--- a/src/HOL/Library/List_Prefix.thy Fri Nov 03 21:34:22 2000 +0100
+++ b/src/HOL/Library/List_Prefix.thy Fri Nov 03 21:35:36 2000 +0100
@@ -15,86 +15,85 @@
instance list :: ("term") ord ..
defs (overloaded)
- prefix_def: "xs \<le> zs == \<exists>ys. zs = xs @ ys"
- strict_prefix_def: "xs < zs == xs \<le> zs \<and> xs \<noteq> (zs::'a list)"
+ prefix_def: "xs \<le> ys == \<exists>zs. ys = xs @ zs"
+ strict_prefix_def: "xs < ys == xs \<le> ys \<and> xs \<noteq> (ys::'a list)"
instance list :: ("term") order
-proof
- fix xs ys zs :: "'a list"
- show "xs \<le> xs" by (simp add: prefix_def)
- { assume "xs \<le> ys" and "ys \<le> zs" thus "xs \<le> zs" by (auto simp add: prefix_def) }
- { assume "xs \<le> ys" and "ys \<le> xs" thus "xs = ys" by (auto simp add: prefix_def) }
- show "(xs < zs) = (xs \<le> zs \<and> xs \<noteq> zs)" by (simp only: strict_prefix_def)
-qed
+ by intro_classes (auto simp add: prefix_def strict_prefix_def)
-constdefs
- parallel :: "'a list => 'a list => bool" (infixl "\<parallel>" 50)
- "xs \<parallel> ys == \<not> (xs \<le> ys) \<and> \<not> (ys \<le> xs)"
+lemma prefixI [intro?]: "ys = xs @ zs ==> xs \<le> ys"
+ by (unfold prefix_def) blast
-lemma parallelI [intro]: "\<not> (xs \<le> ys) ==> \<not> (ys \<le> xs) ==> xs \<parallel> ys"
- by (unfold parallel_def) blast
+lemma prefixE [elim?]: "xs \<le> ys ==> (!!zs. ys = xs @ zs ==> C) ==> C"
+ by (unfold prefix_def) blast
-lemma parellelE [elim]:
- "xs \<parallel> ys ==> (\<not> (xs \<le> ys) ==> \<not> (ys \<le> xs) ==> C) ==> C"
- by (unfold parallel_def) blast
+lemma strict_prefixI [intro?]: "xs \<le> ys ==> xs \<noteq> ys ==> xs < (ys::'a list)"
+ by (unfold strict_prefix_def) blast
-theorem prefix_cases:
- "(xs \<le> ys ==> C) ==>
- (ys \<le> xs ==> C) ==>
- (xs \<parallel> ys ==> C) ==> C"
- by (unfold parallel_def) blast
+lemma strict_prefixE [elim?]:
+ "xs < ys ==> (xs \<le> ys ==> xs \<noteq> (ys::'a list) ==> C) ==> C"
+ by (unfold strict_prefix_def) blast
-subsection {* Recursion equations *}
+subsection {* Basic properties of prefixes *}
theorem Nil_prefix [iff]: "[] \<le> xs"
- apply (simp add: prefix_def)
- done
+ by (simp add: prefix_def)
theorem prefix_Nil [simp]: "(xs \<le> []) = (xs = [])"
- apply (induct_tac xs)
- apply simp
- apply (simp add: prefix_def)
- done
+ by (induct xs) (simp_all add: prefix_def)
lemma prefix_snoc [simp]: "(xs \<le> ys @ [y]) = (xs = ys @ [y] \<or> xs \<le> ys)"
- apply (unfold prefix_def)
- apply (rule iffI)
- apply (erule exE)
- apply (rename_tac zs)
- apply (rule_tac xs = zs in rev_exhaust)
- apply simp
- apply hypsubst
- apply (simp del: append_assoc add: append_assoc [symmetric])
- apply force
- done
+proof
+ assume "xs \<le> ys @ [y]"
+ then obtain zs where zs: "ys @ [y] = xs @ zs" ..
+ show "xs = ys @ [y] \<or> xs \<le> ys"
+ proof (cases zs rule: rev_cases)
+ assume "zs = []"
+ with zs have "xs = ys @ [y]" by simp
+ thus ?thesis ..
+ next
+ fix z zs' assume "zs = zs' @ [z]"
+ with zs have "ys = xs @ zs'" by simp
+ hence "xs \<le> ys" ..
+ thus ?thesis ..
+ qed
+next
+ assume "xs = ys @ [y] \<or> xs \<le> ys"
+ thus "xs \<le> ys @ [y]"
+ proof
+ assume "xs = ys @ [y]"
+ thus ?thesis by simp
+ next
+ assume "xs \<le> ys"
+ then obtain zs where "ys = xs @ zs" ..
+ hence "ys @ [y] = xs @ (zs @ [y])" by simp
+ thus ?thesis ..
+ qed
+qed
lemma Cons_prefix_Cons [simp]: "(x # xs \<le> y # ys) = (x = y \<and> xs \<le> ys)"
- apply (auto simp add: prefix_def)
- done
+ by (auto simp add: prefix_def)
lemma same_prefix_prefix [simp]: "(xs @ ys \<le> xs @ zs) = (ys \<le> zs)"
- apply (induct_tac xs)
- apply simp_all
- done
+ by (induct xs) simp_all
-lemma [iff]: "(xs @ ys \<le> xs) = (ys = [])"
- apply (insert same_prefix_prefix [where ?zs = "[]"])
- apply simp
- apply blast
- done
+lemma same_prefix_nil [iff]: "(xs @ ys \<le> xs) = (ys = [])"
+proof -
+ have "(xs @ ys \<le> xs @ []) = (ys \<le> [])" by (rule same_prefix_prefix)
+ thus ?thesis by simp
+qed
lemma prefix_prefix [simp]: "xs \<le> ys ==> xs \<le> ys @ zs"
- apply (unfold prefix_def)
- apply clarify
- apply simp
- done
+proof -
+ assume "xs \<le> ys"
+ then obtain us where "ys = xs @ us" ..
+ hence "ys @ zs = xs @ (us @ zs)" by simp
+ thus ?thesis ..
+qed
theorem prefix_Cons: "(xs \<le> y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> zs \<le> ys))"
- apply (unfold prefix_def)
- apply (case_tac xs)
- apply auto
- done
+ by (cases xs) (auto simp add: prefix_def)
theorem prefix_append:
"(xs \<le> ys @ zs) = (xs \<le> ys \<or> (\<exists>us. xs = ys @ us \<and> us \<le> zs))"
@@ -109,42 +108,78 @@
"xs \<le> ys ==> length xs < length ys ==> xs @ [ys ! length xs] \<le> ys"
apply (unfold prefix_def)
apply (auto simp add: nth_append)
- apply (case_tac ys)
+ apply (case_tac zs)
apply auto
done
theorem prefix_length_le: "xs \<le> ys ==> length xs \<le> length ys"
- apply (auto simp add: prefix_def)
- done
+ by (auto simp add: prefix_def)
-subsection {* Prefix cases *}
+subsection {* Parallel lists *}
+
+constdefs
+ parallel :: "'a list => 'a list => bool" (infixl "\<parallel>" 50)
+ "xs \<parallel> ys == \<not> xs \<le> ys \<and> \<not> ys \<le> xs"
+
+lemma parallelI [intro]: "\<not> xs \<le> ys ==> \<not> ys \<le> xs ==> xs \<parallel> ys"
+ by (unfold parallel_def) blast
-lemma prefix_Nil_cases [case_names Nil]:
- "xs \<le> [] ==>
- (xs = [] ==> C) ==> C"
- by simp
+lemma parallelE [elim]:
+ "xs \<parallel> ys ==> (\<not> xs \<le> ys ==> \<not> ys \<le> xs ==> C) ==> C"
+ by (unfold parallel_def) blast
-lemma prefix_Cons_cases [case_names Nil Cons]:
- "xs \<le> y # ys ==>
- (xs = [] ==> C) ==>
- (!!zs. xs = y # zs ==> zs \<le> ys ==> C) ==> C"
- by (simp only: prefix_Cons) blast
+theorem prefix_cases:
+ "(xs \<le> ys ==> C) ==>
+ (ys \<le> xs ==> C) ==>
+ (xs \<parallel> ys ==> C) ==> C"
+ by (unfold parallel_def) blast
-lemma prefix_snoc_cases [case_names prefix snoc]:
- "xs \<le> ys @ [y] ==>
- (xs \<le> ys ==> C) ==>
- (xs = ys @ [y] ==> C) ==> C"
- by (simp only: prefix_snoc) blast
-
-lemma prefix_append_cases [case_names prefix append]:
- "xs \<le> ys @ zs ==>
- (xs \<le> ys ==> C) ==>
- (!!us. xs = ys @ us ==> us \<le> zs ==> C) ==> C"
- by (simp only: prefix_append) blast
-
-lemmas prefix_any_cases [cases set: prefix] = (*dummy set name*)
- prefix_Nil_cases prefix_Cons_cases
- prefix_snoc_cases prefix_append_cases
+theorem parallel_decomp:
+ "xs \<parallel> ys ==> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"
+ (concl is "?E xs")
+proof -
+ assume "xs \<parallel> ys"
+ have "?this --> ?E xs" (is "?P xs")
+ proof (induct (stripped) xs rule: rev_induct)
+ assume "[] \<parallel> ys" hence False by auto
+ thus "?E []" ..
+ next
+ fix x xs
+ assume hyp: "?P xs"
+ assume asm: "xs @ [x] \<parallel> ys"
+ show "?E (xs @ [x])"
+ proof (rule prefix_cases)
+ assume le: "xs \<le> ys"
+ then obtain ys' where ys: "ys = xs @ ys'" ..
+ show ?thesis
+ proof (cases ys')
+ assume "ys' = []" with ys have "xs = ys" by simp
+ with asm have "[x] \<parallel> []" by auto
+ hence False by blast
+ thus ?thesis ..
+ next
+ fix c cs assume ys': "ys' = c # cs"
+ with asm ys have "xs @ [x] \<parallel> xs @ c # cs" by (simp only:)
+ hence "x \<noteq> c" by auto
+ moreover have "xs @ [x] = xs @ x # []" by simp
+ moreover from ys ys' have "ys = xs @ c # cs" by (simp only:)
+ ultimately show ?thesis by blast
+ qed
+ next
+ assume "ys \<le> xs" hence "ys \<le> xs @ [x]" by simp
+ with asm have False by blast
+ thus ?thesis ..
+ next
+ assume "xs \<parallel> ys"
+ with hyp obtain as b bs c cs where neq: "(b::'a) \<noteq> c"
+ and xs: "xs = as @ b # bs" and ys: "ys = as @ c # cs"
+ by blast
+ from xs have "xs @ [x] = as @ b # (bs @ [x])" by simp
+ with neq ys show ?thesis by blast
+ qed
+ qed
+ thus ?thesis ..
+qed
end