1797
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1
theory Lambda
2041
+ − 2
imports "../NewParser"
1594
+ − 3
begin
+ − 4
+ − 5
atom_decl name
+ − 6
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 7
nominal_datatype lam =
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 8
Var "name"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 9
| App "lam" "lam"
2041
+ − 10
| Lam x::"name" l::"lam" bind_set x in l
1594
+ − 11
1845
b7423c6b5564
deleted offending [eqvt]-attribute in Abs; Lambda works again, but there is now a problem in CoreHaskell
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 12
thm lam.fv
b7423c6b5564
deleted offending [eqvt]-attribute in Abs; Lambda works again, but there is now a problem in CoreHaskell
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 13
thm lam.supp
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 14
lemmas supp_fn' = lam.fv[simplified lam.supp]
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 15
1805
+ − 16
declare lam.perm[eqvt]
+ − 17
2082
0854af516f14
cleaned up a bit the examples; added equivariance to all examples
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 18
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 19
section {* Strong Induction Principles*}
1594
+ − 20
2041
+ − 21
(*
1797
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 22
Old way of establishing strong induction
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 23
principles by chosing a fresh name.
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 24
*)
1594
+ − 25
lemma
+ − 26
fixes c::"'a::fs"
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 27
assumes a1: "\<And>name c. P c (Var name)"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 28
and a2: "\<And>lam1 lam2 c. \<lbrakk>\<And>d. P d lam1; \<And>d. P d lam2\<rbrakk> \<Longrightarrow> P c (App lam1 lam2)"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 29
and a3: "\<And>name lam c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lam\<rbrakk> \<Longrightarrow> P c (Lam name lam)"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 30
shows "P c lam"
1594
+ − 31
proof -
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 32
have "\<And>p. P c (p \<bullet> lam)"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 33
apply(induct lam arbitrary: c rule: lam.induct)
1805
+ − 34
apply(perm_simp)
1594
+ − 35
apply(rule a1)
1805
+ − 36
apply(perm_simp)
1594
+ − 37
apply(rule a2)
1797
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 38
apply(assumption)
1594
+ − 39
apply(assumption)
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 40
apply(subgoal_tac "\<exists>new::name. (atom new) \<sharp> (c, Lam (p \<bullet> name) (p \<bullet> lam))")
1594
+ − 41
defer
+ − 42
apply(simp add: fresh_def)
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 43
apply(rule_tac X="supp (c, Lam (p \<bullet> name) (p \<bullet> lam))" in obtain_at_base)
1594
+ − 44
apply(simp add: supp_Pair finite_supp)
+ − 45
apply(blast)
+ − 46
apply(erule exE)
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 47
apply(rule_tac t="p \<bullet> Lam name lam" and
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 48
s="(((p \<bullet> name) \<leftrightarrow> new) + p) \<bullet> Lam name lam" in subst)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 49
apply(simp del: lam.perm)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 50
apply(subst lam.perm)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 51
apply(subst (2) lam.perm)
1594
+ − 52
apply(rule flip_fresh_fresh)
+ − 53
apply(simp add: fresh_def)
+ − 54
apply(simp only: supp_fn')
+ − 55
apply(simp)
+ − 56
apply(simp add: fresh_Pair)
+ − 57
apply(simp)
+ − 58
apply(rule a3)
+ − 59
apply(simp add: fresh_Pair)
+ − 60
apply(drule_tac x="((p \<bullet> name) \<leftrightarrow> new) + p" in meta_spec)
+ − 61
apply(simp)
+ − 62
done
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 63
then have "P c (0 \<bullet> lam)" by blast
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 64
then show "P c lam" by simp
1594
+ − 65
qed
+ − 66
1797
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 67
(*
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 68
New way of establishing strong induction
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 69
principles by using a appropriate permutation.
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 70
*)
1594
+ − 71
lemma
+ − 72
fixes c::"'a::fs"
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 73
assumes a1: "\<And>name c. P c (Var name)"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 74
and a2: "\<And>lam1 lam2 c. \<lbrakk>\<And>d. P d lam1; \<And>d. P d lam2\<rbrakk> \<Longrightarrow> P c (App lam1 lam2)"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 75
and a3: "\<And>name lam c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lam\<rbrakk> \<Longrightarrow> P c (Lam name lam)"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 76
shows "P c lam"
1594
+ − 77
proof -
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 78
have "\<And>p. P c (p \<bullet> lam)"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 79
apply(induct lam arbitrary: c rule: lam.induct)
1805
+ − 80
apply(perm_simp)
1594
+ − 81
apply(rule a1)
1805
+ − 82
apply(perm_simp)
1594
+ − 83
apply(rule a2)
+ − 84
apply(assumption)
1797
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 85
apply(assumption)
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 86
apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom name}) \<sharp>* c \<and> supp (p \<bullet> Lam name lam) \<sharp>* q")
1594
+ − 87
apply(erule exE)
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 88
apply(rule_tac t="p \<bullet> Lam name lam" and
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 89
s="q \<bullet> p \<bullet> Lam name lam" in subst)
1594
+ − 90
defer
1805
+ − 91
apply(simp)
1594
+ − 92
apply(rule a3)
+ − 93
apply(simp add: eqvts fresh_star_def)
+ − 94
apply(drule_tac x="q + p" in meta_spec)
+ − 95
apply(simp)
1797
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 96
apply(rule at_set_avoiding2)
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 97
apply(simp add: finite_supp)
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 98
apply(simp add: finite_supp)
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 99
apply(simp add: finite_supp)
1805
+ − 100
apply(perm_simp)
1797
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 101
apply(simp add: fresh_star_def fresh_def supp_fn')
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 102
apply(rule supp_perm_eq)
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 103
apply(simp)
fddb470720f1
renamed ExLam to Lambda and completed the proof of the strong ind principle; tuned paper
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 104
done
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 105
then have "P c (0 \<bullet> lam)" by blast
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 106
then show "P c lam" by simp
1594
+ − 107
qed
+ − 108
1805
+ − 109
section {* Typing *}
+ − 110
+ − 111
nominal_datatype ty =
+ − 112
TVar string
1810
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 113
| TFun ty ty
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 114
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 115
notation
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 116
TFun ("_ \<rightarrow> _")
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 117
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 118
declare ty.perm[eqvt]
1805
+ − 119
+ − 120
inductive
+ − 121
valid :: "(name \<times> ty) list \<Rightarrow> bool"
+ − 122
where
+ − 123
"valid []"
+ − 124
| "\<lbrakk>atom x \<sharp> Gamma; valid Gamma\<rbrakk> \<Longrightarrow> valid ((x, T)#Gamma)"
+ − 125
1828
+ − 126
inductive
+ − 127
typing :: "(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool" ("_ \<turnstile> _ : _" [60,60,60] 60)
+ − 128
where
+ − 129
t_Var[intro]: "\<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Var x : T"
1947
+ − 130
| t_App[intro]: "\<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : T2"
1828
+ − 131
| t_Lam[intro]: "\<lbrakk>atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam x t : T1 \<rightarrow> T2"
+ − 132
1831
+ − 133
equivariance valid
+ − 134
equivariance typing
1816
+ − 135
1831
+ − 136
thm valid.eqvt
+ − 137
thm typing.eqvt
1811
ae176476b525
implemented in thmdecls the case where eqvt-lemmas are of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 138
thm eqvts
ae176476b525
implemented in thmdecls the case where eqvt-lemmas are of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 139
thm eqvts_raw
ae176476b525
implemented in thmdecls the case where eqvt-lemmas are of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 140
1949
+ − 141
thm typing.induct[of "\<Gamma>" "t" "T", no_vars]
+ − 142
+ − 143
lemma
+ − 144
fixes c::"'a::fs"
+ − 145
assumes a: "\<Gamma> \<turnstile> t : T"
+ − 146
and a1: "\<And>\<Gamma> x T c. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P c \<Gamma> (Var x) T"
+ − 147
and a2: "\<And>\<Gamma> t1 T1 T2 t2 c. \<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<And>d. P d \<Gamma> t1 T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1; \<And>d. P d \<Gamma> t2 T1\<rbrakk>
+ − 148
\<Longrightarrow> P c \<Gamma> (App t1 t2) T2"
+ − 149
and a3: "\<And>x \<Gamma> T1 t T2 c. \<lbrakk>atom x \<sharp> c; atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2; \<And>d. P d ((x, T1) # \<Gamma>) t T2\<rbrakk>
+ − 150
\<Longrightarrow> P c \<Gamma> (Lam x t) T1 \<rightarrow> T2"
+ − 151
shows "P c \<Gamma> t T"
+ − 152
proof -
+ − 153
from a have "\<And>p c. P c (p \<bullet> \<Gamma>) (p \<bullet> t) (p \<bullet> T)"
+ − 154
proof (induct)
+ − 155
case (t_Var \<Gamma> x T p c)
+ − 156
then show ?case
+ − 157
apply -
+ − 158
apply(perm_strict_simp)
+ − 159
apply(rule a1)
1954
+ − 160
apply(drule_tac p="p" in permute_boolI)
1949
+ − 161
apply(perm_strict_simp add: permute_minus_cancel)
+ − 162
apply(assumption)
1954
+ − 163
apply(rotate_tac 1)
+ − 164
apply(drule_tac p="p" in permute_boolI)
1949
+ − 165
apply(perm_strict_simp add: permute_minus_cancel)
+ − 166
apply(assumption)
+ − 167
done
+ − 168
next
+ − 169
case (t_App \<Gamma> t1 T1 T2 t2 p c)
+ − 170
then show ?case
+ − 171
apply -
+ − 172
apply(perm_strict_simp)
1954
+ − 173
apply(rule a2)
+ − 174
apply(drule_tac p="p" in permute_boolI)
1949
+ − 175
apply(perm_strict_simp add: permute_minus_cancel)
+ − 176
apply(assumption)
+ − 177
apply(assumption)
1954
+ − 178
apply(rotate_tac 2)
+ − 179
apply(drule_tac p="p" in permute_boolI)
1949
+ − 180
apply(perm_strict_simp add: permute_minus_cancel)
+ − 181
apply(assumption)
+ − 182
apply(assumption)
+ − 183
done
+ − 184
next
+ − 185
case (t_Lam x \<Gamma> T1 t T2 p c)
+ − 186
then show ?case
+ − 187
apply -
+ − 188
apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom x}) \<sharp>* c \<and>
+ − 189
supp (p \<bullet> \<Gamma>, p \<bullet> Lam x t, p \<bullet> (T1 \<rightarrow> T2)) \<sharp>* q")
+ − 190
apply(erule exE)
1954
+ − 191
apply(rule_tac t="p \<bullet> \<Gamma>" and s="(q + p) \<bullet> \<Gamma>" in subst)
+ − 192
apply(simp only: permute_plus)
1949
+ − 193
apply(rule supp_perm_eq)
+ − 194
apply(simp add: supp_Pair fresh_star_union)
1954
+ − 195
apply(rule_tac t="p \<bullet> Lam x t" and s="(q + p) \<bullet> Lam x t" in subst)
+ − 196
apply(simp only: permute_plus)
1949
+ − 197
apply(rule supp_perm_eq)
+ − 198
apply(simp add: supp_Pair fresh_star_union)
1954
+ − 199
apply(rule_tac t="p \<bullet> (T1 \<rightarrow> T2)" and s="(q + p) \<bullet> (T1 \<rightarrow> T2)" in subst)
+ − 200
apply(simp only: permute_plus)
+ − 201
apply(rule supp_perm_eq)
+ − 202
apply(simp add: supp_Pair fresh_star_union)
+ − 203
apply(simp (no_asm) only: eqvts)
1949
+ − 204
apply(rule a3)
1954
+ − 205
apply(simp only: eqvts permute_plus)
1949
+ − 206
apply(simp add: fresh_star_def)
1954
+ − 207
apply(drule_tac p="q + p" in permute_boolI)
1949
+ − 208
apply(perm_strict_simp add: permute_minus_cancel)
+ − 209
apply(assumption)
1954
+ − 210
apply(rotate_tac 1)
+ − 211
apply(drule_tac p="q + p" in permute_boolI)
1949
+ − 212
apply(perm_strict_simp add: permute_minus_cancel)
+ − 213
apply(assumption)
+ − 214
apply(drule_tac x="d" in meta_spec)
+ − 215
apply(drule_tac x="q + p" in meta_spec)
1954
+ − 216
apply(perm_strict_simp add: permute_minus_cancel)
+ − 217
apply(assumption)
1949
+ − 218
apply(rule at_set_avoiding2)
+ − 219
apply(simp add: finite_supp)
+ − 220
apply(simp add: finite_supp)
+ − 221
apply(simp add: finite_supp)
+ − 222
apply(rule_tac p="-p" in permute_boolE)
+ − 223
apply(perm_strict_simp add: permute_minus_cancel)
+ − 224
(* supplied by the user *)
+ − 225
apply(simp add: fresh_star_prod)
+ − 226
apply(simp add: fresh_star_def)
1954
+ − 227
sorry
+ − 228
qed
+ − 229
then have "P c (0 \<bullet> \<Gamma>) (0 \<bullet> t) (0 \<bullet> T)" .
+ − 230
then show "P c \<Gamma> t T" by simp
+ − 231
qed
1949
+ − 232
+ − 233
+ − 234
+ − 235
+ − 236
+ − 237
1833
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 238
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 239
inductive
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 240
tt :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 241
for r :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 242
where
1835
+ − 243
aa: "tt r a a"
+ − 244
| bb: "tt r a b ==> tt r a c"
1833
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 245
1835
+ − 246
(* PROBLEM: derived eqvt-theorem does not conform with [eqvt]
1833
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 247
equivariance tt
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 248
*)
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 249
1810
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 250
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 251
inductive
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 252
alpha_lam_raw'
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 253
where
1861
+ − 254
a1: "name = namea \<Longrightarrow> alpha_lam_raw' (Var_raw name) (Var_raw namea)"
+ − 255
| a2: "\<lbrakk>alpha_lam_raw' lam_raw1 lam_raw1a; alpha_lam_raw' lam_raw2 lam_raw2a\<rbrakk> \<Longrightarrow>
1810
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 256
alpha_lam_raw' (App_raw lam_raw1 lam_raw2) (App_raw lam_raw1a lam_raw2a)"
1865
+ − 257
| a3: "\<exists>pi. ({atom name}, lam_raw) \<approx>gen alpha_lam_raw' fv_lam_raw pi ({atom namea}, lam_rawa) \<Longrightarrow>
1810
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 258
alpha_lam_raw' (Lam_raw name lam_raw) (Lam_raw namea lam_rawa)"
894930834ca8
fixed bug in thmdecls with destructing Trueprop; some initial infrastructure for eqvt-theorems of the form _ ==> _
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 259
1866
+ − 260
equivariance alpha_lam_raw'
1947
+ − 261
1861
+ − 262
thm eqvts_raw
+ − 263
1800
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 264
section {* size function *}
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 265
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 266
lemma size_eqvt_raw:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 267
fixes t::"lam_raw"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 268
shows "size (pi \<bullet> t) = size t"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 269
apply (induct rule: lam_raw.inducts)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 270
apply simp_all
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 271
done
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 272
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 273
instantiation lam :: size
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 274
begin
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 275
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 276
quotient_definition
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 277
"size_lam :: lam \<Rightarrow> nat"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 278
is
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 279
"size :: lam_raw \<Rightarrow> nat"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 280
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 281
lemma size_rsp:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 282
"alpha_lam_raw x y \<Longrightarrow> size x = size y"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 283
apply (induct rule: alpha_lam_raw.inducts)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 284
apply (simp_all only: lam_raw.size)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 285
apply (simp_all only: alphas)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 286
apply clarify
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 287
apply (simp_all only: size_eqvt_raw)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 288
done
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 289
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 290
lemma [quot_respect]:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 291
"(alpha_lam_raw ===> op =) size size"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 292
by (simp_all add: size_rsp)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 293
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 294
lemma [quot_preserve]:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 295
"(rep_lam ---> id) size = size"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 296
by (simp_all add: size_lam_def)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 297
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 298
instance
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 299
by default
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 300
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 301
end
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 302
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 303
lemmas size_lam[simp] =
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 304
lam_raw.size(4)[quot_lifted]
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 305
lam_raw.size(5)[quot_lifted]
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 306
lam_raw.size(6)[quot_lifted]
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 307
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 308
(* is this needed? *)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 309
lemma [measure_function]:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 310
"is_measure (size::lam\<Rightarrow>nat)"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 311
by (rule is_measure_trivial)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 312
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 313
section {* Matching *}
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 314
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 315
definition
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 316
MATCH :: "('c::pt \<Rightarrow> (bool * 'a::pt * 'b::pt)) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 317
where
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 318
"MATCH M d x \<equiv> if (\<exists>!r. \<exists>q. M q = (True, x, r)) then (THE r. \<exists>q. M q = (True, x, r)) else d"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 319
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 320
(*
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 321
lemma MATCH_eqvt:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 322
shows "p \<bullet> (MATCH M d x) = MATCH (p \<bullet> M) (p \<bullet> d) (p \<bullet> x)"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 323
unfolding MATCH_def
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 324
apply(perm_simp the_eqvt)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 325
apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 326
apply(simp)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 327
thm eqvts_raw
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 328
apply(subst if_eqvt)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 329
apply(subst ex1_eqvt)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 330
apply(subst permute_fun_def)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 331
apply(subst ex_eqvt)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 332
apply(subst permute_fun_def)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 333
apply(subst eq_eqvt)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 334
apply(subst permute_fun_app_eq[where f="M"])
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 335
apply(simp only: permute_minus_cancel)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 336
apply(subst permute_prod.simps)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 337
apply(subst permute_prod.simps)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 338
apply(simp only: permute_minus_cancel)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 339
apply(simp only: permute_bool_def)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 340
apply(simp)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 341
apply(subst ex1_eqvt)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 342
apply(subst permute_fun_def)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 343
apply(subst ex_eqvt)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 344
apply(subst permute_fun_def)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 345
apply(subst eq_eqvt)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 346
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 347
apply(simp only: eqvts)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 348
apply(simp)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 349
apply(subgoal_tac "(p \<bullet> (\<exists>!r. \<exists>q. M q = (True, x, r))) = (\<exists>!r. \<exists>q. (p \<bullet> M) q = (True, p \<bullet> x, r))")
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 350
apply(drule sym)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 351
apply(simp)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 352
apply(rule impI)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 353
apply(simp add: perm_bool)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 354
apply(rule trans)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 355
apply(rule pt_the_eqvt[OF pta at])
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 356
apply(assumption)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 357
apply(simp add: pt_ex_eqvt[OF pt at])
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 358
apply(simp add: pt_eq_eqvt[OF ptb at])
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 359
apply(rule cheat)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 360
apply(rule trans)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 361
apply(rule pt_ex1_eqvt)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 362
apply(rule pta)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 363
apply(rule at)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 364
apply(simp add: pt_ex_eqvt[OF pt at])
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 365
apply(simp add: pt_eq_eqvt[OF ptb at])
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 366
apply(subst pt_pi_rev[OF pta at])
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 367
apply(subst pt_fun_app_eq[OF pt at])
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 368
apply(subst pt_pi_rev[OF pt at])
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 369
apply(simp)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 370
done
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 371
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 372
lemma MATCH_cng:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 373
assumes a: "M1 = M2" "d1 = d2"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 374
shows "MATCH M1 d1 x = MATCH M2 d2 x"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 375
using a by simp
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 376
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 377
lemma MATCH_eq:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 378
assumes a: "t = l x" "G x" "\<And>x'. t = l x' \<Longrightarrow> G x' \<Longrightarrow> r x' = r x"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 379
shows "MATCH (\<lambda>x. (G x, l x, r x)) d t = r x"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 380
using a
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 381
unfolding MATCH_def
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 382
apply(subst if_P)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 383
apply(rule_tac a="r x" in ex1I)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 384
apply(rule_tac x="x" in exI)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 385
apply(blast)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 386
apply(erule exE)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 387
apply(drule_tac x="q" in meta_spec)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 388
apply(auto)[1]
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 389
apply(rule the_equality)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 390
apply(blast)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 391
apply(erule exE)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 392
apply(drule_tac x="q" in meta_spec)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 393
apply(auto)[1]
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 394
done
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 395
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 396
lemma MATCH_eq2:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 397
assumes a: "t = l x1 x2" "G x1 x2" "\<And>x1' x2'. t = l x1' x2' \<Longrightarrow> G x1' x2' \<Longrightarrow> r x1' x2' = r x1 x2"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 398
shows "MATCH (\<lambda>(x1,x2). (G x1 x2, l x1 x2, r x1 x2)) d t = r x1 x2"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 399
sorry
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 400
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 401
lemma MATCH_neq:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 402
assumes a: "\<And>x. t = l x \<Longrightarrow> G x \<Longrightarrow> False"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 403
shows "MATCH (\<lambda>x. (G x, l x, r x)) d t = d"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 404
using a
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 405
unfolding MATCH_def
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 406
apply(subst if_not_P)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 407
apply(blast)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 408
apply(rule refl)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 409
done
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 410
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 411
lemma MATCH_neq2:
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 412
assumes a: "\<And>x1 x2. t = l x1 x2 \<Longrightarrow> G x1 x2 \<Longrightarrow> False"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 413
shows "MATCH (\<lambda>(x1,x2). (G x1 x2, l x1 x2, r x1 x2)) d t = d"
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 414
using a
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 415
unfolding MATCH_def
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 416
apply(subst if_not_P)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 417
apply(auto)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 418
done
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 419
*)
78fdc6b36a1c
changed the eqvt-tac to move only outermost permutations inside; added tracing infrastructure for the eqvt-tac
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 420
1954
+ − 421
ML {*
+ − 422
fun mk_avoids ctxt params name set =
+ − 423
let
+ − 424
val (_, ctxt') = ProofContext.add_fixes
+ − 425
(map (fn (s, T) => (Binding.name s, SOME T, NoSyn)) params) ctxt;
+ − 426
fun mk s =
+ − 427
let
+ − 428
val t = Syntax.read_term ctxt' s;
+ − 429
val t' = list_abs_free (params, t) |>
+ − 430
funpow (length params) (fn Abs (_, _, t) => t)
+ − 431
in (t', HOLogic.dest_setT (fastype_of t)) end
+ − 432
handle TERM _ =>
+ − 433
error ("Expression " ^ quote s ^ " to be avoided in case " ^
+ − 434
quote name ^ " is not a set type");
+ − 435
fun add_set p [] = [p]
+ − 436
| add_set (t, T) ((u, U) :: ps) =
+ − 437
if T = U then
+ − 438
let val S = HOLogic.mk_setT T
+ − 439
in (Const (@{const_name sup}, S --> S --> S) $ u $ t, T) :: ps
+ − 440
end
+ − 441
else (u, U) :: add_set (t, T) ps
+ − 442
in
+ − 443
(mk #> add_set) set
+ − 444
end;
+ − 445
*}
+ − 446
+ − 447
+ − 448
ML {*
+ − 449
writeln (commas (map (Syntax.string_of_term @{context} o fst)
+ − 450
(mk_avoids @{context} [] "t_Var" "{x}" [])))
+ − 451
*}
+ − 452
1947
+ − 453
1949
+ − 454
ML {*
+ − 455
+ − 456
fun prove_strong_ind (pred_name, avoids) ctxt =
+ − 457
Proof.theorem NONE (K I) [] ctxt
+ − 458
+ − 459
local structure P = OuterParse and K = OuterKeyword in
+ − 460
+ − 461
val _ =
+ − 462
OuterSyntax.local_theory_to_proof "nominal_inductive"
+ − 463
"proves strong induction theorem for inductive predicate involving nominal datatypes" K.thy_goal
+ − 464
(P.xname -- (Scan.optional (P.$$$ "avoids" |-- P.enum1 "|" (P.name --
+ − 465
(P.$$$ ":" |-- P.and_list1 P.term))) []) >> prove_strong_ind)
+ − 466
+ − 467
end;
+ − 468
+ − 469
*}
+ − 470
1950
+ − 471
(*
1949
+ − 472
nominal_inductive typing
1950
+ − 473
*)
1949
+ − 474
2157
+ − 475
(* Substitution *)
2159
+ − 476
2161
+ − 477
definition new where
2162
+ − 478
"new s = (THE x. \<forall>a \<in> s. atom x \<noteq> a)"
2161
+ − 479
+ − 480
lemma size_no_change: "size (p \<bullet> (t :: lam_raw)) = size t"
+ − 481
by (induct t) simp_all
+ − 482
+ − 483
function
2162
+ − 484
subst_raw :: "lam_raw \<Rightarrow> name \<Rightarrow> lam_raw \<Rightarrow> lam_raw"
2159
+ − 485
where
2162
+ − 486
"subst_raw (Var_raw x) y s = (if x=y then s else (Var_raw x))"
+ − 487
| "subst_raw (App_raw l r) y s = App_raw (subst_raw l y s) (subst_raw r y s)"
+ − 488
| "subst_raw (Lam_raw x t) y s =
+ − 489
Lam_raw (new ({atom y} \<union> fv_lam_raw s \<union> fv_lam_raw t - {atom x}))
+ − 490
(subst_raw ((x \<leftrightarrow> (new ({atom y} \<union> fv_lam_raw s \<union> fv_lam_raw t - {atom x}))) \<bullet> t) y s)"
2161
+ − 491
by (pat_completeness, auto)
+ − 492
termination
+ − 493
apply (relation "measure (\<lambda>(t, y, s). (size t))")
+ − 494
apply (auto simp add: size_no_change)
+ − 495
done
+ − 496
2162
+ − 497
lemma fv_subst[simp]: "fv_lam_raw (subst_raw t y s) =
+ − 498
(if (atom y \<in> fv_lam_raw t) then fv_lam_raw s \<union> (fv_lam_raw t - {atom y}) else fv_lam_raw t)"
+ − 499
apply (induct t arbitrary: s)
+ − 500
apply (auto simp add: supp_at_base)[1]
+ − 501
apply (auto simp add: supp_at_base)[1]
+ − 502
apply (simp only: fv_lam_raw.simps)
+ − 503
apply simp
+ − 504
apply (rule conjI)
+ − 505
apply clarify
+ − 506
sorry
+ − 507
+ − 508
thm supp_at_base
2161
+ − 509
lemma new_eqvt[eqvt]: "p \<bullet> (new s) = new (p \<bullet> s)"
+ − 510
sorry
+ − 511
2162
+ − 512
lemma subst_var_raw_eqvt[eqvt]: "p \<bullet> (subst_raw t y s) = subst_raw (p \<bullet> t) (p \<bullet> y) (p \<bullet> s)"
2161
+ − 513
apply (induct t arbitrary: p y s)
+ − 514
apply simp_all
+ − 515
apply(perm_simp)
+ − 516
apply simp
+ − 517
sorry
+ − 518
2162
+ − 519
lemma subst_id: "alpha_lam_raw (subst_raw x d (Var_raw d)) x"
2161
+ − 520
apply (induct x arbitrary: d)
+ − 521
apply (simp_all add: alpha_lam_raw.intros)
+ − 522
apply (rule alpha_lam_raw.intros)
2162
+ − 523
apply (rule_tac x="(name \<leftrightarrow> new (insert (atom d) (supp d)))" in exI)
+ − 524
apply (simp add: alphas)
+ − 525
oops
2157
+ − 526
+ − 527
quotient_definition
+ − 528
subst ("_ [ _ ::= _ ]" [100,100,100] 100)
+ − 529
where
+ − 530
"subst :: lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" is "subst_raw"
+ − 531
2162
+ − 532
lemmas fv_rsp = quot_respect(10)[simplified]
2157
+ − 533
+ − 534
lemma subst_rsp_pre1:
+ − 535
assumes a: "alpha_lam_raw a b"
+ − 536
shows "alpha_lam_raw (subst_raw a y c) (subst_raw b y c)"
+ − 537
using a
+ − 538
apply (induct a b arbitrary: c y rule: alpha_lam_raw.induct)
+ − 539
apply (simp add: equivp_reflp[OF lam_equivp])
+ − 540
apply (simp add: alpha_lam_raw.intros)
+ − 541
apply (simp only: alphas)
+ − 542
apply clarify
+ − 543
apply (simp only: subst_raw.simps)
2162
+ − 544
apply (rule alpha_lam_raw.intros)
+ − 545
apply (simp only: alphas)
2157
+ − 546
sorry
+ − 547
+ − 548
lemma subst_rsp_pre2:
+ − 549
assumes a: "alpha_lam_raw a b"
+ − 550
shows "alpha_lam_raw (subst_raw c y a) (subst_raw c y b)"
2162
+ − 551
using a
+ − 552
apply (induct c arbitrary: a b y)
+ − 553
apply (simp add: equivp_reflp[OF lam_equivp])
+ − 554
apply (simp add: alpha_lam_raw.intros)
+ − 555
apply simp
+ − 556
apply (rule alpha_lam_raw.intros)
2157
+ − 557
sorry
+ − 558
+ − 559
lemma [quot_respect]:
+ − 560
"(alpha_lam_raw ===> op = ===> alpha_lam_raw ===> alpha_lam_raw) subst_raw subst_raw"
+ − 561
proof (intro fun_relI, simp)
+ − 562
fix a b c d :: lam_raw
+ − 563
fix y :: name
+ − 564
assume a: "alpha_lam_raw a b"
+ − 565
assume b: "alpha_lam_raw c d"
+ − 566
have c: "alpha_lam_raw (subst_raw a y c) (subst_raw b y c)" using subst_rsp_pre1 a by simp
+ − 567
then have d: "alpha_lam_raw (subst_raw b y c) (subst_raw b y d)" using subst_rsp_pre2 b by simp
+ − 568
show "alpha_lam_raw (subst_raw a y c) (subst_raw b y d)"
+ − 569
using c d equivp_transp[OF lam_equivp] by blast
+ − 570
qed
+ − 571
+ − 572
lemma simp3:
2162
+ − 573
"x \<noteq> y \<Longrightarrow> atom x \<notin> fv_lam_raw s \<Longrightarrow> alpha_lam_raw (subst_raw (Lam_raw x t) y s) (Lam_raw x (subst_raw t y s))"
+ − 574
apply simp
+ − 575
apply (rule alpha_lam_raw.intros)
+ − 576
apply (rule_tac x ="(x \<leftrightarrow> (new (insert (atom y) (fv_lam_raw s \<union> fv_lam_raw t) -
+ − 577
{atom x})))" in exI)
+ − 578
apply (simp only: alphas)
+ − 579
apply simp
+ − 580
sorry
2157
+ − 581
+ − 582
lemmas subst_simps = subst_raw.simps(1-2)[quot_lifted,no_vars]
+ − 583
simp3[quot_lifted,simplified lam.supp,simplified fresh_def[symmetric], no_vars]
1947
+ − 584
1594
+ − 585
end
+ − 586
+ − 587
+ − 588