src/HOL/MicroJava/BV/Listn.thy
 author kleing Sun, 24 Mar 2002 14:06:21 +0100 changeset 13066 b57d926d1de2 parent 13006 51c5f3f11d16 child 13074 96bf406fd3e5 permissions -rw-r--r--
cleanup + simpler monotonicity
```
(*  Title:      HOL/MicroJava/BV/Listn.thy
ID:         \$Id\$
Author:     Tobias Nipkow

Lists of a fixed length
*)

theory Listn = Err:

constdefs

list :: "nat \<Rightarrow> 'a set \<Rightarrow> 'a list set"
"list n A == {xs. length xs = n & set xs <= A}"

le :: "'a ord \<Rightarrow> ('a list)ord"
"le r == list_all2 (%x y. x <=_r y)"

syntax "@lesublist" :: "'a list \<Rightarrow> 'a ord \<Rightarrow> 'a list \<Rightarrow> bool"
("(_ /<=[_] _)" [50, 0, 51] 50)
syntax "@lesssublist" :: "'a list \<Rightarrow> 'a ord \<Rightarrow> 'a list \<Rightarrow> bool"
("(_ /<[_] _)" [50, 0, 51] 50)
translations
"x <=[r] y" == "x <=_(Listn.le r) y"
"x <[r] y"  == "x <_(Listn.le r) y"

constdefs
map2 :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'c list"
"map2 f == (%xs ys. map (split f) (zip xs ys))"

syntax "@plussublist" :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'b list \<Rightarrow> 'c list"
("(_ /+[_] _)" [65, 0, 66] 65)
translations  "x +[f] y" == "x +_(map2 f) y"

consts coalesce :: "'a err list \<Rightarrow> 'a list err"
primrec
"coalesce [] = OK[]"
"coalesce (ex#exs) = Err.sup (op #) ex (coalesce exs)"

constdefs
sl :: "nat \<Rightarrow> 'a sl \<Rightarrow> 'a list sl"
"sl n == %(A,r,f). (list n A, le r, map2 f)"

sup :: "('a \<Rightarrow> 'b \<Rightarrow> 'c err) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'c list err"
"sup f == %xs ys. if size xs = size ys then coalesce(xs +[f] ys) else Err"

upto_esl :: "nat \<Rightarrow> 'a esl \<Rightarrow> 'a list esl"
"upto_esl m == %(A,r,f). (Union{list n A |n. n <= m}, le r, sup f)"

lemmas [simp] = set_update_subsetI

lemma unfold_lesub_list:
"xs <=[r] ys == Listn.le r xs ys"

lemma Nil_le_conv [iff]:
"([] <=[r] ys) = (ys = [])"
apply (unfold lesub_def Listn.le_def)
apply simp
done

lemma Cons_notle_Nil [iff]:
"~ x#xs <=[r] []"
apply (unfold lesub_def Listn.le_def)
apply simp
done

lemma Cons_le_Cons [iff]:
"x#xs <=[r] y#ys = (x <=_r y & xs <=[r] ys)"
apply (unfold lesub_def Listn.le_def)
apply simp
done

lemma Cons_less_Conss [simp]:
"order r \<Longrightarrow>
x#xs <_(Listn.le r) y#ys =
(x <_r y & xs <=[r] ys  |  x = y & xs <_(Listn.le r) ys)"
apply (unfold lesssub_def)
apply blast
done

lemma list_update_le_cong:
"\<lbrakk> i<size xs; xs <=[r] ys; x <=_r y \<rbrakk> \<Longrightarrow> xs[i:=x] <=[r] ys[i:=y]";
apply (unfold unfold_lesub_list)
apply (unfold Listn.le_def)
done

lemma le_listD:
"\<lbrakk> xs <=[r] ys; p < size xs \<rbrakk> \<Longrightarrow> xs!p <=_r ys!p"
apply (unfold Listn.le_def lesub_def)
done

lemma le_list_refl:
"!x. x <=_r x \<Longrightarrow> xs <=[r] xs"
apply (unfold unfold_lesub_list)
done

lemma le_list_trans:
"\<lbrakk> order r; xs <=[r] ys; ys <=[r] zs \<rbrakk> \<Longrightarrow> xs <=[r] zs"
apply (unfold unfold_lesub_list)
apply clarify
apply simp
apply (blast intro: order_trans)
done

lemma le_list_antisym:
"\<lbrakk> order r; xs <=[r] ys; ys <=[r] xs \<rbrakk> \<Longrightarrow> xs = ys"
apply (unfold unfold_lesub_list)
apply (rule nth_equalityI)
apply blast
apply clarify
apply simp
apply (blast intro: order_antisym)
done

lemma order_listI [simp, intro!]:
"order r \<Longrightarrow> order(Listn.le r)"
apply (subst order_def)
apply (blast intro: le_list_refl le_list_trans le_list_antisym
dest: order_refl)
done

lemma lesub_list_impl_same_size [simp]:
"xs <=[r] ys \<Longrightarrow> size ys = size xs"
apply (unfold Listn.le_def lesub_def)
done

lemma lesssub_list_impl_same_size:
"xs <_(Listn.le r) ys \<Longrightarrow> size ys = size xs"
apply (unfold lesssub_def)
apply auto
done

lemma le_list_appendI:
"\<And>b c d. a <=[r] b \<Longrightarrow> c <=[r] d \<Longrightarrow> a@c <=[r] b@d"
apply (induct a)
apply simp
apply (case_tac b)
apply auto
done

lemma le_listI:
"length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> a!n <=_r b!n) \<Longrightarrow> a <=[r] b"
apply (unfold lesub_def Listn.le_def)
done

lemma listI:
"\<lbrakk> length xs = n; set xs <= A \<rbrakk> \<Longrightarrow> xs : list n A"
apply (unfold list_def)
apply blast
done

lemma listE_length [simp]:
"xs : list n A \<Longrightarrow> length xs = n"
apply (unfold list_def)
apply blast
done

lemma less_lengthI:
"\<lbrakk> xs : list n A; p < n \<rbrakk> \<Longrightarrow> p < length xs"
by simp

lemma listE_set [simp]:
"xs : list n A \<Longrightarrow> set xs <= A"
apply (unfold list_def)
apply blast
done

lemma list_0 [simp]:
"list 0 A = {[]}"
apply (unfold list_def)
apply auto
done

lemma in_list_Suc_iff:
"(xs : list (Suc n) A) = (? y:A. ? ys:list n A. xs = y#ys)"
apply (unfold list_def)
apply (case_tac "xs")
apply auto
done

lemma Cons_in_list_Suc [iff]:
"(x#xs : list (Suc n) A) = (x:A & xs : list n A)";
done

lemma list_not_empty:
"? a. a:A \<Longrightarrow> ? xs. xs : list n A";
apply (induct "n")
apply simp
apply blast
done

lemma nth_in [rule_format, simp]:
"!i n. length xs = n \<longrightarrow> set xs <= A \<longrightarrow> i < n \<longrightarrow> (xs!i) : A"
apply (induct "xs")
apply simp
apply (simp add: nth_Cons split: nat.split)
done

lemma listE_nth_in:
"\<lbrakk> xs : list n A; i < n \<rbrakk> \<Longrightarrow> (xs!i) : A"
by auto

lemma listn_Cons_Suc [elim!]:
"l#xs \<in> list n A \<Longrightarrow> (\<And>n'. n = Suc n' \<Longrightarrow> l \<in> A \<Longrightarrow> xs \<in> list n' A \<Longrightarrow> P) \<Longrightarrow> P"
by (cases n) auto

lemma listn_appendE [elim!]:
"a@b \<in> list n A \<Longrightarrow> (\<And>n1 n2. n=n1+n2 \<Longrightarrow> a \<in> list n1 A \<Longrightarrow> b \<in> list n2 A \<Longrightarrow> P) \<Longrightarrow> P"
proof -
have "\<And>n. a@b \<in> list n A \<Longrightarrow> \<exists>n1 n2. n=n1+n2 \<and> a \<in> list n1 A \<and> b \<in> list n2 A"
(is "\<And>n. ?list a n \<Longrightarrow> \<exists>n1 n2. ?P a n n1 n2")
proof (induct a)
fix n assume "?list [] n"
hence "?P [] n 0 n" by simp
thus "\<exists>n1 n2. ?P [] n n1 n2" by fast
next
fix n l ls
assume "?list (l#ls) n"
then obtain n' where n: "n = Suc n'" "l \<in> A" and "ls@b \<in> list n' A" by fastsimp
assume "\<And>n. ls @ b \<in> list n A \<Longrightarrow> \<exists>n1 n2. n = n1 + n2 \<and> ls \<in> list n1 A \<and> b \<in> list n2 A"
hence "\<exists>n1 n2. n' = n1 + n2 \<and> ls \<in> list n1 A \<and> b \<in> list n2 A" .
then obtain n1 n2 where "n' = n1 + n2" "ls \<in> list n1 A" "b \<in> list n2 A" by fast
with n have "?P (l#ls) n (n1+1) n2" by simp
thus "\<exists>n1 n2. ?P (l#ls) n n1 n2" by fastsimp
qed
moreover
assume "a@b \<in> list n A" "\<And>n1 n2. n=n1+n2 \<Longrightarrow> a \<in> list n1 A \<Longrightarrow> b \<in> list n2 A \<Longrightarrow> P"
ultimately
show ?thesis by blast
qed

lemma listt_update_in_list [simp, intro!]:
"\<lbrakk> xs : list n A; x:A \<rbrakk> \<Longrightarrow> xs[i := x] : list n A"
apply (unfold list_def)
apply simp
done

lemma plus_list_Nil [simp]:
"[] +[f] xs = []"
apply (unfold plussub_def map2_def)
apply simp
done

lemma plus_list_Cons [simp]:
"(x#xs) +[f] ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x +_f y)#(xs +[f] ys))"
by (simp add: plussub_def map2_def split: list.split)

lemma length_plus_list [rule_format, simp]:
"!ys. length(xs +[f] ys) = min(length xs) (length ys)"
apply (induct xs)
apply simp
apply clarify
apply (simp (no_asm_simp) split: list.split)
done

lemma nth_plus_list [rule_format, simp]:
"!xs ys i. length xs = n \<longrightarrow> length ys = n \<longrightarrow> i<n \<longrightarrow>
(xs +[f] ys)!i = (xs!i) +_f (ys!i)"
apply (induct n)
apply simp
apply clarify
apply (case_tac xs)
apply simp
apply (force simp add: nth_Cons split: list.split nat.split)
done

lemma plus_list_ub1 [rule_format]:
"\<lbrakk> semilat(A,r,f); set xs <= A; set ys <= A; size xs = size ys \<rbrakk>
\<Longrightarrow> xs <=[r] xs +[f] ys"
apply (unfold unfold_lesub_list)
done

lemma plus_list_ub2:
"\<lbrakk> semilat(A,r,f); set xs <= A; set ys <= A; size xs = size ys \<rbrakk>
\<Longrightarrow> ys <=[r] xs +[f] ys"
apply (unfold unfold_lesub_list)
done

lemma plus_list_lub [rule_format]:
"semilat(A,r,f) \<Longrightarrow> !xs ys zs. set xs <= A \<longrightarrow> set ys <= A \<longrightarrow> set zs <= A
\<longrightarrow> size xs = n & size ys = n \<longrightarrow>
xs <=[r] zs & ys <=[r] zs \<longrightarrow> xs +[f] ys <=[r] zs"
apply (unfold unfold_lesub_list)
done

lemma list_update_incr [rule_format]:
"\<lbrakk> semilat(A,r,f); x:A \<rbrakk> \<Longrightarrow> set xs <= A \<longrightarrow>
(!i. i<size xs \<longrightarrow> xs <=[r] xs[i := x +_f xs!i])"
apply (unfold unfold_lesub_list)
apply (induct xs)
apply simp
apply clarify
apply (simp add: nth_Cons split: nat.split)
done

lemma acc_le_listI [intro!]:
"\<lbrakk> order r; acc r \<rbrakk> \<Longrightarrow> acc(Listn.le r)"
apply (unfold acc_def)
apply (subgoal_tac
"wf(UN n. {(ys,xs). size xs = n & size ys = n & xs <_(Listn.le r) ys})")
apply (erule wf_subset)
apply (blast intro: lesssub_list_impl_same_size)
apply (rule wf_UN)
prefer 2
apply clarify
apply (rename_tac m n)
apply (case_tac "m=n")
apply simp
apply (rule conjI)
apply (fast intro!: equals0I dest: not_sym)
apply (fast intro!: equals0I dest: not_sym)
apply clarify
apply (rename_tac n)
apply (induct_tac n)
apply (simp add: lesssub_def cong: conj_cong)
apply (rename_tac k)
apply (simp (no_asm) add: length_Suc_conv cong: conj_cong)
apply clarify
apply (rename_tac M m)
apply (case_tac "? x xs. size xs = k & x#xs : M")
prefer 2
apply (erule thin_rl)
apply (erule thin_rl)
apply blast
apply (erule_tac x = "{a. ? xs. size xs = k & a#xs:M}" in allE)
apply (erule impE)
apply blast
apply (thin_tac "? x xs. ?P x xs")
apply clarify
apply (rename_tac maxA xs)
apply (erule_tac x = "{ys. size ys = size xs & maxA#ys : M}" in allE)
apply (erule impE)
apply blast
apply clarify
apply (thin_tac "m : M")
apply (thin_tac "maxA#xs : M")
apply (rule bexI)
prefer 2
apply assumption
apply clarify
apply simp
apply blast
done

lemma closed_listI:
"closed S f \<Longrightarrow> closed (list n S) (map2 f)"
apply (unfold closed_def)
apply (induct n)
apply simp
apply clarify
apply clarify
apply simp
done

lemma semilat_Listn_sl:
"\<And>L. semilat L \<Longrightarrow> semilat (Listn.sl n L)"
apply (unfold Listn.sl_def)
apply (simp (no_asm_simp) only: split_tupled_all)
apply (simp (no_asm) only: semilat_Def split_conv)
apply (rule conjI)
apply simp
apply (rule conjI)
apply (simp only: semilatDclosedI closed_listI)
apply (simp (no_asm) only: list_def)
apply (simp (no_asm_simp) add: plus_list_ub1 plus_list_ub2 plus_list_lub)
done

lemma coalesce_in_err_list [rule_format]:
"!xes. xes : list n (err A) \<longrightarrow> coalesce xes : err(list n A)"
apply (induct n)
apply simp
apply clarify
apply clarify
apply (simp (no_asm) add: plussub_def Err.sup_def lift2_def split: err.split)
apply force
done

lemma lem: "\<And>x xs. x +_(op #) xs = x#xs"

lemma coalesce_eq_OK1_D [rule_format]:
"semilat(err A, Err.le r, lift2 f) \<Longrightarrow>
!xs. xs : list n A \<longrightarrow> (!ys. ys : list n A \<longrightarrow>
(!zs. coalesce (xs +[f] ys) = OK zs \<longrightarrow> xs <=[r] zs))"
apply (induct n)
apply simp
apply clarify
apply clarify
apply (simp split: err.split_asm add: lem Err.sup_def lift2_def)
done

lemma coalesce_eq_OK2_D [rule_format]:
"semilat(err A, Err.le r, lift2 f) \<Longrightarrow>
!xs. xs : list n A \<longrightarrow> (!ys. ys : list n A \<longrightarrow>
(!zs. coalesce (xs +[f] ys) = OK zs \<longrightarrow> ys <=[r] zs))"
apply (induct n)
apply simp
apply clarify
apply clarify
apply (simp split: err.split_asm add: lem Err.sup_def lift2_def)
done

lemma lift2_le_ub:
"\<lbrakk> semilat(err A, Err.le r, lift2 f); x:A; y:A; x +_f y = OK z;
u:A; x <=_r u; y <=_r u \<rbrakk> \<Longrightarrow> z <=_r u"
apply (unfold semilat_Def plussub_def err_def)
apply clarify
apply (rotate_tac -3)
apply (erule thin_rl)
apply (erule thin_rl)
apply force
done

lemma coalesce_eq_OK_ub_D [rule_format]:
"semilat(err A, Err.le r, lift2 f) \<Longrightarrow>
!xs. xs : list n A \<longrightarrow> (!ys. ys : list n A \<longrightarrow>
(!zs us. coalesce (xs +[f] ys) = OK zs & xs <=[r] us & ys <=[r] us
& us : list n A \<longrightarrow> zs <=[r] us))"
apply (induct n)
apply simp
apply clarify
apply clarify
apply (simp (no_asm_use) split: err.split_asm add: lem Err.sup_def lift2_def)
apply clarify
apply (rule conjI)
apply (blast intro: lift2_le_ub)
apply blast
done

lemma lift2_eq_ErrD:
"\<lbrakk> x +_f y = Err; semilat(err A, Err.le r, lift2 f); x:A; y:A \<rbrakk>
\<Longrightarrow> ~(? u:A. x <=_r u & y <=_r u)"
by (simp add: OK_plus_OK_eq_Err_conv [THEN iffD1])

lemma coalesce_eq_Err_D [rule_format]:
"\<lbrakk> semilat(err A, Err.le r, lift2 f) \<rbrakk>
\<Longrightarrow> !xs. xs:list n A \<longrightarrow> (!ys. ys:list n A \<longrightarrow>
coalesce (xs +[f] ys) = Err \<longrightarrow>
~(? zs:list n A. xs <=[r] zs & ys <=[r] zs))"
apply (induct n)
apply simp
apply clarify
apply clarify
apply (simp split: err.split_asm add: lem Err.sup_def lift2_def)
apply (blast dest: lift2_eq_ErrD)
apply blast
done

lemma closed_err_lift2_conv:
"closed (err A) (lift2 f) = (!x:A. !y:A. x +_f y : err A)"
apply (unfold closed_def)
done

lemma closed_map2_list [rule_format]:
"closed (err A) (lift2 f) \<Longrightarrow>
!xs. xs : list n A \<longrightarrow> (!ys. ys : list n A \<longrightarrow>
map2 f xs ys : list n (err A))"
apply (unfold map2_def)
apply (induct n)
apply simp
apply clarify
apply clarify
done

lemma closed_lift2_sup:
"closed (err A) (lift2 f) \<Longrightarrow>
closed (err (list n A)) (lift2 (sup f))"
by (fastsimp  simp add: closed_def plussub_def sup_def lift2_def
coalesce_in_err_list closed_map2_list
split: err.split)

lemma err_semilat_sup:
"err_semilat (A,r,f) \<Longrightarrow>
err_semilat (list n A, Listn.le r, sup f)"
apply (unfold Err.sl_def)
apply (simp only: split_conv)
apply (simp (no_asm) only: semilat_Def plussub_def)
apply (simp (no_asm_simp) only: semilatDclosedI closed_lift2_sup)
apply (rule conjI)
apply (drule semilatDorderI)
apply simp
apply (simp (no_asm) only: unfold_lesub_err Err.le_def err_def sup_def lift2_def)
apply (simp (no_asm_simp) add: coalesce_eq_OK1_D coalesce_eq_OK2_D split: err.split)
apply (blast intro: coalesce_eq_OK_ub_D dest: coalesce_eq_Err_D)
done

lemma err_semilat_upto_esl:
"\<And>L. err_semilat L \<Longrightarrow> err_semilat(upto_esl m L)"
apply (unfold Listn.upto_esl_def)
apply (simp (no_asm_simp) only: split_tupled_all)
apply simp
apply (fastsimp intro!: err_semilat_UnionI err_semilat_sup
dest: lesub_list_impl_same_size