theory Terms
imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv"
begin
atom_decl name
text {* primrec seems to be genarally faster than fun *}
section {*** lets with binding patterns ***}
datatype rtrm1 =
rVr1 "name"
| rAp1 "rtrm1" "rtrm1"
| rLm1 "name" "rtrm1" --"name is bound in trm1"
| rLt1 "bp" "rtrm1" "rtrm1" --"all variables in bp are bound in the 2nd trm1"
and bp =
BUnit
| BVr "name"
| BPr "bp" "bp"
(* to be given by the user *)
primrec
bv1
where
"bv1 (BUnit) = {}"
| "bv1 (BVr x) = {atom x}"
| "bv1 (BPr bp1 bp2) = (bv1 bp1) \<union> (bv1 bp2)"
setup {* snd o define_raw_perms ["rtrm1", "bp"] ["Terms.rtrm1", "Terms.bp"] *}
local_setup {* snd o define_fv_alpha "Terms.rtrm1"
[[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term bv1}, 0)], [], [(SOME @{term bv1}, 0)]]],
[[], [[]], [[], []]]] *}
print_theorems
notation
alpha_rtrm1 ("_ \<approx>1 _" [100, 100] 100) and
alpha_bp ("_ \<approx>1b _" [100, 100] 100)
thm alpha_rtrm1_alpha_bp.intros
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 alpha_rtrm1 fv_rtrm1 pi ({atom ab}, s)))"
"(rLt1 bp rtrm11 rtrm12 \<approx>1 rLt1 bpa rtrm11a rtrm12a) =
((\<exists>pi. (bv1 bp, bp) \<approx>gen alpha_bp fv_bp pi (bv1 bpa, bpa)) \<and> rtrm11 \<approx>1 rtrm11a \<and>
(\<exists>pi. (bv1 bp, rtrm12) \<approx>gen alpha_rtrm1 fv_rtrm1 pi (bv1 bpa, rtrm12a)))"
"alpha_bp BUnit BUnit"
"(alpha_bp (BVr name) (BVr namea)) = (name = namea)"
"(alpha_bp (BPr bp1 bp2) (BPr bp1a bp2a)) = (alpha_bp bp1 bp1a \<and> alpha_bp bp2 bp2a)"
apply -
apply rule apply (erule alpha_rtrm1.cases) apply (simp_all add: alpha_rtrm1_alpha_bp.intros)
apply rule apply (erule alpha_rtrm1.cases) apply (simp_all add: alpha_rtrm1_alpha_bp.intros)
apply rule apply (erule alpha_rtrm1.cases) apply (simp_all add: alpha_rtrm1_alpha_bp.intros)
apply rule apply (erule alpha_rtrm1.cases) apply (simp_all add: alpha_rtrm1_alpha_bp.intros)
apply rule apply (erule alpha_bp.cases) apply (simp_all add: alpha_rtrm1_alpha_bp.intros)
apply rule apply (erule alpha_bp.cases) apply (simp_all add: alpha_rtrm1_alpha_bp.intros)
done
lemma alpha_bp_refl: "alpha_bp a a"
apply induct
apply (simp_all add: alpha1_inj)
done
lemma alpha_bp_eq_eq: "alpha_bp a b = (a = b)"
apply rule
apply (induct a b rule: alpha_rtrm1_alpha_bp.inducts(2))
apply (simp_all add: alpha_bp_refl)
done
lemma alpha_bp_eq: "alpha_bp = (op =)"
apply (rule ext)+
apply (rule alpha_bp_eq_eq)
done
lemma bv1_eqvt[eqvt]:
shows "(pi \<bullet> bv1 x) = bv1 (pi \<bullet> x)"
apply (induct x)
apply (simp_all add: empty_eqvt insert_eqvt atom_eqvt eqvts)
done
lemma fv_rtrm1_eqvt[eqvt]:
"(pi\<bullet>fv_rtrm1 t) = fv_rtrm1 (pi\<bullet>t)"
"(pi\<bullet>fv_bp b) = fv_bp (pi\<bullet>b)"
apply (induct t and b)
apply (simp_all add: insert_eqvt atom_eqvt empty_eqvt union_eqvt Diff_eqvt bv1_eqvt)
done
lemma alpha1_eqvt:
"t \<approx>1 s \<Longrightarrow> (pi \<bullet> t) \<approx>1 (pi \<bullet> s)"
"alpha_bp a b \<Longrightarrow> alpha_bp (pi \<bullet> a) (pi \<bullet> b)"
apply (induct t s and a b rule: alpha_rtrm1_alpha_bp.inducts)
apply (simp_all add:eqvts alpha1_inj)
apply (erule exE)
apply (rule_tac x="pi \<bullet> pia" in exI)
apply (simp add: alpha_gen)
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 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 fv_rtrm1_eqvt insert_eqvt empty_eqvt)
apply(simp add: permute_eqvt[symmetric])
apply (erule exE)
apply (erule exE)
apply (rule conjI)
apply (rule_tac x="pi \<bullet> pia" in exI)
apply (simp add: alpha_gen)
apply(erule conjE)+
apply(rule conjI)
apply(rule_tac ?p1="- pi" in permute_eq_iff[THEN iffD1])
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 fv_rtrm1_eqvt Diff_eqvt bv1_eqvt)
apply(simp add: permute_eqvt[symmetric])
apply (rule_tac x="pi \<bullet> piaa" in exI)
apply (simp add: alpha_gen)
apply(erule conjE)+
apply(rule conjI)
apply(rule_tac ?p1="- pi" in permute_eq_iff[THEN iffD1])
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 fv_rtrm1_eqvt Diff_eqvt bv1_eqvt)
apply(simp add: permute_eqvt[symmetric])
done
lemma alpha1_equivp: "equivp alpha_rtrm1"
sorry
quotient_type trm1 = rtrm1 / alpha_rtrm1
by (rule alpha1_equivp)
local_setup {*
(fn ctxt => ctxt
|> snd o (Quotient_Def.quotient_lift_const ("Vr1", @{term rVr1}))
|> snd o (Quotient_Def.quotient_lift_const ("Ap1", @{term rAp1}))
|> snd o (Quotient_Def.quotient_lift_const ("Lm1", @{term rLm1}))
|> snd o (Quotient_Def.quotient_lift_const ("Lt1", @{term rLt1}))
|> snd o (Quotient_Def.quotient_lift_const ("fv_trm1", @{term fv_rtrm1})))
*}
print_theorems
lemma alpha_rfv1:
shows "t \<approx>1 s \<Longrightarrow> fv_rtrm1 t = fv_rtrm1 s"
apply(induct rule: alpha_rtrm1_alpha_bp.inducts(1))
apply(simp_all add: alpha_gen.simps)
done
lemma [quot_respect]:
"(op = ===> alpha_rtrm1) rVr1 rVr1"
"(alpha_rtrm1 ===> alpha_rtrm1 ===> alpha_rtrm1) rAp1 rAp1"
"(op = ===> alpha_rtrm1 ===> alpha_rtrm1) rLm1 rLm1"
"(op = ===> alpha_rtrm1 ===> alpha_rtrm1 ===> alpha_rtrm1) rLt1 rLt1"
apply (auto simp add: alpha1_inj)
apply (rule_tac x="0" in exI)
apply (simp add: fresh_star_def fresh_zero_perm alpha_rfv1 alpha_gen)
apply (rule_tac x="0" in exI)
apply (simp add: alpha_gen fresh_star_def fresh_zero_perm alpha_rfv1 alpha_bp_eq)
apply (rule_tac x="0" in exI)
apply (simp add: alpha_gen fresh_star_def fresh_zero_perm alpha_rfv1)
done
lemma [quot_respect]:
"(op = ===> alpha_rtrm1 ===> alpha_rtrm1) permute permute"
by (simp add: alpha1_eqvt)
lemma [quot_respect]:
"(alpha_rtrm1 ===> op =) fv_rtrm1 fv_rtrm1"
by (simp add: alpha_rfv1)
lemmas trm1_bp_induct = rtrm1_bp.induct[quot_lifted]
lemmas trm1_bp_inducts = rtrm1_bp.inducts[quot_lifted]
instantiation trm1 and bp :: pt
begin
quotient_definition
"permute_trm1 :: perm \<Rightarrow> trm1 \<Rightarrow> trm1"
is
"permute :: perm \<Rightarrow> rtrm1 \<Rightarrow> rtrm1"
lemmas permute_trm1[simp] = permute_rtrm1_permute_bp.simps[quot_lifted]
instance
apply default
apply(induct_tac [!] x rule: trm1_bp_inducts(1))
apply(simp_all)
done
end
lemmas fv_trm1 = fv_rtrm1_fv_bp.simps[quot_lifted]
lemmas fv_trm1_eqvt = fv_rtrm1_eqvt[quot_lifted]
lemmas alpha1_INJ = alpha1_inj[unfolded alpha_gen, quot_lifted, folded alpha_gen]
lemma lm1_supp_pre:
shows "(supp (atom x, t)) supports (Lm1 x t) "
apply(simp add: supports_def)
apply(fold fresh_def)
apply(simp add: fresh_Pair swap_fresh_fresh)
apply(clarify)
apply(subst swap_at_base_simps(3))
apply(simp_all add: fresh_atom)
done
lemma lt1_supp_pre:
shows "(supp (x, t, s)) supports (Lt1 t x s) "
apply(simp add: supports_def)
apply(fold fresh_def)
apply(simp add: fresh_Pair swap_fresh_fresh)
done
lemma bp_supp: "finite (supp (bp :: bp))"
apply (induct bp)
apply(simp_all add: supp_def)
apply (fold supp_def)
apply (simp add: supp_at_base)
apply(simp add: Collect_imp_eq)
apply(simp add: Collect_neg_eq[symmetric])
apply (fold supp_def)
apply (simp)
done
instance trm1 :: fs
apply default
apply(induct_tac x rule: trm1_bp_inducts(1))
apply(simp_all)
apply(simp add: supp_def alpha1_INJ eqvts)
apply(simp add: supp_def[symmetric] supp_at_base)
apply(simp only: supp_def alpha1_INJ eqvts permute_trm1)
apply(simp add: Collect_imp_eq Collect_neg_eq)
apply(rule supports_finite)
apply(rule lm1_supp_pre)
apply(simp add: supp_Pair supp_atom)
apply(rule supports_finite)
apply(rule lt1_supp_pre)
apply(simp add: supp_Pair supp_atom bp_supp)
done
lemma fv_eq_bv: "fv_bp bp = bv1 bp"
apply(induct bp rule: trm1_bp_inducts(2))
apply(simp_all)
done
lemma helper: "{b. \<forall>pi. pi \<bullet> (a \<rightleftharpoons> b) \<bullet> bp \<noteq> bp} = {}"
apply auto
apply (rule_tac x="(x \<rightleftharpoons> a)" in exI)
apply auto
done
lemma supp_fv:
shows "supp t = fv_trm1 t"
apply(induct t rule: trm1_bp_inducts(1))
apply(simp_all)
apply(simp add: supp_def permute_trm1 alpha1_INJ fv_trm1)
apply(simp only: supp_at_base[simplified supp_def])
apply(simp add: supp_def permute_trm1 alpha1_INJ fv_trm1)
apply(simp add: Collect_imp_eq Collect_neg_eq)
apply(subgoal_tac "supp (Lm1 name rtrm1) = supp (Abs {atom name} rtrm1)")
apply(simp add: supp_Abs fv_trm1)
apply(simp (no_asm) add: supp_def permute_set_eq atom_eqvt)
apply(simp add: alpha1_INJ)
apply(simp add: Abs_eq_iff)
apply(simp add: alpha_gen.simps)
apply(simp add: supp_eqvt[symmetric] fv_trm1_eqvt[symmetric])
apply(subgoal_tac "supp (Lt1 bp rtrm11 rtrm12) = supp(rtrm11) \<union> supp (Abs (bv1 bp) rtrm12)")
apply(simp add: supp_Abs fv_trm1 fv_eq_bv)
apply(simp (no_asm) add: supp_def)
apply(simp add: alpha1_INJ alpha_bp_eq)
apply(simp add: Abs_eq_iff)
apply(simp add: alpha_gen)
apply(simp add: supp_eqvt[symmetric] fv_trm1_eqvt[symmetric] bv1_eqvt fv_eq_bv)
apply(simp add: Collect_imp_eq Collect_neg_eq fresh_star_def helper)
done
lemma trm1_supp:
"supp (Vr1 x) = {atom x}"
"supp (Ap1 t1 t2) = supp t1 \<union> supp t2"
"supp (Lm1 x t) = (supp t) - {atom x}"
"supp (Lt1 b t s) = supp t \<union> (supp s - bv1 b)"
by (simp_all add: supp_fv fv_trm1 fv_eq_bv)
lemma trm1_induct_strong:
assumes "\<And>name b. P b (Vr1 name)"
and "\<And>rtrm11 rtrm12 b. \<lbrakk>\<And>c. P c rtrm11; \<And>c. P c rtrm12\<rbrakk> \<Longrightarrow> P b (Ap1 rtrm11 rtrm12)"
and "\<And>name rtrm1 b. \<lbrakk>\<And>c. P c rtrm1; (atom name) \<sharp> b\<rbrakk> \<Longrightarrow> P b (Lm1 name rtrm1)"
and "\<And>bp rtrm11 rtrm12 b. \<lbrakk>\<And>c. P c rtrm11; \<And>c. P c rtrm12; bv1 bp \<sharp>* b\<rbrakk> \<Longrightarrow> P b (Lt1 bp rtrm11 rtrm12)"
shows "P a rtrma"
sorry
section {*** lets with single assignments ***}
datatype rtrm2 =
rVr2 "name"
| rAp2 "rtrm2" "rtrm2"
| rLm2 "name" "rtrm2" --"bind (name) in (rtrm2)"
| rLt2 "rassign" "rtrm2" --"bind (bv2 rassign) in (rtrm2)"
and rassign =
rAs "name" "rtrm2"
(* to be given by the user *)
primrec
rbv2
where
"rbv2 (rAs x t) = {atom x}"
setup {* snd o define_raw_perms ["rtrm2", "rassign"] ["Terms.rtrm2", "Terms.rassign"] *}
local_setup {* snd o define_fv_alpha "Terms.rtrm2"
[[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term rbv2}, 0)], [(SOME @{term rbv2}, 0)]]],
[[[], []]]] *}
print_theorems
notation
alpha_rtrm2 ("_ \<approx>2 _" [100, 100] 100) and
alpha_rassign ("_ \<approx>2b _" [100, 100] 100)
thm alpha_rtrm2_alpha_rassign.intros
lemma alpha2_equivp:
"equivp alpha_rtrm2"
"equivp alpha_rassign"
sorry
quotient_type
trm2 = rtrm2 / alpha_rtrm2
and
assign = rassign / alpha_rassign
by (auto intro: alpha2_equivp)
local_setup {*
(fn ctxt => ctxt
|> snd o (Quotient_Def.quotient_lift_const ("Vr2", @{term rVr2}))
|> snd o (Quotient_Def.quotient_lift_const ("Ap2", @{term rAp2}))
|> snd o (Quotient_Def.quotient_lift_const ("Lm2", @{term rLm2}))
|> snd o (Quotient_Def.quotient_lift_const ("Lt2", @{term rLt2}))
|> snd o (Quotient_Def.quotient_lift_const ("As", @{term rAs}))
|> snd o (Quotient_Def.quotient_lift_const ("fv_trm2", @{term fv_rtrm2}))
|> snd o (Quotient_Def.quotient_lift_const ("bv2", @{term rbv2})))
*}
print_theorems
section {*** lets with many assignments ***}
datatype trm3 =
Vr3 "name"
| Ap3 "trm3" "trm3"
| Lm3 "name" "trm3" --"bind (name) in (trm3)"
| Lt3 "assigns" "trm3" --"bind (bv3 assigns) in (trm3)"
and assigns =
ANil
| ACons "name" "trm3" "assigns"
(* to be given by the user *)
primrec
bv3
where
"bv3 ANil = {}"
| "bv3 (ACons x t as) = {atom x} \<union> (bv3 as)"
setup {* snd o define_raw_perms ["rtrm3", "assigns"] ["Terms.trm3", "Terms.assigns"] *}
local_setup {* snd o define_fv_alpha "Terms.trm3"
[[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term bv3}, 0)], [(SOME @{term bv3}, 0)]]],
[[], [[], [], []]]] *}
print_theorems
notation
alpha_trm3 ("_ \<approx>3 _" [100, 100] 100) and
alpha_assigns ("_ \<approx>3a _" [100, 100] 100)
thm alpha_trm3_alpha_assigns.intros
lemma alpha3_equivp:
"equivp alpha_trm3"
"equivp alpha_assigns"
sorry
quotient_type
qtrm3 = trm3 / alpha_trm3
and
qassigns = assigns / alpha_assigns
by (auto intro: alpha3_equivp)
section {*** lam with indirect list recursion ***}
datatype trm4 =
Vr4 "name"
| Ap4 "trm4" "trm4 list"
| Lm4 "name" "trm4" --"bind (name) in (trm)"
print_theorems
thm trm4.recs
(* there cannot be a clause for lists, as *)
(* permutations are already defined in Nominal (also functions, options, and so on) *)
setup {* snd o define_raw_perms ["trm4"] ["Terms.trm4"] *}
(* "repairing" of the permute function *)
lemma repaired:
fixes ts::"trm4 list"
shows "permute_trm4_list p ts = p \<bullet> ts"
apply(induct ts)
apply(simp_all)
done
thm permute_trm4_permute_trm4_list.simps
thm permute_trm4_permute_trm4_list.simps[simplified repaired]
local_setup {* snd o define_fv_alpha "Terms.trm4" [
[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]]], [[], [[], []]] ] *}
print_theorems
notation
alpha_trm4 ("_ \<approx>4 _" [100, 100] 100) and
alpha_trm4_list ("_ \<approx>4l _" [100, 100] 100)
thm alpha_trm4_alpha_trm4_list.intros
lemma alpha4_equivp: "equivp alpha_trm4" sorry
lemma alpha4list_equivp: "equivp alpha_trm4_list" sorry
quotient_type
qtrm4 = trm4 / alpha_trm4 and
qtrm4list = "trm4 list" / alpha_trm4_list
by (simp_all add: alpha4_equivp alpha4list_equivp)
datatype rtrm5 =
rVr5 "name"
| rAp5 "rtrm5" "rtrm5"
| rLt5 "rlts" "rtrm5" --"bind (bv5 lts) in (rtrm5)"
and rlts =
rLnil
| rLcons "name" "rtrm5" "rlts"
primrec
rbv5
where
"rbv5 rLnil = {}"
| "rbv5 (rLcons n t ltl) = {atom n} \<union> (rbv5 ltl)"
setup {* snd o define_raw_perms ["rtrm5", "rlts"] ["Terms.rtrm5", "Terms.rlts"] *}
print_theorems
local_setup {* snd o define_fv_alpha "Terms.rtrm5" [
[[[]], [[], []], [[(SOME @{term rbv5}, 0)], [(SOME @{term rbv5}, 0)]]], [[], [[], [], []]] ] *}
print_theorems
(* Alternate version with additional binding of name in rlts in rLcons *)
(*local_setup {* snd o define_fv_alpha "Terms.rtrm5" [
[[[]], [[], []], [[(SOME @{term rbv5}, 0)], [(SOME @{term rbv5}, 0)]]], [[], [[(NONE,0)], [], [(NONE,0)]]] ] *}
print_theorems*)
notation
alpha_rtrm5 ("_ \<approx>5 _" [100, 100] 100) and
alpha_rlts ("_ \<approx>l _" [100, 100] 100)
thm alpha_rtrm5_alpha_rlts.intros
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 alpha_rtrm5 fv_rtrm5 pi (rbv5 l2, t2))) \<and>
(\<exists>pi. ((rbv5 l1, l1) \<approx>gen alpha_rlts 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 -
apply (simp_all add: alpha_rtrm5_alpha_rlts.intros)
apply rule
apply (erule alpha_rtrm5.cases)
apply (simp_all add: alpha_rtrm5_alpha_rlts.intros)
apply rule
apply (erule alpha_rtrm5.cases)
apply (simp_all add: alpha_rtrm5_alpha_rlts.intros)
apply rule
apply (erule alpha_rtrm5.cases)
apply (simp_all add: alpha_rtrm5_alpha_rlts.intros)
apply rule
apply (erule alpha_rlts.cases)
apply (simp_all add: alpha_rtrm5_alpha_rlts.intros)
done
lemma alpha5_equivps:
shows "equivp alpha_rtrm5"
and "equivp alpha_rlts"
sorry
quotient_type
trm5 = rtrm5 / alpha_rtrm5
and
lts = rlts / alpha_rlts
by (auto intro: alpha5_equivps)
local_setup {*
(fn ctxt => ctxt
|> snd o (Quotient_Def.quotient_lift_const ("Vr5", @{term rVr5}))
|> snd o (Quotient_Def.quotient_lift_const ("Ap5", @{term rAp5}))
|> snd o (Quotient_Def.quotient_lift_const ("Lt5", @{term rLt5}))
|> snd o (Quotient_Def.quotient_lift_const ("Lnil", @{term rLnil}))
|> snd o (Quotient_Def.quotient_lift_const ("Lcons", @{term rLcons}))
|> snd o (Quotient_Def.quotient_lift_const ("fv_trm5", @{term fv_rtrm5}))
|> snd o (Quotient_Def.quotient_lift_const ("fv_lts", @{term fv_rlts}))
|> snd o (Quotient_Def.quotient_lift_const ("bv5", @{term rbv5})))
*}
print_theorems
lemma rbv5_eqvt:
"pi \<bullet> (rbv5 x) = rbv5 (pi \<bullet> x)"
sorry
lemma fv_rtrm5_eqvt:
"pi \<bullet> (fv_rtrm5 x) = fv_rtrm5 (pi \<bullet> x)"
sorry
lemma fv_rlts_eqvt:
"pi \<bullet> (fv_rlts x) = fv_rlts (pi \<bullet> x)"
sorry
lemma alpha5_eqvt:
"xa \<approx>5 y \<Longrightarrow> (x \<bullet> xa) \<approx>5 (x \<bullet> y)"
"xb \<approx>l ya \<Longrightarrow> (x \<bullet> xb) \<approx>l (x \<bullet> ya)"
apply(induct rule: alpha_rtrm5_alpha_rlts.inducts)
apply (simp_all add: alpha5_inj)
apply (erule exE)+
apply(unfold alpha_gen)
apply (erule conjE)+
apply (rule conjI)
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 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 fv_rtrm5_eqvt)
apply (subst permute_eqvt[symmetric])
apply (simp)
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 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 fv_rlts_eqvt)
apply (subst permute_eqvt[symmetric])
apply (simp)
done
lemma alpha5_rfv:
"(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: alpha_rtrm5_alpha_rlts.inducts)
apply(simp_all add: alpha_gen)
done
lemma bv_list_rsp:
shows "x \<approx>l y \<Longrightarrow> rbv5 x = rbv5 y"
apply(induct rule: alpha_rtrm5_alpha_rlts.inducts(2))
apply(simp_all)
done
lemma [quot_respect]:
"(alpha_rlts ===> op =) fv_rlts fv_rlts"
"(alpha_rtrm5 ===> op =) fv_rtrm5 fv_rtrm5"
"(alpha_rlts ===> op =) rbv5 rbv5"
"(op = ===> alpha_rtrm5) rVr5 rVr5"
"(alpha_rtrm5 ===> alpha_rtrm5 ===> alpha_rtrm5) rAp5 rAp5"
"(alpha_rlts ===> alpha_rtrm5 ===> alpha_rtrm5) rLt5 rLt5"
"(alpha_rlts ===> alpha_rtrm5 ===> alpha_rtrm5) rLt5 rLt5"
"(op = ===> alpha_rtrm5 ===> alpha_rlts ===> alpha_rlts) rLcons rLcons"
"(op = ===> alpha_rtrm5 ===> alpha_rtrm5) permute permute"
"(op = ===> alpha_rlts ===> alpha_rlts) permute permute"
apply (simp_all add: alpha5_inj alpha5_rfv alpha5_eqvt bv_list_rsp)
apply (clarify) apply (rule conjI)
apply (rule_tac x="0" in exI) apply (simp add: fresh_star_def fresh_zero_perm alpha_gen alpha5_rfv)
apply (rule_tac x="0" in exI) apply (simp add: fresh_star_def fresh_zero_perm alpha_gen alpha5_rfv)
apply (clarify) apply (rule conjI)
apply (rule_tac x="0" in exI) apply (simp add: fresh_star_def fresh_zero_perm alpha_gen alpha5_rfv)
apply (rule_tac x="0" in exI) apply (simp add: fresh_star_def fresh_zero_perm alpha_gen alpha5_rfv)
done
lemma
shows "(alpha_rlts ===> op =) rbv5 rbv5"
by (simp add: bv_list_rsp)
lemmas trm5_lts_inducts = rtrm5_rlts.inducts[quot_lifted]
instantiation trm5 and lts :: pt
begin
quotient_definition
"permute_trm5 :: perm \<Rightarrow> trm5 \<Rightarrow> trm5"
is
"permute :: perm \<Rightarrow> rtrm5 \<Rightarrow> rtrm5"
quotient_definition
"permute_lts :: perm \<Rightarrow> lts \<Rightarrow> lts"
is
"permute :: perm \<Rightarrow> rlts \<Rightarrow> rlts"
lemma trm5_lts_zero:
"0 \<bullet> (x\<Colon>trm5) = x"
"0 \<bullet> (y\<Colon>lts) = y"
apply(induct x and y rule: trm5_lts_inducts)
apply(simp_all add: permute_rtrm5_permute_rlts.simps[quot_lifted])
done
lemma trm5_lts_plus:
"(p + q) \<bullet> (x\<Colon>trm5) = p \<bullet> q \<bullet> x"
"(p + q) \<bullet> (y\<Colon>lts) = p \<bullet> q \<bullet> y"
apply(induct x and y rule: trm5_lts_inducts)
apply(simp_all add: permute_rtrm5_permute_rlts.simps[quot_lifted])
done
instance
apply default
apply (simp_all add: trm5_lts_zero trm5_lts_plus)
done
end
lemmas permute_trm5_lts = permute_rtrm5_permute_rlts.simps[quot_lifted]
lemmas alpha5_INJ = alpha5_inj[unfolded alpha_gen, quot_lifted, folded alpha_gen]
lemmas bv5[simp] = rbv5.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))"
apply (subst alpha5_INJ)
apply (rule conjI)
apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
apply (simp only: alpha_gen)
apply (simp add: permute_trm5_lts fresh_star_def)
apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
apply (simp only: alpha_gen)
apply (simp add: permute_trm5_lts fresh_star_def)
done
lemma lets_ok2:
"(Lt5 (Lcons x (Vr5 x) (Lcons y (Vr5 y) Lnil)) (Ap5 (Vr5 x) (Vr5 y))) =
(Lt5 (Lcons y (Vr5 y) (Lcons x (Vr5 x) Lnil)) (Ap5 (Vr5 x) (Vr5 y)))"
apply (subst alpha5_INJ)
apply (rule conjI)
apply (rule_tac x="0 :: perm" in exI)
apply (simp only: alpha_gen)
apply (simp add: permute_trm5_lts fresh_star_def)
apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
apply (simp only: alpha_gen)
apply (simp add: permute_trm5_lts fresh_star_def)
done
lemma lets_not_ok1:
"x \<noteq> y \<Longrightarrow> (Lt5 (Lcons x (Vr5 x) (Lcons y (Vr5 y) Lnil)) (Ap5 (Vr5 x) (Vr5 y))) \<noteq>
(Lt5 (Lcons y (Vr5 x) (Lcons x (Vr5 y) Lnil)) (Ap5 (Vr5 x) (Vr5 y)))"
apply (subst alpha5_INJ(3))
apply(clarify)
apply (simp add: alpha_gen)
apply (simp add: permute_trm5_lts fresh_star_def)
apply (simp add: alpha5_INJ(5))
apply(clarify)
apply (simp add: alpha5_INJ(2))
apply (simp only: alpha5_INJ(1))
done
lemma distinct_helper:
shows "\<not>(rVr5 x \<approx>5 rAp5 y z)"
apply auto
apply (erule alpha_rtrm5.cases)
apply (simp_all only: rtrm5.distinct)
done
lemma distinct_helper2:
shows "(Vr5 x) \<noteq> (Ap5 y z)"
by (lifting distinct_helper)
lemma lets_nok:
"x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow>
(Lt5 (Lcons x (Ap5 (Vr5 z) (Vr5 z)) (Lcons y (Vr5 z) Lnil)) (Ap5 (Vr5 x) (Vr5 y))) \<noteq>
(Lt5 (Lcons y (Vr5 z) (Lcons x (Ap5 (Vr5 z) (Vr5 z)) Lnil)) (Ap5 (Vr5 x) (Vr5 y)))"
apply (subst alpha5_INJ)
apply (simp only: alpha_gen permute_trm5_lts fresh_star_def)
apply (subst alpha5_INJ(5))
apply (subst alpha5_INJ(5))
apply (simp add: distinct_helper2)
done
(* example with a bn function defined over the type itself *)
datatype rtrm6 =
rVr6 "name"
| rLm6 "name" "rtrm6"
| rLt6 "rtrm6" "rtrm6" --"bind (bv6 left) in (right)"
primrec
rbv6
where
"rbv6 (rVr6 n) = {}"
| "rbv6 (rLm6 n t) = {atom n} \<union> rbv6 t"
| "rbv6 (rLt6 l r) = rbv6 l \<union> rbv6 r"
setup {* snd o define_raw_perms ["rtrm6"] ["Terms.rtrm6"] *}
print_theorems
local_setup {* snd o define_fv_alpha "Terms.rtrm6" [
[[[]], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term rbv6}, 0)], [(SOME @{term rbv6}, 0)]]]] *}
notation alpha_rtrm6 ("_ \<approx>6a _" [100, 100] 100)
(* HERE THE RULES DIFFER *)
thm alpha_rtrm6.intros
inductive
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 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"
sorry
quotient_type
trm6 = rtrm6 / alpha6
by (auto intro: alpha6_equivps)
local_setup {*
(fn ctxt => ctxt
|> snd o (Quotient_Def.quotient_lift_const ("Vr6", @{term rVr6}))
|> snd o (Quotient_Def.quotient_lift_const ("Lm6", @{term rLm6}))
|> snd o (Quotient_Def.quotient_lift_const ("Lt6", @{term rLt6}))
|> snd o (Quotient_Def.quotient_lift_const ("fv_trm6", @{term fv_rtrm6}))
|> snd o (Quotient_Def.quotient_lift_const ("bv6", @{term rbv6})))
*}
print_theorems
lemma [quot_respect]:
"(op = ===> alpha6 ===> alpha6) permute permute"
apply auto (* will work with eqvt *)
sorry
(* Definitely not true , see lemma below *)
lemma [quot_respect]:"(alpha6 ===> op =) rbv6 rbv6"
apply simp apply clarify
apply (erule alpha6.induct)
oops
lemma "(a :: name) \<noteq> b \<Longrightarrow> \<not> (alpha6 ===> op =) rbv6 rbv6"
apply simp
apply (rule_tac x="rLm6 (a::name) (rVr6 (a :: name))" in exI)
apply (rule_tac x="rLm6 (b::name) (rVr6 (b :: name))" in exI)
apply simp
apply (rule a2)
apply (rule_tac x="(a \<leftrightarrow> b)" in exI)
apply (simp add: alpha_gen fresh_star_def)
apply (rule a1)
apply (rule refl)
done
lemma [quot_respect]:"(alpha6 ===> op =) fv_rtrm6 fv_rtrm6"
apply simp apply clarify
apply (induct_tac x y rule: alpha6.induct)
apply simp_all
apply (erule exE)
apply (simp_all add: alpha_gen)
apply (erule conjE)+
apply (erule exE)
apply (erule conjE)+
apply (simp)
oops
lemma [quot_respect]: "(op = ===> alpha6) rVr6 rVr6"
by (simp_all add: a1)
lemma [quot_respect]:
"(op = ===> alpha6 ===> alpha6) rLm6 rLm6"
"(alpha6 ===> alpha6 ===> alpha6) rLt6 rLt6"
apply simp_all apply (clarify)
apply (rule a2)
apply (rule_tac x="0::perm" in exI)
apply (simp add: alpha_gen)
(* needs rfv6_rsp *) defer
apply clarify
apply (rule a3)
apply (rule_tac x="0::perm" in exI)
apply (simp add: alpha_gen)
(* needs rbv6_rsp *)
oops
instantiation trm6 :: pt begin
quotient_definition
"permute_trm6 :: perm \<Rightarrow> trm6 \<Rightarrow> trm6"
is
"permute :: perm \<Rightarrow> rtrm6 \<Rightarrow> rtrm6"
instance
apply default
sorry
end
lemma lifted_induct:
"\<lbrakk>x1 = x2; \<And>a b. a = b \<Longrightarrow> P (Vr6 a) (Vr6 b);
\<And>a t b s.
\<exists>pi. fv_trm6 t - {atom a} = fv_trm6 s - {atom b} \<and>
(fv_trm6 t - {atom a}) \<sharp>* pi \<and> pi \<bullet> t = s \<and> P (pi \<bullet> t) s \<Longrightarrow>
P (Lm6 a t) (Lm6 b s);
\<And>t1 s1 t2 s2.
\<exists>pi. fv_trm6 s1 - bv6 t1 = fv_trm6 s2 - bv6 t2 \<and>
(fv_trm6 s1 - bv6 t1) \<sharp>* pi \<and> pi \<bullet> s1 = s2 \<and> P (pi \<bullet> s1) s2 \<Longrightarrow>
P (Lt6 t1 s1) (Lt6 t2 s2)\<rbrakk>
\<Longrightarrow> P x1 x2"
unfolding alpha_gen
apply (lifting alpha6.induct[unfolded alpha_gen])
apply injection
(* notice unsolvable goals: (alpha6 ===> op =) rbv6 rbv6 *)
oops
lemma lifted_inject_a3:
"\<exists>pi. fv_trm6 s1 - bv6 t1 = fv_trm6 s2 - bv6 t2 \<and>
(fv_trm6 s1 - bv6 t1) \<sharp>* pi \<and> pi \<bullet> s1 = s2 \<Longrightarrow> Lt6 t1 s1 = Lt6 t2 s2"
apply(lifting a3[unfolded alpha_gen])
apply injection
(* notice unsolvable goals: (alpha6 ===> op =) rbv6 rbv6 *)
oops
(* example with a respectful bn function defined over the type itself *)
datatype rtrm7 =
rVr7 "name"
| rLm7 "name" "rtrm7"
| rLt7 "rtrm7" "rtrm7" --"bind (bv7 left) in (right)"
primrec
rbv7
where
"rbv7 (rVr7 n) = {atom n}"
| "rbv7 (rLm7 n t) = rbv7 t - {atom n}"
| "rbv7 (rLt7 l r) = rbv7 l \<union> rbv7 r"
setup {* snd o define_raw_perms ["rtrm7"] ["Terms.rtrm7"] *}
print_theorems
local_setup {* snd o define_fv_alpha "Terms.rtrm7" [
[[[]], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term rbv7}, 0)], [(SOME @{term rbv7}, 0)]]]] *}
notation
alpha_rtrm7 ("_ \<approx>7a _" [100, 100] 100)
(* HERE THE RULES DIFFER *)
thm alpha_rtrm7.intros
inductive
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 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 "(x::name) \<noteq> y \<Longrightarrow> \<not> (alpha7 ===> op =) rbv7 rbv7"
apply simp
apply (rule_tac x="rLt7 (rVr7 x) (rVr7 x)" in exI)
apply (rule_tac x="rLt7 (rVr7 y) (rVr7 y)" in exI)
apply simp
apply (rule a3)
apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
apply (simp_all add: alpha_gen fresh_star_def)
apply (rule a1)
apply (rule refl)
done
datatype rfoo8 =
Foo0 "name"
| Foo1 "rbar8" "rfoo8" --"bind bv(bar) in foo"
and rbar8 =
Bar0 "name"
| Bar1 "name" "name" "rbar8" --"bind second name in b"
primrec
rbv8
where
"rbv8 (Bar0 x) = {}"
| "rbv8 (Bar1 v x b) = {atom v}"
setup {* snd o define_raw_perms ["rfoo8", "rbar8"] ["Terms.rfoo8", "Terms.rbar8"] *}
print_theorems
local_setup {* snd o define_fv_alpha "Terms.rfoo8" [
[[[]], [[(SOME @{term rbv8}, 0)], [(SOME @{term rbv8}, 0)]]], [[[]], [[], [(NONE, 1)], [(NONE, 1)]]]] *}
notation
alpha_rfoo8 ("_ \<approx>f' _" [100, 100] 100) and
alpha_rbar8 ("_ \<approx>b' _" [100, 100] 100)
(* HERE THE RULE DIFFERS *)
thm alpha_rfoo8_alpha_rbar8.intros
inductive
alpha8f :: "rfoo8 \<Rightarrow> rfoo8 \<Rightarrow> bool" ("_ \<approx>f _" [100, 100] 100)
and
alpha8b :: "rbar8 \<Rightarrow> rbar8 \<Rightarrow> bool" ("_ \<approx>b _" [100, 100] 100)
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 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
apply (erule alpha8f_alpha8b.inducts(2))
apply (simp_all)
done
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 =) fv_rbar8 fv_rbar8"
apply simp apply clarify apply (simp add: fv_rbar8_rsp_hlp)
done
lemma "(alpha8f ===> op =) fv_rfoo8 fv_rfoo8"
apply simp apply clarify
apply (erule alpha8f_alpha8b.inducts(1))
apply (simp_all add: alpha_gen fv_rbar8_rsp_hlp)
apply clarify
apply (erule alpha8f_alpha8b.inducts(2))
apply (simp_all)
done
datatype rlam9 =
Var9 "name"
| Lam9 "name" "rlam9" --"bind name in rlam"
and rbla9 =
Bla9 "rlam9" "rlam9" --"bind bv(first) in second"
primrec
rbv9
where
"rbv9 (Var9 x) = {}"
| "rbv9 (Lam9 x b) = {atom x}"
setup {* snd o define_raw_perms ["rlam9", "rbla9"] ["Terms.rlam9", "Terms.rbla9"] *}
print_theorems
local_setup {* snd o define_fv_alpha "Terms.rlam9" [
[[[]], [[(NONE, 0)], [(NONE, 0)]]], [[[(SOME @{term rbv9}, 0)], [(SOME @{term rbv9}, 0)]]]] *}
notation
alpha_rlam9 ("_ \<approx>9l' _" [100, 100] 100) and
alpha_rbla9 ("_ \<approx>9b' _" [100, 100] 100)
(* HERE THE RULES DIFFER *)
thm alpha_rlam9_alpha_rbla9.intros
inductive
alpha9l :: "rlam9 \<Rightarrow> rlam9 \<Rightarrow> bool" ("_ \<approx>9l _" [100, 100] 100)
and
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 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
sorry
local_setup {*
(fn ctxt => ctxt
|> snd o (Quotient_Def.quotient_lift_const ("qVar9", @{term Var9}))
|> snd o (Quotient_Def.quotient_lift_const ("qLam9", @{term Lam9}))
|> snd o (Quotient_Def.quotient_lift_const ("qBla9", @{term Bla9}))
|> snd o (Quotient_Def.quotient_lift_const ("fv_lam9", @{term fv_rlam9}))
|> snd o (Quotient_Def.quotient_lift_const ("fv_bla9", @{term fv_rbla9}))
|> snd o (Quotient_Def.quotient_lift_const ("bv9", @{term rbv9})))
*}
print_theorems
instantiation lam9 and bla9 :: pt
begin
quotient_definition
"permute_lam9 :: perm \<Rightarrow> lam9 \<Rightarrow> lam9"
is
"permute :: perm \<Rightarrow> rlam9 \<Rightarrow> rlam9"
quotient_definition
"permute_bla9 :: perm \<Rightarrow> bla9 \<Rightarrow> bla9"
is
"permute :: perm \<Rightarrow> rbla9 \<Rightarrow> rbla9"
instance
sorry
end
lemma "\<lbrakk>b1 = b2; \<exists>pi. fv_lam9 t1 - bv9 b1 = fv_lam9 t2 - bv9 b2 \<and> (fv_lam9 t1 - bv9 b1) \<sharp>* pi \<and> pi \<bullet> t1 = t2\<rbrakk>
\<Longrightarrow> qBla9 b1 t1 = qBla9 b2 t2"
apply (lifting a3[unfolded alpha_gen])
apply injection
sorry
text {* type schemes *}
datatype ty =
Var "name"
| Fun "ty" "ty"
setup {* snd o define_raw_perms ["ty"] ["Terms.ty"] *}
print_theorems
datatype tyS =
All "name set" "ty"
setup {* snd o define_raw_perms ["tyS"] ["Terms.tyS"] *}
print_theorems
local_setup {* define_raw_fv "Terms.ty" [[[[]], [[], []]]] *}
print_theorems
(*
Doesnot work yet since we do not refer to fv_ty
local_setup {* define_raw_fv "Terms.tyS" [[[[], []]]] *}
print_theorems
*)
primrec
fv_tyS
where
"fv_tyS (All xs T) = (fv_ty T - atom ` xs)"
inductive
alpha_tyS :: "tyS \<Rightarrow> tyS \<Rightarrow> bool" ("_ \<approx>tyS _" [100, 100] 100)
where
a1: "\<exists>pi. ((atom ` xs1, T1) \<approx>gen (op =) fv_ty pi (atom ` xs2, T2))
\<Longrightarrow> All xs1 T1 \<approx>tyS All xs2 T2"
lemma
shows "All {a, b} (Fun (Var a) (Var b)) \<approx>tyS All {b, a} (Fun (Var a) (Var b))"
apply(rule a1)
apply(simp add: alpha_gen)
apply(rule_tac x="0::perm" in exI)
apply(simp add: fresh_star_def)
done
lemma
shows "All {a, b} (Fun (Var a) (Var b)) \<approx>tyS All {a, b} (Fun (Var b) (Var a))"
apply(rule a1)
apply(simp add: alpha_gen)
apply(rule_tac x="(atom a \<rightleftharpoons> atom b)" in exI)
apply(simp add: fresh_star_def)
done
lemma
shows "All {a, b, c} (Fun (Var a) (Var b)) \<approx>tyS All {a, b} (Fun (Var a) (Var b))"
apply(rule a1)
apply(simp add: alpha_gen)
apply(rule_tac x="0::perm" in exI)
apply(simp add: fresh_star_def)
done
lemma
assumes a: "a \<noteq> b"
shows "\<not>(All {a, b} (Fun (Var a) (Var b)) \<approx>tyS All {c} (Fun (Var c) (Var c)))"
using a
apply(clarify)
apply(erule alpha_tyS.cases)
apply(simp add: alpha_gen)
apply(erule conjE)+
apply(erule exE)
apply(erule conjE)+
apply(clarify)
apply(simp)
apply(simp add: fresh_star_def)
apply(auto)
done
end