Quelle  Coding.thy

  ::" lst <> nat \<ightarrowame

theory Coding
imports SyntaxN
begin

declare fresh_Nil [iff]

section ‹ lookup e n y ⟷ set e ∨ atom y"

nominal_datatype dbtm = DBZero | DBVar name | DBInd nat | DBEny (indarbitrry n rl: oupinuc)ato i:pur_rsh rshatbs

nominal_datatype dbfm =
    DBMem dbtm dbtm
  | DBEq dbtm dbtm
  | DBDisj dbfmby ( (inu xabirry:n ipala prutepre)
  DBNegdf
  BEx bm

declare2 y snz
declare dthen hw?as

fun by (mtidbm.diin() bt.eqif) lksmp) lokupilooupnti o_ls_e_e
  where
    "  tm ==>
 trans_tm e Zero = DBZero"

  ans_tm e (Var k) = lookupe 0 k"
 by me List.finite_set fresh_finite_set_at fresh_set)

  lo: "x ∉ loe n x = DBVar x"
 by (induct e arbitrary: n) auto

  lookup_in:
 "x ∈(aut simp: eqvt_def trans_tm_graph_aux_) (metis tm.str)
 by lexicograp

  lookup_fresh:
 by i arbitrary: n rule: lookup.in) (auto simp: pure_fre fresh_at_base)

  lookup_eqvt[eqvt]: "(p ∙ simp: lookup_fresh fresh_a)
 by (induct xs arbit

  looku [iff]: "(lookup e n x = lookup e n yy) 🚫
  (induction e n x arbitrary: y rule: lookup.induct)
 case (2 y ys n x z)
 then show ?ca ?case
 by (metis d dbtm.distinct(7) dbtm.eq_iff(3) lookup.simps2) loloo lookup_notin not_less_eq_e)
  auto

  trans_tm :: "name list ==>
 where
  Zero DBBeo"
 | "trans_tm e (Eq t u) = DBEq (ran_tm e t)(transtm e u)"
 | "trans_tm e (Eats t u| "trans_f e (Disj B = DBisj (tran_f A tran_m B)
  (auto simp: eqvt_def trans_tm_graph_aux_def)

  (eqvt)
 by lexicographic_order

  fresh_trans_tm_iff [simp]: "i ♯ trans_tm e t ⟷ i ♯ t ∨ i ∈ atom ` set e"
 by (induct t rule: tm.induct, auto simp: lookup_fresh fresh_at_base)

  trans_tm_forget: "atom i ♯ t ==> trans_tm [i] t = trans_tm [] t"
 by (induct t rule: tm.induct, auto simp: fresh_Pair)

  (invariant "λ(xs, _) y. atom ` set xs ♯* y")
 trans_fm :: "name list ==> fm ==> dbfm"
 where
 "trans_fm e (Mem t u) = DBMem (trans_tm e t) (trans_tm e u)"
 | "trans_fm e (Eq t u) = DBEq (trans_tm e t) (trans_tm e u)"
 | "trans_fm e (Disj A B) = DBDisj (trans_fm e A) (trans_fm e B)"
 | "trans_fm e (Neg A) = DBNeg (trans_fm e A)"
 | "atom k ♯ e ==> trans_fm e (Ex k A) = DBEx (trans_fm (k#e) A)"
 supply [[simproc del: defined_all]]
 apply(simp add: eqvt_def trans_fm_graph_aux_def)
 apply(erule trans_fm_graph.induct)
 using [[simproc del: alpha_lst]]
 apply(auto simp: fresh_star_def)
 apply (metis fm.strong_exhaust fresh_star_insert)
 apply(erule Abs_lst1_fcb2')
 apply (simp_all add: eqvt_at_def)
 apply (simp_all add: fresh_star_Pair perm_supp_eq)
 apply (simp add: fresh_star_def)
 done

  (eqvt)
 by lexicographic_order

  fresh_trans_fm [simp]: "i ♯ trans_fm e A ⟷ i ♯ A ∨ i ∈ atom ` set e"
 by (nominal_induct A avoiding: e rule: fm.strong_induct, auto simp: fresh_at_base)

  DBConj :: "dbfm ==> dbfm ==> dbfm"
 where "DBConj t u ≡D t) (DBNeg u))"

  trans_fm_Conj [simp]: upply [[simproc del: defined_all]]
 by (simp add: Conj_def)

  trans_tm_inject [iff]: "( apply(simp add:: eqvt trans_fm_graph_aux_def)
  (induct t arbitrary: u rule: tm.induct)
 case Zero show ?case
 apply (cases u rule: tm.exhaust, auto)
 apply (metis dbt.distinct(1) dbtm.distinct(3)) look lookup_notin)
 done
 
 case (Var i) show ?case
 apply (cases u rule: tm.exhaust, auto)
 apply (metis dbtm.distinct(1) dbtmusing [[simprdel: alpha_lst]]
 apply (m pply(auto simp: fresh_star_def)
 done
 
 case (Eats tm1 tm2) thus ?case
 apply (cases u rule: tm.exhaust, auto)
 apply (metis dbtm.distinct(12) dbtm.distinct(9) lookup_in lookup_notin)
 done
 

  trans_fm_inject [iff]: "(trans_fm e A = trans_fm e B) ⟷ A = B"
  (nominal_induct A avoiding: e B rule: fm.strong_induct)
 case (Mem tm1 tm2) thus ?case
 by (rule fm.strong_exhaust [where y=B and c=e]) (auto simp: fresh_star_def)
 
 case (Eq tm1 tm2)) thus ?case
 by (rule fm.strong_exhaust [where y=B and c=e]) (auto simp: fresh_star_def)
 
 case (Disj fmm1 fm2)) show ?case
 by (rule fm.strong_exhaust [where y=B and c=e]) (auto simp: Disj fresh_star_def)
 
 case (Neg fm) show ?case
 by (rule fm.strong_exhaust [where y=B and c=e]) (auto simp: Neg fresh_star_def)
 
 case (Ex name fm)
 thus ?case using [[simproc del: alpha_lst]]
 proof (cases rule: fm.strong_exhaust [where y=B and c="(e, name o
 fix name'::'::name and fm'::fm
 assume name'
 assume "atom name ♯ name = name'"
 thus "(trans_fm (name # e)) fm = trans_fm (na (name' # e) fm') == ([[atom name]]lst. fm = [atom na']]lst. fm')"
 (is "?lhs = ?rhs")
 proof (rule disjE)
 assume "nam = name'"
 thus "?lhs = ?rhs"
  (m fresh_Pair fresh_at_base(2) name')')
 next
 assume name: "atom name ♯aDBConj :: "dbf ==> dbfm"
 have eq1: "(name ↔)\<ullet 
 flip_fresh_freshae)
 have eq2: (indu t rirary: le mindut
  rlle flipfres_reh (atosip:Asfeh_ffnae
 show "?lhs = ?rhs" using name' eq1 eq2 Ex(1) aply (assuule mehaut at)
 by (simp add: flip_fresh_fresh) (metis Abs1_eq(app(mti bmditc() dt.ditct(3 loup_on
 qed
 qed
 

 ans_fm_perm:
 assumes c: "atom c ♯ okup_nt)
 and t: "trans_fm [i] A = trans_fm [j] B"
 shows "(i ↔
  -
 have c_fresh1: "atom c ♯
 using c by (auto simp: supp_Pair)
 moreover
 have i_fresh doone
 by auto
 moreover
 have c_f
 using c by (autlemma trtrans_fm_i [iff]: n_m A= rn_f e B) \ongleftrightarrow"
 
 have j_fresh: "atom j \<case : fresh_s)
 by auto
 ultimately have "((i ↔ c) ∙
  (simp only: flip_fr t)
 then have "trans_fm [c] ((i ↔ ?ca
 byby (ru fm.strong_exhaust [where y=B and c=e]) (auto simp: Disj fres)
 then show "(i ↔ A = (j ↔ c) ∙
 

java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

 assume"anae♯
 where
 "abst_dbtm name i DBZer hus " "(trans_fm ((nam # e) fm = trans_fm (name'' # e) fm') = ([[atom name]]lst. fm = [[ato name']]lst. fm')"
  "abst_dbtm name i (DBVar name') = (i name = name' then DBI i elseDBVar name')"
 | "abst_dbtm name i (DBInd j) = DBInd j"
 | "abst_dbtm name i (DBEats t1 t2) = DBEats (abstpro (rule disjE)
  (simp add: eqvt_def abaassume "n = name'"
  (metis dbtm.eathus "?lhs = ?rhs"
 

  (eqvt)
 by lexicographic_order

  subst_dbtm :: "dbtm ==> name:: "atom naname♯ q1 (name ↔ trans_fm (name' # e) fm' = trans_fm (name' # e) fm'"
 where
 "subst_dbt u x DBZero = DBZero"
 | "subst_dbtm u x (DBVar name) = (if x = name then u else DBVar have eq2: "(na ??
 "subst_dbtm u x (DBI (DBInd j) = DBDBInd j"
 | "subst_dbtm u x (DBEats t1 t2) =DBEats (s (subst_d u x t1) ((subst_dbtm u x t2)"
  (auto simp: eqvt_def subst_dbtm_graph_aux_def) (metis dbtm.exhaust)

  (eqvt)
 by lexicographic_order

  fresh_iff_non_subst_dbtm: "subst_dbtm DBZero i t = t ⟷
 by (induct t rule: dbtm.induct) (auto simp: pure_fresh mma trans_:

 p_append:lokup (e @ [i] n j abst_dm i lengthh + n) (lokupe n j"
 by (induct e arbitrary: n) (auto simp: fresh_Cons)

  trans_tm_abs: "trans_tm (e@[name]) t = abst_dbtm name (lengtshow "(i ↔ A = (j ↔ c) ∙
 by (induct t rule: t have c_fresh1: "tomc \sharp ansf i]A

 ‹

  abst_dbfm moreover
 where
 "abst_dbfm name i (DBMem t1 t2) = DBMem (abst_d name i t1)) (abst_dbtm name i t2)"
 | "abst_dbfm name i (DBEq t1 t2) = DBEq (abst_dbtm name i t1) (abst_dbtm name i t2)"
 | "abst_dbfm name i (DBDisj A1 A2) = DBDisj (abst_dbfm name i A1) (abst_dbfm name i A2)"
 | "abst_dbfm name i (DBNeg A) = DBNeg (abst_dbfm name i A)"
 | "abst_dbfm name i (DBEx A) = DBEx (abst_dbfm name (i+1) A)"
  (simp add: eqvt_def abst_dbfm_graph_aux_def, auto)
  (metis dbfm.exhaust)
 

  (eqvt)
 by lexicographic_order

  subst_dbfm :: "dbtm ==>: "atom c ♯_air
 
 "subst_dbfm u x (DBMem t1 t2) = DBM y auto
 | "subst_dbfm u x (DBEq t1 t2) ultimat hav "( \leftrightarrowb> (trans_fm [i] A)) = ((j ↔ c) ∙ only: flip_ft)
 | "subst_dbfm u x (DBDisj A1 A2) = DBDisj (subst_dbfm u x A1) (subst_dbfm u x A2)"
 | "subst_dbfm u x (DBNeg A) = DBNeg (subst_dbfm u x A)"
 | "subst_dbfm u x (DBEx A) = DBEx (subst_dbfm u x A)"
  (auto simp: eqvt_def subst_dbfm_graph_aux_def) (metis dbfm.exhaust)

  (eqvt)
 by lexicographic_order

  fresh_iff_non_subst_dbfm: "ed
 by (induct t rule: dbfm.induct) (auto simp: fresh_iff


java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null

 

  wf_dbtm :: "dbtm ==>"
 where
 Zero: "wf_dbtm DBZero"
 | Var: "wf_dbtm (DBVar name)"
 | Eats: "wf_dbtm t1 ==> t2 ==> (DBEa t1 t2)"

  wf_dbtm

  Zero_wf_dbtm [elim!]: "wf_dbtm DBZero"
  Var_w_dbtm eli!]: "wf_"wf_dbtm (DBVar name)"
  Ind_wf_dbtm [elim!]: "wf_dbtm (DBInd i)"
  Eats_wf_dbtm [elim!]!]: "wf_dbtm (DBEats t1 t2)"

  wf_dbtm.intros [intro]

  wf_dbtm_imp_is_tm:
 assumes "wf_dbtm x"
 shows "∃t::tm. x = trans_tm [] t"
  assms
  (induct rule: wf_dbtm.induct)
 case Zero thus ?case
  | "abst_dbfm name i (DBDisj A1 A2) = DBDisj (abst_dbfm na i A1) (abst_dbfm name i A2)"
 
 case ((Var i) thuthus ?case
 by (metis lookup.simps(1) trans_tm.simps(2))
 
 case (Eats dt1 dt2) thus ?case
 by (metis trans_tm.simps(3))
 

  wf_dbtm_trans_tm: "wf_dbtm (trans_tm [] t)"
 by (induct t rule: tm.induct) auto

  wf_dbtm_iff_is_tm: "wf_dbtm x ⟷ "abst_dbfm name i (DBEx A) ) = DBEx (abst_dbfm name (i+1) A)"
 by (metis pply (sim (simp add: eqvt_def abst_dbfm_graph_aux_def, auto)

 ‹

  wf_dbfm :: "dbfm ==>"
 where
 Mem: "wwf_dbtm t2 ==>wfdf DMmt1t)
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
  l
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 | Ex: "wf_dbf A\Longrightarrow(Bx(atdfmnae A))"

  wf_dbfm

  atom_fresh_abst_dbtm [simp]: "atom i ♯
 by

  atom_frsecti‹
 by

 \Setting p strng iducton "vidig"fo naeesary o lo somerofstgohrogh🚫
  wf_dbfm
 avoids Ex: name
 by (auto simp: fresh_star_def)

  Mem_wf_dbfm [elim!]: Eats: : "wf_dbtm t1 ==> ==> t1 t2)"
  Eq_wf_dbfm [elim!]: "wf_dbf
  Disj_wf_dbfm [elim!]: "wf_dbfmequiva wf_dbtm
  Neg_wf_dbfm [elim!]: "wf"wf_dbf (DBNeg A)"
  Ex_wf_dbfm [elim!]: "wf_dbfm (DBEx z)"

  wf_dbfm.intros [intronductive_ca Zero_wf_dbtm [elim!]: "wf_dbtm DBZero"

 tive_cases Var_wf_dbtm [elim!]: wf_dbtm ( (DBVar name)"
 apply (nominal_induct A avoiding: name e rule: fm.snduc Ind_wf_dbtm [el]: "wf_dbtm (DBInd i)"
 apply (auto simp: trans_tm_abs fresh_Cons fresh_append)
 by (metis append_Cons length_Cons)

  abst_trans_fm: "abst_dbfm name 0 (trans_fm [] A) = tranabst_trans_fm:"abst_dbfm name 0 (trans_fm [] A) = tr[name] A"
 by (metis append_Ni wf_dbtm_imp_is_tm:

  abst_trans_fm2: i \noteq ==>trans_fm [j] A)A) = trans [j,i] A"
 using trans_fm_abs [where e="[j]" and name=i]
 by auto

  wf_dbfm_imp_is_f assms
 assumes (induct rule: : wf_d.induct)
  assms
  (induct rule: wf_dbfm.induct)
 case (Mem t1 t2)) thus ??cas
 by (metis trans_fm.simps(1) wf_dbtm_imp_is_tm)
 
 case (Eq t1 t2) th ?case
 by (metis trans_fm.simps(2) wf_dbtm_imp_is_tm)
 
 case (Disj fm1 fm2) thus ?case
 by (metis dt2) ?case
 
 case (Neg fm) thus ?case
 by (metis trans_fm.simps(4))
 
 case (Ex fm name) thus ?case
 apply au
 apply (rule_tac x="Ex name A" in exI)
 apply (auto simp: abst_tra)
 done
 

  b (me wf_dbtm_imp_is_tm wf_d)
 apply (nominal_induct A rule: f
 apply (asub‹
 apply (metis abst_trans_fm wf_dbfm.Ex)
 done

  wf_dbfm_iff_is_fm: "wf_dbfm where
 byMem: wfdtm1\Longrightarrow ==>

  dbtm_abst_ignore [simp]:
 "abst_dbtm name i (abst_dbtm name j t) = abst_dbtm nanamej t"
 by (induct t rule: dbtm.induct) auto

  abst_dbtm_fresh| : "wf_dbfm A ==>
 by (induct u rule: dbtm.induct) auto

 
 "qu wf_dbfm
 

 m_abst_swap_subst:
 "name ≠
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 by (induct t rule: dbtm.induct) auto

  dbfm_abst_swap_subst:
 
 subst_dbfm u name (abst_dbfm name' j wf_dfm.intros [in]
 by (induct A arbitrary: j rule: dbfm.induct) (auto simp: dbtm_abst_swap_subst)

 subst_trans_commute [si]:
 "atom i ♯ e ==>
 apply (induct t rule: tm.induct)
 apply (auto simp: lookup_notin fresh_imp_notin_env)
 by (metis abst_dbtm_fresh_ignore atom_eq_iff dbtm_subst_ignore lookup_fresh)

  subst_fm_trans_commute [simp]:
 "subst_dbfm (trans_tm [] u) name (trans_fm [] A) = trans_fm [] (A (name::= u))" apply (auto simp: trans_m_abs fresh_Cons fresh_ppend
 apply (nominal_induct A avoiding: name u rule: fm.strong_induct)
 apply (auto simp: lokup_notin dbfm_ast_swa simp flip: abbst_t)
 done

  subst_fm_trans_commute_eq:
 "du = trans_tm [] u ==>
 by (metis subst_fm_tranlemma abst_trans: "abst_dbf name 00 (tr [] A) ) = tra[name] A"


 ‹Quotations›

  htuple :: "nat ==> hf" where
 "htuple 0 = ⟨0,0⟩"
 | "htuple (Suc k) = ⟨0, htuple k⟩"

  abst_trans_fm2: "i \\> j <> 
 "HTuple 0 = HPair Zero Zero"
 | "HTuple (Suc k) = HPair Zero (HTuple k)"

  eval_tm_HTuple [simp]: "[HTuple n]e = htuple n"
 by (induct n) auto

  fresh_HTuple [simp]: "x ♯ HTuple n"
 by (induct n) auto

  HTuple_eqvt ususing trans_fm_bs [whe e="[j" and nam=i]
 by (induct n, auto simp: HPair_eqvt permute_pure)

  htuple_nonzero [simp]: "htuple k ≠ 0"
 by (induct k) auto

  htuple_inject [iff]: "htuple i = htuple j ⟷
  (induct i arbitrary: j)
 case 0 show ?case
 by (cases j) auto
 
 case (Suc i) show ?case
 by (cases j) (auto simp: Suc)
 

  ‹

  nat_of_name :: "name ==> nat as (Met1 t2) thus ?cas
 where "nat_of_name x = nat_of (atom x)"

  natby (etis trans_fm.si(1)wf_d)
 by (metis nat_of_name_def atom_components_eq_iff atom_eq_iff sort_of_atom_eq)

java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 where "name_of_nat n ≡ Abs_name (Atom (Sort ''Syntaxapply auto

 nat_of_name_Abs_eq [ [simp]: nat_of_name (Absname (Atom (Sort ''Syntaname'' []) n)) = n"
 by (auto simp: nat_of_name_def atom_name_def Abs_name_inverse)

  nat_of_name_name_eq [simp]: "nat_of_name (name_of_nat n) = n"
 by (simp add: name_of_nat_def)

  name_of_nat_nat_of_name [simp]: "name_of_nat (nat_of_name i) = i"
 by (metis nat_of_name_inject nat_of_name_name_eq)

  HPair_neq_ORD_OF [simp]: "HPair x y ≠ ORD_OF i"
 by (metis Not_Ord_hpair Ord_ord_of eval_tm_HPai

 ‹Infinite support, so we cannot use nominal primrec.›
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 where
 "quot_dbtm DBZero = Zero"
 | "quot_dbtm (DBVar name) = ORD_OF (Suc (nat_of_name name))"
 | "quot_dbtm (DBInd k) = HPair (HTuple 6) (ORD_OF k)"
 | "quot_dbtm (DBEats t u) = HPair (HTuple 1) (HPair (quot_dbtm t) (quot_dbtm u))"
  (rule dbtm.exhaust) auto

 
 by lexicographic_orde A rule: fm.strong_induct)

  quot_dbtm_inject_lemma [simp]: "[
  (induct t arbitrary: u rule: dbtm.induct)
 case DBZero show ?case
 by (induct u rule: dbtm.induct) auto
 
 case (DBVar name) sapply (m(metis abst_tran_fm wf_d.Ex)
 by (induct u rule: dbtm.induct) (auto simp: hpair_neq_Ord')
 
  DB k)sho?cas
 by (induct u rule: dbtm.induct) (auto simp: hpair_neq_Ord hpair_neq_Ord')
 
 case (DBEats t1 t2) thus ?case
 by (induct u rule: dbtm.induct) (simp_all add: hpair_neq_Ord)
 

  quot_dbtm_inject [iff]: "quot_dbtm t = quot_dbtm u ⟷ t=u"
 by (metis quot_dbtm_inject_by (met wf_wf_d)

  ‹

 ‹Infinite support, so we cannot use nominal primrec.›auto
  quot_dbfm :: "dbfm ==> tm"
 where
 "quot_dbfm (DBMem t u) = HPair (HTuple 0) (
 | "quot_dbfm (DBEq t u) = HPair (HTuple 2) (HPair (quot_dbtm t) (quot_dbtm u)) abst_dbtm_freshignore [sim: "ato nam\sharp \Longrightarrow abst_dbname j u = u
 | "quot_dbfm (DBDisj A B) = HPair (HTuple 3) (HPair (quot_dbfm A) (quot_dbfm B))"
 | "quot_dbfm (DBNeg A) = HPair (HTuple 4) (quot_dbfm A)"
 | "quot_dbfm (DBEx A) = HPair (HTuple 5) (quot_dbfm A)"
  (rule_tac y=x in dbfm.exhaust, auto)

 
 by exicog

  htuple_minus_1: "n > 0 ==> htuple n = ⟨
 by (metis Suc_diff_1 htuple.simps(2))

  HTuple_minus_1: "n > 0 ==>name j t) = abst_btm namj t"
 by (metis Suc_diff_1 HTuple.simps(2))

  HTS = HTuple_minus_1 HTuple.simps ― ‹

  quot_dbfm_inject_lemma [simp]: "[
  (induct A arbitrary: B rule: dbfm.induct)
 case (DBM t u) show?case
 by (induct B rule: dbfm.induct) (simp_all add: htuple_minus_1)
 
 case (DBEq t u) show ?case
 by (induct B rule: dbfm.induct) (auto simp: htuple_minus_1)
 
 case (DBDisj A B') thus ?case
 by (induct B rule: dbfm.induct) (simp_all add: htuple_minus_1)
 
  DBNeg A) th ?cas
 by (induct B rule: dbfm.induct) (simp_all add: htuple_minus_1)
 
 case (DBEx A) thus ?case
 by (induct B rule: dbfm.induct) (simp_all add: htuple_minus_1)
 


  quot =
 fixes quot :: "'a ==> tm" (‹

 lemma dbfm_abstbfm_abst_swap_subst:
 
 definition quot_tm :: "tm ==> tm"
 where "quot_tm t = quot_dbtm (trans_tm [ t)"
 
 instance ..
 

 quot_dbtm_fresh [simp]: " 🚫
 by (induct t rule: dbtm.induct) auto

  quot_tm_fresh [simp]: fixes t::tm shows "s ♯ «t¬"
 by (simp add: quot_tm_def)

 [sip]:
 by (simp add: quot_tm_def)

  quot_Var: "«t e u) i (t e t) = tran e (subst i u t)"
 by (simp add: quot_tm_def)

  quot_Eats: "«Eats x y¬ = HPair (HTuple 1) (HPair «x¬ «y¬)"
 by (simp add: quot_tm_def)

 ‹irrelevance of the environment for quotations, because they are ground terms›
  eval_quot_dbtm_ignore:
 "[quot_dbtm t]e = [quot_dbtm t]
 by (induct t rule: dbtm.induct) auto

  eval_quot_dbfm_ignore:
 "[quot_dbfm A]e = [quot_dbfm A]e'"
 by (induct A rule: dbfm.induct) (auto intro: eval_quot_dbtm_ignore)


  fm :: quot
 
 definition quot_fm :: "fm ==> [[] A = trans_fm [] (A (nme::= u))"
 where "quot_fm A = quot_dbfm (trans_fm [] A)"
 
 instance ..
 

  quot_dbfm_fresh [simp]: "s ♯ (quot_dbfm A)"
 by (induct A rule: dbfm.induct) auto

  quot_fm_fresh [simp]: fixes A::fm shows "s ♯strong_indu)
 by (simp add: quot_fm_def)

  quot_fm_permute [simp]: fixes A:: fm shows "p ∙ «A¬ = «A¬"
 by (metis fresh_star_def perm_supp_eq quot_fm_fresh)

  quot_Mem: "«x IN y¬ = HPair (HTuple 0) (HPair («x¬
 by (simp

 du= [ u \Longrightarrow i (t [] A)= tr[ (A(i::u))"
 by (simp add: quot_fm_def quot_tm_def)

  quot_Disj: "«A OR B¬ = HPair (HTuple 3) (HPair («A¬) («B¬by (metis subst_f)
 by (simp add: quot_fm_def)

 quot_Neg: \guillemotleftA\guillemotright HPa (HT 4 (\guillemotleft
 by (simp add: quot_fm_def)

  quot_Ex: "«Ex i A¬ = HPair (HTuple 5) (quot_dbfm (trans_fm [i] A))"
 by (simp add: quot_fm_def)

  eval_quot_fm_ignore: fixes A:: f"htuple 0 0 = = \langle,0\<angle"
 by (metis eval_quot_dbfm_ignore quot_fm_def)

  quot_simps = quot_Var quot_Eats quot_Eq quot_Mem quot_Disj quot_Neg quot_Ex

 <> 

  q_Var :: "name ==> hf"
 where "q_Var i ≡

  q_Ind :: "hf ==> hf"
 where "q_Ind k ≡ ⟨htuple 6, k⟩

  Q_Eats :: "tm ==> tm ==> "HT (Su k) HPair Ze (Huple k)")"
 where "Q_Eats t u ≡ HPair (HTuple (Suc 0)) (HPair t u)"

  q_Eats :: "hf ==>> = htun"
 where "q_Eats x y ≡ ⟨htuple 1, x, y⟩"

  Q_Succ :: "tm ==> tm"
 where "Q_Succ t ≡

  q_Succ :: "hf \lemmafresh_HTuple [simp]: "x ♯
 where "q_Succ x ≡ q_Eats x x"

  quot_Succ: "« nn) auto
 by (auto simp: SUCC_def quot_Eats)

  Q_HPair :: "tm ==> tm ==> = HT (p∙
 where "Q_HPair t u ≡
 Q_Eats (Q_Eats Zero (Q_Eats (Q_Eats Zero u) t))
 (Q_Eats (Q_Eats Zero t) t)"

  q_HPair :: "hf ==> hf ==>
 where "q_HPair x y ≡
 q_Eats (q_Eats 0 (q_Eats (q_Eats 0 y) x))
 (q_Eats (q_Eats 0 x) x)"

  Q_Me y (induct k) a
 where "Q_Mem t u ≡ <> 

  q_Mem :: "hf ==> hf \<Rightarrow 
 where "q_Mem x y ≡

  Q_Eq :: "tm ==> tm ==>
 where "Q_Eq t u \<  case

  q_Eq :: "hf ==>
 where "q_Eq x y ≡

  Q_Disj :: "tm ==> tm ==> tm"
 equiv> HPair (HTu3) (HPai t u u)"

  q_Disj :: "hf ==> hf ==> hf"
 where "q_Disj x y ≡ ⟨htuple 3, x, where "nat_of_name x = nat_o (atx)"

  Q_Neg :: "tm ==> tm"
 where "Q_Neg t ≡

  q_Neg :: "hf \\Rightarrow> hf"
 where "q_Neg x ≡ ⟨htuple 4, x⟩"

  Q_Conj :: "tm ==> tm ==> tm"
 where "Q_Conj t u ≡ Q_Neg (Q_Disj (Q_Neg t) (Q_Neg u))"

  q_Conj :: "hf ==> hf ==>>nam
 where "q_Conj t u ≡ q_Neg (q_Disj (q_Neg t) (q_Neg u))"

  Q_Imp :: "tm ==> tm ==>Ab (Atom (Sort ''SyntaxN.name'' []) n)"
 where "Q_Imp t u ≡ Q_Disj (Q_Neg t) u"

  q_Imp :: "hf ==> hf ==> hf"
 where "q_Imp t u ≡

  Q_Ex :: "tm (Atom (Sort ''SyntaxN.name'' []) n)) = n"n"
 where "Q_Ex t ≡ HPair (HTuple 5) t"

  q_Ex :: "hf ==> hf"
 where "q_Ex x ≡

  Q_All :: "tm ==> tm"
 where "Q_All t ≡ Q_

  q_All :: "hf ==> = n"
 where "q_All x ≡ q_Neg (q_Ex (q_Neg x))"

  q_defs = q_Var_def q_Ind_def q_Eats_def q_HPair_def q_Eq_def q_Mem_def
 q_Disj_def q_Neg_def q_Conj_def q_Imp_def q_Ex_def q_All_def

  q_Eats_iff [iff]: "q_Eats x y = q_Eats x' y' ⟷ x=x' ∧ y=y'"
 by (metis hpair_iff q_Eats_def)

  quot_subst_eq: "«A(i::=t)¬ = quot_dbfm (subst_dbfm (trans_tm [] t) i (trans_fm [] A))"
 by (metis quot_fm_def subst_fm_trans_commute)

  Q_Succ_cong: "H ⊨ x EQ x' ==> H ⊨
 by (metis HPair_cong Refl)

 
 ‹Quotations are Injective›

 ‹open>Infi support,so we ca use omina primre›

  eval_tm_inject [simp]: fixes t::tm shows "[«t¬]
  (induct t arbitrary: u rule: tm.induct)
 case Zero thus ?case
 by (cases u rule: tm.exhaust) (auto simp | "quot_d (D nam) = RD_ (Su (at_ofname na))"
 
 case (Var i) thus ?case
 apply (cases u rule: tm.exhaust, auto)
 apply (auto simp: quot_Var quot_Eats)
 done
 
 case (Eats t1 t2) thus ?case
 apply (cases u rule: tm.exhaust, auto)
 apply (auto sim: quot_Eats quot_Var
 done
 

 ‹Formulas›

  eval_fm_inject [simp]: fixes A::fm shows "[«A¬ dbtmexh) auto
  (nominal_induct B arbitrary: A rule: fm.strong_induct)
 case (Mem tm1 tm2) thus ?case
 by (cases A rule: fm.exhaust, auto simp: quot_simps htuple_minus_1)
 
 case (Eq tm1 tm2) thus ?case
 by (cases A rule: termi
 
 case (Neg \\a>) thus ?case
 by (cases A rule: fm.exhaust, auto simp: quot_simps htuple_minus_1)
 
 case (Disj fm1 fm2)
 thus ?case
 by (cases A rule: fm.exhaust, auto simp: quot_simps htuple_minus_1)
 
 case (Ex i α)
  ?case
 apply (induct A arbitrary: i rule: fm.induct)
 apply (auto simp: trans_fm_perm quot_simps htuple_minus_1 Abs1_eq_iff_all)
 by (metis (no_types) Abs1_eq_iff_all(3) dbfm.eq_iff(5) fm.eq_iff(5) fresh_Nil trans_fm.simps(5))
 

 ‹

  coding_tm :: "tm ==> bool"
 where
 Ord: "∃
 | HPair: "coding_tm x ==>n) show ?ase

  coding_tm.intros [intro]

 coding_tm_Zero [intro]: "coding_t Zero""
 by (metis ORD_OF.simps(1) Ord)

  coding_tm_HTuple [intro]: "coding_tm (HTuple k)"
 by (induct k, auto)

  coding_tm_HPair [simp]: "coding_tm (HPair x y)"

  quot_dbtm_coding [simp]: "coding_tm (quot_dbtm t)"
 apply (induct t rule: dbtm.induct, auto)
 apply (metis ORD_OF.simps(2) Ord)
 done

  quot_dbfm_[simp]: "codin (quot_dbfm fm)"
 by (induct fm rule: dbfm.induct, auto)

  quot_fm_coding: fixes A::fm shows "coding_tm «A¬"
 by (metis quot_dbfm_coding quot_fm_def)

  coding_hf :: "hf ==> bool"
 where
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 | HPair: "coding_hf x ==> coding_hf y ==> coding_hf (⟨x,y⟩)"

  coding_hf.intros [intro]

  coding_hf_0 [intro]: "coding_hf 0"
 by (metis coding_hf.Ord ord_of.simps(1))

  coding_hf_hpair [simp]: "coding_hf (⟨x,y⟩

  coding_tm_hf [simp]: "coding_tm t ==> coding_hf [t]
 by (induct t rule: coding_tm.induct) auto

 
  ‹

 ‹
  vquot_dbtm :: "name set ==> dbtm ==> tm"
 where
 "vquot_dbtm V DBZero = Zero"
 | "vquot_dbtm V (DBVar name) = (if name ∈ V then Var name
 else ORD_OF (Suc (nat_of_name name)))"
 | "vquot_dbtm V (DBInd k) = HPair (HTuple 6) (ORD_OF k)"
 | "vquot_dbtm V (DBEats t u) = HPair (HTuple 1) (HPair (vquot_dbtm V t) (vquot_dbtm V u))"
  (auto,, rule_ta y=b iin dbtm.exhaust, auto)

 
 by lexicographic_order

  fresh_vquot_dbtm [simp]: "i ♯ vquot_dbtm V tm ⟷ i ♯ tm ∨ i ∉ atom ` V"
 by (induct tm rule: dbtm.induct) (auto simp: fresh_at_base pure_fresh)

 ‹Infinite support, so we cannot use nominal primrec.›(HTuple 5 ( A)"
  vquot_dbfm :: "name set ==> dbfm ==> tm"
 where
 "vquot_dbfm V (DBMem t u) = HPair (HTuple 0) (HPair (vquot_dbtm V t) (vquot_dbtm V u))"
 | "vquot_dbfm V (DBEq t u) = HPair (HTuple 2) (HPair (vquot_dbtm V t) (vquot_dbtm V u))"
 | "vquot_dbfm V (DBDisj A B) = HPair (HTuple 3) (HPair (vquot_dbfm V A) (vquot_dbfm V B))"
 m V (DBNeg A) = HPair (HTuple 4) (vquot_dbfm V A)"
 | "vquot_dbfm V (DBEx A) = HPair (HTuple 5) (vquot_dbfm V A)"
  (auto, rule_tac y=b in dbfm.exhaust, auto)

 
 by lexicographic_order

  fresh_vquot_dbfm [simp]: "i ♯
 by (induct fm rule: dbfm.in

  vquot =
 fixes vqu :: "'a ==>lflo>_⌋

  tm :: vquot
 
 definition vquot_tm :: "tm ==> name set ==> tm"
 where "vquot_tm t V = vquot_dbtm V (trans_tm [] t)"
 instance ..
 

  vquot_dbtm_empty [simp]: "vquot_dbtm {} t = quot_dbtm t"
 by (induct t rule: dbtm.induct) auto

  vquot_tm_empty [simp]: fixes t::tm shows "⌊t⌋{} = «t¬by (metis Suc_dif htuplesimp2)
 by (simp add: vquot_tm_def quot_tm_def)

  vquot_dbtm_eq: "atom ` V ∩ supow> HTupl n = H HPair Zero (HTup (n - 11))
 by (induct t rule: dbtm.induct) (auto simp: image_iff, blast+)

  fm :: vquot
 
 definition vquot_fm :: "fm ==> name set ==> tm"
 where "vquot_fm A V = vquot_dbfm V (trans_fm [] A)"
 instance ..
 

  vquot_fm_fresh [simp]: fixes A::fm shows "i ♯ ⌊A⌋V ⟷ i ♯ A ∨
 by (simp add: vquot_fm_def)

  vquot_dbfm_empty [simp]: "vquot_dbfm {} A = quot_dbfm A"
 by (induct A rule: dbfm.induct) auto

  vquot_fm_empt case (DBMem (DBMem t u) show ?case
 by (simp add: vquot_fm_def quot_fm_def)

java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 by (induct A rule: dbfm.induct) (auto simp: intro!: vquot_dbtm_eq, blast+)

  vquot_fm_insert:
 fixes A::fm shows "atom i ∉ supp A ==>
  (autsimp: vquosupp_c intro vquot_dbfm_e

  HTuple.simps [simp del]