(* Title: HOL/MicroJava/BV/Convert.thy
ID: $Id$
Author: Cornelia Pusch and Gerwin Klein
Copyright 1999 Technische Universitaet Muenchen
*)
header "Lifted Type Relations"
theory Convert = JVMExec:
text "The supertype relation lifted to type err, type lists and state types."
datatype 'a err = Err | Ok 'a
types
locvars_type = "ty err list"
opstack_type = "ty list"
state_type = "opstack_type \<times> locvars_type"
consts
strict :: "('a \<Rightarrow> 'b err) \<Rightarrow> ('a err \<Rightarrow> 'b err)"
primrec
"strict f Err = Err"
"strict f (Ok x) = f x"
consts
opt_map :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a option \<Rightarrow> 'b option)"
primrec
"opt_map f None = None"
"opt_map f (Some x) = Some (f x)"
consts
val :: "'a err \<Rightarrow> 'a"
primrec
"val (Ok s) = s"
constdefs
(* lifts a relation to err with Err as top element *)
lift_top :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a err \<Rightarrow> 'b err \<Rightarrow> bool)"
"lift_top P a' a \<equiv> case a of
Err \<Rightarrow> True
| Ok t \<Rightarrow> (case a' of Err \<Rightarrow> False | Ok t' \<Rightarrow> P t' t)"
(* lifts a relation to option with None as bottom element *)
lift_bottom :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a option \<Rightarrow> 'b option \<Rightarrow> bool)"
"lift_bottom P a' a \<equiv> case a' of
None \<Rightarrow> True
| Some t' \<Rightarrow> (case a of None \<Rightarrow> False | Some t \<Rightarrow> P t' t)"
sup_ty_opt :: "['code prog,ty err,ty err] \<Rightarrow> bool" ("_ \<turnstile>_ <=o _")
"sup_ty_opt G \<equiv> lift_top (\<lambda>t t'. G \<turnstile> t \<preceq> t')"
sup_loc :: "['code prog,locvars_type,locvars_type] \<Rightarrow> bool"
("_ \<turnstile> _ <=l _" [71,71] 70)
"G \<turnstile> LT <=l LT' \<equiv> list_all2 (\<lambda>t t'. (G \<turnstile> t <=o t')) LT LT'"
sup_state :: "['code prog,state_type,state_type] \<Rightarrow> bool"
("_ \<turnstile> _ <=s _" [71,71] 70)
"G \<turnstile> s <=s s' \<equiv> (G \<turnstile> map Ok (fst s) <=l map Ok (fst s')) \<and> G \<turnstile> snd s <=l snd s'"
sup_state_opt :: "['code prog,state_type option,state_type option] \<Rightarrow> bool"
("_ \<turnstile> _ <=' _" [71,71] 70)
"sup_state_opt G \<equiv> lift_bottom (\<lambda>t t'. G \<turnstile> t <=s t')"
lemma not_Err_eq [iff]:
"(x \<noteq> Err) = (\<exists>a. x = Ok a)"
by (cases x) auto
lemma not_Some_eq [iff]:
"(\<forall>y. x \<noteq> Ok y) = (x = Err)"
by (cases x) auto
lemma lift_top_refl [simp]:
"[| !!x. P x x |] ==> lift_top P x x"
by (simp add: lift_top_def split: err.splits)
lemma lift_top_trans [trans]:
"[| !!x y z. [| P x y; P y z |] ==> P x z; lift_top P x y; lift_top P y z |]
==> lift_top P x z"
proof -
assume [trans]: "!!x y z. [| P x y; P y z |] ==> P x z"
assume a: "lift_top P x y"
assume b: "lift_top P y z"
{ assume "z = Err"
hence ?thesis by (simp add: lift_top_def)
} note z_none = this
{ assume "x = Err"
with a b
have ?thesis
by (simp add: lift_top_def split: err.splits)
} note x_none = this
{ fix r t
assume x: "x = Ok r" and z: "z = Ok t"
with a b
obtain s where y: "y = Ok s"
by (simp add: lift_top_def split: err.splits)
from a x y
have "P r s" by (simp add: lift_top_def)
also
from b y z
have "P s t" by (simp add: lift_top_def)
finally
have "P r t" .
with x z
have ?thesis by (simp add: lift_top_def)
}
with x_none z_none
show ?thesis by blast
qed
lemma lift_top_Err_any [simp]:
"lift_top P Err any = (any = Err)"
by (simp add: lift_top_def split: err.splits)
lemma lift_top_Ok_Ok [simp]:
"lift_top P (Ok a) (Ok b) = P a b"
by (simp add: lift_top_def split: err.splits)
lemma lift_top_any_Ok [simp]:
"lift_top P any (Ok b) = (\<exists>a. any = Ok a \<and> P a b)"
by (simp add: lift_top_def split: err.splits)
lemma lift_top_Ok_any:
"lift_top P (Ok a) any = (any = Err \<or> (\<exists>b. any = Ok b \<and> P a b))"
by (simp add: lift_top_def split: err.splits)
lemma lift_bottom_refl [simp]:
"[| !!x. P x x |] ==> lift_bottom P x x"
by (simp add: lift_bottom_def split: option.splits)
lemma lift_bottom_trans [trans]:
"[| !!x y z. [| P x y; P y z |] ==> P x z; lift_bottom P x y; lift_bottom P y z |]
==> lift_bottom P x z"
proof -
assume [trans]: "!!x y z. [| P x y; P y z |] ==> P x z"
assume a: "lift_bottom P x y"
assume b: "lift_bottom P y z"
{ assume "x = None"
hence ?thesis by (simp add: lift_bottom_def)
} note z_none = this
{ assume "z = None"
with a b
have ?thesis
by (simp add: lift_bottom_def split: option.splits)
} note x_none = this
{ fix r t
assume x: "x = Some r" and z: "z = Some t"
with a b
obtain s where y: "y = Some s"
by (simp add: lift_bottom_def split: option.splits)
from a x y
have "P r s" by (simp add: lift_bottom_def)
also
from b y z
have "P s t" by (simp add: lift_bottom_def)
finally
have "P r t" .
with x z
have ?thesis by (simp add: lift_bottom_def)
}
with x_none z_none
show ?thesis by blast
qed
lemma lift_bottom_any_None [simp]:
"lift_bottom P any None = (any = None)"
by (simp add: lift_bottom_def split: option.splits)
lemma lift_bottom_Some_Some [simp]:
"lift_bottom P (Some a) (Some b) = P a b"
by (simp add: lift_bottom_def split: option.splits)
lemma lift_bottom_any_Some [simp]:
"lift_bottom P (Some a) any = (\<exists>b. any = Some b \<and> P a b)"
by (simp add: lift_bottom_def split: option.splits)
lemma lift_bottom_Some_any:
"lift_bottom P any (Some b) = (any = None \<or> (\<exists>a. any = Some a \<and> P a b))"
by (simp add: lift_bottom_def split: option.splits)
theorem sup_ty_opt_refl [simp]:
"G \<turnstile> t <=o t"
by (simp add: sup_ty_opt_def)
theorem sup_loc_refl [simp]:
"G \<turnstile> t <=l t"
by (induct t, auto simp add: sup_loc_def)
theorem sup_state_refl [simp]:
"G \<turnstile> s <=s s"
by (simp add: sup_state_def)
theorem sup_state_opt_refl [simp]:
"G \<turnstile> s <=' s"
by (simp add: sup_state_opt_def)
theorem anyConvErr [simp]:
"(G \<turnstile> Err <=o any) = (any = Err)"
by (simp add: sup_ty_opt_def)
theorem OkanyConvOk [simp]:
"(G \<turnstile> (Ok ty') <=o (Ok ty)) = (G \<turnstile> ty' \<preceq> ty)"
by (simp add: sup_ty_opt_def)
theorem sup_ty_opt_Ok:
"G \<turnstile> a <=o (Ok b) \<Longrightarrow> \<exists> x. a = Ok x"
by (clarsimp simp add: sup_ty_opt_def)
lemma widen_PrimT_conv1 [simp]:
"[| G \<turnstile> S \<preceq> T; S = PrimT x|] ==> T = PrimT x"
by (auto elim: widen.elims)
theorem sup_PTS_eq:
"(G \<turnstile> Ok (PrimT p) <=o X) = (X=Err \<or> X = Ok (PrimT p))"
by (auto simp add: sup_ty_opt_def lift_top_Ok_any)
theorem sup_loc_Nil [iff]:
"(G \<turnstile> [] <=l XT) = (XT=[])"
by (simp add: sup_loc_def)
theorem sup_loc_Cons [iff]:
"(G \<turnstile> (Y#YT) <=l XT) = (\<exists>X XT'. XT=X#XT' \<and> (G \<turnstile> Y <=o X) \<and> (G \<turnstile> YT <=l XT'))"
by (simp add: sup_loc_def list_all2_Cons1)
theorem sup_loc_Cons2:
"(G \<turnstile> YT <=l (X#XT)) = (\<exists>Y YT'. YT=Y#YT' \<and> (G \<turnstile> Y <=o X) \<and> (G \<turnstile> YT' <=l XT))"
by (simp add: sup_loc_def list_all2_Cons2)
theorem sup_loc_length:
"G \<turnstile> a <=l b \<Longrightarrow> length a = length b"
proof -
assume G: "G \<turnstile> a <=l b"
have "\<forall> b. (G \<turnstile> a <=l b) \<longrightarrow> length a = length b"
by (induct a, auto)
with G
show ?thesis by blast
qed
theorem sup_loc_nth:
"[| G \<turnstile> a <=l b; n < length a |] ==> G \<turnstile> (a!n) <=o (b!n)"
proof -
assume a: "G \<turnstile> a <=l b" "n < length a"
have "\<forall> n b. (G \<turnstile> a <=l b) \<longrightarrow> n < length a \<longrightarrow> (G \<turnstile> (a!n) <=o (b!n))"
(is "?P a")
proof (induct a)
show "?P []" by simp
fix x xs assume IH: "?P xs"
show "?P (x#xs)"
proof (intro strip)
fix n b
assume "G \<turnstile> (x # xs) <=l b" "n < length (x # xs)"
with IH
show "G \<turnstile> ((x # xs) ! n) <=o (b ! n)"
by - (cases n, auto)
qed
qed
with a
show ?thesis by blast
qed
theorem all_nth_sup_loc:
"\<forall>b. length a = length b \<longrightarrow> (\<forall> n. n < length a \<longrightarrow> (G \<turnstile> (a!n) <=o (b!n))) \<longrightarrow>
(G \<turnstile> a <=l b)" (is "?P a")
proof (induct a)
show "?P []" by simp
fix l ls assume IH: "?P ls"
show "?P (l#ls)"
proof (intro strip)
fix b
assume f: "\<forall>n. n < length (l # ls) \<longrightarrow> (G \<turnstile> ((l # ls) ! n) <=o (b ! n))"
assume l: "length (l#ls) = length b"
then obtain b' bs where b: "b = b'#bs"
by - (cases b, simp, simp add: neq_Nil_conv, rule that)
with f
have "\<forall>n. n < length ls \<longrightarrow> (G \<turnstile> (ls!n) <=o (bs!n))"
by auto
with f b l IH
show "G \<turnstile> (l # ls) <=l b"
by auto
qed
qed
theorem sup_loc_append:
"length a = length b ==>
(G \<turnstile> (a@x) <=l (b@y)) = ((G \<turnstile> a <=l b) \<and> (G \<turnstile> x <=l y))"
proof -
assume l: "length a = length b"
have "\<forall>b. length a = length b \<longrightarrow> (G \<turnstile> (a@x) <=l (b@y)) = ((G \<turnstile> a <=l b) \<and>
(G \<turnstile> x <=l y))" (is "?P a")
proof (induct a)
show "?P []" by simp
fix l ls assume IH: "?P ls"
show "?P (l#ls)"
proof (intro strip)
fix b
assume "length (l#ls) = length (b::ty err list)"
with IH
show "(G \<turnstile> ((l#ls)@x) <=l (b@y)) = ((G \<turnstile> (l#ls) <=l b) \<and> (G \<turnstile> x <=l y))"
by - (cases b, auto)
qed
qed
with l
show ?thesis by blast
qed
theorem sup_loc_rev [simp]:
"(G \<turnstile> (rev a) <=l rev b) = (G \<turnstile> a <=l b)"
proof -
have "\<forall>b. (G \<turnstile> (rev a) <=l rev b) = (G \<turnstile> a <=l b)" (is "\<forall>b. ?Q a b" is "?P a")
proof (induct a)
show "?P []" by simp
fix l ls assume IH: "?P ls"
{
fix b
have "?Q (l#ls) b"
proof (cases (open) b)
case Nil
thus ?thesis by (auto dest: sup_loc_length)
next
case Cons
show ?thesis
proof
assume "G \<turnstile> (l # ls) <=l b"
thus "G \<turnstile> rev (l # ls) <=l rev b"
by (clarsimp simp add: Cons IH sup_loc_length sup_loc_append)
next
assume "G \<turnstile> rev (l # ls) <=l rev b"
hence G: "G \<turnstile> (rev ls @ [l]) <=l (rev list @ [a])"
by (simp add: Cons)
hence "length (rev ls) = length (rev list)"
by (auto dest: sup_loc_length)
from this G
obtain "G \<turnstile> rev ls <=l rev list" "G \<turnstile> l <=o a"
by (simp add: sup_loc_append)
thus "G \<turnstile> (l # ls) <=l b"
by (simp add: Cons IH)
qed
qed
}
thus "?P (l#ls)" by blast
qed
thus ?thesis by blast
qed
theorem sup_loc_update [rulify]:
"\<forall> n y. (G \<turnstile> a <=o b) \<longrightarrow> n < length y \<longrightarrow> (G \<turnstile> x <=l y) \<longrightarrow>
(G \<turnstile> x[n := a] <=l y[n := b])" (is "?P x")
proof (induct x)
show "?P []" by simp
fix l ls assume IH: "?P ls"
show "?P (l#ls)"
proof (intro strip)
fix n y
assume "G \<turnstile>a <=o b" "G \<turnstile> (l # ls) <=l y" "n < length y"
with IH
show "G \<turnstile> (l # ls)[n := a] <=l y[n := b]"
by - (cases n, auto simp add: sup_loc_Cons2 list_all2_Cons1)
qed
qed
theorem sup_state_length [simp]:
"G \<turnstile> s2 <=s s1 ==>
length (fst s2) = length (fst s1) \<and> length (snd s2) = length (snd s1)"
by (auto dest: sup_loc_length simp add: sup_state_def);
theorem sup_state_append_snd:
"length a = length b ==>
(G \<turnstile> (i,a@x) <=s (j,b@y)) = ((G \<turnstile> (i,a) <=s (j,b)) \<and> (G \<turnstile> (i,x) <=s (j,y)))"
by (auto simp add: sup_state_def sup_loc_append)
theorem sup_state_append_fst:
"length a = length b ==>
(G \<turnstile> (a@x,i) <=s (b@y,j)) = ((G \<turnstile> (a,i) <=s (b,j)) \<and> (G \<turnstile> (x,i) <=s (y,j)))"
by (auto simp add: sup_state_def sup_loc_append)
theorem sup_state_Cons1:
"(G \<turnstile> (x#xt, a) <=s (yt, b)) =
(\<exists>y yt'. yt=y#yt' \<and> (G \<turnstile> x \<preceq> y) \<and> (G \<turnstile> (xt,a) <=s (yt',b)))"
by (auto simp add: sup_state_def map_eq_Cons)
theorem sup_state_Cons2:
"(G \<turnstile> (xt, a) <=s (y#yt, b)) =
(\<exists>x xt'. xt=x#xt' \<and> (G \<turnstile> x \<preceq> y) \<and> (G \<turnstile> (xt',a) <=s (yt,b)))"
by (auto simp add: sup_state_def map_eq_Cons sup_loc_Cons2)
theorem sup_state_ignore_fst:
"G \<turnstile> (a, x) <=s (b, y) \<Longrightarrow> G \<turnstile> (c, x) <=s (c, y)"
by (simp add: sup_state_def)
theorem sup_state_rev_fst:
"(G \<turnstile> (rev a, x) <=s (rev b, y)) = (G \<turnstile> (a, x) <=s (b, y))"
proof -
have m: "!!f x. map f (rev x) = rev (map f x)" by (simp add: rev_map)
show ?thesis by (simp add: m sup_state_def)
qed
lemma sup_state_opt_None_any [iff]:
"(G \<turnstile> None <=' any) = True"
by (simp add: sup_state_opt_def lift_bottom_def)
lemma sup_state_opt_any_None [iff]:
"(G \<turnstile> any <=' None) = (any = None)"
by (simp add: sup_state_opt_def)
lemma sup_state_opt_Some_Some [iff]:
"(G \<turnstile> (Some a) <=' (Some b)) = (G \<turnstile> a <=s b)"
by (simp add: sup_state_opt_def del: split_paired_Ex)
lemma sup_state_opt_any_Some [iff]:
"(G \<turnstile> (Some a) <=' any) = (\<exists>b. any = Some b \<and> G \<turnstile> a <=s b)"
by (simp add: sup_state_opt_def)
lemma sup_state_opt_Some_any:
"(G \<turnstile> any <=' (Some b)) = (any = None \<or> (\<exists>a. any = Some a \<and> G \<turnstile> a <=s b))"
by (simp add: sup_state_opt_def lift_bottom_Some_any)
theorem sup_ty_opt_trans [trans]:
"\<lbrakk>G \<turnstile> a <=o b; G \<turnstile> b <=o c\<rbrakk> \<Longrightarrow> G \<turnstile> a <=o c"
by (auto intro: lift_top_trans widen_trans simp add: sup_ty_opt_def)
theorem sup_loc_trans [trans]:
"\<lbrakk>G \<turnstile> a <=l b; G \<turnstile> b <=l c\<rbrakk> \<Longrightarrow> G \<turnstile> a <=l c"
proof -
assume G: "G \<turnstile> a <=l b" "G \<turnstile> b <=l c"
hence "\<forall> n. n < length a \<longrightarrow> (G \<turnstile> (a!n) <=o (c!n))"
proof (intro strip)
fix n
assume n: "n < length a"
with G
have "G \<turnstile> (a!n) <=o (b!n)"
by - (rule sup_loc_nth)
also
from n G
have "G \<turnstile> ... <=o (c!n)"
by - (rule sup_loc_nth, auto dest: sup_loc_length)
finally
show "G \<turnstile> (a!n) <=o (c!n)" .
qed
with G
show ?thesis
by (auto intro!: all_nth_sup_loc [rulify] dest!: sup_loc_length)
qed
theorem sup_state_trans [trans]:
"\<lbrakk>G \<turnstile> a <=s b; G \<turnstile> b <=s c\<rbrakk> \<Longrightarrow> G \<turnstile> a <=s c"
by (auto intro: sup_loc_trans simp add: sup_state_def)
theorem sup_state_opt_trans [trans]:
"\<lbrakk>G \<turnstile> a <=' b; G \<turnstile> b <=' c\<rbrakk> \<Longrightarrow> G \<turnstile> a <=' c"
by (auto intro: lift_bottom_trans sup_state_trans simp add: sup_state_opt_def)
end