author | urbanc |
Wed, 08 Mar 2006 17:55:51 +0100 | |
changeset 19217 | 5caacd449ea4 |
parent 19171 | 17b952ec5641 |
child 19225 | a23af144eb47 |
permissions | -rw-r--r-- |
19157
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1 |
(* $Id$ *) |
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2 |
theory Iteration |
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imports "../nominal" |
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4 |
begin |
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5 |
|
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6 |
atom_decl name |
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7 |
|
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8 |
nominal_datatype lam = Var "name" |
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9 |
| App "lam" "lam" |
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10 |
| Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100) |
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11 |
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12 |
types 'a f1_ty = "name\<Rightarrow>('a::pt_name)" |
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13 |
'a f2_ty = "'a\<Rightarrow>'a\<Rightarrow>('a::pt_name)" |
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14 |
'a f3_ty = "name\<Rightarrow>'a\<Rightarrow>('a::pt_name)" |
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|
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16 |
consts |
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17 |
it :: "'a f1_ty \<Rightarrow> 'a f2_ty \<Rightarrow> 'a f3_ty \<Rightarrow> (lam \<times> (name prm \<Rightarrow> 'a::pt_name)) set" |
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18 |
|
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19 |
inductive "it f1 f2 f3" |
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20 |
intros |
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21 |
it1: "(Var a,\<lambda>pi. f1(pi\<bullet>a)) \<in> it f1 f2 f3" |
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22 |
it2: "\<lbrakk>(t1,r1) \<in> it f1 f2 f3; (t2,r2) \<in> it f1 f2 f3\<rbrakk> \<Longrightarrow> |
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23 |
(App t1 t2,\<lambda>pi. f2 (r1 pi) (r2 pi)) \<in> it f1 f2 f3" |
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it3: "(t,r) \<in> it f1 f2 f3 \<Longrightarrow> |
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25 |
(Lam [a].t,\<lambda>pi. fresh_fun (\<lambda>a'. f3 a' (r (pi@[(a,a')])))) \<in> it f1 f2 f3" |
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26 |
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27 |
constdefs |
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28 |
itfun' :: "'a f1_ty \<Rightarrow> 'a f2_ty \<Rightarrow> 'a f3_ty \<Rightarrow> lam \<Rightarrow> name prm \<Rightarrow> ('a::pt_name)" |
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29 |
"itfun' f1 f2 f3 t \<equiv> (THE y. (t,y) \<in> it f1 f2 f3)" |
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30 |
|
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31 |
itfun :: "'a f1_ty \<Rightarrow> 'a f2_ty \<Rightarrow> 'a f3_ty \<Rightarrow> lam \<Rightarrow> ('a::pt_name)" |
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32 |
"itfun f1 f2 f3 t \<equiv> itfun' f1 f2 f3 t ([]::name prm)" |
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33 |
|
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34 |
lemma it_total: |
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35 |
shows "\<exists>r. (t,r) \<in> it f1 f2 f3" |
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36 |
apply(induct t rule: lam.induct_weak) |
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37 |
apply(auto intro: it.intros) |
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38 |
done |
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39 |
|
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40 |
lemma it_prm_eq: |
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41 |
assumes a: "(t,y) \<in> it f1 f2 f3" |
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42 |
and b: "pi1 \<triangleq> pi2" |
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43 |
shows "y pi1 = y pi2" |
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44 |
using a b |
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45 |
apply(induct fixing: pi1 pi2) |
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46 |
apply(auto simp add: pt3[OF pt_name_inst]) |
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47 |
apply(rule_tac f="fresh_fun" in arg_cong) |
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48 |
apply(auto simp add: expand_fun_eq) |
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49 |
apply(drule_tac x="pi1@[(a,x)]" in meta_spec) |
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50 |
apply(drule_tac x="pi2@[(a,x)]" in meta_spec) |
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51 |
apply(force simp add: prm_eq_def pt2[OF pt_name_inst]) |
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52 |
done |
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53 |
|
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54 |
lemma f3_freshness_conditions_simple: |
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55 |
fixes f3::"('a::pt_name) f3_ty" |
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56 |
and y ::"name prm \<Rightarrow> 'a::pt_name" |
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57 |
and a ::"name" |
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58 |
and pi::"name prm" |
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59 |
assumes a: "finite ((supp f3)::name set)" |
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parents:
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60 |
and b: "finite ((supp y)::name set)" |
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parents:
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61 |
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)" |
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parents:
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62 |
shows "\<exists>(a''::name). a''\<sharp>(\<lambda>a'. f3 a' (y (pi@[(a,a')]))) \<and> a''\<sharp>(\<lambda>a'. f3 a' (y (pi@[(a,a')]))) a''" |
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parents:
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63 |
proof - |
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parents:
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64 |
from c obtain a' where d0: "a'\<sharp>f3" and d1: "\<forall>(y::'a::pt_name). a'\<sharp>f3 a' y" by force |
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parents:
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65 |
have "\<exists>(a''::name). a''\<sharp>(f3,a,a',pi,y)" |
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parents:
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66 |
by (rule at_exists_fresh[OF at_name_inst], simp add: supp_prod fs_name1 a b) |
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67 |
then obtain a'' where d2: "a''\<sharp>f3" and d3: "a''\<noteq>a'" and d3b: "a''\<sharp>(f3,a,pi,y)" |
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parents:
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68 |
by (auto simp add: fresh_prod at_fresh[OF at_name_inst]) |
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parents:
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69 |
have d3c: "a''\<notin>((supp (f3,a,pi,y))::name set)" using d3b by (simp add: fresh_def) |
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parents:
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70 |
have d4: "a''\<sharp>f3 a'' (y (pi@[(a,a'')]))" |
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parents:
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71 |
proof - |
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parents:
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72 |
have d5: "[(a'',a')]\<bullet>f3 = f3" |
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parents:
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73 |
by (rule pt_fresh_fresh[OF pt_name_inst, OF at_name_inst, OF d2, OF d0]) |
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74 |
from d1 have "\<forall>(y::'a::pt_name). ([(a'',a')]\<bullet>a')\<sharp>([(a'',a')]\<bullet>(f3 a' y))" |
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parents:
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75 |
by (simp add: pt_fresh_bij[OF pt_name_inst, OF at_name_inst]) |
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parents:
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76 |
hence "\<forall>(y::'a::pt_name). a''\<sharp>(f3 a'' ([(a'',a')]\<bullet>y))" using d3 d5 |
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parents:
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77 |
by (simp add: at_calc[OF at_name_inst] pt_fun_app_eq[OF pt_name_inst, OF at_name_inst]) |
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parents:
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78 |
hence "a''\<sharp>(f3 a'' ([(a'',a')]\<bullet>((rev [(a'',a')])\<bullet>(y (pi@[(a,a'')])))))" by force |
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parents:
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79 |
thus ?thesis by (simp only: pt_pi_rev[OF pt_name_inst, OF at_name_inst]) |
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parents:
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80 |
qed |
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parents:
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81 |
have d6: "a''\<sharp>(\<lambda>a'. f3 a' (y (pi@[(a,a')])))" |
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parents:
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82 |
proof - |
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parents:
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83 |
from a b have d7: "finite ((supp (f3,a,pi,y))::name set)" by (simp add: supp_prod fs_name1) |
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parents:
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84 |
have "((supp (f3,a,pi,y))::name set) supports (\<lambda>a'. f3 a' (y (pi@[(a,a')])))" |
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parents:
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85 |
by (supports_simp add: perm_append) |
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parents:
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86 |
with d7 d3c show ?thesis by (simp add: supports_fresh) |
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parents:
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87 |
qed |
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parents:
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88 |
from d6 d4 show ?thesis by force |
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parents:
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89 |
qed |
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parents:
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90 |
|
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parents:
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91 |
text {* FIXME: this lemma should be thrown out somehow *} |
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parents:
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92 |
lemma f3_freshness_conditions: |
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parents:
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93 |
fixes f3::"('a::pt_name) f3_ty" |
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parents:
diff
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|
94 |
and y ::"name prm \<Rightarrow> 'a::pt_name" |
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parents:
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|
95 |
and a ::"name" |
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parents:
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96 |
and pi1::"name prm" |
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parents:
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|
97 |
and pi2::"name prm" |
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parents:
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|
98 |
assumes a: "finite ((supp f3)::name set)" |
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parents:
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99 |
and b: "finite ((supp y)::name set)" |
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parents:
diff
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|
100 |
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)" |
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parents:
diff
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|
101 |
shows "\<exists>(a''::name). a''\<sharp>(\<lambda>a'. f3 a' (y (pi1@[(a,a')]@pi2))) \<and> |
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urbanc
parents:
diff
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|
102 |
a''\<sharp>(\<lambda>a'. f3 a' (y (pi1@[(a,a')]@pi2))) a''" |
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parents:
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|
103 |
proof - |
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parents:
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|
104 |
from c obtain a' where d0: "a'\<sharp>f3" and d1: "\<forall>(y::'a::pt_name). a'\<sharp>f3 a' y" by force |
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urbanc
parents:
diff
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|
105 |
have "\<exists>(a''::name). a''\<sharp>(f3,a,a',pi1,pi2,y)" |
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urbanc
parents:
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|
106 |
by (rule at_exists_fresh[OF at_name_inst], simp add: supp_prod fs_name1 a b) |
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parents:
diff
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|
107 |
then obtain a'' where d2: "a''\<sharp>f3" and d3: "a''\<noteq>a'" and d3b: "a''\<sharp>(f3,a,pi1,pi2,y)" |
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parents:
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|
108 |
by (auto simp add: fresh_prod at_fresh[OF at_name_inst]) |
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urbanc
parents:
diff
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|
109 |
have d3c: "a''\<notin>((supp (f3,a,pi1,pi2,y))::name set)" using d3b by (simp add: fresh_def) |
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urbanc
parents:
diff
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|
110 |
have d4: "a''\<sharp>f3 a'' (y (pi1@[(a,a'')]@pi2))" |
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parents:
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|
111 |
proof - |
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parents:
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|
112 |
have d5: "[(a'',a')]\<bullet>f3 = f3" |
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parents:
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|
113 |
by (rule pt_fresh_fresh[OF pt_name_inst, OF at_name_inst, OF d2, OF d0]) |
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parents:
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|
114 |
from d1 have "\<forall>(y::'a::pt_name). ([(a'',a')]\<bullet>a')\<sharp>([(a'',a')]\<bullet>(f3 a' y))" |
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parents:
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|
115 |
by (simp add: pt_fresh_bij[OF pt_name_inst, OF at_name_inst]) |
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parents:
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|
116 |
hence "\<forall>(y::'a::pt_name). a''\<sharp>(f3 a'' ([(a'',a')]\<bullet>y))" using d3 d5 |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
117 |
by (simp add: at_calc[OF at_name_inst] pt_fun_app_eq[OF pt_name_inst, OF at_name_inst]) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
118 |
hence "a''\<sharp>(f3 a'' ([(a'',a')]\<bullet>((rev [(a'',a')])\<bullet>(y (pi1@[(a,a'')]@pi2)))))" by force |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
119 |
thus ?thesis by (simp only: pt_pi_rev[OF pt_name_inst, OF at_name_inst]) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
120 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
121 |
have d6: "a''\<sharp>(\<lambda>a'. f3 a' (y (pi1@[(a,a')]@pi2)))" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
122 |
proof - |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
123 |
from a b have d7: "finite ((supp (f3,a,pi1,pi2,y))::name set)" by (simp add: supp_prod fs_name1) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
124 |
have e: "((supp (f3,a,pi1,pi2,y))::name set) supports (\<lambda>a'. f3 a' (y (pi1@[(a,a')]@pi2)))" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
125 |
by (supports_simp add: perm_append) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
126 |
from e d7 d3c show ?thesis by (rule supports_fresh) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
127 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
128 |
from d6 d4 show ?thesis by force |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
129 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
130 |
|
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
131 |
lemma it_fin_supp: |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
132 |
assumes f: "finite ((supp (f1,f2,f3))::name set)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
133 |
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(r::'a::pt_name). a\<sharp>f3 a r)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
134 |
and a: "(t,r) \<in> it f1 f2 f3" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
135 |
shows "finite ((supp r)::name set)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
136 |
using a |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
137 |
proof (induct) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
138 |
case it1 thus ?case using f by (finite_guess add: supp_prod fs_name1) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
139 |
next |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
140 |
case it2 thus ?case using f by (finite_guess add: supp_prod fs_name1) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
141 |
next |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
142 |
case (it3 c r t) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
143 |
have ih: "finite ((supp r)::name set)" by fact |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
144 |
let ?g' = "\<lambda>pi a'. f3 a' (r (pi@[(c,a')]))" --"helper function" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
145 |
have fact1: "\<forall>pi. finite ((supp (?g' pi))::name set)" using f ih |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
146 |
by (rule_tac allI, finite_guess add: perm_append supp_prod fs_name1) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
147 |
have fact2: "\<forall>pi. \<exists>(a''::name). a''\<sharp>(?g' pi) \<and> a''\<sharp>((?g' pi) a'')" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
148 |
proof |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
149 |
fix pi::"name prm" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
150 |
show "\<exists>(a''::name). a''\<sharp>(?g' pi) \<and> a''\<sharp>((?g' pi) a'')" using f c ih |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
151 |
by (rule_tac f3_freshness_conditions_simple, simp_all add: supp_prod) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
152 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
153 |
show ?case using fact1 fact2 ih f |
19217 | 154 |
by (finite_guess add: fresh_fun_eqvt perm_append supp_prod fs_name1) |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
155 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
156 |
|
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
157 |
lemma it_trans: |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
158 |
shows "\<lbrakk>(t,r)\<in>rec f1 f2 f3; r=r'\<rbrakk> \<Longrightarrow> (t,r')\<in>rec f1 f2 f3" by simp |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
159 |
|
19158 | 160 |
lemma it_perm_aux: |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
161 |
assumes f: "finite ((supp (f1,f2,f3))::name set)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
162 |
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
163 |
and a: "(t,y) \<in> it f1 f2 f3" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
164 |
shows "(pi1\<bullet>t, \<lambda>pi2. y (pi2@pi1)) \<in> it f1 f2 f3" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
165 |
using a |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
166 |
proof (induct) |
19158 | 167 |
case (it1 c) show ?case by (auto simp add: pt_name2, rule it.intros) |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
168 |
next |
19158 | 169 |
case (it2 t1 t2 r1 r2) thus ?case by (auto intro: it.intros) |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
170 |
next |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
171 |
case (it3 c r t) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
172 |
let ?g = "\<lambda>pi' a'. f3 a' (r (pi'@[(pi1\<bullet>c,a')]@pi1))" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
173 |
and ?h = "\<lambda>pi' a'. f3 a' (r ((pi'@pi1)@[(c,a')]))" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
174 |
have ih': "(t,r) \<in> it f1 f2 f3" by fact |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
175 |
hence fin_r: "finite ((supp r)::name set)" using f c by (auto intro: it_fin_supp) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
176 |
have ih: "(pi1\<bullet>t,\<lambda>pi2. r (pi2@pi1)) \<in> it f1 f2 f3" by fact |
19168 | 177 |
hence "(Lam [(pi1\<bullet>c)].(pi1\<bullet>t),\<lambda>pi'. fresh_fun (?g pi')) \<in> it f1 f2 f3" |
178 |
by (auto intro: it_trans it.intros) |
|
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
179 |
moreover |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
180 |
have "\<forall>pi'. (fresh_fun (?g pi')) = (fresh_fun (?h pi'))" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
181 |
proof |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
182 |
fix pi'::"name prm" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
183 |
have fin_g: "finite ((supp (?g pi'))::name set)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
184 |
using f fin_r by (finite_guess add: perm_append supp_prod fs_name1) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
185 |
have fr_g: "\<exists>(a::name). (a\<sharp>(?g pi')\<and> a\<sharp>(?g pi' a))" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
186 |
using f c fin_r by (rule_tac f3_freshness_conditions, simp_all add: supp_prod) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
187 |
have fin_h: "finite ((supp (?h pi'))::name set)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
188 |
using f fin_r by (finite_guess add: perm_append supp_prod fs_name1) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
189 |
have fr_h: "\<exists>(a::name). (a\<sharp>(?h pi')\<and> a\<sharp>(?h pi' a))" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
190 |
using f c fin_r by (rule_tac f3_freshness_conditions_simple, simp_all add: supp_prod) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
191 |
show "fresh_fun (?g pi') = fresh_fun (?h pi')" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
192 |
proof - |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
193 |
have "\<exists>(d::name). d\<sharp>(?g pi', ?h pi', pi1)" using fin_g fin_h |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
194 |
by (rule_tac at_exists_fresh[OF at_name_inst], simp only: supp_prod finite_Un fs_name1, simp) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
195 |
then obtain d::"name" where f1: "d\<sharp>?g pi'" and f2: "d\<sharp>?h pi'" and f3: "d\<sharp>(rev pi1)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
196 |
by (auto simp only: fresh_prod fresh_list_rev) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
197 |
have "?g pi' d = ?h pi' d" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
198 |
proof - |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
199 |
have "r (pi'@[(pi1\<bullet>c,d)]@pi1) = r ((pi'@pi1)@[(c,d)])" using f3 ih' |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
200 |
by (auto intro!: it_prm_eq at_prm_eq_append[OF at_name_inst] |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
201 |
simp only: append_assoc at_ds10[OF at_name_inst]) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
202 |
then show ?thesis by simp |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
203 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
204 |
then show "fresh_fun (?g pi') = fresh_fun (?h pi')" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
205 |
using f1 fin_g fr_g f2 fin_h fr_h |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
206 |
by (simp add: fresh_fun_app[OF pt_name_inst, OF at_name_inst]) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
207 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
208 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
209 |
hence "(\<lambda>pi'. (fresh_fun (?g pi'))) = (\<lambda>pi'. (fresh_fun (?h pi')))" by (simp add: expand_fun_eq) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
210 |
ultimately show ?case by simp |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
211 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
212 |
|
19158 | 213 |
lemma it_perm: |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
214 |
assumes f: "finite ((supp (f1,f2,f3))::name set)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
215 |
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
216 |
and a: "(pi\<bullet>t,y) \<in> it f1 f2 f3" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
217 |
shows "(t, \<lambda>pi2. y (pi2@(rev pi))) \<in> it f1 f2 f3" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
218 |
proof - |
19217 | 219 |
from a have "((rev pi)\<bullet>(pi\<bullet>t),\<lambda>pi2. y (pi2@(rev pi))) \<in> it f1 f2 f3" |
220 |
using f c by (simp add: it_perm_aux) |
|
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
221 |
thus ?thesis by perm_simp |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
222 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
223 |
|
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
224 |
lemma it_unique: |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
225 |
assumes f: "finite ((supp (f1,f2,f3))::name set)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
226 |
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
227 |
and a: "(t,y) \<in> it f1 f2 f3" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
228 |
and b: "(t,y') \<in> it f1 f2 f3" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
229 |
shows "y pi = y' pi" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
230 |
using a b |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
231 |
proof (induct fixing: y' pi) |
19158 | 232 |
case (it1 c) thus ?case by (cases, simp_all add: lam.inject) |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
233 |
next |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
234 |
case (it2 r1 r2 t1 t2) |
19158 | 235 |
with `(App t1 t2, y') \<in> it f1 f2 f3` show ?case |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
236 |
by (cases, simp_all (no_asm_use) add: lam.inject, force) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
237 |
next |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
238 |
case (it3 c r t r') |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
239 |
have "(t,r) \<in> it f1 f2 f3" by fact |
19158 | 240 |
hence fin_r: "finite ((supp r)::name set)" using f c by (simp only: it_fin_supp) |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
241 |
have ih: "\<And>r' pi. (t,r') \<in> it f1 f2 f3 \<Longrightarrow> r pi = r' pi" by fact |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
242 |
have "(Lam [c].t, r') \<in> it f1 f2 f3" by fact |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
243 |
then show "fresh_fun (\<lambda>a'. f3 a' (r (pi@[(c,a')]))) = r' pi" |
19158 | 244 |
proof (cases, auto simp add: lam.inject) |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
245 |
fix a::"name" and t'::"lam" and r''::"name prm\<Rightarrow>'a::pt_name" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
246 |
assume i5: "[c].t = [a].t'" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
247 |
and i6: "(t',r'') \<in> it f1 f2 f3" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
248 |
hence fin_r'': "finite ((supp r'')::name set)" using f c by (auto intro: it_fin_supp) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
249 |
let ?g = "\<lambda>a'. f3 a' (r (pi@[(c,a')]))" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
250 |
and ?h = "\<lambda>a'. f3 a' (r'' (pi@[(a,a')]))" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
251 |
show "fresh_fun ?g = fresh_fun ?h" using i5 |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
252 |
proof (cases "a=c") |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
253 |
case True |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
254 |
have i7: "a=c" by fact |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
255 |
with i5 have i8: "t=t'" by (simp add: alpha) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
256 |
show "fresh_fun ?g = fresh_fun ?h" using i6 i7 i8 ih by simp |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
257 |
next |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
258 |
case False |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
259 |
assume i9: "a\<noteq>c" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
260 |
with i5[symmetric] have i10: "t'=[(a,c)]\<bullet>t" and i11: "a\<sharp>t" by (simp_all add: alpha) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
261 |
have fin_g: "finite ((supp ?g)::name set)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
262 |
using f fin_r by (finite_guess add: perm_append supp_prod fs_name1) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
263 |
have fin_h: "finite ((supp ?h)::name set)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
264 |
using f fin_r'' by (finite_guess add: perm_append supp_prod fs_name1) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
265 |
have fr_g: "\<exists>(a''::name). a''\<sharp>?g \<and> a''\<sharp>(?g a'')" |
19158 | 266 |
using f c fin_r by (simp add: f3_freshness_conditions_simple supp_prod) |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
267 |
have fr_h: "\<exists>(a''::name). a''\<sharp>?h \<and> a''\<sharp>(?h a'')" |
19158 | 268 |
using f c fin_r'' by (simp add: f3_freshness_conditions_simple supp_prod) |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
269 |
have "\<exists>(d::name). d\<sharp>(?g,?h,t,a,c)" using fin_g fin_h |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
270 |
by (rule_tac at_exists_fresh[OF at_name_inst], simp only: finite_Un supp_prod fs_name1, simp) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
271 |
then obtain d::"name" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
272 |
where f1: "d\<sharp>?g" and f2: "d\<sharp>?h" and f3: "d\<sharp>t" and f4: "d\<noteq>a" and f5: "d\<noteq>c" |
19158 | 273 |
by (force simp add: fresh_prod fresh_atm) |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
274 |
have g1: "[(a,d)]\<bullet>t = t" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
275 |
by (rule pt_fresh_fresh[OF pt_name_inst, OF at_name_inst, OF i11, OF f3]) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
276 |
from i6 have "(([(a,c)]@[(a,d)])\<bullet>t,r'') \<in> it f1 f2 f3" using g1 i10 by (simp only: pt_name2) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
277 |
hence "(t, \<lambda>pi2. r'' (pi2@(rev ([(a,c)]@[(a,d)])))) \<in> it f1 f2 f3" |
19158 | 278 |
by (simp only: it_perm[OF f, OF c]) |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
279 |
hence g2: "(t, \<lambda>pi2. r'' (pi2@([(a,d)]@[(a,c)]))) \<in> it f1 f2 f3" by simp |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
280 |
have "distinct [a,c,d]" using i9 f4 f5 by force |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
281 |
hence g3: "(pi@[(c,d)]@[(a,d)]@[(a,c)]) \<triangleq> (pi@[(a,d)])" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
282 |
by (rule_tac at_prm_eq_append[OF at_name_inst], |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
283 |
force simp add: prm_eq_def at_calc[OF at_name_inst]) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
284 |
have "fresh_fun ?g = ?g d" using fin_g fr_g f1 |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
285 |
by (simp add: fresh_fun_app[OF pt_name_inst, OF at_name_inst]) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
286 |
also have "\<dots> = f3 d ((\<lambda>pi2. r'' (pi2@([(a,d)]@[(a,c)]))) (pi@[(c,d)]))" using ih g2 by simp |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
287 |
also have "\<dots> = f3 d (r'' (pi@[(c,d)]@[(a,d)]@[(a,c)]))" by simp |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
288 |
also have "\<dots> = f3 d (r'' (pi@[(a,d)]))" using i6 g3 by (simp add: it_prm_eq) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
289 |
also have "\<dots> = fresh_fun ?h" using fin_h fr_h f2 |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
290 |
by (simp add: fresh_fun_app[OF pt_name_inst, OF at_name_inst]) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
291 |
finally show "fresh_fun ?g = fresh_fun ?h" by simp |
6e4ce7402dbe
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urbanc
parents:
diff
changeset
|
292 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
293 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
294 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
295 |
|
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
296 |
lemma it_function: |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
297 |
assumes f: "finite ((supp (f1,f2,f3))::name set)" |
19158 | 298 |
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(r::'a::pt_name). a\<sharp>f3 a r)" |
299 |
shows "\<exists>!r. (t,r) \<in> it f1 f2 f3" |
|
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
300 |
proof (rule ex_ex1I) |
19158 | 301 |
case goal1 show "\<exists>r. (t,r) \<in> it f1 f2 f3" by (rule it_total) |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
302 |
next |
19158 | 303 |
case (goal2 r1 r2) |
304 |
have a1: "(t,r1) \<in> it f1 f2 f3" and a2: "(t,r2) \<in> it f1 f2 f3" by fact |
|
305 |
hence "\<forall>pi. r1 pi = r2 pi" using it_unique[OF f, OF c] by simp |
|
306 |
thus "r1=r2" by (simp add: expand_fun_eq) |
|
19157
6e4ce7402dbe
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urbanc
parents:
diff
changeset
|
307 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
308 |
|
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
309 |
lemma it_eqvt: |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
310 |
fixes pi::"name prm" |
19158 | 311 |
assumes f: "finite ((supp (f1,f2,f3))::name set)" |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
312 |
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(r::'a::pt_name). a\<sharp>f3 a r)" |
19158 | 313 |
and a: "(t,r) \<in> it f1 f2 f3" |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
314 |
shows "(pi\<bullet>t,pi\<bullet>r) \<in> it (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3)" |
19217 | 315 |
using a proof(induct) |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
316 |
case it1 show ?case by (perm_simp add: it.intros) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
317 |
next |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
318 |
case it2 thus ?case by (perm_simp add: it.intros) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
319 |
next |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
320 |
case (it3 c r t) (* lam case *) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
321 |
let ?g = "pi\<bullet>(\<lambda>pi'. fresh_fun (\<lambda>a'. f3 a' (r (pi'@[(c,a')]))))" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
322 |
and ?h = "\<lambda>pi'. fresh_fun (\<lambda>a'. (pi\<bullet>f3) a' ((pi\<bullet>r) (pi'@[((pi\<bullet>c),a')])))" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
323 |
have "(t,r) \<in> it f1 f2 f3" by fact |
19158 | 324 |
hence fin_r: "finite ((supp r)::name set)" using f c by (auto intro: it_fin_supp) |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
325 |
have ih: "(pi\<bullet>t,pi\<bullet>r) \<in> it (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3)" by fact |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
326 |
hence "(Lam [(pi\<bullet>c)].(pi\<bullet>t),?h) \<in> it (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3)" by (simp add: it.intros) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
327 |
moreover |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
328 |
have "?g = ?h" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
329 |
proof - |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
330 |
let ?g' = "\<lambda>pi a'. f3 a' (r (pi@[(c,a')]))" |
19158 | 331 |
have fact1: "\<forall>pi. finite ((supp (?g' pi))::name set)" using fin_r f |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
332 |
by (rule_tac allI, finite_guess add: perm_append supp_prod fs_name1) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
333 |
have fact2: "\<forall>pi. \<exists>(a''::name). a''\<sharp>(?g' pi) \<and> a''\<sharp>((?g' pi) a'')" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
334 |
proof |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
335 |
fix pi::"name prm" |
19158 | 336 |
show "\<exists>(a''::name). a''\<sharp>(?g' pi) \<and> a''\<sharp>((?g' pi) a'')" using f c fin_r |
337 |
by (simp add: f3_freshness_conditions_simple supp_prod) |
|
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
338 |
qed |
19158 | 339 |
from fact1 fact2 show "?g = ?h" by (perm_simp add: fresh_fun_eqvt perm_append) |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
340 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
341 |
ultimately show "(pi\<bullet>Lam [c].t,?g) \<in> it (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3)" by simp |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
342 |
qed |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
343 |
|
19168 | 344 |
lemma the1_equality': |
345 |
assumes a: "\<exists>!r. P r" |
|
346 |
and b: "P b" |
|
347 |
and c: "b y = a" |
|
348 |
shows "(THE r. P r) y = a" |
|
349 |
apply(simp add: c[symmetric]) |
|
19217 | 350 |
apply(rule fun_cong[OF the1_equality, OF a, OF b]) |
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
351 |
done |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
352 |
|
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
353 |
lemma itfun'_prm: |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
354 |
assumes f: "finite ((supp (f1,f2,f3))::name set)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
355 |
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)" |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
356 |
shows "itfun' f1 f2 f3 (pi1\<bullet>t) pi2 = itfun' f1 f2 f3 t (pi2@pi1)" |
19168 | 357 |
apply(auto simp add: itfun_def itfun'_def) |
358 |
apply(rule the1_equality'[OF it_function, OF f, OF c]) |
|
359 |
apply(rule it_perm_aux[OF f, OF c]) |
|
360 |
apply(rule theI'[OF it_function,OF f, OF c]) |
|
19157
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
361 |
apply(simp) |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
362 |
done |
6e4ce7402dbe
initial commit (especially 2nd half needs to be cleaned up)
urbanc
parents:
diff
changeset
|
363 |
|
19168 | 364 |
lemma itfun'_eqvt: |
365 |
fixes pi1::"name prm" |
|
366 |
assumes f: "finite ((supp (f1,f2,f3))::name set)" |
|
367 |
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)" |
|
368 |
shows "pi1\<bullet>(itfun' f1 f2 f3 t pi2) = itfun' (pi1\<bullet>f1) (pi1\<bullet>f2) (pi1\<bullet>f3) (pi1\<bullet>t) (pi1\<bullet>pi2)" |
|
369 |
proof - |
|
370 |
have f_pi: "finite ((supp (pi1\<bullet>f1,pi1\<bullet>f2,pi1\<bullet>f3))::name set)" using f |
|
371 |
(* FIXME: check why proof (finite_guess_debug add: perm_append fs_name1) does not work *) |
|
372 |
by (simp add: supp_prod pt_supp_finite_pi[OF pt_name_inst, OF at_name_inst]) |
|
373 |
have fs_pi: "\<exists>(a::name). a\<sharp>(pi1\<bullet>f3) \<and> (\<forall>(y::'a::pt_name). a\<sharp>(pi1\<bullet>f3) a y)" |
|
374 |
proof - |
|
375 |
from c obtain a where fs1: "a\<sharp>f3" and fs2: "(\<forall>(y::'a::pt_name). a\<sharp>f3 a y)" by force |
|
376 |
have "(pi1\<bullet>a)\<sharp>(pi1\<bullet>f3)" using fs1 by (simp add: fresh_eqvt) |
|
377 |
moreover |
|
378 |
have "\<forall>(y::'a::pt_name). (pi1\<bullet>a)\<sharp>((pi1\<bullet>f3) (pi1\<bullet>a) y)" |
|
379 |
proof |
|
380 |
fix y::"'a::pt_name" |
|
381 |
from fs2 have "a\<sharp>f3 a ((rev pi1)\<bullet>y)" by simp |
|
382 |
then show "(pi1\<bullet>a)\<sharp>((pi1\<bullet>f3) (pi1\<bullet>a) y)" |
|
383 |
by (perm_simp add: pt_fresh_right[OF pt_name_inst, OF at_name_inst]) |
|
384 |
qed |
|
385 |
ultimately show "\<exists>(a::name). a\<sharp>(pi1\<bullet>f3) \<and> (\<forall>(y::'a::pt_name). a\<sharp>(pi1\<bullet>f3) a y)" by blast |
|
386 |
qed |
|
387 |
show ?thesis |
|
388 |
apply(rule sym) |
|
389 |
apply(auto simp add: itfun_def itfun'_def) |
|
390 |
apply(rule the1_equality'[OF it_function, OF f_pi, OF fs_pi]) |
|
391 |
apply(rule it_eqvt[OF f, OF c]) |
|
392 |
apply(rule theI'[OF it_function,OF f, OF c]) |
|
393 |
apply(rule sym) |
|
394 |
apply(rule pt_bij2[OF pt_name_inst, OF at_name_inst]) |
|
395 |
apply(perm_simp) |
|
396 |
done |
|
397 |
qed |
|
398 |
||
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parents:
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|
399 |
lemma itfun_Var: |
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|
400 |
assumes f: "finite ((supp (f1,f2,f3))::name set)" |
6e4ce7402dbe
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parents:
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|
401 |
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)" |
6e4ce7402dbe
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parents:
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|
402 |
shows "itfun f1 f2 f3 (Var c) = (f1 c)" |
19168 | 403 |
using f c by (auto intro!: the1_equality' it_function it.intros simp add: itfun_def itfun'_def) |
19157
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parents:
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|
404 |
|
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parents:
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|
405 |
lemma itfun_App: |
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|
406 |
assumes f: "finite ((supp (f1,f2,f3))::name set)" |
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parents:
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|
407 |
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)" |
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parents:
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|
408 |
shows "itfun f1 f2 f3 (App t1 t2) = (f2 (itfun f1 f2 f3 t1) (itfun f1 f2 f3 t2))" |
19168 | 409 |
by (auto intro!: the1_equality' it_function[OF f, OF c] it.intros |
410 |
intro: theI'[OF it_function, OF f, OF c] simp add: itfun_def itfun'_def) |
|
19157
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parents:
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411 |
|
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412 |
lemma itfun_Lam_aux1: |
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|
413 |
assumes f: "finite ((supp (f1,f2,f3))::name set)" |
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parents:
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|
414 |
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)" |
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parents:
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|
415 |
shows "itfun f1 f2 f3 (Lam [a].t) = fresh_fun (\<lambda>a'. f3 a' (itfun' f1 f2 f3 t ([]@[(a,a')])))" |
19168 | 416 |
by (auto intro!: the1_equality' it_function[OF f, OF c] it.intros |
417 |
intro: theI'[OF it_function, OF f, OF c] simp add: itfun_def itfun'_def) |
|
19157
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|
418 |
|
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419 |
lemma itfun_Lam_aux2: |
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420 |
assumes f: "finite ((supp (f1,f2,f3))::name set)" |
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parents:
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|
421 |
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)" |
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parents:
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422 |
and a: "b\<sharp>(a,t,f1,f2,f3)" |
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parents:
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changeset
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423 |
shows "itfun f1 f2 f3 (Lam [b].([(b,a)]\<bullet>t)) = f3 b (itfun f1 f2 f3 ([(a,b)]\<bullet>t))" |
6e4ce7402dbe
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424 |
proof - |
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425 |
have eq1: "itfun f1 f2 f3 (Lam [b].([(b,a)]\<bullet>t)) = itfun f1 f2 f3 (Lam [a].t)" |
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426 |
proof - |
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427 |
have "Lam [b].([(b,a)]\<bullet>t) = Lam [a].t" using a |
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428 |
by (simp add: lam.inject alpha fresh_prod fresh_atm) |
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429 |
thus ?thesis by simp |
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430 |
qed |
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parents:
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|
431 |
let ?g = "(\<lambda>a'. f3 a' (itfun' f1 f2 f3 t ([]@[(a,a')])))" |
19168 | 432 |
have fin_g: "finite ((supp ?g)::name set)" |
433 |
using f by (finite_guess add: itfun'_eqvt[OF f, OF c] supp_prod fs_name1) |
|
434 |
have fr_g: "\<exists>(a''::name). a''\<sharp>?g \<and> a''\<sharp>(?g a'')" using f c |
|
435 |
by (rule_tac f3_freshness_conditions_simple, auto simp add: supp_prod, |
|
19217 | 436 |
finite_guess add: itfun'_eqvt[OF f, OF c] supp_prod fs_name1) |
19168 | 437 |
have fresh_b: "b\<sharp>?g" |
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|
438 |
proof - |
19168 | 439 |
have "finite ((supp (a,t,f1,f2,f3))::name set)" using f |
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440 |
by (simp add: supp_prod fs_name1) |
19168 | 441 |
moreover |
442 |
have "((supp (a,t,f1,f2,f3))::name set) supports ?g" |
|
443 |
by (supports_simp add: itfun'_eqvt[OF f, OF c] perm_append) |
|
444 |
ultimately show ?thesis using a |
|
445 |
by (auto intro!: supports_fresh, simp add: fresh_def) |
|
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446 |
qed |
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parents:
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|
447 |
have "itfun f1 f2 f3 (Lam [b].([(b,a)]\<bullet>t)) = itfun f1 f2 f3 (Lam [a].t)" by (simp add: eq1) |
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|
448 |
also have "\<dots> = fresh_fun ?g" by (rule itfun_Lam_aux1[OF f, OF c]) |
19168 | 449 |
also have "\<dots> = ?g b" using fresh_b fin_g fr_g |
450 |
by (simp add: fresh_fun_app[OF pt_name_inst, OF at_name_inst]) |
|
19157
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parents:
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changeset
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451 |
also have "\<dots> = f3 b (itfun f1 f2 f3 ([(a,b)]\<bullet>t))" |
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parents:
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452 |
by (simp add: itfun_def itfun'_prm[OF f, OF c]) |
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parents:
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453 |
finally show "itfun f1 f2 f3 (Lam [b].([(b,a)]\<bullet>t)) = f3 b (itfun f1 f2 f3 ([(a,b)]\<bullet>t))" by simp |
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parents:
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|
454 |
qed |
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parents:
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changeset
|
455 |
|
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|
456 |
lemma itfun_Lam: |
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parents:
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|
457 |
assumes f: "finite ((supp (f1,f2,f3))::name set)" |
6e4ce7402dbe
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parents:
diff
changeset
|
458 |
and c: "\<exists>(a::name). a\<sharp>f3 \<and> (\<forall>(y::'a::pt_name). a\<sharp>f3 a y)" |
6e4ce7402dbe
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parents:
diff
changeset
|
459 |
and a: "b\<sharp>(f1,f2,f3)" |
6e4ce7402dbe
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parents:
diff
changeset
|
460 |
shows "itfun f1 f2 f3 (Lam [b].t) = f3 b (itfun f1 f2 f3 t)" |
6e4ce7402dbe
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parents:
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461 |
proof - |
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parents:
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462 |
have "\<exists>(a::name). a\<sharp>(b,t)" |
6e4ce7402dbe
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parents:
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changeset
|
463 |
by (rule at_exists_fresh[OF at_name_inst], simp add: supp_prod fs_name1) |
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parents:
diff
changeset
|
464 |
then obtain a::"name" where a1: "a\<sharp>b" and a2: "a\<sharp>t" by (force simp add: fresh_prod) |
19168 | 465 |
have fresh_b: "b\<sharp>(a,[(b,a)]\<bullet>t,f1,f2,f3)" using a a1 a2 |
466 |
by (simp add: fresh_prod fresh_atm fresh_left calc_atm) |
|
467 |
have "itfun f1 f2 f3 (Lam [b].t) = itfun f1 f2 f3 (Lam [b].(([(b,a)])\<bullet>([(b,a)]\<bullet>t)))" by (perm_simp) |
|
468 |
also have "\<dots> = f3 b (itfun f1 f2 f3 (([(a,b)])\<bullet>([(b,a)]\<bullet>t)))" using fresh_b |
|
469 |
by (rule itfun_Lam_aux2[OF f, OF c]) |
|
19157
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parents:
diff
changeset
|
470 |
also have "\<dots> = f3 b (itfun f1 f2 f3 t)" by (simp add: pt_swap_bij'[OF pt_name_inst, OF at_name_inst]) |
19168 | 471 |
finally show ?thesis by simp |
19157
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472 |
qed |
19168 | 473 |
|
19157
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parents:
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|
474 |
end |