201
+ − 1
theory LamEx
758
+ − 2
imports Nominal "../QuotMain" "../QuotList"
201
+ − 3
begin
+ − 4
+ − 5
atom_decl name
+ − 6
804
+ − 7
datatype rlam =
201
+ − 8
rVar "name"
+ − 9
| rApp "rlam" "rlam"
+ − 10
| rLam "name" "rlam"
+ − 11
804
+ − 12
fun
243
+ − 13
rfv :: "rlam \<Rightarrow> name set"
+ − 14
where
+ − 15
rfv_var: "rfv (rVar a) = {a}"
+ − 16
| rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)"
+ − 17
| rfv_lam: "rfv (rLam a t) = (rfv t) - {a}"
804
+ − 18
+ − 19
overloading
876
+ − 20
perm_rlam \<equiv> "perm :: 'x prm \<Rightarrow> rlam \<Rightarrow> rlam" (unchecked)
804
+ − 21
begin
243
+ − 22
804
+ − 23
fun
+ − 24
perm_rlam
+ − 25
where
+ − 26
"perm_rlam pi (rVar a) = rVar (pi \<bullet> a)"
+ − 27
| "perm_rlam pi (rApp t1 t2) = rApp (perm_rlam pi t1) (perm_rlam pi t2)"
+ − 28
| "perm_rlam pi (rLam a t) = rLam (pi \<bullet> a) (perm_rlam pi t)"
+ − 29
+ − 30
end
243
+ − 31
271
+ − 32
inductive
246
+ − 33
alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100)
+ − 34
where
+ − 35
a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)"
+ − 36
| a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2"
804
+ − 37
| a3: "\<lbrakk>t \<approx> ([(a,b)] \<bullet> s); a \<notin> rfv (rLam b t)\<rbrakk> \<Longrightarrow> rLam a t \<approx> rLam b s"
246
+ − 38
804
+ − 39
lemma helper:
+ − 40
fixes t::"rlam"
+ − 41
and a::"name"
+ − 42
shows "[(a, a)] \<bullet> t = t"
+ − 43
by (induct t)
+ − 44
(auto simp add: calc_atm)
259
+ − 45
271
+ − 46
lemma alpha_refl:
286
+ − 47
fixes t::"rlam"
+ − 48
shows "t \<approx> t"
+ − 49
apply(induct t rule: rlam.induct)
+ − 50
apply(simp add: a1)
+ − 51
apply(simp add: a2)
+ − 52
apply(rule a3)
804
+ − 53
apply(simp add: helper)
286
+ − 54
apply(simp)
+ − 55
done
+ − 56
534
+ − 57
lemma alpha_equivp:
+ − 58
shows "equivp alpha"
271
+ − 59
sorry
+ − 60
766
df053507edba
renamed "quotient" command to "quotient_type"; needs new keyword file to be installed
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 61
quotient_type lam = rlam / alpha
804
+ − 62
by (rule alpha_equivp)
201
+ − 63
203
+ − 64
767
37285ec4387d
on suggestion of Tobias renamed "quotient_def" to "quotient_definition"; needs new keyword file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 65
quotient_definition
876
+ − 66
"Var :: name \<Rightarrow> lam"
705
+ − 67
as
663
0dd10a900cae
Different syntax for definitions that allows overloading and retrieving of definitions by matching whole constants.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 68
"rVar"
229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 69
767
37285ec4387d
on suggestion of Tobias renamed "quotient_def" to "quotient_definition"; needs new keyword file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 70
quotient_definition
705
+ − 71
"App :: lam \<Rightarrow> lam \<Rightarrow> lam"
+ − 72
as
663
0dd10a900cae
Different syntax for definitions that allows overloading and retrieving of definitions by matching whole constants.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 73
"rApp"
229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 74
767
37285ec4387d
on suggestion of Tobias renamed "quotient_def" to "quotient_definition"; needs new keyword file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 75
quotient_definition
876
+ − 76
"Lam :: name \<Rightarrow> lam \<Rightarrow> lam"
705
+ − 77
as
663
0dd10a900cae
Different syntax for definitions that allows overloading and retrieving of definitions by matching whole constants.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 78
"rLam"
201
+ − 79
767
37285ec4387d
on suggestion of Tobias renamed "quotient_def" to "quotient_definition"; needs new keyword file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 80
quotient_definition
876
+ − 81
"fv :: lam \<Rightarrow> name set"
705
+ − 82
as
663
0dd10a900cae
Different syntax for definitions that allows overloading and retrieving of definitions by matching whole constants.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 83
"rfv"
243
+ − 84
804
+ − 85
thm Var_def
+ − 86
thm App_def
+ − 87
thm Lam_def
243
+ − 88
thm fv_def
+ − 89
229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 90
(* definition of overloaded permutation function *)
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 91
(* for the lifted type lam *)
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 92
overloading
268
4d58c02289ca
simplified the quotient_def code; type of the defined constant must now be given; for-part eliminated
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 93
perm_lam \<equiv> "perm :: 'x prm \<Rightarrow> lam \<Rightarrow> lam" (unchecked)
229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 94
begin
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 95
767
37285ec4387d
on suggestion of Tobias renamed "quotient_def" to "quotient_definition"; needs new keyword file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 96
quotient_definition
705
+ − 97
"perm_lam :: 'x prm \<Rightarrow> lam \<Rightarrow> lam"
+ − 98
as
663
0dd10a900cae
Different syntax for definitions that allows overloading and retrieving of definitions by matching whole constants.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 99
"perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam"
229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 100
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 101
end
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 102
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 103
thm perm_lam_def
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 104
804
+ − 105
(* lemmas that need to be lifted *)
+ − 106
lemma pi_var_eqvt1:
229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 107
fixes pi::"'x prm"
804
+ − 108
shows "(pi \<bullet> rVar a) \<approx> rVar (pi \<bullet> a)"
+ − 109
by (simp add: alpha_refl)
229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 110
804
+ − 111
lemma pi_var_eqvt2:
+ − 112
fixes pi::"'x prm"
+ − 113
shows "(pi \<bullet> rVar a) = rVar (pi \<bullet> a)"
+ − 114
by (simp)
+ − 115
+ − 116
lemma pi_app_eqvt1:
229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 117
fixes pi::"'x prm"
804
+ − 118
shows "(pi \<bullet> rApp t1 t2) \<approx> rApp (pi \<bullet> t1) (pi \<bullet> t2)"
+ − 119
by (simp add: alpha_refl)
+ − 120
+ − 121
lemma pi_app_eqvt2:
+ − 122
fixes pi::"'x prm"
+ − 123
shows "(pi \<bullet> rApp t1 t2) = rApp (pi \<bullet> t1) (pi \<bullet> t2)"
+ − 124
by (simp)
229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 125
804
+ − 126
lemma pi_lam_eqvt1:
229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 127
fixes pi::"'x prm"
804
+ − 128
shows "(pi \<bullet> rLam a t) \<approx> rLam (pi \<bullet> a) (pi \<bullet> t)"
+ − 129
by (simp add: alpha_refl)
229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 130
804
+ − 131
lemma pi_lam_eqvt2:
+ − 132
fixes pi::"'x prm"
+ − 133
shows "(pi \<bullet> rLam a t) = rLam (pi \<bullet> a) (pi \<bullet> t)"
+ − 134
by (simp add: alpha)
451
586e3dc4afdb
Added 'TRY' to refl in clean_tac to get as far as possible. Removed unnecessary [quot_rsp] in FSet. Added necessary [quot_rsp] and one lifted thm in LamEx.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 135
586e3dc4afdb
Added 'TRY' to refl in clean_tac to get as far as possible. Removed unnecessary [quot_rsp] in FSet. Added necessary [quot_rsp] and one lifted thm in LamEx.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 136
201
+ − 137
lemma real_alpha:
451
586e3dc4afdb
Added 'TRY' to refl in clean_tac to get as far as possible. Removed unnecessary [quot_rsp] in FSet. Added necessary [quot_rsp] and one lifted thm in LamEx.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 138
assumes a: "t = [(a,b)]\<bullet>s" "a\<sharp>[b].s"
201
+ − 139
shows "Lam a t = Lam b s"
451
586e3dc4afdb
Added 'TRY' to refl in clean_tac to get as far as possible. Removed unnecessary [quot_rsp] in FSet. Added necessary [quot_rsp] and one lifted thm in LamEx.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 140
using a
586e3dc4afdb
Added 'TRY' to refl in clean_tac to get as far as possible. Removed unnecessary [quot_rsp] in FSet. Added necessary [quot_rsp] and one lifted thm in LamEx.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 141
unfolding fresh_def supp_def
217
+ − 142
sorry
+ − 143
636
+ − 144
lemma perm_rsp[quot_respect]:
286
+ − 145
"(op = ===> alpha ===> alpha) op \<bullet> op \<bullet>"
229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 146
apply(auto)
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 147
(* this is propably true if some type conditions are imposed ;o) *)
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 148
sorry
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 149
451
586e3dc4afdb
Added 'TRY' to refl in clean_tac to get as far as possible. Removed unnecessary [quot_rsp] in FSet. Added necessary [quot_rsp] and one lifted thm in LamEx.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 150
lemma fresh_rsp:
586e3dc4afdb
Added 'TRY' to refl in clean_tac to get as far as possible. Removed unnecessary [quot_rsp] in FSet. Added necessary [quot_rsp] and one lifted thm in LamEx.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 151
"(op = ===> alpha ===> op =) fresh fresh"
229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 152
apply(auto)
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 153
(* this is probably only true if some type conditions are imposed *)
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 154
sorry
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 155
636
+ − 156
lemma rVar_rsp[quot_respect]:
451
586e3dc4afdb
Added 'TRY' to refl in clean_tac to get as far as possible. Removed unnecessary [quot_rsp] in FSet. Added necessary [quot_rsp] and one lifted thm in LamEx.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 157
"(op = ===> alpha) rVar rVar"
876
+ − 158
by (auto intro: a1)
234
+ − 159
636
+ − 160
lemma rApp_rsp[quot_respect]: "(alpha ===> alpha ===> alpha) rApp rApp"
876
+ − 161
by (auto intro: a2)
234
+ − 162
636
+ − 163
lemma rLam_rsp[quot_respect]: "(op = ===> alpha ===> alpha) rLam rLam"
229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 164
apply(auto)
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 165
apply(rule a3)
804
+ − 166
apply(simp add: helper)
+ − 167
apply(simp)
229
13f985a93dbc
fixed the definition of alpha; this *breaks* some of the experiments
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 168
done
217
+ − 169
804
+ − 170
lemma rfv_rsp[quot_respect]:
+ − 171
"(alpha ===> op =) rfv rfv"
247
+ − 172
sorry
217
+ − 173
285
+ − 174
lemma rvar_inject: "rVar a \<approx> rVar b = (a = b)"
876
+ − 175
apply (auto)
+ − 176
apply (erule alpha.cases)
+ − 177
apply (simp_all add: rlam.inject alpha_refl)
+ − 178
done
237
+ − 179
804
+ − 180
lemma pi_var1:
+ − 181
fixes pi::"'x prm"
+ − 182
shows "pi \<bullet> Var a = Var (pi \<bullet> a)"
+ − 183
by (lifting pi_var_eqvt1)
370
+ − 184
804
+ − 185
lemma pi_var2:
+ − 186
fixes pi::"'x prm"
+ − 187
shows "pi \<bullet> Var a = Var (pi \<bullet> a)"
+ − 188
by (lifting pi_var_eqvt2)
+ − 189
376
+ − 190
804
+ − 191
lemma pi_app:
+ − 192
fixes pi::"'x prm"
+ − 193
shows "pi \<bullet> App t1 t2 = App (pi \<bullet> t1) (pi \<bullet> t2)"
+ − 194
by (lifting pi_app_eqvt2)
249
+ − 195
804
+ − 196
lemma pi_lam:
+ − 197
fixes pi::"'x prm"
+ − 198
shows "pi \<bullet> Lam a t = Lam (pi \<bullet> a) (pi \<bullet> t)"
+ − 199
by (lifting pi_lam_eqvt2)
376
+ − 200
804
+ − 201
lemma fv_var:
+ − 202
shows "fv (Var a) = {a}"
+ − 203
by (lifting rfv_var)
249
+ − 204
804
+ − 205
lemma fv_app:
+ − 206
shows "fv (App t1 t2) = fv t1 \<union> fv t2"
+ − 207
by (lifting rfv_app)
+ − 208
+ − 209
lemma fv_lam:
+ − 210
shows "fv (Lam a t) = fv t - {a}"
+ − 211
by (lifting rfv_lam)
376
+ − 212
804
+ − 213
lemma a1:
+ − 214
"a = b \<Longrightarrow> Var a = Var b"
+ − 215
by (lifting a1)
376
+ − 216
804
+ − 217
lemma a2:
+ − 218
"\<lbrakk>x = xa; xb = xc\<rbrakk> \<Longrightarrow> App x xb = App xa xc"
+ − 219
by (lifting a2)
376
+ − 220
804
+ − 221
lemma a3:
+ − 222
"\<lbrakk>x = [(a, b)] \<bullet> xa; a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> Lam a x = Lam b xa"
+ − 223
by (lifting a3)
286
+ − 224
804
+ − 225
lemma alpha_cases:
+ − 226
"\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P;
+ − 227
\<And>x xa xb xc. \<lbrakk>a1 = App x xb; a2 = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P;
+ − 228
\<And>x a b xa. \<lbrakk>a1 = Lam a x; a2 = Lam b xa; x = [(a, b)] \<bullet> xa; a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> P\<rbrakk>
370
+ − 229
\<Longrightarrow> P"
804
+ − 230
by (lifting alpha.cases)
378
+ − 231
804
+ − 232
lemma alpha_induct:
+ − 233
"\<lbrakk>qx = qxa; \<And>a b. a = b \<Longrightarrow> qxb (Var a) (Var b);
+ − 234
\<And>x xa xb xc. \<lbrakk>x = xa; qxb x xa; xb = xc; qxb xb xc\<rbrakk> \<Longrightarrow> qxb (App x xb) (App xa xc);
+ − 235
\<And>x a b xa.
370
+ − 236
\<lbrakk>x = [(a, b)] \<bullet> xa; qxb x ([(a, b)] \<bullet> xa); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> qxb (Lam a x) (Lam b xa)\<rbrakk>
+ − 237
\<Longrightarrow> qxb qx qxa"
804
+ − 238
by (lifting alpha.induct)
+ − 239
+ − 240
lemma var_inject:
+ − 241
"(Var a = Var b) = (a = b)"
+ − 242
by (lifting rvar_inject)
252
e30997c88050
Regularize for equalities and a better tactic. "alpha.cases" now lifts.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 243
804
+ − 244
lemma lam_induct:
877
+ − 245
"\<lbrakk>\<And>name. P (Var name);
804
+ − 246
\<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (App lam1 lam2);
878
+ − 247
\<And>name lam. P lam \<Longrightarrow> P (Lam name lam)\<rbrakk>
+ − 248
\<Longrightarrow> P lam"
804
+ − 249
by (lifting rlam.induct)
451
586e3dc4afdb
Added 'TRY' to refl in clean_tac to get as far as possible. Removed unnecessary [quot_rsp] in FSet. Added necessary [quot_rsp] and one lifted thm in LamEx.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 250
878
+ − 251
instance lam::pt_name
+ − 252
apply(default)
+ − 253
sorry
+ − 254
+ − 255
instance lam::fs_name
+ − 256
apply(default)
877
+ − 257
sorry
+ − 258
881
+ − 259
lemma fresh_lam:
+ − 260
"(a \<sharp> Lam b t) \<longleftrightarrow> (a = b) \<or> (a \<noteq> b \<and> a \<sharp> t)"
+ − 261
sorry
+ − 262
877
+ − 263
lemma lam_induct_strong:
878
+ − 264
fixes a::"'a::fs_name"
879
+ − 265
assumes a1: "\<And>name b. P b (Var name)"
+ − 266
and a2: "\<And>lam1 lam2 b. \<lbrakk>\<And>c. P c lam1; \<And>c. P c lam2\<rbrakk> \<Longrightarrow> P b (App lam1 lam2)"
881
+ − 267
and a3: "\<And>name lam b. \<lbrakk>\<And>c. P c lam; name \<sharp> b\<rbrakk> \<Longrightarrow> P b (Lam name lam)"
879
+ − 268
shows "P a lam"
878
+ − 269
proof -
880
+ − 270
have "\<And>(pi::name prm) a. P a (pi \<bullet> lam)"
878
+ − 271
proof (induct lam rule: lam_induct)
+ − 272
case (1 name pi)
879
+ − 273
show "P a (pi \<bullet> Var name)"
+ − 274
apply (simp only: pi_var1)
+ − 275
apply (rule a1)
+ − 276
done
878
+ − 277
next
+ − 278
case (2 lam1 lam2 pi)
880
+ − 279
have b1: "\<And>(pi::name prm) a. P a (pi \<bullet> lam1)" by fact
+ − 280
have b2: "\<And>(pi::name prm) a. P a (pi \<bullet> lam2)" by fact
879
+ − 281
show "P a (pi \<bullet> App lam1 lam2)"
+ − 282
apply (simp only: pi_app)
+ − 283
apply (rule a2)
880
+ − 284
apply (rule b1)
+ − 285
apply (rule b2)
+ − 286
done
878
+ − 287
next
881
+ − 288
case (3 name lam pi a)
+ − 289
have b: "\<And>(pi::name prm) a. P a (pi \<bullet> lam)" by fact
+ − 290
obtain c::name where fr: "c\<sharp>(a, pi\<bullet>name, pi\<bullet>lam)" sorry
+ − 291
from b fr have p: "P a (Lam c (([(c, pi\<bullet>name)]@pi)\<bullet>lam))"
+ − 292
apply -
+ − 293
apply(rule a3)
+ − 294
apply(blast)
+ − 295
apply(simp)
+ − 296
done
+ − 297
have eq: "[(c, pi\<bullet>name)] \<bullet> Lam (pi \<bullet> name) (pi \<bullet> lam) = Lam (pi \<bullet> name) (pi \<bullet> lam)"
+ − 298
apply(rule perm_fresh_fresh)
+ − 299
using fr
+ − 300
apply(simp add: fresh_lam)
+ − 301
apply(simp add: fresh_lam)
+ − 302
done
879
+ − 303
show "P a (pi \<bullet> Lam name lam)"
+ − 304
apply (simp add: pi_lam)
881
+ − 305
apply(subst eq[symmetric])
+ − 306
using p
+ − 307
apply(simp only: pi_lam pt_name2 swap_simps)
+ − 308
done
878
+ − 309
qed
+ − 310
then have "P a (([]::name prm) \<bullet> lam)" by blast
+ − 311
then show "P a lam" by simp
+ − 312
qed
877
+ − 313
+ − 314
249
+ − 315
lemma var_supp:
+ − 316
shows "supp (Var a) = ((supp a)::name set)"
+ − 317
apply(simp add: supp_def)
804
+ − 318
apply(simp add: pi_var2)
249
+ − 319
apply(simp add: var_inject)
+ − 320
done
+ − 321
+ − 322
lemma var_fresh:
+ − 323
fixes a::"name"
804
+ − 324
shows "(a \<sharp> (Var b)) = (a \<sharp> b)"
249
+ − 325
apply(simp add: fresh_def)
+ − 326
apply(simp add: var_supp)
+ − 327
done
247
+ − 328
663
0dd10a900cae
Different syntax for definitions that allows overloading and retrieving of definitions by matching whole constants.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 329
end