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theory LamEx
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imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs"
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begin
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atom_decl name
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datatype rlam =
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rVar "name"
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| rApp "rlam" "rlam"
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| rLam "name" "rlam"
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fun
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rfv :: "rlam \<Rightarrow> atom set"
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where
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rfv_var: "rfv (rVar a) = {atom a}"
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| rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)"
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| rfv_lam: "rfv (rLam a t) = (rfv t) - {atom a}"
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instantiation rlam :: pt
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begin
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primrec
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permute_rlam
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where
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"permute_rlam pi (rVar a) = rVar (pi \<bullet> a)"
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| "permute_rlam pi (rApp t1 t2) = rApp (permute_rlam pi t1) (permute_rlam pi t2)"
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| "permute_rlam pi (rLam a t) = rLam (pi \<bullet> a) (permute_rlam pi t)"
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instance
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apply default
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apply(induct_tac [!] x)
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apply(simp_all)
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done
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end
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instantiation rlam :: fs
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begin
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lemma neg_conj:
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"\<not>(P \<and> Q) \<longleftrightarrow> (\<not>P) \<or> (\<not>Q)"
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by simp
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instance
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apply default
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apply(induct_tac x)
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(* var case *)
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apply(simp add: supp_def)
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apply(fold supp_def)[1]
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apply(simp add: supp_at_base)
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(* app case *)
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apply(simp only: supp_def)
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apply(simp only: permute_rlam.simps)
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apply(simp only: rlam.inject)
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apply(simp only: neg_conj)
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apply(simp only: Collect_disj_eq)
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apply(simp only: infinite_Un)
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apply(simp only: Collect_disj_eq)
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apply(simp)
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(* lam case *)
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apply(simp only: supp_def)
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apply(simp only: permute_rlam.simps)
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apply(simp only: rlam.inject)
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apply(simp only: neg_conj)
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apply(simp only: Collect_disj_eq)
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apply(simp only: infinite_Un)
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apply(simp only: Collect_disj_eq)
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apply(simp)
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apply(fold supp_def)[1]
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apply(simp add: supp_at_base)
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done
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end
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(* for the eqvt proof of the alpha-equivalence *)
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declare permute_rlam.simps[eqvt]
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lemma rfv_eqvt[eqvt]:
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shows "(pi\<bullet>rfv t) = rfv (pi\<bullet>t)"
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apply(induct t)
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apply(simp_all)
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apply(simp add: permute_set_eq atom_eqvt)
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apply(simp add: union_eqvt)
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apply(simp add: Diff_eqvt)
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apply(simp add: permute_set_eq atom_eqvt)
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done
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inductive
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alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100)
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where
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a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)"
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| a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2"
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| a3: "\<exists>pi. (rfv t - {atom a} = rfv s - {atom b} \<and> (rfv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) \<approx> s)
+ − 95
\<Longrightarrow> rLam a t \<approx> rLam b s"
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989
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lemma a3_inverse:
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assumes "rLam a t \<approx> rLam b s"
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+ − 99
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989
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using assms
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apply(erule_tac alpha.cases)
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apply(auto)
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done
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text {* should be automatic with new version of eqvt-machinery *}
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lemma alpha_eqvt:
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apply(induct rule: alpha.induct)
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apply(simp add: a1)
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apply(simp add: a2)
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apply(simp)
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apply(rule a3)
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apply(erule conjE)
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apply(erule exE)
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apply(erule conjE)
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apply(rule_tac x="pi \<bullet> pia" in exI)
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apply(rule conjI)
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apply(rule_tac ?p1="- pi" in permute_eq_iff[THEN iffD1])
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apply(simp only: Diff_eqvt rfv_eqvt insert_eqvt atom_eqvt empty_eqvt)
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apply(simp)
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apply(rule conjI)
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apply(rule_tac ?p1="- pi" in fresh_star_permute_iff[THEN iffD1])
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apply(simp add: Diff_eqvt rfv_eqvt atom_eqvt insert_eqvt empty_eqvt)
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apply(subst permute_eqvt[symmetric])
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apply(simp)
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done
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lemma alpha_refl:
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shows "t \<approx> t"
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apply(induct t rule: rlam.induct)
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apply(simp add: a1)
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apply(simp add: a2)
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+ − 133
apply(rule a3)
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+ − 134
apply(rule_tac x="0" in exI)
1011
+ − 135
apply(simp_all add: fresh_star_def fresh_zero_perm)
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+ − 136
done
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lemma alpha_sym:
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shows "t \<approx> s \<Longrightarrow> s \<approx> t"
1011
+ − 140
apply(induct rule: alpha.induct)
+ − 141
apply(simp add: a1)
+ − 142
apply(simp add: a2)
+ − 143
apply(rule a3)
+ − 144
apply(erule exE)
+ − 145
apply(rule_tac x="- pi" in exI)
+ − 146
apply(simp)
+ − 147
apply(simp add: fresh_star_def fresh_minus_perm)
+ − 148
apply(erule conjE)+
+ − 149
apply(rotate_tac 3)
+ − 150
apply(drule_tac pi="- pi" in alpha_eqvt)
+ − 151
apply(simp)
+ − 152
done
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lemma alpha_trans:
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shows "t1 \<approx> t2 \<Longrightarrow> t2 \<approx> t3 \<Longrightarrow> t1 \<approx> t3"
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apply(induct arbitrary: t3 rule: alpha.induct)
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apply(erule alpha.cases)
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apply(simp_all)
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+ − 159
apply(simp add: a1)
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apply(rotate_tac 4)
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diff
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+ − 161
apply(erule alpha.cases)
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diff
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+ − 162
apply(simp_all)
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apply(simp add: a2)
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diff
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+ − 164
apply(rotate_tac 1)
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+ − 165
apply(erule alpha.cases)
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apply(simp_all)
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+ − 167
apply(erule conjE)+
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apply(erule exE)+
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apply(erule conjE)+
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+ − 170
apply(rule a3)
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apply(rule_tac x="pia + pi" in exI)
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apply(simp add: fresh_star_plus)
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apply(drule_tac x="- pia \<bullet> sa" in spec)
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apply(drule mp)
989
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apply(rotate_tac 7)
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apply(drule_tac pi="- pia" in alpha_eqvt)
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apply(simp)
989
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apply(rotate_tac 9)
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apply(drule_tac pi="pia" in alpha_eqvt)
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apply(simp)
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done
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+ − 182
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lemma alpha_equivp:
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shows "equivp alpha"
981
+ − 185
apply(rule equivpI)
+ − 186
unfolding reflp_def symp_def transp_def
+ − 187
apply(auto intro: alpha_refl alpha_sym alpha_trans)
+ − 188
done
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+ − 189
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lemma alpha_rfv:
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shows "t \<approx> s \<Longrightarrow> rfv t = rfv s"
981
+ − 192
apply(induct rule: alpha.induct)
1011
+ − 193
apply(simp_all)
981
+ − 194
done
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+ − 195
1011
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inductive
+ − 197
alpha2 :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx>2 _" [100, 100] 100)
+ − 198
where
+ − 199
a21: "a = b \<Longrightarrow> (rVar a) \<approx>2 (rVar b)"
+ − 200
| a22: "\<lbrakk>t1 \<approx>2 t2; s1 \<approx>2 s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx>2 rApp t2 s2"
+ − 201
| a23: "(a = b \<and> t \<approx>2 s) \<or> (a \<noteq> b \<and> ((a \<leftrightarrow> b) \<bullet> t) \<approx>2 s \<and> atom b \<notin> rfv t)\<Longrightarrow> rLam a t \<approx>2 rLam b s"
+ − 202
+ − 203
lemma fv_vars:
+ − 204
fixes a::name
+ − 205
assumes a1: "\<forall>x \<in> rfv t - {atom a}. pi \<bullet> x = x"
+ − 206
shows "(pi \<bullet> t) \<approx>2 ((a \<leftrightarrow> (pi \<bullet> a)) \<bullet> t)"
+ − 207
using a1
+ − 208
apply(induct t)
+ − 209
apply(auto)
+ − 210
apply(rule a21)
+ − 211
apply(case_tac "name = a")
+ − 212
apply(simp)
+ − 213
apply(simp)
+ − 214
defer
+ − 215
apply(rule a22)
+ − 216
apply(simp)
+ − 217
apply(simp)
+ − 218
apply(rule a23)
+ − 219
apply(case_tac "a = name")
+ − 220
apply(simp)
+ − 221
oops
+ − 222
+ − 223
+ − 224
lemma
+ − 225
assumes a1: "t \<approx>2 s"
+ − 226
shows "t \<approx> s"
+ − 227
using a1
+ − 228
apply(induct)
+ − 229
apply(rule alpha.intros)
+ − 230
apply(simp)
+ − 231
apply(rule alpha.intros)
+ − 232
apply(simp)
+ − 233
apply(simp)
+ − 234
apply(rule alpha.intros)
+ − 235
apply(erule disjE)
+ − 236
apply(rule_tac x="0" in exI)
+ − 237
apply(simp add: fresh_star_def fresh_zero_perm)
+ − 238
apply(erule conjE)+
+ − 239
apply(drule alpha_rfv)
+ − 240
apply(simp)
+ − 241
apply(rule_tac x="(a \<leftrightarrow> b)" in exI)
+ − 242
apply(simp)
+ − 243
apply(erule conjE)+
+ − 244
apply(rule conjI)
+ − 245
apply(drule alpha_rfv)
+ − 246
apply(drule sym)
+ − 247
apply(simp)
+ − 248
apply(simp add: rfv_eqvt[symmetric])
+ − 249
defer
+ − 250
apply(subgoal_tac "atom a \<sharp> (rfv t - {atom a})")
+ − 251
apply(subgoal_tac "atom b \<sharp> (rfv t - {atom a})")
+ − 252
+ − 253
defer
+ − 254
sorry
+ − 255
+ − 256
lemma
+ − 257
assumes a1: "t \<approx> s"
+ − 258
shows "t \<approx>2 s"
+ − 259
using a1
+ − 260
apply(induct)
+ − 261
apply(rule alpha2.intros)
+ − 262
apply(simp)
+ − 263
apply(rule alpha2.intros)
+ − 264
apply(simp)
+ − 265
apply(simp)
+ − 266
apply(clarify)
+ − 267
apply(rule alpha2.intros)
+ − 268
apply(frule alpha_rfv)
+ − 269
apply(rotate_tac 4)
+ − 270
apply(drule sym)
+ − 271
apply(simp)
+ − 272
apply(drule sym)
+ − 273
apply(simp)
+ − 274
oops
+ − 275
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quotient_type lam = rlam / alpha
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+ − 277
by (rule alpha_equivp)
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+ − 278
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+ − 279
quotient_definition
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"Var :: name \<Rightarrow> lam"
1139
+ − 281
is
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+ − 282
"rVar"
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+ − 283
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+ − 284
quotient_definition
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+ − 285
"App :: lam \<Rightarrow> lam \<Rightarrow> lam"
1139
+ − 286
is
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+ − 287
"rApp"
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+ − 288
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+ − 289
quotient_definition
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+ − 290
"Lam :: name \<Rightarrow> lam \<Rightarrow> lam"
1139
+ − 291
is
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+ − 292
"rLam"
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diff
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+ − 293
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diff
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+ − 294
quotient_definition
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+ − 295
"fv :: lam \<Rightarrow> atom set"
1139
+ − 296
is
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diff
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+ − 297
"rfv"
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diff
changeset
+ − 298
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+ − 299
lemma perm_rsp[quot_respect]:
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+ − 300
"(op = ===> alpha ===> alpha) permute permute"
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parents:
diff
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+ − 301
apply(auto)
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parents:
diff
changeset
+ − 302
apply(rule alpha_eqvt)
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parents:
diff
changeset
+ − 303
apply(simp)
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parents:
diff
changeset
+ − 304
done
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parents:
diff
changeset
+ − 305
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parents:
diff
changeset
+ − 306
lemma rVar_rsp[quot_respect]:
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parents:
diff
changeset
+ − 307
"(op = ===> alpha) rVar rVar"
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parents:
diff
changeset
+ − 308
by (auto intro: a1)
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parents:
diff
changeset
+ − 309
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parents:
diff
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+ − 310
lemma rApp_rsp[quot_respect]:
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parents:
diff
changeset
+ − 311
"(alpha ===> alpha ===> alpha) rApp rApp"
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parents:
diff
changeset
+ − 312
by (auto intro: a2)
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parents:
diff
changeset
+ − 313
fa810f01f7b5
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parents:
diff
changeset
+ − 314
lemma rLam_rsp[quot_respect]:
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parents:
diff
changeset
+ − 315
"(op = ===> alpha ===> alpha) rLam rLam"
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parents:
diff
changeset
+ − 316
apply(auto)
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parents:
diff
changeset
+ − 317
apply(rule a3)
fa810f01f7b5
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parents:
diff
changeset
+ − 318
apply(rule_tac x="0" in exI)
fa810f01f7b5
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parents:
diff
changeset
+ − 319
unfolding fresh_star_def
1011
+ − 320
apply(simp add: fresh_star_def fresh_zero_perm)
947
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parents:
diff
changeset
+ − 321
apply(simp add: alpha_rfv)
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 322
done
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 323
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 324
lemma rfv_rsp[quot_respect]:
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parents:
diff
changeset
+ − 325
"(alpha ===> op =) rfv rfv"
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 326
apply(simp add: alpha_rfv)
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 327
done
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 328
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 329
fa810f01f7b5
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parents:
diff
changeset
+ − 330
section {* lifted theorems *}
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parents:
diff
changeset
+ − 331
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parents:
diff
changeset
+ − 332
lemma lam_induct:
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parents:
diff
changeset
+ − 333
"\<lbrakk>\<And>name. P (Var name);
fa810f01f7b5
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parents:
diff
changeset
+ − 334
\<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (App lam1 lam2);
fa810f01f7b5
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parents:
diff
changeset
+ − 335
\<And>name lam. P lam \<Longrightarrow> P (Lam name lam)\<rbrakk>
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parents:
diff
changeset
+ − 336
\<Longrightarrow> P lam"
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parents:
diff
changeset
+ − 337
apply (lifting rlam.induct)
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parents:
diff
changeset
+ − 338
done
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 339
fa810f01f7b5
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 340
instantiation lam :: pt
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parents:
diff
changeset
+ − 341
begin
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 342
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 343
quotient_definition
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 344
"permute_lam :: perm \<Rightarrow> lam \<Rightarrow> lam"
1139
+ − 345
is
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 346
"permute :: perm \<Rightarrow> rlam \<Rightarrow> rlam"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 347
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 348
lemma permute_lam [simp]:
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 349
shows "pi \<bullet> Var a = Var (pi \<bullet> a)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 350
and "pi \<bullet> App t1 t2 = App (pi \<bullet> t1) (pi \<bullet> t2)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 351
and "pi \<bullet> Lam a t = Lam (pi \<bullet> a) (pi \<bullet> t)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 352
apply(lifting permute_rlam.simps)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 353
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 354
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 355
instance
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 356
apply default
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 357
apply(induct_tac [!] x rule: lam_induct)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 358
apply(simp_all)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 359
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 360
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 361
end
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 362
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 363
lemma fv_lam [simp]:
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 364
shows "fv (Var a) = {atom a}"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 365
and "fv (App t1 t2) = fv t1 \<union> fv t2"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 366
and "fv (Lam a t) = fv t - {atom a}"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 367
apply(lifting rfv_var rfv_app rfv_lam)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 368
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 369
990
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 370
lemma fv_eqvt:
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 371
shows "(p \<bullet> fv t) = fv (p \<bullet> t)"
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 372
apply(lifting rfv_eqvt)
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 373
done
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 374
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 375
lemma a1:
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 376
"a = b \<Longrightarrow> Var a = Var b"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 377
by (lifting a1)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 378
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 379
lemma a2:
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 380
"\<lbrakk>x = xa; xb = xc\<rbrakk> \<Longrightarrow> App x xb = App xa xc"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 381
by (lifting a2)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 382
1011
+ − 383
lemma a3:
+ − 384
"\<lbrakk>\<exists>pi. (fv t - {atom a} = fv s - {atom b} \<and> (fv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) = s)\<rbrakk>
+ − 385
\<Longrightarrow> Lam a t = Lam b s"
+ − 386
apply (lifting a3)
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 387
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 388
989
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 389
lemma a3_inv:
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 390
assumes "Lam a t = Lam b s"
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 391
shows "\<exists>pi. (fv t - {atom a} = fv s - {atom b} \<and> (fv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) = s)"
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 392
using assms
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 393
apply(lifting a3_inverse)
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 394
done
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 395
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 396
lemma alpha_cases:
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 397
"\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P;
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 398
\<And>x xa xb xc. \<lbrakk>a1 = App x xb; a2 = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P;
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 399
\<And>t a s b. \<lbrakk>a1 = Lam a t; a2 = Lam b s;
989
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 400
\<exists>pi. fv t - {atom a} = fv s - {atom b} \<and> (fv t - {atom a}) \<sharp>* pi \<and> (pi \<bullet> t) = s\<rbrakk>
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 401
\<Longrightarrow> P\<rbrakk>
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 402
\<Longrightarrow> P"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 403
by (lifting alpha.cases)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 404
975
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 405
(* not sure whether needed *)
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 406
lemma alpha_induct:
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 407
"\<lbrakk>qx = qxa; \<And>a b. a = b \<Longrightarrow> qxb (Var a) (Var b);
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 408
\<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);
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 409
\<And>t a s b.
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 410
\<lbrakk>\<exists>pi. fv t - {atom a} = fv s - {atom b} \<and>
989
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 411
(fv t - {atom a}) \<sharp>* pi \<and> ((pi \<bullet> t) = s \<and> qxb (pi \<bullet> t) s)\<rbrakk>
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 412
\<Longrightarrow> qxb (Lam a t) (Lam b s)\<rbrakk>
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 413
\<Longrightarrow> qxb qx qxa"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 414
by (lifting alpha.induct)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 415
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 416
(* should they lift automatically *)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 417
lemma lam_inject [simp]:
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 418
shows "(Var a = Var b) = (a = b)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 419
and "(App t1 t2 = App s1 s2) = (t1 = s1 \<and> t2 = s2)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 420
apply(lifting rlam.inject(1) rlam.inject(2))
981
+ − 421
apply(regularize)
+ − 422
prefer 2
+ − 423
apply(regularize)
+ − 424
prefer 2
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 425
apply(auto)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 426
apply(drule alpha.cases)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 427
apply(simp_all)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 428
apply(simp add: alpha.a1)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 429
apply(drule alpha.cases)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 430
apply(simp_all)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 431
apply(drule alpha.cases)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 432
apply(simp_all)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 433
apply(rule alpha.a2)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 434
apply(simp_all)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 435
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 436
989
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 437
lemma Lam_pseudo_inject:
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 438
shows "(Lam a t = Lam b s) =
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 439
(\<exists>pi. (fv t - {atom a} = fv s - {atom b} \<and> (fv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) = s))"
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 440
apply(rule iffI)
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 441
apply(rule a3_inv)
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 442
apply(assumption)
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 443
apply(rule a3)
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 444
apply(assumption)
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 445
done
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 446
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 447
lemma rlam_distinct:
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 448
shows "\<not>(rVar nam \<approx> rApp rlam1' rlam2')"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 449
and "\<not>(rApp rlam1' rlam2' \<approx> rVar nam)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 450
and "\<not>(rVar nam \<approx> rLam nam' rlam')"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 451
and "\<not>(rLam nam' rlam' \<approx> rVar nam)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 452
and "\<not>(rApp rlam1 rlam2 \<approx> rLam nam' rlam')"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 453
and "\<not>(rLam nam' rlam' \<approx> rApp rlam1 rlam2)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 454
apply auto
981
+ − 455
apply (erule alpha.cases)
+ − 456
apply (simp_all only: rlam.distinct)
+ − 457
apply (erule alpha.cases)
+ − 458
apply (simp_all only: rlam.distinct)
+ − 459
apply (erule alpha.cases)
+ − 460
apply (simp_all only: rlam.distinct)
+ − 461
apply (erule alpha.cases)
+ − 462
apply (simp_all only: rlam.distinct)
+ − 463
apply (erule alpha.cases)
+ − 464
apply (simp_all only: rlam.distinct)
+ − 465
apply (erule alpha.cases)
+ − 466
apply (simp_all only: rlam.distinct)
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 467
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 468
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 469
lemma lam_distinct[simp]:
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 470
shows "Var nam \<noteq> App lam1' lam2'"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 471
and "App lam1' lam2' \<noteq> Var nam"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 472
and "Var nam \<noteq> Lam nam' lam'"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 473
and "Lam nam' lam' \<noteq> Var nam"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 474
and "App lam1 lam2 \<noteq> Lam nam' lam'"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 475
and "Lam nam' lam' \<noteq> App lam1 lam2"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 476
apply(lifting rlam_distinct(1) rlam_distinct(2) rlam_distinct(3) rlam_distinct(4) rlam_distinct(5) rlam_distinct(6))
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 477
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 478
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 479
lemma var_supp1:
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 480
shows "(supp (Var a)) = (supp a)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 481
apply (simp add: supp_def)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 482
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 483
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 484
lemma var_supp:
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 485
shows "(supp (Var a)) = {a:::name}"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 486
using var_supp1 by (simp add: supp_at_base)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 487
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 488
lemma app_supp:
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 489
shows "supp (App t1 t2) = (supp t1) \<union> (supp t2)"
997
b7d259ded92e
Ported LF to the generic lambda and solved the simpler _supp cases.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 490
apply(simp only: supp_def lam_inject)
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 491
apply(simp add: Collect_imp_eq Collect_neg_eq)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 492
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 493
990
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 494
(* supp for lam *)
975
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 495
lemma lam_supp1:
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 496
shows "(supp (atom x, t)) supports (Lam x t) "
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 497
apply(simp add: supports_def)
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 498
apply(fold fresh_def)
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 499
apply(simp add: fresh_Pair swap_fresh_fresh)
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 500
apply(clarify)
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 501
apply(subst swap_at_base_simps(3))
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 502
apply(simp_all add: fresh_atom)
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 503
done
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 504
975
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 505
lemma lam_fsupp1:
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 506
assumes a: "finite (supp t)"
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 507
shows "finite (supp (Lam x t))"
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 508
apply(rule supports_finite)
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 509
apply(rule lam_supp1)
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 510
apply(simp add: a supp_Pair supp_atom)
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 511
done
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 512
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 513
instance lam :: fs
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 514
apply(default)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 515
apply(induct_tac x rule: lam_induct)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 516
apply(simp add: var_supp)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 517
apply(simp add: app_supp)
975
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 518
apply(simp add: lam_fsupp1)
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 519
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 520
990
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 521
lemma supp_fv:
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 522
shows "supp t = fv t"
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 523
apply(induct t rule: lam_induct)
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 524
apply(simp add: var_supp)
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 525
apply(simp add: app_supp)
1016
+ − 526
apply(subgoal_tac "supp (Lam name lam) = supp (Abs {atom name} lam)")
+ − 527
apply(simp add: supp_Abs)
990
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 528
apply(simp (no_asm) add: supp_def permute_set_eq atom_eqvt)
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 529
apply(simp add: Lam_pseudo_inject)
1016
+ − 530
apply(simp add: Abs_eq_iff alpha_gen)
990
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 531
apply(simp add: supp_eqvt[symmetric] fv_eqvt[symmetric])
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 532
done
c25ff084868f
now also final step is proved - the supp of lambdas is now completely characterised
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 533
989
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 534
lemma lam_supp2:
1016
+ − 535
shows "supp (Lam x t) = supp (Abs {atom x} t)"
989
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 536
apply(simp add: supp_def permute_set_eq atom_eqvt)
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 537
apply(simp add: Lam_pseudo_inject)
1016
+ − 538
apply(simp add: Abs_eq_iff supp_fv alpha_gen)
995
ee0619b5adff
introduced a generic alpha (but not sure whether it is helpful)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 539
done
975
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 540
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 541
lemma lam_supp:
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 542
shows "supp (Lam x t) = ((supp t) - {atom x})"
989
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 543
apply(simp add: lam_supp2)
1016
+ − 544
apply(simp add: supp_Abs)
989
af02b193a19a
the supp of a lambda can now be characterised, *provided* the notion of free variables coincides with support on lambda terms
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 545
done
975
513ebe332964
test about supp/freshness for lam (old proofs work in principle - for single binders)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 546
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 547
lemma fresh_lam:
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 548
"(atom a \<sharp> Lam b t) \<longleftrightarrow> (a = b) \<or> (a \<noteq> b \<and> atom a \<sharp> t)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 549
apply(simp add: fresh_def)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 550
apply(simp add: lam_supp)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 551
apply(auto)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 552
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 553
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 554
lemma lam_induct_strong:
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 555
fixes a::"'a::fs"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 556
assumes a1: "\<And>name b. P b (Var name)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 557
and a2: "\<And>lam1 lam2 b. \<lbrakk>\<And>c. P c lam1; \<And>c. P c lam2\<rbrakk> \<Longrightarrow> P b (App lam1 lam2)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 558
and a3: "\<And>name lam b. \<lbrakk>\<And>c. P c lam; (atom name) \<sharp> b\<rbrakk> \<Longrightarrow> P b (Lam name lam)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 559
shows "P a lam"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 560
proof -
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 561
have "\<And>pi a. P a (pi \<bullet> lam)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 562
proof (induct lam rule: lam_induct)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 563
case (1 name pi)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 564
show "P a (pi \<bullet> Var name)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 565
apply (simp)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 566
apply (rule a1)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 567
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 568
next
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 569
case (2 lam1 lam2 pi)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 570
have b1: "\<And>pi a. P a (pi \<bullet> lam1)" by fact
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 571
have b2: "\<And>pi a. P a (pi \<bullet> lam2)" by fact
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 572
show "P a (pi \<bullet> App lam1 lam2)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 573
apply (simp)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 574
apply (rule a2)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 575
apply (rule b1)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 576
apply (rule b2)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 577
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 578
next
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 579
case (3 name lam pi a)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 580
have b: "\<And>pi a. P a (pi \<bullet> lam)" by fact
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 581
obtain c::name where fr: "atom c\<sharp>(a, pi\<bullet>name, pi\<bullet>lam)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 582
apply(rule obtain_atom)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 583
apply(auto)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 584
sorry
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 585
from b fr have p: "P a (Lam c (((c \<leftrightarrow> (pi \<bullet> name)) + pi)\<bullet>lam))"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 586
apply -
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 587
apply(rule a3)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 588
apply(blast)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 589
apply(simp add: fresh_Pair)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 590
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 591
have eq: "(atom c \<rightleftharpoons> atom (pi\<bullet>name)) \<bullet> Lam (pi \<bullet> name) (pi \<bullet> lam) = Lam (pi \<bullet> name) (pi \<bullet> lam)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 592
apply(rule swap_fresh_fresh)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 593
using fr
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 594
apply(simp add: fresh_lam fresh_Pair)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 595
apply(simp add: fresh_lam fresh_Pair)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 596
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 597
show "P a (pi \<bullet> Lam name lam)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 598
apply (simp)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 599
apply(subst eq[symmetric])
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 600
using p
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 601
apply(simp only: permute_lam)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 602
apply(simp add: flip_def)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 603
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 604
qed
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 605
then have "P a (0 \<bullet> lam)" by blast
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 606
then show "P a lam" by simp
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 607
qed
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 608
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 609
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 610
lemma var_fresh:
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 611
fixes a::"name"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 612
shows "(atom a \<sharp> (Var b)) = (atom a \<sharp> b)"
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 613
apply(simp add: fresh_def)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 614
apply(simp add: var_supp1)
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 615
done
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 616
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 617
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 618
1087
+ − 619
end
947
fa810f01f7b5
added an LamEx example together with the new nominal infrastructure
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 620