theory ExCoreHaskell

imports "Parser"

begin

(* core haskell *)

ML {* val _ = recursive := false *}

atom_decl var

atom_decl tvar

(* there are types, coercion types and regular types list-data-structure *)

nominal_datatype tkind =

  KStar

| KFun "tkind" "tkind"

and ckind =

  CKEq "ty" "ty"

and ty =

  TVar "tvar"

| TC "char"

| TApp "ty" "ty"

| TFun "char" "ty_lst"

| TAll tv::"tvar" "tkind" T::"ty"  bind tv in T

| TEq "ty" "ty" "ty"

and ty_lst =

  TsNil

| TsCons "ty" "ty_lst"

and co =

  CC "char"

| CApp "co" "co"

| CFun "char" "co_lst"

| CAll tv::"tvar" "ckind" C::"co"  bind tv in C

| CEq "co" "co" "co"

| CSym "co"

| CLeft "co"

| CRight "co"

| CSim "co"

| CRightc "co"

| CLeftc "co"

| CCoe "co" "co"

and co_lst =

  CsNil

| CsCons "co" "co_lst"

and trm =

  Var "var"

| C "char"

| LAMT tv::"tvar" "tkind" t::"trm" bind tv in t

| LAMC tv::"tvar" "ckind" t::"trm" bind tv in t

| APP "trm" "ty"

| Lam v::"var" "ty" t::"trm"       bind v in t

| App "trm" "trm"

| Let x::"var" "ty" "trm" t::"trm" bind x in t

| Case "trm" "assoc_lst"

| Cast "trm" "ty"                  --"ty is supposed to be a coercion type only"

and assoc_lst =

  ANil

| ACons p::"pat" t::"trm" "assoc_lst" bind "bv p" in t

and pat =

  K "char" "tvtk_lst" "tvck_lst" "vt_lst"

and vt_lst =

  VTNil

| VTCons "var" "ty" "vt_lst"

and tvtk_lst =

  TVTKNil

| TVTKCons "tvar" "tkind" "tvtk_lst"

and tvck_lst =

  TVCKNil

| TVCKCons "tvar" "ckind" "tvck_lst"

binder

    bv :: "pat \<Rightarrow> atom set"

and bv_vt :: "vt_lst \<Rightarrow> atom set"

and bv_tvtk :: "tvtk_lst \<Rightarrow> atom set"

and bv_tvck :: "tvck_lst \<Rightarrow> atom set"

where

  "bv (K s tvts tvcs vs) = (bv_tvtk tvts) \<union> (bv_tvck tvcs) \<union> (bv_vt vs)"

| "bv_vt VTNil = {}"

| "bv_vt (VTCons v k t) = {atom v} \<union> bv_vt t"

| "bv_tvtk TVTKNil = {}"

| "bv_tvtk (TVTKCons v k t) = {atom v} \<union> bv_tvtk t"

| "bv_tvck TVCKNil = {}"

| "bv_tvck (TVCKCons v k t) = {atom v} \<union> bv_tvck t"

lemmas fv_supp=tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vt_lst_tvtk_lst_tvck_lst.supp(1-9,11,13,15)

lemmas supp=tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vt_lst_tvtk_lst_tvck_lst.fv[simplified fv_supp]

lemmas perm=tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vt_lst_tvtk_lst_tvck_lst.perm

lemmas eq_iff=tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vt_lst_tvtk_lst_tvck_lst.eq_iff

lemmas inducts=tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vt_lst_tvtk_lst_tvck_lst.inducts

lemmas alpha_inducts=alpha_tkind_raw_alpha_ckind_raw_alpha_ty_raw_alpha_ty_lst_raw_alpha_co_raw_alpha_co_lst_raw_alpha_trm_raw_alpha_assoc_lst_raw_alpha_pat_raw_alpha_vt_lst_raw_alpha_tvtk_lst_raw_alpha_tvck_lst_raw_alpha_bv_raw_alpha_bv_vt_raw_alpha_bv_tvtk_raw_alpha_bv_tvck_raw.inducts

lemmas alpha_intros=alpha_tkind_raw_alpha_ckind_raw_alpha_ty_raw_alpha_ty_lst_raw_alpha_co_raw_alpha_co_lst_raw_alpha_trm_raw_alpha_assoc_lst_raw_alpha_pat_raw_alpha_vt_lst_raw_alpha_tvtk_lst_raw_alpha_tvck_lst_raw_alpha_bv_raw_alpha_bv_vt_raw_alpha_bv_tvtk_raw_alpha_bv_tvck_raw.intros

lemma fresh_star_minus_perm: "as \<sharp>* - p = as \<sharp>* (p :: perm)"

  unfolding fresh_star_def Ball_def

  by auto (simp_all add: fresh_minus_perm)

primrec permute_bv_vt_raw

where "permute_bv_vt_raw p VTNil_raw = VTNil_raw"

|     "permute_bv_vt_raw p (VTCons_raw v t l) = VTCons_raw (p \<bullet> v) t (permute_bv_vt_raw p l)"

primrec permute_bv_tvck_raw

where "permute_bv_tvck_raw p TVCKNil_raw = TVCKNil_raw"

|     "permute_bv_tvck_raw p (TVCKCons_raw v t l) = TVCKCons_raw (p \<bullet> v) t (permute_bv_tvck_raw p l)"

primrec permute_bv_tvtk_raw

where "permute_bv_tvtk_raw p TVTKNil_raw = TVTKNil_raw"

|     "permute_bv_tvtk_raw p (TVTKCons_raw v t l) = TVTKCons_raw (p \<bullet> v) t (permute_bv_tvtk_raw p l)"

primrec permute_bv_raw

where "permute_bv_raw p (K_raw c l1 l2 l3) =

     K_raw c (permute_bv_tvtk_raw p l1) (permute_bv_tvck_raw p l2) (permute_bv_vt_raw p l3)"

quotient_definition "permute_bv_vt :: perm \<Rightarrow> vt_lst \<Rightarrow> vt_lst"

is "permute_bv_vt_raw"

quotient_definition "permute_bv_tvck :: perm \<Rightarrow> tvck_lst \<Rightarrow> tvck_lst"

is "permute_bv_tvck_raw"

quotient_definition "permute_bv_tvtk :: perm \<Rightarrow> tvtk_lst \<Rightarrow> tvtk_lst"

is "permute_bv_tvtk_raw"

quotient_definition "permute_bv :: perm \<Rightarrow> pat \<Rightarrow> pat"

is "permute_bv_raw"

lemma rsp_pre:

 "alpha_tvtk_lst_raw d a \<Longrightarrow> alpha_tvtk_lst_raw (permute_bv_tvtk_raw x d) (permute_bv_tvtk_raw x a)"

 "alpha_tvck_lst_raw e b \<Longrightarrow> alpha_tvck_lst_raw (permute_bv_tvck_raw x e) (permute_bv_tvck_raw x b)"

 "alpha_vt_lst_raw f c \<Longrightarrow> alpha_vt_lst_raw (permute_bv_vt_raw x f) (permute_bv_vt_raw x c)"

 apply (erule_tac [!] alpha_inducts)

 apply simp_all

 apply (rule_tac [!] alpha_intros)

 apply simp_all

 done

lemma [quot_respect]:

 "(op = ===> alpha_pat_raw ===> alpha_pat_raw) permute_bv_raw permute_bv_raw"

 "(op = ===> alpha_tvtk_lst_raw ===> alpha_tvtk_lst_raw) permute_bv_tvtk_raw permute_bv_tvtk_raw"

 "(op = ===> alpha_tvck_lst_raw ===> alpha_tvck_lst_raw) permute_bv_tvck_raw permute_bv_tvck_raw"

 "(op = ===> alpha_vt_lst_raw ===> alpha_vt_lst_raw) permute_bv_vt_raw permute_bv_vt_raw"

 apply (simp_all add: rsp_pre)

 apply clarify

 apply (erule_tac alpha_inducts)

 apply (simp_all)

 apply (rule alpha_intros)

 apply (simp_all add: rsp_pre)

 done

lemmas permute_bv[simp] = permute_bv_raw.simps[quot_lifted]

 permute_bv_vt_raw.simps[quot_lifted]

 permute_bv_tvck_raw.simps[quot_lifted]

 permute_bv_tvtk_raw.simps[quot_lifted]

lemma perm_bv1:

  "p \<bullet> bv_tvck b = bv_tvck (permute_bv_tvck p b)"

  "p \<bullet> bv_tvtk c = bv_tvtk (permute_bv_tvtk p c)"

  "p \<bullet> bv_vt d = bv_vt (permute_bv_vt p d)"

  apply(induct b rule: inducts(12))

  apply(rule TrueI)

  apply(simp_all add:permute_bv eqvts)

  apply(induct c rule: inducts(11))

  apply(rule TrueI)

  apply(simp_all add:permute_bv eqvts)

  apply(induct d rule: inducts(10))

  apply(rule TrueI)

  apply(simp_all add:permute_bv eqvts)

  done

lemma perm_bv2:

  "p \<bullet> bv l = bv (permute_bv p l)"

  apply(induct l rule: inducts(9))

  apply(rule TrueI)

  apply(simp_all add:permute_bv)

  apply(simp add: perm_bv1[symmetric])

  apply(simp add: eqvts)

  done

lemma alpha_perm_bn1:

 " alpha_bv_tvtk tvtk_lst (permute_bv_tvtk q tvtk_lst)"

 "alpha_bv_tvck tvck_lst (permute_bv_tvck q tvck_lst)"

 "alpha_bv_vt vt_lst (permute_bv_vt q vt_lst)"

  apply(induct tvtk_lst rule: inducts(11))

  apply(simp_all add:permute_bv eqvts eq_iff)

  apply(induct tvck_lst rule: inducts(12))

  apply(simp_all add:permute_bv eqvts eq_iff)

  apply(induct vt_lst rule: inducts(10))

  apply(simp_all add:permute_bv eqvts eq_iff)

  done

lemma alpha_perm_bn:

  "alpha_bv pat (permute_bv q pat)"

  apply(induct pat rule: inducts(9))

  apply(simp_all add:permute_bv eqvts eq_iff alpha_perm_bn1)

  done

lemma ACons_subst:

  "supp (Abs (bv pat) trm) \<sharp>* q \<Longrightarrow> (ACons pat trm al) = ACons (permute_bv q pat) (q \<bullet> trm) al"

  apply (simp only: eq_iff)

  apply (rule_tac x="q" in exI)

  apply (simp add: alphas)

  apply (simp add: perm_bv2[symmetric])

  apply (simp add: eqvts[symmetric])

  apply (simp add: supp_abs)

  apply (simp add: fv_supp)

  apply (simp add: alpha_perm_bn)

  apply (rule supp_perm_eq[symmetric])

  apply (subst supp_finite_atom_set)

  apply (rule finite_Diff)

  apply (simp add: finite_supp)

  apply (assumption)

  done

lemma permute_bv_zero1:

  "permute_bv_tvck 0 b = b"

  "permute_bv_tvtk 0 c = c"

  "permute_bv_vt 0 d = d"

  apply(induct b rule: inducts(12))

  apply(rule TrueI)

  apply(simp_all add:permute_bv eqvts)

  apply(induct c rule: inducts(11))

  apply(rule TrueI)

  apply(simp_all add:permute_bv eqvts)

  apply(induct d rule: inducts(10))

  apply(rule TrueI)

  apply(simp_all add:permute_bv eqvts)

  done

lemma permute_bv_zero2:

  "permute_bv 0 a = a"

  apply(induct a rule: inducts(9))

  apply(rule TrueI)

  apply(simp_all add:permute_bv eqvts permute_bv_zero1)

  done

lemma fv_alpha1: "fv_bv_tvtk x \<sharp>* pa \<Longrightarrow> alpha_bv_tvtk (pa \<bullet> x) x"

apply (induct x rule: inducts(11))

apply (tactic {* ALLGOALS (TRY o rtac @{thm TrueI}) *})

apply (simp_all add: eq_iff fresh_star_union)

apply (subst supp_perm_eq)

apply (simp_all add: fv_supp)

done

lemma fv_alpha2: "fv_bv_tvck x \<sharp>* pa \<Longrightarrow> alpha_bv_tvck (pa \<bullet> x) x"

apply (induct x rule: inducts(12))

apply (tactic {* ALLGOALS (TRY o rtac @{thm TrueI}) *})

apply (simp_all add: eq_iff fresh_star_union)

apply (subst supp_perm_eq)

apply (simp_all add: fv_supp)

done

lemma fv_alpha3: "fv_bv_vt x \<sharp>* pa \<Longrightarrow> alpha_bv_vt (pa \<bullet> x) x"

apply (induct x rule: inducts(10))

apply (tactic {* ALLGOALS (TRY o rtac @{thm TrueI}) *})

apply (simp_all add: eq_iff fresh_star_union)

apply (subst supp_perm_eq)

apply (simp_all add: fv_supp)

done

lemma fv_alpha: "fv_bv x \<sharp>* pa \<Longrightarrow> alpha_bv (pa \<bullet> x) x"

apply (induct x rule: inducts(9))

apply (tactic {* ALLGOALS (TRY o rtac @{thm TrueI}) *})

apply (simp_all add: eq_iff fresh_star_union)

apply (simp add: fv_alpha1 fv_alpha2 fv_alpha3)

apply (simp add: eqvts)

done

lemma fin1: "finite (fv_bv_tvtk x)"

apply (induct x rule: inducts(11))

apply (tactic {* ALLGOALS (TRY o rtac @{thm TrueI}) *})

apply (simp_all add: fv_supp finite_supp)

done

lemma fin2: "finite (fv_bv_tvck x)"

apply (induct x rule: inducts(12))

apply (tactic {* ALLGOALS (TRY o rtac @{thm TrueI}) *})

apply (simp_all add: fv_supp finite_supp)

done

lemma fin3: "finite (fv_bv_vt x)"

apply (induct x rule: inducts(10))

apply (tactic {* ALLGOALS (TRY o rtac @{thm TrueI}) *})

apply (simp_all add: fv_supp finite_supp)

done

lemma fin_fv_bv: "finite (fv_bv x)"

apply (induct x rule: inducts(9))

apply (tactic {* ALLGOALS (TRY o rtac @{thm TrueI}) *})

apply (simp add: fin1 fin2 fin3)

done

lemma finb1: "finite (bv_tvtk x)"

apply (induct x rule: inducts(11))

apply (tactic {* ALLGOALS (TRY o rtac @{thm TrueI}) *})

apply (simp_all add: fv_supp finite_supp)

done

lemma finb2: "finite (bv_tvck x)"

apply (induct x rule: inducts(12))

apply (tactic {* ALLGOALS (TRY o rtac @{thm TrueI}) *})

apply (simp_all add: fv_supp finite_supp)

done

lemma finb3: "finite (bv_vt x)"

apply (induct x rule: inducts(10))

apply (tactic {* ALLGOALS (TRY o rtac @{thm TrueI}) *})

apply (simp_all add: fv_supp finite_supp)

done

lemma fin_bv: "finite (bv x)"

apply (induct x rule: inducts(9))

apply (tactic {* ALLGOALS (TRY o rtac @{thm TrueI}) *})

apply (simp add: finb1 finb2 finb3)

done

lemma "bv x \<sharp>* fv_bv x"

apply (induct x rule: inducts(9))

apply (tactic {* ALLGOALS (TRY o rtac @{thm TrueI}) *})

apply (simp)

oops

lemma strong_inudction_principle:

  assumes a01: "\<And>b. P1 b KStar"

  and a02: "\<And>tkind1 tkind2 b. \<lbrakk>\<And>c. P1 c tkind1; \<And>c. P1 c tkind2\<rbrakk> \<Longrightarrow> P1 b (KFun tkind1 tkind2)"

  and a03: "\<And>ty1 ty2 b. \<lbrakk>\<And>c. P3 c ty1; \<And>c. P3 c ty2\<rbrakk> \<Longrightarrow> P2 b (CKEq ty1 ty2)"

  and a04: "\<And>tvar b. P3 b (TVar tvar)"

  and a05: "\<And>char b. P3 b (TC char)"

  and a06: "\<And>ty1 ty2 b. \<lbrakk>\<And>c. P3 c ty1; \<And>c. P3 c ty2\<rbrakk> \<Longrightarrow> P3 b (TApp ty1 ty2)"

  and a07: "\<And>char ty_lst b. \<lbrakk>\<And>c. P4 c ty_lst\<rbrakk> \<Longrightarrow> P3 b (TFun char ty_lst)"

  and a08: "\<And>tvar tkind ty b. \<lbrakk>\<And>c. P1 c tkind; \<And>c. P3 c ty; atom tvar \<sharp> b\<rbrakk>

    \<Longrightarrow> P3 b (TAll tvar tkind ty)"

  and a09: "\<And>ty1 ty2 ty3 b. \<lbrakk>\<And>c. P3 c ty1; \<And>c. P3 c ty2; \<And>c. P3 c ty3\<rbrakk> \<Longrightarrow> P3 b (TEq ty1 ty2 ty3)"

  and a10: "\<And>b. P4 b TsNil"

  and a11: "\<And>ty ty_lst b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P4 c ty_lst\<rbrakk> \<Longrightarrow> P4 b (TsCons ty ty_lst)"

  and a12: "\<And>char b. P5 b (CC char)"

  and a13: "\<And>co1 co2 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2\<rbrakk> \<Longrightarrow> P5 b (CApp co1 co2)"

  and a14: "\<And>char co_lst b. \<lbrakk>\<And>c. P6 c co_lst\<rbrakk> \<Longrightarrow> P5 b (CFun char co_lst)"

  and a15: "\<And>tvar ckind co b. \<lbrakk>\<And>c. P2 c ckind; \<And>c. P5 c co; atom tvar \<sharp> b\<rbrakk>

    \<Longrightarrow> P5 b (CAll tvar ckind co)"

  and a16: "\<And>co1 co2 co3 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2; \<And>c. P5 c co3\<rbrakk> \<Longrightarrow> P5 b (CEq co1 co2 co3)"

  and a17: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CSym co)"

  and a18: "\<And>co1 co2 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2\<rbrakk> \<Longrightarrow> P5 b (CCir co1 co2)"

  and a19: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CLeft co)"

  and a20: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CRight co)"

  and a21: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CSim co)"

  and a22: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CRightc co)"

  and a23: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CLeftc co)"

  and a24: "\<And>co1 co2 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2\<rbrakk> \<Longrightarrow> P5 b (CCoe co1 co2)"

  and a25: "\<And>b. P6 b CsNil"

  and a26: "\<And>co co_lst b. \<lbrakk>\<And>c. P5 c co; \<And>c. P6 c co_lst\<rbrakk> \<Longrightarrow> P6 b (CsCons co co_lst)"

  and a27: "\<And>var b. P7 b (Var var)"

  and a28: "\<And>char b. P7 b (C char)"

  and a29: "\<And>tvar tkind trm b. \<lbrakk>\<And>c. P1 c tkind; \<And>c. P7 c trm; atom tvar \<sharp> b\<rbrakk>

    \<Longrightarrow> P7 b (LAMT tvar tkind trm)"

  and a30: "\<And>tvar ckind trm b. \<lbrakk>\<And>c. P2 c ckind; \<And>c. P7 c trm; atom tvar \<sharp> b\<rbrakk>

    \<Longrightarrow> P7 b (LAMC tvar ckind trm)"

  and a31: "\<And>trm ty b. \<lbrakk>\<And>c. P7 c trm; \<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P7 b (APP trm ty)"

  and a32: "\<And>var ty trm b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P7 c trm; atom var \<sharp> b\<rbrakk> \<Longrightarrow> P7 b (Lam var ty trm)"

  and a33: "\<And>trm1 trm2 b. \<lbrakk>\<And>c. P7 c trm1; \<And>c. P7 c trm2\<rbrakk> \<Longrightarrow> P7 b (App trm1 trm2)"

  and a34: "\<And>var ty trm1 trm2 b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P7 c trm1; \<And>c. P7 c trm2; atom var \<sharp> b\<rbrakk>

    \<Longrightarrow> P7 b (Let var ty trm1 trm2)"

  and a35: "\<And>trm assoc_lst b. \<lbrakk>\<And>c. P7 c trm; \<And>c. P8 c assoc_lst\<rbrakk> \<Longrightarrow> P7 b (Case trm assoc_lst)"

  and a36: "\<And>trm ty b. \<lbrakk>\<And>c. P7 c trm; \<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P7 b (Cast trm ty)"

  and a37: "\<And>b. P8 b ANil"

  and a38: "\<And>pat trm assoc_lst b. \<lbrakk>\<And>c. P9 c pat; \<And>c. P7 c trm; \<And>c. P8 c assoc_lst; bv(pat) \<sharp>* b\<rbrakk>

    \<Longrightarrow> P8 b (ACons pat trm assoc_lst)"

  and a39: "\<And>char tvtk_lst tvck_lst vt_lst b. \<lbrakk>\<And>c. P11 c tvtk_lst; \<And>c. P12 c tvck_lst; \<And>c. P10 c vt_lst\<rbrakk>

    \<Longrightarrow> P9 b (K char tvtk_lst tvck_lst vt_lst)"

  and a40: "\<And>b. P10 b VTNil"

  and a41: "\<And>var ty vt_lst b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P10 c vt_lst\<rbrakk> \<Longrightarrow> P10 b (VTCons var ty vt_lst)"

  and a42: "\<And>b. P11 b TVTKNil"

  and a43: "\<And>tvar tkind tvtk_lst b. \<lbrakk>\<And>c. P1 c tkind; \<And>c. P11 c tvtk_lst\<rbrakk>

    \<Longrightarrow> P11 b (TVTKCons tvar tkind tvtk_lst)"

  and a44: "\<And>b. P12 b TVCKNil"

  and a45: "\<And>tvar ckind tvck_lst b. \<lbrakk>\<And>c. P2 c ckind; \<And>c. P12 c tvck_lst\<rbrakk>

    \<Longrightarrow> P12 b (TVCKCons tvar ckind tvck_lst)"

  shows "P1 (a :: 'a :: pt) tkind \<and>

         P2 (b :: 'b :: pt) ckind \<and>

         P3 (c :: 'c :: {pt,fs}) ty \<and>

         P4 (d :: 'd :: pt) ty_lst \<and>

         P5 (e :: 'e :: {pt,fs}) co \<and>

         P6 (f :: 'f :: pt) co_lst \<and>

         P7 (g :: 'g :: {pt,fs}) trm \<and>

         P8 (h :: 'h :: {pt,fs}) assoc_lst \<and>

         P9 (i :: 'i :: pt) pat \<and>

         P10 (j :: 'j :: pt) vt_lst \<and>

         P11 (k :: 'k :: pt) tvtk_lst \<and>

         P12 (l :: 'l :: pt) tvck_lst"

proof -

  have a1: "(\<forall>p a. P1 a (p \<bullet> tkind))" and "(\<forall>p b. P2 b (p \<bullet> ckind))" and "(\<forall>p c. P3 c (p \<bullet> ty))" and "(\<forall>p d. P4 d (p \<bullet> ty_lst))" and "(\<forall>p e. P5 e (p \<bullet> co))" and " (\<forall>p f. P6 f (p \<bullet> co_lst))" and "(\<forall>p g. P7 g (p \<bullet> trm))" and "(\<forall>p h. P8 h (p \<bullet> assoc_lst))" and a1:"(\<forall>p q i. P9 i (permute_bv p (q \<bullet> pat)))" and a2:"(\<forall>p q j. P10 j (permute_bv_vt q (p \<bullet> vt_lst)))" and a3:"(\<forall>p q k. P11 k ( permute_bv_tvtk q (p \<bullet> tvtk_lst)))" and a4:"(\<forall>p q l. P12 l (permute_bv_tvck q (p \<bullet> tvck_lst)))"

    apply (induct rule: tkind_ckind_ty_ty_lst_co_co_lst_trm_assoc_lst_pat_vt_lst_tvtk_lst_tvck_lst.inducts)

    apply (tactic {* ALLGOALS (REPEAT o rtac allI) *})

    apply (tactic {* ALLGOALS (TRY o SOLVED' (simp_tac @{simpset} THEN_ALL_NEW resolve_tac @{thms assms} THEN_ALL_NEW asm_full_simp_tac @{simpset})) *})

(* GOAL1 *)

    apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (atom (p \<bullet> tvar))) \<sharp> c \<and>

                       supp (Abs (p \<bullet> {atom tvar}) (p \<bullet> ty)) \<sharp>* pa)")

    apply clarify

    apply (simp only: perm)

    apply(rule_tac t="TAll (p \<bullet> tvar) (p \<bullet> tkind) (p \<bullet> ty)"

               and s="TAll (pa \<bullet> p \<bullet> tvar) (p \<bullet> tkind) (pa \<bullet> p \<bullet> ty)" in subst)

    apply (simp only: eq_iff)

    apply (rule_tac x="-pa" in exI)

    apply (simp add: alphas eqvts eqvts_raw supp_abs fv_supp)

    apply (rule_tac t="supp (pa \<bullet> p \<bullet> ty) - {atom (pa \<bullet> p \<bullet> tvar)}"

                and s="pa \<bullet> (p \<bullet> supp ty - {p \<bullet> atom tvar})" in subst)

    apply (simp add: eqvts)

    apply (simp add: eqvts[symmetric])

    apply (rule conjI)

    apply (rule supp_perm_eq)

    apply (simp add: eqvts)

    apply (subst supp_finite_atom_set)

    apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)

    apply (simp add: eqvts)

    apply (simp add: eqvts)

    apply (subst supp_perm_eq)

    apply (subst supp_finite_atom_set)

    apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)

    apply (simp add: eqvts)

    apply assumption

    apply (simp add: fresh_star_minus_perm)

    apply (rule a08)

    apply simp

    apply(rotate_tac 1)

    apply(erule_tac x="(pa + p)" in allE)

    apply simp

    apply (simp add: eqvts eqvts_raw)

    apply (rule at_set_avoiding2_atom)

    apply (simp add: finite_supp)

    apply (simp add: finite_supp)

    apply (simp add: fresh_def)

    apply (simp only: supp_abs eqvts)

    apply blast

(* GOAL2 *)

    apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (atom (p \<bullet> tvar))) \<sharp> e \<and>

                       supp (Abs (p \<bullet> {atom tvar}) (p \<bullet> co)) \<sharp>* pa)")

    apply clarify

    apply (simp only: perm)

    apply(rule_tac t="CAll (p \<bullet> tvar) (p \<bullet> ckind) (p \<bullet> co)"

               and s="CAll (pa \<bullet> p \<bullet> tvar) (p \<bullet> ckind) (pa \<bullet> p \<bullet> co)" in subst)

    apply (simp only: eq_iff)

    apply (rule_tac x="-pa" in exI)

    apply (simp add: alphas eqvts eqvts_raw supp_abs fv_supp)

    apply (rule_tac t="supp (pa \<bullet> p \<bullet> co) - {atom (pa \<bullet> p \<bullet> tvar)}"

                and s="pa \<bullet> (p \<bullet> supp co - {p \<bullet> atom tvar})" in subst)

    apply (simp add: eqvts)

    apply (simp add: eqvts[symmetric])

    apply (rule conjI)

    apply (rule supp_perm_eq)

    apply (simp add: eqvts)

    apply (subst supp_finite_atom_set)

    apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)

    apply (simp add: eqvts)

    apply (simp add: eqvts)

    apply (subst supp_perm_eq)

    apply (subst supp_finite_atom_set)

    apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)

    apply (simp add: eqvts)

    apply assumption

    apply (simp add: fresh_star_minus_perm)

    apply (rule a15)

    apply simp

    apply(rotate_tac 1)

    apply(erule_tac x="(pa + p)" in allE)

    apply simp

    apply (simp add: eqvts eqvts_raw)

    apply (rule at_set_avoiding2_atom)

    apply (simp add: finite_supp)

    apply (simp add: finite_supp)

    apply (simp add: fresh_def)

    apply (simp only: supp_abs eqvts)

    apply blast

(* GOAL3 a copy-and-paste of Goal2 with consts and variable names changed *)

    apply(subgoal_tac "\<exists>pa. ((pa \<bullet> (atom (p \<bullet> tvar))) \<sharp> g \<and>

                       supp (Abs (p \<bullet> {atom tvar}) (p \<bullet> trm)) \<sharp>* pa)")

    apply clarify

    apply (simp only: perm)

    apply(rule_tac t="LAMT (p \<bullet> tvar) (p \<bullet> tkind) (p \<bullet> trm)"

               and s="LAMT (pa \<bullet> p \<bullet> tvar) (p \<bullet> tkind) (pa \<bullet> p \<bullet> trm)" in subst)

    apply (simp only: eq_iff)

    apply (rule_tac x="-pa" in exI)

    apply (simp add: alphas eqvts eqvts_raw supp_abs fv_supp)

    apply (rule_tac t="supp (pa \<bullet> p \<bullet> trm) - {atom (pa \<bullet> p \<bullet> tvar)}"

                and s="pa \<bullet> (p \<bullet> supp trm - {p \<bullet> atom tvar})" in subst)

    apply (simp add: eqvts)

    apply (simp add: eqvts[symmetric])

    apply (rule conjI)

    apply (rule supp_perm_eq)

    apply (simp add: eqvts)

    apply (subst supp_finite_atom_set)

    apply (simp add: eqvts[symmetric] finite_eqvt[symmetric] finite_supp)

    apply (simp add: eqvts)

    apply (simp add: eqvts)

    apply (subst supp_perm_eq)