author | kuncar |
Thu, 24 May 2012 14:20:23 +0200 | |
changeset 47982 | 7aa35601ff65 |
parent 47936 | 756f30eac792 |
child 51377 | 7da251a6c16e |
permissions | -rw-r--r-- |
47455 | 1 |
(* Title: HOL/Library/Quotient_Set.thy |
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Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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Author: Cezary Kaliszyk and Christian Urban |
80d460bc6fa8
Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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*) |
80d460bc6fa8
Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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4 |
|
80d460bc6fa8
Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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header {* Quotient infrastructure for the set type *} |
80d460bc6fa8
Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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6 |
|
80d460bc6fa8
Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
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theory Quotient_Set |
80d460bc6fa8
Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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8 |
imports Main Quotient_Syntax |
80d460bc6fa8
Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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begin |
80d460bc6fa8
Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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|
47648
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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subsection {* Relator for set type *} |
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parents:
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12 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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parents:
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definition set_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" |
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where "set_rel R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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15 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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parents:
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16 |
lemma set_relI: |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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parents:
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17 |
assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
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18 |
assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
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|
19 |
shows "set_rel R A B" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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|
20 |
using assms unfolding set_rel_def by simp |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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21 |
|
6b9d20a095ae
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parents:
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22 |
lemma set_rel_conversep: "set_rel (conversep R) = conversep (set_rel R)" |
6b9d20a095ae
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parents:
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23 |
unfolding set_rel_def by auto |
6b9d20a095ae
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huffman
parents:
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24 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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parents:
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25 |
lemma set_rel_OO: "set_rel (R OO S) = set_rel R OO set_rel S" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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26 |
apply (intro ext, rename_tac X Z) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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27 |
apply (rule iffI) |
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parents:
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28 |
apply (rule_tac b="{y. (\<exists>x\<in>X. R x y) \<and> (\<exists>z\<in>Z. S y z)}" in relcomppI) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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29 |
apply (simp add: set_rel_def, fast) |
6b9d20a095ae
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huffman
parents:
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diff
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30 |
apply (simp add: set_rel_def, fast) |
6b9d20a095ae
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huffman
parents:
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|
31 |
apply (simp add: set_rel_def, fast) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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|
32 |
done |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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|
33 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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|
34 |
lemma set_rel_eq [relator_eq]: "set_rel (op =) = (op =)" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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parents:
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35 |
unfolding set_rel_def fun_eq_iff by auto |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
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|
36 |
|
47982
7aa35601ff65
prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents:
47936
diff
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37 |
lemma reflp_set_rel[reflexivity_rule]: "reflp R \<Longrightarrow> reflp (set_rel R)" |
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6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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parents:
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38 |
unfolding reflp_def set_rel_def by fast |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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changeset
|
39 |
|
47982
7aa35601ff65
prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents:
47936
diff
changeset
|
40 |
lemma left_total_set_rel[reflexivity_rule]: |
7aa35601ff65
prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents:
47936
diff
changeset
|
41 |
assumes lt_R: "left_total R" |
7aa35601ff65
prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents:
47936
diff
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42 |
shows "left_total (set_rel R)" |
7aa35601ff65
prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents:
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diff
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|
43 |
proof - |
7aa35601ff65
prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents:
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diff
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|
44 |
{ |
7aa35601ff65
prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents:
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45 |
fix A |
7aa35601ff65
prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents:
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46 |
let ?B = "{y. \<exists>x \<in> A. R x y}" |
7aa35601ff65
prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents:
47936
diff
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|
47 |
have "(\<forall>x\<in>A. \<exists>y\<in>?B. R x y) \<and> (\<forall>y\<in>?B. \<exists>x\<in>A. R x y)" using lt_R by(elim left_totalE) blast |
7aa35601ff65
prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents:
47936
diff
changeset
|
48 |
} |
7aa35601ff65
prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents:
47936
diff
changeset
|
49 |
then have "\<And>A. \<exists>B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y)" by blast |
7aa35601ff65
prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents:
47936
diff
changeset
|
50 |
then show ?thesis by (auto simp: set_rel_def intro: left_totalI) |
7aa35601ff65
prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents:
47936
diff
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|
51 |
qed |
7aa35601ff65
prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar
parents:
47936
diff
changeset
|
52 |
|
47648
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
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|
53 |
lemma symp_set_rel: "symp R \<Longrightarrow> symp (set_rel R)" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
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|
54 |
unfolding symp_def set_rel_def by fast |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
55 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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parents:
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diff
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|
56 |
lemma transp_set_rel: "transp R \<Longrightarrow> transp (set_rel R)" |
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added covariant relator set_rel, with transfer rules for set operations
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parents:
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diff
changeset
|
57 |
unfolding transp_def set_rel_def by fast |
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added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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|
58 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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|
59 |
lemma equivp_set_rel: "equivp R \<Longrightarrow> equivp (set_rel R)" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
60 |
by (blast intro: equivpI reflp_set_rel symp_set_rel transp_set_rel |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
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|
61 |
elim: equivpE) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
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|
62 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
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|
63 |
lemma right_total_set_rel [transfer_rule]: |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
64 |
"right_total A \<Longrightarrow> right_total (set_rel A)" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
65 |
unfolding right_total_def set_rel_def |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
66 |
by (rule allI, rename_tac Y, rule_tac x="{x. \<exists>y\<in>Y. A x y}" in exI, fast) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
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|
67 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
68 |
lemma right_unique_set_rel [transfer_rule]: |
6b9d20a095ae
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huffman
parents:
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diff
changeset
|
69 |
"right_unique A \<Longrightarrow> right_unique (set_rel A)" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
70 |
unfolding right_unique_def set_rel_def by fast |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
71 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
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|
72 |
lemma bi_total_set_rel [transfer_rule]: |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
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|
73 |
"bi_total A \<Longrightarrow> bi_total (set_rel A)" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
74 |
unfolding bi_total_def set_rel_def |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
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|
75 |
apply safe |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
76 |
apply (rename_tac X, rule_tac x="{y. \<exists>x\<in>X. A x y}" in exI, fast) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
77 |
apply (rename_tac Y, rule_tac x="{x. \<exists>y\<in>Y. A x y}" in exI, fast) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
78 |
done |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
79 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
80 |
lemma bi_unique_set_rel [transfer_rule]: |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
81 |
"bi_unique A \<Longrightarrow> bi_unique (set_rel A)" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
82 |
unfolding bi_unique_def set_rel_def by fast |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
83 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
84 |
subsection {* Transfer rules for transfer package *} |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
85 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
86 |
subsubsection {* Unconditional transfer rules *} |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
87 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
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|
88 |
lemma empty_transfer [transfer_rule]: "(set_rel A) {} {}" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
89 |
unfolding set_rel_def by simp |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
90 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
91 |
lemma insert_transfer [transfer_rule]: |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
92 |
"(A ===> set_rel A ===> set_rel A) insert insert" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
93 |
unfolding fun_rel_def set_rel_def by auto |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
94 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
95 |
lemma union_transfer [transfer_rule]: |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
96 |
"(set_rel A ===> set_rel A ===> set_rel A) union union" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
97 |
unfolding fun_rel_def set_rel_def by auto |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
98 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
99 |
lemma Union_transfer [transfer_rule]: |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
100 |
"(set_rel (set_rel A) ===> set_rel A) Union Union" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
101 |
unfolding fun_rel_def set_rel_def by simp fast |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
102 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
103 |
lemma image_transfer [transfer_rule]: |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
104 |
"((A ===> B) ===> set_rel A ===> set_rel B) image image" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
105 |
unfolding fun_rel_def set_rel_def by simp fast |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
106 |
|
47660 | 107 |
lemma UNION_transfer [transfer_rule]: |
108 |
"(set_rel A ===> (A ===> set_rel B) ===> set_rel B) UNION UNION" |
|
109 |
unfolding SUP_def [abs_def] by transfer_prover |
|
110 |
||
47648
6b9d20a095ae
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huffman
parents:
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diff
changeset
|
111 |
lemma Ball_transfer [transfer_rule]: |
6b9d20a095ae
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huffman
parents:
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diff
changeset
|
112 |
"(set_rel A ===> (A ===> op =) ===> op =) Ball Ball" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
113 |
unfolding set_rel_def fun_rel_def by fast |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
114 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
115 |
lemma Bex_transfer [transfer_rule]: |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
116 |
"(set_rel A ===> (A ===> op =) ===> op =) Bex Bex" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
117 |
unfolding set_rel_def fun_rel_def by fast |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
118 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
119 |
lemma Pow_transfer [transfer_rule]: |
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120 |
"(set_rel A ===> set_rel (set_rel A)) Pow Pow" |
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121 |
apply (rule fun_relI, rename_tac X Y, rule set_relI) |
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122 |
apply (rename_tac X', rule_tac x="{y\<in>Y. \<exists>x\<in>X'. A x y}" in rev_bexI, clarsimp) |
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|
123 |
apply (simp add: set_rel_def, fast) |
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|
124 |
apply (rename_tac Y', rule_tac x="{x\<in>X. \<exists>y\<in>Y'. A x y}" in rev_bexI, clarsimp) |
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|
125 |
apply (simp add: set_rel_def, fast) |
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126 |
done |
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|
127 |
|
47922 | 128 |
lemma set_rel_transfer [transfer_rule]: |
129 |
"((A ===> B ===> op =) ===> set_rel A ===> set_rel B ===> op =) |
|
130 |
set_rel set_rel" |
|
131 |
unfolding fun_rel_def set_rel_def by fast |
|
132 |
||
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133 |
subsubsection {* Rules requiring bi-unique or bi-total relations *} |
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134 |
|
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|
135 |
lemma member_transfer [transfer_rule]: |
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|
136 |
assumes "bi_unique A" |
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|
137 |
shows "(A ===> set_rel A ===> op =) (op \<in>) (op \<in>)" |
6b9d20a095ae
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changeset
|
138 |
using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast |
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|
139 |
|
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changeset
|
140 |
lemma Collect_transfer [transfer_rule]: |
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|
141 |
assumes "bi_total A" |
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changeset
|
142 |
shows "((A ===> op =) ===> set_rel A) Collect Collect" |
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changeset
|
143 |
using assms unfolding fun_rel_def set_rel_def bi_total_def by fast |
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|
144 |
|
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changeset
|
145 |
lemma inter_transfer [transfer_rule]: |
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|
146 |
assumes "bi_unique A" |
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changeset
|
147 |
shows "(set_rel A ===> set_rel A ===> set_rel A) inter inter" |
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changeset
|
148 |
using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast |
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changeset
|
149 |
|
47680 | 150 |
lemma Diff_transfer [transfer_rule]: |
151 |
assumes "bi_unique A" |
|
152 |
shows "(set_rel A ===> set_rel A ===> set_rel A) (op -) (op -)" |
|
153 |
using assms unfolding fun_rel_def set_rel_def bi_unique_def |
|
154 |
unfolding Ball_def Bex_def Diff_eq |
|
155 |
by (safe, simp, metis, simp, metis) |
|
156 |
||
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|
157 |
lemma subset_transfer [transfer_rule]: |
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|
158 |
assumes [transfer_rule]: "bi_unique A" |
6b9d20a095ae
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changeset
|
159 |
shows "(set_rel A ===> set_rel A ===> op =) (op \<subseteq>) (op \<subseteq>)" |
6b9d20a095ae
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changeset
|
160 |
unfolding subset_eq [abs_def] by transfer_prover |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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changeset
|
161 |
|
6b9d20a095ae
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changeset
|
162 |
lemma UNIV_transfer [transfer_rule]: |
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changeset
|
163 |
assumes "bi_total A" |
6b9d20a095ae
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parents:
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changeset
|
164 |
shows "(set_rel A) UNIV UNIV" |
6b9d20a095ae
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changeset
|
165 |
using assms unfolding set_rel_def bi_total_def by simp |
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changeset
|
166 |
|
6b9d20a095ae
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changeset
|
167 |
lemma Compl_transfer [transfer_rule]: |
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changeset
|
168 |
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A" |
6b9d20a095ae
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changeset
|
169 |
shows "(set_rel A ===> set_rel A) uminus uminus" |
6b9d20a095ae
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changeset
|
170 |
unfolding Compl_eq [abs_def] by transfer_prover |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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changeset
|
171 |
|
6b9d20a095ae
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changeset
|
172 |
lemma Inter_transfer [transfer_rule]: |
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changeset
|
173 |
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A" |
6b9d20a095ae
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changeset
|
174 |
shows "(set_rel (set_rel A) ===> set_rel A) Inter Inter" |
6b9d20a095ae
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changeset
|
175 |
unfolding Inter_eq [abs_def] by transfer_prover |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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diff
changeset
|
176 |
|
6b9d20a095ae
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changeset
|
177 |
lemma finite_transfer [transfer_rule]: |
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changeset
|
178 |
assumes "bi_unique A" |
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changeset
|
179 |
shows "(set_rel A ===> op =) finite finite" |
6b9d20a095ae
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parents:
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changeset
|
180 |
apply (rule fun_relI, rename_tac X Y) |
6b9d20a095ae
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parents:
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changeset
|
181 |
apply (rule iffI) |
6b9d20a095ae
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changeset
|
182 |
apply (subgoal_tac "Y \<subseteq> (\<lambda>x. THE y. A x y) ` X") |
6b9d20a095ae
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changeset
|
183 |
apply (erule finite_subset, erule finite_imageI) |
6b9d20a095ae
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changeset
|
184 |
apply (rule subsetI, rename_tac y) |
6b9d20a095ae
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diff
changeset
|
185 |
apply (clarsimp simp add: set_rel_def) |
6b9d20a095ae
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parents:
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diff
changeset
|
186 |
apply (drule (1) bspec, clarify) |
6b9d20a095ae
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changeset
|
187 |
apply (rule image_eqI) |
6b9d20a095ae
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changeset
|
188 |
apply (rule the_equality [symmetric]) |
6b9d20a095ae
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changeset
|
189 |
apply assumption |
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parents:
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diff
changeset
|
190 |
apply (simp add: assms [unfolded bi_unique_def]) |
6b9d20a095ae
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parents:
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diff
changeset
|
191 |
apply assumption |
6b9d20a095ae
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diff
changeset
|
192 |
apply (subgoal_tac "X \<subseteq> (\<lambda>y. THE x. A x y) ` Y") |
6b9d20a095ae
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huffman
parents:
47647
diff
changeset
|
193 |
apply (erule finite_subset, erule finite_imageI) |
6b9d20a095ae
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huffman
parents:
47647
diff
changeset
|
194 |
apply (rule subsetI, rename_tac x) |
6b9d20a095ae
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huffman
parents:
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diff
changeset
|
195 |
apply (clarsimp simp add: set_rel_def) |
6b9d20a095ae
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parents:
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diff
changeset
|
196 |
apply (drule (1) bspec, clarify) |
6b9d20a095ae
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parents:
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diff
changeset
|
197 |
apply (rule image_eqI) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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47647
diff
changeset
|
198 |
apply (rule the_equality [symmetric]) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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parents:
47647
diff
changeset
|
199 |
apply assumption |
6b9d20a095ae
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parents:
47647
diff
changeset
|
200 |
apply (simp add: assms [unfolded bi_unique_def]) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
201 |
apply assumption |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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parents:
47647
diff
changeset
|
202 |
done |
6b9d20a095ae
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huffman
parents:
47647
diff
changeset
|
203 |
|
6b9d20a095ae
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parents:
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diff
changeset
|
204 |
subsection {* Setup for lifting package *} |
6b9d20a095ae
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huffman
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diff
changeset
|
205 |
|
47777
f29e7dcd7c40
use a quot_map theorem attribute instead of the complicated map attribute
kuncar
parents:
47680
diff
changeset
|
206 |
lemma Quotient_set[quot_map]: |
47648
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
207 |
assumes "Quotient R Abs Rep T" |
6b9d20a095ae
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huffman
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diff
changeset
|
208 |
shows "Quotient (set_rel R) (image Abs) (image Rep) (set_rel T)" |
6b9d20a095ae
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parents:
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diff
changeset
|
209 |
using assms unfolding Quotient_alt_def4 |
6b9d20a095ae
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parents:
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diff
changeset
|
210 |
apply (simp add: set_rel_OO set_rel_conversep) |
6b9d20a095ae
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huffman
parents:
47647
diff
changeset
|
211 |
apply (simp add: set_rel_def, fast) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
212 |
done |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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parents:
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diff
changeset
|
213 |
|
6b9d20a095ae
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parents:
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diff
changeset
|
214 |
lemma set_invariant_commute [invariant_commute]: |
6b9d20a095ae
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huffman
parents:
47647
diff
changeset
|
215 |
"set_rel (Lifting.invariant P) = Lifting.invariant (\<lambda>A. Ball A P)" |
6b9d20a095ae
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huffman
parents:
47647
diff
changeset
|
216 |
unfolding fun_eq_iff set_rel_def Lifting.invariant_def Ball_def by fast |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
217 |
|
6b9d20a095ae
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parents:
47647
diff
changeset
|
218 |
subsection {* Contravariant set map (vimage) and set relator *} |
47626
f7b1034cb9ce
move definition of set_rel into Library/Quotient_Set.thy
huffman
parents:
47455
diff
changeset
|
219 |
|
47647
ec29cc09599d
renamed contravariant relator set_rel to vset_rel, to make room for new covariant relator
huffman
parents:
47626
diff
changeset
|
220 |
definition "vset_rel R xs ys \<equiv> \<forall>x y. R x y \<longrightarrow> x \<in> xs \<longleftrightarrow> y \<in> ys" |
47626
f7b1034cb9ce
move definition of set_rel into Library/Quotient_Set.thy
huffman
parents:
47455
diff
changeset
|
221 |
|
47647
ec29cc09599d
renamed contravariant relator set_rel to vset_rel, to make room for new covariant relator
huffman
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47626
diff
changeset
|
222 |
lemma vset_rel_eq [id_simps]: |
ec29cc09599d
renamed contravariant relator set_rel to vset_rel, to make room for new covariant relator
huffman
parents:
47626
diff
changeset
|
223 |
"vset_rel op = = op =" |
ec29cc09599d
renamed contravariant relator set_rel to vset_rel, to make room for new covariant relator
huffman
parents:
47626
diff
changeset
|
224 |
by (subst fun_eq_iff, subst fun_eq_iff) (simp add: set_eq_iff vset_rel_def) |
47626
f7b1034cb9ce
move definition of set_rel into Library/Quotient_Set.thy
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parents:
47455
diff
changeset
|
225 |
|
47647
ec29cc09599d
renamed contravariant relator set_rel to vset_rel, to make room for new covariant relator
huffman
parents:
47626
diff
changeset
|
226 |
lemma vset_rel_equivp: |
47626
f7b1034cb9ce
move definition of set_rel into Library/Quotient_Set.thy
huffman
parents:
47455
diff
changeset
|
227 |
assumes e: "equivp R" |
47647
ec29cc09599d
renamed contravariant relator set_rel to vset_rel, to make room for new covariant relator
huffman
parents:
47626
diff
changeset
|
228 |
shows "vset_rel R xs ys \<longleftrightarrow> xs = ys \<and> (\<forall>x y. x \<in> xs \<longrightarrow> R x y \<longrightarrow> y \<in> xs)" |
ec29cc09599d
renamed contravariant relator set_rel to vset_rel, to make room for new covariant relator
huffman
parents:
47626
diff
changeset
|
229 |
unfolding vset_rel_def |
47626
f7b1034cb9ce
move definition of set_rel into Library/Quotient_Set.thy
huffman
parents:
47455
diff
changeset
|
230 |
using equivp_reflp[OF e] |
f7b1034cb9ce
move definition of set_rel into Library/Quotient_Set.thy
huffman
parents:
47455
diff
changeset
|
231 |
by auto (metis, metis equivp_symp[OF e]) |
f7b1034cb9ce
move definition of set_rel into Library/Quotient_Set.thy
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parents:
47455
diff
changeset
|
232 |
|
44413
80d460bc6fa8
Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
233 |
lemma set_quotient [quot_thm]: |
47308 | 234 |
assumes "Quotient3 R Abs Rep" |
47647
ec29cc09599d
renamed contravariant relator set_rel to vset_rel, to make room for new covariant relator
huffman
parents:
47626
diff
changeset
|
235 |
shows "Quotient3 (vset_rel R) (vimage Rep) (vimage Abs)" |
47308 | 236 |
proof (rule Quotient3I) |
237 |
from assms have "\<And>x. Abs (Rep x) = x" by (rule Quotient3_abs_rep) |
|
44413
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Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
238 |
then show "\<And>xs. Rep -` (Abs -` xs) = xs" |
80d460bc6fa8
Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
239 |
unfolding vimage_def by auto |
80d460bc6fa8
Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
240 |
next |
47647
ec29cc09599d
renamed contravariant relator set_rel to vset_rel, to make room for new covariant relator
huffman
parents:
47626
diff
changeset
|
241 |
show "\<And>xs. vset_rel R (Abs -` xs) (Abs -` xs)" |
ec29cc09599d
renamed contravariant relator set_rel to vset_rel, to make room for new covariant relator
huffman
parents:
47626
diff
changeset
|
242 |
unfolding vset_rel_def vimage_def |
47308 | 243 |
by auto (metis Quotient3_rel_abs[OF assms])+ |
44413
80d460bc6fa8
Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
244 |
next |
80d460bc6fa8
Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
245 |
fix r s |
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show "vset_rel R r s = (vset_rel R r r \<and> vset_rel R s s \<and> Rep -` r = Rep -` s)" |
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unfolding vset_rel_def vimage_def set_eq_iff |
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by auto (metis rep_abs_rsp[OF assms] assms[simplified Quotient3_def])+ |
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qed |
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declare [[mapQ3 set = (vset_rel, set_quotient)]] |
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|
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lemma empty_set_rsp[quot_respect]: |
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"vset_rel R {} {}" |
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unfolding vset_rel_def by simp |
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lemma collect_rsp[quot_respect]: |
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assumes "Quotient3 R Abs Rep" |
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259 |
shows "((R ===> op =) ===> vset_rel R) Collect Collect" |
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by (intro fun_relI) (simp add: fun_rel_def vset_rel_def) |
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lemma collect_prs[quot_preserve]: |
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assumes "Quotient3 R Abs Rep" |
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shows "((Abs ---> id) ---> op -` Rep) Collect = Collect" |
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unfolding fun_eq_iff |
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by (simp add: Quotient3_abs_rep[OF assms]) |
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lemma union_rsp[quot_respect]: |
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assumes "Quotient3 R Abs Rep" |
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shows "(vset_rel R ===> vset_rel R ===> vset_rel R) op \<union> op \<union>" |
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by (intro fun_relI) (simp add: vset_rel_def) |
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lemma union_prs[quot_preserve]: |
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assumes "Quotient3 R Abs Rep" |
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shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op \<union> = op \<union>" |
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unfolding fun_eq_iff |
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by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]]) |
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lemma diff_rsp[quot_respect]: |
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assumes "Quotient3 R Abs Rep" |
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shows "(vset_rel R ===> vset_rel R ===> vset_rel R) op - op -" |
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by (intro fun_relI) (simp add: vset_rel_def) |
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lemma diff_prs[quot_preserve]: |
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assumes "Quotient3 R Abs Rep" |
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shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op - = op -" |
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unfolding fun_eq_iff |
47308 | 288 |
by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]] vimage_Diff) |
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lemma inter_rsp[quot_respect]: |
47308 | 291 |
assumes "Quotient3 R Abs Rep" |
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shows "(vset_rel R ===> vset_rel R ===> vset_rel R) op \<inter> op \<inter>" |
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by (intro fun_relI) (auto simp add: vset_rel_def) |
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lemma inter_prs[quot_preserve]: |
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assumes "Quotient3 R Abs Rep" |
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shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op \<inter> = op \<inter>" |
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unfolding fun_eq_iff |
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by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]]) |
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|
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lemma mem_prs[quot_preserve]: |
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assumes "Quotient3 R Abs Rep" |
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shows "(Rep ---> op -` Abs ---> id) op \<in> = op \<in>" |
47308 | 304 |
by (simp add: fun_eq_iff Quotient3_abs_rep[OF assms]) |
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|
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306 |
lemma mem_rsp[quot_respect]: |
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shows "(R ===> vset_rel R ===> op =) op \<in> op \<in>" |
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by (intro fun_relI) (simp add: vset_rel_def) |
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|
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310 |
end |