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Quellcode-Bibliothek
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Datei:
Fsub.thy
Sprache: Isabelle
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(*<*)
theory Fsub
imports "HOL-Nominal.Nominal"
begin
(*>*)
text\<open>Authors: Christian Urban,
Benjamin Pierce,
Dimitrios Vytiniotis
Stephanie Weirich
Steve Zdancewic
Julien Narboux
Stefan Berghofer
with great help from Markus Wenzel.\<close>
section \<open>Types for Names, Nominal Datatype Declaration for Types and Terms\<close>
no_syntax
"_Map" :: "maplets => 'a \ 'b" ("(1[_])")
text \<open>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 \<open>tyvrs\<close> and \<open>vrs\<close>:\<close>
atom_decl tyvrs vrs
text\<open>There are numerous facts that come with this declaration: for example that
there are infinitely many elements in \<open>tyvrs\<close> and \<open>vrs\<close>.\<close>
text\<open>The constructors for types and terms in System \FSUB{} contain abstractions
over type-variables and term-variables. The nominal datatype package uses
\<open>\<guillemotleft>\<dots>\<guillemotright>\<dots>\<close> to indicate where abstractions occur.\<close>
nominal_datatype ty =
Tvar "tyvrs"
| Top
| Arrow "ty" "ty" (infixr "\" 200)
| Forall "\tyvrs\ty" "ty"
nominal_datatype trm =
Var "vrs"
| Abs "\vrs\trm" "ty"
| TAbs "\tyvrs\trm" "ty"
| App "trm" "trm" (infixl "\" 200)
| TApp "trm" "ty" (infixl "\\<^sub>\" 200)
text \<open>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>\<open>Arrow\<close>.\<close>
abbreviation
Forall_syn :: "tyvrs \ ty \ ty \ ty" ("(3\_<:_./ _)" [0, 0, 10] 10)
where
"\X<:T\<^sub>1. T\<^sub>2 \ ty.Forall X T\<^sub>2 T\<^sub>1"
abbreviation
Abs_syn :: "vrs \ ty \ trm \ trm" ("(3\_:_./ _)" [0, 0, 10] 10)
where
"\x:T. t \ trm.Abs x t T"
abbreviation
TAbs_syn :: "tyvrs \ ty \ trm \ trm" ("(3\_<:_./ _)" [0, 0, 10] 10)
where
"\X<:T. t \ trm.TAbs X t T"
text \<open>Again there are numerous facts that are proved automatically for \<^typ>\<open>ty\<close>
and \<^typ>\<open>trm\<close>: for example that the set of free variables, i.e.~the \<open>support\<close>,
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>\<open>ty\<close>s
and \<^typ>\<open>trm\<close>s are equal:\<close>
lemma alpha_illustration:
shows "(\X<:T. Tvar X) = (\Y<:T. Tvar Y)"
and "(\x:T. Var x) = (\y:T. Var y)"
by (simp_all add: ty.inject trm.inject alpha calc_atm fresh_atm)
section \<open>SubTyping Contexts\<close>
nominal_datatype binding =
VarB vrs ty
| TVarB tyvrs ty
type_synonym env = "binding list"
text \<open>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).\<close>
text \<open>In order to state validity-conditions for typing-contexts, the notion of
a \<open>dom\<close> of a typing-context is handy.\<close>
nominal_primrec
"tyvrs_of" :: "binding \ tyvrs set"
where
"tyvrs_of (VarB x y) = {}"
| "tyvrs_of (TVarB x y) = {x}"
by auto
nominal_primrec
"vrs_of" :: "binding \ vrs set"
where
"vrs_of (VarB x y) = {x}"
| "vrs_of (TVarB x y) = {}"
by auto
primrec
"ty_dom" :: "env \ tyvrs set"
where
"ty_dom [] = {}"
| "ty_dom (X#\) = (tyvrs_of X)\(ty_dom \)"
primrec
"trm_dom" :: "env \ vrs set"
where
"trm_dom [] = {}"
| "trm_dom (X#\) = (vrs_of X)\(trm_dom \)"
lemma vrs_of_eqvt[eqvt]:
fixes pi ::"tyvrs prm"
and pi'::"vrs prm"
shows "pi \(tyvrs_of x) = tyvrs_of (pi\x)"
and "pi'\(tyvrs_of x) = tyvrs_of (pi'\x)"
and "pi \(vrs_of x) = vrs_of (pi\x)"
and "pi'\(vrs_of x) = vrs_of (pi'\x)"
by (nominal_induct x rule: binding.strong_induct) (simp_all add: tyvrs_of.simps eqvts)
lemma doms_eqvt[eqvt]:
fixes pi::"tyvrs prm"
and pi'::"vrs prm"
shows "pi \(ty_dom \) = ty_dom (pi\\)"
and "pi'\(ty_dom \) = ty_dom (pi'\\)"
and "pi \(trm_dom \) = trm_dom (pi\\)"
and "pi'\(trm_dom \) = trm_dom (pi'\\)"
by (induct \<Gamma>) (simp_all add: eqvts)
lemma finite_vrs:
shows "finite (tyvrs_of x)"
and "finite (vrs_of x)"
by (nominal_induct rule:binding.strong_induct) auto
lemma finite_doms:
shows "finite (ty_dom \)"
and "finite (trm_dom \)"
by (induct \<Gamma>) (auto simp add: finite_vrs)
lemma ty_dom_supp:
shows "(supp (ty_dom \)) = (ty_dom \)"
and "(supp (trm_dom \)) = (trm_dom \)"
by (simp only: at_fin_set_supp at_tyvrs_inst at_vrs_inst finite_doms)+
lemma ty_dom_inclusion:
assumes a: "(TVarB X T)\set \"
shows "X\(ty_dom \)"
using a by (induct \<Gamma>) (auto)
lemma ty_binding_existence:
assumes "X \ (tyvrs_of a)"
shows "\T.(TVarB X T=a)"
using assms
by (nominal_induct a rule: binding.strong_induct) (auto)
lemma ty_dom_existence:
assumes a: "X\(ty_dom \)"
shows "\T.(TVarB X T)\set \"
using a
apply (induct \<Gamma>, auto)
apply (rename_tac a \<Gamma>')
apply (subgoal_tac "\T.(TVarB X T=a)")
apply (auto)
apply (auto simp add: ty_binding_existence)
done
lemma doms_append:
shows "ty_dom (\@\) = ((ty_dom \) \ (ty_dom \))"
and "trm_dom (\@\) = ((trm_dom \) \ (trm_dom \))"
by (induct \<Gamma>) (auto)
lemma ty_vrs_prm_simp:
fixes pi::"vrs prm"
and S::"ty"
shows "pi\S = S"
by (induct S rule: ty.induct) (auto simp add: calc_atm)
lemma fresh_ty_dom_cons:
fixes X::"tyvrs"
shows "X\(ty_dom (Y#\)) = (X\(tyvrs_of Y) \ X\(ty_dom \))"
apply (nominal_induct rule:binding.strong_induct)
apply (auto)
apply (simp add: fresh_def supp_def eqvts)
apply (simp add: fresh_fin_insert [OF pt_tyvrs_inst at_tyvrs_inst fs_tyvrs_inst] finite_doms)
apply (simp add: fresh_def supp_def eqvts)
apply (simp add: fresh_fin_insert [OF pt_tyvrs_inst at_tyvrs_inst fs_tyvrs_inst] finite_doms)+
done
lemma tyvrs_fresh:
fixes X::"tyvrs"
assumes "X \ a"
shows "X \ tyvrs_of a"
and "X \ vrs_of a"
using assms
apply (nominal_induct a rule:binding.strong_induct)
apply (auto)
apply (fresh_guess)+
done
lemma fresh_dom:
fixes X::"tyvrs"
assumes a: "X\\"
shows "X\(ty_dom \)"
using a
apply(induct \<Gamma>)
apply(simp add: fresh_set_empty)
apply(simp only: fresh_ty_dom_cons)
apply(auto simp add: fresh_prod fresh_list_cons tyvrs_fresh)
done
text \<open>Not all lists of type \<^typ>\<open>env\<close> are well-formed. One condition
requires that in \<^term>\<open>TVarB X S#\<Gamma>\<close> all free variables of \<^term>\<open>S\<close> must be
in the \<^term>\<open>ty_dom\<close> of \<^term>\<open>\<Gamma>\<close>, that is \<^term>\<open>S\<close> must be \<open>closed\<close>
in \<^term>\<open>\<Gamma>\<close>. The set of free variables of \<^term>\<open>S\<close> is the
\<open>support\<close> of \<^term>\<open>S\<close>.\<close>
definition "closed_in" :: "ty \ env \ bool" ("_ closed'_in _" [100,100] 100) where
"S closed_in \ \ (supp S)\(ty_dom \)"
lemma closed_in_eqvt[eqvt]:
fixes pi::"tyvrs prm"
assumes a: "S closed_in \"
shows "(pi\S) closed_in (pi\\)"
using a
proof -
from a have "pi\(S closed_in \)" by (simp add: perm_bool)
then show "(pi\S) closed_in (pi\\)" by (simp add: closed_in_def eqvts)
qed
lemma tyvrs_vrs_prm_simp:
fixes pi::"vrs prm"
shows "tyvrs_of (pi\a) = tyvrs_of a"
apply (nominal_induct rule:binding.strong_induct)
apply (simp_all add: eqvts)
apply (simp add: dj_perm_forget[OF dj_tyvrs_vrs])
done
lemma ty_vrs_fresh:
fixes x::"vrs"
and T::"ty"
shows "x \ T"
by (simp add: fresh_def supp_def ty_vrs_prm_simp)
lemma ty_dom_vrs_prm_simp:
fixes pi::"vrs prm"
and \<Gamma>::"env"
shows "(ty_dom (pi\\)) = (ty_dom \)"
apply(induct \<Gamma>)
apply (simp add: eqvts)
apply(simp add: tyvrs_vrs_prm_simp)
done
lemma closed_in_eqvt'[eqvt]:
fixes pi::"vrs prm"
assumes a: "S closed_in \"
shows "(pi\S) closed_in (pi\\)"
using a
by (simp add: closed_in_def ty_dom_vrs_prm_simp ty_vrs_prm_simp)
lemma fresh_vrs_of:
fixes x::"vrs"
shows "x\vrs_of b = x\b"
by (nominal_induct b rule: binding.strong_induct)
(simp_all add: fresh_singleton fresh_set_empty ty_vrs_fresh fresh_atm)
lemma fresh_trm_dom:
fixes x::"vrs"
shows "x\ trm_dom \ = x\\"
by (induct \<Gamma>)
(simp_all add: fresh_set_empty fresh_list_cons
fresh_fin_union [OF pt_vrs_inst at_vrs_inst fs_vrs_inst]
finite_doms finite_vrs fresh_vrs_of fresh_list_nil)
lemma closed_in_fresh: "(X::tyvrs) \ ty_dom \ \ T closed_in \ \ X \ T"
by (auto simp add: closed_in_def fresh_def ty_dom_supp)
text \<open>Now validity of a context is a straightforward inductive definition.\<close>
inductive
valid_rel :: "env \ bool" ("\ _ ok" [100] 100)
where
valid_nil[simp]: "\ [] ok"
| valid_consT[simp]: "\\ \ ok; X\(ty_dom \); T closed_in \\ \ \ (TVarB X T#\) ok"
| valid_cons [simp]: "\\ \ ok; x\(trm_dom \); T closed_in \\ \ \ (VarB x T#\) ok"
equivariance valid_rel
declare binding.inject [simp add]
declare trm.inject [simp add]
inductive_cases validE[elim]:
"\ (TVarB X T#\) ok"
"\ (VarB x T#\) ok"
"\ (b#\) ok"
declare binding.inject [simp del]
declare trm.inject [simp del]
lemma validE_append:
assumes a: "\ (\@\) ok"
shows "\ \ ok"
using a
proof (induct \<Delta>)
case (Cons a \<Gamma>')
then show ?case
by (nominal_induct a rule:binding.strong_induct)
(auto elim: validE)
qed (auto)
lemma replace_type:
assumes a: "\ (\@(TVarB X T)#\) ok"
and b: "S closed_in \"
shows "\ (\@(TVarB X S)#\) ok"
using a b
proof(induct \<Delta>)
case Nil
then show ?case by (auto elim: validE intro: valid_cons simp add: doms_append closed_in_def)
next
case (Cons a \<Gamma>')
then show ?case
by (nominal_induct a rule:binding.strong_induct)
(auto elim: validE intro!: valid_cons simp add: doms_append closed_in_def)
qed
text \<open>Well-formed contexts have a unique type-binding for a type-variable.\<close>
lemma uniqueness_of_ctxt:
fixes \<Gamma>::"env"
assumes a: "\ \ ok"
and b: "(TVarB X T)\set \"
and c: "(TVarB X S)\set \"
shows "T=S"
using a b c
proof (induct)
case (valid_consT \<Gamma> X' T')
moreover
{ fix \<Gamma>'::"env"
assume a: "X'\(ty_dom \')"
have "\(\T.(TVarB X' T)\(set \'))" using a
proof (induct \<Gamma>')
case (Cons Y \<Gamma>')
thus "\ (\T.(TVarB X' T)\set(Y#\'))"
by (simp add: fresh_ty_dom_cons
fresh_fin_union[OF pt_tyvrs_inst at_tyvrs_inst fs_tyvrs_inst]
finite_vrs finite_doms,
auto simp add: fresh_atm fresh_singleton)
qed (simp)
}
ultimately show "T=S" by (auto simp add: binding.inject)
qed (auto)
lemma uniqueness_of_ctxt':
fixes \<Gamma>::"env"
assumes a: "\ \ ok"
and b: "(VarB x T)\set \"
and c: "(VarB x S)\set \"
shows "T=S"
using a b c
proof (induct)
case (valid_cons \<Gamma> x' T')
moreover
{ fix \<Gamma>'::"env"
assume a: "x'\(trm_dom \')"
have "\(\T.(VarB x' T)\(set \'))" using a
proof (induct \<Gamma>')
case (Cons y \<Gamma>')
thus "\ (\T.(VarB x' T)\set(y#\'))"
by (simp add: fresh_fin_union[OF pt_vrs_inst at_vrs_inst fs_vrs_inst]
finite_vrs finite_doms,
auto simp add: fresh_atm fresh_singleton)
qed (simp)
}
ultimately show "T=S" by (auto simp add: binding.inject)
qed (auto)
section \<open>Size and Capture-Avoiding Substitution for Types\<close>
nominal_primrec
size_ty :: "ty \ nat"
where
"size_ty (Tvar X) = 1"
| "size_ty (Top) = 1"
| "size_ty (T1 \ T2) = (size_ty T1) + (size_ty T2) + 1"
| "X \ T1 \ size_ty (\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
nominal_primrec
subst_ty :: "ty \ tyvrs \ ty \ ty" ("_[_ \ _]\<^sub>\" [300, 0, 0] 300)
where
"(Tvar X)[Y \ T]\<^sub>\ = (if X=Y then T else Tvar X)"
| "(Top)[Y \ T]\<^sub>\ = Top"
| "(T\<^sub>1 \ T\<^sub>2)[Y \ T]\<^sub>\ = T\<^sub>1[Y \ T]\<^sub>\ \ T\<^sub>2[Y \ T]\<^sub>\"
| "X\(Y,T,T\<^sub>1) \ (\X<:T\<^sub>1. T\<^sub>2)[Y \ T]\<^sub>\ = (\X<:T\<^sub>1[Y \ T]\<^sub>\. T\<^sub>2[Y \ T]\<^sub>\)"
apply (finite_guess)+
apply (rule TrueI)+
apply (simp add: abs_fresh)
apply (fresh_guess)+
done
lemma subst_eqvt[eqvt]:
fixes pi::"tyvrs prm"
and T::"ty"
shows "pi\(T[X \ T']\<^sub>\) = (pi\T)[(pi\X) \ (pi\T')]\<^sub>\"
by (nominal_induct T avoiding: X T' rule: ty.strong_induct)
(perm_simp add: fresh_bij)+
lemma subst_eqvt'[eqvt]:
fixes pi::"vrs prm"
and T::"ty"
shows "pi\(T[X \ T']\<^sub>\) = (pi\T)[(pi\X) \ (pi\T')]\<^sub>\"
by (nominal_induct T avoiding: X T' rule: ty.strong_induct)
(perm_simp add: fresh_left)+
lemma type_subst_fresh:
fixes X::"tyvrs"
assumes "X\T" and "X\P"
shows "X\T[Y \ P]\<^sub>\"
using assms
by (nominal_induct T avoiding: X Y P rule:ty.strong_induct)
(auto simp add: abs_fresh)
lemma fresh_type_subst_fresh:
assumes "X\T'"
shows "X\T[X \ T']\<^sub>\"
using assms
by (nominal_induct T avoiding: X T' rule: ty.strong_induct)
(auto simp add: fresh_atm abs_fresh fresh_nat)
lemma type_subst_identity:
"X\T \ T[X \ U]\<^sub>\ = T"
by (nominal_induct T avoiding: X U rule: ty.strong_induct)
(simp_all add: fresh_atm abs_fresh)
lemma type_substitution_lemma:
"X \ Y \ X\L \ M[X \ N]\<^sub>\[Y \ L]\<^sub>\ = M[Y \ L]\<^sub>\[X \ N[Y \ L]\<^sub>\]\<^sub>\"
by (nominal_induct M avoiding: X Y N L rule: ty.strong_induct)
(auto simp add: type_subst_fresh type_subst_identity)
lemma type_subst_rename:
"Y\T \ ([(Y,X)]\T)[Y \ U]\<^sub>\ = T[X \ U]\<^sub>\"
by (nominal_induct T avoiding: X Y U rule: ty.strong_induct)
(simp_all add: fresh_atm calc_atm abs_fresh fresh_aux)
nominal_primrec
subst_tyb :: "binding \ tyvrs \ ty \ binding" ("_[_ \ _]\<^sub>b" [100,100,100] 100)
where
"(TVarB X U)[Y \ T]\<^sub>b = TVarB X (U[Y \ T]\<^sub>\)"
| "(VarB X U)[Y \ T]\<^sub>b = VarB X (U[Y \ T]\<^sub>\)"
by auto
lemma binding_subst_fresh:
fixes X::"tyvrs"
assumes "X\a"
and "X\P"
shows "X\a[Y \ P]\<^sub>b"
using assms
by (nominal_induct a rule: binding.strong_induct)
(auto simp add: type_subst_fresh)
lemma binding_subst_identity:
shows "X\B \ B[X \ U]\<^sub>b = B"
by (induct B rule: binding.induct)
(simp_all add: fresh_atm type_subst_identity)
primrec subst_tyc :: "env \ tyvrs \ ty \ env" ("_[_ \ _]\<^sub>e" [100,100,100] 100) where
"([])[Y \ T]\<^sub>e= []"
| "(B#\)[Y \ T]\<^sub>e = (B[Y \ T]\<^sub>b)#(\[Y \ T]\<^sub>e)"
lemma ctxt_subst_fresh':
fixes X::"tyvrs"
assumes "X\\"
and "X\P"
shows "X\\[Y \ P]\<^sub>e"
using assms
by (induct \<Gamma>)
(auto simp add: fresh_list_cons binding_subst_fresh)
lemma ctxt_subst_mem_TVarB: "TVarB X T \ set \ \ TVarB X (T[Y \ U]\<^sub>\) \ set (\[Y \ U]\<^sub>e)"
by (induct \<Gamma>) auto
lemma ctxt_subst_mem_VarB: "VarB x T \ set \ \ VarB x (T[Y \ U]\<^sub>\) \ set (\[Y \ U]\<^sub>e)"
by (induct \<Gamma>) auto
lemma ctxt_subst_identity: "X\\ \ \[X \ U]\<^sub>e = \"
by (induct \<Gamma>) (simp_all add: fresh_list_cons binding_subst_identity)
lemma ctxt_subst_append: "(\ @ \)[X \ T]\<^sub>e = \[X \ T]\<^sub>e @ \[X \ T]\<^sub>e"
by (induct \<Delta>) simp_all
nominal_primrec
subst_trm :: "trm \ vrs \ trm \ trm" ("_[_ \ _]" [300, 0, 0] 300)
where
"(Var x)[y \ t'] = (if x=y then t' else (Var x))"
| "(t1 \ t2)[y \ t'] = t1[y \ t'] \ t2[y \ t']"
| "(t \\<^sub>\ T)[y \ t'] = t[y \ t'] \\<^sub>\ T"
| "X\(T,t') \ (\X<:T. t)[y \ t'] = (\X<:T. t[y \ t'])"
| "x\(y,t') \ (\x:T. t)[y \ t'] = (\x:T. t[y \ t'])"
apply(finite_guess)+
apply(rule TrueI)+
apply(simp add: abs_fresh)+
apply(fresh_guess add: ty_vrs_fresh abs_fresh)+
done
lemma subst_trm_fresh_tyvar:
fixes X::"tyvrs"
shows "X\t \ X\u \ X\t[x \ u]"
by (nominal_induct t avoiding: x u rule: trm.strong_induct)
(auto simp add: trm.fresh abs_fresh)
lemma subst_trm_fresh_var:
"x\u \ x\t[x \ u]"
by (nominal_induct t avoiding: x u rule: trm.strong_induct)
(simp_all add: abs_fresh fresh_atm ty_vrs_fresh)
lemma subst_trm_eqvt[eqvt]:
fixes pi::"tyvrs prm"
and t::"trm"
shows "pi\(t[x \ u]) = (pi\t)[(pi\x) \ (pi\u)]"
by (nominal_induct t avoiding: x u rule: trm.strong_induct)
(perm_simp add: fresh_left)+
lemma subst_trm_eqvt'[eqvt]:
fixes pi::"vrs prm"
and t::"trm"
shows "pi\(t[x \ u]) = (pi\t)[(pi\x) \ (pi\u)]"
by (nominal_induct t avoiding: x u rule: trm.strong_induct)
(perm_simp add: fresh_left)+
lemma subst_trm_rename:
"y\t \ ([(y, x)] \ t)[y \ u] = t[x \ u]"
by (nominal_induct t avoiding: x y u rule: trm.strong_induct)
(simp_all add: fresh_atm calc_atm abs_fresh fresh_aux ty_vrs_fresh perm_fresh_fresh)
nominal_primrec (freshness_context: "T2::ty")
subst_trm_ty :: "trm \ tyvrs \ ty \ trm" ("_[_ \\<^sub>\ _]" [300, 0, 0] 300)
where
"(Var x)[Y \\<^sub>\ T2] = Var x"
| "(t1 \ t2)[Y \\<^sub>\ T2] = t1[Y \\<^sub>\ T2] \ t2[Y \\<^sub>\ T2]"
| "(t1 \\<^sub>\ T)[Y \\<^sub>\ T2] = t1[Y \\<^sub>\ T2] \\<^sub>\ T[Y \ T2]\<^sub>\"
| "X\(Y,T,T2) \ (\X<:T. t)[Y \\<^sub>\ T2] = (\X<:T[Y \ T2]\<^sub>\. t[Y \\<^sub>\ T2])"
| "(\x:T. t)[Y \\<^sub>\ T2] = (\x:T[Y \ T2]\<^sub>\. t[Y \\<^sub>\ T2])"
apply(finite_guess)+
apply(rule TrueI)+
apply(simp add: abs_fresh ty_vrs_fresh)+
apply(simp add: type_subst_fresh)
apply(fresh_guess add: ty_vrs_fresh abs_fresh)+
done
lemma subst_trm_ty_fresh:
fixes X::"tyvrs"
shows "X\t \ X\T \ X\t[Y \\<^sub>\ T]"
by (nominal_induct t avoiding: Y T rule: trm.strong_induct)
(auto simp add: abs_fresh type_subst_fresh)
lemma subst_trm_ty_fresh':
"X\T \ X\t[X \\<^sub>\ T]"
by (nominal_induct t avoiding: X T rule: trm.strong_induct)
(simp_all add: abs_fresh fresh_type_subst_fresh fresh_atm)
lemma subst_trm_ty_eqvt[eqvt]:
fixes pi::"tyvrs prm"
and t::"trm"
shows "pi\(t[X \\<^sub>\ T]) = (pi\t)[(pi\X) \\<^sub>\ (pi\T)]"
by (nominal_induct t avoiding: X T rule: trm.strong_induct)
(perm_simp add: fresh_bij subst_eqvt)+
lemma subst_trm_ty_eqvt'[eqvt]:
fixes pi::"vrs prm"
and t::"trm"
shows "pi\(t[X \\<^sub>\ T]) = (pi\t)[(pi\X) \\<^sub>\ (pi\T)]"
by (nominal_induct t avoiding: X T rule: trm.strong_induct)
(perm_simp add: fresh_left subst_eqvt')+
lemma subst_trm_ty_rename:
"Y\t \ ([(Y, X)] \ t)[Y \\<^sub>\ U] = t[X \\<^sub>\ U]"
by (nominal_induct t avoiding: X Y U rule: trm.strong_induct)
(simp_all add: fresh_atm calc_atm abs_fresh fresh_aux type_subst_rename)
section \<open>Subtyping-Relation\<close>
text \<open>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>\<open>X\<sharp>\<Gamma>\<close> in the \<open>S_Forall\<close>-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.)\<close>
inductive
subtype_of :: "env \ ty \ ty \ bool" ("_\_<:_" [100,100,100] 100)
where
SA_Top[intro]: "\\ \ ok; S closed_in \\ \ \ \ S <: Top"
| SA_refl_TVar[intro]: "\\ \ ok; X \ ty_dom \\\ \ \ Tvar X <: Tvar X"
| SA_trans_TVar[intro]: "\(TVarB X S) \ set \; \ \ S <: T\ \ \ \ (Tvar X) <: T"
| SA_arrow[intro]: "\\ \ T\<^sub>1 <: S\<^sub>1; \ \ S\<^sub>2 <: T\<^sub>2\ \ \ \ (S\<^sub>1 \ S\<^sub>2) <: (T\<^sub>1 \ T\<^sub>2)"
| SA_all[intro]: "\\ \ T\<^sub>1 <: S\<^sub>1; ((TVarB X T\<^sub>1)#\) \ S\<^sub>2 <: T\<^sub>2\ \ \ \ (\X<:S\<^sub>1. S\<^sub>2) <: (\X<:T\<^sub>1. T\<^sub>2)"
lemma subtype_implies_ok:
fixes X::"tyvrs"
assumes a: "\ \ S <: T"
shows "\ \ ok"
using a by (induct) (auto)
lemma subtype_implies_closed:
assumes a: "\ \ S <: T"
shows "S closed_in \ \ T closed_in \"
using a
proof (induct)
case (SA_Top \<Gamma> S)
have "Top closed_in \" by (simp add: closed_in_def ty.supp)
moreover
have "S closed_in \" by fact
ultimately show "S closed_in \ \ Top closed_in \" by simp
next
case (SA_trans_TVar X S \<Gamma> T)
have "(TVarB X S)\set \" by fact
hence "X \ ty_dom \" by (rule ty_dom_inclusion)
hence "(Tvar X) closed_in \" by (simp add: closed_in_def ty.supp supp_atm)
moreover
have "S closed_in \ \ T closed_in \" by fact
hence "T closed_in \" by force
ultimately show "(Tvar X) closed_in \ \ T closed_in \" by simp
qed (auto simp add: closed_in_def ty.supp supp_atm abs_supp)
lemma subtype_implies_fresh:
fixes X::"tyvrs"
assumes a1: "\ \ S <: T"
and a2: "X\\"
shows "X\S \ X\T"
proof -
from a1 have "\ \ ok" by (rule subtype_implies_ok)
with a2 have "X\ty_dom(\)" by (simp add: fresh_dom)
moreover
from a1 have "S closed_in \ \ T closed_in \" by (rule subtype_implies_closed)
hence "supp S \ ((supp (ty_dom \))::tyvrs set)"
and "supp T \ ((supp (ty_dom \))::tyvrs set)" by (simp_all add: ty_dom_supp closed_in_def)
ultimately show "X\S \ X\T" by (force simp add: supp_prod fresh_def)
qed
lemma valid_ty_dom_fresh:
fixes X::"tyvrs"
assumes valid: "\ \ ok"
shows "X\(ty_dom \) = X\\"
using valid
apply induct
apply (simp add: fresh_list_nil fresh_set_empty)
apply (simp_all add: binding.fresh fresh_list_cons
fresh_fin_insert [OF pt_tyvrs_inst at_tyvrs_inst fs_tyvrs_inst] finite_doms fresh_atm)
apply (auto simp add: closed_in_fresh)
done
equivariance subtype_of
nominal_inductive subtype_of
apply (simp_all add: abs_fresh)
apply (fastforce simp add: valid_ty_dom_fresh dest: subtype_implies_ok)
apply (force simp add: closed_in_fresh dest: subtype_implies_closed subtype_implies_ok)+
done
section \<open>Reflexivity of Subtyping\<close>
lemma subtype_reflexivity:
assumes a: "\ \ ok"
and b: "T closed_in \"
shows "\ \ T <: T"
using a b
proof(nominal_induct T avoiding: \<Gamma> rule: ty.strong_induct)
case (Forall X T\<^sub>1 T\<^sub>2)
have ih_T\<^sub>1: "\<And>\<Gamma>. \<lbrakk>\<turnstile> \<Gamma> ok; T\<^sub>1 closed_in \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> T\<^sub>1 <: T\<^sub>1" by fact
have ih_T\<^sub>2: "\<And>\<Gamma>. \<lbrakk>\<turnstile> \<Gamma> ok; T\<^sub>2 closed_in \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> T\<^sub>2 <: T\<^sub>2" by fact
have fresh_cond: "X\\" by fact
hence fresh_ty_dom: "X\(ty_dom \)" by (simp add: fresh_dom)
have "(\X<:T\<^sub>2. T\<^sub>1) closed_in \" by fact
hence closed\<^sub>T2: "T\<^sub>2 closed_in \<Gamma>" and closed\<^sub>T1: "T\<^sub>1 closed_in ((TVarB X T\<^sub>2)#\<Gamma>)"
by (auto simp add: closed_in_def ty.supp abs_supp)
have ok: "\ \ ok" by fact
hence ok': "\ ((TVarB X T\<^sub>2)#\) ok" using closed\<^sub>T2 fresh_ty_dom by simp
have "\ \ T\<^sub>2 <: T\<^sub>2" using ih_T\<^sub>2 closed\<^sub>T2 ok by simp
moreover
have "((TVarB X T\<^sub>2)#\) \ T\<^sub>1 <: T\<^sub>1" using ih_T\<^sub>1 closed\<^sub>T1 ok' by simp
ultimately show "\ \ (\X<:T\<^sub>2. T\<^sub>1) <: (\X<:T\<^sub>2. T\<^sub>1)" using fresh_cond
by (simp add: subtype_of.SA_all)
qed (auto simp add: closed_in_def ty.supp supp_atm)
lemma subtype_reflexivity_semiautomated:
assumes a: "\ \ ok"
and b: "T closed_in \"
shows "\ \ T <: T"
using a b
apply(nominal_induct T avoiding: \<Gamma> rule: ty.strong_induct)
apply(auto simp add: ty.supp abs_supp supp_atm closed_in_def)
\<comment> \<open>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 \<open>closed_in_def\<close>.\<close>
apply(drule_tac x="(TVarB tyvrs ty2)#\" in meta_spec)
apply(force dest: fresh_dom simp add: closed_in_def)
done
section \<open>Weakening\<close>
text \<open>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.\<close>
definition extends :: "env \ env \ bool" ("_ extends _" [100,100] 100) where
"\ extends \ \ \X Q. (TVarB X Q)\set \ \ (TVarB X Q)\set \"
lemma extends_ty_dom:
assumes a: "\ extends \"
shows "ty_dom \ \ ty_dom \"
using a
apply (auto simp add: extends_def)
apply (drule ty_dom_existence)
apply (force simp add: ty_dom_inclusion)
done
lemma extends_closed:
assumes a1: "T closed_in \"
and a2: "\ extends \"
shows "T closed_in \"
using a1 a2
by (auto dest: extends_ty_dom simp add: closed_in_def)
lemma extends_memb:
assumes a: "\ extends \"
and b: "(TVarB X T) \ set \"
shows "(TVarB X T) \ set \"
using a b by (simp add: extends_def)
lemma weakening:
assumes a: "\ \ S <: T"
and b: "\ \ ok"
and c: "\ extends \"
shows "\ \ S <: T"
using a b c
proof (nominal_induct \<Gamma> S T avoiding: \<Delta> rule: subtype_of.strong_induct)
case (SA_Top \<Gamma> S)
have lh_drv_prem: "S closed_in \" by fact
have "\ \ ok" by fact
moreover
have "\ extends \" by fact
hence "S closed_in \" using lh_drv_prem by (simp only: extends_closed)
ultimately show "\ \ S <: Top" by force
next
case (SA_trans_TVar X S \<Gamma> T)
have lh_drv_prem: "(TVarB X S) \ set \" by fact
have ih: "\\. \ \ ok \ \ extends \ \ \ \ S <: T" by fact
have ok: "\ \ ok" by fact
have extends: "\ extends \" by fact
have "(TVarB X S) \ set \" using lh_drv_prem extends by (simp only: extends_memb)
moreover
have "\ \ S <: T" using ok extends ih by simp
ultimately show "\ \ Tvar X <: T" using ok by force
next
case (SA_refl_TVar \<Gamma> X)
have lh_drv_prem: "X \ ty_dom \" by fact
have "\ \ ok" by fact
moreover
have "\ extends \" by fact
hence "X \ ty_dom \" using lh_drv_prem by (force dest: extends_ty_dom)
ultimately show "\ \ Tvar X <: Tvar X" by force
next
case (SA_arrow \<Gamma> T\<^sub>1 S\<^sub>1 S\<^sub>2 T\<^sub>2) thus "\<Delta> \<turnstile> S\<^sub>1 \<rightarrow> S\<^sub>2 <: T\<^sub>1 \<rightarrow> T\<^sub>2" by blast
next
case (SA_all \<Gamma> T\<^sub>1 S\<^sub>1 X S\<^sub>2 T\<^sub>2)
have fresh_cond: "X\\" by fact
hence fresh_dom: "X\(ty_dom \)" by (simp add: fresh_dom)
have ih\<^sub>1: "\<And>\<Delta>. \<turnstile> \<Delta> ok \<Longrightarrow> \<Delta> extends \<Gamma> \<Longrightarrow> \<Delta> \<turnstile> T\<^sub>1 <: S\<^sub>1" by fact
have ih\<^sub>2: "\<And>\<Delta>. \<turnstile> \<Delta> ok \<Longrightarrow> \<Delta> extends ((TVarB X T\<^sub>1)#\<Gamma>) \<Longrightarrow> \<Delta> \<turnstile> S\<^sub>2 <: T\<^sub>2" by fact
have lh_drv_prem: "\ \ T\<^sub>1 <: S\<^sub>1" by fact
hence closed\<^sub>T1: "T\<^sub>1 closed_in \<Gamma>" by (simp add: subtype_implies_closed)
have ok: "\ \ ok" by fact
have ext: "\ extends \" by fact
have "T\<^sub>1 closed_in \" using ext closed\<^sub>T1 by (simp only: extends_closed)
hence "\ ((TVarB X T\<^sub>1)#\) ok" using fresh_dom ok by force
moreover
have "((TVarB X T\<^sub>1)#\) extends ((TVarB X T\<^sub>1)#\)" using ext by (force simp add: extends_def)
ultimately have "((TVarB X T\<^sub>1)#\) \ S\<^sub>2 <: T\<^sub>2" using ih\<^sub>2 by simp
moreover
have "\ \ T\<^sub>1 <: S\<^sub>1" using ok ext ih\<^sub>1 by simp
ultimately show "\ \ (\X<:S\<^sub>1. S\<^sub>2) <: (\X<:T\<^sub>1. T\<^sub>2)" using ok by (force intro: SA_all)
qed
text \<open>In fact all ``non-binding" cases can be solved automatically:\<close>
lemma weakening_more_automated:
assumes a: "\ \ S <: T"
and b: "\ \ ok"
and c: "\ extends \"
shows "\ \ S <: T"
using a b c
proof (nominal_induct \<Gamma> S T avoiding: \<Delta> rule: subtype_of.strong_induct)
case (SA_all \<Gamma> T\<^sub>1 S\<^sub>1 X S\<^sub>2 T\<^sub>2)
have fresh_cond: "X\\" by fact
hence fresh_dom: "X\(ty_dom \)" by (simp add: fresh_dom)
have ih\<^sub>1: "\<And>\<Delta>. \<turnstile> \<Delta> ok \<Longrightarrow> \<Delta> extends \<Gamma> \<Longrightarrow> \<Delta> \<turnstile> T\<^sub>1 <: S\<^sub>1" by fact
have ih\<^sub>2: "\<And>\<Delta>. \<turnstile> \<Delta> ok \<Longrightarrow> \<Delta> extends ((TVarB X T\<^sub>1)#\<Gamma>) \<Longrightarrow> \<Delta> \<turnstile> S\<^sub>2 <: T\<^sub>2" by fact
have lh_drv_prem: "\ \ T\<^sub>1 <: S\<^sub>1" by fact
hence closed\<^sub>T1: "T\<^sub>1 closed_in \<Gamma>" by (simp add: subtype_implies_closed)
have ok: "\ \ ok" by fact
have ext: "\ extends \" by fact
have "T\<^sub>1 closed_in \" using ext closed\<^sub>T1 by (simp only: extends_closed)
hence "\ ((TVarB X T\<^sub>1)#\) ok" using fresh_dom ok by force
moreover
have "((TVarB X T\<^sub>1)#\) extends ((TVarB X T\<^sub>1)#\)" using ext by (force simp add: extends_def)
ultimately have "((TVarB X T\<^sub>1)#\) \ S\<^sub>2 <: T\<^sub>2" using ih\<^sub>2 by simp
moreover
have "\ \ T\<^sub>1 <: S\<^sub>1" using ok ext ih\<^sub>1 by simp
ultimately show "\ \ (\X<:S\<^sub>1. S\<^sub>2) <: (\X<:T\<^sub>1. T\<^sub>2)" using ok by (force intro: SA_all)
qed (blast intro: extends_closed extends_memb dest: extends_ty_dom)+
section \<open>Transitivity and Narrowing\<close>
text \<open>Some inversion lemmas that are needed in the transitivity and narrowing proof.\<close>
declare ty.inject [simp add]
inductive_cases S_TopE: "\ \ Top <: T"
inductive_cases S_ArrowE_left: "\ \ S\<^sub>1 \ S\<^sub>2 <: T"
declare ty.inject [simp del]
lemma S_ForallE_left:
shows "\\ \ (\X<:S\<^sub>1. S\<^sub>2) <: T; X\\; X\S\<^sub>1; X\T\
\<Longrightarrow> T = Top \<or> (\<exists>T\<^sub>1 T\<^sub>2. T = (\<forall>X<:T\<^sub>1. T\<^sub>2) \<and> \<Gamma> \<turnstile> T\<^sub>1 <: S\<^sub>1 \<and> ((TVarB X T\<^sub>1)#\<Gamma>) \<turnstile> S\<^sub>2 <: T\<^sub>2)"
apply(erule subtype_of.strong_cases[where X="X"])
apply(auto simp add: abs_fresh ty.inject alpha)
done
text \<open>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>\<open>\<Gamma> \<turnstile> S<:Q\<close> and \<^term>\<open>\<Gamma> \<turnstile> Q<:T\<close>, then \<^term>\<open>\<Gamma> \<turnstile> S<:T\<close>.
\item If \<open>\<Gamma>,X<:Q,\<Delta> \<turnstile> M<:N\<close> and \<^term>\<open>\<Gamma> \<turnstile> P<:Q\<close> then \<open>\<Gamma>,X<:P,\<Delta> \<turnstile> M<:N\<close>.
\end{enumerate}
\end{lemma}
The two parts are proved simultaneously, by induction on the size
of \<^term>\<open>Q\<close>. The argument for part (2) assumes that part (1) has
been established already for the \<^term>\<open>Q\<close> in question; part (1) uses
part (2) only for strictly smaller \<^term>\<open>Q\<close>.
\end{quote}
For the induction on the size of \<^term>\<open>Q\<close>, we use the induction-rule
\<open>measure_induct_rule\<close>:
\begin{center}
@{thm measure_induct_rule[of "size_ty",no_vars]}
\end{center}
That means in order to show a property \<^term>\<open>P a\<close> for all \<^term>\<open>a\<close>,
the induct-rule requires to prove that for all \<^term>\<open>x\<close> \<^term>\<open>P x\<close> holds using the
assumption that for all \<^term>\<open>y\<close> whose size is strictly smaller than
that of \<^term>\<open>x\<close> the property \<^term>\<open>P y\<close> holds.\<close>
lemma
shows subtype_transitivity: "\\S<:Q \ \\Q<:T \ \\S<:T"
and subtype_narrow: "(\@[(TVarB X Q)]@\)\M<:N \ \\P<:Q \ (\@[(TVarB X P)]@\)\M<:N"
proof (induct Q arbitrary: \<Gamma> S T \<Delta> X P M N taking: "size_ty" rule: measure_induct_rule)
case (less Q)
have IH_trans:
"\Q' \ S T. \size_ty Q' < size_ty Q; \\S<:Q'; \\Q'<:T\ \ \\S<:T" by fact
have IH_narrow:
"\Q' \ \ X M N P. \size_ty Q' < size_ty Q; (\@[(TVarB X Q')]@\)\M<:N; \\P<:Q'\
\<Longrightarrow> (\<Delta>@[(TVarB X P)]@\<Gamma>)\<turnstile>M<:N" by fact
{ fix \<Gamma> S T
have "\\ \ S <: Q; \ \ Q <: T\ \ \ \ S <: T"
proof (induct \<Gamma> S Q\<equiv>Q rule: subtype_of.induct)
case (SA_Top \<Gamma> S)
then have rh_drv: "\ \ Top <: T" by simp
then have T_inst: "T = Top" by (auto elim: S_TopE)
from \<open>\<turnstile> \<Gamma> ok\<close> and \<open>S closed_in \<Gamma>\<close>
have "\ \ S <: Top" by auto
then show "\ \ S <: T" using T_inst by simp
next
case (SA_trans_TVar Y U \<Gamma>)
then have IH_inner: "\ \ U <: T" by simp
have "(TVarB Y U) \ set \" by fact
with IH_inner show "\ \ Tvar Y <: T" by auto
next
case (SA_refl_TVar \<Gamma> X)
then show "\ \ Tvar X <: T" by simp
next
case (SA_arrow \<Gamma> Q\<^sub>1 S\<^sub>1 S\<^sub>2 Q\<^sub>2)
then have rh_drv: "\ \ Q\<^sub>1 \ Q\<^sub>2 <: T" by simp
from \<open>Q\<^sub>1 \<rightarrow> Q\<^sub>2 = Q\<close>
have Q\<^sub>12_less: "size_ty Q\<^sub>1 < size_ty Q" "size_ty Q\<^sub>2 < size_ty Q" by auto
have lh_drv_prm\<^sub>1: "\<Gamma> \<turnstile> Q\<^sub>1 <: S\<^sub>1" by fact
have lh_drv_prm\<^sub>2: "\<Gamma> \<turnstile> S\<^sub>2 <: Q\<^sub>2" by fact
from rh_drv have "T=Top \ (\T\<^sub>1 T\<^sub>2. T=T\<^sub>1\T\<^sub>2 \ \\T\<^sub>1<:Q\<^sub>1 \ \\Q\<^sub>2<:T\<^sub>2)"
by (auto elim: S_ArrowE_left)
moreover
have "S\<^sub>1 closed_in \" and "S\<^sub>2 closed_in \"
using lh_drv_prm\<^sub>1 lh_drv_prm\<^sub>2 by (simp_all add: subtype_implies_closed)
hence "(S\<^sub>1 \ S\<^sub>2) closed_in \" by (simp add: closed_in_def ty.supp)
moreover
have "\ \ ok" using rh_drv by (rule subtype_implies_ok)
moreover
{ assume "\T\<^sub>1 T\<^sub>2. T = T\<^sub>1\T\<^sub>2 \ \ \ T\<^sub>1 <: Q\<^sub>1 \ \ \ Q\<^sub>2 <: T\<^sub>2"
then obtain T\<^sub>1 T\<^sub>2
where T_inst: "T = T\<^sub>1 \ T\<^sub>2"
and rh_drv_prm\<^sub>1: "\<Gamma> \<turnstile> T\<^sub>1 <: Q\<^sub>1"
and rh_drv_prm\<^sub>2: "\<Gamma> \<turnstile> Q\<^sub>2 <: T\<^sub>2" by force
from IH_trans[of "Q\<^sub>1"]
have "\ \ T\<^sub>1 <: S\<^sub>1" using Q\<^sub>12_less rh_drv_prm\<^sub>1 lh_drv_prm\<^sub>1 by simp
moreover
from IH_trans[of "Q\<^sub>2"]
have "\ \ S\<^sub>2 <: T\<^sub>2" using Q\<^sub>12_less rh_drv_prm\<^sub>2 lh_drv_prm\<^sub>2 by simp
ultimately have "\ \ S\<^sub>1 \ S\<^sub>2 <: T\<^sub>1 \ T\<^sub>2" by auto
then have "\ \ S\<^sub>1 \ S\<^sub>2 <: T" using T_inst by simp
}
ultimately show "\ \ S\<^sub>1 \ S\<^sub>2 <: T" by blast
next
case (SA_all \<Gamma> Q\<^sub>1 S\<^sub>1 X S\<^sub>2 Q\<^sub>2)
then have rh_drv: "\ \ (\X<:Q\<^sub>1. Q\<^sub>2) <: T" by simp
have lh_drv_prm\<^sub>1: "\<Gamma> \<turnstile> Q\<^sub>1 <: S\<^sub>1" by fact
have lh_drv_prm\<^sub>2: "((TVarB X Q\<^sub>1)#\<Gamma>) \<turnstile> S\<^sub>2 <: Q\<^sub>2" by fact
then have "X\\" by (force dest: subtype_implies_ok simp add: valid_ty_dom_fresh)
then have fresh_cond: "X\\" "X\Q\<^sub>1" "X\T" using rh_drv lh_drv_prm\<^sub>1
by (simp_all add: subtype_implies_fresh)
from rh_drv
have "T = Top \
(\<exists>T\<^sub>1 T\<^sub>2. T = (\<forall>X<:T\<^sub>1. T\<^sub>2) \<and> \<Gamma> \<turnstile> T\<^sub>1 <: Q\<^sub>1 \<and> ((TVarB X T\<^sub>1)#\<Gamma>) \<turnstile> Q\<^sub>2 <: T\<^sub>2)"
using fresh_cond by (simp add: S_ForallE_left)
moreover
have "S\<^sub>1 closed_in \" and "S\<^sub>2 closed_in ((TVarB X Q\<^sub>1)#\)"
using lh_drv_prm\<^sub>1 lh_drv_prm\<^sub>2 by (simp_all add: subtype_implies_closed)
then have "(\X<:S\<^sub>1. S\<^sub>2) closed_in \" by (force simp add: closed_in_def ty.supp abs_supp)
moreover
have "\ \ ok" using rh_drv by (rule subtype_implies_ok)
moreover
{ assume "\T\<^sub>1 T\<^sub>2. T=(\X<:T\<^sub>1. T\<^sub>2) \ \\T\<^sub>1<:Q\<^sub>1 \ ((TVarB X T\<^sub>1)#\)\Q\<^sub>2<:T\<^sub>2"
then obtain T\<^sub>1 T\<^sub>2
where T_inst: "T = (\X<:T\<^sub>1. T\<^sub>2)"
and rh_drv_prm\<^sub>1: "\<Gamma> \<turnstile> T\<^sub>1 <: Q\<^sub>1"
and rh_drv_prm\<^sub>2:"((TVarB X T\<^sub>1)#\<Gamma>) \<turnstile> Q\<^sub>2 <: T\<^sub>2" by force
have "(\X<:Q\<^sub>1. Q\<^sub>2) = Q" by fact
then have Q\<^sub>12_less: "size_ty Q\<^sub>1 < size_ty Q" "size_ty Q\<^sub>2 < size_ty Q"
using fresh_cond by auto
from IH_trans[of "Q\<^sub>1"]
have "\ \ T\<^sub>1 <: S\<^sub>1" using lh_drv_prm\<^sub>1 rh_drv_prm\<^sub>1 Q\<^sub>12_less by blast
moreover
from IH_narrow[of "Q\<^sub>1" "[]"]
have "((TVarB X T\<^sub>1)#\) \ S\<^sub>2 <: Q\<^sub>2" using Q\<^sub>12_less lh_drv_prm\<^sub>2 rh_drv_prm\<^sub>1 by simp
with IH_trans[of "Q\<^sub>2"]
have "((TVarB X T\<^sub>1)#\) \ S\<^sub>2 <: T\<^sub>2" using Q\<^sub>12_less rh_drv_prm\<^sub>2 by simp
ultimately have "\ \ (\X<:S\<^sub>1. S\<^sub>2) <: (\X<:T\<^sub>1. T\<^sub>2)"
using fresh_cond by (simp add: subtype_of.SA_all)
hence "\ \ (\X<:S\<^sub>1. S\<^sub>2) <: T" using T_inst by simp
}
ultimately show "\ \ (\X<:S\<^sub>1. S\<^sub>2) <: T" by blast
qed
} note transitivity_lemma = this
{ \<comment> \<open>The transitivity proof is now by the auxiliary lemma.\<close>
case 1
from \<open>\<Gamma> \<turnstile> S <: Q\<close> and \<open>\<Gamma> \<turnstile> Q <: T\<close>
show "\ \ S <: T" by (rule transitivity_lemma)
next
case 2
from \<open>(\<Delta>@[(TVarB X Q)]@\<Gamma>) \<turnstile> M <: N\<close>
and \<open>\<Gamma> \<turnstile> P<:Q\<close>
show "(\@[(TVarB X P)]@\) \ M <: N"
proof (induct "\@[(TVarB X Q)]@\" M N arbitrary: \ X \ rule: subtype_of.induct)
case (SA_Top S \<Gamma> X \<Delta>)
from \<open>\<Gamma> \<turnstile> P <: Q\<close>
have "P closed_in \" by (simp add: subtype_implies_closed)
with \<open>\<turnstile> (\<Delta>@[(TVarB X Q)]@\<Gamma>) ok\<close> have "\<turnstile> (\<Delta>@[(TVarB X P)]@\<Gamma>) ok"
by (simp add: replace_type)
moreover
from \<open>S closed_in (\<Delta>@[(TVarB X Q)]@\<Gamma>)\<close> have "S closed_in (\<Delta>@[(TVarB X P)]@\<Gamma>)"
by (simp add: closed_in_def doms_append)
ultimately show "(\@[(TVarB X P)]@\) \ S <: Top" by (simp add: subtype_of.SA_Top)
next
case (SA_trans_TVar Y S N \<Gamma> X \<Delta>)
then have IH_inner: "(\@[(TVarB X P)]@\) \ S <: N"
and lh_drv_prm: "(TVarB Y S) \ set (\@[(TVarB X Q)]@\)"
and rh_drv: "\ \ P<:Q"
and ok\<^sub>Q: "\<turnstile> (\<Delta>@[(TVarB X Q)]@\<Gamma>) ok" by (simp_all add: subtype_implies_ok)
then have ok\<^sub>P: "\<turnstile> (\<Delta>@[(TVarB X P)]@\<Gamma>) ok" by (simp add: subtype_implies_ok)
show "(\@[(TVarB X P)]@\) \ Tvar Y <: N"
proof (cases "X=Y")
case False
have "X\Y" by fact
hence "(TVarB Y S)\set (\@[(TVarB X P)]@\)" using lh_drv_prm by (simp add:binding.inject)
with IH_inner show "(\@[(TVarB X P)]@\) \ Tvar Y <: N" by (simp add: subtype_of.SA_trans_TVar)
next
case True
have memb\<^sub>XQ: "(TVarB X Q)\<in>set (\<Delta>@[(TVarB X Q)]@\<Gamma>)" by simp
have memb\<^sub>XP: "(TVarB X P)\<in>set (\<Delta>@[(TVarB X P)]@\<Gamma>)" by simp
have eq: "X=Y" by fact
hence "S=Q" using ok\<^sub>Q lh_drv_prm memb\<^sub>XQ by (simp only: uniqueness_of_ctxt)
hence "(\@[(TVarB X P)]@\) \ Q <: N" using IH_inner by simp
moreover
have "(\@[(TVarB X P)]@\) extends \" by (simp add: extends_def)
hence "(\@[(TVarB X P)]@\) \ P <: Q" using rh_drv ok\<^sub>P by (simp only: weakening)
ultimately have "(\@[(TVarB X P)]@\) \ P <: N" by (simp add: transitivity_lemma)
then show "(\@[(TVarB X P)]@\) \ Tvar Y <: N" using memb\<^sub>XP eq by auto
qed
next
case (SA_refl_TVar Y \<Gamma> X \<Delta>)
from \<open>\<Gamma> \<turnstile> P <: Q\<close>
have "P closed_in \" by (simp add: subtype_implies_closed)
with \<open>\<turnstile> (\<Delta>@[(TVarB X Q)]@\<Gamma>) ok\<close> have "\<turnstile> (\<Delta>@[(TVarB X P)]@\<Gamma>) ok"
by (simp add: replace_type)
moreover
from \<open>Y \<in> ty_dom (\<Delta>@[(TVarB X Q)]@\<Gamma>)\<close> have "Y \<in> ty_dom (\<Delta>@[(TVarB X P)]@\<Gamma>)"
by (simp add: doms_append)
ultimately show "(\@[(TVarB X P)]@\) \ Tvar Y <: Tvar Y" by (simp add: subtype_of.SA_refl_TVar)
next
case (SA_arrow S\<^sub>1 Q\<^sub>1 Q\<^sub>2 S\<^sub>2 \<Gamma> X \<Delta>)
then show "(\@[(TVarB X P)]@\) \ Q\<^sub>1 \ Q\<^sub>2 <: S\<^sub>1 \ S\<^sub>2" by blast
next
case (SA_all T\<^sub>1 S\<^sub>1 Y S\<^sub>2 T\<^sub>2 \<Gamma> X \<Delta>)
have IH_inner\<^sub>1: "(\<Delta>@[(TVarB X P)]@\<Gamma>) \<turnstile> T\<^sub>1 <: S\<^sub>1"
and IH_inner\<^sub>2: "(((TVarB Y T\<^sub>1)#\<Delta>)@[(TVarB X P)]@\<Gamma>) \<turnstile> S\<^sub>2 <: T\<^sub>2"
by (fastforce intro: SA_all)+
then show "(\@[(TVarB X P)]@\) \ (\Y<:S\<^sub>1. S\<^sub>2) <: (\Y<:T\<^sub>1. T\<^sub>2)" by auto
qed
}
qed
section \<open>Typing\<close>
inductive
typing :: "env \ trm \ ty \ bool" ("_ \ _ : _" [60,60,60] 60)
where
T_Var[intro]: "\ VarB x T \ set \; \ \ ok \ \ \ \ Var x : T"
| T_App[intro]: "\ \ \ t\<^sub>1 : T\<^sub>1 \ T\<^sub>2; \ \ t\<^sub>2 : T\<^sub>1 \ \ \ \ t\<^sub>1 \ t\<^sub>2 : T\<^sub>2"
| T_Abs[intro]: "\ VarB x T\<^sub>1 # \ \ t\<^sub>2 : T\<^sub>2 \ \ \ \ (\x:T\<^sub>1. t\<^sub>2) : T\<^sub>1 \ T\<^sub>2"
| T_Sub[intro]: "\ \ \ t : S; \ \ S <: T \ \ \ \ t : T"
| T_TAbs[intro]:"\ TVarB X T\<^sub>1 # \ \ t\<^sub>2 : T\<^sub>2 \ \ \ \ (\X<:T\<^sub>1. t\<^sub>2) : (\X<:T\<^sub>1. T\<^sub>2)"
| T_TApp[intro]:"\X\(\,t\<^sub>1,T\<^sub>2); \ \ t\<^sub>1 : (\X<:T\<^sub>11. T\<^sub>12); \ \ T\<^sub>2 <: T\<^sub>11\ \ \ \ t\<^sub>1 \\<^sub>\ T\<^sub>2 : (T\<^sub>12[X \ T\<^sub>2]\<^sub>\)"
equivariance typing
lemma better_T_TApp:
assumes H1: "\ \ t\<^sub>1 : (\X<:T11. T12)"
and H2: "\ \ T2 <: T11"
shows "\ \ t\<^sub>1 \\<^sub>\ T2 : (T12[X \ T2]\<^sub>\)"
proof -
obtain Y::tyvrs where Y: "Y \ (X, T12, \, t\<^sub>1, T2)"
by (rule exists_fresh) (rule fin_supp)
then have "Y \ (\, t\<^sub>1, T2)" by simp
moreover from Y have "(\X<:T11. T12) = (\Y<:T11. [(Y, X)] \ T12)"
by (auto simp add: ty.inject alpha' fresh_prod fresh_atm)
with H1 have "\ \ t\<^sub>1 : (\Y<:T11. [(Y, X)] \ T12)" by simp
ultimately have "\ \ t\<^sub>1 \\<^sub>\ T2 : (([(Y, X)] \ T12)[Y \ T2]\<^sub>\)" using H2
by (rule T_TApp)
with Y show ?thesis by (simp add: type_subst_rename)
qed
lemma typing_ok:
assumes "\ \ t : T"
shows "\ \ ok"
using assms by (induct) (auto)
nominal_inductive typing
by (auto dest!: typing_ok intro: closed_in_fresh fresh_dom type_subst_fresh
simp: abs_fresh fresh_type_subst_fresh ty_vrs_fresh valid_ty_dom_fresh fresh_trm_dom)
lemma ok_imp_VarB_closed_in:
assumes ok: "\ \ ok"
shows "VarB x T \ set \ \ T closed_in \" using ok
by induct (auto simp add: binding.inject closed_in_def)
lemma tyvrs_of_subst: "tyvrs_of (B[X \ T]\<^sub>b) = tyvrs_of B"
by (nominal_induct B rule: binding.strong_induct) simp_all
lemma ty_dom_subst: "ty_dom (\[X \ T]\<^sub>e) = ty_dom \"
by (induct \<Gamma>) (simp_all add: tyvrs_of_subst)
lemma vrs_of_subst: "vrs_of (B[X \ T]\<^sub>b) = vrs_of B"
by (nominal_induct B rule: binding.strong_induct) simp_all
lemma trm_dom_subst: "trm_dom (\[X \ T]\<^sub>e) = trm_dom \"
by (induct \<Gamma>) (simp_all add: vrs_of_subst)
lemma subst_closed_in:
"T closed_in (\ @ TVarB X S # \) \ U closed_in \ \ T[X \ U]\<^sub>\ closed_in (\[X \ U]\<^sub>e @ \)"
apply (nominal_induct T avoiding: X U \<Gamma> rule: ty.strong_induct)
apply (simp add: closed_in_def ty.supp supp_atm doms_append ty_dom_subst)
apply blast
apply (simp add: closed_in_def ty.supp)
apply (simp add: closed_in_def ty.supp)
apply (simp add: closed_in_def ty.supp abs_supp)
apply (drule_tac x = X in meta_spec)
apply (drule_tac x = U in meta_spec)
apply (drule_tac x = "(TVarB tyvrs ty2) # \" in meta_spec)
apply (simp add: doms_append ty_dom_subst)
apply blast
done
lemmas subst_closed_in' = subst_closed_in [where \="[]", simplified]
lemma typing_closed_in:
assumes "\ \ t : T"
shows "T closed_in \"
using assms
proof induct
case (T_Var x T \<Gamma>)
from \<open>\<turnstile> \<Gamma> ok\<close> and \<open>VarB x T \<in> set \<Gamma>\<close>
show ?case by (rule ok_imp_VarB_closed_in)
next
case (T_App \<Gamma> t\<^sub>1 T\<^sub>1 T\<^sub>2 t\<^sub>2)
then show ?case by (auto simp add: ty.supp closed_in_def)
next
case (T_Abs x T\<^sub>1 \<Gamma> t\<^sub>2 T\<^sub>2)
from \<open>VarB x T\<^sub>1 # \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2\<close>
have "T\<^sub>1 closed_in \" by (auto dest: typing_ok)
with T_Abs show ?case by (auto simp add: ty.supp closed_in_def)
next
case (T_Sub \<Gamma> t S T)
from \<open>\<Gamma> \<turnstile> S <: T\<close> show ?case by (simp add: subtype_implies_closed)
next
case (T_TAbs X T\<^sub>1 \<Gamma> t\<^sub>2 T\<^sub>2)
from \<open>TVarB X T\<^sub>1 # \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2\<close>
have "T\<^sub>1 closed_in \" by (auto dest: typing_ok)
with T_TAbs show ?case by (auto simp add: ty.supp closed_in_def abs_supp)
next
case (T_TApp X \<Gamma> t\<^sub>1 T2 T11 T12)
then have "T12 closed_in (TVarB X T11 # \)"
by (auto simp add: closed_in_def ty.supp abs_supp)
moreover from T_TApp have "T2 closed_in \"
by (simp add: subtype_implies_closed)
ultimately show ?case by (rule subst_closed_in')
qed
subsection \<open>Evaluation\<close>
inductive
val :: "trm \ bool"
where
Abs[intro]: "val (\x:T. t)"
| TAbs[intro]: "val (\X<:T. t)"
equivariance val
inductive_cases val_inv_auto[elim]:
"val (Var x)"
"val (t1 \ t2)"
"val (t1 \\<^sub>\ t2)"
inductive
eval :: "trm \ trm \ bool" ("_ \ _" [60,60] 60)
where
E_Abs : "\ x \ v\<^sub>2; val v\<^sub>2 \ \ (\x:T\<^sub>11. t\<^sub>12) \ v\<^sub>2 \ t\<^sub>12[x \ v\<^sub>2]"
| E_App1 [intro]: "t \ t' \ t \ u \ t' \ u"
| E_App2 [intro]: "\ val v; t \ t' \ \ v \ t \ v \ t'"
| E_TAbs : "X \ (T\<^sub>11, T\<^sub>2) \ (\X<:T\<^sub>11. t\<^sub>12) \\<^sub>\ T\<^sub>2 \ t\<^sub>12[X \\<^sub>\ T\<^sub>2]"
| E_TApp [intro]: "t \ t' \ t \\<^sub>\ T \ t' \\<^sub>\ T"
lemma better_E_Abs[intro]:
assumes H: "val v2"
shows "(\x:T11. t12) \ v2 \ t12[x \ v2]"
proof -
obtain y::vrs where y: "y \ (x, t12, v2)" by (rule exists_fresh) (rule fin_supp)
then have "y \ v2" by simp
then have "(\y:T11. [(y, x)] \ t12) \ v2 \ ([(y, x)] \ t12)[y \ v2]" using H
by (rule E_Abs)
moreover from y have "(\x:T11. t12) \ v2 = (\y:T11. [(y, x)] \ t12) \ v2"
by (auto simp add: trm.inject alpha' fresh_prod fresh_atm)
ultimately have "(\x:T11. t12) \ v2 \ ([(y, x)] \ t12)[y \ v2]"
by simp
with y show ?thesis by (simp add: subst_trm_rename)
qed
lemma better_E_TAbs[intro]: "(\X<:T11. t12) \\<^sub>\ T2 \ t12[X \\<^sub>\ T2]"
proof -
obtain Y::tyvrs where Y: "Y \ (X, t12, T11, T2)" by (rule exists_fresh) (rule fin_supp)
then have "Y \ (T11, T2)" by simp
then have "(\Y<:T11. [(Y, X)] \ t12) \\<^sub>\ T2 \ ([(Y, X)] \ t12)[Y \\<^sub>\ T2]"
by (rule E_TAbs)
moreover from Y have "(\X<:T11. t12) \\<^sub>\ T2 = (\Y<:T11. [(Y, X)] \ t12) \\<^sub>\ T2"
by (auto simp add: trm.inject alpha' fresh_prod fresh_atm)
ultimately have "(\X<:T11. t12) \\<^sub>\ T2 \ ([(Y, X)] \ t12)[Y \\<^sub>\ T2]"
by simp
with Y show ?thesis by (simp add: subst_trm_ty_rename)
qed
equivariance eval
nominal_inductive eval
by (simp_all add: abs_fresh ty_vrs_fresh subst_trm_fresh_tyvar
subst_trm_fresh_var subst_trm_ty_fresh')
inductive_cases eval_inv_auto[elim]:
"Var x \ t'"
"(\x:T. t) \ t'"
"(\X<:T. t) \ t'"
lemma ty_dom_cons:
shows "ty_dom (\@[VarB X Q]@\) = ty_dom (\@\)"
by (induct \<Gamma>) (auto)
lemma closed_in_cons:
assumes "S closed_in (\ @ VarB X Q # \)"
shows "S closed_in (\@\)"
using assms ty_dom_cons closed_in_def by auto
lemma closed_in_weaken: "T closed_in (\ @ \) \ T closed_in (\ @ B # \)"
by (auto simp add: closed_in_def doms_append)
lemma closed_in_weaken': "T closed_in \ \ T closed_in (\ @ \)"
by (auto simp add: closed_in_def doms_append)
lemma valid_subst:
assumes ok: "\ (\ @ TVarB X Q # \) ok"
and closed: "P closed_in \"
shows "\ (\[X \ P]\<^sub>e @ \) ok" using ok closed
apply (induct \<Delta>)
apply simp_all
apply (erule validE)
apply assumption
apply (erule validE)
apply simp
apply (rule valid_consT)
apply assumption
apply (simp add: doms_append ty_dom_subst)
apply (simp add: fresh_fin_insert [OF pt_tyvrs_inst at_tyvrs_inst fs_tyvrs_inst] finite_doms)
apply (rule_tac S=Q in subst_closed_in')
apply (simp add: closed_in_def doms_append ty_dom_subst)
apply (simp add: closed_in_def doms_append)
apply blast
apply simp
apply (rule valid_cons)
apply assumption
apply (simp add: doms_append trm_dom_subst)
apply (rule_tac S=Q in subst_closed_in')
apply (simp add: closed_in_def doms_append ty_dom_subst)
apply (simp add: closed_in_def doms_append)
apply blast
done
lemma ty_dom_vrs:
shows "ty_dom (G @ [VarB x Q] @ D) = ty_dom (G @ D)"
by (induct G) (auto)
lemma valid_cons':
assumes "\ (\ @ VarB x Q # \) ok"
shows "\ (\ @ \) ok"
using assms
proof (induct "\ @ VarB x Q # \" arbitrary: \ \)
case valid_nil
have "[] = \ @ VarB x Q # \" by fact
then have "False" by auto
then show ?case by auto
next
case (valid_consT G X T)
then show ?case
proof (cases \<Gamma>)
case Nil
with valid_consT show ?thesis by simp
next
case (Cons b bs)
with valid_consT
have "\ (bs @ \) ok" by simp
moreover from Cons and valid_consT have "X \ ty_dom (bs @ \)"
by (simp add: doms_append)
moreover from Cons and valid_consT have "T closed_in (bs @ \)"
by (simp add: closed_in_def doms_append)
ultimately have "\ (TVarB X T # bs @ \) ok"
by (rule valid_rel.valid_consT)
with Cons and valid_consT show ?thesis by simp
qed
next
case (valid_cons G x T)
then show ?case
proof (cases \<Gamma>)
case Nil
with valid_cons show ?thesis by simp
next
case (Cons b bs)
with valid_cons
have "\ (bs @ \) ok" by simp
moreover from Cons and valid_cons have "x \ trm_dom (bs @ \)"
by (simp add: doms_append finite_doms
fresh_fin_insert [OF pt_vrs_inst at_vrs_inst fs_vrs_inst])
moreover from Cons and valid_cons have "T closed_in (bs @ \)"
by (simp add: closed_in_def doms_append)
ultimately have "\ (VarB x T # bs @ \) ok"
by (rule valid_rel.valid_cons)
with Cons and valid_cons show ?thesis by simp
qed
qed
text \<open>A.5(6)\<close>
lemma type_weaken:
assumes "(\@\) \ t : T"
and "\ (\ @ B # \) ok"
shows "(\ @ B # \) \ t : T"
using assms
proof(nominal_induct "\ @ \" t T avoiding: \ \ B rule: typing.strong_induct)
case (T_Var x T)
then show ?case by auto
next
case (T_App X t\<^sub>1 T\<^sub>2 T\<^sub>11 T\<^sub>12)
then show ?case by force
next
case (T_Abs y T\<^sub>1 t\<^sub>2 T\<^sub>2 \<Delta> \<Gamma>)
then have "VarB y T\<^sub>1 # \ @ \ \ t\<^sub>2 : T\<^sub>2" by simp
then have closed: "T\<^sub>1 closed_in (\ @ \)"
by (auto dest: typing_ok)
have "\ (VarB y T\<^sub>1 # \ @ B # \) ok"
apply (rule valid_cons)
apply (rule T_Abs)
apply (simp add: doms_append
fresh_fin_insert [OF pt_vrs_inst at_vrs_inst fs_vrs_inst]
fresh_fin_union [OF pt_vrs_inst at_vrs_inst fs_vrs_inst]
finite_doms finite_vrs fresh_vrs_of T_Abs fresh_trm_dom)
apply (rule closed_in_weaken)
apply (rule closed)
done
then have "\ ((VarB y T\<^sub>1 # \) @ B # \) ok" by simp
with _ have "(VarB y T\<^sub>1 # \) @ B # \ \ t\<^sub>2 : T\<^sub>2"
by (rule T_Abs) simp
then have "VarB y T\<^sub>1 # \ @ B # \ \ t\<^sub>2 : T\<^sub>2" by simp
then show ?case by (rule typing.T_Abs)
next
case (T_Sub t S T \<Delta> \<Gamma>)
from refl and \<open>\<turnstile> (\<Delta> @ B # \<Gamma>) ok\<close>
have "\ @ B # \ \ t : S" by (rule T_Sub)
moreover from \<open>(\<Delta> @ \<Gamma>)\<turnstile>S<:T\<close> and \<open>\<turnstile> (\<Delta> @ B # \<Gamma>) ok\<close>
have "(\ @ B # \)\S<:T"
by (rule weakening) (simp add: extends_def T_Sub)
ultimately show ?case by (rule typing.T_Sub)
next
case (T_TAbs X T\<^sub>1 t\<^sub>2 T\<^sub>2 \<Delta> \<Gamma>)
from \<open>TVarB X T\<^sub>1 # \<Delta> @ \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2\<close>
have closed: "T\<^sub>1 closed_in (\ @ \)"
by (auto dest: typing_ok)
have "\ (TVarB X T\<^sub>1 # \ @ B # \) ok"
apply (rule valid_consT)
apply (rule T_TAbs)
apply (simp add: doms_append
fresh_fin_insert [OF pt_tyvrs_inst at_tyvrs_inst fs_tyvrs_inst]
fresh_fin_union [OF pt_tyvrs_inst at_tyvrs_inst fs_tyvrs_inst]
finite_doms finite_vrs tyvrs_fresh T_TAbs fresh_dom)
apply (rule closed_in_weaken)
apply (rule closed)
done
then have "\ ((TVarB X T\<^sub>1 # \) @ B # \) ok" by simp
with _ have "(TVarB X T\<^sub>1 # \) @ B # \ \ t\<^sub>2 : T\<^sub>2"
by (rule T_TAbs) simp
then have "TVarB X T\<^sub>1 # \ @ B # \ \ t\<^sub>2 : T\<^sub>2" by simp
then show ?case by (rule typing.T_TAbs)
next
case (T_TApp X t\<^sub>1 T2 T11 T12 \<Delta> \<Gamma>)
have "\ @ B # \ \ t\<^sub>1 : (\X<:T11. T12)"
by (rule T_TApp refl)+
moreover from \<open>(\<Delta> @ \<Gamma>)\<turnstile>T2<:T11\<close> and \<open>\<turnstile> (\<Delta> @ B # \<Gamma>) ok\<close>
have "(\ @ B # \)\T2<:T11"
by (rule weakening) (simp add: extends_def T_TApp)
ultimately show ?case by (rule better_T_TApp)
qed
lemma type_weaken':
"\ \ t : T \ \ (\@\) ok \ (\@\) \ t : T"
apply (induct \<Delta>)
apply simp_all
apply (erule validE)
--> --------------------
--> maximum size reached
--> --------------------
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