--- a/src/HOL/Library/Sublist.thy Sun Nov 03 21:12:50 2024 +0100
+++ b/src/HOL/Library/Sublist.thy Sun Nov 03 22:29:07 2024 +0100
@@ -80,7 +80,7 @@
next
assume "xs = ys @ [y] \<or> prefix xs ys"
then show "prefix xs (ys @ [y])"
- by auto (metis append.assoc prefix_def)
+ by auto (metis append.assoc prefix_def)
qed
lemma Cons_prefix_Cons [simp]: "prefix (x # xs) (y # ys) = (x = y \<and> prefix xs ys)"
@@ -96,7 +96,7 @@
by (induct xs) simp_all
lemma same_prefix_nil [simp]: "prefix (xs @ ys) xs = (ys = [])"
- by (simp add: prefix_def)
+ by (simp add: prefix_def)
lemma prefix_prefix [simp]: "prefix xs ys \<Longrightarrow> prefix xs (ys @ zs)"
unfolding prefix_def by fastforce
@@ -295,7 +295,7 @@
lemma length_prefixes[simp]: "length (prefixes xs) = length xs+1"
by (induction xs) auto
-
+
lemma distinct_prefixes [intro]: "distinct (prefixes xs)"
by (induction xs) (auto simp: distinct_map)
@@ -310,8 +310,8 @@
lemma last_prefixes [simp]: "last (prefixes xs) = xs"
by (induction xs) (simp_all add: last_map)
-
-lemma prefixes_append:
+
+lemma prefixes_append:
"prefixes (xs @ ys) = prefixes xs @ map (\<lambda>ys'. xs @ ys') (tl (prefixes ys))"
proof (induction xs)
case Nil
@@ -323,7 +323,7 @@
(ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = zs@[z] \<and> xs = prefixes zs)) \<and> x = ys"
by (cases ys rule: rev_cases) auto
-lemma prefixes_tailrec [code]:
+lemma prefixes_tailrec [code]:
"prefixes xs = rev (snd (foldl (\<lambda>(acc1, acc2) x. (x#acc1, rev (x#acc1)#acc2)) ([],[[]]) xs))"
proof -
have "foldl (\<lambda>(acc1, acc2) x. (x#acc1, rev (x#acc1)#acc2)) (ys, rev ys # zs) xs =
@@ -335,14 +335,14 @@
qed simp_all
from this [of "[]" "[]"] show ?thesis by simp
qed
-
+
lemma set_prefixes_eq: "set (prefixes xs) = {ys. prefix ys xs}"
by auto
lemma card_set_prefixes [simp]: "card (set (prefixes xs)) = Suc (length xs)"
by (subst distinct_card) auto
-lemma set_prefixes_append:
+lemma set_prefixes_append:
"set (prefixes (xs @ ys)) = set (prefixes xs) \<union> {xs @ ys' |ys'. ys' \<in> set (prefixes ys)}"
by (subst prefixes_append, cases ys) auto
@@ -376,13 +376,14 @@
by - (rule Least_equality, fastforce+)
have 2: "?L \<noteq> {}" using \<open>x # xs \<in> L\<close> by auto
from Suc.hyps(1)[OF 1[symmetric] 2] obtain ps where IH: "?P ?L ps" ..
- { fix qs
- assume "\<forall>qs. (\<forall>xa. x # xa \<in> L \<longrightarrow> prefix qs xa) \<longrightarrow> length qs \<le> length ps"
- and "\<forall>xs\<in>L. prefix qs xs"
- hence "length (tl qs) \<le> length ps"
- by (metis Cons_prefix_Cons hd_Cons_tl list.sel(2) Nil_prefix)
- hence "length qs \<le> Suc (length ps)" by auto
- }
+ have "length qs \<le> Suc (length ps)"
+ if "\<forall>qs. (\<forall>xa. x # xa \<in> L \<longrightarrow> prefix qs xa) \<longrightarrow> length qs \<le> length ps"
+ and "\<forall>xs\<in>L. prefix qs xs" for qs
+ proof -
+ from that have "length (tl qs) \<le> length ps"
+ by (metis Cons_prefix_Cons hd_Cons_tl list.sel(2) Nil_prefix)
+ thus ?thesis by auto
+ qed
hence "?P L (x#ps)" using True IH by auto
thus ?thesis ..
next
@@ -544,7 +545,7 @@
assumes "suffix xs ys"
obtains zs where "ys = zs @ xs"
using assms unfolding suffix_def by blast
-
+
lemma suffix_tl [simp]: "suffix (tl xs) xs"
by (induct xs) (auto simp: suffix_def)
@@ -599,7 +600,7 @@
then have "ys = rev zs @ xs" by simp
then show "suffix xs ys" ..
qed
-
+
lemma strict_suffix_to_prefix [code]: "strict_suffix xs ys \<longleftrightarrow> strict_prefix (rev xs) (rev ys)"
by (auto simp: suffix_to_prefix strict_suffix_def strict_prefix_def)
@@ -645,7 +646,7 @@
theorem suffix_Cons: "suffix xs (y # ys) \<longleftrightarrow> xs = y # ys \<or> suffix xs ys"
unfolding suffix_def by (auto simp: Cons_eq_append_conv)
-theorem suffix_append:
+theorem suffix_append:
"suffix xs (ys @ zs) \<longleftrightarrow> suffix xs zs \<or> (\<exists>xs'. xs = xs' @ zs \<and> suffix xs' ys)"
by (auto simp: suffix_def append_eq_append_conv2)
@@ -672,7 +673,7 @@
then show ?case by (cases ys) simp_all
next
case (Suc n)
- then show ?case
+ then show ?case
by (cases xs) (auto intro: Suc dest: suffix_ConsD' suffix_order.less_imp_le)
qed
@@ -808,7 +809,7 @@
lemma last_suffixes [simp]: "last (suffixes xs) = xs"
by (cases xs) simp_all
-lemma suffixes_append:
+lemma suffixes_append:
"suffixes (xs @ ys) = suffixes ys @ map (\<lambda>xs'. xs' @ ys) (tl (suffixes xs))"
proof (induction ys rule: rev_induct)
case Nil
@@ -824,7 +825,7 @@
(ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = z#zs \<and> xs = suffixes zs)) \<and> x = ys"
by (cases ys) auto
-lemma suffixes_tailrec [code]:
+lemma suffixes_tailrec [code]:
"suffixes xs = rev (snd (foldl (\<lambda>(acc1, acc2) x. (x#acc1, (x#acc1)#acc2)) ([],[[]]) (rev xs)))"
proof -
have "foldl (\<lambda>(acc1, acc2) x. (x#acc1, (x#acc1)#acc2)) (ys, ys # zs) (rev xs) =
@@ -836,14 +837,14 @@
qed simp_all
from this [of "[]" "[]"] show ?thesis by simp
qed
-
+
lemma set_suffixes_eq: "set (suffixes xs) = {ys. suffix ys xs}"
by auto
-
+
lemma card_set_suffixes [simp]: "card (set (suffixes xs)) = Suc (length xs)"
by (subst distinct_card) auto
-
-lemma set_suffixes_append:
+
+lemma set_suffixes_append:
"set (suffixes (xs @ ys)) = set (suffixes ys) \<union> {xs' @ ys |xs'. xs' \<in> set (suffixes xs)}"
by (subst suffixes_append, cases xs rule: rev_cases) auto
@@ -853,10 +854,10 @@
lemma prefixes_conv_suffixes: "prefixes xs = map rev (suffixes (rev xs))"
by (induction xs) auto
-
+
lemma prefixes_rev: "prefixes (rev xs) = map rev (suffixes xs)"
by (induction xs) auto
-
+
lemma suffixes_rev: "suffixes (rev xs) = map rev (prefixes xs)"
by (induction xs) auto
@@ -870,13 +871,13 @@
| list_emb_Cons [intro] : "list_emb P xs ys \<Longrightarrow> list_emb P xs (y#ys)"
| list_emb_Cons2 [intro]: "P x y \<Longrightarrow> list_emb P xs ys \<Longrightarrow> list_emb P (x#xs) (y#ys)"
-lemma list_emb_mono:
+lemma list_emb_mono:
assumes "\<And>x y. P x y \<longrightarrow> Q x y"
shows "list_emb P xs ys \<longrightarrow> list_emb Q xs ys"
-proof
- assume "list_emb P xs ys"
+proof
+ assume "list_emb P xs ys"
then show "list_emb Q xs ys" by (induct) (auto simp: assms)
-qed
+qed
lemma list_emb_Nil2 [simp]:
assumes "list_emb P xs []" shows "xs = []"
@@ -888,13 +889,11 @@
using assms by (induct xs) auto
lemma list_emb_Cons_Nil [simp]: "list_emb P (x#xs) [] = False"
-proof -
- { assume "list_emb P (x#xs) []"
- from list_emb_Nil2 [OF this] have False by simp
- } moreover {
- assume False
- then have "list_emb P (x#xs) []" by simp
- } ultimately show ?thesis by blast
+proof
+ show False if "list_emb P (x#xs) []"
+ using list_emb_Nil2 [OF that] by simp
+ show "list_emb P (x#xs) []" if False
+ using that ..
qed
lemma list_emb_append2 [intro]: "list_emb P xs ys \<Longrightarrow> list_emb P xs (zs @ ys)"
@@ -1002,13 +1001,13 @@
"list_emb P (x#xs) [] \<longleftrightarrow> False"
"list_emb P (x#xs) (y#ys) \<longleftrightarrow> (if P x y then list_emb P xs ys else list_emb P (x#xs) ys)"
by simp_all
-
+
subsection \<open>Subsequences (special case of homeomorphic embedding)\<close>
abbreviation subseq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
where "subseq xs ys \<equiv> list_emb (=) xs ys"
-
+
definition strict_subseq where "strict_subseq xs ys \<longleftrightarrow> xs \<noteq> ys \<and> subseq xs ys"
lemma subseq_Cons2: "subseq xs ys \<Longrightarrow> subseq (x#xs) (x#ys)" by auto
@@ -1073,7 +1072,7 @@
thus "subseq xs ys"
by (induction ys arbitrary: xs) (auto simp: Let_def)
next
- have [simp]: "[] \<in> set (subseqs ys)" for ys :: "'a list"
+ have [simp]: "[] \<in> set (subseqs ys)" for ys :: "'a list"
by (induction ys) (auto simp: Let_def)
assume "subseq xs ys"
thus "xs \<in> set (subseqs ys)"
@@ -1105,33 +1104,39 @@
lemma subseq_append [simp]:
"subseq (xs @ zs) (ys @ zs) \<longleftrightarrow> subseq xs ys" (is "?l = ?r")
proof
- { fix xs' ys' xs ys zs :: "'a list" assume "subseq xs' ys'"
- then have "xs' = xs @ zs \<and> ys' = ys @ zs \<longrightarrow> subseq xs ys"
- proof (induct arbitrary: xs ys zs)
- case list_emb_Nil show ?case by simp
- next
- case (list_emb_Cons xs' ys' x)
- { assume "ys=[]" then have ?case using list_emb_Cons(1) by auto }
- moreover
- { fix us assume "ys = x#us"
- then have ?case using list_emb_Cons(2) by(simp add: list_emb.list_emb_Cons) }
- ultimately show ?case by (auto simp:Cons_eq_append_conv)
- next
- case (list_emb_Cons2 x y xs' ys')
- { assume "xs=[]" then have ?case using list_emb_Cons2(1) by auto }
- moreover
- { fix us vs assume "xs=x#us" "ys=x#vs" then have ?case using list_emb_Cons2 by auto}
- moreover
- { fix us assume "xs=x#us" "ys=[]" then have ?case using list_emb_Cons2(2) by bestsimp }
- ultimately show ?case using \<open>(=) x y\<close> by (auto simp: Cons_eq_append_conv)
- qed }
- moreover assume ?l
- ultimately show ?r by blast
-next
- assume ?r then show ?l by (metis list_emb_append_mono subseq_order.order_refl)
+ have "xs' = xs @ zs \<and> ys' = ys @ zs \<longrightarrow> subseq xs ys"
+ if "subseq xs' ys'" for xs' ys' xs ys zs :: "'a list"
+ using that
+ proof (induct arbitrary: xs ys zs)
+ case list_emb_Nil
+ show ?case by simp
+ next
+ case (list_emb_Cons xs' ys' x)
+ have ?case if "ys = []"
+ using list_emb_Cons(1) that by auto
+ moreover
+ have ?case if "ys = x#us" for us
+ using list_emb_Cons(2) that by (simp add: list_emb.list_emb_Cons)
+ ultimately show ?case
+ by (auto simp: Cons_eq_append_conv)
+ next
+ case (list_emb_Cons2 x y xs' ys')
+ have ?case if "xs = []"
+ using list_emb_Cons2(1) that by auto
+ moreover
+ have ?case if "xs = x#us" "ys = x#vs" for us vs
+ using list_emb_Cons2 that by auto
+ moreover
+ have ?case if "xs = x#us" "ys = []" for us
+ using list_emb_Cons2(2) that by bestsimp
+ ultimately show ?case
+ using \<open>x = y\<close> by (auto simp: Cons_eq_append_conv)
+ qed
+ then show "?l \<Longrightarrow> ?r" by blast
+ show "?r \<Longrightarrow> ?l" by (metis list_emb_append_mono subseq_order.order_refl)
qed
-lemma subseq_append_iff:
+lemma subseq_append_iff:
"subseq xs (ys @ zs) \<longleftrightarrow> (\<exists>xs1 xs2. xs = xs1 @ xs2 \<and> subseq xs1 ys \<and> subseq xs2 zs)"
(is "?lhs = ?rhs")
proof
@@ -1139,7 +1144,7 @@
proof (induction xs "ys @ zs" arbitrary: ys zs rule: list_emb.induct)
case (list_emb_Cons xs ws y ys zs)
from list_emb_Cons(2)[of "tl ys" zs] and list_emb_Cons(2)[of "[]" "tl zs"] and list_emb_Cons(1,3)
- show ?case by (cases ys) auto
+ show ?case by (cases ys) auto
next
case (list_emb_Cons2 x y xs ws ys zs)
from list_emb_Cons2(3)[of "tl ys" zs] and list_emb_Cons2(3)[of "[]" "tl zs"]
@@ -1148,7 +1153,7 @@
qed auto
qed (auto intro: list_emb_append_mono)
-lemma subseq_appendE [case_names append]:
+lemma subseq_appendE [case_names append]:
assumes "subseq xs (ys @ zs)"
obtains xs1 xs2 where "xs = xs1 @ xs2" "subseq xs1 ys" "subseq xs2 zs"
using assms by (subst (asm) subseq_append_iff) auto
@@ -1173,13 +1178,13 @@
assumes "subseq xs ys" shows "subseq (filter P xs) (filter P ys)"
using assms by induct auto
-lemma subseq_conv_nths:
- "subseq xs ys \<longleftrightarrow> (\<exists>N. xs = nths ys N)" (is "?L = ?R")
+lemma subseq_conv_nths: "subseq xs ys \<longleftrightarrow> (\<exists>N. xs = nths ys N)"
+ (is "?L = ?R")
proof
- assume ?L
- then show ?R
+ show ?R if ?L using that
proof (induct)
- case list_emb_Nil show ?case by (metis nths_empty)
+ case list_emb_Nil
+ show ?case by (metis nths_empty)
next
case (list_emb_Cons xs ys x)
then obtain N where "xs = nths ys N" by blast
@@ -1194,27 +1199,30 @@
moreover from list_emb_Cons2 have "x = y" by simp
ultimately show ?case by blast
qed
-next
- assume ?R
- then obtain N where "xs = nths ys N" ..
- moreover have "subseq (nths ys N) ys"
- proof (induct ys arbitrary: N)
- case Nil show ?case by simp
- next
- case Cons then show ?case by (auto simp: nths_Cons)
+ show ?L if ?R
+ proof -
+ from that obtain N where "xs = nths ys N" ..
+ moreover have "subseq (nths ys N) ys"
+ proof (induct ys arbitrary: N)
+ case Nil
+ show ?case by simp
+ next
+ case Cons
+ then show ?case by (auto simp: nths_Cons)
+ qed
+ ultimately show ?thesis by simp
qed
- ultimately show ?L by simp
qed
-
-
+
+
subsection \<open>Contiguous sublists\<close>
subsubsection \<open>\<open>sublist\<close>\<close>
-definition sublist :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
+definition sublist :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
"sublist xs ys = (\<exists>ps ss. ys = ps @ xs @ ss)"
-
-definition strict_sublist :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
+
+definition strict_sublist :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
"strict_sublist xs ys \<longleftrightarrow> sublist xs ys \<and> xs \<noteq> ys"
interpretation sublist_order: order sublist strict_sublist
@@ -1227,38 +1235,37 @@
thus "sublist xs zs" unfolding sublist_def by blast
next
fix xs ys :: "'a list"
- {
- assume "sublist xs ys" "sublist ys xs"
- then obtain as bs cs ds
- where xs: "xs = as @ ys @ bs" and ys: "ys = cs @ xs @ ds"
+ show "xs = ys" if "sublist xs ys" "sublist ys xs"
+ proof -
+ from that obtain as bs cs ds where xs: "xs = as @ ys @ bs" and ys: "ys = cs @ xs @ ds"
by (auto simp: sublist_def)
have "xs = as @ cs @ xs @ ds @ bs" by (subst xs, subst ys) auto
- also have "length \<dots> = length as + length cs + length xs + length bs + length ds"
+ also have "length \<dots> = length as + length cs + length xs + length bs + length ds"
by simp
finally have "as = []" "bs = []" by simp_all
- with xs show "xs = ys" by simp
- }
- thus "strict_sublist xs ys \<longleftrightarrow> (sublist xs ys \<and> \<not>sublist ys xs)"
+ with xs show ?thesis by simp
+ qed
+ thus "strict_sublist xs ys \<longleftrightarrow> (sublist xs ys \<and> \<not> sublist ys xs)"
by (auto simp: strict_sublist_def)
qed (auto simp: strict_sublist_def sublist_def intro: exI[of _ "[]"])
-
+
lemma sublist_Nil_left [simp, intro]: "sublist [] ys"
by (auto simp: sublist_def)
-
+
lemma sublist_Cons_Nil [simp]: "\<not>sublist (x#xs) []"
by (auto simp: sublist_def)
-
+
lemma sublist_Nil_right [simp]: "sublist xs [] \<longleftrightarrow> xs = []"
by (cases xs) auto
-
+
lemma sublist_appendI [simp, intro]: "sublist xs (ps @ xs @ ss)"
by (auto simp: sublist_def)
-
+
lemma sublist_append_leftI [simp, intro]: "sublist xs (ps @ xs)"
by (auto simp: sublist_def intro: exI[of _ "[]"])
-
+
lemma sublist_append_rightI [simp, intro]: "sublist xs (xs @ ss)"
- by (auto simp: sublist_def intro: exI[of _ "[]"])
+ by (auto simp: sublist_def intro: exI[of _ "[]"])
lemma sublist_altdef: "sublist xs ys \<longleftrightarrow> (\<exists>ys'. prefix ys' ys \<and> suffix xs ys')"
proof safe
@@ -1271,7 +1278,7 @@
assume "prefix ys' ys" "suffix xs ys'"
thus "sublist xs ys" by (auto simp: prefix_def suffix_def)
qed
-
+
lemma sublist_altdef': "sublist xs ys \<longleftrightarrow> (\<exists>ys'. suffix ys' ys \<and> prefix xs ys')"
proof safe
assume "sublist xs ys"
@@ -1286,7 +1293,7 @@
lemma sublist_Cons_right: "sublist xs (y # ys) \<longleftrightarrow> prefix xs (y # ys) \<or> sublist xs ys"
by (auto simp: sublist_def prefix_def Cons_eq_append_conv)
-
+
lemma sublist_code [code]:
"sublist [] ys \<longleftrightarrow> True"
"sublist (x # xs) [] \<longleftrightarrow> False"
@@ -1294,7 +1301,7 @@
by (simp_all add: sublist_Cons_right)
lemma sublist_append:
- "sublist xs (ys @ zs) \<longleftrightarrow>
+ "sublist xs (ys @ zs) \<longleftrightarrow>
sublist xs ys \<or> sublist xs zs \<or> (\<exists>xs1 xs2. xs = xs1 @ xs2 \<and> suffix xs1 ys \<and> prefix xs2 zs)"
by (auto simp: sublist_altdef prefix_append suffix_append)
@@ -1315,10 +1322,10 @@
lemma set_mono_sublist: "sublist xs ys \<Longrightarrow> set xs \<subseteq> set ys"
by (auto simp add: sublist_def)
-
+
lemma prefix_imp_sublist [simp, intro]: "prefix xs ys \<Longrightarrow> sublist xs ys"
by (auto simp: sublist_def prefix_def intro: exI[of _ "[]"])
-
+
lemma suffix_imp_sublist [simp, intro]: "suffix xs ys \<Longrightarrow> sublist xs ys"
by (auto simp: sublist_def suffix_def intro: exI[of _ "[]"])
@@ -1333,13 +1340,13 @@
lemma sublist_dropWhile [simp, intro]: "sublist (dropWhile P xs) xs"
by (rule suffix_imp_sublist[OF suffix_dropWhile])
-
+
lemma sublist_tl [simp, intro]: "sublist (tl xs) xs"
by (rule suffix_imp_sublist) (simp_all add: suffix_drop)
-
+
lemma sublist_butlast [simp, intro]: "sublist (butlast xs) xs"
by (rule prefix_imp_sublist) (simp_all add: prefixeq_butlast)
-
+
lemma sublist_rev [simp]: "sublist (rev xs) (rev ys) = sublist xs ys"
proof
assume "sublist (rev xs) (rev ys)"
@@ -1354,15 +1361,15 @@
also have "rev \<dots> = rev bs @ rev xs @ rev as" by simp
finally show "sublist (rev xs) (rev ys)" by simp
qed
-
+
lemma sublist_rev_left: "sublist (rev xs) ys = sublist xs (rev ys)"
by (subst sublist_rev [symmetric]) (simp only: rev_rev_ident)
-
+
lemma sublist_rev_right: "sublist xs (rev ys) = sublist (rev xs) ys"
by (subst sublist_rev [symmetric]) (simp only: rev_rev_ident)
-lemma snoc_sublist_snoc:
- "sublist (xs @ [x]) (ys @ [y]) \<longleftrightarrow>
+lemma snoc_sublist_snoc:
+ "sublist (xs @ [x]) (ys @ [y]) \<longleftrightarrow>
(x = y \<and> suffix xs ys \<or> sublist (xs @ [x]) ys) "
by (subst (1 2) sublist_rev [symmetric])
(simp del: sublist_rev add: sublist_Cons_right suffix_to_prefix)
@@ -1370,8 +1377,8 @@
lemma sublist_snoc:
"sublist xs (ys @ [y]) \<longleftrightarrow> suffix xs (ys @ [y]) \<or> sublist xs ys"
by (subst (1 2) sublist_rev [symmetric])
- (simp del: sublist_rev add: sublist_Cons_right suffix_to_prefix)
-
+ (simp del: sublist_rev add: sublist_Cons_right suffix_to_prefix)
+
lemma sublist_imp_subseq [intro]: "sublist xs ys \<Longrightarrow> subseq xs ys"
by (auto simp: sublist_def)
@@ -1415,7 +1422,7 @@
"sublists [] = [[]]"
| "sublists (x # xs) = sublists xs @ map ((#) x) (prefixes xs)"
-lemma in_set_sublists [simp]: "xs \<in> set (sublists ys) \<longleftrightarrow> sublist xs ys"
+lemma in_set_sublists [simp]: "xs \<in> set (sublists ys) \<longleftrightarrow> sublist xs ys"
by (induction ys arbitrary: xs) (auto simp: sublist_Cons_right prefix_Cons)
lemma set_sublists_eq: "set (sublists xs) = {ys. sublist ys xs}"
@@ -1428,8 +1435,8 @@
subsection \<open>Parametricity\<close>
context includes lifting_syntax
-begin
-
+begin
+
private lemma prefix_primrec:
"prefix = rec_list (\<lambda>xs. True) (\<lambda>x xs xsa ys.
case ys of [] \<Rightarrow> False | y # ys \<Rightarrow> x = y \<and> xsa ys)"
@@ -1446,7 +1453,7 @@
qed
private lemma list_emb_primrec:
- "list_emb = (\<lambda>uu uua uuaa. rec_list (\<lambda>P xs. List.null xs) (\<lambda>y ys ysa P xs. case xs of [] \<Rightarrow> True
+ "list_emb = (\<lambda>uu uua uuaa. rec_list (\<lambda>P xs. List.null xs) (\<lambda>y ys ysa P xs. case xs of [] \<Rightarrow> True
| x # xs \<Rightarrow> if P x y then ysa P xs else ysa P (x # xs)) uuaa uu uua)"
proof (intro ext, goal_cases)
case (1 P xs ys)
@@ -1457,12 +1464,12 @@
lemma prefix_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
- shows "(list_all2 A ===> list_all2 A ===> (=)) prefix prefix"
+ shows "(list_all2 A ===> list_all2 A ===> (=)) prefix prefix"
unfolding prefix_primrec by transfer_prover
-
+
lemma suffix_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
- shows "(list_all2 A ===> list_all2 A ===> (=)) suffix suffix"
+ shows "(list_all2 A ===> list_all2 A ===> (=)) suffix suffix"
unfolding suffix_to_prefix [abs_def] by transfer_prover
lemma sublist_transfer [transfer_rule]:
@@ -1474,7 +1481,7 @@
assumes [transfer_rule]: "bi_unique A"
shows "(list_all2 A ===> list_all2 A ===> (=)) parallel parallel"
unfolding parallel_def by transfer_prover
-
+
lemma list_emb_transfer [transfer_rule]:
@@ -1483,34 +1490,34 @@
lemma strict_prefix_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
- shows "(list_all2 A ===> list_all2 A ===> (=)) strict_prefix strict_prefix"
+ shows "(list_all2 A ===> list_all2 A ===> (=)) strict_prefix strict_prefix"
unfolding strict_prefix_def by transfer_prover
-
+
lemma strict_suffix_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
- shows "(list_all2 A ===> list_all2 A ===> (=)) strict_suffix strict_suffix"
+ shows "(list_all2 A ===> list_all2 A ===> (=)) strict_suffix strict_suffix"
unfolding strict_suffix_def by transfer_prover
-
+
lemma strict_subseq_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
- shows "(list_all2 A ===> list_all2 A ===> (=)) strict_subseq strict_subseq"
+ shows "(list_all2 A ===> list_all2 A ===> (=)) strict_subseq strict_subseq"
unfolding strict_subseq_def by transfer_prover
-
+
lemma strict_sublist_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
- shows "(list_all2 A ===> list_all2 A ===> (=)) strict_sublist strict_sublist"
+ shows "(list_all2 A ===> list_all2 A ===> (=)) strict_sublist strict_sublist"
unfolding strict_sublist_def by transfer_prover
lemma prefixes_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
shows "(list_all2 A ===> list_all2 (list_all2 A)) prefixes prefixes"
unfolding prefixes_def by transfer_prover
-
+
lemma suffixes_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
shows "(list_all2 A ===> list_all2 (list_all2 A)) suffixes suffixes"
unfolding suffixes_def by transfer_prover
-
+
lemma sublists_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
shows "(list_all2 A ===> list_all2 (list_all2 A)) sublists sublists"