theory Terms
imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv" "../../Attic/Prove"
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"

print_theorems

(* 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"] *}
thm permute_rtrm1_permute_bp.simps

local_setup {* 
  snd o define_fv_alpha "Terms.rtrm1"
  [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term bv1}, 0)], [], [(SOME @{term bv1}, 0)]]],
  [[], [[]], [[], []]]] *}

notation
  alpha_rtrm1 ("_ \<approx>1 _" [100, 100] 100) and
  alpha_bp ("_ \<approx>1b _" [100, 100] 100)
thm alpha_rtrm1_alpha_bp.intros
thm fv_rtrm1_fv_bp.simps

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha1_inj}, []), (build_alpha_inj @{thms alpha_rtrm1_alpha_bp.intros} @{thms rtrm1.distinct rtrm1.inject bp.distinct bp.inject} @{thms alpha_rtrm1.cases alpha_bp.cases} ctxt)) ctxt)) *}
thm alpha1_inj

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: 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

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha1_equivp}, []),
  (build_equivps [@{term alpha_rtrm1}, @{term alpha_bp}] @{thm rtrm1_bp.induct} @{thm alpha_rtrm1_alpha_bp.induct} @{thms rtrm1.inject bp.inject} @{thms alpha1_inj} @{thms rtrm1.distinct bp.distinct} @{thms alpha_rtrm1.cases alpha_bp.cases} @{thms alpha1_eqvt} ctxt)) ctxt)) *}
thm alpha1_equivp

ML {* 
fun define_quotient_type args tac ctxt =
let
  val mthd = Method.SIMPLE_METHOD tac
  val mthdt = Method.Basic (fn _ => mthd)
  val bymt = Proof.global_terminal_proof (mthdt, NONE)
in
  bymt (Quotient_Type.quotient_type args ctxt)
end
*}

local_setup  {* define_quotient_type [(([], @{binding trm1}, NoSyn), (@{typ rtrm1}, @{term alpha_rtrm1}))] 
  (rtac @{thm alpha1_equivp(1)} 1)
*}

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

prove fv_rtrm1_rsp': {*
 (@{term Trueprop} $ (Quotient_Term.equiv_relation_chk @{context} (fastype_of @{term fv_rtrm1}, fastype_of @{term fv_trm1}) $ @{term fv_rtrm1} $ @{term fv_rtrm1})) *}
by (tactic {*
  (rtac @{thm fun_rel_id} THEN'
  eresolve_tac @{thms alpha_rtrm1_alpha_bp.inducts} THEN_ALL_NEW
  asm_full_simp_tac (HOL_ss addsimps @{thms alpha_gen fv_rtrm1_fv_bp.simps})) 1 *})


lemmas fv_rtrm1_rsp = apply_rsp'[OF fv_rtrm1_rsp']

(* We need this since 'prove' doesn't support attributes *)
lemma [quot_respect]: "(alpha_rtrm1 ===> op =) fv_rtrm1 fv_rtrm1"
  by (rule fv_rtrm1_rsp')

ML {*
fun contr_rsp_tac inj rsp equivps =
let
  val reflps = map (fn x => @{thm equivp_reflp} OF [x]) equivps
in
  REPEAT o rtac @{thm fun_rel_id} THEN'
  simp_tac (HOL_ss addsimps inj) THEN'
  (TRY o REPEAT_ALL_NEW (CHANGED o rtac conjI)) THEN_ALL_NEW
  (asm_simp_tac HOL_ss THEN_ALL_NEW (
   rtac @{thm exI[of _ "0 :: perm"]} THEN'
   asm_full_simp_tac (HOL_ss addsimps (rsp @ reflps @
     @{thms alpha_gen fresh_star_def fresh_zero_perm permute_zero ball_triv}))
  ))
end
*}

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 (tactic {* contr_rsp_tac @{thms alpha1_inj} @{thms fv_rtrm1_rsp} @{thms alpha1_equivp} 1 *})+
done


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"

lemma [quot_respect]:
  "(op = ===> alpha_rtrm1 ===> alpha_rtrm1) permute permute"
  by (simp add: alpha1_eqvt)

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(simp add: supp_at_base supp_def[symmetric])
  apply(simp add: Collect_imp_eq Collect_neg_eq[symmetric] supp_def)
  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 helper2: "{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:
  "supp t = fv_trm1 t"
  "supp b = fv_bp b"
apply(induct t and b rule: trm1_bp_inducts)
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 helper2)
apply(simp (no_asm) add: supp_def permute_set_eq atom_eqvt)
apply(simp (no_asm) add: supp_def eqvts)
apply(fold supp_def)
apply(simp add: supp_at_base)
apply(simp (no_asm) add: supp_def Collect_imp_eq Collect_neg_eq)
apply(simp add: Collect_imp_eq[symmetric] Collect_neg_eq[symmetric] supp_def[symmetric])
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)]]],
   [[[], []]]] *}

notation
  alpha_rtrm2 ("_ \<approx>2 _" [100, 100] 100) and
  alpha_rassign ("_ \<approx>2b _" [100, 100] 100)
thm alpha_rtrm2_alpha_rassign.intros

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha2_inj}, []), (build_alpha_inj @{thms alpha_rtrm2_alpha_rassign.intros} @{thms rtrm2.distinct rtrm2.inject rassign.distinct rassign.inject} @{thms alpha_rtrm2.cases alpha_rassign.cases} ctxt)) ctxt)) *}
thm alpha2_inj

lemma alpha2_eqvt:
  "t \<approx>2 s \<Longrightarrow> (pi \<bullet> t) \<approx>2 (pi \<bullet> s)"
  "a \<approx>2b b \<Longrightarrow> (pi \<bullet> a) \<approx>2b (pi \<bullet> b)"
sorry

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha2_equivp}, []),
  (build_equivps [@{term alpha_rtrm2}, @{term alpha_rassign}] @{thm rtrm2_rassign.induct} @{thm alpha_rtrm2_alpha_rassign.induct} @{thms rtrm2.inject rassign.inject} @{thms alpha2_inj} @{thms rtrm2.distinct rassign.distinct} @{thms alpha_rtrm2.cases alpha_rassign.cases} @{thms alpha2_eqvt} ctxt)) ctxt)) *}
thm alpha2_equivp


quotient_type
  trm2 = rtrm2 / alpha_rtrm2
and
  assign = rassign / alpha_rassign
  by (rule alpha2_equivp(1)) (rule alpha2_equivp(2))

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 rtrm3 =
  rVr3 "name"
| rAp3 "rtrm3" "rtrm3"
| rLm3 "name" "rtrm3" --"bind (name) in (trm3)"
| rLt3 "rassigns" "rtrm3" --"bind (bv3 assigns) in (trm3)"
and rassigns =
  rANil
| rACons "name" "rtrm3" "rassigns"

(* to be given by the user *)
primrec 
  bv3
where
  "bv3 rANil = {}"
| "bv3 (rACons x t as) = {atom x} \<union> (bv3 as)"

setup {* snd o define_raw_perms ["rtrm3", "rassigns"] ["Terms.rtrm3", "Terms.rassigns"] *}

local_setup {* snd o define_fv_alpha "Terms.rtrm3"
  [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[], [(SOME @{term bv3}, 0)]]],
   [[], [[], [], []]]] *}
print_theorems

notation
  alpha_rtrm3 ("_ \<approx>3 _" [100, 100] 100) and
  alpha_rassigns ("_ \<approx>3a _" [100, 100] 100)
thm alpha_rtrm3_alpha_rassigns.intros

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha3_inj}, []), (build_alpha_inj @{thms alpha_rtrm3_alpha_rassigns.intros} @{thms rtrm3.distinct rtrm3.inject rassigns.distinct rassigns.inject} @{thms alpha_rtrm3.cases alpha_rassigns.cases} ctxt)) ctxt)) *}
thm alpha3_inj

lemma alpha3_eqvt:
  "t \<approx>3 s \<Longrightarrow> (pi \<bullet> t) \<approx>3 (pi \<bullet> s)"
  "a \<approx>3a b \<Longrightarrow> (pi \<bullet> a) \<approx>3a (pi \<bullet> b)"
sorry

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha3_equivp}, []),
  (build_equivps [@{term alpha_rtrm3}, @{term alpha_rassigns}] @{thm rtrm3_rassigns.induct} @{thm alpha_rtrm3_alpha_rassigns.induct} @{thms rtrm3.inject rassigns.inject} @{thms alpha3_inj} @{thms rtrm3.distinct rassigns.distinct} @{thms alpha_rtrm3.cases alpha_rassigns.cases} @{thms alpha3_eqvt} ctxt)) ctxt)) *}
thm alpha3_equivp

quotient_type
  trm3 = rtrm3 / alpha_rtrm3
and
  assigns = rassigns / alpha_rassigns
  by (rule alpha3_equivp(1)) (rule alpha3_equivp(2))


section {*** lam with indirect list recursion ***}

datatype rtrm4 =
  rVr4 "name"
| rAp4 "rtrm4" "rtrm4 list"
| rLm4 "name" "rtrm4"  --"bind (name) in (trm)"
print_theorems

thm rtrm4.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 ["rtrm4"] ["Terms.rtrm4"] *}

(* "repairing" of the permute function *)
lemma repaired:
  fixes ts::"rtrm4 list"
  shows "permute_rtrm4_list p ts = p \<bullet> ts"
  apply(induct ts)
  apply(simp_all)
  done

thm permute_rtrm4_permute_rtrm4_list.simps
thm permute_rtrm4_permute_rtrm4_list.simps[simplified repaired]

local_setup {* snd o define_fv_alpha "Terms.rtrm4" [
  [[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]]], [[], [[], []]]  ] *}
print_theorems

notation
  alpha_rtrm4 ("_ \<approx>4 _" [100, 100] 100) and
  alpha_rtrm4_list ("_ \<approx>4l _" [100, 100] 100)
thm alpha_rtrm4_alpha_rtrm4_list.intros

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha4_inj}, []), (build_alpha_inj @{thms alpha_rtrm4_alpha_rtrm4_list.intros} @{thms rtrm4.distinct rtrm4.inject list.distinct list.inject} @{thms alpha_rtrm4.cases alpha_rtrm4_list.cases} ctxt)) ctxt)) *}
thm alpha4_inj

lemma alpha4_eqvt:
  "t \<approx>4 s \<Longrightarrow> (pi \<bullet> t) \<approx>4 (pi \<bullet> s)"
  "a \<approx>4l b \<Longrightarrow> (pi \<bullet> a) \<approx>4l (pi \<bullet> b)"
sorry

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha4_equivp}, []),
  (build_equivps [@{term alpha_rtrm4}, @{term alpha_rtrm4_list}] @{thm rtrm4.induct} @{thm alpha_rtrm4_alpha_rtrm4_list.induct} @{thms rtrm4.inject list.inject} @{thms alpha4_inj} @{thms rtrm4.distinct list.distinct} @{thms alpha_rtrm4_list.cases alpha_rtrm4.cases} @{thms alpha4_eqvt} ctxt)) ctxt)) *}
thm alpha4_equivp

quotient_type 
  qrtrm4 = rtrm4 / alpha_rtrm4 and
  qrtrm4list = "rtrm4 list" / alpha_rtrm4_list
  by (simp_all add: alpha4_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

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha5_inj}, []), (build_alpha_inj @{thms alpha_rtrm5_alpha_rlts.intros} @{thms rtrm5.distinct rtrm5.inject rlts.distinct rlts.inject} @{thms alpha_rtrm5.cases alpha_rlts.cases} ctxt)) ctxt)) *}
thm alpha5_inj

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> 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)
  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)
  done

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha5_equivp}, []),
  (build_equivps [@{term alpha_rtrm5}, @{term alpha_rlts}] @{thm rtrm5_rlts.induct} @{thm alpha_rtrm5_alpha_rlts.induct} @{thms rtrm5.inject rlts.inject} @{thms alpha5_inj} @{thms rtrm5.distinct rlts.distinct} @{thms alpha_rtrm5.cases alpha_rlts.cases} @{thms alpha5_eqvt} ctxt)) ctxt)) *}
thm alpha5_equivp

quotient_type
  trm5 = rtrm5 / alpha_rtrm5
and
  lts = rlts / alpha_rlts
  by (auto intro: alpha5_equivp)

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 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"
  "(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)
  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]
and
  alpha5_INJ = alpha5_inj[unfolded alpha_gen, quot_lifted, folded alpha_gen]
and
  bv5[simp] = rbv5.simps[quot_lifted]
and
  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="(x \<leftrightarrow> y)" in exI)
apply (simp only: alpha_gen)
apply (simp add: permute_trm5_lts fresh_star_def)
apply (rule_tac x="0 :: perm" 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 (simp add: alpha5_INJ(3) alpha_gen)
apply (simp add: permute_trm5_lts fresh_star_def alpha5_INJ(5) alpha5_INJ(2) 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 (simp only: alpha5_INJ(3) alpha5_INJ(5) alpha_gen permute_trm5_lts fresh_star_def)
apply (simp add: distinct_helper2)
done


(* example with a bn function defined over the type itself *)
datatype rtrm6 =
  rVr6 "name"
| rLm6 "name" "rtrm6" --"bind name in 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)]]]] *}
notation alpha_rtrm6 ("_ \<approx>6 _" [100, 100] 100)
(* HERE THE RULES DIFFER *)
thm alpha_rtrm6.intros

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha6_inj}, []), (build_alpha_inj @{thms alpha_rtrm6.intros} @{thms rtrm6.distinct rtrm6.inject} @{thms alpha_rtrm6.cases} ctxt)) ctxt)) *}
thm alpha6_inj

lemma alpha6_eqvt:
  "xa \<approx>6 y \<Longrightarrow> (x \<bullet> xa) \<approx>6 (x \<bullet> y)"
sorry

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha6_equivp}, []),
  (build_equivps [@{term alpha_rtrm6}] @{thm rtrm6.induct} @{thm alpha_rtrm6.induct} @{thms rtrm6.inject} @{thms alpha6_inj} @{thms rtrm6.distinct} @{thms alpha_rtrm6.cases} @{thms alpha6_eqvt} ctxt)) ctxt)) *}
thm alpha6_equivp

quotient_type
  trm6 = rtrm6 / alpha_rtrm6
  by (auto intro: alpha6_equivp)

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 = ===> alpha_rtrm6 ===> alpha_rtrm6) permute permute"
by (auto simp add: alpha6_eqvt)

(* Definitely not true , see lemma below *)
lemma [quot_respect]:"(alpha_rtrm6 ===> op =) rbv6 rbv6"
apply simp apply clarify
apply (erule alpha_rtrm6.induct)
oops

lemma "(a :: name) \<noteq> b \<Longrightarrow> \<not> (alpha_rtrm6 ===> 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 (simp add: alpha6_inj)
apply (rule_tac x="(a \<leftrightarrow> b)" in  exI)
apply (simp add: alpha_gen fresh_star_def)
apply (simp add: alpha6_inj)
done

lemma fv6_rsp: "x \<approx>6 y \<Longrightarrow> fv_rtrm6 x = fv_rtrm6 y"
apply (induct_tac x y rule: alpha_rtrm6.induct)
apply simp_all
apply (erule exE)
apply (simp_all add: alpha_gen)
done

lemma [quot_respect]:"(alpha_rtrm6 ===> op =) fv_rtrm6 fv_rtrm6"
by (simp add: fv6_rsp)

lemma [quot_respect]:
 "(op = ===> alpha_rtrm6) rVr6 rVr6"
 "(op = ===> alpha_rtrm6 ===> alpha_rtrm6) rLm6 rLm6"
apply auto
apply (simp_all add: alpha6_inj)
apply (rule_tac x="0::perm" in exI)
apply (simp add: alpha_gen fv6_rsp fresh_star_def fresh_zero_perm)
done

lemma [quot_respect]:
 "(alpha_rtrm6 ===> alpha_rtrm6 ===> alpha_rtrm6) rLt6 rLt6"
apply auto
apply (simp_all add: alpha6_inj)
apply (rule_tac [!] x="0::perm" in exI)
apply (simp_all add: alpha_gen fresh_star_def fresh_zero_perm)
(* 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>name namea. name = namea \<Longrightarrow> P (Vr6 name) (Vr6 namea);
 \<And>name rtrm6 namea rtrm6a.
    \<lbrakk>True;
     \<exists>pi. fv_trm6 rtrm6 - {atom name} = fv_trm6 rtrm6a - {atom namea} \<and>
          (fv_trm6 rtrm6 - {atom name}) \<sharp>* pi \<and> pi \<bullet> rtrm6 = rtrm6a \<and> P (pi \<bullet> rtrm6) rtrm6a\<rbrakk>
    \<Longrightarrow> P (Lm6 name rtrm6) (Lm6 namea rtrm6a);
 \<And>rtrm61 rtrm61a rtrm62 rtrm62a.
    \<lbrakk>rtrm61 = rtrm61a; P rtrm61 rtrm61a;
     \<exists>pi. fv_trm6 rtrm62 - bv6 rtrm61 = fv_trm6 rtrm62a - bv6 rtrm61a \<and>
          (fv_trm6 rtrm62 - bv6 rtrm61) \<sharp>* pi \<and> pi \<bullet> rtrm62 = rtrm62a \<and> P (pi \<bullet> rtrm62) rtrm62a\<rbrakk>
    \<Longrightarrow> P (Lt6 rtrm61 rtrm62) (Lt6 rtrm61a rtrm62a)\<rbrakk>
\<Longrightarrow> P x1 x2"
apply (lifting alpha_rtrm6.induct[unfolded alpha_gen])
apply injection
(* notice unsolvable goals: (alpha_rtrm6 ===> op =) rbv6 rbv6 *)
oops

lemma lifted_inject_a3:
"(Lt6 rtrm61 rtrm62 = Lt6 rtrm61a rtrm62a) =
(rtrm61 = rtrm61a \<and>
 (\<exists>pi. fv_trm6 rtrm62 - bv6 rtrm61 = fv_trm6 rtrm62a - bv6 rtrm61a \<and>
       (fv_trm6 rtrm62 - bv6 rtrm61) \<sharp>* pi \<and> pi \<bullet> rtrm62 = rtrm62a))"
apply(lifting alpha6_inj(3)[unfolded alpha_gen])
apply injection
(* notice unsolvable goals: (alpha_rtrm6 ===> op =) rbv6 rbv6 *)
oops




(* example with a respectful bn function defined over the type itself *)

datatype rtrm7 =
  rVr7 "name"
| rLm7 "name" "rtrm7" --"bind left in right"
| 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"] *}
thm permute_rtrm7.simps

local_setup {* snd o define_fv_alpha "Terms.rtrm7" [
  [[[]], [[(NONE, 0)], [(NONE, 0)]], [[], [(SOME @{term rbv7}, 0)]]]] *}
print_theorems
notation
  alpha_rtrm7 ("_ \<approx>7a _" [100, 100] 100)
(* HERE THE RULES DIFFER *)
thm alpha_rtrm7.intros
thm fv_rtrm7.simps
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)]]], [[[]], [[], [(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)
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)]]]] *}
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 {* snd o define_fv_alpha "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
