diff changeset
    75
  by (auto simp add: fun_eq_iff)
f559866a7aa2 marked candidates for rule declarations
haftmann
parents: 46664
diff changeset
    76
46883
eec472dae593 tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents: 46882
diff changeset
    77
lemma bot_empty_eq2 [pred_set_conv]: "\<bottom> = (\<lambda>x y. (x, y) \<in> {})"
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    78
  by (auto simp add: fun_eq_iff)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    79
46883
eec472dae593 tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents: 46882
diff changeset
    80
lemma top_empty_eq [pred_set_conv]: "\<top> = (\<lambda>x. x \<in> UNIV)"
eec472dae593 tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents: 46882
diff changeset
    81
  by (auto simp add: fun_eq_iff)
46689
f559866a7aa2 marked candidates for rule declarations
haftmann
parents: 46664
diff changeset
    82
46883
eec472dae593 tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents: 46882
diff changeset
    83
lemma top_empty_eq2 [pred_set_conv]: "\<top> = (\<lambda>x y. (x, y) \<in> UNIV)"
eec472dae593 tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents: 46882
diff changeset
    84
  by (auto simp add: fun_eq_iff)
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    85
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    86
lemma inf_Int_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<sqinter> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    87
  by (simp add: inf_fun_def)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    88
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    89
lemma inf_Int_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<sqinter> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    90
  by (simp add: inf_fun_def)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    91
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    92
lemma sup_Un_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<squnion> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    93
  by (simp add: sup_fun_def)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    94
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    95
lemma sup_Un_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<squnion> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    96
  by (simp add: sup_fun_def)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
    97
46981
d54cea5b64e4 generalized INF_INT_eq, SUP_UN_eq
haftmann
parents: 46884
diff changeset
    98
lemma INF_INT_eq [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Inter>i\<in>S. r i))"
d54cea5b64e4 generalized INF_INT_eq, SUP_UN_eq
haftmann
parents: 46884
diff changeset
    99
  by (simp add: fun_eq_iff)
d54cea5b64e4 generalized INF_INT_eq, SUP_UN_eq
haftmann
parents: 46884
diff changeset
   100
d54cea5b64e4 generalized INF_INT_eq, SUP_UN_eq
haftmann
parents: 46884
diff changeset
   101
lemma INF_INT_eq2 [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Inter>i\<in>S. r i))"
d54cea5b64e4 generalized INF_INT_eq, SUP_UN_eq
haftmann
parents: 46884
diff changeset
   102
  by (simp add: fun_eq_iff)
d54cea5b64e4 generalized INF_INT_eq, SUP_UN_eq
haftmann
parents: 46884
diff changeset
   103
d54cea5b64e4 generalized INF_INT_eq, SUP_UN_eq
haftmann
parents: 46884
diff changeset
   104
lemma SUP_UN_eq [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Union>i\<in>S. r i))"
d54cea5b64e4 generalized INF_INT_eq, SUP_UN_eq
haftmann
parents: 46884
diff changeset
   105
  by (simp add: fun_eq_iff)
d54cea5b64e4 generalized INF_INT_eq, SUP_UN_eq
haftmann
parents: 46884
diff changeset
   106
d54cea5b64e4 generalized INF_INT_eq, SUP_UN_eq
haftmann
parents: 46884
diff changeset
   107
lemma SUP_UN_eq2 [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Union>i\<in>S. r i))"
d54cea5b64e4 generalized INF_INT_eq, SUP_UN_eq
haftmann
parents: 46884
diff changeset
   108
  by (simp add: fun_eq_iff)
d54cea5b64e4 generalized INF_INT_eq, SUP_UN_eq
haftmann
parents: 46884
diff changeset
   109
46833
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   110
lemma Inf_INT_eq [pred_set_conv]: "\<Sqinter>S = (\<lambda>x. x \<in> INTER S Collect)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   111
  by (simp add: fun_eq_iff)
46833
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   112
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   113
lemma INF_Int_eq [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x. x \<in> i)) = (\<lambda>x. x \<in> \<Inter>S)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   114
  by (simp add: fun_eq_iff)
46833
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   115
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   116
lemma Inf_INT_eq2 [pred_set_conv]: "\<Sqinter>S = (\<lambda>x y. (x, y) \<in> INTER (prod_case ` S) Collect)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   117
  by (simp add: fun_eq_iff)
46833
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   118
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   119
lemma INF_Int_eq2 [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x y. (x, y) \<in> i)) = (\<lambda>x y. (x, y) \<in> \<Inter>S)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   120
  by (simp add: fun_eq_iff)
46833
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   121
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   122
lemma Sup_SUP_eq [pred_set_conv]: "\<Squnion>S = (\<lambda>x. x \<in> UNION S Collect)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   123
  by (simp add: fun_eq_iff)
46833
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   124
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   125
lemma SUP_Sup_eq [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x. x \<in> i)) = (\<lambda>x. x \<in> \<Union>S)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   126
  by (simp add: fun_eq_iff)
46833
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   127
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   128
lemma Sup_SUP_eq2 [pred_set_conv]: "\<Squnion>S = (\<lambda>x y. (x, y) \<in> UNION (prod_case ` S) Collect)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   129
  by (simp add: fun_eq_iff)
46833
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   130
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   131
lemma SUP_Sup_eq2 [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x y. (x, y) \<in> i)) = (\<lambda>x y. (x, y) \<in> \<Union>S)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   132
  by (simp add: fun_eq_iff)
46833
85619a872ab5 tuned syntax; more candidates
haftmann
parents: 46767
diff changeset
   133
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   134
subsection {* Properties of relations *}
5978
fa2c2dd74f8c moved diag (diagonal relation) from Univ to Relation
paulson
parents: 5608
diff changeset
   135
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   136
subsubsection {* Reflexivity *}
10786
04ee73606993 Field of a relation, and some Domain/Range rules
paulson
parents: 10358
diff changeset
   137
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   138
definition refl_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   139
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   140
  "refl_on A r \<longleftrightarrow> r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. (x, x) \<in> r)"
6806
43c081a0858d new preficates refl, sym [from Integ/Equiv], antisym
paulson
parents: 5978
diff changeset
   141
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   142
abbreviation refl :: "'a rel \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   143
where -- {* reflexivity over a type *}
45137
6e422d180de8 modernized definitions
haftmann
parents: 45012
diff changeset
   144
  "refl \<equiv> refl_on UNIV"
26297
74012d599204 added lemmas
nipkow
parents: 26271
diff changeset
   145
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   146
definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   147
where
47375
8e6a45f1bf8f define reflp directly, in the manner of symp and transp
huffman
parents: 47087
diff changeset
   148
  "reflp r \<longleftrightarrow> (\<forall>x. r x x)"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   149
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   150
lemma reflp_refl_eq [pred_set_conv]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   151
  "reflp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> refl r" 
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   152
  by (simp add: refl_on_def reflp_def)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   153
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   154
lemma refl_onI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl_on A r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   155
  by (unfold refl_on_def) (iprover intro!: ballI)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   156
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   157
lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   158
  by (unfold refl_on_def) blast
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   159
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   160
lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   161
  by (unfold refl_on_def) blast
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   162
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   163
lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   164
  by (unfold refl_on_def) blast
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   165
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   166
lemma reflpI:
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   167
  "(\<And>x. r x x) \<Longrightarrow> reflp r"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   168
  by (auto intro: refl_onI simp add: reflp_def)
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   169
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   170
lemma reflpE:
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   171
  assumes "reflp r"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   172
  obtains "r x x"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   173
  using assms by (auto dest: refl_onD simp add: reflp_def)
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   174
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: 47436
diff changeset
   175
lemma reflpD:
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: 47436
diff changeset
   176
  assumes "reflp r"
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: 47436
diff changeset
   177
  shows "r 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: 47436
diff changeset
   178
  using assms by (auto elim: reflpE)
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: 47436
diff changeset
   179
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   180
lemma refl_on_Int: "refl_on A r ==> refl_on B s ==> refl_on (A \<inter> B) (r \<inter> s)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   181
  by (unfold refl_on_def) blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   182
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   183
lemma reflp_inf:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   184
  "reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<sqinter> s)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   185
  by (auto intro: reflpI elim: reflpE)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   186
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   187
lemma refl_on_Un: "refl_on A r ==> refl_on B s ==> refl_on (A \<union> B) (r \<union> s)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   188
  by (unfold refl_on_def) blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   189
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   190
lemma reflp_sup:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   191
  "reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<squnion> s)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   192
  by (auto intro: reflpI elim: reflpE)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   193
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   194
lemma refl_on_INTER:
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   195
  "ALL x:S. refl_on (A x) (r x) ==> refl_on (INTER S A) (INTER S r)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   196
  by (unfold refl_on_def) fast
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   197
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   198
lemma refl_on_UNION:
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   199
  "ALL x:S. refl_on (A x) (r x) \<Longrightarrow> refl_on (UNION S A) (UNION S r)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   200
  by (unfold refl_on_def) blast
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   201
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   202
lemma refl_on_empty [simp]: "refl_on {} {}"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   203
  by (simp add:refl_on_def)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   204
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   205
lemma refl_on_def' [nitpick_unfold, code]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   206
  "refl_on A r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<in> A \<and> y \<in> A) \<and> (\<forall>x \<in> A. (x, x) \<in> r)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   207
  by (auto intro: refl_onI dest: refl_onD refl_onD1 refl_onD2)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   208
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   209
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   210
subsubsection {* Irreflexivity *}
6806
43c081a0858d new preficates refl, sym [from Integ/Equiv], antisym
paulson
parents: 5978
diff changeset
   211
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   212
definition irrefl :: "'a rel \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   213
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   214
  "irrefl r \<longleftrightarrow> (\<forall>x. (x, x) \<notin> r)"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   215
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   216
lemma irrefl_distinct [code]:
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   217
  "irrefl r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<noteq> y)"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   218
  by (auto simp add: irrefl_def)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   219
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   220
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   221
subsubsection {* Symmetry *}
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   222
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   223
definition sym :: "'a rel \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   224
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   225
  "sym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   226
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   227
definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   228
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   229
  "symp r \<longleftrightarrow> (\<forall>x y. r x y \<longrightarrow> r y x)"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   230
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   231
lemma symp_sym_eq [pred_set_conv]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   232
  "symp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> sym r" 
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   233
  by (simp add: sym_def symp_def)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   234
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   235
lemma symI:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   236
  "(\<And>a b. (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r) \<Longrightarrow> sym r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   237
  by (unfold sym_def) iprover
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   238
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   239
lemma sympI:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   240
  "(\<And>a b. r a b \<Longrightarrow> r b a) \<Longrightarrow> symp r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   241
  by (fact symI [to_pred])
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   242
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   243
lemma symE:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   244
  assumes "sym r" and "(b, a) \<in> r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   245
  obtains "(a, b) \<in> r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   246
  using assms by (simp add: sym_def)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   247
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   248
lemma sympE:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   249
  assumes "symp r" and "r b a"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   250
  obtains "r a b"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   251
  using assms by (rule symE [to_pred])
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   252
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   253
lemma symD:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   254
  assumes "sym r" and "(b, a) \<in> r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   255
  shows "(a, b) \<in> r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   256
  using assms by (rule symE)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   257
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   258
lemma sympD:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   259
  assumes "symp r" and "r b a"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   260
  shows "r a b"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   261
  using assms by (rule symD [to_pred])
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   262
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   263
lemma sym_Int:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   264
  "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<inter> s)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   265
  by (fast intro: symI elim: symE)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   266
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   267
lemma symp_inf:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   268
  "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<sqinter> s)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   269
  by (fact sym_Int [to_pred])
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   270
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   271
lemma sym_Un:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   272
  "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<union> s)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   273
  by (fast intro: symI elim: symE)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   274
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   275
lemma symp_sup:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   276
  "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<squnion> s)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   277
  by (fact sym_Un [to_pred])
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   278
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   279
lemma sym_INTER:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   280
  "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (INTER S r)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   281
  by (fast intro: symI elim: symE)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   282
46982
144d94446378 spelt out missing colemmas
haftmann
parents: 46981
diff changeset
   283
lemma symp_INF:
144d94446378 spelt out missing colemmas
haftmann
parents: 46981
diff changeset
   284
  "\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (INFI S r)"
144d94446378 spelt out missing colemmas
haftmann
parents: 46981
diff changeset
   285
  by (fact sym_INTER [to_pred])
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   286
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   287
lemma sym_UNION:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   288
  "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (UNION S r)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   289
  by (fast intro: symI elim: symE)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   290
46982
144d94446378 spelt out missing colemmas
haftmann
parents: 46981
diff changeset
   291
lemma symp_SUP:
144d94446378 spelt out missing colemmas
haftmann
parents: 46981
diff changeset
   292
  "\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (SUPR S r)"
144d94446378 spelt out missing colemmas
haftmann
parents: 46981
diff changeset
   293
  by (fact sym_UNION [to_pred])
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   294
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   295
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   296
subsubsection {* Antisymmetry *}
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   297
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   298
definition antisym :: "'a rel \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   299
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   300
  "antisym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   301
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   302
abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   303
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   304
  "antisymP r \<equiv> antisym {(x, y). r x y}"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   305
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   306
lemma antisymI:
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   307
  "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   308
  by (unfold antisym_def) iprover
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   309
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   310
lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   311
  by (unfold antisym_def) iprover
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   312
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   313
lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   314
  by (unfold antisym_def) blast
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   315
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   316
lemma antisym_empty [simp]: "antisym {}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   317
  by (unfold antisym_def) blast
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   318
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   319
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   320
subsubsection {* Transitivity *}
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   321
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   322
definition trans :: "'a rel \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   323
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   324
  "trans r \<longleftrightarrow> (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   325
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   326
definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   327
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   328
  "transp r \<longleftrightarrow> (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   329
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   330
lemma transp_trans_eq [pred_set_conv]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   331
  "transp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> trans r" 
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   332
  by (simp add: trans_def transp_def)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   333
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   334
abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   335
where -- {* FIXME drop *}
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   336
  "transP r \<equiv> trans {(x, y). r x y}"
5978
fa2c2dd74f8c moved diag (diagonal relation) from Univ to Relation
paulson
parents: 5608
diff changeset
   337
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   338
lemma transI:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   339
  "(\<And>x y z. (x, y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r) \<Longrightarrow> trans r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   340
  by (unfold trans_def) iprover
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   341
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   342
lemma transpI:
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   343
  "(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   344
  by (fact transI [to_pred])
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   345
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   346
lemma transE:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   347
  assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   348
  obtains "(x, z) \<in> r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   349
  using assms by (unfold trans_def) iprover
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   350
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   351
lemma transpE:
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   352
  assumes "transp r" and "r x y" and "r y z"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   353
  obtains "r x z"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   354
  using assms by (rule transE [to_pred])
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   355
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   356
lemma transD:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   357
  assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   358
  shows "(x, z) \<in> r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   359
  using assms by (rule transE)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   360
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   361
lemma transpD:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   362
  assumes "transp r" and "r x y" and "r y z"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   363
  shows "r x z"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   364
  using assms by (rule transD [to_pred])
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   365
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   366
lemma trans_Int:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   367
  "trans r \<Longrightarrow> trans s \<Longrightarrow> trans (r \<inter> s)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   368
  by (fast intro: transI elim: transE)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   369
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   370
lemma transp_inf:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   371
  "transp r \<Longrightarrow> transp s \<Longrightarrow> transp (r \<sqinter> s)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   372
  by (fact trans_Int [to_pred])
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   373
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   374
lemma trans_INTER:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   375
  "\<forall>x\<in>S. trans (r x) \<Longrightarrow> trans (INTER S r)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   376
  by (fast intro: transI elim: transD)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   377
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   378
(* FIXME thm trans_INTER [to_pred] *)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   379
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   380
lemma trans_join [code]:
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   381
  "trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   382
  by (auto simp add: trans_def)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   383
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   384
lemma transp_trans:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   385
  "transp r \<longleftrightarrow> trans {(x, y). r x y}"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   386
  by (simp add: trans_def transp_def)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   387
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   388
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   389
subsubsection {* Totality *}
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   390
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   391
definition total_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   392
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   393
  "total_on A r \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r)"
29859
33bff35f1335 Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents: 29609
diff changeset
   394
33bff35f1335 Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents: 29609
diff changeset
   395
abbreviation "total \<equiv> total_on UNIV"
33bff35f1335 Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents: 29609
diff changeset
   396
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   397
lemma total_on_empty [simp]: "total_on {} r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   398
  by (simp add: total_on_def)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   399
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   400
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   401
subsubsection {* Single valued relations *}
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   402
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   403
definition single_valued :: "('a \<times> 'b) set \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   404
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   405
  "single_valued r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   406
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   407
abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   408
  "single_valuedP r \<equiv> single_valued {(x, y). r x y}"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   409
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   410
lemma single_valuedI:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   411
  "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   412
  by (unfold single_valued_def)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   413
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   414
lemma single_valuedD:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   415
  "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   416
  by (simp add: single_valued_def)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   417
52392
ee996ca08de3 added lemma
nipkow
parents: 50420
diff changeset
   418
lemma simgle_valued_empty[simp]: "single_valued {}"
ee996ca08de3 added lemma
nipkow
parents: 50420
diff changeset
   419
by(simp add: single_valued_def)
ee996ca08de3 added lemma
nipkow
parents: 50420
diff changeset
   420
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   421
lemma single_valued_subset:
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   422
  "r \<subseteq> s ==> single_valued s ==> single_valued r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   423
  by (unfold single_valued_def) blast
11136
e34e7f6d9b57 moved inv_image to Relation
oheimb
parents: 10832
diff changeset
   424
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   425
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   426
subsection {* Relation operations *}
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   427
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   428
subsubsection {* The identity relation *}
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   429
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   430
definition Id :: "'a rel"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   431
where
48253
4410a709913c a first guess to avoid the Codegenerator_Test to loop infinitely
bulwahn
parents: 47937
diff changeset
   432
  [code del]: "Id = {p. \<exists>x. p = (x, x)}"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   433
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   434
lemma IdI [intro]: "(a, a) : Id"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   435
  by (simp add: Id_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   436
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   437
lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   438
  by (unfold Id_def) (iprover elim: CollectE)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   439
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   440
lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   441
  by (unfold Id_def) blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   442
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   443
lemma refl_Id: "refl Id"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   444
  by (simp add: refl_on_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   445
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   446
lemma antisym_Id: "antisym Id"
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   447
  -- {* A strange result, since @{text Id} is also symmetric. *}
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   448
  by (simp add: antisym_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   449
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   450
lemma sym_Id: "sym Id"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   451
  by (simp add: sym_def)
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   452
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   453
lemma trans_Id: "trans Id"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   454
  by (simp add: trans_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   455
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   456
lemma single_valued_Id [simp]: "single_valued Id"
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   457
  by (unfold single_valued_def) blast
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   458
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   459
lemma irrefl_diff_Id [simp]: "irrefl (r - Id)"
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   460
  by (simp add:irrefl_def)
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   461
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   462
lemma trans_diff_Id: "trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r - Id)"
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   463
  unfolding antisym_def trans_def by blast
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   464
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   465
lemma total_on_diff_Id [simp]: "total_on A (r - Id) = total_on A r"
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   466
  by (simp add: total_on_def)
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   467
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   468
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   469
subsubsection {* Diagonal: identity over a set *}
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   470
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   471
definition Id_on  :: "'a set \<Rightarrow> 'a rel"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   472
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   473
  "Id_on A = (\<Union>x\<in>A. {(x, x)})"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   474
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   475
lemma Id_on_empty [simp]: "Id_on {} = {}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   476
  by (simp add: Id_on_def) 
13812
91713a1915ee converting HOL/UNITY to use unconditional fairness
paulson
parents: 13639
diff changeset
   477
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   478
lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   479
  by (simp add: Id_on_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   480
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 53680
diff changeset
   481
lemma Id_onI [intro!]: "a : A ==> (a, a) : Id_on A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   482
  by (rule Id_on_eqI) (rule refl)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   483
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   484
lemma Id_onE [elim!]:
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   485
  "c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   486
  -- {* The general elimination rule. *}
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   487
  by (unfold Id_on_def) (iprover elim!: UN_E singletonE)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   488
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   489
lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   490
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   491
45967
76cf71ed15c7 dropped obsolete code equation for Id
haftmann
parents: 45139
diff changeset
   492
lemma Id_on_def' [nitpick_unfold]:
44278
1220ecb81e8f observe distinction between sets and predicates more properly
haftmann
parents: 41792
diff changeset
   493
  "Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   494
  by auto
40923
be80c93ac0a2 adding a nice definition of Id_on for quickcheck and nitpick
bulwahn
parents: 36772
diff changeset
   495
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   496
lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   497
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   498
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   499
lemma refl_on_Id_on: "refl_on A (Id_on A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   500
  by (rule refl_onI [OF Id_on_subset_Times Id_onI])
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   501
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   502
lemma antisym_Id_on [simp]: "antisym (Id_on A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   503
  by (unfold antisym_def) blast
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   504
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   505
lemma sym_Id_on [simp]: "sym (Id_on A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   506
  by (rule symI) clarify
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   507
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   508
lemma trans_Id_on [simp]: "trans (Id_on A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   509
  by (fast intro: transI elim: transD)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   510
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   511
lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   512
  by (unfold single_valued_def) blast
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   513
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   514
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   515
subsubsection {* Composition *}
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   516
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   517
inductive_set relcomp  :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a \<times> 'c) set" (infixr "O" 75)
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   518
  for r :: "('a \<times> 'b) set" and s :: "('b \<times> 'c) set"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   519
where
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   520
  relcompI [intro]: "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> s \<Longrightarrow> (a, c) \<in> r O s"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   521
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   522
notation relcompp (infixr "OO" 75)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   523
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   524
lemmas relcomppI = relcompp.intros
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   525
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   526
text {*
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   527
  For historic reasons, the elimination rules are not wholly corresponding.
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   528
  Feel free to consolidate this.
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   529
*}
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   530
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   531
inductive_cases relcompEpair: "(a, c) \<in> r O s"
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   532
inductive_cases relcomppE [elim!]: "(r OO s) a c"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   533
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   534
lemma relcompE [elim!]: "xz \<in> r O s \<Longrightarrow>
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   535
  (\<And>x y z. xz = (x, z) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (y, z) \<in> s  \<Longrightarrow> P) \<Longrightarrow> P"
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   536
  by (cases xz) (simp, erule relcompEpair, iprover)
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   537
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   538
lemma R_O_Id [simp]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   539
  "R O Id = R"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   540
  by fast
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   541
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   542
lemma Id_O_R [simp]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   543
  "Id O R = R"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   544
  by fast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   545
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   546
lemma relcomp_empty1 [simp]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   547
  "{} O R = {}"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   548
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   549
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   550
lemma relcompp_bot1 [simp]:
46883
eec472dae593 tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents: 46882
diff changeset
   551
  "\<bottom> OO R = \<bottom>"
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   552
  by (fact relcomp_empty1 [to_pred])
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   553
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   554
lemma relcomp_empty2 [simp]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   555
  "R O {} = {}"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   556
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   557
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   558
lemma relcompp_bot2 [simp]:
46883
eec472dae593 tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents: 46882
diff changeset
   559
  "R OO \<bottom> = \<bottom>"
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   560
  by (fact relcomp_empty2 [to_pred])
23185
1fa87978cf27 Added simp-rules: "R O {} = {}" and "{} O R = {}"
krauss
parents: 22172
diff changeset
   561
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   562
lemma O_assoc:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   563
  "(R O S) O T = R O (S O T)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   564
  by blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   565
46883
eec472dae593 tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents: 46882
diff changeset
   566
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   567
lemma relcompp_assoc:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   568
  "(r OO s) OO t = r OO (s OO t)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   569
  by (fact O_assoc [to_pred])
23185
1fa87978cf27 Added simp-rules: "R O {} = {}" and "{} O R = {}"
krauss
parents: 22172
diff changeset
   570
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   571
lemma trans_O_subset:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   572
  "trans r \<Longrightarrow> r O r \<subseteq> r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   573
  by (unfold trans_def) blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   574
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   575
lemma transp_relcompp_less_eq:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   576
  "transp r \<Longrightarrow> r OO r \<le> r "
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   577
  by (fact trans_O_subset [to_pred])
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   578
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   579
lemma relcomp_mono:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   580
  "r' \<subseteq> r \<Longrightarrow> s' \<subseteq> s \<Longrightarrow> r' O s' \<subseteq> r O s"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   581
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   582
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   583
lemma relcompp_mono:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   584
  "r' \<le> r \<Longrightarrow> s' \<le> s \<Longrightarrow> r' OO s' \<le> r OO s "
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   585
  by (fact relcomp_mono [to_pred])
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   586
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   587
lemma relcomp_subset_Sigma:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   588
  "r \<subseteq> A \<times> B \<Longrightarrow> s \<subseteq> B \<times> C \<Longrightarrow> r O s \<subseteq> A \<times> C"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   589
  by blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   590
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   591
lemma relcomp_distrib [simp]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   592
  "R O (S \<union> T) = (R O S) \<union> (R O T)" 
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   593
  by auto
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   594
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   595
lemma relcompp_distrib [simp]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   596
  "R OO (S \<squnion> T) = R OO S \<squnion> R OO T"
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   597
  by (fact relcomp_distrib [to_pred])
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   598
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   599
lemma relcomp_distrib2 [simp]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   600
  "(S \<union> T) O R = (S O R) \<union> (T O R)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   601
  by auto
28008
f945f8d9ad4d added distributivity of relation composition over union [simp]
krauss
parents: 26297
diff changeset
   602
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   603
lemma relcompp_distrib2 [simp]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   604
  "(S \<squnion> T) OO R = S OO R \<squnion> T OO R"
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   605
  by (fact relcomp_distrib2 [to_pred])
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   606
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   607
lemma relcomp_UNION_distrib:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   608
  "s O UNION I r = (\<Union>i\<in>I. s O r i) "
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   609
  by auto
28008
f945f8d9ad4d added distributivity of relation composition over union [simp]
krauss
parents: 26297
diff changeset
   610
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   611
(* FIXME thm relcomp_UNION_distrib [to_pred] *)
36772
ef97c5006840 added lemmas rel_comp_UNION_distrib(2)
krauss
parents: 36729
diff changeset
   612
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   613
lemma relcomp_UNION_distrib2:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   614
  "UNION I r O s = (\<Union>i\<in>I. r i O s) "
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   615
  by auto
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   616
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   617
(* FIXME thm relcomp_UNION_distrib2 [to_pred] *)
36772
ef97c5006840 added lemmas rel_comp_UNION_distrib(2)
krauss
parents: 36729
diff changeset
   618
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   619
lemma single_valued_relcomp:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   620
  "single_valued r \<Longrightarrow> single_valued s \<Longrightarrow> single_valued (r O s)"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   621
  by (unfold single_valued_def) blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   622
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   623
lemma relcomp_unfold:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   624
  "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   625
  by (auto simp add: set_eq_iff)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   626
55083
0a689157e3ce move BNF_LFP up the dependency chain
blanchet
parents: 54611
diff changeset
   627
lemma eq_OO: "op= OO R = R"
0a689157e3ce move BNF_LFP up the dependency chain
blanchet
parents: 54611
diff changeset
   628
by blast
0a689157e3ce move BNF_LFP up the dependency chain
blanchet
parents: 54611
diff changeset
   629
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   630
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   631
subsubsection {* Converse *}
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   632
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   633
inductive_set converse :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'a) set" ("(_^-1)" [1000] 999)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   634
  for r :: "('a \<times> 'b) set"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   635
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   636
  "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r^-1"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   637
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   638
notation (xsymbols)
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   639
  converse  ("(_\<inverse>)" [1000] 999)
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   640
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   641
notation
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   642
  conversep ("(_^--1)" [1000] 1000)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   643
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   644
notation (xsymbols)
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   645
  conversep  ("(_\<inverse>\<inverse>)" [1000] 1000)
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   646
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   647
lemma converseI [sym]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   648
  "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   649
  by (fact converse.intros)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   650
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   651
lemma conversepI (* CANDIDATE [sym] *):
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   652
  "r a b \<Longrightarrow> r\<inverse>\<inverse> b a"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   653
  by (fact conversep.intros)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   654
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   655
lemma converseD [sym]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   656
  "(a, b) \<in> r\<inverse> \<Longrightarrow> (b, a) \<in> r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   657
  by (erule converse.cases) iprover
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   658
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   659
lemma conversepD (* CANDIDATE [sym] *):
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   660
  "r\<inverse>\<inverse> b a \<Longrightarrow> r a b"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   661
  by (fact converseD [to_pred])
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   662
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   663
lemma converseE [elim!]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   664
  -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   665
  "yx \<in> r\<inverse> \<Longrightarrow> (\<And>x y. yx = (y, x) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> P) \<Longrightarrow> P"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   666
  by (cases yx) (simp, erule converse.cases, iprover)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   667
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46833
diff changeset
   668
lemmas conversepE [elim!] = conversep.cases
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   669
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   670
lemma converse_iff [iff]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   671
  "(a, b) \<in> r\<inverse> \<longleftrightarrow> (b, a) \<in> r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   672
  by (auto intro: converseI)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   673
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   674
lemma conversep_iff [iff]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   675
  "r\<inverse>\<inverse> a b = r b a"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   676
  by (fact converse_iff [to_pred])
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   677
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   678
lemma converse_converse [simp]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   679
  "(r\<inverse>)\<inverse> = r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   680
  by (simp add: set_eq_iff)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   681
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   682
lemma conversep_conversep [simp]:
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   683
  "(r\<inverse>\<inverse>)\<inverse>\<inverse> = r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   684
  by (fact converse_converse [to_pred])
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   685
53680
c5096c22892b added lemmas and made concerse executable
nipkow
parents: 52749
diff changeset
   686
lemma converse_empty[simp]: "{}\<inverse> = {}"
c5096c22892b added lemmas and made concerse executable
nipkow
parents: 52749
diff changeset
   687
by auto
c5096c22892b added lemmas and made concerse executable
nipkow
parents: 52749
diff changeset
   688
c5096c22892b added lemmas and made concerse executable
nipkow
parents: 52749
diff changeset
   689
lemma converse_UNIV[simp]: "UNIV\<inverse> = UNIV"
c5096c22892b added lemmas and made concerse executable
nipkow
parents: 52749
diff changeset
   690
by auto
c5096c22892b added lemmas and made concerse executable
nipkow
parents: 52749
diff changeset
   691
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   692
lemma converse_relcomp: "(r O s)^-1 = s^-1 O r^-1"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   693
  by blast
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   694
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   695
lemma converse_relcompp: "(r OO s)^--1 = s^--1 OO r^--1"
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   696
  by (iprover intro: order_antisym conversepI relcomppI
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   697
    elim: relcomppE dest: conversepD)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   698
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   699
lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   700
  by blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   701
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   702
lemma converse_meet: "(r \<sqinter> s)^--1 = r^--1 \<sqinter> s^--1"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   703
  by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   704
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   705
lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   706
  by blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   707
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   708
lemma converse_join: "(r \<squnion> s)^--1 = r^--1 \<squnion> s^--1"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   709
  by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   710
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   711
lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   712
  by fast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   713
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   714
lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   715
  by blast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   716
52749
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   717
lemma converse_mono[simp]: "r^-1 \<subseteq> s ^-1 \<longleftrightarrow> r \<subseteq> s"
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   718
  by auto
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   719
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   720
lemma conversep_mono[simp]: "r^--1 \<le> s ^--1 \<longleftrightarrow> r \<le> s"
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   721
  by (fact converse_mono[to_pred])
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   722
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   723
lemma converse_inject[simp]: "r^-1 = s ^-1 \<longleftrightarrow> r = s"
52730
6bf02eb4ddf7 two useful relation theorems
traytel
parents: 52392
diff changeset
   724
  by auto
6bf02eb4ddf7 two useful relation theorems
traytel
parents: 52392
diff changeset
   725
52749
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   726
lemma conversep_inject[simp]: "r^--1 = s ^--1 \<longleftrightarrow> r = s"
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   727
  by (fact converse_inject[to_pred])
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   728
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   729
lemma converse_subset_swap: "r \<subseteq> s ^-1 = (r ^-1 \<subseteq> s)"
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   730
  by auto
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   731
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   732
lemma conversep_le_swap: "r \<le> s ^--1 = (r ^--1 \<le> s)"
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   733
  by (fact converse_subset_swap[to_pred])
52730
6bf02eb4ddf7 two useful relation theorems
traytel
parents: 52392
diff changeset
   734
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   735
lemma converse_Id [simp]: "Id^-1 = Id"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   736
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   737
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   738
lemma converse_Id_on [simp]: "(Id_on A)^-1 = Id_on A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   739
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   740
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   741
lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   742
  by (unfold refl_on_def) auto
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   743
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   744
lemma sym_converse [simp]: "sym (converse r) = sym r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   745
  by (unfold sym_def) blast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   746
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   747
lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   748
  by (unfold antisym_def) blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   749
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   750
lemma trans_converse [simp]: "trans (converse r) = trans r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   751
  by (unfold trans_def) blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   752
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   753
lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   754
  by (unfold sym_def) fast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   755
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   756
lemma sym_Un_converse: "sym (r \<union> r^-1)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   757
  by (unfold sym_def) blast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   758
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   759
lemma sym_Int_converse: "sym (r \<inter> r^-1)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   760
  by (unfold sym_def) blast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   761
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   762
lemma total_on_converse [simp]: "total_on A (r^-1) = total_on A r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   763
  by (auto simp: total_on_def)
29859
33bff35f1335 Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents: 29609
diff changeset
   764
52749
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   765
lemma finite_converse [iff]: "finite (r^-1) = finite r"  
54611
31afce809794 set_comprehension_pointfree simproc causes to many surprises if enabled by default
traytel
parents: 54555
diff changeset
   766
  unfolding converse_def conversep_iff using [[simproc add: finite_Collect]]
31afce809794 set_comprehension_pointfree simproc causes to many surprises if enabled by default
traytel
parents: 54555
diff changeset
   767
  by (auto elim: finite_imageD simp: inj_on_def)
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   768
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   769
lemma conversep_noteq [simp]: "(op \<noteq>)^--1 = op \<noteq>"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   770
  by (auto simp add: fun_eq_iff)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   771
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   772
lemma conversep_eq [simp]: "(op =)^--1 = op ="
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   773
  by (auto simp add: fun_eq_iff)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   774
53680
c5096c22892b added lemmas and made concerse executable
nipkow
parents: 52749
diff changeset
   775
lemma converse_unfold [code]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   776
  "r\<inverse> = {(y, x). (x, y) \<in> r}"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   777
  by (simp add: set_eq_iff)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   778
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   779
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   780
subsubsection {* Domain, range and field *}
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   781
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   782
inductive_set Domain :: "('a \<times> 'b) set \<Rightarrow> 'a set"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   783
  for r :: "('a \<times> 'b) set"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   784
where
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   785
  DomainI [intro]: "(a, b) \<in> r \<Longrightarrow> a \<in> Domain r"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   786
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   787
abbreviation (input) "DomainP \<equiv> Domainp"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   788
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   789
lemmas DomainPI = Domainp.DomainI
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   790
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   791
inductive_cases DomainE [elim!]: "a \<in> Domain r"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   792
inductive_cases DomainpE [elim!]: "Domainp r a"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   793
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   794
inductive_set Range :: "('a \<times> 'b) set \<Rightarrow> 'b set"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   795
  for r :: "('a \<times> 'b) set"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   796
where
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   797
  RangeI [intro]: "(a, b) \<in> r \<Longrightarrow> b \<in> Range r"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   798
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   799
abbreviation (input) "RangeP \<equiv> Rangep"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   800
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   801
lemmas RangePI = Rangep.RangeI
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   802
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   803
inductive_cases RangeE [elim!]: "b \<in> Range r"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   804
inductive_cases RangepE [elim!]: "Rangep r b"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   805
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   806
definition Field :: "'a rel \<Rightarrow> 'a set"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   807
where
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   808
  "Field r = Domain r \<union> Range r"
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   809
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   810
lemma Domain_fst [code]:
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   811
  "Domain r = fst ` r"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   812
  by force
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   813
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   814
lemma Range_snd [code]:
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   815
  "Range r = snd ` r"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   816
  by force
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   817
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   818
lemma fst_eq_Domain: "fst ` R = Domain R"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   819
  by force
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   820
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   821
lemma snd_eq_Range: "snd ` R = Range R"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   822
  by force
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   823
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   824
lemma Domain_empty [simp]: "Domain {} = {}"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   825
  by auto
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   826
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   827
lemma Range_empty [simp]: "Range {} = {}"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   828
  by auto
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   829
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   830
lemma Field_empty [simp]: "Field {} = {}"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   831
  by (simp add: Field_def)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   832
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   833
lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   834
  by auto
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   835
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   836
lemma Range_empty_iff: "Range r = {} \<longleftrightarrow> r = {}"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   837
  by auto
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   838
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46833
diff changeset
   839
lemma Domain_insert [simp]: "Domain (insert (a, b) r) = insert a (Domain r)"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   840
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   841
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46833
diff changeset
   842
lemma Range_insert [simp]: "Range (insert (a, b) r) = insert b (Range r)"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   843
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   844
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   845
lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} \<union> Field r"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   846
  by (auto simp add: Field_def)
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   847
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   848
lemma Domain_iff: "a \<in> Domain r \<longleftrightarrow> (\<exists>y. (a, y) \<in> r)"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   849
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   850
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   851
lemma Range_iff: "a \<in> Range r \<longleftrightarrow> (\<exists>y. (y, a) \<in> r)"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   852
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   853
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   854
lemma Domain_Id [simp]: "Domain Id = UNIV"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   855
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   856
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   857
lemma Range_Id [simp]: "Range Id = UNIV"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   858
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   859
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   860
lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   861
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   862
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   863
lemma Range_Id_on [simp]: "Range (Id_on A) = A"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   864
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   865
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   866
lemma Domain_Un_eq: "Domain (A \<union> B) = Domain A \<union> Domain B"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   867
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   868
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   869
lemma Range_Un_eq: "Range (A \<union> B) = Range A \<union> Range B"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   870
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   871
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   872
lemma Field_Un [simp]: "Field (r \<union> s) = Field r \<union> Field s"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   873
  by (auto simp: Field_def)
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   874
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   875
lemma Domain_Int_subset: "Domain (A \<inter> B) \<subseteq> Domain A \<inter> Domain B"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   876
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   877
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   878
lemma Range_Int_subset: "Range (A \<inter> B) \<subseteq> Range A \<inter> Range B"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   879
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   880
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   881
lemma Domain_Diff_subset: "Domain A - Domain B \<subseteq> Domain (A - B)"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   882
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   883
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   884
lemma Range_Diff_subset: "Range A - Range B \<subseteq> Range (A - B)"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   885
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   886
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   887
lemma Domain_Union: "Domain (\<Union>S) = (\<Union>A\<in>S. Domain A)"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   888
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   889
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   890
lemma Range_Union: "Range (\<Union>S) = (\<Union>A\<in>S. Range A)"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   891
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   892
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   893
lemma Field_Union [simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   894
  by (auto simp: Field_def)
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   895
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   896
lemma Domain_converse [simp]: "Domain (r\<inverse>) = Range r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   897
  by auto
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   898
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   899
lemma Range_converse [simp]: "Range (r\<inverse>) = Domain r"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   900
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   901
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   902
lemma Field_converse [simp]: "Field (r\<inverse>) = Field r"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   903
  by (auto simp: Field_def)
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   904
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   905
lemma Domain_Collect_split [simp]: "Domain {(x, y). P x y} = {x. EX y. P x y}"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   906
  by auto
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   907
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   908
lemma Range_Collect_split [simp]: "Range {(x, y). P x y} = {y. EX x. P x y}"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   909
  by auto
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   910
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   911
lemma finite_Domain: "finite r \<Longrightarrow> finite (Domain r)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   912
  by (induct set: finite) auto
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   913
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   914
lemma finite_Range: "finite r \<Longrightarrow> finite (Range r)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   915
  by (induct set: finite) auto
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   916
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   917
lemma finite_Field: "finite r \<Longrightarrow> finite (Field r)"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   918
  by (simp add: Field_def finite_Domain finite_Range)
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   919
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   920
lemma Domain_mono: "r \<subseteq> s \<Longrightarrow> Domain r \<subseteq> Domain s"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   921
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   922
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   923
lemma Range_mono: "r \<subseteq> s \<Longrightarrow> Range r \<subseteq> Range s"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   924
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   925
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   926
lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   927
  by (auto simp: Field_def Domain_def Range_def)
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   928
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   929
lemma Domain_unfold:
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   930
  "Domain r = {x. \<exists>y. (x, y) \<in> r}"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   931
  by blast
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   932
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   933
lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   934
  by auto
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   935
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   936
lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   937
  by auto
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   938
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   939
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   940
subsubsection {* Image of a set under a relation *}
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   941
50420
f1a27e82af16 corrected nonsensical associativity of `` and dvd
nipkow
parents: 48620
diff changeset
   942
definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "``" 90)
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   943
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   944
  "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   945
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   946
lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   947
  by (simp add: Image_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   948
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   949
lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   950
  by (simp add: Image_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   951
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   952
lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   953
  by (rule Image_iff [THEN trans]) simp
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   954
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 53680
diff changeset
   955
lemma ImageI [intro]: "(a, b) : r ==> a : A ==> b : r``A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   956
  by (unfold Image_def) blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   957
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   958
lemma ImageE [elim!]:
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   959
  "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   960
  by (unfold Image_def) (iprover elim!: CollectE bexE)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   961
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   962
lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   963
  -- {* This version's more effective when we already have the required @{text a} *}
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   964
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   965
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   966
lemma Image_empty [simp]: "R``{} = {}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   967
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   968
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   969
lemma Image_Id [simp]: "Id `` A = A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   970
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   971
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   972
lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   973
  by blast
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   974
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   975
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   976
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   977
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   978
lemma Image_Int_eq:
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   979
  "single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   980
  by (simp add: single_valued_def, blast) 
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   981
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   982
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   983
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   984
13812
91713a1915ee converting HOL/UNITY to use unconditional fairness
paulson
parents: 13639
diff changeset
   985
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   986
  by blast
13812
91713a1915ee converting HOL/UNITY to use unconditional fairness
paulson
parents: 13639
diff changeset
   987
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   988
lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   989
  by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   990
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   991
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   992
  -- {* NOT suitable for rewriting *}
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   993
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   994
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   995
lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   996
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   997
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   998
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   999
  by blast
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
  1000
54410
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1001
lemma UN_Image: "(\<Union>i\<in>I. X i) `` S = (\<Union>i\<in>I. X i `` S)"
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1002
  by auto
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1003
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
  1004
lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1005
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1006
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
  1007
text{*Converse inclusion requires some assumptions*}
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
  1008
lemma Image_INT_eq:
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
  1009
     "[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
  1010
apply (rule equalityI)
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
  1011
 apply (rule Image_INT_subset) 
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
  1012
apply  (simp add: single_valued_def, blast)
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
  1013
done
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1014
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
  1015
lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1016
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1017
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
  1018
lemma Image_Collect_split [simp]: "{(x, y). P x y} `` A = {y. EX x:A. P x y}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1019
  by auto
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1020
54410
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1021
lemma Sigma_Image: "(SIGMA x:A. B x) `` X = (\<Union>x\<in>X \<inter> A. B x)"
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1022
  by auto
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1023
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1024
lemma relcomp_Image: "(X O Y) `` Z = Y `` (X `` Z)"
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1025
  by auto
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1026
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1027
subsubsection {* Inverse image *}
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1028
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1029
definition inv_image :: "'b rel \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a rel"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1030
where
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1031
  "inv_image r f = {(x, y). (f x, f y) \<in> r}"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
  1032
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1033
definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1034
where
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
  1035
  "inv_imagep r f = (\<lambda>x y. r (f x) (f y))"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
  1036
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
  1037
lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
  1038
  by (simp add: inv_image_def inv_imagep_def)
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
  1039
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
  1040
lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1041
  by (unfold sym_def inv_image_def) blast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
  1042
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
  1043
lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1044
  apply (unfold trans_def inv_image_def)
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1045
  apply (simp (no_asm))
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1046
  apply blast
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1047
  done
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1048
32463
3a0a65ca2261 moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents: 32235
diff changeset
  1049
lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)"
3a0a65ca2261 moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents: 32235
diff changeset
  1050
  by (auto simp:inv_image_def)
3a0a65ca2261 moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents: 32235
diff changeset
  1051
33218
ecb5cd453ef2 lemma converse_inv_image
krauss
parents: 32876
diff changeset
  1052
lemma converse_inv_image[simp]: "(inv_image R f)^-1 = inv_image (R^-1) f"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1053
  unfolding inv_image_def converse_unfold by auto
33218
ecb5cd453ef2 lemma converse_inv_image
krauss
parents: 32876
diff changeset
  1054
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1055
lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1056
  by (simp add: inv_imagep_def)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1057
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1058
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1059
subsubsection {* Powerset *}
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1060
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1061
definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1062
where
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1063
  "Powp A = (\<lambda>B. \<forall>x \<in> B. A x)"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1064
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1065
lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1066
  by (auto simp add: Powp_def fun_eq_iff)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1067
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1068
lemmas Powp_mono [mono] = Pow_mono [to_pred]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1069
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1070
subsubsection {* Expressing relation operations via @{const Finite_Set.fold} *}
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1071
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1072
lemma Id_on_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1073
  assumes "finite A"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1074
  shows "Id_on A = Finite_Set.fold (\<lambda>x. Set.insert (Pair x x)) {} A"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1075
proof -
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1076
  interpret comp_fun_commute "\<lambda>x. Set.insert (Pair x x)" by default auto
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1077
  show ?thesis using assms unfolding Id_on_def by (induct A) simp_all
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1078
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1079
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1080
lemma comp_fun_commute_Image_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1081
  "comp_fun_commute (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1082
proof -
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1083
  interpret comp_fun_idem Set.insert
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1084
      by (fact comp_fun_idem_insert)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1085
  show ?thesis 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1086
  by default (auto simp add: fun_eq_iff comp_fun_commute split:prod.split)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1087
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1088
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1089
lemma Image_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1090
  assumes "finite R"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1091
  shows "R `` S = Finite_Set.fold (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A) {} R"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1092
proof -
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1093
  interpret comp_fun_commute "(\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)" 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1094
    by (rule comp_fun_commute_Image_fold)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1095
  have *: "\<And>x F. Set.insert x F `` S = (if fst x \<in> S then Set.insert (snd x) (F `` S) else (F `` S))"
52749
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
  1096
    by (force intro: rev_ImageI)
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1097
  show ?thesis using assms by (induct R) (auto simp: *)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1098
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1099
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1100
lemma insert_relcomp_union_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1101
  assumes "finite S"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1102
  shows "{x} O S \<union> X = Finite_Set.fold (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') X S"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1103
proof -
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1104
  interpret comp_fun_commute "\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A'"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1105
  proof - 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1106
    interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1107
    show "comp_fun_commute (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A')"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1108
    by default (auto simp add: fun_eq_iff split:prod.split)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1109
  qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1110
  have *: "{x} O S = {(x', z). x' = fst x \<and> (snd x,z) \<in> S}" by (auto simp: relcomp_unfold intro!: exI)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1111
  show ?thesis unfolding *
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1112
  using `finite S` by (induct S) (auto split: prod.split)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1113
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1114
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1115
lemma insert_relcomp_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1116
  assumes "finite S"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1117
  shows "Set.insert x R O S = 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1118
    Finite_Set.fold (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') (R O S) S"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1119
proof -
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1120
  have "Set.insert x R O S = ({x} O S) \<union> (R O S)" by auto
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1121
  then show ?thesis by (auto simp: insert_relcomp_union_fold[OF assms])
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1122
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1123
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1124
lemma comp_fun_commute_relcomp_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1125
  assumes "finite S"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1126
  shows "comp_fun_commute (\<lambda>(x,y) A. 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1127
    Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S)"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1128
proof -
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1129
  have *: "\<And>a b A. 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1130
    Finite_Set.fold (\<lambda>(w, z) A'. if b = w then Set.insert (a, z) A' else A') A S = {(a,b)} O S \<union> A"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1131
    by (auto simp: insert_relcomp_union_fold[OF assms] cong: if_cong)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1132
  show ?thesis by default (auto simp: *)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1133
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1134
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1135
lemma relcomp_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1136
  assumes "finite R"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1137
  assumes "finite S"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1138
  shows "R O S = Finite_Set.fold 
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1139
    (\<lambda>(x,y) A. Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S) {} R"
52749
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
  1140
  using assms by (induct R)
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
  1141
    (auto simp: comp_fun_commute.fold_insert comp_fun_commute_relcomp_fold insert_relcomp_fold
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1142
      cong: if_cong)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1143
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1144
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1145
1128
64b30e3cc6d4 Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff changeset
  1146
end
46689
f559866a7aa2 marked candidates for rule declarations
haftmann
parents: 46664
diff changeset
  1147