src/HOL/Nominal/Examples/Fsub.thy

more markup for inner and outer syntax;
added enclose;

(* $Id$ *)

(*<*)
theory Fsub
imports "../Nominal" 
begin
(*>*)

text{* Authors: Christian Urban,
                Benjamin Pierce,
                Dimitrios Vytiniotis
                Stephanie Weirich and
                Steve Zdancewic

       with great help from Stefan Berghofer and  Markus Wenzel. *}

section {* Types for Names, Nominal Datatype Declaration for Types and Terms *}

text {* The main point of this solution is to use names everywhere (be they bound, 
  binding or free). In System \FSUB{} there are two kinds of names corresponding to 
  type-variables and to term-variables. These two kinds of names are represented in 
  the nominal datatype package as atom-types @{text "tyvrs"} and @{text "vrs"}: *}

atom_decl tyvrs vrs

text{* There are numerous facts that come with this declaration: for example that 
  there are infinitely many elements in @{text "tyvrs"} and @{text "vrs"}. *}

text{* The constructors for types and terms in System \FSUB{} contain abstractions 
  over type-variables and term-variables. The nominal datatype-package uses 
  @{text "\<guillemotleft>\<dots>\<guillemotright>\<dots>"} to indicate where abstractions occur. *}

nominal_datatype ty = 
    Tvar   "tyvrs"
  | Top
  | Arrow  "ty" "ty"          ("_ \<rightarrow> _" [100,100] 100)
  | Forall "\<guillemotleft>tyvrs\<guillemotright>ty" "ty" 

nominal_datatype trm = 
    Var   "vrs"
  | Lam   "\<guillemotleft>vrs\<guillemotright>trm" "ty" 
  | Tabs  "\<guillemotleft>tyvrs\<guillemotright>trm" "ty"
  | App   "trm" "trm"
  | Tapp  "trm" "ty"

text {* To be polite to the eye, some more familiar notation is introduced. 
  Because of the change in the order of arguments, one needs to use 
  translation rules, instead of syntax annotations at the term-constructors
  as given above for @{term "Arrow"}. *}

syntax
  Forall_syn :: "tyvrs \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> ty" ("\<forall>[_<:_]._" [100,100,100] 100)
  Lam_syn    :: "vrs \<Rightarrow> ty \<Rightarrow> trm \<Rightarrow> trm"   ("Lam [_:_]._" [100,100,100] 100)
  Tabs_syn   :: "tyvrs \<Rightarrow> ty \<Rightarrow> trm \<Rightarrow> trm" ("Tabs [_<:_]._" [100,100,100] 100)

translations 
  "\<forall>[X<:T\<^isub>1].T\<^isub>2" \<rightleftharpoons> "ty.Forall X T\<^isub>2 T\<^isub>1"
  "Lam [x:T].t" \<rightleftharpoons> "trm.Lam x t T"
  "Tabs [X<:T].t" \<rightleftharpoons> "trm.Tabs X t T"

text {* Again there are numerous facts that are proved automatically for @{typ "ty"} 
  and @{typ "trm"}: for example that the set of free variables, i.e.~the @{text "support"}, 
  is finite. However note that nominal-datatype declarations do \emph{not} define 
  ``classical" constructor-based datatypes, but rather define $\alpha$-equivalence 
  classes---we can for example show that $\alpha$-equivalent @{typ "ty"}s 
  and @{typ "trm"}s are equal: *}

lemma alpha_illustration:
  shows "\<forall>[X<:T].(Tvar X) = \<forall>[Y<:T].(Tvar Y)" 
  and "Lam [x:T].(Var x) = Lam [y:T].(Var y)"
  by (simp_all add: ty.inject trm.inject alpha calc_atm fresh_atm)

section {* SubTyping Contexts *}

types ty_context = "(tyvrs\<times>ty) list"

text {* Typing contexts are represented as lists that ``grow" on the left; we
  thereby deviating from the convention in the POPLmark-paper. The lists contain
  pairs of type-variables and types (this is sufficient for Part 1A). *}

text {* In order to state validity-conditions for typing-contexts, the notion of
  a @{text "domain"} of a typing-context is handy. *}

consts
  "domain" :: "ty_context \<Rightarrow> tyvrs set"
primrec
  "domain [] = {}"
  "domain (X#\<Gamma>) = {fst X}\<union>(domain \<Gamma>)" 

lemma domain_eqvt[eqvt]:
  fixes pi::"tyvrs prm"
  and   pi'::"vrs prm"
  shows "pi\<bullet>(domain \<Gamma>) = domain (pi\<bullet>\<Gamma>)"
  and   "pi'\<bullet>(domain \<Gamma>) = domain (pi'\<bullet>\<Gamma>)"
  by (induct \<Gamma>) (simp_all add: eqvts)

lemma finite_domain:
  shows "finite (domain \<Gamma>)"
  by (induct \<Gamma>, auto)

lemma domain_supp:
  shows "(supp (domain \<Gamma>)) = (domain \<Gamma>)"
  by (simp only: at_fin_set_supp at_tyvrs_inst finite_domain)

lemma domain_inclusion:
  assumes a: "(X,T)\<in>set \<Gamma>" 
  shows "X\<in>(domain \<Gamma>)"
  using a by (induct \<Gamma>, auto)

lemma domain_existence:
  assumes a: "X\<in>(domain \<Gamma>)" 
  shows "\<exists>T.(X,T)\<in>set \<Gamma>"
  using a by (induct \<Gamma>, auto)

lemma domain_append:
  shows "domain (\<Gamma>@\<Delta>) = ((domain \<Gamma>) \<union> (domain \<Delta>))"
  by (induct \<Gamma>, auto)

lemma fresh_domain_cons:
  fixes X::"tyvrs"
  shows "X\<sharp>(domain (Y#\<Gamma>)) = (X\<sharp>(fst Y) \<and> X\<sharp>(domain \<Gamma>))"
  by (simp add: fresh_fin_insert pt_tyvrs_inst at_tyvrs_inst fs_tyvrs_inst finite_domain)

lemma fresh_domain:
  fixes X::"tyvrs"
  assumes a: "X\<sharp>\<Gamma>" 
  shows "X\<sharp>(domain \<Gamma>)"
using a
apply(induct \<Gamma>)
apply(simp add: fresh_set_empty) 
apply(simp only: fresh_domain_cons)
apply(auto simp add: fresh_prod fresh_list_cons) 
done

text {* Not all lists of type @{typ "ty_context"} are well-formed. One condition
  requires that in @{term "(X,S)#\<Gamma>"} all free variables of @{term "S"} must be 
  in the @{term "domain"} of @{term "\<Gamma>"}, that is @{term "S"} must be @{text "closed"} 
  in @{term "\<Gamma>"}. The set of free variables of @{term "S"} is the 
  @{text "support"} of @{term "S"}. *}

constdefs
  "closed_in" :: "ty \<Rightarrow> ty_context \<Rightarrow> bool" ("_ closed'_in _" [100,100] 100)
  "S closed_in \<Gamma> \<equiv> (supp S)\<subseteq>(domain \<Gamma>)"

lemma closed_in_eqvt[eqvt]:
  fixes pi::"tyvrs prm"
  assumes a: "S closed_in \<Gamma>" 
  shows "(pi\<bullet>S) closed_in (pi\<bullet>\<Gamma>)"
  using a
proof (unfold "closed_in_def")
  assume "supp S \<subseteq> (domain \<Gamma>)" 
  hence "pi\<bullet>(supp S) \<subseteq> pi\<bullet>(domain \<Gamma>)"
    by (simp add: pt_subseteq_eqvt[OF pt_tyvrs_inst, OF at_tyvrs_inst])
  thus "(supp (pi\<bullet>S)) \<subseteq> (domain (pi\<bullet>\<Gamma>))"
    by (simp add: domain_eqvt pt_perm_supp[OF pt_tyvrs_inst, OF at_tyvrs_inst])
qed

lemma ty_vrs_prm_simp:
  fixes pi::"vrs prm"
  and   S::"ty"
  shows "pi\<bullet>S = S"
by (induct S rule: ty.weak_induct) (auto simp add: calc_atm)

lemma ty_context_vrs_prm_simp:
  fixes pi::"vrs prm"
  and   \<Gamma>::"ty_context"
  shows "pi\<bullet>\<Gamma> = \<Gamma>"
by (induct \<Gamma>) 
   (auto simp add: calc_atm ty_vrs_prm_simp)

lemma closed_in_eqvt'[eqvt]:
  fixes pi::"vrs prm"
  assumes a: "S closed_in \<Gamma>" 
  shows "(pi\<bullet>S) closed_in (pi\<bullet>\<Gamma>)"
using a
by (simp add: ty_vrs_prm_simp ty_context_vrs_prm_simp)

text {* Now validity of a context is a straightforward inductive definition. *}
  
inductive2 
  valid_rel :: "ty_context \<Rightarrow> bool" ("\<turnstile> _ ok" [100] 100)
where
  valid_nil[simp]:  "\<turnstile> [] ok"
| valid_cons[simp]: "\<lbrakk>\<turnstile> \<Gamma> ok; X\<sharp>(domain \<Gamma>); T closed_in \<Gamma>\<rbrakk>  \<Longrightarrow>  \<turnstile> ((X,T)#\<Gamma>) ok"

equivariance valid_rel

lemma validE:
  assumes a: "\<turnstile> ((X,T)#\<Gamma>) ok"
  shows "\<turnstile> \<Gamma> ok \<and> X\<sharp>(domain \<Gamma>) \<and> T closed_in \<Gamma>"
using a by (cases, auto)

lemma validE_append:
  assumes a: "\<turnstile> (\<Delta>@\<Gamma>) ok" 
  shows "\<turnstile> \<Gamma> ok"
  using a by (induct \<Delta>, auto dest: validE)

lemma replace_type:
  assumes a: "\<turnstile> (\<Delta>@(X,T)#\<Gamma>) ok"
  and     b: "S closed_in \<Gamma>"
  shows "\<turnstile> (\<Delta>@(X,S)#\<Gamma>) ok"
using a b
apply(induct \<Delta>)
apply(auto dest!: validE intro!: valid_cons simp add: domain_append closed_in_def)
done

text {* Well-formed contexts have a unique type-binding for a type-variable. *} 

lemma uniqueness_of_ctxt:
  fixes \<Gamma>::"ty_context"
  assumes a: "\<turnstile> \<Gamma> ok"
  and     b: "(X,T)\<in>set \<Gamma>"
  and     c: "(X,S)\<in>set \<Gamma>"
  shows "T=S"
using a b c
proof (induct)
  case valid_nil thus "T=S" by simp
next
  case valid_cons
  moreover
  { fix \<Gamma>::"ty_context"
    assume a: "X\<sharp>(domain \<Gamma>)" 
    have "\<not>(\<exists>T.(X,T)\<in>(set \<Gamma>))" using a 
    proof (induct \<Gamma>)
      case (Cons Y \<Gamma>)
      thus "\<not> (\<exists>T.(X,T)\<in>set(Y#\<Gamma>))" 
	by (simp only: fresh_domain_cons, auto simp add: fresh_atm)
    qed (simp)
  }
  ultimately show "T=S" by auto
qed 

section {* Size and Capture-Avoiding Substitution for Types *}

consts size_ty :: "ty \<Rightarrow> nat"

nominal_primrec
  "size_ty (Tvar X) = 1"
  "size_ty (Top) = 1"
  "size_ty (T1 \<rightarrow> T2) = (size_ty T1) + (size_ty T2) + 1"
  "X\<sharp>T1 \<Longrightarrow> size_ty (\<forall>[X<:T1].T2) = (size_ty T1) + (size_ty T2) + 1"
  apply (finite_guess)+
  apply (rule TrueI)+
  apply (simp add: fresh_nat)
  apply (fresh_guess)+
  done

consts subst_ty :: "tyvrs \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> ty"

syntax 
 subst_ty_syn :: "ty \<Rightarrow> tyvrs \<Rightarrow> ty \<Rightarrow> ty" ("_[_:=_]\<^isub>t\<^isub>y" [100,100,100] 100)

translations 
  "T1[Y:=T2]\<^isub>t\<^isub>y" \<rightleftharpoons> "subst_ty Y T2 T1"

nominal_primrec
  "(Tvar X)[Y:=T]\<^isub>t\<^isub>y= (if X=Y then T else (Tvar X))"
  "(Top)[Y:=T]\<^isub>t\<^isub>y = Top"
  "(T\<^isub>1 \<rightarrow> T\<^isub>2)[Y:=T]\<^isub>t\<^isub>y = (T\<^isub>1[Y:=T]\<^isub>t\<^isub>y) \<rightarrow> (T\<^isub>2[Y:=T]\<^isub>t\<^isub>y)"
  "\<lbrakk>X\<sharp>(Y,T); X\<sharp>T\<^isub>1\<rbrakk> \<Longrightarrow> (\<forall>[X<:T\<^isub>1].T\<^isub>2)[Y:=T]\<^isub>t\<^isub>y = (\<forall>[X<:(T\<^isub>1[Y:=T]\<^isub>t\<^isub>y)].(T\<^isub>2[Y:=T]\<^isub>t\<^isub>y))"
  apply (finite_guess)+
  apply (rule TrueI)+
  apply (simp add: abs_fresh)
  apply (fresh_guess)+
  done

consts 
  subst_tyc :: "ty_context \<Rightarrow> tyvrs \<Rightarrow> ty \<Rightarrow> ty_context" ("_[_:=_]\<^isub>t\<^isub>y\<^isub>c" [100,100,100] 100)
primrec
"([])[Y:=T]\<^isub>t\<^isub>y\<^isub>c= []"
"(XT#\<Gamma>)[Y:=T]\<^isub>t\<^isub>y\<^isub>c = (fst XT,(snd XT)[Y:=T]\<^isub>t\<^isub>y)#(\<Gamma>[Y:=T]\<^isub>t\<^isub>y\<^isub>c)"
 
section {* Subtyping-Relation *}

text {* The definition for the subtyping-relation follows quite closely what is written 
  in the POPLmark-paper, except for the premises dealing with well-formed contexts and 
  the freshness constraint @{term "X\<sharp>\<Gamma>"} in the @{text "S_Forall"}-rule. (The freshness
  constraint is specific to the \emph{nominal approach}. Note, however, that the constraint
  does \emph{not} make the subtyping-relation ``partial"\ldots because we work over
  $\alpha$-equivalence classes.) *}

inductive2 
  subtype_of :: "ty_context \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   ("_\<turnstile>_<:_" [100,100,100] 100)
where
  S_Top[intro]:    "\<lbrakk>\<turnstile> \<Gamma> ok; S closed_in \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> S <: Top"
| S_Var[intro]:    "\<lbrakk>(X,S) \<in> set \<Gamma>; \<Gamma> \<turnstile> S <: T\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> (Tvar X) <: T"
| S_Refl[intro]:   "\<lbrakk>\<turnstile> \<Gamma> ok; X \<in> domain \<Gamma>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> Tvar X <: Tvar X"
| S_Arrow[intro]:  "\<lbrakk>\<Gamma> \<turnstile> T\<^isub>1 <: S\<^isub>1; \<Gamma> \<turnstile> S\<^isub>2 <: T\<^isub>2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> (S\<^isub>1 \<rightarrow> S\<^isub>2) <: (T\<^isub>1 \<rightarrow> T\<^isub>2)" 
| S_Forall[intro]: "\<lbrakk>\<Gamma> \<turnstile> T\<^isub>1 <: S\<^isub>1; X\<sharp>\<Gamma>; ((X,T\<^isub>1)#\<Gamma>) \<turnstile> S\<^isub>2 <: T\<^isub>2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> \<forall>[X<:S\<^isub>1].S\<^isub>2 <: \<forall>[X<:T\<^isub>1].T\<^isub>2"

lemma subtype_implies_ok:
  fixes X::"tyvrs"
  assumes a: "\<Gamma> \<turnstile> S <: T"
  shows "\<turnstile> \<Gamma> ok"  
using a by (induct) (auto)

lemma subtype_implies_closed:
  assumes a: "\<Gamma> \<turnstile> S <: T"
  shows "S closed_in \<Gamma> \<and> T closed_in \<Gamma>"
using a
proof (induct)
  case (S_Top \<Gamma> S)
  have "Top closed_in \<Gamma>" by (simp add: closed_in_def ty.supp)
  moreover
  have "S closed_in \<Gamma>" by fact
  ultimately show "S closed_in \<Gamma> \<and> Top closed_in \<Gamma>" by simp
next
  case (S_Var X S \<Gamma> T)
  have "(X,S)\<in>set \<Gamma>" by fact
  hence "X \<in> domain \<Gamma>" by (rule domain_inclusion)
  hence "(Tvar X) closed_in \<Gamma>" by (simp add: closed_in_def ty.supp supp_atm)
  moreover
  have "S closed_in \<Gamma> \<and> T closed_in \<Gamma>" by fact
  hence "T closed_in \<Gamma>" by force
  ultimately show "(Tvar X) closed_in \<Gamma> \<and> T closed_in \<Gamma>" by simp
qed (auto simp add: closed_in_def ty.supp supp_atm abs_supp)

lemma subtype_implies_fresh:
  fixes X::"tyvrs"
  assumes a1: "\<Gamma> \<turnstile> S <: T"
  and     a2: "X\<sharp>\<Gamma>"
  shows "X\<sharp>S \<and> X\<sharp>T"  
proof -
  from a1 have "\<turnstile> \<Gamma> ok" by (rule subtype_implies_ok)
  with a2 have "X\<sharp>domain(\<Gamma>)" by (simp add: fresh_domain)
  moreover
  from a1 have "S closed_in \<Gamma> \<and> T closed_in \<Gamma>" by (rule subtype_implies_closed)
  hence "supp S \<subseteq> ((supp (domain \<Gamma>))::tyvrs set)" 
    and "supp T \<subseteq> ((supp (domain \<Gamma>))::tyvrs set)" by (simp_all add: domain_supp closed_in_def)
  ultimately show "X\<sharp>S \<and> X\<sharp>T" by (force simp add: supp_prod fresh_def)
qed

equivariance subtype_of

nominal_inductive subtype_of  
  by (simp_all add: abs_fresh subtype_implies_fresh)

thm subtype_of.strong_induct

section {* Reflexivity of Subtyping *}

lemma subtype_reflexivity:
  assumes a: "\<turnstile> \<Gamma> ok"
  and b: "T closed_in \<Gamma>"
  shows "\<Gamma> \<turnstile> T <: T"
using a b
proof(nominal_induct T avoiding: \<Gamma> rule: ty.induct)
  case (Forall X T\<^isub>1 T\<^isub>2)
  have ih_T\<^isub>1: "\<And>\<Gamma>. \<lbrakk>\<turnstile> \<Gamma> ok; T\<^isub>1 closed_in \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> T\<^isub>1 <: T\<^isub>1" by fact 
  have ih_T\<^isub>2: "\<And>\<Gamma>. \<lbrakk>\<turnstile> \<Gamma> ok; T\<^isub>2 closed_in \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> T\<^isub>2 <: T\<^isub>2" by fact
  have fresh_cond: "X\<sharp>\<Gamma>" by fact
  hence fresh_domain: "X\<sharp>(domain \<Gamma>)" by (simp add: fresh_domain)
  have "(\<forall>[X<:T\<^isub>2].T\<^isub>1) closed_in \<Gamma>" by fact
  hence closed\<^isub>T\<^isub>2: "T\<^isub>2 closed_in \<Gamma>" and closed\<^isub>T\<^isub>1: "T\<^isub>1 closed_in ((X,T\<^isub>2)#\<Gamma>)" 
    by (auto simp add: closed_in_def ty.supp abs_supp)
  have ok: "\<turnstile> \<Gamma> ok" by fact  
  hence ok': "\<turnstile> ((X,T\<^isub>2)#\<Gamma>) ok" using closed\<^isub>T\<^isub>2 fresh_domain by simp
  have "\<Gamma> \<turnstile> T\<^isub>2 <: T\<^isub>2" using ih_T\<^isub>2 closed\<^isub>T\<^isub>2 ok by simp
  moreover
  have "((X,T\<^isub>2)#\<Gamma>) \<turnstile> T\<^isub>1 <: T\<^isub>1" using ih_T\<^isub>1 closed\<^isub>T\<^isub>1 ok' by simp
  ultimately show "\<Gamma> \<turnstile> \<forall>[X<:T\<^isub>2].T\<^isub>1 <: \<forall>[X<:T\<^isub>2].T\<^isub>1" using fresh_cond 
    by (simp add: subtype_of.S_Forall)
qed (auto simp add: closed_in_def ty.supp supp_atm)

lemma subtype_reflexivity_semiautomated:
  assumes a: "\<turnstile> \<Gamma> ok"
  and     b: "T closed_in \<Gamma>"
  shows "\<Gamma> \<turnstile> T <: T"
using a b
apply(nominal_induct T avoiding: \<Gamma> rule: ty.induct)
apply(auto simp add: ty.supp abs_supp supp_atm closed_in_def)
  --{* Too bad that this instantiation cannot be found automatically by
  \isakeyword{auto}; \isakeyword{blast} would find it if we had not used 
  an explicit definition for @{text "closed_in_def"}. *}
apply(drule_tac x="(tyvrs, ty2)#\<Gamma>" in meta_spec)
apply(force dest: fresh_domain simp add: closed_in_def)
done


section {* Weakening *}

text {* In order to prove weakening we introduce the notion of a type-context extending 
  another. This generalization seems to make the proof for weakening to be
  smoother than if we had strictly adhered to the version in the POPLmark-paper. *}

constdefs 
  extends :: "ty_context \<Rightarrow> ty_context \<Rightarrow> bool" ("_ extends _" [100,100] 100)
  "\<Delta> extends \<Gamma> \<equiv> \<forall>X Q. (X,Q)\<in>set \<Gamma> \<longrightarrow> (X,Q)\<in>set \<Delta>"

lemma extends_domain:
  assumes a: "\<Delta> extends \<Gamma>"
  shows "domain \<Gamma> \<subseteq> domain \<Delta>"
  using a 
  apply (auto simp add: extends_def)
  apply (drule domain_existence)
  apply (force simp add: domain_inclusion)
  done

lemma extends_closed:
  assumes a1: "T closed_in \<Gamma>"
  and     a2: "\<Delta> extends \<Gamma>"
  shows "T closed_in \<Delta>"
  using a1 a2
  by (auto dest: extends_domain simp add: closed_in_def)

lemma extends_memb:
  assumes a: "\<Delta> extends \<Gamma>"
  and b: "(X,T) \<in> set \<Gamma>"
  shows "(X,T) \<in> set \<Delta>"
  using a b by (simp add: extends_def)

lemma weakening:
  assumes a: "\<Gamma> \<turnstile> S <: T"
  and b: "\<turnstile> \<Delta> ok"
  and c: "\<Delta> extends \<Gamma>"
  shows "\<Delta> \<turnstile> S <: T"
  using a b c 
proof (nominal_induct \<Gamma> S T avoiding: \<Delta> rule: subtype_of.strong_induct)
  case (S_Top \<Gamma> S) 
  have lh_drv_prem: "S closed_in \<Gamma>" by fact
  have "\<turnstile> \<Delta> ok" by fact
  moreover
  have "\<Delta> extends \<Gamma>" by fact
  hence "S closed_in \<Delta>" using lh_drv_prem by (simp only: extends_closed)
  ultimately show "\<Delta> \<turnstile> S <: Top" by force
next 
  case (S_Var X S \<Gamma> T)
  have lh_drv_prem: "(X,S) \<in> set \<Gamma>" by fact
  have ih: "\<And>\<Delta>. \<turnstile> \<Delta> ok \<Longrightarrow> \<Delta> extends \<Gamma> \<Longrightarrow> \<Delta> \<turnstile> S <: T" by fact
  have ok: "\<turnstile> \<Delta> ok" by fact
  have extends: "\<Delta> extends \<Gamma>" by fact
  have "(X,S) \<in> set \<Delta>" using lh_drv_prem extends by (simp only: extends_memb)
  moreover
  have "\<Delta> \<turnstile> S <: T" using ok extends ih by simp
  ultimately show "\<Delta> \<turnstile> Tvar X <: T" using ok by force
next
  case (S_Refl \<Gamma> X)
  have lh_drv_prem: "X \<in> domain \<Gamma>" by fact
  have "\<turnstile> \<Delta> ok" by fact
  moreover
  have "\<Delta> extends \<Gamma>" by fact
  hence "X \<in> domain \<Delta>" using lh_drv_prem by (force dest: extends_domain)
  ultimately show "\<Delta> \<turnstile> Tvar X <: Tvar X" by force
next 
  case (S_Arrow \<Gamma> T\<^isub>1 S\<^isub>1 S\<^isub>2 T\<^isub>2) thus "\<Delta> \<turnstile> S\<^isub>1 \<rightarrow> S\<^isub>2 <: T\<^isub>1 \<rightarrow> T\<^isub>2" by blast
next
  case (S_Forall \<Gamma> T\<^isub>1 S\<^isub>1 X S\<^isub>2 T\<^isub>2)
  have fresh_cond: "X\<sharp>\<Delta>" by fact
  hence fresh_domain: "X\<sharp>(domain \<Delta>)" by (simp add: fresh_domain)
  have ih\<^isub>1: "\<And>\<Delta>. \<turnstile> \<Delta> ok \<Longrightarrow> \<Delta> extends \<Gamma> \<Longrightarrow> \<Delta> \<turnstile> T\<^isub>1 <: S\<^isub>1" by fact
  have ih\<^isub>2: "\<And>\<Delta>. \<turnstile> \<Delta> ok \<Longrightarrow> \<Delta> extends ((X,T\<^isub>1)#\<Gamma>) \<Longrightarrow> \<Delta> \<turnstile> S\<^isub>2 <: T\<^isub>2" by fact
  have lh_drv_prem: "\<Gamma> \<turnstile> T\<^isub>1 <: S\<^isub>1" by fact
  hence closed\<^isub>T\<^isub>1: "T\<^isub>1 closed_in \<Gamma>" by (simp add: subtype_implies_closed) 
  have ok: "\<turnstile> \<Delta> ok" by fact
  have ext: "\<Delta> extends \<Gamma>" by fact
  have "T\<^isub>1 closed_in \<Delta>" using ext closed\<^isub>T\<^isub>1 by (simp only: extends_closed)
  hence "\<turnstile> ((X,T\<^isub>1)#\<Delta>) ok" using fresh_domain ok by force   
  moreover 
  have "((X,T\<^isub>1)#\<Delta>) extends ((X,T\<^isub>1)#\<Gamma>)" using ext by (force simp add: extends_def)
  ultimately have "((X,T\<^isub>1)#\<Delta>) \<turnstile> S\<^isub>2 <: T\<^isub>2" using ih\<^isub>2 by simp
  moreover
  have "\<Delta> \<turnstile> T\<^isub>1 <: S\<^isub>1" using ok ext ih\<^isub>1 by simp 
  ultimately show "\<Delta> \<turnstile> \<forall>[X<:S\<^isub>1].S\<^isub>2 <: \<forall>[X<:T\<^isub>1].T\<^isub>2" using ok by (force intro: S_Forall)
qed

text {* In fact all ``non-binding" cases can be solved automatically: *}

lemma weakening_more_automated:
  assumes a: "\<Gamma> \<turnstile> S <: T"
  and b: "\<turnstile> \<Delta> ok"
  and c: "\<Delta> extends \<Gamma>"
  shows "\<Delta> \<turnstile> S <: T"
  using a b c 
proof (nominal_induct \<Gamma> S T avoiding: \<Delta> rule: subtype_of.strong_induct)
  case (S_Forall \<Gamma> T\<^isub>1 S\<^isub>1 X S\<^isub>2 T\<^isub>2)
  have fresh_cond: "X\<sharp>\<Delta>" by fact
  hence fresh_domain: "X\<sharp>(domain \<Delta>)" by (simp add: fresh_domain)
  have ih\<^isub>1: "\<And>\<Delta>. \<turnstile> \<Delta> ok \<Longrightarrow> \<Delta> extends \<Gamma> \<Longrightarrow> \<Delta> \<turnstile> T\<^isub>1 <: S\<^isub>1" by fact
  have ih\<^isub>2: "\<And>\<Delta>. \<turnstile> \<Delta> ok \<Longrightarrow> \<Delta> extends ((X,T\<^isub>1)#\<Gamma>) \<Longrightarrow> \<Delta> \<turnstile> S\<^isub>2 <: T\<^isub>2" by fact
  have lh_drv_prem: "\<Gamma> \<turnstile> T\<^isub>1 <: S\<^isub>1" by fact
  hence closed\<^isub>T\<^isub>1: "T\<^isub>1 closed_in \<Gamma>" by (simp add: subtype_implies_closed) 
  have ok: "\<turnstile> \<Delta> ok" by fact
  have ext: "\<Delta> extends \<Gamma>" by fact
  have "T\<^isub>1 closed_in \<Delta>" using ext closed\<^isub>T\<^isub>1 by (simp only: extends_closed)
  hence "\<turnstile> ((X,T\<^isub>1)#\<Delta>) ok" using fresh_domain ok by force   
  moreover
  have "((X,T\<^isub>1)#\<Delta>) extends ((X,T\<^isub>1)#\<Gamma>)" using ext by (force simp add: extends_def)
  ultimately have "((X,T\<^isub>1)#\<Delta>) \<turnstile> S\<^isub>2 <: T\<^isub>2" using ih\<^isub>2 by simp
  moreover
  have "\<Delta> \<turnstile> T\<^isub>1 <: S\<^isub>1" using ok ext ih\<^isub>1 by simp 
  ultimately show "\<Delta> \<turnstile> \<forall>[X<:S\<^isub>1].S\<^isub>2 <: \<forall>[X<:T\<^isub>1].T\<^isub>2" using ok by (force intro: S_Forall)
qed (blast intro: extends_closed extends_memb dest: extends_domain)+

section {* Transitivity and Narrowing *}

text {* Some inversion lemmas that are needed in the transitivity and narrowing proof.*}

lemma S_TopE:
  assumes a: "\<Gamma> \<turnstile> Top <: T"
  shows "T = Top"
using a by (cases, auto) 

lemma S_ArrowE_left:
  assumes a: "\<Gamma> \<turnstile> S\<^isub>1 \<rightarrow> S\<^isub>2 <: T" 
  shows "T = Top \<or> (\<exists>T\<^isub>1 T\<^isub>2. T = T\<^isub>1 \<rightarrow> T\<^isub>2 \<and> \<Gamma> \<turnstile> T\<^isub>1 <: S\<^isub>1 \<and> \<Gamma> \<turnstile> S\<^isub>2 <: T\<^isub>2)"
using a by (cases, auto simp add: ty.inject)

lemma S_ForallE_left:
  shows "\<lbrakk>\<Gamma> \<turnstile> \<forall>[X<:S\<^isub>1].S\<^isub>2 <: T; X\<sharp>\<Gamma>; X\<sharp>S\<^isub>1\<rbrakk>
         \<Longrightarrow> T = Top \<or> (\<exists>T\<^isub>1 T\<^isub>2. T = \<forall>[X<:T\<^isub>1].T\<^isub>2 \<and> \<Gamma> \<turnstile> T\<^isub>1 <: S\<^isub>1 \<and> ((X,T\<^isub>1)#\<Gamma>) \<turnstile> S\<^isub>2 <: T\<^isub>2)"
  apply(frule subtype_implies_ok)
  apply(ind_cases2 "\<Gamma> \<turnstile> \<forall>[X<:S\<^isub>1].S\<^isub>2 <: T")
  apply(auto simp add: ty.inject alpha)
  apply(rule_tac x="[(X,Xa)]\<bullet>T\<^isub>2" in exI)
  apply(rule conjI)
  apply(rule sym)
  apply(rule pt_bij2[OF pt_tyvrs_inst, OF at_tyvrs_inst])
  apply(rule pt_tyvrs3)
  apply(simp)
  apply(rule at_ds5[OF at_tyvrs_inst])
  apply(rule conjI)
  apply(simp add: pt_fresh_left[OF pt_tyvrs_inst, OF at_tyvrs_inst] calc_atm)
  apply(drule_tac \<Gamma>="((Xa,T\<^isub>1)#\<Gamma>)" in  subtype_implies_closed)+
  apply(simp add: closed_in_def)
  apply(drule fresh_domain)+
  apply(simp add: fresh_def)
  apply(subgoal_tac "X \<notin> (insert Xa (domain \<Gamma>))")(*A*)
  apply(force)
  (*A*)apply(simp add: at_fin_set_supp[OF at_tyvrs_inst, OF finite_domain])
  (* 2nd conjunct *)apply(frule_tac X="X" in subtype_implies_fresh)
  apply(assumption)
  apply(drule_tac X="Xa" in subtype_implies_fresh)
  apply(assumption)
  apply(simp add: fresh_prod)
  apply(drule_tac pi="[(X,Xa)]" in subtype_of.eqvt(2))
  apply(simp add: calc_atm)
  apply(simp add: pt_fresh_fresh[OF pt_tyvrs_inst, OF at_tyvrs_inst])
  done

text {* Next we prove the transitivity and narrowing for the subtyping-relation. 
The POPLmark-paper says the following:

\begin{quote}
\begin{lemma}[Transitivity and Narrowing] \
\begin{enumerate}
\item If @{term "\<Gamma> \<turnstile> S<:Q"} and @{term "\<Gamma> \<turnstile> Q<:T"}, then @{term "\<Gamma> \<turnstile> S<:T"}.
\item If @{text "\<Gamma>,X<:Q,\<Delta> \<turnstile> M<:N"} and @{term "\<Gamma> \<turnstile> P<:Q"} then @{text "\<Gamma>,X<:P,\<Delta> \<turnstile> M<:N"}.
\end{enumerate}
\end{lemma}

The two parts are proved simultaneously, by induction on the size
of @{term "Q"}.  The argument for part (2) assumes that part (1) has 
been established already for the @{term "Q"} in question; part (1) uses 
part (2) only for strictly smaller @{term "Q"}.
\end{quote}

For the induction on the size of @{term "Q"}, we use the induction-rule 
@{text "measure_induct_rule"}:

\begin{center}
@{thm measure_induct_rule[of "size_ty",no_vars]}
\end{center}

That means in order to show a property @{term "P a"} for all @{term "a"}, 
the induct-rule requires to prove that for all @{term x} @{term "P x"} holds using the 
assumption that for all @{term y} whose size is strictly smaller than 
that of @{term x} the property @{term "P y"} holds. *}

lemma 
  shows trans: "\<Gamma>\<turnstile>S<:Q \<Longrightarrow> \<Gamma>\<turnstile>Q<:T \<Longrightarrow> \<Gamma>\<turnstile>S<:T" 
  and narrow: "(\<Delta>@[(X,Q)]@\<Gamma>)\<turnstile>M<:N \<Longrightarrow> \<Gamma>\<turnstile>P<:Q \<Longrightarrow> (\<Delta>@[(X,P)]@\<Gamma>)\<turnstile>M<:N"
proof (induct Q arbitrary: \<Gamma> S T \<Delta> X P M N taking: "size_ty" rule: measure_induct_rule)
  case (less Q)
    --{* \begin{minipage}[t]{0.9\textwidth}
    First we mention the induction hypotheses of the outer induction for later
    reference:\end{minipage}*}
  have IH_trans:  
    "\<And>Q' \<Gamma> S T. \<lbrakk>size_ty Q' < size_ty Q; \<Gamma>\<turnstile>S<:Q'; \<Gamma>\<turnstile>Q'<:T\<rbrakk> \<Longrightarrow> \<Gamma>\<turnstile>S<:T" by fact
  have IH_narrow:
    "\<And>Q' \<Delta> \<Gamma> X M N P. \<lbrakk>size_ty Q' < size_ty Q; (\<Delta>@[(X,Q')]@\<Gamma>)\<turnstile>M<:N; \<Gamma>\<turnstile>P<:Q'\<rbrakk> 
    \<Longrightarrow> (\<Delta>@[(X,P)]@\<Gamma>)\<turnstile>M<:N" by fact
    --{* \begin{minipage}[t]{0.9\textwidth}
    We proceed with the transitivity proof as an auxiliary lemma, because it needs 
    to be referenced in the narrowing proof.\end{minipage}*}
  have transitivity_aux:
    "\<And>\<Gamma> S T. \<lbrakk>\<Gamma> \<turnstile> S <: Q; \<Gamma> \<turnstile> Q <: T\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> S <: T"
  proof - 
    fix \<Gamma>' S' T
    assume "\<Gamma>' \<turnstile> S' <: Q" --{* left-hand derivation *}
      and  "\<Gamma>' \<turnstile> Q <: T"  --{* right-hand derivation *}
    thus "\<Gamma>' \<turnstile> S' <: T"
    proof (nominal_induct \<Gamma>' S' Q\<equiv>Q rule: subtype_of.strong_induct) 
      case (S_Top \<Gamma> S) 
	--{* \begin{minipage}[t]{0.9\textwidth}
	In this case the left-hand derivation is @{term "\<Gamma> \<turnstile> S <: Top"}, giving
	us @{term "\<turnstile> \<Gamma> ok"} and @{term "S closed_in \<Gamma>"}. This case is straightforward, 
	because the right-hand derivation must be of the form @{term "\<Gamma> \<turnstile> Top <: Top"} 
	giving us the equation @{term "T = Top"}.\end{minipage}*}
      hence rh_drv: "\<Gamma> \<turnstile> Top <: T" by simp
      hence T_inst: "T = Top" by (simp add: S_TopE)
      have "\<turnstile> \<Gamma> ok" 
	and "S closed_in \<Gamma>" by fact+
      hence "\<Gamma> \<turnstile> S <: Top" by (simp add: subtype_of.S_Top)
      thus "\<Gamma> \<turnstile> S <: T" using T_inst by simp
    next
      case (S_Var Y U \<Gamma>) 
	-- {* \begin{minipage}[t]{0.9\textwidth}
	In this case the left-hand derivation is @{term "\<Gamma> \<turnstile> Tvar Y <: Q"} 
	with @{term "S = Tvar Y"}. We have therefore @{term "(Y,U)"} 
	is in @{term "\<Gamma>"} and by inner induction hypothesis that @{term "\<Gamma> \<turnstile> U <: T"}. 
	By @{text "S_Var"} follows @{term "\<Gamma> \<turnstile> Tvar Y <: T"}.\end{minipage}*}
      hence IH_inner: "\<Gamma> \<turnstile> U <: T" by simp
      have "(Y,U) \<in> set \<Gamma>" by fact
      with IH_inner show "\<Gamma> \<turnstile> Tvar Y <: T" by (simp add: subtype_of.S_Var)
    next
      case (S_Refl \<Gamma> X) 
	--{* \begin{minipage}[t]{0.9\textwidth}
        In this case the left-hand derivation is @{term "\<Gamma>\<turnstile>(Tvar X) <: (Tvar X)"} with
        @{term "Q=Tvar X"}. The goal then follows immediately from the right-hand 
	derivation.\end{minipage}*}
      thus "\<Gamma> \<turnstile> Tvar X <: T" by simp
    next
      case (S_Arrow \<Gamma> Q\<^isub>1 S\<^isub>1 S\<^isub>2 Q\<^isub>2) 
	--{* \begin{minipage}[t]{0.9\textwidth}
	In this case the left-hand derivation is @{term "\<Gamma> \<turnstile> S\<^isub>1 \<rightarrow> S\<^isub>2 <: Q\<^isub>1 \<rightarrow> Q\<^isub>2"} with
        @{term "S\<^isub>1\<rightarrow>S\<^isub>2=S"} and @{term "Q\<^isub>1\<rightarrow>Q\<^isub>2=Q"}. We know that the @{text "size_ty"} of 
	@{term Q\<^isub>1} and @{term Q\<^isub>2} is smaller than that of @{term Q};
	so we can apply the outer induction hypotheses for @{term Q\<^isub>1} and @{term Q\<^isub>2}. 
	We also have the sub-derivations  @{term "\<Gamma>\<turnstile>Q\<^isub>1<:S\<^isub>1"} and @{term "\<Gamma>\<turnstile>S\<^isub>2<:Q\<^isub>2"}.
	The right-hand derivation is @{term "\<Gamma> \<turnstile> Q\<^isub>1 \<rightarrow> Q\<^isub>2 <: T"}. There exist types 
	@{text "T\<^isub>1,T\<^isub>2"} such that @{term "T=Top \<or> T=T\<^isub>1\<rightarrow>T\<^isub>2"}. The @{term "Top"}-case is 
	straightforward once we know @{term "(S\<^isub>1 \<rightarrow> S\<^isub>2) closed_in \<Gamma>"} and @{term "\<turnstile> \<Gamma> ok"}. 
	In the other case we have the sub-derivations @{term "\<Gamma>\<turnstile>T\<^isub>1<:Q\<^isub>1"} and @{term "\<Gamma>\<turnstile>Q\<^isub>2<:T\<^isub>2"}. 
	Using the outer induction hypothesis for transitivity we can derive @{term "\<Gamma>\<turnstile>T\<^isub>1<:S\<^isub>1"} 
	and @{term "\<Gamma>\<turnstile>S\<^isub>2<:T\<^isub>2"}. By rule @{text "S_Arrow"} follows @{term "\<Gamma> \<turnstile> S\<^isub>1 \<rightarrow> S\<^isub>2 <: T\<^isub>1 \<rightarrow> T\<^isub>2"},
	which is @{term "\<Gamma> \<turnstile> S\<^isub>1 \<rightarrow> S\<^isub>2 <: T\<^isub>"}.\end{minipage}*}
      hence rh_drv: "\<Gamma> \<turnstile> Q\<^isub>1 \<rightarrow> Q\<^isub>2 <: T" by simp
      from `Q\<^isub>1 \<rightarrow> Q\<^isub>2 = Q` 
      have Q\<^isub>1\<^isub>2_less: "size_ty Q\<^isub>1 < size_ty Q" "size_ty Q\<^isub>2 < size_ty Q" by auto
      have lh_drv_prm\<^isub>1: "\<Gamma> \<turnstile> Q\<^isub>1 <: S\<^isub>1" by fact
      have lh_drv_prm\<^isub>2: "\<Gamma> \<turnstile> S\<^isub>2 <: Q\<^isub>2" by fact      
      from rh_drv have "T=Top \<or> (\<exists>T\<^isub>1 T\<^isub>2. T=T\<^isub>1\<rightarrow>T\<^isub>2 \<and> \<Gamma>\<turnstile>T\<^isub>1<:Q\<^isub>1 \<and> \<Gamma>\<turnstile>Q\<^isub>2<:T\<^isub>2)" 
	by (simp add: S_ArrowE_left)  
      moreover
      have "S\<^isub>1 closed_in \<Gamma>" and "S\<^isub>2 closed_in \<Gamma>" 
	using lh_drv_prm\<^isub>1 lh_drv_prm\<^isub>2 by (simp_all add: subtype_implies_closed)
      hence "(S\<^isub>1 \<rightarrow> S\<^isub>2) closed_in \<Gamma>" by (simp add: closed_in_def ty.supp)
      moreover
      have "\<turnstile> \<Gamma> ok" using rh_drv by (rule subtype_implies_ok)
      moreover
      { assume "\<exists>T\<^isub>1 T\<^isub>2. T=T\<^isub>1\<rightarrow>T\<^isub>2 \<and> \<Gamma>\<turnstile>T\<^isub>1<:Q\<^isub>1 \<and> \<Gamma>\<turnstile>Q\<^isub>2<:T\<^isub>2"
	then obtain T\<^isub>1 T\<^isub>2 
	  where T_inst: "T = T\<^isub>1 \<rightarrow> T\<^isub>2" 
	  and   rh_drv_prm\<^isub>1: "\<Gamma> \<turnstile> T\<^isub>1 <: Q\<^isub>1"
	  and   rh_drv_prm\<^isub>2: "\<Gamma> \<turnstile> Q\<^isub>2 <: T\<^isub>2" by force
	from IH_trans[of "Q\<^isub>1"] 
	have "\<Gamma> \<turnstile> T\<^isub>1 <: S\<^isub>1" using Q\<^isub>1\<^isub>2_less rh_drv_prm\<^isub>1 lh_drv_prm\<^isub>1 by simp 
	moreover
	from IH_trans[of "Q\<^isub>2"] 
	have "\<Gamma> \<turnstile> S\<^isub>2 <: T\<^isub>2" using Q\<^isub>1\<^isub>2_less rh_drv_prm\<^isub>2 lh_drv_prm\<^isub>2 by simp
	ultimately have "\<Gamma> \<turnstile> S\<^isub>1 \<rightarrow> S\<^isub>2 <: T\<^isub>1 \<rightarrow> T\<^isub>2" by (simp add: subtype_of.S_Arrow)
	hence "\<Gamma> \<turnstile> S\<^isub>1 \<rightarrow> S\<^isub>2 <: T" using T_inst by simp
      }
      ultimately show "\<Gamma> \<turnstile> S\<^isub>1 \<rightarrow> S\<^isub>2 <: T" by blast
    next
      case (S_Forall \<Gamma> Q\<^isub>1 S\<^isub>1 X S\<^isub>2 Q\<^isub>2) 
	--{* \begin{minipage}[t]{0.9\textwidth}
	In this case the left-hand derivation is @{text "\<Gamma>\<turnstile>\<forall>[X<:S\<^isub>1].S\<^isub>2 <: \<forall>[X<:Q\<^isub>1].Q\<^isub>2"} with 
	@{text "\<forall>[X<:S\<^isub>1].S\<^isub>2=S"} and @{text "\<forall>[X<:Q\<^isub>1].Q\<^isub>2=Q"}. We therefore have the sub-derivations  
	@{term "\<Gamma>\<turnstile>Q\<^isub>1<:S\<^isub>1"} and @{term "((X,Q\<^isub>1)#\<Gamma>)\<turnstile>S\<^isub>2<:Q\<^isub>2"}. Since @{term "X"} is a binder, we
	assume that it is sufficiently fresh; in particular we have the freshness conditions
	@{term "X\<sharp>\<Gamma>"} and @{term "X\<sharp>Q\<^isub>1"} (these assumptions are provided by the strong 
	induction-rule @{text "subtype_of_induct"}). We know that the @{text "size_ty"} of 
	@{term Q\<^isub>1} and @{term Q\<^isub>2} is smaller than that of @{term Q};
	so we can apply the outer induction hypotheses for @{term Q\<^isub>1} and @{term Q\<^isub>2}. 
	The right-hand derivation is @{text "\<Gamma> \<turnstile> \<forall>[X<:Q\<^isub>1].Q\<^isub>2 <: T"}. Since @{term "X\<sharp>\<Gamma>"} 
	and @{term "X\<sharp>Q\<^isub>1"} there exists types @{text "T\<^isub>1,T\<^isub>2"} such that 
	@{text "T=Top \<or> T=\<forall>[X<:T\<^isub>1].T\<^isub>2"}. The @{term "Top"}-case is straightforward once we know 
	@{text "(\<forall>[X<:S\<^isub>1].S\<^isub>2) closed_in \<Gamma>"} and @{term "\<turnstile> \<Gamma> ok"}. In the other case we have 
	the sub-derivations @{term "\<Gamma>\<turnstile>T\<^isub>1<:Q\<^isub>1"} and @{term "((X,T\<^isub>1)#\<Gamma>)\<turnstile>Q\<^isub>2<:T\<^isub>2"}. Using the outer 
	induction hypothesis for transitivity we can derive @{term "\<Gamma>\<turnstile>T\<^isub>1<:S\<^isub>1"}. From the outer 
	induction for narrowing we get @{term "((X,T\<^isub>1)#\<Gamma>) \<turnstile> S\<^isub>2 <: Q\<^isub>2"} and then using again 
	induction for transitivity we obtain @{term "((X,T\<^isub>1)#\<Gamma>) \<turnstile> S\<^isub>2 <: T\<^isub>2"}. By rule 
	@{text "S_Forall"} and the freshness condition @{term "X\<sharp>\<Gamma>"} follows 
	@{text "\<Gamma> \<turnstile> \<forall>[X<:S\<^isub>1].S\<^isub>2 <: \<forall>[X<:T\<^isub>1].T\<^isub>2"}, which is @{text "\<Gamma> \<turnstile>  \<forall>[X<:S\<^isub>1].S\<^isub>2 <: T\<^isub>"}.
	\end{minipage}*}
      hence rh_drv: "\<Gamma> \<turnstile> \<forall>[X<:Q\<^isub>1].Q\<^isub>2 <: T" by simp
      have lh_drv_prm\<^isub>1: "\<Gamma> \<turnstile> Q\<^isub>1 <: S\<^isub>1" by fact
      have lh_drv_prm\<^isub>2: "((X,Q\<^isub>1)#\<Gamma>) \<turnstile> S\<^isub>2 <: Q\<^isub>2" by fact
      have "X\<sharp>\<Gamma>" by fact
      then have fresh_cond: "X\<sharp>\<Gamma>" "X\<sharp>Q\<^isub>1" using lh_drv_prm\<^isub>1 by (simp_all add: subtype_implies_fresh)
      from `\<forall>[X<:Q\<^isub>1].Q\<^isub>2 = Q` 
      have Q\<^isub>1\<^isub>2_less: "size_ty Q\<^isub>1 < size_ty Q" "size_ty Q\<^isub>2 < size_ty Q " using fresh_cond by auto
      from rh_drv 
      have "T=Top \<or> (\<exists>T\<^isub>1 T\<^isub>2. T=\<forall>[X<:T\<^isub>1].T\<^isub>2 \<and> \<Gamma>\<turnstile>T\<^isub>1<:Q\<^isub>1 \<and> ((X,T\<^isub>1)#\<Gamma>)\<turnstile>Q\<^isub>2<:T\<^isub>2)" 
	using fresh_cond by (simp add: S_ForallE_left)
      moreover
      have "S\<^isub>1 closed_in \<Gamma>" and "S\<^isub>2 closed_in ((X,Q\<^isub>1)#\<Gamma>)" 
	using lh_drv_prm\<^isub>1 lh_drv_prm\<^isub>2 by (simp_all add: subtype_implies_closed)
      hence "(\<forall>[X<:S\<^isub>1].S\<^isub>2) closed_in \<Gamma>" by (force simp add: closed_in_def ty.supp abs_supp)
      moreover
      have "\<turnstile> \<Gamma> ok" using rh_drv by (rule subtype_implies_ok)
      moreover
      { assume "\<exists>T\<^isub>1 T\<^isub>2. T=\<forall>[X<:T\<^isub>1].T\<^isub>2 \<and> \<Gamma>\<turnstile>T\<^isub>1<:Q\<^isub>1 \<and> ((X,T\<^isub>1)#\<Gamma>)\<turnstile>Q\<^isub>2<:T\<^isub>2"
	then obtain T\<^isub>1 T\<^isub>2 
	  where T_inst: "T = \<forall>[X<:T\<^isub>1].T\<^isub>2" 
	  and   rh_drv_prm\<^isub>1: "\<Gamma> \<turnstile> T\<^isub>1 <: Q\<^isub>1" 
	  and   rh_drv_prm\<^isub>2:"((X,T\<^isub>1)#\<Gamma>) \<turnstile> Q\<^isub>2 <: T\<^isub>2" by force
	from IH_trans[of "Q\<^isub>1"] 
	have "\<Gamma> \<turnstile> T\<^isub>1 <: S\<^isub>1" using lh_drv_prm\<^isub>1 rh_drv_prm\<^isub>1 Q\<^isub>1\<^isub>2_less by blast
	moreover
	from IH_narrow[of "Q\<^isub>1" "[]"] 
	have "((X,T\<^isub>1)#\<Gamma>) \<turnstile> S\<^isub>2 <: Q\<^isub>2" using Q\<^isub>1\<^isub>2_less lh_drv_prm\<^isub>2 rh_drv_prm\<^isub>1 by simp
	with IH_trans[of "Q\<^isub>2"] 
	have "((X,T\<^isub>1)#\<Gamma>) \<turnstile> S\<^isub>2 <: T\<^isub>2" using Q\<^isub>1\<^isub>2_less rh_drv_prm\<^isub>2 by simp
	ultimately have "\<Gamma> \<turnstile> \<forall>[X<:S\<^isub>1].S\<^isub>2 <: \<forall>[X<:T\<^isub>1].T\<^isub>2"
	  using fresh_cond by (simp add: subtype_of.S_Forall)
	hence "\<Gamma> \<turnstile> \<forall>[X<:S\<^isub>1].S\<^isub>2 <: T" using T_inst by simp
      }
      ultimately show "\<Gamma> \<turnstile> \<forall>[X<:S\<^isub>1].S\<^isub>2 <: T" by blast
    qed
  qed

  { --{* The transitivity proof is now by the auxiliary lemma. *}
    case 1 
    have  "\<Gamma> \<turnstile> S <: Q" 
      and "\<Gamma> \<turnstile> Q <: T" by fact+
    thus "\<Gamma> \<turnstile> S <: T" by (rule transitivity_aux) 
  next 
    --{* The narrowing proof proceeds by an induction over @{term "(\<Delta>@[(X,Q)]@\<Gamma>) \<turnstile> M <: N"}. *}
    case 2
    have  "(\<Delta>@[(X,Q)]@\<Gamma>) \<turnstile> M <: N" --{* left-hand derivation *}
      and "\<Gamma> \<turnstile> P<:Q" by fact+ --{* right-hand derivation *}
    thus "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> M <: N" 
    proof (nominal_induct \<Gamma>\<equiv>"\<Delta>@[(X,Q)]@\<Gamma>" M N avoiding: \<Delta> \<Gamma> X rule: subtype_of.strong_induct) 
      case (S_Top _ S \<Delta> \<Gamma> X)
	--{* \begin{minipage}[t]{0.9\textwidth}
	In this case the left-hand derivation is @{term "(\<Delta>@[(X,Q)]@\<Gamma>) \<turnstile> S <: Top"}. We show
	that the context @{term "\<Delta>@[(X,P)]@\<Gamma>"} is ok and that @{term S} is closed in 
	@{term "\<Delta>@[(X,P)]@\<Gamma>"}. Then we can apply the @{text "S_Top"}-rule.\end{minipage}*}
      hence lh_drv_prm\<^isub>1: "\<turnstile> (\<Delta>@[(X,Q)]@\<Gamma>) ok" 
	and lh_drv_prm\<^isub>2: "S closed_in (\<Delta>@[(X,Q)]@\<Gamma>)" by simp_all
      have rh_drv: "\<Gamma> \<turnstile> P <: Q" by fact
      hence "P closed_in \<Gamma>" by (simp add: subtype_implies_closed)
      with lh_drv_prm\<^isub>1 have "\<turnstile> (\<Delta>@[(X,P)]@\<Gamma>) ok" by (simp add: replace_type)
      moreover
      from lh_drv_prm\<^isub>2 have "S closed_in (\<Delta>@[(X,P)]@\<Gamma>)" 
	by (simp add: closed_in_def domain_append)
      ultimately show "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> S <: Top" by (simp add: subtype_of.S_Top)
    next
      case (S_Var Y S _ N \<Delta> \<Gamma> X) 
	--{* \begin{minipage}[t]{0.9\textwidth}
	In this case the left-hand derivation is @{term "(\<Delta>@[(X,Q)]@\<Gamma>) \<turnstile> Tvar Y <: N"} and
	by inner induction hypothesis we have @{term "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> S <: N"}. We therefore 
	know that the contexts @{term "\<Delta>@[(X,Q)]@\<Gamma>"} and @{term "\<Delta>@[(X,P)]@\<Gamma>"} are ok, and that 
	@{term "(Y,S)"} is in @{term "\<Delta>@[(X,Q)]@\<Gamma>"}. We need to show that 
	@{term "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> Tvar Y <: N"}  holds. In case @{term "X\<noteq>Y"} we know that 
	@{term "(Y,S)"} is in @{term "\<Delta>@[(X,P)]@\<Gamma>"} and can use the inner induction hypothesis 
	and rule @{text "S_Var"} to conclude. In case @{term "X=Y"} we can infer that 
	@{term "S=Q"}; moreover we have that  @{term "(\<Delta>@[(X,P)]@\<Gamma>) extends \<Gamma>"} and therefore 
	by @{text "weakening"} that @{term "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> P <: Q"} holds. By transitivity we 
	obtain then @{term "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> P <: N"} and can conclude by applying rule 
	@{text "S_Var"}.\end{minipage}*}
      hence IH_inner: "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> S <: N"
	and lh_drv_prm: "(Y,S) \<in> set (\<Delta>@[(X,Q)]@\<Gamma>)"
	and rh_drv: "\<Gamma> \<turnstile> P<:Q"
	and ok\<^isub>Q: "\<turnstile> (\<Delta>@[(X,Q)]@\<Gamma>) ok" by (simp_all add: subtype_implies_ok)
      hence ok\<^isub>P: "\<turnstile> (\<Delta>@[(X,P)]@\<Gamma>) ok" by (simp add: subtype_implies_ok) 
      show "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> Tvar Y <: N"
      proof (cases "X=Y")
	case False
	have "X\<noteq>Y" by fact
	hence "(Y,S)\<in>set (\<Delta>@[(X,P)]@\<Gamma>)" using lh_drv_prm by simp
	with IH_inner show "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> Tvar Y <: N" by (simp add: subtype_of.S_Var)
      next
	case True
	have memb\<^isub>X\<^isub>Q: "(X,Q)\<in>set (\<Delta>@[(X,Q)]@\<Gamma>)" by simp
	have memb\<^isub>X\<^isub>P: "(X,P)\<in>set (\<Delta>@[(X,P)]@\<Gamma>)" by simp
	have eq: "X=Y" by fact 
	hence "S=Q" using ok\<^isub>Q lh_drv_prm memb\<^isub>X\<^isub>Q by (simp only: uniqueness_of_ctxt)
	hence "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> Q <: N" using IH_inner by simp
	moreover
	have "(\<Delta>@[(X,P)]@\<Gamma>) extends \<Gamma>" by (simp add: extends_def)
	hence "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> P <: Q" using rh_drv ok\<^isub>P by (simp only: weakening)
	ultimately have "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> P <: N" by (simp add: transitivity_aux) 
	thus "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> Tvar Y <: N" using memb\<^isub>X\<^isub>P eq by (simp only: subtype_of.S_Var)
      qed
    next
      case (S_Refl _ Y \<Delta> \<Gamma> X)
	--{* \begin{minipage}[t]{0.9\textwidth}
	In this case the left-hand derivation is @{term "(\<Delta>@[(X,Q)]@\<Gamma>) \<turnstile> Tvar Y <: Tvar Y"} and we
	therefore know that @{term "\<Delta>@[(X,Q)]@\<Gamma>"} is ok and that @{term "Y"} is in 
	the domain of @{term "\<Delta>@[(X,Q)]@\<Gamma>"}. We therefore know that @{term "\<Delta>@[(X,P)]@\<Gamma>"} is ok
	and that @{term Y} is in the domain of @{term "\<Delta>@[(X,P)]@\<Gamma>"}. We can conclude by applying 
	rule @{text "S_Refl"}.\end{minipage}*}
      hence lh_drv_prm\<^isub>1: "\<turnstile> (\<Delta>@[(X,Q)]@\<Gamma>) ok" 
	and lh_drv_prm\<^isub>2: "Y \<in> domain (\<Delta>@[(X,Q)]@\<Gamma>)" by simp_all
      have "\<Gamma> \<turnstile> P <: Q" by fact
      hence "P closed_in \<Gamma>" by (simp add: subtype_implies_closed)
      with lh_drv_prm\<^isub>1 have "\<turnstile> (\<Delta>@[(X,P)]@\<Gamma>) ok" by (simp add: replace_type)
      moreover
      from lh_drv_prm\<^isub>2 have "Y \<in> domain (\<Delta>@[(X,P)]@\<Gamma>)" by (simp add: domain_append)
      ultimately show "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> Tvar Y <: Tvar Y" by (simp add: subtype_of.S_Refl)
    next
      case (S_Arrow _ S\<^isub>1 Q\<^isub>1 Q\<^isub>2 S\<^isub>2 \<Delta> \<Gamma> X) 
	--{* \begin{minipage}[t]{0.9\textwidth}
	In this case the left-hand derivation is @{term "(\<Delta>@[(X,Q)]@\<Gamma>) \<turnstile> Q\<^isub>1 \<rightarrow> Q\<^isub>2 <: S\<^isub>1 \<rightarrow> S\<^isub>2"} 
	and the proof is trivial.\end{minipage}*}
      thus "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> Q\<^isub>1 \<rightarrow> Q\<^isub>2 <: S\<^isub>1 \<rightarrow> S\<^isub>2" by blast 
    next
      case (S_Forall _ T\<^isub>1 S\<^isub>1 Y S\<^isub>2 T\<^isub>2 \<Delta> \<Gamma> X)
	--{* \begin{minipage}[t]{0.9\textwidth}
	In this case the left-hand derivation is @{text "(\<Delta>@[(X,Q)]@\<Gamma>) \<turnstile> \<forall>[Y<:S\<^isub>1].S\<^isub>2 <: \<forall>[Y<:T\<^isub>1].T\<^isub>2"}
	and therfore we know that the binder @{term Y} is fresh for @{term "\<Delta>@[(X,Q)]@\<Gamma>"}. By
	the inner induction hypothesis we have that @{term "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> T\<^isub>1 <: S\<^isub>1"} and 
	@{term "((Y,T\<^isub>1)#\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> S\<^isub>2 <: T\<^isub>2"}. Since @{term P} is a subtype of @{term Q}
	we can infer that @{term Y} is fresh for @{term P} and thus also fresh for 
	@{term "\<Delta>@[(X,P)]@\<Gamma>"}. We can then conclude by applying rule @{text "S_Forall"}.
	\end{minipage}*}
      hence IH_inner\<^isub>1: "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> T\<^isub>1 <: S\<^isub>1" 
	and IH_inner\<^isub>2: "((Y,T\<^isub>1)#\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> S\<^isub>2 <: T\<^isub>2" 
	and lh_drv_prm: "Y\<sharp>(\<Delta>@[(X,Q)]@\<Gamma>)" by force+
      have rh_drv: "\<Gamma> \<turnstile> P <: Q" by fact
      hence "Y\<sharp>P" using lh_drv_prm by (simp only: fresh_list_append subtype_implies_fresh)
      hence "Y\<sharp>(\<Delta>@[(X,P)]@\<Gamma>)" using lh_drv_prm 
	by (simp add: fresh_list_append fresh_list_cons fresh_prod)
      with IH_inner\<^isub>1 IH_inner\<^isub>2 
      show "(\<Delta>@[(X,P)]@\<Gamma>) \<turnstile> \<forall>[Y<:S\<^isub>1].S\<^isub>2 <: \<forall>[Y<:T\<^isub>1].T\<^isub>2" by (simp add: subtype_of.S_Forall)
    qed
  } 
qed

end