author | haftmann |
Wed, 02 Jan 2008 15:14:02 +0100 | |
changeset 25762 | c03e9d04b3e4 |
parent 25622 | 6067d838041a |
child 26193 | 37a7eb7fd5f7 |
permissions | -rw-r--r-- |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
1 |
(* Title : HOL/Hyperreal/StarDef.thy |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
2 |
ID : $Id$ |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
3 |
Author : Jacques D. Fleuriot and Brian Huffman |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
4 |
*) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
5 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
6 |
header {* Construction of Star Types Using Ultrafilters *} |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
7 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
8 |
theory StarDef |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
9 |
imports Filter |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
10 |
uses ("transfer.ML") |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
11 |
begin |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
12 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
13 |
subsection {* A Free Ultrafilter over the Naturals *} |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
14 |
|
19765 | 15 |
definition |
21404
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more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
16 |
FreeUltrafilterNat :: "nat set set" ("\<U>") where |
19765 | 17 |
"\<U> = (SOME U. freeultrafilter U)" |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
18 |
|
21787
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
19 |
lemma freeultrafilter_FreeUltrafilterNat: "freeultrafilter \<U>" |
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
20 |
apply (unfold FreeUltrafilterNat_def) |
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
21 |
apply (rule someI_ex) |
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
22 |
apply (rule freeultrafilter_Ex) |
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
23 |
apply (rule nat_infinite) |
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
24 |
done |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
25 |
|
21787
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
26 |
interpretation FreeUltrafilterNat: freeultrafilter [FreeUltrafilterNat] |
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
27 |
by (rule freeultrafilter_FreeUltrafilterNat) |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
28 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
29 |
text {* This rule takes the place of the old ultra tactic *} |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
30 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
31 |
lemma ultra: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
32 |
"\<lbrakk>{n. P n} \<in> \<U>; {n. P n \<longrightarrow> Q n} \<in> \<U>\<rbrakk> \<Longrightarrow> {n. Q n} \<in> \<U>" |
21787
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
33 |
by (simp add: Collect_imp_eq |
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
34 |
FreeUltrafilterNat.Un_iff FreeUltrafilterNat.Compl_iff) |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
35 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
36 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
37 |
subsection {* Definition of @{text star} type constructor *} |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
38 |
|
19765 | 39 |
definition |
21404
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more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
40 |
starrel :: "((nat \<Rightarrow> 'a) \<times> (nat \<Rightarrow> 'a)) set" where |
19765 | 41 |
"starrel = {(X,Y). {n. X n = Y n} \<in> \<U>}" |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
42 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
43 |
typedef 'a star = "(UNIV :: (nat \<Rightarrow> 'a) set) // starrel" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
44 |
by (auto intro: quotientI) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
45 |
|
19765 | 46 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
47 |
star_n :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a star" where |
19765 | 48 |
"star_n X = Abs_star (starrel `` {X})" |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
49 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
50 |
theorem star_cases [case_names star_n, cases type: star]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
51 |
"(\<And>X. x = star_n X \<Longrightarrow> P) \<Longrightarrow> P" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
52 |
by (cases x, unfold star_n_def star_def, erule quotientE, fast) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
53 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
54 |
lemma all_star_eq: "(\<forall>x. P x) = (\<forall>X. P (star_n X))" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
55 |
by (auto, rule_tac x=x in star_cases, simp) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
56 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
57 |
lemma ex_star_eq: "(\<exists>x. P x) = (\<exists>X. P (star_n X))" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
58 |
by (auto, rule_tac x=x in star_cases, auto) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
59 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
60 |
text {* Proving that @{term starrel} is an equivalence relation *} |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
61 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
62 |
lemma starrel_iff [iff]: "((X,Y) \<in> starrel) = ({n. X n = Y n} \<in> \<U>)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
63 |
by (simp add: starrel_def) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
64 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
65 |
lemma equiv_starrel: "equiv UNIV starrel" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
66 |
proof (rule equiv.intro) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
67 |
show "reflexive starrel" by (simp add: refl_def) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
68 |
show "sym starrel" by (simp add: sym_def eq_commute) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
69 |
show "trans starrel" by (auto intro: transI elim!: ultra) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
70 |
qed |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
71 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
72 |
lemmas equiv_starrel_iff = |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
73 |
eq_equiv_class_iff [OF equiv_starrel UNIV_I UNIV_I] |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
74 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
75 |
lemma starrel_in_star: "starrel``{x} \<in> star" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
76 |
by (simp add: star_def quotientI) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
77 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
78 |
lemma star_n_eq_iff: "(star_n X = star_n Y) = ({n. X n = Y n} \<in> \<U>)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
79 |
by (simp add: star_n_def Abs_star_inject starrel_in_star equiv_starrel_iff) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
80 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
81 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
82 |
subsection {* Transfer principle *} |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
83 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
84 |
text {* This introduction rule starts each transfer proof. *} |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
85 |
lemma transfer_start: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
86 |
"P \<equiv> {n. Q} \<in> \<U> \<Longrightarrow> Trueprop P \<equiv> Trueprop Q" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
87 |
by (subgoal_tac "P \<equiv> Q", simp, simp add: atomize_eq) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
88 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
89 |
text {*Initialize transfer tactic.*} |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
90 |
use "transfer.ML" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
91 |
setup Transfer.setup |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
92 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
93 |
text {* Transfer introduction rules. *} |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
94 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
95 |
lemma transfer_ex [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
96 |
"\<lbrakk>\<And>X. p (star_n X) \<equiv> {n. P n (X n)} \<in> \<U>\<rbrakk> |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
97 |
\<Longrightarrow> \<exists>x::'a star. p x \<equiv> {n. \<exists>x. P n x} \<in> \<U>" |
21787
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
98 |
by (simp only: ex_star_eq FreeUltrafilterNat.Collect_ex) |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
99 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
100 |
lemma transfer_all [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
101 |
"\<lbrakk>\<And>X. p (star_n X) \<equiv> {n. P n (X n)} \<in> \<U>\<rbrakk> |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
102 |
\<Longrightarrow> \<forall>x::'a star. p x \<equiv> {n. \<forall>x. P n x} \<in> \<U>" |
21787
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
103 |
by (simp only: all_star_eq FreeUltrafilterNat.Collect_all) |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
104 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
105 |
lemma transfer_not [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
106 |
"\<lbrakk>p \<equiv> {n. P n} \<in> \<U>\<rbrakk> \<Longrightarrow> \<not> p \<equiv> {n. \<not> P n} \<in> \<U>" |
21787
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
107 |
by (simp only: FreeUltrafilterNat.Collect_not) |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
108 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
109 |
lemma transfer_conj [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
110 |
"\<lbrakk>p \<equiv> {n. P n} \<in> \<U>; q \<equiv> {n. Q n} \<in> \<U>\<rbrakk> |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
111 |
\<Longrightarrow> p \<and> q \<equiv> {n. P n \<and> Q n} \<in> \<U>" |
21787
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
112 |
by (simp only: FreeUltrafilterNat.Collect_conj) |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
113 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
114 |
lemma transfer_disj [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
115 |
"\<lbrakk>p \<equiv> {n. P n} \<in> \<U>; q \<equiv> {n. Q n} \<in> \<U>\<rbrakk> |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
116 |
\<Longrightarrow> p \<or> q \<equiv> {n. P n \<or> Q n} \<in> \<U>" |
21787
9edd495b6330
consistent naming for FreeUltrafilterNat lemmas; cleaned up
huffman
parents:
21404
diff
changeset
|
117 |
by (simp only: FreeUltrafilterNat.Collect_disj) |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
118 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
119 |
lemma transfer_imp [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
120 |
"\<lbrakk>p \<equiv> {n. P n} \<in> \<U>; q \<equiv> {n. Q n} \<in> \<U>\<rbrakk> |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
121 |
\<Longrightarrow> p \<longrightarrow> q \<equiv> {n. P n \<longrightarrow> Q n} \<in> \<U>" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
122 |
by (simp only: imp_conv_disj transfer_disj transfer_not) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
123 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
124 |
lemma transfer_iff [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
125 |
"\<lbrakk>p \<equiv> {n. P n} \<in> \<U>; q \<equiv> {n. Q n} \<in> \<U>\<rbrakk> |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
126 |
\<Longrightarrow> p = q \<equiv> {n. P n = Q n} \<in> \<U>" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
127 |
by (simp only: iff_conv_conj_imp transfer_conj transfer_imp) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
128 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
129 |
lemma transfer_if_bool [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
130 |
"\<lbrakk>p \<equiv> {n. P n} \<in> \<U>; x \<equiv> {n. X n} \<in> \<U>; y \<equiv> {n. Y n} \<in> \<U>\<rbrakk> |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
131 |
\<Longrightarrow> (if p then x else y) \<equiv> {n. if P n then X n else Y n} \<in> \<U>" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
132 |
by (simp only: if_bool_eq_conj transfer_conj transfer_imp transfer_not) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
133 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
134 |
lemma transfer_eq [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
135 |
"\<lbrakk>x \<equiv> star_n X; y \<equiv> star_n Y\<rbrakk> \<Longrightarrow> x = y \<equiv> {n. X n = Y n} \<in> \<U>" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
136 |
by (simp only: star_n_eq_iff) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
137 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
138 |
lemma transfer_if [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
139 |
"\<lbrakk>p \<equiv> {n. P n} \<in> \<U>; x \<equiv> star_n X; y \<equiv> star_n Y\<rbrakk> |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
140 |
\<Longrightarrow> (if p then x else y) \<equiv> star_n (\<lambda>n. if P n then X n else Y n)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
141 |
apply (rule eq_reflection) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
142 |
apply (auto simp add: star_n_eq_iff transfer_not elim!: ultra) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
143 |
done |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
144 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
145 |
lemma transfer_fun_eq [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
146 |
"\<lbrakk>\<And>X. f (star_n X) = g (star_n X) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
147 |
\<equiv> {n. F n (X n) = G n (X n)} \<in> \<U>\<rbrakk> |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
148 |
\<Longrightarrow> f = g \<equiv> {n. F n = G n} \<in> \<U>" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
149 |
by (simp only: expand_fun_eq transfer_all) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
150 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
151 |
lemma transfer_star_n [transfer_intro]: "star_n X \<equiv> star_n (\<lambda>n. X n)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
152 |
by (rule reflexive) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
153 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
154 |
lemma transfer_bool [transfer_intro]: "p \<equiv> {n. p} \<in> \<U>" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
155 |
by (simp add: atomize_eq) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
156 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
157 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
158 |
subsection {* Standard elements *} |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
159 |
|
19765 | 160 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
161 |
star_of :: "'a \<Rightarrow> 'a star" where |
19765 | 162 |
"star_of x == star_n (\<lambda>n. x)" |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
163 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
164 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
165 |
Standard :: "'a star set" where |
20719 | 166 |
"Standard = range star_of" |
167 |
||
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
168 |
text {* Transfer tactic should remove occurrences of @{term star_of} *} |
18708 | 169 |
setup {* Transfer.add_const "StarDef.star_of" *} |
20719 | 170 |
|
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
171 |
declare star_of_def [transfer_intro] |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
172 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
173 |
lemma star_of_inject: "(star_of x = star_of y) = (x = y)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
174 |
by (transfer, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
175 |
|
20719 | 176 |
lemma Standard_star_of [simp]: "star_of x \<in> Standard" |
177 |
by (simp add: Standard_def) |
|
178 |
||
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
179 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
180 |
subsection {* Internal functions *} |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
181 |
|
19765 | 182 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
183 |
Ifun :: "('a \<Rightarrow> 'b) star \<Rightarrow> 'a star \<Rightarrow> 'b star" ("_ \<star> _" [300,301] 300) where |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
184 |
"Ifun f \<equiv> \<lambda>x. Abs_star |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
185 |
(\<Union>F\<in>Rep_star f. \<Union>X\<in>Rep_star x. starrel``{\<lambda>n. F n (X n)})" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
186 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
187 |
lemma Ifun_congruent2: |
19980
dc17fd6c0908
replaced respects2 by congruent2 because of type problem
nipkow
parents:
19765
diff
changeset
|
188 |
"congruent2 starrel starrel (\<lambda>F X. starrel``{\<lambda>n. F n (X n)})" |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
189 |
by (auto simp add: congruent2_def equiv_starrel_iff elim!: ultra) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
190 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
191 |
lemma Ifun_star_n: "star_n F \<star> star_n X = star_n (\<lambda>n. F n (X n))" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
192 |
by (simp add: Ifun_def star_n_def Abs_star_inverse starrel_in_star |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
193 |
UN_equiv_class2 [OF equiv_starrel equiv_starrel Ifun_congruent2]) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
194 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
195 |
text {* Transfer tactic should remove occurrences of @{term Ifun} *} |
18708 | 196 |
setup {* Transfer.add_const "StarDef.Ifun" *} |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
197 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
198 |
lemma transfer_Ifun [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
199 |
"\<lbrakk>f \<equiv> star_n F; x \<equiv> star_n X\<rbrakk> \<Longrightarrow> f \<star> x \<equiv> star_n (\<lambda>n. F n (X n))" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
200 |
by (simp only: Ifun_star_n) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
201 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
202 |
lemma Ifun_star_of [simp]: "star_of f \<star> star_of x = star_of (f x)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
203 |
by (transfer, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
204 |
|
20719 | 205 |
lemma Standard_Ifun [simp]: |
206 |
"\<lbrakk>f \<in> Standard; x \<in> Standard\<rbrakk> \<Longrightarrow> f \<star> x \<in> Standard" |
|
207 |
by (auto simp add: Standard_def) |
|
208 |
||
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
209 |
text {* Nonstandard extensions of functions *} |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
210 |
|
19765 | 211 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
212 |
starfun :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a star \<Rightarrow> 'b star)" ("*f* _" [80] 80) where |
19765 | 213 |
"starfun f == \<lambda>x. star_of f \<star> x" |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
214 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
215 |
definition |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
216 |
starfun2 :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a star \<Rightarrow> 'b star \<Rightarrow> 'c star)" |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
217 |
("*f2* _" [80] 80) where |
19765 | 218 |
"starfun2 f == \<lambda>x y. star_of f \<star> x \<star> y" |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
219 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
220 |
declare starfun_def [transfer_unfold] |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
221 |
declare starfun2_def [transfer_unfold] |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
222 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
223 |
lemma starfun_star_n: "( *f* f) (star_n X) = star_n (\<lambda>n. f (X n))" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
224 |
by (simp only: starfun_def star_of_def Ifun_star_n) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
225 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
226 |
lemma starfun2_star_n: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
227 |
"( *f2* f) (star_n X) (star_n Y) = star_n (\<lambda>n. f (X n) (Y n))" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
228 |
by (simp only: starfun2_def star_of_def Ifun_star_n) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
229 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
230 |
lemma starfun_star_of [simp]: "( *f* f) (star_of x) = star_of (f x)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
231 |
by (transfer, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
232 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
233 |
lemma starfun2_star_of [simp]: "( *f2* f) (star_of x) = *f* f x" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
234 |
by (transfer, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
235 |
|
20719 | 236 |
lemma Standard_starfun [simp]: "x \<in> Standard \<Longrightarrow> starfun f x \<in> Standard" |
237 |
by (simp add: starfun_def) |
|
238 |
||
239 |
lemma Standard_starfun2 [simp]: |
|
240 |
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> starfun2 f x y \<in> Standard" |
|
241 |
by (simp add: starfun2_def) |
|
242 |
||
21887 | 243 |
lemma Standard_starfun_iff: |
244 |
assumes inj: "\<And>x y. f x = f y \<Longrightarrow> x = y" |
|
245 |
shows "(starfun f x \<in> Standard) = (x \<in> Standard)" |
|
246 |
proof |
|
247 |
assume "x \<in> Standard" |
|
248 |
thus "starfun f x \<in> Standard" by simp |
|
249 |
next |
|
250 |
have inj': "\<And>x y. starfun f x = starfun f y \<Longrightarrow> x = y" |
|
251 |
using inj by transfer |
|
252 |
assume "starfun f x \<in> Standard" |
|
253 |
then obtain b where b: "starfun f x = star_of b" |
|
254 |
unfolding Standard_def .. |
|
255 |
hence "\<exists>x. starfun f x = star_of b" .. |
|
256 |
hence "\<exists>a. f a = b" by transfer |
|
257 |
then obtain a where "f a = b" .. |
|
258 |
hence "starfun f (star_of a) = star_of b" by transfer |
|
259 |
with b have "starfun f x = starfun f (star_of a)" by simp |
|
260 |
hence "x = star_of a" by (rule inj') |
|
261 |
thus "x \<in> Standard" |
|
262 |
unfolding Standard_def by auto |
|
263 |
qed |
|
264 |
||
265 |
lemma Standard_starfun2_iff: |
|
266 |
assumes inj: "\<And>a b a' b'. f a b = f a' b' \<Longrightarrow> a = a' \<and> b = b'" |
|
267 |
shows "(starfun2 f x y \<in> Standard) = (x \<in> Standard \<and> y \<in> Standard)" |
|
268 |
proof |
|
269 |
assume "x \<in> Standard \<and> y \<in> Standard" |
|
270 |
thus "starfun2 f x y \<in> Standard" by simp |
|
271 |
next |
|
272 |
have inj': "\<And>x y z w. starfun2 f x y = starfun2 f z w \<Longrightarrow> x = z \<and> y = w" |
|
273 |
using inj by transfer |
|
274 |
assume "starfun2 f x y \<in> Standard" |
|
275 |
then obtain c where c: "starfun2 f x y = star_of c" |
|
276 |
unfolding Standard_def .. |
|
277 |
hence "\<exists>x y. starfun2 f x y = star_of c" by auto |
|
278 |
hence "\<exists>a b. f a b = c" by transfer |
|
279 |
then obtain a b where "f a b = c" by auto |
|
280 |
hence "starfun2 f (star_of a) (star_of b) = star_of c" |
|
281 |
by transfer |
|
282 |
with c have "starfun2 f x y = starfun2 f (star_of a) (star_of b)" |
|
283 |
by simp |
|
284 |
hence "x = star_of a \<and> y = star_of b" |
|
285 |
by (rule inj') |
|
286 |
thus "x \<in> Standard \<and> y \<in> Standard" |
|
287 |
unfolding Standard_def by auto |
|
288 |
qed |
|
289 |
||
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
290 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
291 |
subsection {* Internal predicates *} |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
292 |
|
19765 | 293 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
294 |
unstar :: "bool star \<Rightarrow> bool" where |
19765 | 295 |
"unstar b = (b = star_of True)" |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
296 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
297 |
lemma unstar_star_n: "unstar (star_n P) = ({n. P n} \<in> \<U>)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
298 |
by (simp add: unstar_def star_of_def star_n_eq_iff) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
299 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
300 |
lemma unstar_star_of [simp]: "unstar (star_of p) = p" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
301 |
by (simp add: unstar_def star_of_inject) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
302 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
303 |
text {* Transfer tactic should remove occurrences of @{term unstar} *} |
18708 | 304 |
setup {* Transfer.add_const "StarDef.unstar" *} |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
305 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
306 |
lemma transfer_unstar [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
307 |
"p \<equiv> star_n P \<Longrightarrow> unstar p \<equiv> {n. P n} \<in> \<U>" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
308 |
by (simp only: unstar_star_n) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
309 |
|
19765 | 310 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
311 |
starP :: "('a \<Rightarrow> bool) \<Rightarrow> 'a star \<Rightarrow> bool" ("*p* _" [80] 80) where |
19765 | 312 |
"*p* P = (\<lambda>x. unstar (star_of P \<star> x))" |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
313 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
314 |
definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
315 |
starP2 :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a star \<Rightarrow> 'b star \<Rightarrow> bool" ("*p2* _" [80] 80) where |
19765 | 316 |
"*p2* P = (\<lambda>x y. unstar (star_of P \<star> x \<star> y))" |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
317 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
318 |
declare starP_def [transfer_unfold] |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
319 |
declare starP2_def [transfer_unfold] |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
320 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
321 |
lemma starP_star_n: "( *p* P) (star_n X) = ({n. P (X n)} \<in> \<U>)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
322 |
by (simp only: starP_def star_of_def Ifun_star_n unstar_star_n) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
323 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
324 |
lemma starP2_star_n: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
325 |
"( *p2* P) (star_n X) (star_n Y) = ({n. P (X n) (Y n)} \<in> \<U>)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
326 |
by (simp only: starP2_def star_of_def Ifun_star_n unstar_star_n) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
327 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
328 |
lemma starP_star_of [simp]: "( *p* P) (star_of x) = P x" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
329 |
by (transfer, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
330 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
331 |
lemma starP2_star_of [simp]: "( *p2* P) (star_of x) = *p* P x" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
332 |
by (transfer, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
333 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
334 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
335 |
subsection {* Internal sets *} |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
336 |
|
19765 | 337 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
338 |
Iset :: "'a set star \<Rightarrow> 'a star set" where |
19765 | 339 |
"Iset A = {x. ( *p2* op \<in>) x A}" |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
340 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
341 |
lemma Iset_star_n: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
342 |
"(star_n X \<in> Iset (star_n A)) = ({n. X n \<in> A n} \<in> \<U>)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
343 |
by (simp add: Iset_def starP2_star_n) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
344 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
345 |
text {* Transfer tactic should remove occurrences of @{term Iset} *} |
18708 | 346 |
setup {* Transfer.add_const "StarDef.Iset" *} |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
347 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
348 |
lemma transfer_mem [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
349 |
"\<lbrakk>x \<equiv> star_n X; a \<equiv> Iset (star_n A)\<rbrakk> |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
350 |
\<Longrightarrow> x \<in> a \<equiv> {n. X n \<in> A n} \<in> \<U>" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
351 |
by (simp only: Iset_star_n) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
352 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
353 |
lemma transfer_Collect [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
354 |
"\<lbrakk>\<And>X. p (star_n X) \<equiv> {n. P n (X n)} \<in> \<U>\<rbrakk> |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
355 |
\<Longrightarrow> Collect p \<equiv> Iset (star_n (\<lambda>n. Collect (P n)))" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
356 |
by (simp add: atomize_eq expand_set_eq all_star_eq Iset_star_n) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
357 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
358 |
lemma transfer_set_eq [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
359 |
"\<lbrakk>a \<equiv> Iset (star_n A); b \<equiv> Iset (star_n B)\<rbrakk> |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
360 |
\<Longrightarrow> a = b \<equiv> {n. A n = B n} \<in> \<U>" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
361 |
by (simp only: expand_set_eq transfer_all transfer_iff transfer_mem) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
362 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
363 |
lemma transfer_ball [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
364 |
"\<lbrakk>a \<equiv> Iset (star_n A); \<And>X. p (star_n X) \<equiv> {n. P n (X n)} \<in> \<U>\<rbrakk> |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
365 |
\<Longrightarrow> \<forall>x\<in>a. p x \<equiv> {n. \<forall>x\<in>A n. P n x} \<in> \<U>" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
366 |
by (simp only: Ball_def transfer_all transfer_imp transfer_mem) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
367 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
368 |
lemma transfer_bex [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
369 |
"\<lbrakk>a \<equiv> Iset (star_n A); \<And>X. p (star_n X) \<equiv> {n. P n (X n)} \<in> \<U>\<rbrakk> |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
370 |
\<Longrightarrow> \<exists>x\<in>a. p x \<equiv> {n. \<exists>x\<in>A n. P n x} \<in> \<U>" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
371 |
by (simp only: Bex_def transfer_ex transfer_conj transfer_mem) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
372 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
373 |
lemma transfer_Iset [transfer_intro]: |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
374 |
"\<lbrakk>a \<equiv> star_n A\<rbrakk> \<Longrightarrow> Iset a \<equiv> Iset (star_n (\<lambda>n. A n))" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
375 |
by simp |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
376 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
377 |
text {* Nonstandard extensions of sets. *} |
19765 | 378 |
|
379 |
definition |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20719
diff
changeset
|
380 |
starset :: "'a set \<Rightarrow> 'a star set" ("*s* _" [80] 80) where |
19765 | 381 |
"starset A = Iset (star_of A)" |
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
382 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
383 |
declare starset_def [transfer_unfold] |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
384 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
385 |
lemma starset_mem: "(star_of x \<in> *s* A) = (x \<in> A)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
386 |
by (transfer, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
387 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
388 |
lemma starset_UNIV: "*s* (UNIV::'a set) = (UNIV::'a star set)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
389 |
by (transfer UNIV_def, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
390 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
391 |
lemma starset_empty: "*s* {} = {}" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
392 |
by (transfer empty_def, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
393 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
394 |
lemma starset_insert: "*s* (insert x A) = insert (star_of x) ( *s* A)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
395 |
by (transfer insert_def Un_def, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
396 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
397 |
lemma starset_Un: "*s* (A \<union> B) = *s* A \<union> *s* B" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
398 |
by (transfer Un_def, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
399 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
400 |
lemma starset_Int: "*s* (A \<inter> B) = *s* A \<inter> *s* B" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
401 |
by (transfer Int_def, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
402 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
403 |
lemma starset_Compl: "*s* -A = -( *s* A)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
404 |
by (transfer Compl_def, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
405 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
406 |
lemma starset_diff: "*s* (A - B) = *s* A - *s* B" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
407 |
by (transfer set_diff_def, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
408 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
409 |
lemma starset_image: "*s* (f ` A) = ( *f* f) ` ( *s* A)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
410 |
by (transfer image_def, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
411 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
412 |
lemma starset_vimage: "*s* (f -` A) = ( *f* f) -` ( *s* A)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
413 |
by (transfer vimage_def, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
414 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
415 |
lemma starset_subset: "( *s* A \<subseteq> *s* B) = (A \<subseteq> B)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
416 |
by (transfer subset_def, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
417 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
418 |
lemma starset_eq: "( *s* A = *s* B) = (A = B)" |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
419 |
by (transfer, rule refl) |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
420 |
|
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
421 |
lemmas starset_simps [simp] = |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
422 |
starset_mem starset_UNIV |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
423 |
starset_empty starset_insert |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
424 |
starset_Un starset_Int |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
425 |
starset_Compl starset_diff |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
426 |
starset_image starset_vimage |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
427 |
starset_subset starset_eq |
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
428 |
|
25601 | 429 |
|
430 |
subsection {* Syntactic classes *} |
|
431 |
||
432 |
instantiation star :: (zero) zero |
|
433 |
begin |
|
434 |
||
435 |
definition |
|
436 |
star_zero_def: "0 \<equiv> star_of 0" |
|
437 |
||
438 |
instance .. |
|
439 |
||
17429
e8d6ed3aacfe
merged Transfer.thy and StarType.thy into StarDef.thy; renamed Ifun2_of to starfun2; cleaned up
huffman
parents:
diff
changeset
|
440 |
end |
25601 | 441 |
|
442 |
instantiation star :: (one) one |
|
443 |
begin |
|
444 |
||
445 |
definition |
|
446 |
star_one_def: "1 \<equiv> star_of 1" |
|
447 |
||
448 |
instance .. |
|
449 |
||
450 |
end |
|
451 |
||
452 |
instantiation star :: (plus) plus |
|
453 |
begin |
|
454 |
||
455 |
definition |
|
456 |
star_add_def: "(op +) \<equiv> *f2* (op +)" |
|
457 |
||
458 |
instance .. |
|
459 |
||
460 |
end |
|
461 |
||
462 |
instantiation star :: (times) times |
|
463 |
begin |
|
464 |
||
465 |
definition |
|
466 |
star_mult_def: "(op *) \<equiv> *f2* (op *)" |
|
467 |
||
468 |
instance .. |
|
469 |
||
470 |
end |
|
471 |
||
25762 | 472 |
instantiation star :: (uminus) uminus |
25601 | 473 |
begin |
474 |
||
475 |
definition |
|
476 |
star_minus_def: "uminus \<equiv> *f* uminus" |
|
477 |
||
25762 | 478 |
instance .. |
479 |
||
480 |
end |
|
481 |
||
482 |
instantiation star :: (minus) minus |
|
483 |
begin |
|
484 |
||
25601 | 485 |
definition |
486 |
star_diff_def: "(op -) \<equiv> *f2* (op -)" |
|
487 |
||
488 |
instance .. |
|
489 |
||
490 |
end |
|
491 |
||
492 |
instantiation star :: (abs) abs |
|
493 |
begin |
|
494 |
||
495 |
definition |
|
496 |
star_abs_def: "abs \<equiv> *f* abs" |
|
497 |
||
498 |
instance .. |
|
499 |
||
500 |
end |
|
501 |
||
502 |
instantiation star :: (sgn) sgn |
|
503 |
begin |
|
504 |
||
505 |
definition |
|
506 |
star_sgn_def: "sgn \<equiv> *f* sgn" |
|
507 |
||
508 |
instance .. |
|
509 |
||
510 |
end |
|
511 |
||
512 |
instantiation star :: (inverse) inverse |
|
513 |
begin |
|
514 |
||
515 |
definition |
|
516 |
star_divide_def: "(op /) \<equiv> *f2* (op /)" |
|
517 |
||
518 |
definition |
|
519 |
star_inverse_def: "inverse \<equiv> *f* inverse" |
|
520 |
||
521 |
instance .. |
|
522 |
||
523 |
end |
|
524 |
||
525 |
instantiation star :: (number) number |
|
526 |
begin |
|
527 |
||
528 |
definition |
|
529 |
star_number_def: "number_of b \<equiv> star_of (number_of b)" |
|
530 |
||
531 |
instance .. |
|
532 |
||
533 |
end |
|
534 |
||
535 |
instantiation star :: (Divides.div) Divides.div |
|
536 |
begin |
|
537 |
||
538 |
definition |
|
539 |
star_div_def: "(op div) \<equiv> *f2* (op div)" |
|
540 |
||
541 |
definition |
|
542 |
star_mod_def: "(op mod) \<equiv> *f2* (op mod)" |
|
543 |
||
544 |
instance .. |
|
545 |
||
546 |
end |
|
547 |
||
548 |
instantiation star :: (power) power |
|
549 |
begin |
|
550 |
||
551 |
definition |
|
552 |
star_power_def: "(op ^) \<equiv> \<lambda>x n. ( *f* (\<lambda>x. x ^ n)) x" |
|
553 |
||
554 |
instance .. |
|
555 |
||
556 |
end |
|
557 |
||
558 |
instantiation star :: (ord) ord |
|
559 |
begin |
|
560 |
||
561 |
definition |
|
562 |
star_le_def: "(op \<le>) \<equiv> *p2* (op \<le>)" |
|
563 |
||
564 |
definition |
|
565 |
star_less_def: "(op <) \<equiv> *p2* (op <)" |
|
566 |
||
567 |
instance .. |
|
568 |
||
569 |
end |
|
570 |
||
571 |
lemmas star_class_defs [transfer_unfold] = |
|
572 |
star_zero_def star_one_def star_number_def |
|
573 |
star_add_def star_diff_def star_minus_def |
|
574 |
star_mult_def star_divide_def star_inverse_def |
|
575 |
star_le_def star_less_def star_abs_def star_sgn_def |
|
576 |
star_div_def star_mod_def star_power_def |
|
577 |
||
578 |
text {* Class operations preserve standard elements *} |
|
579 |
||
580 |
lemma Standard_zero: "0 \<in> Standard" |
|
581 |
by (simp add: star_zero_def) |
|
582 |
||
583 |
lemma Standard_one: "1 \<in> Standard" |
|
584 |
by (simp add: star_one_def) |
|
585 |
||
586 |
lemma Standard_number_of: "number_of b \<in> Standard" |
|
587 |
by (simp add: star_number_def) |
|
588 |
||
589 |
lemma Standard_add: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x + y \<in> Standard" |
|
590 |
by (simp add: star_add_def) |
|
591 |
||
592 |
lemma Standard_diff: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x - y \<in> Standard" |
|
593 |
by (simp add: star_diff_def) |
|
594 |
||
595 |
lemma Standard_minus: "x \<in> Standard \<Longrightarrow> - x \<in> Standard" |
|
596 |
by (simp add: star_minus_def) |
|
597 |
||
598 |
lemma Standard_mult: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x * y \<in> Standard" |
|
599 |
by (simp add: star_mult_def) |
|
600 |
||
601 |
lemma Standard_divide: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x / y \<in> Standard" |
|
602 |
by (simp add: star_divide_def) |
|
603 |
||
604 |
lemma Standard_inverse: "x \<in> Standard \<Longrightarrow> inverse x \<in> Standard" |
|
605 |
by (simp add: star_inverse_def) |
|
606 |
||
607 |
lemma Standard_abs: "x \<in> Standard \<Longrightarrow> abs x \<in> Standard" |
|
608 |
by (simp add: star_abs_def) |
|
609 |
||
610 |
lemma Standard_div: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x div y \<in> Standard" |
|
611 |
by (simp add: star_div_def) |
|
612 |
||
613 |
lemma Standard_mod: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x mod y \<in> Standard" |
|
614 |
by (simp add: star_mod_def) |
|
615 |
||
616 |
lemma Standard_power: "x \<in> Standard \<Longrightarrow> x ^ n \<in> Standard" |
|
617 |
by (simp add: star_power_def) |
|
618 |
||
619 |
lemmas Standard_simps [simp] = |
|
620 |
Standard_zero Standard_one Standard_number_of |
|
621 |
Standard_add Standard_diff Standard_minus |
|
622 |
Standard_mult Standard_divide Standard_inverse |
|
623 |
Standard_abs Standard_div Standard_mod |
|
624 |
Standard_power |
|
625 |
||
626 |
text {* @{term star_of} preserves class operations *} |
|
627 |
||
628 |
lemma star_of_add: "star_of (x + y) = star_of x + star_of y" |
|
629 |
by transfer (rule refl) |
|
630 |
||
631 |
lemma star_of_diff: "star_of (x - y) = star_of x - star_of y" |
|
632 |
by transfer (rule refl) |
|
633 |
||
634 |
lemma star_of_minus: "star_of (-x) = - star_of x" |
|
635 |
by transfer (rule refl) |
|
636 |
||
637 |
lemma star_of_mult: "star_of (x * y) = star_of x * star_of y" |
|
638 |
by transfer (rule refl) |
|
639 |
||
640 |
lemma star_of_divide: "star_of (x / y) = star_of x / star_of y" |
|
641 |
by transfer (rule refl) |
|
642 |
||
643 |
lemma star_of_inverse: "star_of (inverse x) = inverse (star_of x)" |
|
644 |
by transfer (rule refl) |
|
645 |
||
646 |
lemma star_of_div: "star_of (x div y) = star_of x div star_of y" |
|
647 |
by transfer (rule refl) |
|
648 |
||
649 |
lemma star_of_mod: "star_of (x mod y) = star_of x mod star_of y" |
|
650 |
by transfer (rule refl) |
|
651 |
||
652 |
lemma star_of_power: "star_of (x ^ n) = star_of x ^ n" |
|
653 |
by transfer (rule refl) |
|
654 |
||
655 |
lemma star_of_abs: "star_of (abs x) = abs (star_of x)" |
|
656 |
by transfer (rule refl) |
|
657 |
||
658 |
text {* @{term star_of} preserves numerals *} |
|
659 |
||
660 |
lemma star_of_zero: "star_of 0 = 0" |
|
661 |
by transfer (rule refl) |
|
662 |
||
663 |
lemma star_of_one: "star_of 1 = 1" |
|
664 |
by transfer (rule refl) |
|
665 |
||
666 |
lemma star_of_number_of: "star_of (number_of x) = number_of x" |
|
667 |
by transfer (rule refl) |
|
668 |
||
669 |
text {* @{term star_of} preserves orderings *} |
|
670 |
||
671 |
lemma star_of_less: "(star_of x < star_of y) = (x < y)" |
|
672 |
by transfer (rule refl) |
|
673 |
||
674 |
lemma star_of_le: "(star_of x \<le> star_of y) = (x \<le> y)" |
|
675 |
by transfer (rule refl) |
|
676 |
||
677 |
lemma star_of_eq: "(star_of x = star_of y) = (x = y)" |
|
678 |
by transfer (rule refl) |
|
679 |
||
680 |
text{*As above, for 0*} |
|
681 |
||
682 |
lemmas star_of_0_less = star_of_less [of 0, simplified star_of_zero] |
|
683 |
lemmas star_of_0_le = star_of_le [of 0, simplified star_of_zero] |
|
684 |
lemmas star_of_0_eq = star_of_eq [of 0, simplified star_of_zero] |
|
685 |
||
686 |
lemmas star_of_less_0 = star_of_less [of _ 0, simplified star_of_zero] |
|
687 |
lemmas star_of_le_0 = star_of_le [of _ 0, simplified star_of_zero] |
|
688 |
lemmas star_of_eq_0 = star_of_eq [of _ 0, simplified star_of_zero] |
|
689 |
||
690 |
text{*As above, for 1*} |
|
691 |
||
692 |
lemmas star_of_1_less = star_of_less [of 1, simplified star_of_one] |
|
693 |
lemmas star_of_1_le = star_of_le [of 1, simplified star_of_one] |
|
694 |
lemmas star_of_1_eq = star_of_eq [of 1, simplified star_of_one] |
|
695 |
||
696 |
lemmas star_of_less_1 = star_of_less [of _ 1, simplified star_of_one] |
|
697 |
lemmas star_of_le_1 = star_of_le [of _ 1, simplified star_of_one] |
|
698 |
lemmas star_of_eq_1 = star_of_eq [of _ 1, simplified star_of_one] |
|
699 |
||
700 |
text{*As above, for numerals*} |
|
701 |
||
702 |
lemmas star_of_number_less = |
|
703 |
star_of_less [of "number_of w", standard, simplified star_of_number_of] |
|
704 |
lemmas star_of_number_le = |
|
705 |
star_of_le [of "number_of w", standard, simplified star_of_number_of] |
|
706 |
lemmas star_of_number_eq = |
|
707 |
star_of_eq [of "number_of w", standard, simplified star_of_number_of] |
|
708 |
||
709 |
lemmas star_of_less_number = |
|
710 |
star_of_less [of _ "number_of w", standard, simplified star_of_number_of] |
|
711 |
lemmas star_of_le_number = |
|
712 |
star_of_le [of _ "number_of w", standard, simplified star_of_number_of] |
|
713 |
lemmas star_of_eq_number = |
|
714 |
star_of_eq [of _ "number_of w", standard, simplified star_of_number_of] |
|
715 |
||
716 |
lemmas star_of_simps [simp] = |
|
717 |
star_of_add star_of_diff star_of_minus |
|
718 |
star_of_mult star_of_divide star_of_inverse |
|
719 |
star_of_div star_of_mod |
|
720 |
star_of_power star_of_abs |
|
721 |
star_of_zero star_of_one star_of_number_of |
|
722 |
star_of_less star_of_le star_of_eq |
|
723 |
star_of_0_less star_of_0_le star_of_0_eq |
|
724 |
star_of_less_0 star_of_le_0 star_of_eq_0 |
|
725 |
star_of_1_less star_of_1_le star_of_1_eq |
|
726 |
star_of_less_1 star_of_le_1 star_of_eq_1 |
|
727 |
star_of_number_less star_of_number_le star_of_number_eq |
|
728 |
star_of_less_number star_of_le_number star_of_eq_number |
|
729 |
||
730 |
subsection {* Ordering and lattice classes *} |
|
731 |
||
732 |
instance star :: (order) order |
|
733 |
apply (intro_classes) |
|
734 |
apply (transfer, rule order_less_le) |
|
735 |
apply (transfer, rule order_refl) |
|
736 |
apply (transfer, erule (1) order_trans) |
|
737 |
apply (transfer, erule (1) order_antisym) |
|
738 |
done |
|
739 |
||
740 |
instantiation star :: (lower_semilattice) lower_semilattice |
|
741 |
begin |
|
742 |
||
743 |
definition |
|
744 |
star_inf_def [transfer_unfold]: "inf \<equiv> *f2* inf" |
|
745 |
||
746 |
instance |
|
747 |
by default (transfer star_inf_def, auto)+ |
|
748 |
||
749 |
end |
|
750 |
||
751 |
instantiation star :: (upper_semilattice) upper_semilattice |
|
752 |
begin |
|
753 |
||
754 |
definition |
|
755 |
star_sup_def [transfer_unfold]: "sup \<equiv> *f2* sup" |
|
756 |
||
757 |
instance |
|
758 |
by default (transfer star_sup_def, auto)+ |
|
759 |
||
760 |
end |
|
761 |
||
762 |
instance star :: (lattice) lattice .. |
|
763 |
||
764 |
instance star :: (distrib_lattice) distrib_lattice |
|
765 |
by default (transfer, auto simp add: sup_inf_distrib1) |
|
766 |
||
767 |
lemma Standard_inf [simp]: |
|
768 |
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> inf x y \<in> Standard" |
|
769 |
by (simp add: star_inf_def) |
|
770 |
||
771 |
lemma Standard_sup [simp]: |
|
772 |
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> sup x y \<in> Standard" |
|
773 |
by (simp add: star_sup_def) |
|
774 |
||
775 |
lemma star_of_inf [simp]: "star_of (inf x y) = inf (star_of x) (star_of y)" |
|
776 |
by transfer (rule refl) |
|
777 |
||
778 |
lemma star_of_sup [simp]: "star_of (sup x y) = sup (star_of x) (star_of y)" |
|
779 |
by transfer (rule refl) |
|
780 |
||
781 |
instance star :: (linorder) linorder |
|
782 |
by (intro_classes, transfer, rule linorder_linear) |
|
783 |
||
784 |
lemma star_max_def [transfer_unfold]: "max = *f2* max" |
|
785 |
apply (rule ext, rule ext) |
|
786 |
apply (unfold max_def, transfer, fold max_def) |
|
787 |
apply (rule refl) |
|
788 |
done |
|
789 |
||
790 |
lemma star_min_def [transfer_unfold]: "min = *f2* min" |
|
791 |
apply (rule ext, rule ext) |
|
792 |
apply (unfold min_def, transfer, fold min_def) |
|
793 |
apply (rule refl) |
|
794 |
done |
|
795 |
||
796 |
lemma Standard_max [simp]: |
|
797 |
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> max x y \<in> Standard" |
|
798 |
by (simp add: star_max_def) |
|
799 |
||
800 |
lemma Standard_min [simp]: |
|
801 |
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> min x y \<in> Standard" |
|
802 |
by (simp add: star_min_def) |
|
803 |
||
804 |
lemma star_of_max [simp]: "star_of (max x y) = max (star_of x) (star_of y)" |
|
805 |
by transfer (rule refl) |
|
806 |
||
807 |
lemma star_of_min [simp]: "star_of (min x y) = min (star_of x) (star_of y)" |
|
808 |
by transfer (rule refl) |
|
809 |
||
810 |
||
811 |
subsection {* Ordered group classes *} |
|
812 |
||
813 |
instance star :: (semigroup_add) semigroup_add |
|
814 |
by (intro_classes, transfer, rule add_assoc) |
|
815 |
||
816 |
instance star :: (ab_semigroup_add) ab_semigroup_add |
|
817 |
by (intro_classes, transfer, rule add_commute) |
|
818 |
||
819 |
instance star :: (semigroup_mult) semigroup_mult |
|
820 |
by (intro_classes, transfer, rule mult_assoc) |
|
821 |
||
822 |
instance star :: (ab_semigroup_mult) ab_semigroup_mult |
|
823 |
by (intro_classes, transfer, rule mult_commute) |
|
824 |
||
825 |
instance star :: (comm_monoid_add) comm_monoid_add |
|
826 |
by (intro_classes, transfer, rule comm_monoid_add_class.zero_plus.add_0) |
|
827 |
||
828 |
instance star :: (monoid_mult) monoid_mult |
|
829 |
apply (intro_classes) |
|
830 |
apply (transfer, rule mult_1_left) |
|
831 |
apply (transfer, rule mult_1_right) |
|
832 |
done |
|
833 |
||
834 |
instance star :: (comm_monoid_mult) comm_monoid_mult |
|
835 |
by (intro_classes, transfer, rule mult_1) |
|
836 |
||
837 |
instance star :: (cancel_semigroup_add) cancel_semigroup_add |
|
838 |
apply (intro_classes) |
|
839 |
apply (transfer, erule add_left_imp_eq) |
|
840 |
apply (transfer, erule add_right_imp_eq) |
|
841 |
done |
|
842 |
||
843 |
instance star :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add |
|
844 |
by (intro_classes, transfer, rule add_imp_eq) |
|
845 |
||
846 |
instance star :: (ab_group_add) ab_group_add |
|
847 |
apply (intro_classes) |
|
848 |
apply (transfer, rule left_minus) |
|
849 |
apply (transfer, rule diff_minus) |
|
850 |
done |
|
851 |
||
852 |
instance star :: (pordered_ab_semigroup_add) pordered_ab_semigroup_add |
|
853 |
by (intro_classes, transfer, rule add_left_mono) |
|
854 |
||
855 |
instance star :: (pordered_cancel_ab_semigroup_add) pordered_cancel_ab_semigroup_add .. |
|
856 |
||
857 |
instance star :: (pordered_ab_semigroup_add_imp_le) pordered_ab_semigroup_add_imp_le |
|
858 |
by (intro_classes, transfer, rule add_le_imp_le_left) |
|
859 |
||
860 |
instance star :: (pordered_comm_monoid_add) pordered_comm_monoid_add .. |
|
861 |
instance star :: (pordered_ab_group_add) pordered_ab_group_add .. |
|
862 |
||
863 |
instance star :: (pordered_ab_group_add_abs) pordered_ab_group_add_abs |
|
864 |
by intro_classes (transfer, |
|
865 |
simp add: abs_ge_self abs_leI abs_triangle_ineq)+ |
|
866 |
||
867 |
instance star :: (ordered_cancel_ab_semigroup_add) ordered_cancel_ab_semigroup_add .. |
|
868 |
instance star :: (lordered_ab_group_add_meet) lordered_ab_group_add_meet .. |
|
869 |
instance star :: (lordered_ab_group_add_meet) lordered_ab_group_add_meet .. |
|
870 |
instance star :: (lordered_ab_group_add) lordered_ab_group_add .. |
|
871 |
||
872 |
instance star :: (lordered_ab_group_add_abs) lordered_ab_group_add_abs |
|
873 |
by (intro_classes, transfer, rule abs_lattice) |
|
874 |
||
875 |
subsection {* Ring and field classes *} |
|
876 |
||
877 |
instance star :: (semiring) semiring |
|
878 |
apply (intro_classes) |
|
879 |
apply (transfer, rule left_distrib) |
|
880 |
apply (transfer, rule right_distrib) |
|
881 |
done |
|
882 |
||
883 |
instance star :: (semiring_0) semiring_0 |
|
884 |
by intro_classes (transfer, simp)+ |
|
885 |
||
886 |
instance star :: (semiring_0_cancel) semiring_0_cancel .. |
|
887 |
||
888 |
instance star :: (comm_semiring) comm_semiring |
|
889 |
by (intro_classes, transfer, rule left_distrib) |
|
890 |
||
891 |
instance star :: (comm_semiring_0) comm_semiring_0 .. |
|
892 |
instance star :: (comm_semiring_0_cancel) comm_semiring_0_cancel .. |
|
893 |
||
894 |
instance star :: (zero_neq_one) zero_neq_one |
|
895 |
by (intro_classes, transfer, rule zero_neq_one) |
|
896 |
||
897 |
instance star :: (semiring_1) semiring_1 .. |
|
898 |
instance star :: (comm_semiring_1) comm_semiring_1 .. |
|
899 |
||
900 |
instance star :: (no_zero_divisors) no_zero_divisors |
|
901 |
by (intro_classes, transfer, rule no_zero_divisors) |
|
902 |
||
903 |
instance star :: (semiring_1_cancel) semiring_1_cancel .. |
|
904 |
instance star :: (comm_semiring_1_cancel) comm_semiring_1_cancel .. |
|
905 |
instance star :: (ring) ring .. |
|
906 |
instance star :: (comm_ring) comm_ring .. |
|
907 |
instance star :: (ring_1) ring_1 .. |
|
908 |
instance star :: (comm_ring_1) comm_ring_1 .. |
|
909 |
instance star :: (ring_no_zero_divisors) ring_no_zero_divisors .. |
|
910 |
instance star :: (ring_1_no_zero_divisors) ring_1_no_zero_divisors .. |
|
911 |
instance star :: (idom) idom .. |
|
912 |
||
913 |
instance star :: (division_ring) division_ring |
|
914 |
apply (intro_classes) |
|
915 |
apply (transfer, erule left_inverse) |
|
916 |
apply (transfer, erule right_inverse) |
|
917 |
done |
|
918 |
||
919 |
instance star :: (field) field |
|
920 |
apply (intro_classes) |
|
921 |
apply (transfer, erule left_inverse) |
|
922 |
apply (transfer, rule divide_inverse) |
|
923 |
done |
|
924 |
||
925 |
instance star :: (division_by_zero) division_by_zero |
|
926 |
by (intro_classes, transfer, rule inverse_zero) |
|
927 |
||
928 |
instance star :: (pordered_semiring) pordered_semiring |
|
929 |
apply (intro_classes) |
|
930 |
apply (transfer, erule (1) mult_left_mono) |
|
931 |
apply (transfer, erule (1) mult_right_mono) |
|
932 |
done |
|
933 |
||
934 |
instance star :: (pordered_cancel_semiring) pordered_cancel_semiring .. |
|
935 |
||
936 |
instance star :: (ordered_semiring_strict) ordered_semiring_strict |
|
937 |
apply (intro_classes) |
|
938 |
apply (transfer, erule (1) mult_strict_left_mono) |
|
939 |
apply (transfer, erule (1) mult_strict_right_mono) |
|
940 |
done |
|
941 |
||
942 |
instance star :: (pordered_comm_semiring) pordered_comm_semiring |
|
25622 | 943 |
by (intro_classes, transfer, rule mult_mono1_class.less_eq_less_times_zero.mult_mono1) |
25601 | 944 |
|
945 |
instance star :: (pordered_cancel_comm_semiring) pordered_cancel_comm_semiring .. |
|
946 |
||
947 |
instance star :: (ordered_comm_semiring_strict) ordered_comm_semiring_strict |
|
25622 | 948 |
by (intro_classes, transfer, rule ordered_comm_semiring_strict_class.plus_less_eq_less_zero_times.mult_strict_mono) |
25601 | 949 |
|
950 |
instance star :: (pordered_ring) pordered_ring .. |
|
951 |
instance star :: (pordered_ring_abs) pordered_ring_abs |
|
952 |
by intro_classes (transfer, rule abs_eq_mult) |
|
953 |
instance star :: (lordered_ring) lordered_ring .. |
|
954 |
||
955 |
instance star :: (abs_if) abs_if |
|
956 |
by (intro_classes, transfer, rule abs_if) |
|
957 |
||
958 |
instance star :: (sgn_if) sgn_if |
|
959 |
by (intro_classes, transfer, rule sgn_if) |
|
960 |
||
961 |
instance star :: (ordered_ring_strict) ordered_ring_strict .. |
|
962 |
instance star :: (pordered_comm_ring) pordered_comm_ring .. |
|
963 |
||
964 |
instance star :: (ordered_semidom) ordered_semidom |
|
965 |
by (intro_classes, transfer, rule zero_less_one) |
|
966 |
||
967 |
instance star :: (ordered_idom) ordered_idom .. |
|
968 |
instance star :: (ordered_field) ordered_field .. |
|
969 |
||
970 |
subsection {* Power classes *} |
|
971 |
||
972 |
text {* |
|
973 |
Proving the class axiom @{thm [source] power_Suc} for type |
|
974 |
@{typ "'a star"} is a little tricky, because it quantifies |
|
975 |
over values of type @{typ nat}. The transfer principle does |
|
976 |
not handle quantification over non-star types in general, |
|
977 |
but we can work around this by fixing an arbitrary @{typ nat} |
|
978 |
value, and then applying the transfer principle. |
|
979 |
*} |
|
980 |
||
981 |
instance star :: (recpower) recpower |
|
982 |
proof |
|
983 |
show "\<And>a::'a star. a ^ 0 = 1" |
|
984 |
by transfer (rule power_0) |
|
985 |
next |
|
986 |
fix n show "\<And>a::'a star. a ^ Suc n = a * a ^ n" |
|
987 |
by transfer (rule power_Suc) |
|
988 |
qed |
|
989 |
||
990 |
subsection {* Number classes *} |
|
991 |
||
992 |
lemma star_of_nat_def [transfer_unfold]: "of_nat n = star_of (of_nat n)" |
|
993 |
by (induct n, simp_all) |
|
994 |
||
995 |
lemma Standard_of_nat [simp]: "of_nat n \<in> Standard" |
|
996 |
by (simp add: star_of_nat_def) |
|
997 |
||
998 |
lemma star_of_of_nat [simp]: "star_of (of_nat n) = of_nat n" |
|
999 |
by transfer (rule refl) |
|
1000 |
||
1001 |
lemma star_of_int_def [transfer_unfold]: "of_int z = star_of (of_int z)" |
|
1002 |
by (rule_tac z=z in int_diff_cases, simp) |
|
1003 |
||
1004 |
lemma Standard_of_int [simp]: "of_int z \<in> Standard" |
|
1005 |
by (simp add: star_of_int_def) |
|
1006 |
||
1007 |
lemma star_of_of_int [simp]: "star_of (of_int z) = of_int z" |
|
1008 |
by transfer (rule refl) |
|
1009 |
||
1010 |
instance star :: (semiring_char_0) semiring_char_0 |
|
1011 |
by intro_classes (simp only: star_of_nat_def star_of_eq of_nat_eq_iff) |
|
1012 |
||
1013 |
instance star :: (ring_char_0) ring_char_0 .. |
|
1014 |
||
1015 |
instance star :: (number_ring) number_ring |
|
1016 |
by (intro_classes, simp only: star_number_def star_of_int_def number_of_eq) |
|
1017 |
||
1018 |
subsection {* Finite class *} |
|
1019 |
||
1020 |
lemma starset_finite: "finite A \<Longrightarrow> *s* A = star_of ` A" |
|
1021 |
by (erule finite_induct, simp_all) |
|
1022 |
||
1023 |
instance star :: (finite) finite |
|
1024 |
apply (intro_classes) |
|
1025 |
apply (subst starset_UNIV [symmetric]) |
|
1026 |
apply (subst starset_finite [OF finite]) |
|
1027 |
apply (rule finite_imageI [OF finite]) |
|
1028 |
done |
|
1029 |
||
1030 |
end |