(* Title: HOL/Library/List_Prefix.thy ID: $Id: List_Prefix.thy,v 1.16 2005/08/31 13:46:38 wenzelm Exp $ Author: Tobias Nipkow and Markus Wenzel, TU Muenchen *) header {* List prefixes and postfixes *} theory List_Prefix imports Main begin subsection {* Prefix order on lists *} instance list :: (type) ord .. defs (overloaded) prefix_def: "xs ≤ ys == ∃zs. ys = xs @ zs" strict_prefix_def: "xs < ys == xs ≤ ys ∧ xs ≠ (ys::'a list)" instance list :: (type) order by intro_classes (auto simp add: prefix_def strict_prefix_def) lemma prefixI [intro?]: "ys = xs @ zs ==> xs ≤ ys" by (unfold prefix_def) blast lemma prefixE [elim?]: "xs ≤ ys ==> (!!zs. ys = xs @ zs ==> C) ==> C" by (unfold prefix_def) blast lemma strict_prefixI' [intro?]: "ys = xs @ z # zs ==> xs < ys" by (unfold strict_prefix_def prefix_def) blast lemma strict_prefixE' [elim?]: assumes lt: "xs < ys" and r: "!!z zs. ys = xs @ z # zs ==> C" shows C proof - from lt obtain us where "ys = xs @ us" and "xs ≠ ys" by (unfold strict_prefix_def prefix_def) blast with r show ?thesis by (auto simp add: neq_Nil_conv) qed lemma strict_prefixI [intro?]: "xs ≤ ys ==> xs ≠ ys ==> xs < (ys::'a list)" by (unfold strict_prefix_def) blast lemma strict_prefixE [elim?]: "xs < ys ==> (xs ≤ ys ==> xs ≠ (ys::'a list) ==> C) ==> C" by (unfold strict_prefix_def) blast subsection {* Basic properties of prefixes *} theorem Nil_prefix [iff]: "[] ≤ xs" by (simp add: prefix_def) theorem prefix_Nil [simp]: "(xs ≤ []) = (xs = [])" by (induct xs) (simp_all add: prefix_def) lemma prefix_snoc [simp]: "(xs ≤ ys @ [y]) = (xs = ys @ [y] ∨ xs ≤ ys)" proof assume "xs ≤ ys @ [y]" then obtain zs where zs: "ys @ [y] = xs @ zs" .. show "xs = ys @ [y] ∨ xs ≤ 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 ≤ ys" .. thus ?thesis .. qed next assume "xs = ys @ [y] ∨ xs ≤ ys" thus "xs ≤ ys @ [y]" proof assume "xs = ys @ [y]" thus ?thesis by simp next assume "xs ≤ 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 ≤ y # ys) = (x = y ∧ xs ≤ ys)" by (auto simp add: prefix_def) lemma same_prefix_prefix [simp]: "(xs @ ys ≤ xs @ zs) = (ys ≤ zs)" by (induct xs) simp_all lemma same_prefix_nil [iff]: "(xs @ ys ≤ xs) = (ys = [])" proof - have "(xs @ ys ≤ xs @ []) = (ys ≤ [])" by (rule same_prefix_prefix) thus ?thesis by simp qed lemma prefix_prefix [simp]: "xs ≤ ys ==> xs ≤ ys @ zs" proof - assume "xs ≤ ys" then obtain us where "ys = xs @ us" .. hence "ys @ zs = xs @ (us @ zs)" by simp thus ?thesis .. qed lemma append_prefixD: "xs @ ys ≤ zs ==> xs ≤ zs" by (auto simp add: prefix_def) theorem prefix_Cons: "(xs ≤ y # ys) = (xs = [] ∨ (∃zs. xs = y # zs ∧ zs ≤ ys))" by (cases xs) (auto simp add: prefix_def) theorem prefix_append: "(xs ≤ ys @ zs) = (xs ≤ ys ∨ (∃us. xs = ys @ us ∧ us ≤ zs))" apply (induct zs rule: rev_induct) apply force apply (simp del: append_assoc add: append_assoc [symmetric]) apply simp apply blast done lemma append_one_prefix: "xs ≤ ys ==> length xs < length ys ==> xs @ [ys ! length xs] ≤ ys" apply (unfold prefix_def) apply (auto simp add: nth_append) apply (case_tac zs) apply auto done theorem prefix_length_le: "xs ≤ ys ==> length xs ≤ length ys" by (auto simp add: prefix_def) lemma prefix_same_cases: "(xs1::'a list) ≤ ys ==> xs2 ≤ ys ==> xs1 ≤ xs2 ∨ xs2 ≤ xs1" apply (simp add: prefix_def) apply (erule exE)+ apply (simp add: append_eq_append_conv_if split: if_splits) apply (rule disjI2) apply (rule_tac x = "drop (size xs2) xs1" in exI) apply clarify apply (drule sym) apply (insert append_take_drop_id [of "length xs2" xs1]) apply simp apply (rule disjI1) apply (rule_tac x = "drop (size xs1) xs2" in exI) apply clarify apply (insert append_take_drop_id [of "length xs1" xs2]) apply simp done lemma set_mono_prefix: "xs ≤ ys ==> set xs ⊆ set ys" by (auto simp add: prefix_def) subsection {* Parallel lists *} constdefs parallel :: "'a list => 'a list => bool" (infixl "\<parallel>" 50) "xs \<parallel> ys == ¬ xs ≤ ys ∧ ¬ ys ≤ xs" lemma parallelI [intro]: "¬ xs ≤ ys ==> ¬ ys ≤ xs ==> xs \<parallel> ys" by (unfold parallel_def) blast lemma parallelE [elim]: "xs \<parallel> ys ==> (¬ xs ≤ ys ==> ¬ ys ≤ xs ==> C) ==> C" by (unfold parallel_def) blast theorem prefix_cases: "(xs ≤ ys ==> C) ==> (ys < xs ==> C) ==> (xs \<parallel> ys ==> C) ==> C" by (unfold parallel_def strict_prefix_def) blast theorem parallel_decomp: "xs \<parallel> ys ==> ∃as b bs c cs. b ≠ c ∧ xs = as @ b # bs ∧ ys = as @ c # cs" proof (induct xs rule: rev_induct) case Nil hence False by auto thus ?case .. next case (snoc x xs) show ?case proof (rule prefix_cases) assume le: "xs ≤ ys" then obtain ys' where ys: "ys = xs @ ys'" .. show ?thesis proof (cases ys') assume "ys' = []" with ys have "xs = ys" by simp with snoc have "[x] \<parallel> []" by auto hence False by blast thus ?thesis .. next fix c cs assume ys': "ys' = c # cs" with snoc ys have "xs @ [x] \<parallel> xs @ c # cs" by (simp only:) hence "x ≠ 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 < xs" hence "ys ≤ xs @ [x]" by (simp add: strict_prefix_def) with snoc have False by blast thus ?thesis .. next assume "xs \<parallel> ys" with snoc obtain as b bs c cs where neq: "(b::'a) ≠ 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 subsection {* Postfix order on lists *} constdefs postfix :: "'a list => 'a list => bool" ("(_/ >>= _)" [51, 50] 50) "xs >>= ys == ∃zs. xs = zs @ ys" lemma postfix_refl [simp, intro!]: "xs >>= xs" by (auto simp add: postfix_def) lemma postfix_trans: "[|xs >>= ys; ys >>= zs|] ==> xs >>= zs" by (auto simp add: postfix_def) lemma postfix_antisym: "[|xs >>= ys; ys >>= xs|] ==> xs = ys" by (auto simp add: postfix_def) lemma Nil_postfix [iff]: "xs >>= []" by (simp add: postfix_def) lemma postfix_Nil [simp]: "([] >>= xs) = (xs = [])" by (auto simp add:postfix_def) lemma postfix_ConsI: "xs >>= ys ==> x#xs >>= ys" by (auto simp add: postfix_def) lemma postfix_ConsD: "xs >>= y#ys ==> xs >>= ys" by (auto simp add: postfix_def) lemma postfix_appendI: "xs >>= ys ==> zs @ xs >>= ys" by (auto simp add: postfix_def) lemma postfix_appendD: "xs >>= zs @ ys ==> xs >>= ys" by(auto simp add: postfix_def) lemma postfix_is_subset_lemma: "xs = zs @ ys ==> set ys ⊆ set xs" by (induct zs, auto) lemma postfix_is_subset: "xs >>= ys ==> set ys ⊆ set xs" by (unfold postfix_def, erule exE, erule postfix_is_subset_lemma) lemma postfix_ConsD2_lemma [rule_format]: "x#xs = zs @ y#ys --> xs >>= ys" by (induct zs, auto intro!: postfix_appendI postfix_ConsI) lemma postfix_ConsD2: "x#xs >>= y#ys ==> xs >>= ys" by (auto simp add: postfix_def dest!: postfix_ConsD2_lemma) lemma postfix2prefix: "(xs >>= ys) = (rev ys <= rev xs)" apply (unfold prefix_def postfix_def, safe) apply (rule_tac x = "rev zs" in exI, simp) apply (rule_tac x = "rev zs" in exI) apply (rule rev_is_rev_conv [THEN iffD1], simp) done end
lemma prefixI:
ys = xs @ zs ==> xs ≤ ys
lemma prefixE:
xs ≤ ys ==> (!!zs. ys = xs @ zs ==> C) ==> C
lemma strict_prefixI':
ys = xs @ z # zs ==> xs < ys
lemma strict_prefixE':
xs < ys ==> (!!z zs. ys = xs @ z # zs ==> C) ==> C
lemma strict_prefixI:
xs ≤ ys ==> xs ≠ ys ==> xs < ys
lemma strict_prefixE:
xs < ys ==> (xs ≤ ys ==> xs ≠ ys ==> C) ==> C
theorem Nil_prefix:
[] ≤ xs
theorem prefix_Nil:
(xs ≤ []) = (xs = [])
lemma prefix_snoc:
(xs ≤ ys @ [y]) = (xs = ys @ [y] ∨ xs ≤ ys)
lemma Cons_prefix_Cons:
(x # xs ≤ y # ys) = (x = y ∧ xs ≤ ys)
lemma same_prefix_prefix:
(xs @ ys ≤ xs @ zs) = (ys ≤ zs)
lemma same_prefix_nil:
(xs @ ys ≤ xs) = (ys = [])
lemma prefix_prefix:
xs ≤ ys ==> xs ≤ ys @ zs
lemma append_prefixD:
xs @ ys ≤ zs ==> xs ≤ zs
theorem prefix_Cons:
(xs ≤ y # ys) = (xs = [] ∨ (∃zs. xs = y # zs ∧ zs ≤ ys))
theorem prefix_append:
(xs ≤ ys @ zs) = (xs ≤ ys ∨ (∃us. xs = ys @ us ∧ us ≤ zs))
lemma append_one_prefix:
xs ≤ ys ==> length xs < length ys ==> xs @ [ys ! length xs] ≤ ys
theorem prefix_length_le:
xs ≤ ys ==> length xs ≤ length ys
lemma prefix_same_cases:
xs1 ≤ ys ==> xs2 ≤ ys ==> xs1 ≤ xs2 ∨ xs2 ≤ xs1
lemma set_mono_prefix:
xs ≤ ys ==> set xs ⊆ set ys
lemma parallelI:
¬ xs ≤ ys ==> ¬ ys ≤ xs ==> xs \<parallel> ys
lemma parallelE:
xs \<parallel> ys ==> (¬ xs ≤ ys ==> ¬ ys ≤ xs ==> C) ==> C
theorem prefix_cases:
(xs ≤ ys ==> C) ==> (ys < xs ==> C) ==> (xs \<parallel> ys ==> C) ==> C
theorem parallel_decomp:
xs \<parallel> ys ==> ∃as b bs c cs. b ≠ c ∧ xs = as @ b # bs ∧ ys = as @ c # cs
lemma postfix_refl:
xs >>= xs
lemma postfix_trans:
xs >>= ys ==> ys >>= zs ==> xs >>= zs
lemma postfix_antisym:
xs >>= ys ==> ys >>= xs ==> xs = ys
lemma Nil_postfix:
xs >>= []
lemma postfix_Nil:
([] >>= xs) = (xs = [])
lemma postfix_ConsI:
xs >>= ys ==> x # xs >>= ys
lemma postfix_ConsD:
xs >>= y # ys ==> xs >>= ys
lemma postfix_appendI:
xs >>= ys ==> zs @ xs >>= ys
lemma postfix_appendD:
xs >>= zs @ ys ==> xs >>= ys
lemma postfix_is_subset_lemma:
xs = zs @ ys ==> set ys ⊆ set xs
lemma postfix_is_subset:
xs >>= ys ==> set ys ⊆ set xs
lemma postfix_ConsD2_lemma:
x # xs = zs @ y # ys ==> xs >>= ys
lemma postfix_ConsD2:
x # xs >>= y # ys ==> xs >>= ys
lemma postfix2prefix:
(xs >>= ys) = (rev ys ≤ rev xs)