author | Christian Urban <urbanc@in.tum.de> |
Wed, 03 Feb 2010 12:44:29 +0100 | |
changeset 1043 | 534d4c604f80 |
parent 1013 | e63838c26f28 |
child 1061 | 8de99358f309 |
permissions | -rw-r--r-- |
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1 |
(* Title: Nominal2_Base |
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2 |
Authors: Brian Huffman, Christian Urban |
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3 |
|
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4 |
Basic definitions and lemma infrastructure for |
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5 |
Nominal Isabelle. |
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6 |
*) |
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7 |
theory Nominal2_Base |
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8 |
imports Main Infinite_Set |
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9 |
begin |
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10 |
|
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11 |
section {* Atoms and Sorts *} |
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12 |
|
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13 |
text {* A simple implementation for atom_sorts is strings. *} |
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14 |
(* types atom_sort = string *) |
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15 |
|
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16 |
text {* To deal with Church-like binding we use trees of |
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17 |
strings as sorts. *} |
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18 |
|
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19 |
datatype atom_sort = Sort "string" "atom_sort list" |
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20 |
|
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21 |
datatype atom = Atom atom_sort nat |
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22 |
|
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23 |
|
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24 |
text {* Basic projection function. *} |
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25 |
|
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26 |
primrec |
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27 |
sort_of :: "atom \<Rightarrow> atom_sort" |
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28 |
where |
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29 |
"sort_of (Atom s i) = s" |
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30 |
|
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31 |
|
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text {* There are infinitely many atoms of each sort. *} |
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33 |
lemma INFM_sort_of_eq: |
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34 |
shows "INFM a. sort_of a = s" |
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35 |
proof - |
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36 |
have "INFM i. sort_of (Atom s i) = s" by simp |
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37 |
moreover have "inj (Atom s)" by (simp add: inj_on_def) |
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38 |
ultimately show "INFM a. sort_of a = s" by (rule INFM_inj) |
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39 |
qed |
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40 |
|
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41 |
lemma infinite_sort_of_eq: |
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42 |
shows "infinite {a. sort_of a = s}" |
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43 |
using INFM_sort_of_eq unfolding INFM_iff_infinite . |
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44 |
|
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45 |
lemma atom_infinite [simp]: |
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46 |
shows "infinite (UNIV :: atom set)" |
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47 |
using subset_UNIV infinite_sort_of_eq |
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48 |
by (rule infinite_super) |
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49 |
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lemma obtain_atom: |
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51 |
fixes X :: "atom set" |
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52 |
assumes X: "finite X" |
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53 |
obtains a where "a \<notin> X" "sort_of a = s" |
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54 |
proof - |
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55 |
from X have "MOST a. a \<notin> X" |
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56 |
unfolding MOST_iff_cofinite by simp |
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57 |
with INFM_sort_of_eq |
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58 |
have "INFM a. sort_of a = s \<and> a \<notin> X" |
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59 |
by (rule INFM_conjI) |
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60 |
then obtain a where "a \<notin> X" "sort_of a = s" |
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61 |
by (auto elim: INFM_E) |
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62 |
then show ?thesis .. |
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63 |
qed |
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64 |
|
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65 |
section {* Sort-Respecting Permutations *} |
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66 |
|
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67 |
typedef perm = |
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68 |
"{f. bij f \<and> finite {a. f a \<noteq> a} \<and> (\<forall>a. sort_of (f a) = sort_of a)}" |
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69 |
proof |
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70 |
show "id \<in> ?perm" by simp |
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71 |
qed |
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72 |
|
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73 |
lemma permI: |
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74 |
assumes "bij f" and "MOST x. f x = x" and "\<And>a. sort_of (f a) = sort_of a" |
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75 |
shows "f \<in> perm" |
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76 |
using assms unfolding perm_def MOST_iff_cofinite by simp |
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77 |
|
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78 |
lemma perm_is_bij: "f \<in> perm \<Longrightarrow> bij f" |
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79 |
unfolding perm_def by simp |
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80 |
|
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81 |
lemma perm_is_finite: "f \<in> perm \<Longrightarrow> finite {a. f a \<noteq> a}" |
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82 |
unfolding perm_def by simp |
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83 |
|
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84 |
lemma perm_is_sort_respecting: "f \<in> perm \<Longrightarrow> sort_of (f a) = sort_of a" |
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85 |
unfolding perm_def by simp |
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86 |
|
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87 |
lemma perm_MOST: "f \<in> perm \<Longrightarrow> MOST x. f x = x" |
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88 |
unfolding perm_def MOST_iff_cofinite by simp |
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89 |
|
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90 |
lemma perm_id: "id \<in> perm" |
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91 |
unfolding perm_def by simp |
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92 |
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93 |
lemma perm_comp: |
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94 |
assumes f: "f \<in> perm" and g: "g \<in> perm" |
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95 |
shows "(f \<circ> g) \<in> perm" |
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96 |
apply (rule permI) |
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97 |
apply (rule bij_comp) |
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98 |
apply (rule perm_is_bij [OF g]) |
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99 |
apply (rule perm_is_bij [OF f]) |
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100 |
apply (rule MOST_rev_mp [OF perm_MOST [OF g]]) |
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101 |
apply (rule MOST_rev_mp [OF perm_MOST [OF f]]) |
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102 |
apply (simp) |
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103 |
apply (simp add: perm_is_sort_respecting [OF f]) |
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104 |
apply (simp add: perm_is_sort_respecting [OF g]) |
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105 |
done |
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106 |
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107 |
lemma perm_inv: |
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108 |
assumes f: "f \<in> perm" |
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109 |
shows "(inv f) \<in> perm" |
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110 |
apply (rule permI) |
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111 |
apply (rule bij_imp_bij_inv) |
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112 |
apply (rule perm_is_bij [OF f]) |
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113 |
apply (rule MOST_mono [OF perm_MOST [OF f]]) |
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114 |
apply (erule subst, rule inv_f_f) |
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115 |
apply (rule bij_is_inj [OF perm_is_bij [OF f]]) |
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116 |
apply (rule perm_is_sort_respecting [OF f, THEN sym, THEN trans]) |
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117 |
apply (simp add: surj_f_inv_f [OF bij_is_surj [OF perm_is_bij [OF f]]]) |
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118 |
done |
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119 |
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lemma bij_Rep_perm: "bij (Rep_perm p)" |
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using Rep_perm [of p] unfolding perm_def by simp |
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lemma finite_Rep_perm: "finite {a. Rep_perm p a \<noteq> a}" |
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using Rep_perm [of p] unfolding perm_def by simp |
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lemma sort_of_Rep_perm: "sort_of (Rep_perm p a) = sort_of a" |
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using Rep_perm [of p] unfolding perm_def by simp |
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lemma Rep_perm_ext: |
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"Rep_perm p1 = Rep_perm p2 \<Longrightarrow> p1 = p2" |
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by (simp add: expand_fun_eq Rep_perm_inject [symmetric]) |
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|
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subsection {* Permutations form a group *} |
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instantiation perm :: group_add |
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begin |
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|
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definition |
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"0 = Abs_perm id" |
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|
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definition |
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"- p = Abs_perm (inv (Rep_perm p))" |
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|
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definition |
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"p + q = Abs_perm (Rep_perm p \<circ> Rep_perm q)" |
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|
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definition |
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"(p1::perm) - p2 = p1 + - p2" |
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|
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lemma Rep_perm_0: "Rep_perm 0 = id" |
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unfolding zero_perm_def |
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by (simp add: Abs_perm_inverse perm_id) |
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lemma Rep_perm_add: |
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"Rep_perm (p1 + p2) = Rep_perm p1 \<circ> Rep_perm p2" |
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unfolding plus_perm_def |
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by (simp add: Abs_perm_inverse perm_comp Rep_perm) |
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lemma Rep_perm_uminus: |
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"Rep_perm (- p) = inv (Rep_perm p)" |
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unfolding uminus_perm_def |
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by (simp add: Abs_perm_inverse perm_inv Rep_perm) |
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|
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instance |
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apply default |
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unfolding Rep_perm_inject [symmetric] |
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unfolding minus_perm_def |
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unfolding Rep_perm_add |
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unfolding Rep_perm_uminus |
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unfolding Rep_perm_0 |
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by (simp_all add: o_assoc inv_o_cancel [OF bij_is_inj [OF bij_Rep_perm]]) |
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|
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end |
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|
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|
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section {* Implementation of swappings *} |
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|
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definition |
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swap :: "atom \<Rightarrow> atom \<Rightarrow> perm" ("'(_ \<rightleftharpoons> _')") |
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where |
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"(a \<rightleftharpoons> b) = |
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Abs_perm (if sort_of a = sort_of b |
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then (\<lambda>c. if a = c then b else if b = c then a else c) |
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else id)" |
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lemma Rep_perm_swap: |
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"Rep_perm (a \<rightleftharpoons> b) = |
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(if sort_of a = sort_of b |
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then (\<lambda>c. if a = c then b else if b = c then a else c) |
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else id)" |
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192 |
unfolding swap_def |
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apply (rule Abs_perm_inverse) |
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apply (rule permI) |
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apply (auto simp add: bij_def inj_on_def surj_def)[1] |
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apply (rule MOST_rev_mp [OF MOST_neq(1) [of a]]) |
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apply (rule MOST_rev_mp [OF MOST_neq(1) [of b]]) |
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apply (simp) |
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199 |
apply (simp) |
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done |
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|
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lemmas Rep_perm_simps = |
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Rep_perm_0 |
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Rep_perm_add |
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Rep_perm_uminus |
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Rep_perm_swap |
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lemma swap_different_sorts [simp]: |
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"sort_of a \<noteq> sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) = 0" |
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by (rule Rep_perm_ext) (simp add: Rep_perm_simps) |
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|
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lemma swap_cancel: |
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"(a \<rightleftharpoons> b) + (a \<rightleftharpoons> b) = 0" |
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by (rule Rep_perm_ext) |
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(simp add: Rep_perm_simps expand_fun_eq) |
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|
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lemma swap_self [simp]: |
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"(a \<rightleftharpoons> a) = 0" |
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by (rule Rep_perm_ext, simp add: Rep_perm_simps expand_fun_eq) |
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|
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lemma minus_swap [simp]: |
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"- (a \<rightleftharpoons> b) = (a \<rightleftharpoons> b)" |
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by (rule minus_unique [OF swap_cancel]) |
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|
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lemma swap_commute: |
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"(a \<rightleftharpoons> b) = (b \<rightleftharpoons> a)" |
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by (rule Rep_perm_ext) |
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(simp add: Rep_perm_swap expand_fun_eq) |
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|
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lemma swap_triple: |
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assumes "a \<noteq> b" and "c \<noteq> b" |
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assumes "sort_of a = sort_of b" "sort_of b = sort_of c" |
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shows "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)" |
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using assms |
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by (rule_tac Rep_perm_ext) |
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(auto simp add: Rep_perm_simps expand_fun_eq) |
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|
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|
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239 |
section {* Permutation Types *} |
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|
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text {* |
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Infix syntax for @{text permute} has higher precedence than |
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addition, but lower than unary minus. |
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*} |
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|
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class pt = |
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fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a" ("_ \<bullet> _" [76, 75] 75) |
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assumes permute_zero [simp]: "0 \<bullet> x = x" |
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assumes permute_plus [simp]: "(p + q) \<bullet> x = p \<bullet> (q \<bullet> x)" |
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begin |
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lemma permute_diff [simp]: |
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shows "(p - q) \<bullet> x = p \<bullet> - q \<bullet> x" |
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unfolding diff_minus by simp |
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|
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lemma permute_minus_cancel [simp]: |
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shows "p \<bullet> - p \<bullet> x = x" |
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and "- p \<bullet> p \<bullet> x = x" |
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unfolding permute_plus [symmetric] by simp_all |
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|
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lemma permute_swap_cancel [simp]: |
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shows "(a \<rightleftharpoons> b) \<bullet> (a \<rightleftharpoons> b) \<bullet> x = x" |
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unfolding permute_plus [symmetric] |
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by (simp add: swap_cancel) |
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|
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lemma permute_swap_cancel2 [simp]: |
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shows "(a \<rightleftharpoons> b) \<bullet> (b \<rightleftharpoons> a) \<bullet> x = x" |
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unfolding permute_plus [symmetric] |
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by (simp add: swap_commute) |
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|
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lemma inj_permute [simp]: |
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shows "inj (permute p)" |
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by (rule inj_on_inverseI) |
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(rule permute_minus_cancel) |
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|
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lemma surj_permute [simp]: |
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shows "surj (permute p)" |
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by (rule surjI, rule permute_minus_cancel) |
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lemma bij_permute [simp]: |
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shows "bij (permute p)" |
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by (rule bijI [OF inj_permute surj_permute]) |
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|
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lemma inv_permute: |
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shows "inv (permute p) = permute (- p)" |
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by (rule inv_equality) (simp_all) |
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|
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lemma permute_minus: |
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shows "permute (- p) = inv (permute p)" |
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by (simp add: inv_permute) |
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lemma permute_eq_iff [simp]: |
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shows "p \<bullet> x = p \<bullet> y \<longleftrightarrow> x = y" |
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by (rule inj_permute [THEN inj_eq]) |
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|
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end |
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|
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subsection {* Permutations for atoms *} |
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|
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instantiation atom :: pt |
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begin |
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|
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definition |
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"p \<bullet> a = Rep_perm p a" |
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|
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instance |
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apply(default) |
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apply(simp_all add: permute_atom_def Rep_perm_simps) |
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done |
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|
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end |
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lemma sort_of_permute [simp]: |
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shows "sort_of (p \<bullet> a) = sort_of a" |
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unfolding permute_atom_def by (rule sort_of_Rep_perm) |
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|
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lemma swap_atom: |
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shows "(a \<rightleftharpoons> b) \<bullet> c = |
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(if sort_of a = sort_of b |
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then (if c = a then b else if c = b then a else c) else c)" |
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unfolding permute_atom_def |
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by (simp add: Rep_perm_swap) |
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|
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lemma swap_atom_simps [simp]: |
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"sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> a = b" |
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"sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> b = a" |
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"c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> c = c" |
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unfolding swap_atom by simp_all |
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|
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lemma expand_perm_eq: |
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fixes p q :: "perm" |
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shows "p = q \<longleftrightarrow> (\<forall>a::atom. p \<bullet> a = q \<bullet> a)" |
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unfolding permute_atom_def |
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by (metis Rep_perm_ext ext) |
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|
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|
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subsection {* Permutations for permutations *} |
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|
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instantiation perm :: pt |
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begin |
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|
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definition |
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"p \<bullet> q = p + q - p" |
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|
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instance |
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apply default |
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apply (simp add: permute_perm_def) |
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apply (simp add: permute_perm_def diff_minus minus_add add_assoc) |
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done |
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|
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end |
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lemma permute_self: "p \<bullet> p = p" |
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unfolding permute_perm_def by (simp add: diff_minus add_assoc) |
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|
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lemma permute_eqvt: |
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shows "p \<bullet> (q \<bullet> x) = (p \<bullet> q) \<bullet> (p \<bullet> x)" |
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unfolding permute_perm_def by simp |
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|
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lemma zero_perm_eqvt: |
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shows "p \<bullet> (0::perm) = 0" |
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unfolding permute_perm_def by simp |
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changeset
|
363 |
|
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diff
changeset
|
364 |
lemma add_perm_eqvt: |
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changeset
|
365 |
fixes p p1 p2 :: perm |
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diff
changeset
|
366 |
shows "p \<bullet> (p1 + p2) = p \<bullet> p1 + p \<bullet> p2" |
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parents:
diff
changeset
|
367 |
unfolding permute_perm_def |
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changeset
|
368 |
by (simp add: expand_perm_eq) |
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diff
changeset
|
369 |
|
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parents:
diff
changeset
|
370 |
lemma swap_eqvt: |
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parents:
diff
changeset
|
371 |
shows "p \<bullet> (a \<rightleftharpoons> b) = (p \<bullet> a \<rightleftharpoons> p \<bullet> b)" |
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parents:
diff
changeset
|
372 |
unfolding permute_perm_def |
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changeset
|
373 |
by (auto simp add: swap_atom expand_perm_eq) |
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diff
changeset
|
374 |
|
fa810f01f7b5
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parents:
diff
changeset
|
375 |
|
fa810f01f7b5
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parents:
diff
changeset
|
376 |
subsection {* Permutations for functions *} |
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diff
changeset
|
377 |
|
fa810f01f7b5
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parents:
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changeset
|
378 |
instantiation "fun" :: (pt, pt) pt |
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changeset
|
379 |
begin |
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changeset
|
380 |
|
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changeset
|
381 |
definition |
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|
382 |
"p \<bullet> f = (\<lambda>x. p \<bullet> (f (- p \<bullet> x)))" |
fa810f01f7b5
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changeset
|
383 |
|
fa810f01f7b5
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parents:
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changeset
|
384 |
instance |
fa810f01f7b5
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changeset
|
385 |
apply default |
fa810f01f7b5
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parents:
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changeset
|
386 |
apply (simp add: permute_fun_def) |
fa810f01f7b5
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parents:
diff
changeset
|
387 |
apply (simp add: permute_fun_def minus_add) |
fa810f01f7b5
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parents:
diff
changeset
|
388 |
done |
fa810f01f7b5
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parents:
diff
changeset
|
389 |
|
fa810f01f7b5
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parents:
diff
changeset
|
390 |
end |
fa810f01f7b5
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parents:
diff
changeset
|
391 |
|
fa810f01f7b5
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parents:
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changeset
|
392 |
lemma permute_fun_app_eq: |
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changeset
|
393 |
shows "p \<bullet> (f x) = (p \<bullet> f) (p \<bullet> x)" |
fa810f01f7b5
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parents:
diff
changeset
|
394 |
unfolding permute_fun_def by simp |
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diff
changeset
|
395 |
|
fa810f01f7b5
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parents:
diff
changeset
|
396 |
|
fa810f01f7b5
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parents:
diff
changeset
|
397 |
subsection {* Permutations for booleans *} |
fa810f01f7b5
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parents:
diff
changeset
|
398 |
|
fa810f01f7b5
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parents:
diff
changeset
|
399 |
instantiation bool :: pt |
fa810f01f7b5
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parents:
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changeset
|
400 |
begin |
fa810f01f7b5
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parents:
diff
changeset
|
401 |
|
fa810f01f7b5
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parents:
diff
changeset
|
402 |
definition "p \<bullet> (b::bool) = b" |
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
403 |
|
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
404 |
instance |
fa810f01f7b5
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parents:
diff
changeset
|
405 |
apply(default) |
fa810f01f7b5
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parents:
diff
changeset
|
406 |
apply(simp_all add: permute_bool_def) |
fa810f01f7b5
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parents:
diff
changeset
|
407 |
done |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
408 |
|
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
409 |
end |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
410 |
|
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
411 |
lemma Not_eqvt: |
fa810f01f7b5
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parents:
diff
changeset
|
412 |
shows "p \<bullet> (\<not> A) = (\<not> (p \<bullet> A))" |
fa810f01f7b5
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parents:
diff
changeset
|
413 |
by (simp add: permute_bool_def) |
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
414 |
|
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
415 |
|
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
416 |
subsection {* Permutations for sets *} |
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
417 |
|
fa810f01f7b5
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parents:
diff
changeset
|
418 |
lemma permute_set_eq: |
fa810f01f7b5
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parents:
diff
changeset
|
419 |
fixes x::"'a::pt" |
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
420 |
and p::"perm" |
fa810f01f7b5
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parents:
diff
changeset
|
421 |
shows "(p \<bullet> X) = {p \<bullet> x | x. x \<in> X}" |
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
422 |
apply(auto simp add: permute_fun_def permute_bool_def mem_def) |
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
423 |
apply(rule_tac x="- p \<bullet> x" in exI) |
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
424 |
apply(simp) |
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
425 |
done |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
426 |
|
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
427 |
lemma permute_set_eq_image: |
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
428 |
shows "p \<bullet> X = permute p ` X" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
429 |
unfolding permute_set_eq by auto |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
430 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
431 |
lemma permute_set_eq_vimage: |
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
432 |
shows "p \<bullet> X = permute (- p) -` X" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
433 |
unfolding permute_fun_def permute_bool_def |
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
434 |
unfolding vimage_def Collect_def mem_def .. |
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
435 |
|
1013 | 436 |
lemma swap_set_not_in: |
437 |
assumes a: "a \<notin> S" "b \<notin> S" |
|
438 |
shows "(a \<rightleftharpoons> b) \<bullet> S = S" |
|
439 |
using a by (auto simp add: permute_set_eq swap_atom) |
|
440 |
||
441 |
lemma swap_set_in: |
|
442 |
assumes a: "a \<in> S" "b \<notin> S" "sort_of a = sort_of b" |
|
443 |
shows "(a \<rightleftharpoons> b) \<bullet> S \<noteq> S" |
|
444 |
using a by (auto simp add: permute_set_eq swap_atom) |
|
445 |
||
446 |
||
947
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
447 |
subsection {* Permutations for units *} |
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
448 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
449 |
instantiation unit :: pt |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
450 |
begin |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
451 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
452 |
definition "p \<bullet> (u::unit) = u" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
453 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
454 |
instance proof |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
455 |
qed (simp_all add: permute_unit_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
456 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
457 |
end |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
458 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
459 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
460 |
subsection {* Permutations for products *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
461 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
462 |
instantiation "*" :: (pt, pt) pt |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
463 |
begin |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
464 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
465 |
primrec |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
466 |
permute_prod |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
467 |
where |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
468 |
Pair_eqvt: "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
469 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
470 |
instance |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
471 |
by default auto |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
472 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
473 |
end |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
474 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
475 |
subsection {* Permutations for sums *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
476 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
477 |
instantiation "+" :: (pt, pt) pt |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
478 |
begin |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
479 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
480 |
primrec |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
481 |
permute_sum |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
482 |
where |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
483 |
"p \<bullet> (Inl x) = Inl (p \<bullet> x)" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
484 |
| "p \<bullet> (Inr y) = Inr (p \<bullet> y)" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
485 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
486 |
instance proof |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
487 |
qed (case_tac [!] x, simp_all) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
488 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
489 |
end |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
490 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
491 |
subsection {* Permutations for lists *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
492 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
493 |
instantiation list :: (pt) pt |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
494 |
begin |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
495 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
496 |
primrec |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
497 |
permute_list |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
498 |
where |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
499 |
"p \<bullet> [] = []" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
500 |
| "p \<bullet> (x # xs) = p \<bullet> x # p \<bullet> xs" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
501 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
502 |
instance proof |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
503 |
qed (induct_tac [!] x, simp_all) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
504 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
505 |
end |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
506 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
507 |
subsection {* Permutations for options *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
508 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
509 |
instantiation option :: (pt) pt |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
510 |
begin |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
511 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
512 |
primrec |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
513 |
permute_option |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
514 |
where |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
515 |
"p \<bullet> None = None" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
516 |
| "p \<bullet> (Some x) = Some (p \<bullet> x)" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
517 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
518 |
instance proof |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
519 |
qed (induct_tac [!] x, simp_all) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
520 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
521 |
end |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
522 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
523 |
subsection {* Permutations for @{typ char}, @{typ nat}, and @{typ int} *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
524 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
525 |
instantiation char :: pt |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
526 |
begin |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
527 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
528 |
definition "p \<bullet> (c::char) = c" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
529 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
530 |
instance proof |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
531 |
qed (simp_all add: permute_char_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
532 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
533 |
end |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
534 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
535 |
instantiation nat :: pt |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
536 |
begin |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
537 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
538 |
definition "p \<bullet> (n::nat) = n" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
539 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
540 |
instance proof |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
541 |
qed (simp_all add: permute_nat_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
542 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
543 |
end |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
544 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
545 |
instantiation int :: pt |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
546 |
begin |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
547 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
548 |
definition "p \<bullet> (i::int) = i" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
549 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
550 |
instance proof |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
551 |
qed (simp_all add: permute_int_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
552 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
553 |
end |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
554 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
555 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
556 |
section {* Pure types *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
557 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
558 |
text {* Pure types will have always empty support. *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
559 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
560 |
class pure = pt + |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
561 |
assumes permute_pure: "p \<bullet> x = x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
562 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
563 |
text {* Types @{typ unit} and @{typ bool} are pure. *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
564 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
565 |
instance unit :: pure |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
566 |
proof qed (rule permute_unit_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
567 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
568 |
instance bool :: pure |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
569 |
proof qed (rule permute_bool_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
570 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
571 |
text {* Other type constructors preserve purity. *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
572 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
573 |
instance "fun" :: (pure, pure) pure |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
574 |
by default (simp add: permute_fun_def permute_pure) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
575 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
576 |
instance "*" :: (pure, pure) pure |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
577 |
by default (induct_tac x, simp add: permute_pure) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
578 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
579 |
instance "+" :: (pure, pure) pure |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
580 |
by default (induct_tac x, simp_all add: permute_pure) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
581 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
582 |
instance list :: (pure) pure |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
583 |
by default (induct_tac x, simp_all add: permute_pure) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
584 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
585 |
instance option :: (pure) pure |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
586 |
by default (induct_tac x, simp_all add: permute_pure) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
587 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
588 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
589 |
subsection {* Types @{typ char}, @{typ nat}, and @{typ int} *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
590 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
591 |
instance char :: pure |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
592 |
proof qed (rule permute_char_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
593 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
594 |
instance nat :: pure |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
595 |
proof qed (rule permute_nat_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
596 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
597 |
instance int :: pure |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
598 |
proof qed (rule permute_int_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
599 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
600 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
601 |
subsection {* Supp, Freshness and Supports *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
602 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
603 |
context pt |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
604 |
begin |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
605 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
606 |
definition |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
607 |
supp :: "'a \<Rightarrow> atom set" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
608 |
where |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
609 |
"supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
610 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
611 |
end |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
612 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
613 |
definition |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
614 |
fresh :: "atom \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp> _" [55, 55] 55) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
615 |
where |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
616 |
"a \<sharp> x \<equiv> a \<notin> supp x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
617 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
618 |
lemma supp_conv_fresh: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
619 |
shows "supp x = {a. \<not> a \<sharp> x}" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
620 |
unfolding fresh_def by simp |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
621 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
622 |
lemma swap_rel_trans: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
623 |
assumes "sort_of a = sort_of b" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
624 |
assumes "sort_of b = sort_of c" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
625 |
assumes "(a \<rightleftharpoons> c) \<bullet> x = x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
626 |
assumes "(b \<rightleftharpoons> c) \<bullet> x = x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
627 |
shows "(a \<rightleftharpoons> b) \<bullet> x = x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
628 |
proof (cases) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
629 |
assume "a = b \<or> c = b" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
630 |
with assms show "(a \<rightleftharpoons> b) \<bullet> x = x" by auto |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
631 |
next |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
632 |
assume *: "\<not> (a = b \<or> c = b)" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
633 |
have "((a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c)) \<bullet> x = x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
634 |
using assms by simp |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
635 |
also have "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
636 |
using assms * by (simp add: swap_triple) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
637 |
finally show "(a \<rightleftharpoons> b) \<bullet> x = x" . |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
638 |
qed |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
639 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
640 |
lemma swap_fresh_fresh: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
641 |
assumes a: "a \<sharp> x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
642 |
and b: "b \<sharp> x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
643 |
shows "(a \<rightleftharpoons> b) \<bullet> x = x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
644 |
proof (cases) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
645 |
assume asm: "sort_of a = sort_of b" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
646 |
have "finite {c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x}" "finite {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x}" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
647 |
using a b unfolding fresh_def supp_def by simp_all |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
648 |
then have "finite ({c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x} \<union> {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x})" by simp |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
649 |
then obtain c |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
650 |
where "(a \<rightleftharpoons> c) \<bullet> x = x" "(b \<rightleftharpoons> c) \<bullet> x = x" "sort_of c = sort_of b" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
651 |
by (rule obtain_atom) (auto) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
652 |
then show "(a \<rightleftharpoons> b) \<bullet> x = x" using asm by (rule_tac swap_rel_trans) (simp_all) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
653 |
next |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
654 |
assume "sort_of a \<noteq> sort_of b" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
655 |
then show "(a \<rightleftharpoons> b) \<bullet> x = x" by simp |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
656 |
qed |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
657 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
658 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
659 |
subsection {* supp and fresh are equivariant *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
660 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
661 |
lemma finite_Collect_bij: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
662 |
assumes a: "bij f" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
663 |
shows "finite {x. P (f x)} = finite {x. P x}" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
664 |
by (metis a finite_vimage_iff vimage_Collect_eq) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
665 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
666 |
lemma fresh_permute_iff: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
667 |
shows "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
668 |
proof - |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
669 |
have "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
670 |
unfolding fresh_def supp_def by simp |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
671 |
also have "\<dots> \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> p \<bullet> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
672 |
using bij_permute by (rule finite_Collect_bij [symmetric]) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
673 |
also have "\<dots> \<longleftrightarrow> finite {b. p \<bullet> (a \<rightleftharpoons> b) \<bullet> x \<noteq> p \<bullet> x}" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
674 |
by (simp only: permute_eqvt [of p] swap_eqvt) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
675 |
also have "\<dots> \<longleftrightarrow> finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
676 |
by (simp only: permute_eq_iff) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
677 |
also have "\<dots> \<longleftrightarrow> a \<sharp> x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
678 |
unfolding fresh_def supp_def by simp |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
679 |
finally show ?thesis . |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
680 |
qed |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
681 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
682 |
lemma fresh_eqvt: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
683 |
shows "p \<bullet> (a \<sharp> x) = (p \<bullet> a) \<sharp> (p \<bullet> x)" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
684 |
by (simp add: permute_bool_def fresh_permute_iff) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
685 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
686 |
lemma supp_eqvt: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
687 |
fixes p :: "perm" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
688 |
and x :: "'a::pt" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
689 |
shows "p \<bullet> (supp x) = supp (p \<bullet> x)" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
690 |
unfolding supp_conv_fresh |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
691 |
unfolding permute_fun_def Collect_def |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
692 |
by (simp add: Not_eqvt fresh_eqvt) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
693 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
694 |
subsection {* supports *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
695 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
696 |
definition |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
697 |
supports :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" (infixl "supports" 80) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
698 |
where |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
699 |
"S supports x \<equiv> \<forall>a b. (a \<notin> S \<and> b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x)" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
700 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
701 |
lemma supp_is_subset: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
702 |
fixes S :: "atom set" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
703 |
and x :: "'a::pt" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
704 |
assumes a1: "S supports x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
705 |
and a2: "finite S" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
706 |
shows "(supp x) \<subseteq> S" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
707 |
proof (rule ccontr) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
708 |
assume "\<not>(supp x \<subseteq> S)" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
709 |
then obtain a where b1: "a \<in> supp x" and b2: "a \<notin> S" by auto |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
710 |
from a1 b2 have "\<forall>b. b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x" by (unfold supports_def) (auto) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
711 |
hence "{b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x} \<subseteq> S" by auto |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
712 |
with a2 have "finite {b. (a \<rightleftharpoons> b)\<bullet>x \<noteq> x}" by (simp add: finite_subset) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
713 |
then have "a \<notin> (supp x)" unfolding supp_def by simp |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
714 |
with b1 show False by simp |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
715 |
qed |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
716 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
717 |
lemma supports_finite: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
718 |
fixes S :: "atom set" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
719 |
and x :: "'a::pt" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
720 |
assumes a1: "S supports x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
721 |
and a2: "finite S" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
722 |
shows "finite (supp x)" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
723 |
proof - |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
724 |
have "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
725 |
then show "finite (supp x)" using a2 by (simp add: finite_subset) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
726 |
qed |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
727 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
728 |
lemma supp_supports: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
729 |
fixes x :: "'a::pt" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
730 |
shows "(supp x) supports x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
731 |
proof (unfold supports_def, intro strip) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
732 |
fix a b |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
733 |
assume "a \<notin> (supp x) \<and> b \<notin> (supp x)" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
734 |
then have "a \<sharp> x" and "b \<sharp> x" by (simp_all add: fresh_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
735 |
then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (rule swap_fresh_fresh) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
736 |
qed |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
737 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
738 |
lemma supp_is_least_supports: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
739 |
fixes S :: "atom set" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
740 |
and x :: "'a::pt" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
741 |
assumes a1: "S supports x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
742 |
and a2: "finite S" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
743 |
and a3: "\<And>S'. finite S' \<Longrightarrow> (S' supports x) \<Longrightarrow> S \<subseteq> S'" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
744 |
shows "(supp x) = S" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
745 |
proof (rule equalityI) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
746 |
show "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
747 |
with a2 have fin: "finite (supp x)" by (rule rev_finite_subset) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
748 |
have "(supp x) supports x" by (rule supp_supports) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
749 |
with fin a3 show "S \<subseteq> supp x" by blast |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
750 |
qed |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
751 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
752 |
lemma subsetCI: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
753 |
shows "(\<And>x. x \<in> A \<Longrightarrow> x \<notin> B \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> B" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
754 |
by auto |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
755 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
756 |
lemma finite_supp_unique: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
757 |
assumes a1: "S supports x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
758 |
assumes a2: "finite S" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
759 |
assumes a3: "\<And>a b. \<lbrakk>a \<in> S; b \<notin> S; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x \<noteq> x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
760 |
shows "(supp x) = S" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
761 |
using a1 a2 |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
762 |
proof (rule supp_is_least_supports) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
763 |
fix S' |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
764 |
assume "finite S'" and "S' supports x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
765 |
show "S \<subseteq> S'" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
766 |
proof (rule subsetCI) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
767 |
fix a |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
768 |
assume "a \<in> S" and "a \<notin> S'" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
769 |
have "finite (S \<union> S')" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
770 |
using `finite S` `finite S'` by simp |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
771 |
then obtain b where "b \<notin> S \<union> S'" and "sort_of b = sort_of a" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
772 |
by (rule obtain_atom) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
773 |
then have "b \<notin> S" and "b \<notin> S'" and "sort_of a = sort_of b" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
774 |
by simp_all |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
775 |
then have "(a \<rightleftharpoons> b) \<bullet> x = x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
776 |
using `a \<notin> S'` `S' supports x` by (simp add: supports_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
777 |
moreover have "(a \<rightleftharpoons> b) \<bullet> x \<noteq> x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
778 |
using `a \<in> S` `b \<notin> S` `sort_of a = sort_of b` |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
779 |
by (rule a3) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
780 |
ultimately show "False" by simp |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
781 |
qed |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
782 |
qed |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
783 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
784 |
section {* Finitely-supported types *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
785 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
786 |
class fs = pt + |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
787 |
assumes finite_supp: "finite (supp x)" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
788 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
789 |
lemma pure_supp: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
790 |
shows "supp (x::'a::pure) = {}" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
791 |
unfolding supp_def by (simp add: permute_pure) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
792 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
793 |
lemma pure_fresh: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
794 |
fixes x::"'a::pure" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
795 |
shows "a \<sharp> x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
796 |
unfolding fresh_def by (simp add: pure_supp) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
797 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
798 |
instance pure < fs |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
799 |
by default (simp add: pure_supp) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
800 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
801 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
802 |
subsection {* Type @{typ atom} is finitely-supported. *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
803 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
804 |
lemma supp_atom: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
805 |
shows "supp a = {a}" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
806 |
apply (rule finite_supp_unique) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
807 |
apply (clarsimp simp add: supports_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
808 |
apply simp |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
809 |
apply simp |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
810 |
done |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
811 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
812 |
lemma fresh_atom: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
813 |
shows "a \<sharp> b \<longleftrightarrow> a \<noteq> b" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
814 |
unfolding fresh_def supp_atom by simp |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
815 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
816 |
instance atom :: fs |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
817 |
by default (simp add: supp_atom) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
818 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
819 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
820 |
section {* Type @{typ perm} is finitely-supported. *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
821 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
822 |
lemma perm_swap_eq: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
823 |
shows "(a \<rightleftharpoons> b) \<bullet> p = p \<longleftrightarrow> (p \<bullet> (a \<rightleftharpoons> b)) = (a \<rightleftharpoons> b)" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
824 |
unfolding permute_perm_def |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
825 |
by (metis add_diff_cancel minus_perm_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
826 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
827 |
lemma supports_perm: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
828 |
shows "{a. p \<bullet> a \<noteq> a} supports p" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
829 |
unfolding supports_def |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
830 |
by (simp add: perm_swap_eq swap_eqvt) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
831 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
832 |
lemma finite_perm_lemma: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
833 |
shows "finite {a::atom. p \<bullet> a \<noteq> a}" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
834 |
using finite_Rep_perm [of p] |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
835 |
unfolding permute_atom_def . |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
836 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
837 |
lemma supp_perm: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
838 |
shows "supp p = {a. p \<bullet> a \<noteq> a}" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
839 |
apply (rule finite_supp_unique) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
840 |
apply (rule supports_perm) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
841 |
apply (rule finite_perm_lemma) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
842 |
apply (simp add: perm_swap_eq swap_eqvt) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
843 |
apply (auto simp add: expand_perm_eq swap_atom) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
844 |
done |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
845 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
846 |
lemma fresh_perm: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
847 |
shows "a \<sharp> p \<longleftrightarrow> p \<bullet> a = a" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
848 |
unfolding fresh_def by (simp add: supp_perm) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
849 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
850 |
lemma supp_swap: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
851 |
shows "supp (a \<rightleftharpoons> b) = (if a = b \<or> sort_of a \<noteq> sort_of b then {} else {a, b})" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
852 |
by (auto simp add: supp_perm swap_atom) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
853 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
854 |
lemma fresh_zero_perm: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
855 |
shows "a \<sharp> (0::perm)" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
856 |
unfolding fresh_perm by simp |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
857 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
858 |
lemma supp_zero_perm: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
859 |
shows "supp (0::perm) = {}" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
860 |
unfolding supp_perm by simp |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
861 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
862 |
lemma supp_plus_perm: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
863 |
fixes p q::perm |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
864 |
shows "supp (p + q) \<subseteq> supp p \<union> supp q" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
865 |
by (auto simp add: supp_perm) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
866 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
867 |
lemma supp_minus_perm: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
868 |
fixes p::perm |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
869 |
shows "supp (- p) = supp p" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
870 |
apply(auto simp add: supp_perm) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
871 |
apply(metis permute_minus_cancel)+ |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
872 |
done |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
873 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
874 |
instance perm :: fs |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
875 |
by default (simp add: supp_perm finite_perm_lemma) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
876 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
877 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
878 |
section {* Finite Support instances for other types *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
879 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
880 |
subsection {* Type @{typ "'a \<times> 'b"} is finitely-supported. *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
881 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
882 |
lemma supp_Pair: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
883 |
shows "supp (x, y) = supp x \<union> supp y" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
884 |
by (simp add: supp_def Collect_imp_eq Collect_neg_eq) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
885 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
886 |
lemma fresh_Pair: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
887 |
shows "a \<sharp> (x, y) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> y" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
888 |
by (simp add: fresh_def supp_Pair) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
889 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
890 |
instance "*" :: (fs, fs) fs |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
891 |
apply default |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
892 |
apply (induct_tac x) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
893 |
apply (simp add: supp_Pair finite_supp) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
894 |
done |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
895 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
896 |
subsection {* Type @{typ "'a + 'b"} is finitely supported *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
897 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
898 |
lemma supp_Inl: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
899 |
shows "supp (Inl x) = supp x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
900 |
by (simp add: supp_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
901 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
902 |
lemma supp_Inr: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
903 |
shows "supp (Inr x) = supp x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
904 |
by (simp add: supp_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
905 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
906 |
lemma fresh_Inl: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
907 |
shows "a \<sharp> Inl x \<longleftrightarrow> a \<sharp> x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
908 |
by (simp add: fresh_def supp_Inl) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
909 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
910 |
lemma fresh_Inr: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
911 |
shows "a \<sharp> Inr y \<longleftrightarrow> a \<sharp> y" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
912 |
by (simp add: fresh_def supp_Inr) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
913 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
914 |
instance "+" :: (fs, fs) fs |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
915 |
apply default |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
916 |
apply (induct_tac x) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
917 |
apply (simp_all add: supp_Inl supp_Inr finite_supp) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
918 |
done |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
919 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
920 |
subsection {* Type @{typ "'a option"} is finitely supported *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
921 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
922 |
lemma supp_None: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
923 |
shows "supp None = {}" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
924 |
by (simp add: supp_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
925 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
926 |
lemma supp_Some: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
927 |
shows "supp (Some x) = supp x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
928 |
by (simp add: supp_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
929 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
930 |
lemma fresh_None: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
931 |
shows "a \<sharp> None" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
932 |
by (simp add: fresh_def supp_None) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
933 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
934 |
lemma fresh_Some: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
935 |
shows "a \<sharp> Some x \<longleftrightarrow> a \<sharp> x" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
936 |
by (simp add: fresh_def supp_Some) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
937 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
938 |
instance option :: (fs) fs |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
939 |
apply default |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
940 |
apply (induct_tac x) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
941 |
apply (simp_all add: supp_None supp_Some finite_supp) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
942 |
done |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
943 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
944 |
subsubsection {* Type @{typ "'a list"} is finitely supported *} |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
945 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
946 |
lemma supp_Nil: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
947 |
shows "supp [] = {}" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
948 |
by (simp add: supp_def) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
949 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
950 |
lemma supp_Cons: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
951 |
shows "supp (x # xs) = supp x \<union> supp xs" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
952 |
by (simp add: supp_def Collect_imp_eq Collect_neg_eq) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
953 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
954 |
lemma fresh_Nil: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
955 |
shows "a \<sharp> []" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
956 |
by (simp add: fresh_def supp_Nil) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
957 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
958 |
lemma fresh_Cons: |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
959 |
shows "a \<sharp> (x # xs) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> xs" |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
960 |
by (simp add: fresh_def supp_Cons) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
961 |
|
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
962 |
instance list :: (fs) fs |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
963 |
apply default |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
964 |
apply (induct_tac x) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
965 |
apply (simp_all add: supp_Nil supp_Cons finite_supp) |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
966 |
done |
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
967 |
|
1013 | 968 |
section {* Support and freshness for applications *} |
1008
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
947
diff
changeset
|
969 |
|
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
947
diff
changeset
|
970 |
lemma supp_fun_app: |
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
947
diff
changeset
|
971 |
shows "supp (f x) \<subseteq> (supp f) \<union> (supp x)" |
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
947
diff
changeset
|
972 |
proof (rule subsetCI) |
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
947
diff
changeset
|
973 |
fix a::"atom" |
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
947
diff
changeset
|
974 |
assume a: "a \<in> supp (f x)" |
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
947
diff
changeset
|
975 |
assume b: "a \<notin> supp f \<union> supp x" |
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
947
diff
changeset
|
976 |
then have "finite {b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f}" "finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}" |
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
947
diff
changeset
|
977 |
unfolding supp_def by auto |
7c633507a809
updated from huffman - repo
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parents:
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|
978 |
then have "finite ({b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f} \<union> {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x})" by simp |
7c633507a809
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Christian Urban <urbanc@in.tum.de>
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|
979 |
moreover |
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parents:
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|
980 |
have "{b. ((a \<rightleftharpoons> b) \<bullet> f) ((a \<rightleftharpoons> b) \<bullet> x) \<noteq> f x} \<subseteq> ({b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f} \<union> {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x})" |
7c633507a809
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parents:
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diff
changeset
|
981 |
by auto |
7c633507a809
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parents:
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diff
changeset
|
982 |
ultimately have "finite {b. ((a \<rightleftharpoons> b) \<bullet> f) ((a \<rightleftharpoons> b) \<bullet> x) \<noteq> f x}" |
7c633507a809
updated from huffman - repo
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parents:
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diff
changeset
|
983 |
using finite_subset by auto |
7c633507a809
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parents:
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diff
changeset
|
984 |
then have "a \<notin> supp (f x)" unfolding supp_def |
7c633507a809
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parents:
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diff
changeset
|
985 |
by (simp add: permute_fun_app_eq) |
7c633507a809
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Christian Urban <urbanc@in.tum.de>
parents:
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diff
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|
986 |
with a show "False" by simp |
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
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diff
changeset
|
987 |
qed |
7c633507a809
updated from huffman - repo
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parents:
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diff
changeset
|
988 |
|
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
947
diff
changeset
|
989 |
lemma fresh_fun_app: |
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
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diff
changeset
|
990 |
shows "a \<sharp> (f, x) \<Longrightarrow> a \<sharp> f x" |
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
947
diff
changeset
|
991 |
unfolding fresh_def |
7c633507a809
updated from huffman - repo
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parents:
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diff
changeset
|
992 |
using supp_fun_app |
7c633507a809
updated from huffman - repo
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parents:
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diff
changeset
|
993 |
by (auto simp add: supp_Pair) |
7c633507a809
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Christian Urban <urbanc@in.tum.de>
parents:
947
diff
changeset
|
994 |
|
7c633507a809
updated from huffman - repo
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parents:
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diff
changeset
|
995 |
lemma fresh_fun_eqvt_app: |
7c633507a809
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parents:
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diff
changeset
|
996 |
assumes a: "\<forall>p. p \<bullet> f = f" |
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
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diff
changeset
|
997 |
shows "a \<sharp> x \<Longrightarrow> a \<sharp> f x" |
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
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diff
changeset
|
998 |
proof - |
7c633507a809
updated from huffman - repo
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parents:
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diff
changeset
|
999 |
from a have b: "supp f = {}" |
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
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diff
changeset
|
1000 |
unfolding supp_def by simp |
7c633507a809
updated from huffman - repo
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parents:
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diff
changeset
|
1001 |
show "a \<sharp> x \<Longrightarrow> a \<sharp> f x" |
7c633507a809
updated from huffman - repo
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parents:
947
diff
changeset
|
1002 |
unfolding fresh_def |
7c633507a809
updated from huffman - repo
Christian Urban <urbanc@in.tum.de>
parents:
947
diff
changeset
|
1003 |
using supp_fun_app b |
7c633507a809
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Christian Urban <urbanc@in.tum.de>
parents:
947
diff
changeset
|
1004 |
by auto |
7c633507a809
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Christian Urban <urbanc@in.tum.de>
parents:
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diff
changeset
|
1005 |
qed |
7c633507a809
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Christian Urban <urbanc@in.tum.de>
parents:
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diff
changeset
|
1006 |
|
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
1007 |
end |