cleaned up all examples so that they work with the
current nominal-setting.
theory fsub
imports "../nominal"
begin
atom_decl tyvrs vrs
section {* Types *}
nominal_datatype ty = TyVar "tyvrs"
| Top
| Arrow "ty" "ty" ("_ \<rightarrow> _" [900,900] 900)
| TAll "\<guillemotleft>tyvrs\<guillemotright>ty" "ty"
syntax
TAll_syn :: "tyvrs \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> ty" ("\<forall> [_<:_]._" [900,900,900] 900)
translations
"(\<forall>[a<:ty1].ty2)" \<rightleftharpoons> "TAll a ty2 ty1"
section {* typing contexts and their domains *}
types ty_context = "(tyvrs\<times>ty) list"
text {* domain of a context --- (the name "dom" is already used elsewhere) *}
consts
"domain" :: "ty_context \<Rightarrow> tyvrs set"
primrec
"domain [] = {}"
"domain (X#\<Gamma>) = {fst X}\<union>(domain \<Gamma>)"
lemma domain_eqvt:
fixes pi::"tyvrs prm"
shows "pi\<bullet>(domain \<Gamma>) = domain (pi\<bullet>\<Gamma>)"
by (induct \<Gamma>, auto simp add: perm_set_def)
lemma finite_domain:
fixes \<Gamma>::"ty_context"
shows "finite (domain \<Gamma>)"
by (induct \<Gamma>, auto)
lemma domain_inclusion[rule_format]:
shows "(X,T)\<in>set \<Gamma> \<longrightarrow> X\<in>(domain \<Gamma>)"
by (induct \<Gamma>, auto)
lemma domain_existence[rule_format]:
shows "X\<in>(domain \<Gamma>) \<longrightarrow> (\<exists>T.(X,T)\<in>set \<Gamma>)"
by (induct \<Gamma>, auto)
lemma domain_append:
shows "domain (\<Gamma>@\<Delta>) = ((domain \<Gamma>) \<union> (domain \<Delta>))"
by (induct \<Gamma>, auto)
section {* Two functions over types *}
text {* they cannot yet be defined conveniently unless we have a recursion combinator *}
consts size_ty :: "ty \<Rightarrow> nat"
lemma size_ty[simp]:
shows "size_ty (TyVar X) = 1"
and "size_ty (Top) = 1"
and "\<lbrakk>size_ty t1 = m ; size_ty t2 = n\<rbrakk> \<Longrightarrow> size_ty (t1 \<rightarrow> t2) = m + n + 1"
and "\<lbrakk>size_ty t1 = m ; size_ty t2 = n\<rbrakk> \<Longrightarrow> size_ty (\<forall> [a<:t1].t2) = m + n + 1"
sorry
consts subst_ty :: "ty \<Rightarrow> tyvrs \<Rightarrow> ty \<Rightarrow> ty"
lemma subst_ty[simp]:
shows "subst_ty (TyVar X) Y T = (if X=Y then T else (TyVar X))"
and "subst_ty Top Y T = Top"
and "subst_ty (T1 \<rightarrow> T2) Y T = (subst_ty T1 Y T) \<rightarrow> (subst_ty T2 Y T)"
and "X\<sharp>(Y,T) \<Longrightarrow> subst_ty (\<forall>[X<:T1].T2) Y T = (\<forall>[X<:(subst_ty T1 Y T)].(subst_ty T2 Y T))"
and "\<lbrakk>X\<sharp>Y; X\<sharp>T\<rbrakk> \<Longrightarrow> subst_ty (\<forall>[X<:T1].T2) Y T = (\<forall>[X<:(subst_ty T1 Y T)].(subst_ty T2 Y T))"
sorry
consts subst_ctxt :: "ty_context \<Rightarrow> tyvrs \<Rightarrow> ty \<Rightarrow> ty_context"
primrec
"subst_ctxt [] Y T = []"
"subst_ctxt (XT#Xs) Y T = (fst XT, subst_ty (snd XT) Y T)#(subst_ctxt Xs Y T)"
text {* valid contexts *}
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_def2:
shows "(S closed_in \<Gamma>) = ((supp S)\<subseteq>((supp (domain \<Gamma>)):: tyvrs set))"
apply(simp add: closed_in_def)
apply(simp add: at_fin_set_supp[OF at_tyvrs_inst, OF finite_domain])
done
lemma closed_in_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
consts
valid_rel :: "ty_context set"
valid_sym :: "ty_context \<Rightarrow> bool" ("\<turnstile> _ ok" [100] 100)
translations
"\<turnstile> \<Gamma> ok" \<rightleftharpoons> "\<Gamma> \<in> valid_rel"
inductive valid_rel
intros
v1[intro]: "\<turnstile> [] ok"
v2[intro]: "\<lbrakk>\<turnstile> \<Gamma> ok; X\<sharp>\<Gamma>; T closed_in \<Gamma>\<rbrakk> \<Longrightarrow> \<turnstile> ((X,T)#\<Gamma>) ok"
lemma valid_eqvt:
fixes pi::"tyvrs prm"
assumes a: "\<turnstile> \<Gamma> ok"
shows "\<turnstile> (pi\<bullet>\<Gamma>) ok"
using a
proof (induct)
case v1
show ?case by force
next
case (v2 T X \<Gamma>)
have a1: "\<turnstile> (pi \<bullet> \<Gamma>) ok" by fact
have a2: "X\<sharp>\<Gamma>" by fact
have a3: "T closed_in \<Gamma>" by fact
show ?case
proof (simp, rule valid_rel.v2)
show "\<turnstile> (pi \<bullet> \<Gamma>) ok" using a1 by simp
next
show "(pi\<bullet>X)\<sharp>(pi\<bullet>\<Gamma>)" using a2 by (simp add: pt_fresh_bij[OF pt_tyvrs_inst, OF at_tyvrs_inst])
next
show "(pi\<bullet>T) closed_in (pi\<bullet>\<Gamma>)" using a3 by (rule closed_in_eqvt)
qed
qed
lemma validE:
assumes a: "\<turnstile> ((X,T)#\<Gamma>) ok"
shows "\<turnstile> \<Gamma> ok \<and> X\<sharp>\<Gamma> \<and> T closed_in \<Gamma>"
using a by (cases, auto)
lemma validE_append[rule_format]:
shows "\<turnstile> (\<Delta>@\<Gamma>) ok \<longrightarrow> \<turnstile> \<Gamma> ok"
by (induct \<Delta>, auto dest: validE)
lemma domain_fresh[rule_format]:
fixes X::"tyvrs"
shows "\<turnstile> \<Gamma> ok \<longrightarrow> (X\<sharp>(domain \<Gamma>) = X\<sharp>\<Gamma>)"
apply(induct \<Gamma>)
apply(simp add: fresh_list_nil fresh_set_empty)
apply(simp add: fresh_list_cons fresh_prod
fresh_fin_insert[OF pt_tyvrs_inst, OF at_tyvrs_inst, OF fs_tyvrs_inst, OF finite_domain])
apply(rule impI)
apply(clarify)
apply(simp add: fresh_prod)
apply(drule validE)
apply(simp)
apply(simp add: closed_in_def2 fresh_def)
apply(force)
done
lemma closed_in_fresh:
fixes X::"tyvrs"
assumes a1: "S closed_in \<Gamma>"
and a2: "X\<sharp>\<Gamma>"
and a3: "\<turnstile> \<Gamma> ok"
shows "X\<sharp>S"
using a1 a2 a3
apply(simp add: closed_in_def2)
apply(simp add: domain_fresh[symmetric])
apply(simp add: fresh_def)
apply(force)
done
lemma replace_type[rule_format]:
shows "\<turnstile> (\<Delta>@(X,T)#\<Gamma>) ok \<longrightarrow> S closed_in \<Gamma> \<longrightarrow> \<turnstile> (\<Delta>@(X,S)#\<Gamma>) ok"
apply(induct \<Delta>)
apply(auto dest!: validE intro!: v2 simp add: fresh_list_append fresh_list_cons fresh_prod)
apply(drule validE_append)
apply(drule validE)
apply(drule_tac S="S" in closed_in_fresh)
apply(simp)
apply(force)+
apply(simp add: closed_in_def2)
apply(simp add: domain_append)
done
lemma fresh_implies_not_member[rule_format]:
fixes \<Gamma>::"ty_context"
shows "X\<sharp>\<Gamma> \<longrightarrow> \<not>(\<exists>T.(X,T)\<in>(set \<Gamma>))"
apply (induct \<Gamma>)
apply (auto dest: validE simp add: fresh_list_cons fresh_prod fresh_atm)
done
lemma uniqueness_of_ctxt:
fixes \<Gamma>::"ty_context"
shows "\<turnstile> \<Gamma> ok \<longrightarrow> (X,T)\<in>(set \<Gamma>) \<longrightarrow> (X,S)\<in>(set \<Gamma>) \<longrightarrow> (T = S)"
apply (induct \<Gamma>)
apply (auto dest!: validE fresh_implies_not_member)
done
section {* Subtyping *}
consts
subtype_of_rel :: "(ty_context \<times> ty \<times> ty) set"
subtype_of_sym :: "ty_context \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool" ("_ \<turnstile> _ <: _" [100,100,100] 100)
translations
"\<Gamma> \<turnstile> S <: T" \<rightleftharpoons> "(\<Gamma>,S,T) \<in> subtype_of_rel"
inductive subtype_of_rel
intros
S_Top[intro]: "\<lbrakk>\<turnstile> \<Gamma> ok; S closed_in \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> S <: Top"
S_Var[intro]: "\<lbrakk>\<turnstile> \<Gamma> ok; (X,S) \<in> (set \<Gamma>); \<Gamma> \<turnstile> S <: T\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> (TyVar X) <: T"
S_Refl[intro]: "\<lbrakk>\<turnstile> \<Gamma> ok; X \<in> (domain \<Gamma>)\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> TyVar X <: TyVar X"
S_Arrow[intro]: "\<lbrakk>\<Gamma> \<turnstile> T1 <: S1; \<Gamma> \<turnstile> S2 <: T2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> (S1 \<rightarrow> S2) <: (T1 \<rightarrow> T2)"
S_All[intro]: "\<lbrakk>\<Gamma> \<turnstile> T1 <: S1; X\<sharp>\<Gamma>; ((X,T1)#\<Gamma>) \<turnstile> S2 <: T2\<rbrakk>
\<Longrightarrow> \<Gamma> \<turnstile> \<forall> [X<:S1].S2 <: \<forall> [X<:T1].T2"
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 S \<Gamma>)
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 S T X \<Gamma>)
have "(X,S)\<in>set \<Gamma>" by fact
hence "X\<in>(domain \<Gamma>)" by (rule domain_inclusion)
hence "(TyVar 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 "(TyVar X) closed_in \<Gamma> \<and> T closed_in \<Gamma>" by simp
next
case S_Refl thus ?case
by (simp add: closed_in_def ty.supp supp_atm)
next
case S_Arrow thus ?case by (force simp add: closed_in_def ty.supp)
next
case S_All thus ?case by (auto simp add: closed_in_def ty.supp abs_supp)
qed
text {* completely automated proof *}
lemma subtype_implies_closed:
assumes a: "\<Gamma> \<turnstile> S <: T"
shows "S closed_in \<Gamma> \<and> T closed_in \<Gamma>"
using a
apply (induct)
apply (auto simp add: closed_in_def ty.supp abs_supp domain_inclusion supp_atm)
done
lemma subtype_implies_ok:
fixes X::"tyvrs"
assumes a1: "\<Gamma> \<turnstile> S <: T"
shows "\<turnstile> \<Gamma> ok"
using a1 by (induct, auto)
lemma subtype_implies_fresh:
fixes X::"tyvrs"
assumes a1: "\<Gamma> \<turnstile> S <: T"
and a2: "X\<sharp>\<Gamma>"
shows "X\<sharp>(S,T)"
proof -
from a1 have "\<turnstile> \<Gamma> ok" by (rule subtype_implies_ok)
with a2 have b0: "X\<sharp>domain(\<Gamma>)" by (simp add: domain_fresh)
from a1 have "S closed_in \<Gamma> \<and> T closed_in \<Gamma>" by (rule subtype_implies_closed)
hence b1: "supp S \<subseteq> ((supp (domain \<Gamma>))::tyvrs set)"
and b2: "supp T \<subseteq> ((supp (domain \<Gamma>))::tyvrs set)" by (simp_all add: closed_in_def2)
thus "X\<sharp>(S,T)" using b0 b1 b2 by (force simp add: supp_prod fresh_def)
qed
lemma silly_eqvt1:
fixes X::"'a::pt_tyvrs"
and S::"'b::pt_tyvrs"
and pi::"tyvrs prm"
shows "(X,S) \<in> set \<Gamma> \<Longrightarrow> (pi\<bullet>X,pi\<bullet>S) \<in> set (pi\<bullet>\<Gamma>)"
apply(drule_tac pi="pi" in pt_set_bij2[OF pt_tyvrs_inst, OF at_tyvrs_inst])
apply(simp add: pt_list_set_pi[OF pt_tyvrs_inst])
done
lemma silly_eqvt2:
fixes X::"tyvrs"
and pi::"tyvrs prm"
shows "X \<in> (domain \<Gamma>) \<Longrightarrow> (pi\<bullet>X) \<in> (domain (pi\<bullet>\<Gamma>))"
apply(drule_tac pi="pi" in pt_set_bij2[OF pt_tyvrs_inst, OF at_tyvrs_inst])
apply(simp add: domain_eqvt pt_list_set_pi[OF pt_tyvrs_inst] )
done
lemma subtype_eqvt:
fixes pi::"tyvrs prm"
shows "\<Gamma> \<turnstile> S <: T \<Longrightarrow> (pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>S) <: (pi\<bullet>T)"
apply(erule subtype_of_rel.induct)
apply(force intro: valid_eqvt closed_in_eqvt)
(*
apply(simp)
apply(rule S_Var)
apply(rule valid_eqvt)
apply(assumption)
*)
(* FIXME: here *)
(* apply(auto intro: closed_in_eqvt valid_eqvt dest: pt_set_bij2[OF pt_tyvrs_inst, OF at_tyvrs_inst]) *)
apply(force intro: closed_in_eqvt valid_eqvt silly_eqvt1)
apply(force intro: valid_eqvt silly_eqvt2)
apply(force)
apply(force intro!: S_All simp add: fresh_prod pt_fresh_bij1[OF pt_tyvrs_inst, OF at_tyvrs_inst])
done
lemma subtype_of_rel_induct_aux[rule_format]:
fixes P :: "ty_context \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow>'a::fs_tyvrs\<Rightarrow>bool"
assumes a: "\<Gamma> \<turnstile> S <: T"
and a1: "\<And>x \<Gamma> S. (\<turnstile> \<Gamma> ok) \<Longrightarrow> (S closed_in \<Gamma>) \<Longrightarrow> P \<Gamma> S Top x"
and a2: "\<And>x \<Gamma> X S T. (\<turnstile> \<Gamma> ok) \<Longrightarrow> ((X,S)\<in>set \<Gamma>) \<Longrightarrow> (\<Gamma> \<turnstile> S <: T) \<Longrightarrow> (\<forall>z. P \<Gamma> S T z)
\<Longrightarrow> P \<Gamma> (TyVar X) T x"
and a3: "\<And>x \<Gamma> X. (\<turnstile> \<Gamma> ok) \<Longrightarrow> X\<in>(domain \<Gamma>) \<Longrightarrow> P \<Gamma> (TyVar X) (TyVar X) x"
and a4: "\<And>x \<Gamma> S1 S2 T1 T2. \<Gamma> \<turnstile> T1 <: S1 \<Longrightarrow> (\<forall>z. P \<Gamma> T1 S1 z) \<Longrightarrow> \<Gamma> \<turnstile> S2 <: T2 \<Longrightarrow>
(\<forall>z. P \<Gamma> S2 T2 z) \<Longrightarrow> P \<Gamma> (S1 \<rightarrow> S2) (T1 \<rightarrow> T2) x"
and a5: "\<And>x \<Gamma> X S1 S2 T1 T2.
X\<sharp>x \<Longrightarrow> X\<sharp>(\<Gamma>,S1,T1) \<Longrightarrow> \<Gamma> \<turnstile> T1 <: S1 \<Longrightarrow> (\<forall>z. P \<Gamma> T1 S1 z) \<Longrightarrow> ((X,T1)#\<Gamma>) \<turnstile> S2 <: T2
\<Longrightarrow> (\<forall>z. P ((X,T1)#\<Gamma>) S2 T2 z) \<Longrightarrow> P \<Gamma> (\<forall> [X<:S1].S2) (\<forall> [X<:T1].T2) x"
shows "\<forall>(pi::tyvrs prm) (x::'a::fs_tyvrs). P (pi\<bullet>\<Gamma>) (pi\<bullet>S) (pi\<bullet>T) x"
using a
proof (induct)
case (S_Top S \<Gamma>)
have b1: "\<turnstile> \<Gamma> ok" by fact
have b2: "S closed_in \<Gamma>" by fact
show ?case
proof (intro strip, simp)
fix pi::"tyvrs prm" and x::"'a::fs_tyvrs"
from b1 have "\<turnstile> (pi\<bullet>\<Gamma>) ok" by (rule valid_eqvt)
moreover
from b2 have "(pi\<bullet>S) closed_in (pi\<bullet>\<Gamma>)" by (rule closed_in_eqvt)
ultimately show "P (pi \<bullet> \<Gamma>) (pi \<bullet> S) Top x" by (rule a1)
qed
next
case (S_Var S T X \<Gamma>)
have b1: "\<turnstile> \<Gamma> ok" by fact
have b2: "(X,S) \<in> set \<Gamma>" by fact
have b3: "\<Gamma> \<turnstile> S <: T" by fact
have b4: "\<forall>(pi::tyvrs prm) x. P (pi\<bullet>\<Gamma>) (pi\<bullet>S) (pi\<bullet>T) x" by fact
show ?case
proof (intro strip, simp)
fix pi::"tyvrs prm" and x::"'a::fs_tyvrs"
from b1 have "\<turnstile> (pi\<bullet>\<Gamma>) ok" by (rule valid_eqvt)
moreover
from b2 have "pi\<bullet>(X,S)\<in>pi\<bullet>(set \<Gamma>)" by (rule pt_set_bij2[OF pt_tyvrs_inst, OF at_tyvrs_inst])
hence "(pi\<bullet>X,pi\<bullet>S)\<in>set (pi\<bullet>\<Gamma>)" by (simp add: pt_list_set_pi[OF pt_tyvrs_inst])
moreover
from b3 have "(pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>S) <: (pi\<bullet>T)" by (rule subtype_eqvt)
moreover
from b4 have "\<forall>x. P (pi\<bullet>\<Gamma>) (pi\<bullet>S) (pi\<bullet>T) x" by simp
ultimately
show "P (pi\<bullet>\<Gamma>) (TyVar (pi\<bullet>X)) (pi\<bullet>T) x" by (rule a2)
qed
next
case (S_Refl X \<Gamma>)
have b1: "\<turnstile> \<Gamma> ok" by fact
have b2: "X \<in> domain \<Gamma>" by fact
show ?case
proof (intro strip, simp)
fix pi::"tyvrs prm" and x::"'a::fs_tyvrs"
from b1 have "\<turnstile> (pi\<bullet>\<Gamma>) ok" by (rule valid_eqvt)
moreover
from b2 have "(pi\<bullet>X)\<in>pi\<bullet>domain \<Gamma>" by (rule pt_set_bij2[OF pt_tyvrs_inst, OF at_tyvrs_inst])
hence "(pi\<bullet>X)\<in>domain (pi\<bullet>\<Gamma>)" by (simp add: domain_eqvt pt_list_set_pi[OF pt_tyvrs_inst])
ultimately show "P (pi\<bullet>\<Gamma>) (TyVar (pi\<bullet>X)) (TyVar (pi\<bullet>X)) x" by (rule a3)
qed
next
case S_Arrow thus ?case by (simp, auto intro!: a4 subtype_eqvt)
next
case (S_All S1 S2 T1 T2 X \<Gamma>)
have b1: "\<Gamma> \<turnstile> T1 <: S1" by fact
have b2: "\<forall>(pi::tyvrs prm) x. P (pi\<bullet>\<Gamma>) (pi\<bullet>T1) (pi\<bullet>S1) x" by fact
have b4: "((X,T1)#\<Gamma>) \<turnstile> S2 <: T2" by fact
have b5: "\<forall>(pi::tyvrs prm) x. P (pi\<bullet>((X,T1)#\<Gamma>)) (pi\<bullet>S2) (pi\<bullet>T2) x" by fact
have b3': "X\<sharp>\<Gamma>" by fact
have b3'': "X\<sharp>(T1,S1)" using b1 b3' by (rule subtype_implies_fresh)
have b3: "X\<sharp>(\<Gamma>,S1,T1)" using b3' b3'' by (simp add: fresh_prod)
show ?case
proof (intro strip)
fix pi::"tyvrs prm" and x::"'a::fs_tyvrs"
have "\<exists>C::tyvrs. C\<sharp>(pi\<bullet>X,pi\<bullet>S2,pi\<bullet>T2,pi\<bullet>S1,pi\<bullet>T1,pi\<bullet>\<Gamma>,x)"
by (rule at_exists_fresh[OF at_tyvrs_inst], simp add: fs_tyvrs1)
then obtain C::"tyvrs" where f1: "C\<noteq>(pi\<bullet>X)" and f2: "C\<sharp>(pi\<bullet>S1)" and f3: "C\<sharp>(pi\<bullet>T1)"
and f4: "C\<sharp>(pi\<bullet>S2)" and f5: "C\<sharp>(pi\<bullet>T2)" and f6: "C\<sharp>(pi\<bullet>\<Gamma>)" and f7: "C\<sharp>x"
by (auto simp add: fresh_prod fresh_atm)
let ?pi' = "[(C,pi\<bullet>X)]@pi"
from b2 have c1: "\<forall>x. P (?pi'\<bullet>\<Gamma>) (?pi'\<bullet>T1) (?pi'\<bullet>S1) x" by simp
from b5 have "\<forall>x. P (?pi'\<bullet>((X,T1)#\<Gamma>)) (?pi'\<bullet>S2) (?pi'\<bullet>T2) x" by simp
hence "\<forall>x. P ((?pi'\<bullet>X,?pi'\<bullet>T1)#(?pi'\<bullet>\<Gamma>)) (?pi'\<bullet>S2) (?pi'\<bullet>T2) x" by simp
hence c2: "\<forall>x. P ((C,?pi'\<bullet>T1)#(?pi'\<bullet>\<Gamma>)) (?pi'\<bullet>S2) (?pi'\<bullet>T2) x" using f1
by (simp only: pt_tyvrs2 calc_atm, simp)
from b3 have "(pi\<bullet>X)\<sharp>(pi\<bullet>\<Gamma>)"
by (simp add: fresh_prod pt_fresh_bij[OF pt_tyvrs_inst, OF at_tyvrs_inst])
with f6 have f6a: "?pi'\<bullet>\<Gamma>=pi\<bullet>\<Gamma>"
by (simp only: pt_tyvrs2 pt_fresh_fresh[OF pt_tyvrs_inst, OF at_tyvrs_inst])
hence f6': "C\<sharp>(?pi'\<bullet>\<Gamma>)" using f6 by simp
from b3 have "(pi\<bullet>X)\<sharp>(pi\<bullet>S1)"
by (simp add: fresh_prod pt_fresh_bij[OF pt_tyvrs_inst, OF at_tyvrs_inst])
with f2 have f2a: "?pi'\<bullet>S1=pi\<bullet>S1"
by (simp only: pt_tyvrs2 pt_fresh_fresh[OF pt_tyvrs_inst, OF at_tyvrs_inst])
hence f2': "C\<sharp>(?pi'\<bullet>S1)" using f2 by simp
from b3 have "(pi\<bullet>X)\<sharp>(pi\<bullet>T1)"
by (simp add: fresh_prod pt_fresh_bij[OF pt_tyvrs_inst, OF at_tyvrs_inst])
with f3 have f3a: "?pi'\<bullet>T1=pi\<bullet>T1"
by (simp only: pt_tyvrs2 pt_fresh_fresh[OF pt_tyvrs_inst, OF at_tyvrs_inst])
hence f3': "C\<sharp>(?pi'\<bullet>T1)" using f3 by simp
from b1 have e1: "(?pi'\<bullet>\<Gamma>) \<turnstile> (?pi'\<bullet>T1) <: (?pi'\<bullet>S1)" by (rule subtype_eqvt)
from b4 have "(?pi'\<bullet>((X,T1)#\<Gamma>)) \<turnstile> (?pi'\<bullet>S2) <: (?pi'\<bullet>T2)" by (rule subtype_eqvt)
hence "((?pi'\<bullet>X,?pi'\<bullet>T1)#(?pi'\<bullet>\<Gamma>)) \<turnstile> (?pi'\<bullet>S2) <: (?pi'\<bullet>T2)" by simp
hence e2: "((C,?pi'\<bullet>T1)#(?pi'\<bullet>\<Gamma>)) \<turnstile> (?pi'\<bullet>S2) <: (?pi'\<bullet>T2)" using f1
by (simp only: pt_tyvrs2 calc_atm, simp)
have fnew: "C\<sharp>(?pi'\<bullet>\<Gamma>,?pi'\<bullet>S1,?pi'\<bullet>T1)" using f6' f2' f3' by (simp add: fresh_prod)
have main: "P (?pi'\<bullet>\<Gamma>) (\<forall> [C<:(?pi'\<bullet>S1)].(?pi'\<bullet>S2)) (\<forall> [C<:(?pi'\<bullet>T1)].(?pi'\<bullet>T2)) x"
using f7 fnew e1 c1 e2 c2 by (rule a5)
have alpha1: "(\<forall> [C<:(?pi'\<bullet>S1)].(?pi'\<bullet>S2)) = (pi\<bullet>(\<forall> [X<:S1].S2))"
using f1 f4 f2a[symmetric] by (simp add: ty.inject alpha pt_tyvrs2[symmetric])
have alpha2: "(\<forall> [C<:(?pi'\<bullet>T1)].(?pi'\<bullet>T2)) = (pi\<bullet>(\<forall> [X<:T1].T2))"
using f1 f5 f3a[symmetric] by (simp add: ty.inject alpha pt_tyvrs2[symmetric])
show "P (pi\<bullet>\<Gamma>) (pi\<bullet>(\<forall> [X<:S1].S2)) (pi\<bullet>(\<forall> [X<:T1].T2)) x"
using alpha1 alpha2 f6a main by simp
qed
qed
lemma subtype_of_rel_induct[case_names S_Top S_Var S_Refl S_Arrow S_All]:
fixes P :: "ty_context \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow>'a::fs_tyvrs\<Rightarrow>bool"
assumes a: "\<Gamma> \<turnstile> S <: T"
and a1: "\<And>x \<Gamma> S. (\<turnstile> \<Gamma> ok) \<Longrightarrow> (S closed_in \<Gamma>) \<Longrightarrow> P \<Gamma> S Top x"
and a2: "\<And>x \<Gamma> X S T. (\<turnstile> \<Gamma> ok) \<Longrightarrow> ((X,S)\<in>set \<Gamma>) \<Longrightarrow> (\<Gamma> \<turnstile> S <: T) \<Longrightarrow> (\<forall>z. P \<Gamma> S T z)
\<Longrightarrow> P \<Gamma> (TyVar X) T x"
and a3: "\<And>x \<Gamma> X. (\<turnstile> \<Gamma> ok) \<Longrightarrow> X\<in>domain \<Gamma> \<Longrightarrow> P \<Gamma> (TyVar X) (TyVar X) x"
and a4: "\<And>x \<Gamma> S1 S2 T1 T2. \<Gamma> \<turnstile> T1 <: S1 \<Longrightarrow> (\<forall>z. P \<Gamma> T1 S1 z) \<Longrightarrow> \<Gamma> \<turnstile> S2 <: T2 \<Longrightarrow>
(\<forall>z. P \<Gamma> S2 T2 z) \<Longrightarrow> P \<Gamma> (S1 \<rightarrow> S2) (T1 \<rightarrow> T2) x"
and a5: "\<And>x \<Gamma> X S1 S2 T1 T2.
X\<sharp>x \<Longrightarrow> X\<sharp>(\<Gamma>,S1,T1) \<Longrightarrow> \<Gamma> \<turnstile> T1 <: S1 \<Longrightarrow> (\<forall>z. P \<Gamma> T1 S1 z) \<Longrightarrow> ((X,T1)#\<Gamma>) \<turnstile> S2 <: T2
\<Longrightarrow> (\<forall>z. P ((X,T1)#\<Gamma>) S2 T2 z) \<Longrightarrow> P \<Gamma> (\<forall> [X<:S1].S2) (\<forall> [X<:T1].T2) x"
shows "P \<Gamma> S T x"
using a a1 a2 a3 a4 a5 subtype_of_rel_induct_aux[of "\<Gamma>" "S" "T" "P" "[]" "x", simplified] by force
section {* Two proofs for reflexivity of subtyping *}
lemma subtype_reflexivity:
shows "\<turnstile> \<Gamma> ok \<longrightarrow> T closed_in \<Gamma> \<longrightarrow> \<Gamma> \<turnstile> T <: T"
proof(nominal_induct T rule: ty.induct_unsafe)
case (TAll \<Gamma> X T1 T2)
have i1: "\<And>\<Gamma>. \<turnstile> \<Gamma> ok \<longrightarrow> T1 closed_in \<Gamma> \<longrightarrow> \<Gamma> \<turnstile> T1 <: T1" by fact
have i2: "\<And>\<Gamma>. \<turnstile> \<Gamma> ok \<longrightarrow> T2 closed_in \<Gamma> \<longrightarrow> \<Gamma> \<turnstile> T2 <: T2" by fact
have f: "X\<sharp>\<Gamma>" by fact
show "\<turnstile> \<Gamma> ok \<longrightarrow> (\<forall>[X<:T2].T1) closed_in \<Gamma> \<longrightarrow> \<Gamma> \<turnstile> \<forall>[X<:T2].T1 <: \<forall>[X<:T2].T1"
proof (intro strip)
assume "(\<forall>[X<:T2].T1) closed_in \<Gamma>"
hence b1: "T2 closed_in \<Gamma>" and b2: "T1 closed_in ((X,T2)#\<Gamma>)"
by (auto simp add: closed_in_def ty.supp abs_supp)
assume c1: "\<turnstile> \<Gamma> ok"
hence c2: "\<turnstile> ((X,T2)#\<Gamma>) ok" using b1 f by force
have "\<Gamma> \<turnstile> T2 <: T2" using i2 b1 c1 by simp
moreover
have "((X,T2)#\<Gamma>) \<turnstile> T1 <: T1" using i1 b2 c2 by simp
ultimately show "\<Gamma> \<turnstile> \<forall>[X<:T2].T1 <: \<forall>[X<:T2].T1" using f by force
qed
qed (auto simp add: closed_in_def ty.supp supp_atm)
lemma subtype_refl:
shows "\<turnstile> \<Gamma> ok \<longrightarrow> T closed_in \<Gamma> \<longrightarrow> \<Gamma> \<turnstile> T <: T"
apply(nominal_induct T rule: ty.induct_unsafe)
apply(auto simp add: ty.supp abs_supp closed_in_def supp_atm)
(* FIXME: the new induction method from Markus will fix this uglyness *)
apply(atomize)
apply(drule_tac x="(tyvrs, ty2)#z" in spec)
apply(force simp add: closed_in_def)
done
text {* Inversion lemmas\<dots> *}
lemma S_TopE:
shows "\<Gamma> \<turnstile> Top <: T \<Longrightarrow> T = Top"
apply(ind_cases "\<Gamma> \<turnstile> Top <: T", auto)
done
lemma S_ArrowE_left:
assumes a: "\<Gamma> \<turnstile> S1 \<rightarrow> S2 <: T"
shows "T = Top \<or> (\<exists>T1 T2. T = T1 \<rightarrow> T2 \<and> \<Gamma> \<turnstile> T1 <: S1 \<and> \<Gamma> \<turnstile> S2 <: T2)"
using a by (cases, auto simp add: ty.inject)
lemma S_AllE_left:
shows "\<lbrakk>\<Gamma> \<turnstile> \<forall>[X<:S1].S2 <: T; X\<sharp>(\<Gamma>,S1)\<rbrakk>
\<Longrightarrow> T = Top \<or> (\<exists>T1 T2. T = \<forall>[X<:T1].T2 \<and> \<Gamma> \<turnstile> T1 <: S1 \<and> ((X,T1)#\<Gamma>) \<turnstile> S2 <: T2)"
apply(frule subtype_implies_ok)
apply(ind_cases "\<Gamma> \<turnstile> \<forall>[X<:S1].S2 <: T")
apply(auto simp add: ty.inject alpha)
apply(rule_tac x="[(X,Xa)]\<bullet>T2" in exI)
(* term *)
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])
(* 1st conjunct *)
apply(rule conjI)
apply(simp add: pt_fresh_left[OF pt_tyvrs_inst, OF at_tyvrs_inst] calc_atm)
apply(simp add: fresh_prod)
apply(drule_tac \<Gamma>="((Xa,T1)#\<Gamma>)" in subtype_implies_closed)+
apply(simp add: closed_in_def)
apply(auto simp add: domain_fresh[of "\<Gamma>" "X", symmetric])
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(simp add: fresh_prod)
apply(drule_tac X="Xa" in subtype_implies_fresh)
apply(assumption)
apply(simp add: fresh_prod)
apply(drule_tac pi="[(X,Xa)]" in subtype_eqvt)
apply(simp add: calc_atm)
apply(simp add: pt_fresh_fresh[OF pt_tyvrs_inst, OF at_tyvrs_inst])
done
section {* Type substitution *}
lemma subst_ty_fresh[rule_format]:
fixes P :: "ty"
and X :: "tyvrs"
shows "X\<sharp>(T,P) \<longrightarrow> X\<sharp>(subst_ty T Y P)"
apply (nominal_induct T rule: ty.induct_unsafe)
apply (auto simp add: fresh_prod abs_fresh)
done
lemma subst_ctxt_fresh[rule_format]:
fixes X::"tyvrs"
shows "X\<sharp>(\<Gamma>,P) \<longrightarrow> X\<sharp>(subst_ctxt \<Gamma> Y P)"
apply (induct \<Gamma>)
apply (auto intro: subst_ty_fresh simp add: fresh_prod fresh_list_cons)
done
(*
lemma subst_ctxt_closed:
shows "subst_ty b X P closed_in (subst_ctxt \<Delta> X P @ \<Gamma>)"
lemma substitution_ok:
shows "\<turnstile> (\<Delta>@(X,Q)#\<Gamma>) ok \<longrightarrow> \<Gamma> \<turnstile> P <: Q \<longrightarrow> \<turnstile> ((subst_ctxt \<Delta> X P)@\<Gamma>) ok"
apply (induct \<Delta>)
apply (auto dest: validE)
apply (rule v2)
apply assumption
apply (drule validE)
apply (auto simp add: fresh_list_append)
apply (rule subst_ctxt_fresh)
apply (simp add: fresh_prod)
apply (drule_tac X = "a" in subtype_implies_fresh)
apply (simp add: fresh_list_cons)
apply (simp add: fresh_prod)
apply (simp add: fresh_list_cons)
apply (drule validE)
done
*)
(* note: What should nominal induct do if the context is compound? *)
(*
lemma type_substitution:
assumes a0: "(\<Delta>@(X,Q)#\<Gamma>) \<turnstile> S <: T"
shows "\<turnstile> (\<Delta>@(X,Q)#\<Gamma>) ok \<longrightarrow> \<Gamma> \<turnstile> P <: Q
\<longrightarrow> ((subst_ctxt \<Delta> X P)@\<Gamma>) \<turnstile> (subst_ty S X P) <: (subst_ty T X P)"
using a0
thm subtype_of_rel_induct
apply(rule subtype_of_rel_induct[of "\<Delta>@(X,Q)#\<Gamma>" "S" "T" _ "P"])
apply(auto)
apply(rule S_Top)
defer
defer
defer
apply(rule S_Var)
defer
defer
defer
defer
apply(rule S_Refl)
defer
defer
apply (nominal_induct \<Delta> X Q \<Gamma> S T rule: subtype_of_rel_induct)
*)
section {* Weakening *}
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 weakening:
assumes a1: "\<Gamma> \<turnstile> S <: T"
shows "\<turnstile> \<Delta> ok \<longrightarrow> \<Delta> extends \<Gamma> \<longrightarrow> \<Delta> \<turnstile> S <: T"
using a1
proof (nominal_induct \<Gamma> S T rule: subtype_of_rel_induct)
case (S_Top \<Delta> \<Gamma> S)
have lh_drv_prem: "S closed_in \<Gamma>" by fact
show "\<turnstile> \<Delta> ok \<longrightarrow> \<Delta> extends \<Gamma> \<longrightarrow> \<Delta> \<turnstile> S <: Top"
proof (intro strip)
assume "\<turnstile> \<Delta> ok"
moreover
assume "\<Delta> extends \<Gamma>"
hence "S closed_in \<Delta>" using lh_drv_prem by (rule_tac extends_closed)
ultimately show "\<Delta> \<turnstile> S <: Top" by force
qed
next
case (S_Var \<Delta> \<Gamma> X S T)
have lh_drv_prem: "(X,S) \<in> set \<Gamma>" by fact
have ih: "\<forall>\<Delta>. \<turnstile> \<Delta> ok \<longrightarrow> \<Delta> extends \<Gamma> \<longrightarrow> \<Delta> \<turnstile> S <: T" by fact
show "\<turnstile> \<Delta> ok \<longrightarrow> \<Delta> extends \<Gamma> \<longrightarrow> \<Delta> \<turnstile> TyVar X <: T"
proof (intro strip)
assume ok: "\<turnstile> \<Delta> ok"
and extends: "\<Delta> extends \<Gamma>"
have "(X,S) \<in> set \<Delta>" using lh_drv_prem extends by (simp add: extends_def)
moreover
have "\<Delta> \<turnstile> S <: T" using ok extends ih by simp
ultimately show "\<Delta> \<turnstile> TyVar X <: T" using ok by force
qed
next
case (S_Refl \<Delta> \<Gamma> X)
have lh_drv_prem: "X \<in> domain \<Gamma>" by fact
show ?case
proof (intro strip)
assume "\<turnstile> \<Delta> ok"
moreover
assume "\<Delta> extends \<Gamma>"
hence "X \<in> domain \<Delta>" using lh_drv_prem by (force dest: extends_domain)
ultimately show "\<Delta> \<turnstile> TyVar X <: TyVar X" by force
qed
next
case S_Arrow thus ?case by force
next
case (S_All \<Delta> \<Gamma> X S1 S2 T1 T2)
have fresh: "X\<sharp>\<Delta>" by fact
have ih1: "\<forall>\<Delta>. \<turnstile> \<Delta> ok \<longrightarrow> \<Delta> extends \<Gamma> \<longrightarrow> \<Delta> \<turnstile> T1 <: S1" by fact
have ih2: "\<forall>\<Delta>. \<turnstile> \<Delta> ok \<longrightarrow> \<Delta> extends ((X,T1)#\<Gamma>) \<longrightarrow> \<Delta> \<turnstile> S2 <: T2" by fact
have lh_drv_prem: "\<Gamma> \<turnstile> T1 <: S1" by fact
hence b5: "T1 closed_in \<Gamma>" by (simp add: subtype_implies_closed)
show "\<turnstile> \<Delta> ok \<longrightarrow> \<Delta> extends \<Gamma> \<longrightarrow> \<Delta> \<turnstile> \<forall> [X<:S1].S2 <: \<forall> [X<:T1].T2"
proof (intro strip)
assume ok: "\<turnstile> \<Delta> ok"
and extends: "\<Delta> extends \<Gamma>"
have "T1 closed_in \<Delta>" using extends b5 by (simp only: extends_closed)
hence "\<turnstile> ((X,T1)#\<Delta>) ok" using fresh ok by force
moreover
have "((X,T1)#\<Delta>) extends ((X,T1)#\<Gamma>)" using extends by (force simp add: extends_def)
ultimately have "((X,T1)#\<Delta>) \<turnstile> S2 <: T2" using ih2 by simp
moreover
have "\<Delta> \<turnstile> T1 <: S1" using ok extends ih1 by simp
ultimately show "\<Delta> \<turnstile> \<forall> [X<:S1].S2 <: \<forall> [X<:T1].T2" using ok by (force intro: S_All)
qed
qed
text {* more automated proof *}
lemma weakening:
assumes a1: "\<Gamma> \<turnstile> S <: T"
shows "\<turnstile> \<Delta> ok \<longrightarrow> \<Delta> extends \<Gamma> \<longrightarrow> \<Delta> \<turnstile> S <: T"
using a1
proof (nominal_induct \<Gamma> S T rule: subtype_of_rel_induct)
case (S_Top \<Delta> \<Gamma> S) thus ?case by (force intro: extends_closed)
next
case (S_Var \<Delta> \<Gamma> X S T) thus ?case by (force simp add: extends_def)
next
case (S_Refl \<Delta> \<Gamma> X) thus ?case by (force dest: extends_domain)
next
case S_Arrow thus ?case by force
next
case (S_All \<Delta> \<Gamma> X S1 S2 T1 T2)
have fresh: "X\<sharp>\<Delta>" by fact
have ih1: "\<forall>\<Delta>. \<turnstile> \<Delta> ok \<longrightarrow> \<Delta> extends \<Gamma> \<longrightarrow> \<Delta> \<turnstile> T1 <: S1" by fact
have ih2: "\<forall>\<Delta>. \<turnstile> \<Delta> ok \<longrightarrow> \<Delta> extends ((X,T1)#\<Gamma>) \<longrightarrow> \<Delta> \<turnstile> S2 <: T2" by fact
have lh_drv_prem: "\<Gamma> \<turnstile> T1 <: S1" by fact
hence b5: "T1 closed_in \<Gamma>" by (simp add: subtype_implies_closed)
show "\<turnstile> \<Delta> ok \<longrightarrow> \<Delta> extends \<Gamma> \<longrightarrow> \<Delta> \<turnstile> \<forall> [X<:S1].S2 <: \<forall> [X<:T1].T2"
proof (intro strip)
assume ok: "\<turnstile> \<Delta> ok"
and extends: "\<Delta> extends \<Gamma>"
have "T1 closed_in \<Delta>" using extends b5 by (simp only: extends_closed)
hence "\<turnstile> ((X,T1)#\<Delta>) ok" using fresh ok by force
moreover
have "((X,T1)#\<Delta>) extends ((X,T1)#\<Gamma>)" using extends by (force simp add: extends_def)
ultimately have "((X,T1)#\<Delta>) \<turnstile> S2 <: T2" using ih2 by simp
moreover
have "\<Delta> \<turnstile> T1 <: S1" using ok extends ih1 by simp
ultimately show "\<Delta> \<turnstile> \<forall> [X<:S1].S2 <: \<forall> [X<:T1].T2" using ok by (force intro: S_All)
qed
qed
(* helper lemma to calculate the measure of the induction *)
lemma measure_eq [simp]: "(x, y) \<in> measure f = (f x < f y)"
by (simp add: measure_def inv_image_def)
(* HERE *)
lemma transitivity_and_narrowing:
"(\<forall>\<Gamma> S T. \<Gamma> \<turnstile> S <: Q \<longrightarrow> \<Gamma> \<turnstile> Q <: T \<longrightarrow> \<Gamma> \<turnstile> S <: T) \<and>
(\<forall>\<Delta> \<Gamma> X P M N. (\<Delta>@(X,Q)#\<Gamma>) \<turnstile> M <: N \<longrightarrow> \<Gamma> \<turnstile> P <: Q \<longrightarrow> (\<Delta>@(X,P)#\<Gamma>) \<turnstile> M <: N)"
proof (rule wf_induct [of "measure size_ty" _ Q, rule_format])
show "wf (measure size_ty)" by simp
next
case (goal2 Q)
note IH1_outer = goal2[THEN conjunct1]
and IH2_outer = goal2[THEN conjunct2, THEN spec, of _ "[]", simplified]
(* CHECK: Not clear whether the condition size_ty Q = size_ty Q' is needed, or whether
just doing it for Q'=Q is enough *)
have transitivity:
"\<And>\<Gamma> S Q' T. \<Gamma> \<turnstile> S <: Q' \<Longrightarrow> size_ty Q = size_ty Q' \<longrightarrow> \<Gamma> \<turnstile> Q' <: T \<longrightarrow> \<Gamma> \<turnstile> S <: T"
proof -
(* We first handle the case where T = Top once and for all; this saves some
repeated argument below (just like on paper :-). We establish a little lemma
that is invoked where needed in each case of the induction. *)
have top_case:
"\<And>\<Gamma> S T' P. \<lbrakk>\<turnstile> \<Gamma> ok; S closed_in \<Gamma>; P \<longrightarrow> \<Gamma> \<turnstile> S <: T'; T'=Top \<or> P\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> S <: T'"
proof -
fix \<Gamma> S T' P
assume S_Top_prem1: "S closed_in \<Gamma>"
and S_Top_prem2: "\<turnstile> \<Gamma> ok"
and alternative: "P \<longrightarrow> \<Gamma> \<turnstile> S <: T'"
and "T'=Top \<or> P"
moreover
{ assume "T'=Top"
hence "\<Gamma> \<turnstile> S <: T'" using S_Top_prem1 S_Top_prem2 by (simp add: S_Top)
}
moreover
{ assume P
with alternative have "\<Gamma> \<turnstile> S <: T'" by simp
}
ultimately show "\<Gamma> \<turnstile> S <: T'" by blast
qed
(* Now proceed by induction on the left-hand derivation *)
fix \<Gamma> S Q' T
assume a: "\<Gamma> \<turnstile> S <: Q'"
from a show "size_ty Q = size_ty Q' \<longrightarrow> \<Gamma> \<turnstile> Q' <: T \<longrightarrow> \<Gamma> \<turnstile> S <: T"
(* FIXME : nominal induct does not work here because Gamma S and Q are fixed variables *)
(* FIX: It would be better to use S',\<Gamma>',etc., instead of S1,\<Gamma>1, for consistency, in the cases
where these do not refer to sub-phrases of S, etc. *)
proof(rule subtype_of_rel_induct[of "\<Gamma>" "S" "Q'" _ "T"])
case (goal1 T' \<Gamma>1 S1) --"lh drv.: \<Gamma> \<turnstile> S <: Q' \<equiv> \<Gamma>1 \<turnstile> S1 <: Top"
--"automatic proof: thus ?case proof (auto intro!: S_Top dest: S_TopE)"
assume lh_drv_prem1: "\<turnstile> \<Gamma>1 ok"
and lh_drv_prem2: "S1 closed_in \<Gamma>1"
show "size_ty Q = size_ty Top \<longrightarrow> \<Gamma>1 \<turnstile> Top <: T' \<longrightarrow> \<Gamma>1 \<turnstile> S1 <: T'"
proof (intro strip)
assume "\<Gamma>1 \<turnstile> Top <: T'" --"rh drv."
hence "T' = Top" by (rule S_TopE)
thus "\<Gamma>1 \<turnstile> S1 <: T'" using top_case[of "\<Gamma>1" "S1" "False" "T'"] lh_drv_prem1 lh_drv_prem2
by simp
qed
next
case (goal2 T' \<Gamma>1 X1 S1 T1) --"lh drv.: \<Gamma> \<turnstile> S <: Q' \<equiv> \<Gamma>1 \<turnstile> TVar(X1) <: S1"
--"automatic proof: thus ?case proof (auto intro!: S_Var)"
have lh_drv_prem1: "\<turnstile> \<Gamma>1 ok" by fact
have lh_drv_prem2: "(X1,S1)\<in>set \<Gamma>1" by fact
have IH_inner: "\<forall>T. size_ty Q = size_ty T1 \<longrightarrow> \<Gamma>1 \<turnstile> T1 <: T \<longrightarrow> \<Gamma>1 \<turnstile> S1 <: T" by fact
show "size_ty Q = size_ty T1 \<longrightarrow> \<Gamma>1 \<turnstile> T1 <: T' \<longrightarrow> \<Gamma>1 \<turnstile> TyVar X1 <: T'"
proof (intro strip)
assume "\<Gamma>1 \<turnstile> T1 <: T'" --"right-hand drv."
and "size_ty Q = size_ty T1"
with IH_inner have "\<Gamma>1 \<turnstile> S1 <: T'" by simp
thus "\<Gamma>1 \<turnstile> TyVar X1 <: T'" using lh_drv_prem1 lh_drv_prem2 by (simp add: S_Var)
qed
next
case goal3 --"S_Refl case" show ?case by simp
next
case (goal4 T' \<Gamma>1 S1 S2 T1 T2) --"lh drv.: \<Gamma> \<turnstile> S <: Q' == \<Gamma>1 \<turnstile> S1 \<rightarrow> S2 <: T1 \<rightarrow> T2"
have lh_drv_prem1: "\<Gamma>1 \<turnstile> T1 <: S1" by fact
have lh_drv_prem2: "\<Gamma>1 \<turnstile> S2 <: T2" by fact
show "size_ty Q = size_ty (T1 \<rightarrow> T2) \<longrightarrow> \<Gamma>1 \<turnstile> T1 \<rightarrow> T2 <: T' \<longrightarrow> \<Gamma>1 \<turnstile> S1 \<rightarrow> S2 <: T'"
proof (intro strip)
assume measure: "size_ty Q = size_ty (T1 \<rightarrow> T2)"
and rh_deriv: "\<Gamma>1 \<turnstile> T1 \<rightarrow> T2 <: T'"
have "S1 closed_in \<Gamma>1" and "S2 closed_in \<Gamma>1"
using lh_drv_prem1 lh_drv_prem2 by (simp_all add: subtype_implies_closed)
hence "(S1 \<rightarrow> S2) closed_in \<Gamma>1" by (simp add: closed_in_def ty.supp)
moreover
have "\<turnstile> \<Gamma>1 ok" using rh_deriv by (rule subtype_implies_ok)
moreover
have "T' = Top \<or> (\<exists>T3 T4. T'= T3 \<rightarrow> T4 \<and> \<Gamma>1 \<turnstile> T3 <: T1 \<and> \<Gamma>1 \<turnstile> T2 <: T4)"
using rh_deriv by (rule S_ArrowE_left)
moreover
{ assume "\<exists>T3 T4. T'= T3 \<rightarrow> T4 \<and> \<Gamma>1 \<turnstile> T3 <: T1 \<and> \<Gamma>1 \<turnstile> T2 <: T4"
then obtain T3 T4
where T'_inst: "T'= T3 \<rightarrow> T4"
and rh_drv_prem1: "\<Gamma>1 \<turnstile> T3 <: T1"
and rh_drv_prem2: "\<Gamma>1 \<turnstile> T2 <: T4" by force
from IH1_outer[of "T1"] have "\<Gamma>1 \<turnstile> T3 <: S1"
using measure rh_drv_prem1 lh_drv_prem1 by (force simp add: ty.inject)
moreover
from IH1_outer[of "T2"] have "\<Gamma>1 \<turnstile> S2 <: T4"
using measure rh_drv_prem2 lh_drv_prem2 by (force simp add: ty.inject)
ultimately have "\<Gamma>1 \<turnstile> S1 \<rightarrow> S2 <: T'" using T'_inst by force
}
ultimately show "\<Gamma>1 \<turnstile> S1 \<rightarrow> S2 <: T'" using top_case[of "\<Gamma>1" "S1\<rightarrow>S2" _ "T'"] by blast
qed
next
case (goal5 T' \<Gamma>1 X S1 S2 T1 T2) --"lh drv.: \<Gamma> \<turnstile> S <: Q' \<equiv> \<Gamma>1 \<turnstile> \<forall>[X:S1].S2 <: \<forall>[X:T1].T2"
have lh_drv_prem1: "\<Gamma>1 \<turnstile> T1 <: S1" by fact
have lh_drv_prem2: "((X,T1)#\<Gamma>1) \<turnstile> S2 <: T2" by fact
have fresh_cond: "X\<sharp>(\<Gamma>1,S1,T1)" by fact
show "size_ty Q = size_ty (\<forall>[X<:T1].T2) \<longrightarrow> \<Gamma>1 \<turnstile> \<forall>[X<:T1].T2 <: T' \<longrightarrow> \<Gamma>1 \<turnstile> \<forall>[X<:S1].S2 <: T'"
proof (intro strip)
assume measure: "size_ty Q = size_ty (\<forall>[X<:T1].T2)"
and rh_deriv: "\<Gamma>1 \<turnstile> \<forall>[X<:T1].T2 <: T'"
have "S1 closed_in \<Gamma>1" and "S2 closed_in ((X,T1)#\<Gamma>1)"
using lh_drv_prem1 lh_drv_prem2 by (simp_all add: subtype_implies_closed)
hence "(\<forall>[X<:S1].S2) closed_in \<Gamma>1" by (force simp add: closed_in_def ty.supp abs_supp)
moreover
have "\<turnstile> \<Gamma>1 ok" using rh_deriv by (rule subtype_implies_ok)
moreover
(* FIX: Maybe T3,T4 could be T1',T2' *)
have "T' = Top \<or> (\<exists>T3 T4. T'=\<forall>[X<:T3].T4 \<and> \<Gamma>1 \<turnstile> T3 <: T1 \<and> ((X,T3)#\<Gamma>1) \<turnstile> T2 <: T4)"
using rh_deriv fresh_cond by (auto dest: S_AllE_left simp add: fresh_prod)
moreover
{ assume "\<exists>T3 T4. T'=\<forall>[X<:T3].T4 \<and> \<Gamma>1 \<turnstile> T3 <: T1 \<and> ((X,T3)#\<Gamma>1) \<turnstile> T2 <: T4"
then obtain T3 T4
where T'_inst: "T'= \<forall>[X<:T3].T4"
and rh_drv_prem1: "\<Gamma>1 \<turnstile> T3 <: T1"
and rh_drv_prem2:"((X,T3)#\<Gamma>1) \<turnstile> T2 <: T4" by force
from IH1_outer[of "T1"] have "\<Gamma>1 \<turnstile> T3 <: S1"
using lh_drv_prem1 rh_drv_prem1 measure by (force simp add: ty.inject)
moreover
from IH2_outer[of "T1"] have "((X,T3)#\<Gamma>1) \<turnstile> S2 <: T2"
using measure lh_drv_prem2 rh_drv_prem1 by force
with IH1_outer[of "T2"] have "((X,T3)#\<Gamma>1) \<turnstile> S2 <: T4"
using measure rh_drv_prem2 by force
moreover
ultimately have "\<Gamma>1 \<turnstile> \<forall>[X<:S1].S2 <: T'"
using fresh_cond T'_inst by (simp add: fresh_prod S_All)
}
ultimately show "\<Gamma>1 \<turnstile> \<forall>[X<:S1].S2 <: T'" using top_case[of "\<Gamma>1" "\<forall>[X<:S1].S2" _ "T'"] by blast
qed
qed
qed
(* test
have narrowing:
"\<And>\<Delta> \<Gamma> X M N. (\<Delta>@(X,Q)#\<Gamma>) \<turnstile> M <: N \<Longrightarrow> (\<forall>P. \<Gamma> \<turnstile> P<:Q \<longrightarrow> (\<Delta>@(X,P)#\<Gamma>) \<turnstile> M <: N)"
proof -
fix \<Delta> \<Gamma> X M N
assume a: "(\<Delta>@(X,Q)#\<Gamma>) \<turnstile> M <: N"
thus "\<forall>P. \<Gamma> \<turnstile> P <: Q \<longrightarrow> (\<Delta>@(X,P)#\<Gamma>) \<turnstile> M <: N"
thm subtype_of_rel_induct
thm subtype_of_rel.induct
using a proof (induct)
using a proof (induct rule: subtype_of_rel_induct[of "\<Delta>@(X,Q)#\<Gamma>" "M" "N" _ "()"])
*)
have narrowing:
"\<And>\<Delta> \<Gamma> \<Gamma>' X M N. \<Gamma>' \<turnstile> M <: N \<Longrightarrow> \<Gamma>' = \<Delta>@(X,Q)#\<Gamma> \<longrightarrow> (\<forall>P. \<Gamma> \<turnstile> P<:Q \<longrightarrow> (\<Delta>@(X,P)#\<Gamma>) \<turnstile> M <: N)"
proof -
fix \<Delta> \<Gamma> \<Gamma>' X M N
assume "\<Gamma>' \<turnstile> M <: N"
thus "\<Gamma>' = \<Delta>@(X,Q)#\<Gamma> \<longrightarrow> (\<forall>P. \<Gamma> \<turnstile> P <: Q \<longrightarrow> (\<Delta>@(X,P)#\<Gamma>) \<turnstile> M <: N)"
(* FIXME : nominal induct does not work here because Gamma' M and N are fixed variables *)
(* FIX: Same comment about S1,\<Gamma>1 *)
proof (rule subtype_of_rel_induct[of "\<Gamma>'" "M" "N" "\<lambda>\<Gamma>' M N (\<Delta>,\<Gamma>,X).
\<Gamma>' = \<Delta>@(X,Q)#\<Gamma> \<longrightarrow> (\<forall>P. \<Gamma> \<turnstile> P <: Q \<longrightarrow> (\<Delta>@(X,P)#\<Gamma>) \<turnstile> M <: N)" "(\<Delta>,\<Gamma>,X)", simplified],
simp_all only: split_paired_all split_conv)
case (goal1 \<Delta>1 \<Gamma>1 X1 \<Gamma>2 S1)
have lh_drv_prem1: "\<turnstile> \<Gamma>2 ok" by fact
have lh_drv_prem2: "S1 closed_in \<Gamma>2" by fact
show "\<Gamma>2 = \<Delta>1@(X1,Q)#\<Gamma>1 \<longrightarrow> (\<forall>P. \<Gamma>1 \<turnstile> P <: Q \<longrightarrow> (\<Delta>1@(X1,P)#\<Gamma>1) \<turnstile> S1 <: Top)"
proof (intro strip)
fix P
assume a1: "\<Gamma>2 = \<Delta>1@(X1,Q)#\<Gamma>1"
assume a2: "\<Gamma>1 \<turnstile> P <: Q"
from a2 have "P closed_in \<Gamma>1" by (simp add: subtype_implies_closed)
hence a3: "\<turnstile> (\<Delta>1@(X1,P)#\<Gamma>1) ok" using lh_drv_prem1 a1 by (rule_tac replace_type, simp_all)
show "(\<Delta>1@(X1,P)#\<Gamma>1) \<turnstile> S1 <: Top"
using a1 a3 lh_drv_prem2 by (force simp add: closed_in_def domain_append)
qed
next
case (goal2 \<Delta>1 \<Gamma>1 X1 \<Gamma>2 X2 S1 T1)
have lh_drv_prem1: "\<turnstile> \<Gamma>2 ok" by fact
have lh_drv_prem2: "(X2,S1)\<in>set \<Gamma>2" by fact
have lh_drv_prem3: "\<Gamma>2 \<turnstile> S1 <: T1" by fact
have IH_inner:
"\<forall>\<Delta>1 \<Gamma>1 X1. \<Gamma>2 = \<Delta>1@(X1,Q)#\<Gamma>1 \<longrightarrow> (\<forall>P. \<Gamma>1 \<turnstile> P <: Q \<longrightarrow> (\<Delta>1@(X1,P)#\<Gamma>1) \<turnstile> S1 <: T1)" by fact
show "\<Gamma>2 = (\<Delta>1@(X1,Q)#\<Gamma>1) \<longrightarrow> (\<forall>P. \<Gamma>1 \<turnstile> P <: Q \<longrightarrow> (\<Delta>1@(X1,P)#\<Gamma>1) \<turnstile> TyVar X2 <: T1)"
proof (intro strip)
fix P
assume a1: "\<Gamma>2 = \<Delta>1@(X1,Q)#\<Gamma>1"
and a2: "\<Gamma>1 \<turnstile> P <: Q"
from a2 have "P closed_in \<Gamma>1" by (simp add: subtype_implies_closed)
hence a3: "\<turnstile> (\<Delta>1@(X1,P)#\<Gamma>1) ok"
using lh_drv_prem1 a1 by (rule_tac replace_type, simp_all)
show "(\<Delta>1@(X1,P)#\<Gamma>1) \<turnstile> TyVar X2 <: T1"
proof (cases "X1=X2")
case False
have b0: "X1\<noteq>X2" by fact
from lh_drv_prem3 a1 a2 IH_inner
have b1: "(\<Delta>1@(X1,P)#\<Gamma>1) \<turnstile> S1 <: T1" by simp
from lh_drv_prem2 a1 b0 have b2: "(X2,S1)\<in>set (\<Delta>1@(X1,P)#\<Gamma>1)" by simp
show "(\<Delta>1@(X1,P)#\<Gamma>1) \<turnstile> TyVar X2 <: T1" using a3 b2 b1 by (rule S_Var)
next
case True
have b0: "X1=X2" by fact
have a4: "S1=Q"
proof -
have c0: "(X2,Q)\<in>set \<Gamma>2" using b0 a1 by simp
with lh_drv_prem1 lh_drv_prem2 show "S1=Q" by (simp add: uniqueness_of_ctxt)
qed
have a5: "(\<Delta>1@(X1,P)#\<Gamma>1) extends \<Gamma>1" by (force simp add: extends_def)
have a6: "(\<Delta>1@(X2,P)#\<Gamma>1) \<turnstile> P <: Q" using b0 a5 a2 a3 by (simp add: weakening)
have a7: "(\<Delta>1@(X2,P)#\<Gamma>1) \<turnstile> Q <: T1" using b0 IH_inner a1 lh_drv_prem3 a2 a4 by simp
have a8: "(\<Delta>1@(X2,P)#\<Gamma>1) \<turnstile> P <: T1" using a6 a7 transitivity by blast
have a9: "(X2,P)\<in>set (\<Delta>1@(X1,P)#\<Gamma>1)" using b0 by simp
show "(\<Delta>1@(X1,P)#\<Gamma>1) \<turnstile> TyVar X2 <: T1" using a3 b0 a8 a9 by (force simp add: S_Var)
qed
qed
next
case (goal3 \<Delta>1 \<Gamma>1 X1 \<Gamma>2 X2)
have lh_drv_prem1: "\<turnstile> \<Gamma>2 ok" by fact
have lh_drv_prem2: "X2 \<in> domain \<Gamma>2" by fact
show "\<Gamma>2 = (\<Delta>1@(X1,Q)#\<Gamma>1) \<longrightarrow> (\<forall>P. \<Gamma>1 \<turnstile> P <: Q \<longrightarrow> (\<Delta>1@(X1,P)#\<Gamma>1) \<turnstile> TyVar X2 <: TyVar X2)"
proof (intro strip)
fix P
assume a1: "\<Gamma>2 = \<Delta>1@(X1,Q)#\<Gamma>1"
and a2: "\<Gamma>1 \<turnstile> P <: Q"
from a2 have "P closed_in \<Gamma>1" by (simp add: subtype_implies_closed)
hence a3: "\<turnstile> (\<Delta>1@(X1,P)#\<Gamma>1) ok"
using lh_drv_prem1 a1 by (rule_tac replace_type, simp_all)
have b0: "X2 \<in> domain (\<Delta>1@(X1,P)#\<Gamma>1)" using lh_drv_prem2 a1 by (simp add: domain_append)
show "(\<Delta>1@(X1,P)#\<Gamma>1) \<turnstile> TyVar X2 <: TyVar X2" using a3 b0 by (rule S_Refl)
qed
next
case goal4 thus ?case by blast
next
case (goal5 \<Delta>1 \<Gamma>1 X1 \<Gamma>2 X2 S1 S2 T1 T2)
have IH_inner1:
"\<forall>\<Delta>1 \<Gamma>1 X1. \<Gamma>2 = \<Delta>1@(X1,Q)#\<Gamma>1 \<longrightarrow> (\<forall>P. \<Gamma>1 \<turnstile> P <: Q \<longrightarrow> (\<Delta>1@(X1,P)#\<Gamma>1) \<turnstile> T1 <: S1)" by fact
have IH_inner2:
"\<forall>\<Delta>1 \<Gamma>1 X1. (X2,T1)#\<Gamma>2 = \<Delta>1@(X1,Q)#\<Gamma>1 \<longrightarrow> (\<forall>P. \<Gamma>1 \<turnstile> P <: Q \<longrightarrow> (\<Delta>1@(X1,P)#\<Gamma>1) \<turnstile> S2 <: T2)"
by fact
have lh_drv_prem1: "\<Gamma>2 \<turnstile> T1 <: S1" by fact
have lh_drv_prem2: "X2 \<sharp> (\<Gamma>2,S1, T1)" by fact
have lh_drv_prem3: "((X2,T1) # \<Gamma>2) \<turnstile> S2 <: T2" by fact
have freshness: "X2\<sharp>(\<Delta>1, \<Gamma>1, X1)" by fact
show "\<Gamma>2 = (\<Delta>1@(X1,Q)#\<Gamma>1) \<longrightarrow>
(\<forall>P. \<Gamma>1 \<turnstile> P <: Q \<longrightarrow> (\<Delta>1@(X1,P)#\<Gamma>1) \<turnstile> \<forall> [X2<:S1].S2 <: \<forall> [X2<:T1].T2)"
proof (intro strip)
fix P
assume a1: "\<Gamma>2 = \<Delta>1@(X1,Q)#\<Gamma>1"
and a2: "\<Gamma>1 \<turnstile> P <: Q"
have b0: "(\<Delta>1@(X1,P)#\<Gamma>1) \<turnstile> T1 <: S1" using a1 a2 lh_drv_prem1 IH_inner1 by simp
have b1: "(((X2,T1)#\<Delta>1)@(X1,P)#\<Gamma>1) \<turnstile> S2 <: T2" using a1 a2 lh_drv_prem3 IH_inner2
apply auto
apply (drule_tac x="(X2,T1)#\<Delta>1" in spec)
apply(simp)
done
have b3: "X2\<sharp>(\<Delta>1@(X1,P)#\<Gamma>1)" using lh_drv_prem2 freshness a2
by (auto simp add: fresh_prod fresh_list_append fresh_list_cons dest: subtype_implies_fresh)
show "(\<Delta>1@(X1,P)#\<Gamma>1) \<turnstile> \<forall> [X2<:S1].S2 <: \<forall> [X2<:T1].T2" using b0 b3 b1 by force
qed
qed
qed
from transitivity narrowing show ?case by blast
qed
section {* Terms *}
nominal_datatype trm = Var "vrs"
| Lam "\<guillemotleft>vrs\<guillemotright>trm" "ty"
| TAbs "\<guillemotleft>tyvrs\<guillemotright>trm" "ty"
| App "trm" "trm"
| TApp "trm" "ty"
consts
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
"Lam [a:tys].t" \<rightleftharpoons> "trm.Lam a t tys"
"TAbs [a<:tys].t" \<rightleftharpoons> "trm.TAbs a t tys"