--- a/Quot/Nominal/Terms.thy Wed Feb 17 15:52:08 2010 +0100
+++ b/Quot/Nominal/Terms.thy Wed Feb 17 16:22:16 2010 +0100
@@ -1,5 +1,5 @@
theory Terms
-imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm"
+imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv"
begin
atom_decl name
@@ -27,17 +27,10 @@
| "bv1 (BVr x) = {atom x}"
| "bv1 (BPr bp1 bp2) = (bv1 bp1) \<union> (bv1 bp1)"
-(* needs to be calculated by the package *)
-primrec
- rfv_trm1 and rfv_bp
-where
- "rfv_trm1 (rVr1 x) = {atom x}"
-| "rfv_trm1 (rAp1 t1 t2) = (rfv_trm1 t1) \<union> (rfv_trm1 t2)"
-| "rfv_trm1 (rLm1 x t) = (rfv_trm1 t) - {atom x}"
-| "rfv_trm1 (rLt1 bp t1 t2) = (rfv_trm1 t1) \<union> (rfv_trm1 t2 - bv1 bp)"
-| "rfv_bp (BUnit) = {}"
-| "rfv_bp (BVr x) = {atom x}"
-| "rfv_bp (BPr b1 b2) = (rfv_bp b1) \<union> (rfv_bp b2)"
+local_setup {* define_raw_fv "Terms.rtrm1"
+ [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(NONE, 0)], [], [(SOME @{term bv1}, 0)]]],
+ [[], [[]], [[], []]]] *}
+print_theorems
setup {* snd o define_raw_perms ["rtrm1", "bp"] ["Terms.rtrm1", "Terms.bp"] *}
@@ -46,14 +39,14 @@
where
a1: "a = b \<Longrightarrow> (rVr1 a) \<approx>1 (rVr1 b)"
| a2: "\<lbrakk>t1 \<approx>1 t2; s1 \<approx>1 s2\<rbrakk> \<Longrightarrow> rAp1 t1 s1 \<approx>1 rAp1 t2 s2"
-| a3: "(\<exists>pi. (({atom aa}, t) \<approx>gen alpha1 rfv_trm1 pi ({atom ab}, s))) \<Longrightarrow> rLm1 aa t \<approx>1 rLm1 ab s"
-| a4: "t1 \<approx>1 t2 \<Longrightarrow> (\<exists>pi. (((bv1 b1), s1) \<approx>gen alpha1 rfv_trm1 pi ((bv1 b2), s2))) \<Longrightarrow> rLt1 b1 t1 s1 \<approx>1 rLt1 b2 t2 s2"
+| a3: "(\<exists>pi. (({atom aa}, t) \<approx>gen alpha1 fv_rtrm1 pi ({atom ab}, s))) \<Longrightarrow> rLm1 aa t \<approx>1 rLm1 ab s"
+| a4: "t1 \<approx>1 t2 \<Longrightarrow> (\<exists>pi. (((bv1 b1), s1) \<approx>gen alpha1 fv_rtrm1 pi ((bv1 b2), s2))) \<Longrightarrow> rLt1 b1 t1 s1 \<approx>1 rLt1 b2 t2 s2"
lemma alpha1_inj:
"(rVr1 a \<approx>1 rVr1 b) = (a = b)"
"(rAp1 t1 s1 \<approx>1 rAp1 t2 s2) = (t1 \<approx>1 t2 \<and> s1 \<approx>1 s2)"
-"(rLm1 aa t \<approx>1 rLm1 ab s) = (\<exists>pi. (({atom aa}, t) \<approx>gen alpha1 rfv_trm1 pi ({atom ab}, s)))"
-"(rLt1 b1 t1 s1 \<approx>1 rLt1 b2 t2 s2) = (t1 \<approx>1 t2 \<and> (\<exists>pi. (((bv1 b1), s1) \<approx>gen alpha1 rfv_trm1 pi ((bv1 b2), s2))))"
+"(rLm1 aa t \<approx>1 rLm1 ab s) = (\<exists>pi. (({atom aa}, t) \<approx>gen alpha1 fv_rtrm1 pi ({atom ab}, s)))"
+"(rLt1 b1 t1 s1 \<approx>1 rLt1 b2 t2 s2) = (t1 \<approx>1 t2 \<and> (\<exists>pi. (((bv1 b1), s1) \<approx>gen alpha1 fv_rtrm1 pi ((bv1 b2), s2))))"
apply -
apply rule apply (erule alpha1.cases) apply (simp_all add: alpha1.intros)
apply rule apply (erule alpha1.cases) apply (simp_all add: alpha1.intros)
@@ -68,8 +61,8 @@
apply (simp_all add: empty_eqvt insert_eqvt atom_eqvt)
done
-lemma rfv_trm1_eqvt[eqvt]:
- shows "(pi\<bullet>rfv_trm1 t) = rfv_trm1 (pi\<bullet>t)"
+lemma fv_rtrm1_eqvt[eqvt]:
+ shows "(pi\<bullet>fv_rtrm1 t) = fv_rtrm1 (pi\<bullet>t)"
apply (induct t)
apply (simp_all add: insert_eqvt atom_eqvt empty_eqvt union_eqvt Diff_eqvt bv1_eqvt)
done
@@ -85,10 +78,10 @@
apply(erule conjE)+
apply(rule conjI)
apply(rule_tac ?p1="- pi" in permute_eq_iff[THEN iffD1])
- apply(simp add: atom_eqvt Diff_eqvt insert_eqvt empty_eqvt rfv_trm1_eqvt)
+ apply(simp add: atom_eqvt Diff_eqvt insert_eqvt empty_eqvt fv_rtrm1_eqvt)
apply(rule conjI)
apply(rule_tac ?p1="- pi" in fresh_star_permute_iff[THEN iffD1])
- apply(simp add: atom_eqvt Diff_eqvt rfv_trm1_eqvt insert_eqvt empty_eqvt)
+ apply(simp add: atom_eqvt Diff_eqvt fv_rtrm1_eqvt insert_eqvt empty_eqvt)
apply(simp add: permute_eqvt[symmetric])
apply (erule exE)
apply (rule_tac x="pi \<bullet> pia" in exI)
@@ -96,10 +89,10 @@
apply(erule conjE)+
apply(rule conjI)
apply(rule_tac ?p1="- pi" in permute_eq_iff[THEN iffD1])
- apply(simp add: rfv_trm1_eqvt Diff_eqvt bv1_eqvt)
+ apply(simp add: fv_rtrm1_eqvt Diff_eqvt bv1_eqvt)
apply(rule conjI)
apply(rule_tac ?p1="- pi" in fresh_star_permute_iff[THEN iffD1])
- apply(simp add: atom_eqvt rfv_trm1_eqvt Diff_eqvt bv1_eqvt)
+ apply(simp add: atom_eqvt fv_rtrm1_eqvt Diff_eqvt bv1_eqvt)
apply(simp add: permute_eqvt[symmetric])
done
@@ -132,10 +125,10 @@
quotient_definition
"fv_trm1 :: trm1 \<Rightarrow> atom set"
is
- "rfv_trm1"
+ "fv_rtrm1"
lemma alpha_rfv1:
- shows "t \<approx>1 s \<Longrightarrow> rfv_trm1 t = rfv_trm1 s"
+ shows "t \<approx>1 s \<Longrightarrow> fv_rtrm1 t = fv_rtrm1 s"
apply(induct rule: alpha1.induct)
apply(simp_all add: alpha_gen.simps)
done
@@ -160,7 +153,7 @@
done
lemma [quot_respect]:
- "(alpha1 ===> op =) rfv_trm1 rfv_trm1"
+ "(alpha1 ===> op =) fv_rtrm1 fv_rtrm1"
apply (simp add: alpha_rfv1)
done
@@ -185,9 +178,9 @@
end
-lemmas fv_trm1 = rfv_trm1_rfv_bp.simps[quot_lifted]
+lemmas fv_trm1 = fv_rtrm1_fv_bp.simps[quot_lifted]
-lemmas fv_trm1_eqvt = rfv_trm1_eqvt[quot_lifted]
+lemmas fv_trm1_eqvt = fv_rtrm1_eqvt[quot_lifted]
lemmas alpha1_INJ = alpha1_inj[unfolded alpha_gen, quot_lifted, folded alpha_gen]
@@ -291,15 +284,10 @@
where
"rbv2 (rAs x t) = {atom x}"
-(* needs to be calculated by the package *)
-primrec
- fv_rtrm2 and fv_rassign
-where
- "fv_rtrm2 (rVr2 x) = {atom x}"
-| "fv_rtrm2 (rAp2 t1 t2) = (fv_rtrm2 t1) \<union> (fv_rtrm2 t2)"
-| "fv_rtrm2 (rLm2 x t) = (fv_rtrm2 t) - {atom x}"
-| "fv_rtrm2 (rLt2 as t) = (fv_rtrm2 t - rbv2 as) \<union> (fv_rassign as)"
-| "fv_rassign (rAs x t) = fv_rtrm2 t"
+local_setup {* define_raw_fv "Terms.rtrm2"
+ [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[], [(SOME @{term rbv2}, 0)]]],
+ [[[(NONE, 0)], []]]] *}
+print_theorems
setup {* snd o define_raw_perms ["rtrm2", "rassign"] ["Terms.rtrm2", "Terms.rassign"] *}
@@ -347,15 +335,10 @@
"bv3 ANil = {}"
| "bv3 (ACons x t as) = {atom x} \<union> (bv3 as)"
-primrec
- fv_trm3 and fv_assigns
-where
- "fv_trm3 (Vr3 x) = {atom x}"
-| "fv_trm3 (Ap3 t1 t2) = (fv_trm3 t1) \<union> (fv_trm3 t2)"
-| "fv_trm3 (Lm3 x t) = (fv_trm3 t) - {atom x}"
-| "fv_trm3 (Lt3 as t) = (fv_trm3 t - bv3 as) \<union> (fv_assigns as)"
-| "fv_assigns (ANil) = {}"
-| "fv_assigns (ACons x t as) = (fv_trm3 t) \<union> (fv_assigns as)"
+local_setup {* define_raw_fv "Terms.trm3"
+ [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[], [(SOME @{term bv3}, 0)]]],
+ [[], [[(NONE, 0)], [], []]]] *}
+print_theorems
setup {* snd o define_raw_perms ["rtrm3", "assigns"] ["Terms.trm3", "Terms.assigns"] *}
@@ -391,17 +374,13 @@
Vr4 "name"
| Ap4 "trm4" "trm4 list"
| Lm4 "name" "trm4" --"bind (name) in (trm)"
+print_theorems
thm trm4.recs
-primrec
- fv_trm4 and fv_trm4_list
-where
- "fv_trm4 (Vr4 x) = {atom x}"
-| "fv_trm4 (Ap4 t ts) = (fv_trm4 t) \<union> (fv_trm4_list ts)"
-| "fv_trm4 (Lm4 x t) = (fv_trm4 t) - {atom x}"
-| "fv_trm4_list ([]) = {}"
-| "fv_trm4_list (t#ts) = (fv_trm4 t) \<union> (fv_trm4_list ts)"
+local_setup {* define_raw_fv "Terms.trm4" [
+ [[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]]], [[], [[], []]] ] *}
+print_theorems
(* there cannot be a clause for lists, as *)
(* permutations are already defined in Nominal (also functions, options, and so on) *)
@@ -451,14 +430,9 @@
"rbv5 rLnil = {}"
| "rbv5 (rLcons n t ltl) = {atom n} \<union> (rbv5 ltl)"
-primrec
- rfv_trm5 and rfv_lts
-where
- "rfv_trm5 (rVr5 n) = {atom n}"
-| "rfv_trm5 (rAp5 t s) = (rfv_trm5 t) \<union> (rfv_trm5 s)"
-| "rfv_trm5 (rLt5 lts t) = (rfv_trm5 t - rbv5 lts) \<union> (rfv_lts lts - rbv5 lts)"
-| "rfv_lts (rLnil) = {}"
-| "rfv_lts (rLcons n t ltl) = (rfv_trm5 t) \<union> (rfv_lts ltl)"
+local_setup {* define_raw_fv "Terms.rtrm5" [
+ [[[]], [[], []], [[(SOME @{term rbv5}, 0)], [(SOME @{term rbv5}, 0)]]], [[], [[(NONE, 0)], [], []]] ] *}
+print_theorems
setup {* snd o define_raw_perms ["rtrm5", "rlts"] ["Terms.rtrm5", "Terms.rlts"] *}
print_theorems
@@ -470,8 +444,8 @@
where
a1: "a = b \<Longrightarrow> (rVr5 a) \<approx>5 (rVr5 b)"
| a2: "\<lbrakk>t1 \<approx>5 t2; s1 \<approx>5 s2\<rbrakk> \<Longrightarrow> rAp5 t1 s1 \<approx>5 rAp5 t2 s2"
-| a3: "\<lbrakk>\<exists>pi. ((rbv5 l1, t1) \<approx>gen alpha5 rfv_trm5 pi (rbv5 l2, t2));
- \<exists>pi. ((rbv5 l1, l1) \<approx>gen alphalts rfv_lts pi (rbv5 l2, l2))\<rbrakk>
+| a3: "\<lbrakk>\<exists>pi. ((rbv5 l1, t1) \<approx>gen alpha5 fv_rtrm5 pi (rbv5 l2, t2));
+ \<exists>pi. ((rbv5 l1, l1) \<approx>gen alphalts fv_rlts pi (rbv5 l2, l2))\<rbrakk>
\<Longrightarrow> rLt5 l1 t1 \<approx>5 rLt5 l2 t2"
| a4: "rLnil \<approx>l rLnil"
| a5: "ls1 \<approx>l ls2 \<Longrightarrow> t1 \<approx>5 t2 \<Longrightarrow> n1 = n2 \<Longrightarrow> rLcons n1 t1 ls1 \<approx>l rLcons n2 t2 ls2"
@@ -481,8 +455,8 @@
lemma alpha5_inj:
"((rVr5 a) \<approx>5 (rVr5 b)) = (a = b)"
"(rAp5 t1 s1 \<approx>5 rAp5 t2 s2) = (t1 \<approx>5 t2 \<and> s1 \<approx>5 s2)"
- "(rLt5 l1 t1 \<approx>5 rLt5 l2 t2) = ((\<exists>pi. ((rbv5 l1, t1) \<approx>gen alpha5 rfv_trm5 pi (rbv5 l2, t2))) \<and>
- (\<exists>pi. ((rbv5 l1, l1) \<approx>gen alphalts rfv_lts pi (rbv5 l2, l2))))"
+ "(rLt5 l1 t1 \<approx>5 rLt5 l2 t2) = ((\<exists>pi. ((rbv5 l1, t1) \<approx>gen alpha5 fv_rtrm5 pi (rbv5 l2, t2))) \<and>
+ (\<exists>pi. ((rbv5 l1, l1) \<approx>gen alphalts fv_rlts pi (rbv5 l2, l2))))"
"rLnil \<approx>l rLnil"
"(rLcons n1 t1 ls1 \<approx>l rLcons n2 t2 ls2) = (n1 = n2 \<and> ls1 \<approx>l ls2 \<and> t1 \<approx>5 t2)"
apply -
@@ -540,12 +514,12 @@
quotient_definition
"fv_trm5 :: trm5 \<Rightarrow> atom set"
is
- "rfv_trm5"
+ "fv_rtrm5"
quotient_definition
"fv_lts :: lts \<Rightarrow> atom set"
is
- "rfv_lts"
+ "fv_rlts"
quotient_definition
"bv5 :: lts \<Rightarrow> atom set"
@@ -556,12 +530,12 @@
"pi \<bullet> (rbv5 x) = rbv5 (pi \<bullet> x)"
sorry
-lemma rfv_trm5_eqvt:
- "pi \<bullet> (rfv_trm5 x) = rfv_trm5 (pi \<bullet> x)"
+lemma fv_rtrm5_eqvt:
+ "pi \<bullet> (fv_rtrm5 x) = fv_rtrm5 (pi \<bullet> x)"
sorry
-lemma rfv_lts_eqvt:
- "pi \<bullet> (rfv_lts x) = rfv_lts (pi \<bullet> x)"
+lemma fv_rlts_eqvt:
+ "pi \<bullet> (fv_rlts x) = fv_rlts (pi \<bullet> x)"
sorry
lemma alpha5_eqvt:
@@ -576,26 +550,26 @@
apply (rule_tac x="x \<bullet> pi" in exI)
apply (rule conjI)
apply(rule_tac ?p1="- x" in permute_eq_iff[THEN iffD1])
- apply(simp add: atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt rbv5_eqvt rfv_trm5_eqvt)
+ apply(simp add: atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt rbv5_eqvt fv_rtrm5_eqvt)
apply(rule conjI)
apply(rule_tac ?p1="- x" in fresh_star_permute_iff[THEN iffD1])
- apply(simp add: atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt rbv5_eqvt rfv_trm5_eqvt)
+ apply(simp add: atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt rbv5_eqvt fv_rtrm5_eqvt)
apply (subst permute_eqvt[symmetric])
apply (simp)
apply (rule_tac x="x \<bullet> pia" in exI)
apply (rule conjI)
apply(rule_tac ?p1="- x" in permute_eq_iff[THEN iffD1])
- apply(simp add: atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt rbv5_eqvt rfv_lts_eqvt)
+ apply(simp add: atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt rbv5_eqvt fv_rlts_eqvt)
apply(rule conjI)
apply(rule_tac ?p1="- x" in fresh_star_permute_iff[THEN iffD1])
- apply(simp add: atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt rbv5_eqvt rfv_lts_eqvt)
+ apply(simp add: atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt rbv5_eqvt fv_rlts_eqvt)
apply (subst permute_eqvt[symmetric])
apply (simp)
done
lemma alpha5_rfv:
- "(t \<approx>5 s \<Longrightarrow> rfv_trm5 t = rfv_trm5 s)"
- "(l \<approx>l m \<Longrightarrow> rfv_lts l = rfv_lts m)"
+ "(t \<approx>5 s \<Longrightarrow> fv_rtrm5 t = fv_rtrm5 s)"
+ "(l \<approx>l m \<Longrightarrow> fv_rlts l = fv_rlts m)"
apply(induct rule: alpha5_alphalts.inducts)
apply(simp_all add: alpha_gen)
done
@@ -607,8 +581,8 @@
done
lemma [quot_respect]:
- "(alphalts ===> op =) rfv_lts rfv_lts"
- "(alpha5 ===> op =) rfv_trm5 rfv_trm5"
+ "(alphalts ===> op =) fv_rlts fv_rlts"
+ "(alpha5 ===> op =) fv_rtrm5 fv_rtrm5"
"(alphalts ===> op =) rbv5 rbv5"
"(op = ===> alpha5) rVr5 rVr5"
"(alpha5 ===> alpha5 ===> alpha5) rAp5 rAp5"
@@ -671,7 +645,7 @@
lemmas bv5[simp] = rbv5.simps[quot_lifted]
-lemmas fv_trm5_lts[simp] = rfv_trm5_rfv_lts.simps[quot_lifted]
+lemmas fv_trm5_lts[simp] = fv_rtrm5_fv_rlts.simps[quot_lifted]
lemma lets_ok:
"(Lt5 (Lcons x (Vr5 x) Lnil) (Vr5 x)) = (Lt5 (Lcons y (Vr5 y) Lnil) (Vr5 y))"
@@ -748,12 +722,9 @@
| "rbv6 (rLm6 n t) = {atom n} \<union> rbv6 t"
| "rbv6 (rLt6 l r) = rbv6 l \<union> rbv6 r"
-primrec
- rfv_trm6
-where
- "rfv_trm6 (rVr6 n) = {atom n}"
-| "rfv_trm6 (rLm6 n t) = (rfv_trm6 t) - {atom n}"
-| "rfv_trm6 (rLt6 l r) = (rfv_trm6 r - rbv6 l) \<union> rfv_trm6 l"
+local_setup {* define_raw_fv "Terms.rtrm6" [
+ [[[]], [[(NONE, 0)], [(NONE, 0)]], [[], [(SOME @{term rbv6}, 0)]]]] *}
+print_theorems
setup {* snd o define_raw_perms ["rtrm6"] ["Terms.rtrm6"] *}
print_theorems
@@ -762,8 +733,8 @@
alpha6 :: "rtrm6 \<Rightarrow> rtrm6 \<Rightarrow> bool" ("_ \<approx>6 _" [100, 100] 100)
where
a1: "a = b \<Longrightarrow> (rVr6 a) \<approx>6 (rVr6 b)"
-| a2: "(\<exists>pi. (({atom a}, t) \<approx>gen alpha6 rfv_trm6 pi ({atom b}, s))) \<Longrightarrow> rLm6 a t \<approx>6 rLm6 b s"
-| a3: "(\<exists>pi. (((rbv6 t1), s1) \<approx>gen alpha6 rfv_trm6 pi ((rbv6 t2), s2))) \<Longrightarrow> rLt6 t1 s1 \<approx>6 rLt6 t2 s2"
+| a2: "(\<exists>pi. (({atom a}, t) \<approx>gen alpha6 fv_rtrm6 pi ({atom b}, s))) \<Longrightarrow> rLm6 a t \<approx>6 rLm6 b s"
+| a3: "(\<exists>pi. (((rbv6 t1), s1) \<approx>gen alpha6 fv_rtrm6 pi ((rbv6 t2), s2))) \<Longrightarrow> rLt6 t1 s1 \<approx>6 rLt6 t2 s2"
lemma alpha6_equivps:
shows "equivp alpha6"
@@ -791,7 +762,7 @@
quotient_definition
"fv_trm6 :: trm6 \<Rightarrow> atom set"
is
- "rfv_trm6"
+ "fv_rtrm6"
quotient_definition
"bv6 :: trm6 \<Rightarrow> atom set"
@@ -822,7 +793,7 @@
apply (rule refl)
done
-lemma [quot_respect]:"(alpha6 ===> op =) rfv_trm6 rfv_trm6"
+lemma [quot_respect]:"(alpha6 ===> op =) fv_rtrm6 fv_rtrm6"
apply simp apply clarify
apply (induct_tac x y rule: alpha6.induct)
apply simp_all
@@ -907,12 +878,9 @@
| "rbv7 (rLm7 n t) = rbv7 t - {atom n}"
| "rbv7 (rLt7 l r) = rbv7 l \<union> rbv7 r"
-primrec
- rfv_trm7
-where
- "rfv_trm7 (rVr7 n) = {atom n}"
-| "rfv_trm7 (rLm7 n t) = (rfv_trm7 t) - {atom n}"
-| "rfv_trm7 (rLt7 l r) = (rfv_trm7 l) \<union> (rfv_trm7 r - rbv7 l)"
+local_setup {* define_raw_fv "Terms.rtrm7" [
+ [[[]], [[(NONE, 0)], [(NONE, 0)]], [[], [(SOME @{term rbv7}, 0)]]]] *}
+print_theorems
setup {* snd o define_raw_perms ["rtrm7"] ["Terms.rtrm7"] *}
print_theorems
@@ -921,10 +889,10 @@
alpha7 :: "rtrm7 \<Rightarrow> rtrm7 \<Rightarrow> bool" ("_ \<approx>7 _" [100, 100] 100)
where
a1: "a = b \<Longrightarrow> (rVr7 a) \<approx>7 (rVr7 b)"
-| a2: "(\<exists>pi. (({atom a}, t) \<approx>gen alpha7 rfv_trm7 pi ({atom b}, s))) \<Longrightarrow> rLm7 a t \<approx>7 rLm7 b s"
-| a3: "(\<exists>pi. (((rbv7 t1), s1) \<approx>gen alpha7 rfv_trm7 pi ((rbv7 t2), s2))) \<Longrightarrow> rLt7 t1 s1 \<approx>7 rLt7 t2 s2"
+| a2: "(\<exists>pi. (({atom a}, t) \<approx>gen alpha7 fv_rtrm7 pi ({atom b}, s))) \<Longrightarrow> rLm7 a t \<approx>7 rLm7 b s"
+| a3: "(\<exists>pi. (((rbv7 t1), s1) \<approx>gen alpha7 fv_rtrm7 pi ((rbv7 t2), s2))) \<Longrightarrow> rLt7 t1 s1 \<approx>7 rLt7 t2 s2"
-lemma bvfv7: "rbv7 x = rfv_trm7 x"
+lemma bvfv7: "rbv7 x = fv_rtrm7 x"
apply induct
apply simp_all
done
@@ -958,14 +926,9 @@
"rbv8 (Bar0 x) = {}"
| "rbv8 (Bar1 v x b) = {atom v}"
-primrec
- rfv_foo8 and rfv_bar8
-where
- "rfv_foo8 (Foo0 x) = {atom x}"
-| "rfv_foo8 (Foo1 b t) = (rfv_foo8 t - rbv8 b) \<union> (rfv_bar8 b)"
-| "rfv_bar8 (Bar0 x) = {atom x}"
-| "rfv_bar8 (Bar1 v x t) = {atom v} \<union> (rfv_bar8 t - {atom x})"
-print_theorems
+local_setup {* define_raw_fv "Terms.rfoo8" [
+ [[[]], [[], [(SOME @{term rbv8}, 0)]]], [[[]], [[], [(NONE, 1)], [(NONE, 1)]]]] *}
+print_theorems
setup {* snd o define_raw_perms ["rfoo8", "rbar8"] ["Terms.rfoo8", "Terms.rbar8"] *}
print_theorems
@@ -977,8 +940,8 @@
where
a1: "a = b \<Longrightarrow> (Foo0 a) \<approx>f (Foo0 b)"
| a2: "a = b \<Longrightarrow> (Bar0 a) \<approx>b (Bar0 b)"
-| a3: "b1 \<approx>b b2 \<Longrightarrow> (\<exists>pi. (((rbv8 b1), t1) \<approx>gen alpha8f rfv_foo8 pi ((rbv8 b2), t2))) \<Longrightarrow> Foo1 b1 t1 \<approx>f Foo1 b2 t2"
-| a4: "v1 = v2 \<Longrightarrow> (\<exists>pi. (({atom x1}, t1) \<approx>gen alpha8b rfv_bar8 pi ({atom x2}, t2))) \<Longrightarrow> Bar1 v1 x1 t1 \<approx>b Bar1 v2 x2 t2"
+| a3: "b1 \<approx>b b2 \<Longrightarrow> (\<exists>pi. (((rbv8 b1), t1) \<approx>gen alpha8f fv_rfoo8 pi ((rbv8 b2), t2))) \<Longrightarrow> Foo1 b1 t1 \<approx>f Foo1 b2 t2"
+| a4: "v1 = v2 \<Longrightarrow> (\<exists>pi. (({atom x1}, t1) \<approx>gen alpha8b fv_rbar8 pi ({atom x2}, t2))) \<Longrightarrow> Bar1 v1 x1 t1 \<approx>b Bar1 v2 x2 t2"
lemma "(alpha8b ===> op =) rbv8 rbv8"
apply simp apply clarify
@@ -986,18 +949,18 @@
apply (simp_all)
done
-lemma rfv_bar8_rsp_hlp: "x \<approx>b y \<Longrightarrow> rfv_bar8 x = rfv_bar8 y"
+lemma fv_rbar8_rsp_hlp: "x \<approx>b y \<Longrightarrow> fv_rbar8 x = fv_rbar8 y"
apply (erule alpha8f_alpha8b.inducts(2))
apply (simp_all add: alpha_gen)
done
-lemma "(alpha8b ===> op =) rfv_bar8 rfv_bar8"
- apply simp apply clarify apply (simp add: rfv_bar8_rsp_hlp)
+lemma "(alpha8b ===> op =) fv_rbar8 fv_rbar8"
+ apply simp apply clarify apply (simp add: fv_rbar8_rsp_hlp)
done
-lemma "(alpha8f ===> op =) rfv_foo8 rfv_foo8"
+lemma "(alpha8f ===> op =) fv_rfoo8 fv_rfoo8"
apply simp apply clarify
apply (erule alpha8f_alpha8b.inducts(1))
- apply (simp_all add: alpha_gen rfv_bar8_rsp_hlp)
+ apply (simp_all add: alpha_gen fv_rbar8_rsp_hlp)
done
@@ -1017,12 +980,9 @@
"rbv9 (Var9 x) = {}"
| "rbv9 (Lam9 x b) = {atom x}"
-primrec
- rfv_lam9 and rfv_bla9
-where
- "rfv_lam9 (Var9 x) = {atom x}"
-| "rfv_lam9 (Lam9 b t) = (rfv_lam9 t - {atom b})"
-| "rfv_bla9 (Bla9 l r) = (rfv_lam9 r - rbv9 l) \<union> rfv_lam9 l"
+local_setup {* define_raw_fv "Terms.rlam9" [
+ [[[]], [[(NONE, 0)], [(NONE, 0)]]], [[[], [(SOME @{term rbv9}, 0)]]]] *}
+print_theorems
setup {* snd o define_raw_perms ["rlam9", "rbla9"] ["Terms.rlam9", "Terms.rbla9"] *}
print_theorems
@@ -1033,8 +993,8 @@
alpha9b :: "rbla9 \<Rightarrow> rbla9 \<Rightarrow> bool" ("_ \<approx>9b _" [100, 100] 100)
where
a1: "a = b \<Longrightarrow> (Var9 a) \<approx>9l (Var9 b)"
-| a4: "(\<exists>pi. (({atom x1}, t1) \<approx>gen alpha9l rfv_lam9 pi ({atom x2}, t2))) \<Longrightarrow> Lam9 x1 t1 \<approx>9l Lam9 x2 t2"
-| a3: "b1 \<approx>9l b2 \<Longrightarrow> (\<exists>pi. (((rbv9 b1), t1) \<approx>gen alpha9l rfv_lam9 pi ((rbv9 b2), t2))) \<Longrightarrow> Bla9 b1 t1 \<approx>9b Bla9 b2 t2"
+| a4: "(\<exists>pi. (({atom x1}, t1) \<approx>gen alpha9l fv_rlam9 pi ({atom x2}, t2))) \<Longrightarrow> Lam9 x1 t1 \<approx>9l Lam9 x2 t2"
+| a3: "b1 \<approx>9l b2 \<Longrightarrow> (\<exists>pi. (((rbv9 b1), t1) \<approx>gen alpha9l fv_rlam9 pi ((rbv9 b2), t2))) \<Longrightarrow> Bla9 b1 t1 \<approx>9b Bla9 b2 t2"
quotient_type
lam9 = rlam9 / alpha9l and bla9 = rbla9 / alpha9b
@@ -1058,12 +1018,12 @@
quotient_definition
"fv_lam9 :: lam9 \<Rightarrow> atom set"
is
- "rfv_lam9"
+ "fv_rlam9"
quotient_definition
"fv_bla9 :: bla9 \<Rightarrow> atom set"
is
- "rfv_bla9"
+ "fv_rbla9"
quotient_definition
"bv9 :: lam9 \<Rightarrow> atom set"
@@ -1118,21 +1078,24 @@
abbreviation
"atoms xs \<equiv> {atom x| x. x \<in> xs}"
-primrec
- rfv_ty
-where
- "rfv_ty (Var n) = {atom n}"
-| "rfv_ty (Fun T1 T2) = (rfv_ty T1) \<union> (rfv_ty T2)"
+local_setup {* define_raw_fv "Terms.ty" [[[[]], [[], []]]] *}
+print_theorems
+
+(*
+doesn't work yet
+local_setup {* define_raw_fv "Terms.tyS" [[[[], []]]] *}
+print_theorems
+*)
primrec
- rfv_tyS
+ fv_tyS
where
- "rfv_tyS (All xs T) = (rfv_ty T - atoms xs)"
+ "fv_tyS (All xs T) = (fv_ty T - atoms xs)"
inductive
alpha_tyS :: "tyS \<Rightarrow> tyS \<Rightarrow> bool" ("_ \<approx>tyS _" [100, 100] 100)
where
- a1: "\<exists>pi. ((atoms xs1, T1) \<approx>gen (op =) rfv_ty pi (atoms xs2, T2))
+ a1: "\<exists>pi. ((atoms xs1, T1) \<approx>gen (op =) fv_ty pi (atoms xs2, T2))
\<Longrightarrow> All xs1 T1 \<approx>tyS All xs2 T2"
lemma