(* Title: HOL/Library/List_Prefix.thy ID: $Id: List_Prefix.thy,v 1.34 2008/03/27 18:04:39 haftmann Exp $ Author: Tobias Nipkow and Markus Wenzel, TU Muenchen *) header {* List prefixes and postfixes *} theory List_Prefix imports List begin subsection {* Prefix order on lists *} instantiation list :: (type) order begin definition prefix_def [code func del]: "xs ≤ ys = (∃zs. ys = xs @ zs)" definition strict_prefix_def [code func del]: "xs < ys = (xs ≤ ys ∧ xs ≠ (ys::'a list))" instance by intro_classes (auto simp add: prefix_def strict_prefix_def) end lemma prefixI [intro?]: "ys = xs @ zs ==> xs ≤ ys" unfolding prefix_def by blast lemma prefixE [elim?]: assumes "xs ≤ ys" obtains zs where "ys = xs @ zs" using assms unfolding prefix_def by blast lemma strict_prefixI' [intro?]: "ys = xs @ z # zs ==> xs < ys" unfolding strict_prefix_def prefix_def by blast lemma strict_prefixE' [elim?]: assumes "xs < ys" obtains z zs where "ys = xs @ z # zs" proof - from `xs < ys` obtain us where "ys = xs @ us" and "xs ≠ ys" unfolding strict_prefix_def prefix_def by blast with that show ?thesis by (auto simp add: neq_Nil_conv) qed lemma strict_prefixI [intro?]: "xs ≤ ys ==> xs ≠ ys ==> xs < (ys::'a list)" unfolding strict_prefix_def by blast lemma strict_prefixE [elim?]: fixes xs ys :: "'a list" assumes "xs < ys" obtains "xs ≤ ys" and "xs ≠ ys" using assms unfolding strict_prefix_def by 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" by (metis append_Nil2 butlast_append butlast_snoc prefixI zs) next assume "xs = ys @ [y] ∨ xs ≤ ys" then show "xs ≤ ys @ [y]" by (metis order_eq_iff strict_prefixE strict_prefixI' xt1(7)) 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 = [])" by (metis append_Nil2 append_self_conv order_eq_iff prefixI) lemma prefix_prefix [simp]: "xs ≤ ys ==> xs ≤ ys @ zs" by (metis order_le_less_trans prefixI strict_prefixE strict_prefixI) 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 (metis append_eq_appendI) done lemma append_one_prefix: "xs ≤ ys ==> length xs < length ys ==> xs @ [ys ! length xs] ≤ ys" unfolding prefix_def by (metis Cons_eq_appendI append_eq_appendI append_eq_conv_conj eq_Nil_appendI nth_drop') 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" unfolding prefix_def by (metis append_eq_append_conv2) lemma set_mono_prefix: "xs ≤ ys ==> set xs ⊆ set ys" by (auto simp add: prefix_def) lemma take_is_prefix: "take n xs ≤ xs" unfolding prefix_def by (metis append_take_drop_id) lemma map_prefixI: "xs ≤ ys ==> map f xs ≤ map f ys" by (auto simp: prefix_def) lemma prefix_length_less: "xs < ys ==> length xs < length ys" by (auto simp: strict_prefix_def prefix_def) lemma strict_prefix_simps [simp]: "xs < [] = False" "[] < (x # xs) = True" "(x # xs) < (y # ys) = (x = y ∧ xs < ys)" by (simp_all add: strict_prefix_def cong: conj_cong) lemma take_strict_prefix: "xs < ys ==> take n xs < ys" apply (induct n arbitrary: xs ys) apply (case_tac ys, simp_all)[1] apply (metis order_less_trans strict_prefixI take_is_prefix) done lemma not_prefix_cases: assumes pfx: "¬ ps ≤ ls" obtains (c1) "ps ≠ []" and "ls = []" | (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "¬ as ≤ xs" | (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x ≠ a" proof (cases ps) case Nil then show ?thesis using pfx by simp next case (Cons a as) note c = `ps = a#as` show ?thesis proof (cases ls) case Nil then show ?thesis by (metis append_Nil2 pfx c1 same_prefix_nil) next case (Cons x xs) show ?thesis proof (cases "x = a") case True have "¬ as ≤ xs" using pfx c Cons True by simp with c Cons True show ?thesis by (rule c2) next case False with c Cons show ?thesis by (rule c3) qed qed qed lemma not_prefix_induct [consumes 1, case_names Nil Neq Eq]: assumes np: "¬ ps ≤ ls" and base: "!!x xs. P (x#xs) []" and r1: "!!x xs y ys. x ≠ y ==> P (x#xs) (y#ys)" and r2: "!!x xs y ys. [| x = y; ¬ xs ≤ ys; P xs ys |] ==> P (x#xs) (y#ys)" shows "P ps ls" using np proof (induct ls arbitrary: ps) case Nil then show ?case by (auto simp: neq_Nil_conv elim!: not_prefix_cases intro!: base) next case (Cons y ys) then have npfx: "¬ ps ≤ (y # ys)" by simp then obtain x xs where pv: "ps = x # xs" by (rule not_prefix_cases) auto show ?case by (metis Cons.hyps Cons_prefix_Cons npfx pv r1 r2) qed subsection {* Parallel lists *} definition parallel :: "'a list => 'a list => bool" (infixl "\<parallel>" 50) where "(xs \<parallel> ys) = (¬ xs ≤ ys ∧ ¬ ys ≤ xs)" lemma parallelI [intro]: "¬ xs ≤ ys ==> ¬ ys ≤ xs ==> xs \<parallel> ys" unfolding parallel_def by blast lemma parallelE [elim]: assumes "xs \<parallel> ys" obtains "¬ xs ≤ ys ∧ ¬ ys ≤ xs" using assms unfolding parallel_def by blast theorem prefix_cases: obtains "xs ≤ ys" | "ys < xs" | "xs \<parallel> ys" unfolding parallel_def strict_prefix_def by 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 then have False by auto then show ?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' = []" then show ?thesis by (metis append_Nil2 parallelE prefixI snoc.prems ys) next fix c cs assume ys': "ys' = c # cs" then show ?thesis by (metis Cons_eq_appendI eq_Nil_appendI parallelE prefixI same_prefix_prefix snoc.prems ys) qed next assume "ys < xs" then have "ys ≤ xs @ [x]" by (simp add: strict_prefix_def) with snoc have False by blast then show ?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 lemma parallel_append: "a \<parallel> b ==> a @ c \<parallel> b @ d" apply (rule parallelI) apply (erule parallelE, erule conjE, induct rule: not_prefix_induct, simp+)+ done lemma parallel_appendI: "xs \<parallel> ys ==> x = xs @ xs' ==> y = ys @ ys' ==> x \<parallel> y" by (simp add: parallel_append) lemma parallel_commute: "a \<parallel> b <-> b \<parallel> a" unfolding parallel_def by auto subsection {* Postfix order on lists *} definition postfix :: "'a list => 'a list => bool" ("(_/ >>= _)" [51, 50] 50) where "(xs >>= ys) = (∃zs. xs = zs @ ys)" lemma postfixI [intro?]: "xs = zs @ ys ==> xs >>= ys" unfolding postfix_def by blast lemma postfixE [elim?]: assumes "xs >>= ys" obtains zs where "xs = zs @ ys" using assms unfolding postfix_def by blast lemma postfix_refl [iff]: "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: "xs >>= ys ==> set ys ⊆ set xs" proof - assume "xs >>= ys" then obtain zs where "xs = zs @ ys" .. then show ?thesis by (induct zs) auto qed lemma postfix_ConsD2: "x#xs >>= y#ys ==> xs >>= ys" proof - assume "x#xs >>= y#ys" then obtain zs where "x#xs = zs @ y#ys" .. then show ?thesis by (induct zs) (auto intro!: postfix_appendI postfix_ConsI) qed lemma postfix_to_prefix: "xs >>= ys <-> rev ys ≤ rev xs" proof assume "xs >>= ys" then obtain zs where "xs = zs @ ys" .. then have "rev xs = rev ys @ rev zs" by simp then show "rev ys <= rev xs" .. next assume "rev ys <= rev xs" then obtain zs where "rev xs = rev ys @ zs" .. then have "rev (rev xs) = rev zs @ rev (rev ys)" by simp then have "xs = rev zs @ ys" by simp then show "xs >>= ys" .. qed lemma distinct_postfix: "distinct xs ==> xs >>= ys ==> distinct ys" by (clarsimp elim!: postfixE) lemma postfix_map: "xs >>= ys ==> map f xs >>= map f ys" by (auto elim!: postfixE intro: postfixI) lemma postfix_drop: "as >>= drop n as" unfolding postfix_def apply (rule exI [where x = "take n as"]) apply simp done lemma postfix_take: "xs >>= ys ==> xs = take (length xs - length ys) xs @ ys" by (clarsimp elim!: postfixE) lemma parallelD1: "x \<parallel> y ==> ¬ x ≤ y" by blast lemma parallelD2: "x \<parallel> y ==> ¬ y ≤ x" by blast lemma parallel_Nil1 [simp]: "¬ x \<parallel> []" unfolding parallel_def by simp lemma parallel_Nil2 [simp]: "¬ [] \<parallel> x" unfolding parallel_def by simp lemma Cons_parallelI1: "a ≠ b ==> a # as \<parallel> b # bs" by auto lemma Cons_parallelI2: "[| a = b; as \<parallel> bs |] ==> a # as \<parallel> b # bs" by (metis Cons_prefix_Cons parallelE parallelI) lemma not_equal_is_parallel: assumes neq: "xs ≠ ys" and len: "length xs = length ys" shows "xs \<parallel> ys" using len neq proof (induct rule: list_induct2) case Nil then show ?case by simp next case (Cons a as b bs) have ih: "as ≠ bs ==> as \<parallel> bs" by fact show ?case proof (cases "a = b") case True then have "as ≠ bs" using Cons by simp then show ?thesis by (rule Cons_parallelI2 [OF True ih]) next case False then show ?thesis by (rule Cons_parallelI1) qed qed subsection {* Executable code *} lemma less_eq_code [code func]: "([]::'a::{eq, ord} list) ≤ xs <-> True" "(x::'a::{eq, ord}) # xs ≤ [] <-> False" "(x::'a::{eq, ord}) # xs ≤ y # ys <-> x = y ∧ xs ≤ ys" by simp_all lemma less_code [code func]: "xs < ([]::'a::{eq, ord} list) <-> False" "[] < (x::'a::{eq, ord})# xs <-> True" "(x::'a::{eq, ord}) # xs < y # ys <-> x = y ∧ xs < ys" unfolding strict_prefix_def by auto lemmas [code func] = postfix_to_prefix end
lemma prefixI:
ys = xs @ zs ==> xs ≤ ys
lemma prefixE:
[| xs ≤ ys; !!zs. ys = xs @ zs ==> thesis |] ==> thesis
lemma strict_prefixI':
ys = xs @ z # zs ==> xs < ys
lemma strict_prefixE':
[| xs < ys; !!z zs. ys = xs @ z # zs ==> thesis |] ==> thesis
lemma strict_prefixI:
[| xs ≤ ys; xs ≠ ys |] ==> xs < ys
lemma strict_prefixE:
[| xs < ys; [| xs ≤ ys; xs ≠ ys |] ==> thesis |] ==> thesis
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 take_is_prefix:
take n xs ≤ xs
lemma map_prefixI:
xs ≤ ys ==> map f xs ≤ map f ys
lemma prefix_length_less:
xs < ys ==> length xs < length ys
lemma strict_prefix_simps:
(xs < []) = False
([] < x # xs) = True
(x # xs < y # ys) = (x = y ∧ xs < ys)
lemma take_strict_prefix:
xs < ys ==> take n xs < ys
lemma not_prefix_cases:
[| ¬ ps ≤ ls; [| ps ≠ []; ls = [] |] ==> thesis;
!!a as x xs. [| ps = a # as; ls = x # xs; x = a; ¬ as ≤ xs |] ==> thesis;
!!a as x xs. [| ps = a # as; ls = x # xs; x ≠ a |] ==> thesis |]
==> thesis
lemma not_prefix_induct:
[| ¬ ps ≤ ls; !!x xs. P (x # xs) []; !!x xs y ys. x ≠ y ==> P (x # xs) (y # ys);
!!x xs y ys. [| x = y; ¬ xs ≤ ys; P xs ys |] ==> P (x # xs) (y # ys) |]
==> P ps ls
lemma parallelI:
[| ¬ xs ≤ ys; ¬ ys ≤ xs |] ==> xs \<parallel> ys
lemma parallelE:
[| xs \<parallel> ys; ¬ xs ≤ ys ∧ ¬ ys ≤ xs ==> thesis |] ==> thesis
theorem prefix_cases:
[| xs ≤ ys ==> thesis; ys < xs ==> thesis; xs \<parallel> ys ==> thesis |]
==> thesis
theorem parallel_decomp:
xs \<parallel> ys ==> ∃as b bs c cs. b ≠ c ∧ xs = as @ b # bs ∧ ys = as @ c # cs
lemma parallel_append:
a \<parallel> b ==> a @ c \<parallel> b @ d
lemma parallel_appendI:
[| xs \<parallel> ys; x = xs @ xs'; y = ys @ ys' |] ==> x \<parallel> y
lemma parallel_commute:
(a \<parallel> b) = (b \<parallel> a)
lemma postfixI:
xs = zs @ ys ==> xs >>= ys
lemma postfixE:
[| xs >>= ys; !!zs. xs = zs @ ys ==> thesis |] ==> thesis
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:
xs >>= ys ==> set ys ⊆ set xs
lemma postfix_ConsD2:
x # xs >>= y # ys ==> xs >>= ys
lemma postfix_to_prefix:
(xs >>= ys) = (rev ys ≤ rev xs)
lemma distinct_postfix:
[| distinct xs; xs >>= ys |] ==> distinct ys
lemma postfix_map:
xs >>= ys ==> map f xs >>= map f ys
lemma postfix_drop:
as >>= drop n as
lemma postfix_take:
xs >>= ys ==> xs = take (length xs - length ys) xs @ ys
lemma parallelD1:
x \<parallel> y ==> ¬ x ≤ y
lemma parallelD2:
x \<parallel> y ==> ¬ y ≤ x
lemma parallel_Nil1:
¬ x \<parallel> []
lemma parallel_Nil2:
¬ [] \<parallel> x
lemma Cons_parallelI1:
a ≠ b ==> a # as \<parallel> b # bs
lemma Cons_parallelI2:
[| a = b; as \<parallel> bs |] ==> a # as \<parallel> b # bs
lemma not_equal_is_parallel:
[| xs ≠ ys; length xs = length ys |] ==> xs \<parallel> ys
lemma less_eq_code:
([] ≤ xs) = True
(x # xs ≤ []) = False
(x # xs ≤ y # ys) = (x = y ∧ xs ≤ ys)
lemma less_code:
(xs < []) = False
([] < x # xs) = True
(x # xs < y # ys) = (x = y ∧ xs < ys)
lemma
(xs >>= ys) = (rev ys ≤ rev xs)