author | berghofe |
Wed, 11 Jul 2007 11:14:51 +0200 | |
changeset 23746 | a455e69c31cc |
parent 21404 | eb85850d3eb7 |
child 26793 | e36a92ff543e |
permissions | -rw-r--r-- |
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(* Title: HOL/Induct/LList.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Shares NIL, CONS, List_case with List.thy |
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Still needs flatten function -- hard because it need |
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bounds on the amount of lookahead required. |
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Could try (but would it work for the gfp analogue of term?) |
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LListD_Fun_def "LListD_Fun(A) == (%Z. diag({Numb(0)}) <++> diag(A) <**> Z)" |
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A nice but complex example would be [ML for the Working Programmer, page 176] |
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from(1) = enumerate (Lmap (Lmap(pack), makeqq(from(1),from(1)))) |
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Previous definition of llistD_Fun was explicit: |
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llistD_Fun_def |
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"llistD_Fun(r) == |
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{(LNil,LNil)} Un |
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(UN x. (split(%l1 l2.(LCons(x,l1),LCons(x,l2))))`r)" |
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*) |
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header {*Definition of type llist by a greatest fixed point*} |
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theory LList imports SList begin |
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coinductive_set |
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llist :: "'a item set => 'a item set" |
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for A :: "'a item set" |
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where |
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NIL_I: "NIL \<in> llist(A)" |
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| CONS_I: "[| a \<in> A; M \<in> llist(A) |] ==> CONS a M \<in> llist(A)" |
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coinductive_set |
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LListD :: "('a item * 'a item)set => ('a item * 'a item)set" |
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for r :: "('a item * 'a item)set" |
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where |
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NIL_I: "(NIL, NIL) \<in> LListD(r)" |
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| CONS_I: "[| (a,b) \<in> r; (M,N) \<in> LListD(r) |] |
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==> (CONS a M, CONS b N) \<in> LListD(r)" |
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typedef (LList) |
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'a llist = "llist(range Leaf) :: 'a item set" |
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by (blast intro: llist.NIL_I) |
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list_Fun :: "['a item set, 'a item set] => 'a item set" where |
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--{*Now used exclusively for abbreviating the coinduction rule*} |
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"list_Fun A X = {z. z = NIL | (\<exists>M a. z = CONS a M & a \<in> A & M \<in> X)}" |
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LListD_Fun :: |
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"[('a item * 'a item)set, ('a item * 'a item)set] => |
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('a item * 'a item)set" where |
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"LListD_Fun r X = |
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{z. z = (NIL, NIL) | |
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(\<exists>M N a b. z = (CONS a M, CONS b N) & (a, b) \<in> r & (M, N) \<in> X)}" |
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LNil :: "'a llist" where |
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--{*abstract constructor*} |
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"LNil = Abs_LList NIL" |
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LCons :: "['a, 'a llist] => 'a llist" where |
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--{*abstract constructor*} |
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"LCons x xs = Abs_LList(CONS (Leaf x) (Rep_LList xs))" |
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llist_case :: "['b, ['a, 'a llist]=>'b, 'a llist] => 'b" where |
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"llist_case c d l = |
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List_case c (%x y. d (inv Leaf x) (Abs_LList y)) (Rep_LList l)" |
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LList_corec_fun :: "[nat, 'a=> ('b item * 'a) option, 'a] => 'b item" where |
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"LList_corec_fun k f == |
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nat_rec (%x. {}) |
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(%j r x. case f x of None => NIL |
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| Some(z,w) => CONS z (r w)) |
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k" |
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LList_corec :: "['a, 'a => ('b item * 'a) option] => 'b item" where |
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"LList_corec a f = (\<Union>k. LList_corec_fun k f a)" |
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llist_corec :: "['a, 'a => ('b * 'a) option] => 'b llist" where |
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"llist_corec a f = |
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Abs_LList(LList_corec a |
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(%z. case f z of None => None |
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| Some(v,w) => Some(Leaf(v), w)))" |
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llistD_Fun :: "('a llist * 'a llist)set => ('a llist * 'a llist)set" where |
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"llistD_Fun(r) = |
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prod_fun Abs_LList Abs_LList ` |
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LListD_Fun (diag(range Leaf)) |
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(prod_fun Rep_LList Rep_LList ` r)" |
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text{* The case syntax for type @{text "'a llist"} *} |
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syntax (* FIXME proper case syntax!? *) |
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LNil :: logic |
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LCons :: logic |
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translations |
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"case p of LNil => a | LCons x l => b" == "CONST llist_case a (%x l. b) p" |
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subsubsection{* Sample function definitions. Item-based ones start with @{text L} *} |
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definition |
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Lmap :: "('a item => 'b item) => ('a item => 'b item)" where |
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"Lmap f M = LList_corec M (List_case None (%x M'. Some((f(x), M'))))" |
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lmap :: "('a=>'b) => ('a llist => 'b llist)" where |
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"lmap f l = llist_corec l (%z. case z of LNil => None |
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| LCons y z => Some(f(y), z))" |
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iterates :: "['a => 'a, 'a] => 'a llist" where |
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"iterates f a = llist_corec a (%x. Some((x, f(x))))" |
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Lconst :: "'a item => 'a item" where |
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"Lconst(M) == lfp(%N. CONS M N)" |
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definition |
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Lappend :: "['a item, 'a item] => 'a item" where |
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"Lappend M N = LList_corec (M,N) |
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(split(List_case (List_case None (%N1 N2. Some((N1, (NIL,N2))))) |
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(%M1 M2 N. Some((M1, (M2,N))))))" |
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definition |
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lappend :: "['a llist, 'a llist] => 'a llist" where |
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"lappend l n = llist_corec (l,n) |
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(split(llist_case (llist_case None (%n1 n2. Some((n1, (LNil,n2))))) |
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(%l1 l2 n. Some((l1, (l2,n))))))" |
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text{*Append generates its result by applying f, where |
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f((NIL,NIL)) = None |
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f((NIL, CONS N1 N2)) = Some((N1, (NIL,N2)) |
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f((CONS M1 M2, N)) = Some((M1, (M2,N)) |
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*} |
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|
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text{* |
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SHOULD @{text LListD_Fun_CONS_I}, etc., be equations (for rewriting)? |
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*} |
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lemmas UN1_I = UNIV_I [THEN UN_I, standard] |
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|
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subsubsection{* Simplification *} |
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|
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declare option.split [split] |
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|
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text{*This justifies using llist in other recursive type definitions*} |
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lemma llist_mono: |
161 |
assumes subset: "A \<subseteq> B" |
|
162 |
shows "llist A \<subseteq> llist B" |
|
163 |
proof |
|
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fix x |
|
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assume "x \<in> llist A" |
|
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then show "x \<in> llist B" |
|
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proof coinduct |
|
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case llist |
|
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then show ?case using subset |
|
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by cases blast+ |
|
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qed |
|
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qed |
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|
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|
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lemma llist_unfold: "llist(A) = usum {Numb(0)} (uprod A (llist A))" |
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by (fast intro!: llist.intros [unfolded NIL_def CONS_def] |
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elim: llist.cases [unfolded NIL_def CONS_def]) |
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|
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|
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subsection{* Type checking by coinduction *} |
181 |
||
182 |
text {* |
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{\dots} using @{text list_Fun} THE COINDUCTIVE DEFINITION PACKAGE |
|
184 |
COULD DO THIS! |
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*} |
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|
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lemma llist_coinduct: |
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"[| M \<in> X; X \<subseteq> list_Fun A (X Un llist(A)) |] ==> M \<in> llist(A)" |
189 |
apply (simp add: list_Fun_def) |
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apply (erule llist.coinduct) |
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apply (blast intro: elim:); |
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192 |
done |
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193 |
|
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lemma list_Fun_NIL_I [iff]: "NIL \<in> list_Fun A X" |
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by (simp add: list_Fun_def NIL_def) |
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196 |
|
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lemma list_Fun_CONS_I [intro!,simp]: |
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"[| M \<in> A; N \<in> X |] ==> CONS M N \<in> list_Fun A X" |
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by (simp add: list_Fun_def CONS_def) |
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200 |
|
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|
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text{*Utilise the ``strong'' part, i.e. @{text "gfp(f)"}*} |
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203 |
lemma list_Fun_llist_I: "M \<in> llist(A) ==> M \<in> list_Fun A (X Un llist(A))" |
23746 | 204 |
apply (unfold list_Fun_def) |
205 |
apply (erule llist.cases) |
|
206 |
apply auto |
|
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207 |
done |
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208 |
|
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subsection{* @{text LList_corec} satisfies the desired recurion equation *} |
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210 |
|
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text{*A continuity result?*} |
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lemma CONS_UN1: "CONS M (\<Union>x. f(x)) = (\<Union>x. CONS M (f x))" |
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apply (simp add: CONS_def In1_UN1 Scons_UN1_y) |
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214 |
done |
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215 |
|
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lemma CONS_mono: "[| M\<subseteq>M'; N\<subseteq>N' |] ==> CONS M N \<subseteq> CONS M' N'" |
217 |
apply (simp add: CONS_def In1_mono Scons_mono) |
|
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218 |
done |
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219 |
|
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220 |
declare LList_corec_fun_def [THEN def_nat_rec_0, simp] |
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LList_corec_fun_def [THEN def_nat_rec_Suc, simp] |
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222 |
|
19736 | 223 |
|
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224 |
subsubsection{* The directions of the equality are proved separately *} |
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225 |
|
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226 |
lemma LList_corec_subset1: |
17841 | 227 |
"LList_corec a f \<subseteq> |
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(case f a of None => NIL | Some(z,w) => CONS z (LList_corec w f))" |
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229 |
apply (unfold LList_corec_def) |
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230 |
apply (rule UN_least) |
17841 | 231 |
apply (case_tac k) |
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apply (simp_all (no_asm_simp)) |
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233 |
apply (rule allI impI subset_refl [THEN CONS_mono] UNIV_I [THEN UN_upper])+ |
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234 |
done |
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235 |
|
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236 |
lemma LList_corec_subset2: |
17841 | 237 |
"(case f a of None => NIL | Some(z,w) => CONS z (LList_corec w f)) \<subseteq> |
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238 |
LList_corec a f" |
17841 | 239 |
apply (simp add: LList_corec_def) |
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240 |
apply (simp add: CONS_UN1, safe) |
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241 |
apply (rule_tac a="Suc(?k)" in UN_I, simp, simp)+ |
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242 |
done |
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243 |
|
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text{*the recursion equation for @{text LList_corec} -- NOT SUITABLE FOR REWRITING!*} |
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245 |
lemma LList_corec: |
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246 |
"LList_corec a f = |
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247 |
(case f a of None => NIL | Some(z,w) => CONS z (LList_corec w f))" |
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by (rule equalityI LList_corec_subset1 LList_corec_subset2)+ |
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249 |
|
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250 |
text{*definitional version of same*} |
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251 |
lemma def_LList_corec: |
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"[| !!x. h(x) = LList_corec x f |] |
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253 |
==> h(a) = (case f a of None => NIL | Some(z,w) => CONS z (h w))" |
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254 |
by (simp add: LList_corec) |
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255 |
|
13107 | 256 |
text{*A typical use of co-induction to show membership in the @{text gfp}. |
257 |
Bisimulation is @{text "range(%x. LList_corec x f)"} *} |
|
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258 |
lemma LList_corec_type: "LList_corec a f \<in> llist UNIV" |
17841 | 259 |
apply (rule_tac X = "range (%x. LList_corec x ?g)" in llist_coinduct) |
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260 |
apply (rule rangeI, safe) |
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261 |
apply (subst LList_corec, simp) |
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262 |
done |
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263 |
|
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264 |
|
13107 | 265 |
subsection{* @{text llist} equality as a @{text gfp}; the bisimulation principle *} |
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266 |
|
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267 |
text{*This theorem is actually used, unlike the many similar ones in ZF*} |
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268 |
lemma LListD_unfold: "LListD r = dsum (diag {Numb 0}) (dprod r (LListD r))" |
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by (fast intro!: LListD.intros [unfolded NIL_def CONS_def] |
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elim: LListD.cases [unfolded NIL_def CONS_def]) |
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271 |
|
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272 |
lemma LListD_implies_ntrunc_equality [rule_format]: |
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273 |
"\<forall>M N. (M,N) \<in> LListD(diag A) --> ntrunc k M = ntrunc k N" |
17841 | 274 |
apply (induct_tac "k" rule: nat_less_induct) |
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275 |
apply (safe del: equalityI) |
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276 |
apply (erule LListD.cases) |
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277 |
apply (safe del: equalityI) |
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278 |
apply (case_tac "n", simp) |
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279 |
apply (rename_tac "n'") |
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280 |
apply (case_tac "n'") |
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281 |
apply (simp_all add: CONS_def less_Suc_eq) |
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282 |
done |
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283 |
|
13107 | 284 |
text{*The domain of the @{text LListD} relation*} |
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285 |
lemma Domain_LListD: |
17841 | 286 |
"Domain (LListD(diag A)) \<subseteq> llist(A)" |
23746 | 287 |
apply (rule subsetI) |
288 |
apply (erule llist.coinduct) |
|
289 |
apply (simp add: NIL_def CONS_def) |
|
290 |
apply (drule_tac P = "%x. xa \<in> Domain x" in LListD_unfold [THEN subst], auto) |
|
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291 |
done |
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292 |
|
13107 | 293 |
text{*This inclusion justifies the use of coinduction to show @{text "M = N"}*} |
17841 | 294 |
lemma LListD_subset_diag: "LListD(diag A) \<subseteq> diag(llist(A))" |
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295 |
apply (rule subsetI) |
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apply (rule_tac p = x in PairE, safe) |
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297 |
apply (rule diag_eqI) |
17841 | 298 |
apply (rule LListD_implies_ntrunc_equality [THEN ntrunc_equality], assumption) |
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299 |
apply (erule DomainI [THEN Domain_LListD [THEN subsetD]]) |
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300 |
done |
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301 |
|
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302 |
|
13107 | 303 |
subsubsection{* Coinduction, using @{text LListD_Fun} *} |
304 |
||
305 |
text {* THE COINDUCTIVE DEFINITION PACKAGE COULD DO THIS! *} |
|
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306 |
|
17841 | 307 |
lemma LListD_Fun_mono: "A\<subseteq>B ==> LListD_Fun r A \<subseteq> LListD_Fun r B" |
308 |
apply (simp add: LListD_Fun_def) |
|
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309 |
apply (assumption | rule basic_monos)+ |
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|
310 |
done |
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|
311 |
|
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|
312 |
lemma LListD_coinduct: |
17841 | 313 |
"[| M \<in> X; X \<subseteq> LListD_Fun r (X Un LListD(r)) |] ==> M \<in> LListD(r)" |
23746 | 314 |
apply (cases M) |
17841 | 315 |
apply (simp add: LListD_Fun_def) |
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316 |
apply (erule LListD.coinduct) |
17841 | 317 |
apply (auto ); |
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|
318 |
done |
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|
319 |
|
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|
320 |
lemma LListD_Fun_NIL_I: "(NIL,NIL) \<in> LListD_Fun r s" |
17841 | 321 |
by (simp add: LListD_Fun_def NIL_def) |
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|
322 |
|
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|
323 |
lemma LListD_Fun_CONS_I: |
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|
324 |
"[| x\<in>A; (M,N):s |] ==> (CONS x M, CONS x N) \<in> LListD_Fun (diag A) s" |
17841 | 325 |
by (simp add: LListD_Fun_def CONS_def, blast) |
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|
326 |
|
13107 | 327 |
text{*Utilise the "strong" part, i.e. @{text "gfp(f)"}*} |
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|
328 |
lemma LListD_Fun_LListD_I: |
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|
329 |
"M \<in> LListD(r) ==> M \<in> LListD_Fun r (X Un LListD(r))" |
23746 | 330 |
apply (cases M) |
331 |
apply (simp add: LListD_Fun_def) |
|
332 |
apply (erule LListD.cases) |
|
333 |
apply auto |
|
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|
334 |
done |
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|
335 |
|
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|
336 |
|
13107 | 337 |
text{*This converse inclusion helps to strengthen @{text LList_equalityI}*} |
17841 | 338 |
lemma diag_subset_LListD: "diag(llist(A)) \<subseteq> LListD(diag A)" |
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339 |
apply (rule subsetI) |
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|
340 |
apply (erule LListD_coinduct) |
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|
341 |
apply (rule subsetI) |
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|
342 |
apply (erule diagE) |
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|
343 |
apply (erule ssubst) |
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|
344 |
apply (erule llist.cases) |
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|
345 |
apply (simp_all add: diagI LListD_Fun_NIL_I LListD_Fun_CONS_I) |
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|
346 |
done |
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|
347 |
|
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|
348 |
lemma LListD_eq_diag: "LListD(diag A) = diag(llist(A))" |
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|
349 |
apply (rule equalityI LListD_subset_diag diag_subset_LListD)+ |
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|
350 |
done |
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|
351 |
|
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|
352 |
lemma LListD_Fun_diag_I: "M \<in> llist(A) ==> (M,M) \<in> LListD_Fun (diag A) (X Un diag(llist(A)))" |
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|
353 |
apply (rule LListD_eq_diag [THEN subst]) |
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|
354 |
apply (rule LListD_Fun_LListD_I) |
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|
355 |
apply (simp add: LListD_eq_diag diagI) |
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|
356 |
done |
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|
357 |
|
d3e1d554cd6d
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|
358 |
|
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|
359 |
subsubsection{* To show two LLists are equal, exhibit a bisimulation! |
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360 |
[also admits true equality] |
13107 | 361 |
Replace @{text A} by some particular set, like @{text "{x. True}"}??? *} |
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|
362 |
lemma LList_equalityI: |
17841 | 363 |
"[| (M,N) \<in> r; r \<subseteq> LListD_Fun (diag A) (r Un diag(llist(A))) |] |
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364 |
==> M=N" |
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|
365 |
apply (rule LListD_subset_diag [THEN subsetD, THEN diagE]) |
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|
366 |
apply (erule LListD_coinduct) |
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|
367 |
apply (simp add: LListD_eq_diag, safe) |
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|
368 |
done |
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changeset
|
369 |
|
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|
370 |
|
13107 | 371 |
subsection{* Finality of @{text "llist(A)"}: Uniqueness of functions defined by corecursion *} |
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|
372 |
|
13107 | 373 |
text{*We must remove @{text Pair_eq} because it may turn an instance of reflexivity |
374 |
@{text "(h1 b, h2 b) = (h1 ?x17, h2 ?x17)"} into a conjunction! |
|
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|
375 |
(or strengthen the Solver?) |
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|
376 |
*} |
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|
377 |
declare Pair_eq [simp del] |
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|
378 |
|
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|
379 |
text{*abstract proof using a bisimulation*} |
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|
380 |
lemma LList_corec_unique: |
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|
381 |
"[| !!x. h1(x) = (case f x of None => NIL | Some(z,w) => CONS z (h1 w)); |
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|
382 |
!!x. h2(x) = (case f x of None => NIL | Some(z,w) => CONS z (h2 w)) |] |
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|
383 |
==> h1=h2" |
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|
384 |
apply (rule ext) |
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|
385 |
txt{*next step avoids an unknown (and flexflex pair) in simplification*} |
17841 | 386 |
apply (rule_tac A = UNIV and r = "range(%u. (h1 u, h2 u))" |
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|
387 |
in LList_equalityI) |
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|
388 |
apply (rule rangeI, safe) |
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|
389 |
apply (simp add: LListD_Fun_NIL_I UNIV_I [THEN LListD_Fun_CONS_I]) |
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|
390 |
done |
d3e1d554cd6d
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paulson
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changeset
|
391 |
|
d3e1d554cd6d
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|
392 |
lemma equals_LList_corec: |
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|
393 |
"[| !!x. h(x) = (case f x of None => NIL | Some(z,w) => CONS z (h w)) |] |
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|
394 |
==> h = (%x. LList_corec x f)" |
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|
395 |
by (simp add: LList_corec_unique LList_corec) |
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changeset
|
396 |
|
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changeset
|
397 |
|
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|
398 |
subsubsection{*Obsolete proof of @{text LList_corec_unique}: |
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|
399 |
complete induction, not coinduction *} |
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|
400 |
|
d3e1d554cd6d
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changeset
|
401 |
lemma ntrunc_one_CONS [simp]: "ntrunc (Suc 0) (CONS M N) = {}" |
d3e1d554cd6d
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|
402 |
by (simp add: CONS_def ntrunc_one_In1) |
d3e1d554cd6d
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paulson
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changeset
|
403 |
|
d3e1d554cd6d
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changeset
|
404 |
lemma ntrunc_CONS [simp]: |
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changeset
|
405 |
"ntrunc (Suc(Suc(k))) (CONS M N) = CONS (ntrunc k M) (ntrunc k N)" |
d3e1d554cd6d
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|
406 |
by (simp add: CONS_def) |
d3e1d554cd6d
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changeset
|
407 |
|
d3e1d554cd6d
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paulson
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changeset
|
408 |
|
d3e1d554cd6d
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|
409 |
lemma |
d3e1d554cd6d
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|
410 |
assumes prem1: |
d3e1d554cd6d
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changeset
|
411 |
"!!x. h1 x = (case f x of None => NIL | Some(z,w) => CONS z (h1 w))" |
d3e1d554cd6d
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|
412 |
and prem2: |
d3e1d554cd6d
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changeset
|
413 |
"!!x. h2 x = (case f x of None => NIL | Some(z,w) => CONS z (h2 w))" |
d3e1d554cd6d
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changeset
|
414 |
shows "h1=h2" |
d3e1d554cd6d
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paulson
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changeset
|
415 |
apply (rule ntrunc_equality [THEN ext]) |
17841 | 416 |
apply (rule_tac x = x in spec) |
13075
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changeset
|
417 |
apply (induct_tac "k" rule: nat_less_induct) |
d3e1d554cd6d
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paulson
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changeset
|
418 |
apply (rename_tac "n") |
d3e1d554cd6d
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paulson
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10834
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changeset
|
419 |
apply (rule allI) |
d3e1d554cd6d
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paulson
parents:
10834
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changeset
|
420 |
apply (subst prem1) |
d3e1d554cd6d
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paulson
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changeset
|
421 |
apply (subst prem2, simp) |
d3e1d554cd6d
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paulson
parents:
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changeset
|
422 |
apply (intro strip) |
d3e1d554cd6d
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paulson
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10834
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changeset
|
423 |
apply (case_tac "n") |
d3e1d554cd6d
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paulson
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changeset
|
424 |
apply (rename_tac [2] "m") |
17841 | 425 |
apply (case_tac [2] "m", simp_all) |
13075
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changeset
|
426 |
done |
d3e1d554cd6d
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paulson
parents:
10834
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changeset
|
427 |
|
d3e1d554cd6d
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changeset
|
428 |
|
13107 | 429 |
subsection{*Lconst: defined directly by @{text lfp} *} |
13075
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|
430 |
|
d3e1d554cd6d
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|
431 |
text{*But it could be defined by corecursion.*} |
d3e1d554cd6d
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changeset
|
432 |
|
d3e1d554cd6d
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|
433 |
lemma Lconst_fun_mono: "mono(CONS(M))" |
d3e1d554cd6d
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|
434 |
apply (rule monoI subset_refl CONS_mono)+ |
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paulson
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changeset
|
435 |
apply assumption |
d3e1d554cd6d
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paulson
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changeset
|
436 |
done |
d3e1d554cd6d
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changeset
|
437 |
|
13107 | 438 |
text{* @{text "Lconst(M) = CONS M (Lconst M)"} *} |
13075
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|
439 |
lemmas Lconst = Lconst_fun_mono [THEN Lconst_def [THEN def_lfp_unfold]] |
d3e1d554cd6d
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changeset
|
440 |
|
d3e1d554cd6d
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paulson
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changeset
|
441 |
text{*A typical use of co-induction to show membership in the gfp. |
13107 | 442 |
The containing set is simply the singleton @{text "{Lconst(M)}"}. *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
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changeset
|
443 |
lemma Lconst_type: "M\<in>A ==> Lconst(M): llist(A)" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
444 |
apply (rule singletonI [THEN llist_coinduct], safe) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
445 |
apply (rule_tac P = "%u. u \<in> ?A" in Lconst [THEN ssubst]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
446 |
apply (assumption | rule list_Fun_CONS_I singletonI UnI1)+ |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
447 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
448 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
449 |
lemma Lconst_eq_LList_corec: "Lconst(M) = LList_corec M (%x. Some(x,x))" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
450 |
apply (rule equals_LList_corec [THEN fun_cong], simp) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
451 |
apply (rule Lconst) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
452 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
453 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
454 |
text{*Thus we could have used gfp in the definition of Lconst*} |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
455 |
lemma gfp_Lconst_eq_LList_corec: "gfp(%N. CONS M N) = LList_corec M (%x. Some(x,x))" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
456 |
apply (rule equals_LList_corec [THEN fun_cong], simp) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
457 |
apply (rule Lconst_fun_mono [THEN gfp_unfold]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
458 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
459 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
460 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
461 |
subsection{* Isomorphisms *} |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
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changeset
|
462 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
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changeset
|
463 |
lemma LListI: "x \<in> llist (range Leaf) ==> x \<in> LList" |
17841 | 464 |
by (simp add: LList_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
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changeset
|
465 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
466 |
lemma LListD: "x \<in> LList ==> x \<in> llist (range Leaf)" |
17841 | 467 |
by (simp add: LList_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
468 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
469 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
470 |
subsubsection{* Distinctness of constructors *} |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
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changeset
|
471 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
472 |
lemma LCons_not_LNil [iff]: "~ LCons x xs = LNil" |
17841 | 473 |
apply (simp add: LNil_def LCons_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
474 |
apply (subst Abs_LList_inject) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
475 |
apply (rule llist.intros CONS_not_NIL rangeI LListI Rep_LList [THEN LListD])+ |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
476 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
477 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
478 |
lemmas LNil_not_LCons [iff] = LCons_not_LNil [THEN not_sym, standard] |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
479 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
480 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
481 |
subsubsection{* llist constructors *} |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
482 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
483 |
lemma Rep_LList_LNil: "Rep_LList LNil = NIL" |
17841 | 484 |
apply (simp add: LNil_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
485 |
apply (rule llist.NIL_I [THEN LListI, THEN Abs_LList_inverse]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
486 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
487 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
488 |
lemma Rep_LList_LCons: "Rep_LList(LCons x l) = CONS (Leaf x) (Rep_LList l)" |
17841 | 489 |
apply (simp add: LCons_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
490 |
apply (rule llist.CONS_I [THEN LListI, THEN Abs_LList_inverse] |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
491 |
rangeI Rep_LList [THEN LListD])+ |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
492 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
493 |
|
13107 | 494 |
subsubsection{* Injectiveness of @{text CONS} and @{text LCons} *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
495 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
496 |
lemma CONS_CONS_eq2: "(CONS M N=CONS M' N') = (M=M' & N=N')" |
17841 | 497 |
apply (simp add: CONS_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
498 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
499 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
500 |
lemmas CONS_inject = CONS_CONS_eq [THEN iffD1, THEN conjE, standard] |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
501 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
502 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
503 |
text{*For reasoning about abstract llist constructors*} |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
504 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
505 |
declare Rep_LList [THEN LListD, intro] LListI [intro] |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
506 |
declare llist.intros [intro] |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
507 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
508 |
lemma LCons_LCons_eq [iff]: "(LCons x xs=LCons y ys) = (x=y & xs=ys)" |
17841 | 509 |
apply (simp add: LCons_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
510 |
apply (subst Abs_LList_inject) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
511 |
apply (auto simp add: Rep_LList_inject) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
512 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
513 |
|
13524 | 514 |
lemma CONS_D2: "CONS M N \<in> llist(A) ==> M \<in> A & N \<in> llist(A)" |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
515 |
apply (erule llist.cases) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
516 |
apply (erule CONS_neq_NIL, fast) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
517 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
518 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
519 |
|
13107 | 520 |
subsection{* Reasoning about @{text "llist(A)"} *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
521 |
|
13107 | 522 |
text{*A special case of @{text list_equality} for functions over lazy lists*} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
523 |
lemma LList_fun_equalityI: |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
524 |
"[| M \<in> llist(A); g(NIL): llist(A); |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
525 |
f(NIL)=g(NIL); |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
526 |
!!x l. [| x\<in>A; l \<in> llist(A) |] ==> |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
527 |
(f(CONS x l),g(CONS x l)) \<in> |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
528 |
LListD_Fun (diag A) ((%u.(f(u),g(u)))`llist(A) Un |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
529 |
diag(llist(A))) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
530 |
|] ==> f(M) = g(M)" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
531 |
apply (rule LList_equalityI) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
532 |
apply (erule imageI) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
533 |
apply (rule image_subsetI) |
23746 | 534 |
apply (erule_tac a=x in llist.cases) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
535 |
apply (erule ssubst, erule ssubst, erule LListD_Fun_diag_I, blast) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
536 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
537 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
538 |
|
13107 | 539 |
subsection{* The functional @{text Lmap} *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
540 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
541 |
lemma Lmap_NIL [simp]: "Lmap f NIL = NIL" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
542 |
by (rule Lmap_def [THEN def_LList_corec, THEN trans], simp) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
543 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
544 |
lemma Lmap_CONS [simp]: "Lmap f (CONS M N) = CONS (f M) (Lmap f N)" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
545 |
by (rule Lmap_def [THEN def_LList_corec, THEN trans], simp) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
546 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
547 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
548 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
549 |
text{*Another type-checking proof by coinduction*} |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
550 |
lemma Lmap_type: |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
551 |
"[| M \<in> llist(A); !!x. x\<in>A ==> f(x):B |] ==> Lmap f M \<in> llist(B)" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
552 |
apply (erule imageI [THEN llist_coinduct], safe) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
553 |
apply (erule llist.cases, simp_all) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
554 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
555 |
|
13107 | 556 |
text{*This type checking rule synthesises a sufficiently large set for @{text f}*} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
557 |
lemma Lmap_type2: "M \<in> llist(A) ==> Lmap f M \<in> llist(f`A)" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
558 |
apply (erule Lmap_type) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
559 |
apply (erule imageI) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
560 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
561 |
|
13107 | 562 |
subsubsection{* Two easy results about @{text Lmap} *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
563 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
564 |
lemma Lmap_compose: "M \<in> llist(A) ==> Lmap (f o g) M = Lmap f (Lmap g M)" |
17841 | 565 |
apply (simp add: o_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
566 |
apply (drule imageI) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
567 |
apply (erule LList_equalityI, safe) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
568 |
apply (erule llist.cases, simp_all) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
569 |
apply (rule LListD_Fun_NIL_I imageI UnI1 rangeI [THEN LListD_Fun_CONS_I])+ |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
570 |
apply assumption |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
571 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
572 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
573 |
lemma Lmap_ident: "M \<in> llist(A) ==> Lmap (%x. x) M = M" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
574 |
apply (drule imageI) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
575 |
apply (erule LList_equalityI, safe) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
576 |
apply (erule llist.cases, simp_all) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
577 |
apply (rule LListD_Fun_NIL_I imageI UnI1 rangeI [THEN LListD_Fun_CONS_I])+ |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
578 |
apply assumption |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
579 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
580 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
581 |
|
13107 | 582 |
subsection{* @{text Lappend} -- its two arguments cause some complications! *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
583 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
584 |
lemma Lappend_NIL_NIL [simp]: "Lappend NIL NIL = NIL" |
17841 | 585 |
apply (simp add: Lappend_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
586 |
apply (rule LList_corec [THEN trans], simp) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
587 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
588 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
589 |
lemma Lappend_NIL_CONS [simp]: |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
590 |
"Lappend NIL (CONS N N') = CONS N (Lappend NIL N')" |
17841 | 591 |
apply (simp add: Lappend_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
592 |
apply (rule LList_corec [THEN trans], simp) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
593 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
594 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
595 |
lemma Lappend_CONS [simp]: |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
596 |
"Lappend (CONS M M') N = CONS M (Lappend M' N)" |
17841 | 597 |
apply (simp add: Lappend_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
598 |
apply (rule LList_corec [THEN trans], simp) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
599 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
600 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
601 |
declare llist.intros [simp] LListD_Fun_CONS_I [simp] |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
602 |
range_eqI [simp] image_eqI [simp] |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
603 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
604 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
605 |
lemma Lappend_NIL [simp]: "M \<in> llist(A) ==> Lappend NIL M = M" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
606 |
by (erule LList_fun_equalityI, simp_all) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
607 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
608 |
lemma Lappend_NIL2: "M \<in> llist(A) ==> Lappend M NIL = M" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
609 |
by (erule LList_fun_equalityI, simp_all) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
610 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
611 |
|
13107 | 612 |
subsubsection{* Alternative type-checking proofs for @{text Lappend} *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
613 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
614 |
text{*weak co-induction: bisimulation and case analysis on both variables*} |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
615 |
lemma Lappend_type: "[| M \<in> llist(A); N \<in> llist(A) |] ==> Lappend M N \<in> llist(A)" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
616 |
apply (rule_tac X = "\<Union>u\<in>llist (A) . \<Union>v \<in> llist (A) . {Lappend u v}" in llist_coinduct) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
617 |
apply fast |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
618 |
apply safe |
23746 | 619 |
apply (erule_tac a = u in llist.cases) |
620 |
apply (erule_tac a = v in llist.cases, simp_all, blast) |
|
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
621 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
622 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
623 |
text{*strong co-induction: bisimulation and case analysis on one variable*} |
13524 | 624 |
lemma Lappend_type': "[| M \<in> llist(A); N \<in> llist(A) |] ==> Lappend M N \<in> llist(A)" |
17841 | 625 |
apply (rule_tac X = "(%u. Lappend u N) `llist (A)" in llist_coinduct) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
626 |
apply (erule imageI) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
627 |
apply (rule image_subsetI) |
23746 | 628 |
apply (erule_tac a = x in llist.cases) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
629 |
apply (simp add: list_Fun_llist_I, simp) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
630 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
631 |
|
13107 | 632 |
subsection{* Lazy lists as the type @{text "'a llist"} -- strongly typed versions of above *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
633 |
|
13107 | 634 |
subsubsection{* @{text llist_case}: case analysis for @{text "'a llist"} *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
635 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
636 |
declare LListI [THEN Abs_LList_inverse, simp] |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
637 |
declare Rep_LList_inverse [simp] |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
638 |
declare Rep_LList [THEN LListD, simp] |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
639 |
declare rangeI [simp] inj_Leaf [simp] |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
640 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
641 |
lemma llist_case_LNil [simp]: "llist_case c d LNil = c" |
17841 | 642 |
by (simp add: llist_case_def LNil_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
643 |
|
17841 | 644 |
lemma llist_case_LCons [simp]: "llist_case c d (LCons M N) = d M N" |
645 |
by (simp add: llist_case_def LCons_def) |
|
646 |
||
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
647 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
648 |
text{*Elimination is case analysis, not induction.*} |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
649 |
lemma llistE: "[| l=LNil ==> P; !!x l'. l=LCons x l' ==> P |] ==> P" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
650 |
apply (rule Rep_LList [THEN LListD, THEN llist.cases]) |
17841 | 651 |
apply (simp add: Rep_LList_LNil [symmetric] Rep_LList_inject, blast) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
652 |
apply (erule LListI [THEN Rep_LList_cases], clarify) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
653 |
apply (simp add: Rep_LList_LCons [symmetric] Rep_LList_inject, blast) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
654 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
655 |
|
13107 | 656 |
subsubsection{* @{text llist_corec}: corecursion for @{text "'a llist"} *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
657 |
|
13107 | 658 |
text{*Lemma for the proof of @{text llist_corec}*} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
659 |
lemma LList_corec_type2: |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
660 |
"LList_corec a |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
661 |
(%z. case f z of None => None | Some(v,w) => Some(Leaf(v),w)) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
662 |
\<in> llist(range Leaf)" |
17841 | 663 |
apply (rule_tac X = "range (%x. LList_corec x ?g)" in llist_coinduct) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
664 |
apply (rule rangeI, safe) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
665 |
apply (subst LList_corec, force) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
666 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
667 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
668 |
lemma llist_corec: |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
669 |
"llist_corec a f = |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
670 |
(case f a of None => LNil | Some(z,w) => LCons z (llist_corec w f))" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
671 |
apply (unfold llist_corec_def LNil_def LCons_def) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
672 |
apply (subst LList_corec) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
673 |
apply (case_tac "f a") |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
674 |
apply (simp add: LList_corec_type2) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
675 |
apply (force simp add: LList_corec_type2) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
676 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
677 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
678 |
text{*definitional version of same*} |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
679 |
lemma def_llist_corec: |
19736 | 680 |
"[| !!x. h(x) = llist_corec x f |] ==> |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
681 |
h(a) = (case f a of None => LNil | Some(z,w) => LCons z (h w))" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
682 |
by (simp add: llist_corec) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
683 |
|
13107 | 684 |
subsection{* Proofs about type @{text "'a llist"} functions *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
685 |
|
13107 | 686 |
subsection{* Deriving @{text llist_equalityI} -- @{text llist} equality is a bisimulation *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
687 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
688 |
lemma LListD_Fun_subset_Times_llist: |
17841 | 689 |
"r \<subseteq> (llist A) <*> (llist A) |
690 |
==> LListD_Fun (diag A) r \<subseteq> (llist A) <*> (llist A)" |
|
15481 | 691 |
by (auto simp add: LListD_Fun_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
692 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
693 |
lemma subset_Times_llist: |
17841 | 694 |
"prod_fun Rep_LList Rep_LList ` r \<subseteq> |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
695 |
(llist(range Leaf)) <*> (llist(range Leaf))" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
696 |
by (blast intro: Rep_LList [THEN LListD]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
697 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
698 |
lemma prod_fun_lemma: |
17841 | 699 |
"r \<subseteq> (llist(range Leaf)) <*> (llist(range Leaf)) |
700 |
==> prod_fun (Rep_LList o Abs_LList) (Rep_LList o Abs_LList) ` r \<subseteq> r" |
|
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
701 |
apply safe |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
702 |
apply (erule subsetD [THEN SigmaE2], assumption) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
703 |
apply (simp add: LListI [THEN Abs_LList_inverse]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
704 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
705 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
706 |
lemma prod_fun_range_eq_diag: |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
707 |
"prod_fun Rep_LList Rep_LList ` range(%x. (x, x)) = |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
708 |
diag(llist(range Leaf))" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
709 |
apply (rule equalityI, blast) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
710 |
apply (fast elim: LListI [THEN Abs_LList_inverse, THEN subst]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
711 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
712 |
|
13107 | 713 |
text{*Used with @{text lfilter}*} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
714 |
lemma llistD_Fun_mono: |
17841 | 715 |
"A\<subseteq>B ==> llistD_Fun A \<subseteq> llistD_Fun B" |
716 |
apply (simp add: llistD_Fun_def prod_fun_def, auto) |
|
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
717 |
apply (rule image_eqI) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
718 |
prefer 2 apply (blast intro: rev_subsetD [OF _ LListD_Fun_mono], force) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
719 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
720 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
721 |
subsubsection{* To show two llists are equal, exhibit a bisimulation! |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
722 |
[also admits true equality] *} |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
723 |
lemma llist_equalityI: |
17841 | 724 |
"[| (l1,l2) \<in> r; r \<subseteq> llistD_Fun(r Un range(%x.(x,x))) |] ==> l1=l2" |
725 |
apply (simp add: llistD_Fun_def) |
|
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
726 |
apply (rule Rep_LList_inject [THEN iffD1]) |
17841 | 727 |
apply (rule_tac r = "prod_fun Rep_LList Rep_LList `r" and A = "range (Leaf)" in LList_equalityI) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
728 |
apply (erule prod_fun_imageI) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
729 |
apply (erule image_mono [THEN subset_trans]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
730 |
apply (rule image_compose [THEN subst]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
731 |
apply (rule prod_fun_compose [THEN subst]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
732 |
apply (subst image_Un) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
733 |
apply (subst prod_fun_range_eq_diag) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
734 |
apply (rule LListD_Fun_subset_Times_llist [THEN prod_fun_lemma]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
735 |
apply (rule subset_Times_llist [THEN Un_least]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
736 |
apply (rule diag_subset_Times) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
737 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
738 |
|
13107 | 739 |
subsubsection{* Rules to prove the 2nd premise of @{text llist_equalityI} *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
740 |
lemma llistD_Fun_LNil_I [simp]: "(LNil,LNil) \<in> llistD_Fun(r)" |
17841 | 741 |
apply (simp add: llistD_Fun_def LNil_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
742 |
apply (rule LListD_Fun_NIL_I [THEN prod_fun_imageI]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
743 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
744 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
745 |
lemma llistD_Fun_LCons_I [simp]: |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
746 |
"(l1,l2):r ==> (LCons x l1, LCons x l2) \<in> llistD_Fun(r)" |
17841 | 747 |
apply (simp add: llistD_Fun_def LCons_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
748 |
apply (rule rangeI [THEN LListD_Fun_CONS_I, THEN prod_fun_imageI]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
749 |
apply (erule prod_fun_imageI) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
750 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
751 |
|
13107 | 752 |
text{*Utilise the "strong" part, i.e. @{text "gfp(f)"}*} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
753 |
lemma llistD_Fun_range_I: "(l,l) \<in> llistD_Fun(r Un range(%x.(x,x)))" |
17841 | 754 |
apply (simp add: llistD_Fun_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
755 |
apply (rule Rep_LList_inverse [THEN subst]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
756 |
apply (rule prod_fun_imageI) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
757 |
apply (subst image_Un) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
758 |
apply (subst prod_fun_range_eq_diag) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
759 |
apply (rule Rep_LList [THEN LListD, THEN LListD_Fun_diag_I]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
760 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
761 |
|
13107 | 762 |
text{*A special case of @{text list_equality} for functions over lazy lists*} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
763 |
lemma llist_fun_equalityI: |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
764 |
"[| f(LNil)=g(LNil); |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
765 |
!!x l. (f(LCons x l),g(LCons x l)) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
766 |
\<in> llistD_Fun(range(%u. (f(u),g(u))) Un range(%v. (v,v))) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
767 |
|] ==> f(l) = (g(l :: 'a llist) :: 'b llist)" |
17841 | 768 |
apply (rule_tac r = "range (%u. (f (u),g (u)))" in llist_equalityI) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
769 |
apply (rule rangeI, clarify) |
17841 | 770 |
apply (rule_tac l = u in llistE) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
771 |
apply (simp_all add: llistD_Fun_range_I) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
772 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
773 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
774 |
|
13107 | 775 |
subsection{* The functional @{text lmap} *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
776 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
777 |
lemma lmap_LNil [simp]: "lmap f LNil = LNil" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
778 |
by (rule lmap_def [THEN def_llist_corec, THEN trans], simp) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
779 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
780 |
lemma lmap_LCons [simp]: "lmap f (LCons M N) = LCons (f M) (lmap f N)" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
781 |
by (rule lmap_def [THEN def_llist_corec, THEN trans], simp) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
782 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
783 |
|
13107 | 784 |
subsubsection{* Two easy results about @{text lmap} *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
785 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
786 |
lemma lmap_compose [simp]: "lmap (f o g) l = lmap f (lmap g l)" |
17841 | 787 |
by (rule_tac l = l in llist_fun_equalityI, simp_all) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
788 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
789 |
lemma lmap_ident [simp]: "lmap (%x. x) l = l" |
17841 | 790 |
by (rule_tac l = l in llist_fun_equalityI, simp_all) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
791 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
792 |
|
13107 | 793 |
subsection{* iterates -- @{text llist_fun_equalityI} cannot be used! *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
794 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
795 |
lemma iterates: "iterates f x = LCons x (iterates f (f x))" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
796 |
by (rule iterates_def [THEN def_llist_corec, THEN trans], simp) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
797 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
798 |
lemma lmap_iterates [simp]: "lmap f (iterates f x) = iterates f (f x)" |
17841 | 799 |
apply (rule_tac r = "range (%u. (lmap f (iterates f u),iterates f (f u)))" in llist_equalityI) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
800 |
apply (rule rangeI, safe) |
17841 | 801 |
apply (rule_tac x1 = "f (u)" in iterates [THEN ssubst]) |
802 |
apply (rule_tac x1 = u in iterates [THEN ssubst], simp) |
|
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
803 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
804 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
805 |
lemma iterates_lmap: "iterates f x = LCons x (lmap f (iterates f x))" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
806 |
apply (subst lmap_iterates) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
807 |
apply (rule iterates) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
808 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
809 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
810 |
subsection{* A rather complex proof about iterates -- cf Andy Pitts *} |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
811 |
|
13107 | 812 |
subsubsection{* Two lemmas about @{text "natrec n x (%m. g)"}, which is essentially |
813 |
@{text "(g^n)(x)"} *} |
|
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
814 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
815 |
lemma fun_power_lmap: "nat_rec (LCons b l) (%m. lmap(f)) n = |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
816 |
LCons (nat_rec b (%m. f) n) (nat_rec l (%m. lmap(f)) n)" |
17841 | 817 |
by (induct_tac "n", simp_all) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
818 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
819 |
lemma fun_power_Suc: "nat_rec (g x) (%m. g) n = nat_rec x (%m. g) (Suc n)" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
820 |
by (induct_tac "n", simp_all) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
821 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
822 |
lemmas Pair_cong = refl [THEN cong, THEN cong, of concl: Pair] |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
823 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
824 |
|
13107 | 825 |
text{*The bisimulation consists of @{text "{(lmap(f)^n (h(u)), lmap(f)^n (iterates(f,u)))}"} |
826 |
for all @{text u} and all @{text "n::nat"}.*} |
|
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
827 |
lemma iterates_equality: |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
828 |
"(!!x. h(x) = LCons x (lmap f (h x))) ==> h = iterates(f)" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
829 |
apply (rule ext) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
830 |
apply (rule_tac |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
831 |
r = "\<Union>u. range (%n. (nat_rec (h u) (%m y. lmap f y) n, |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
832 |
nat_rec (iterates f u) (%m y. lmap f y) n))" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
833 |
in llist_equalityI) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
834 |
apply (rule UN1_I range_eqI Pair_cong nat_rec_0 [symmetric])+ |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
835 |
apply clarify |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
836 |
apply (subst iterates, atomize) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
837 |
apply (drule_tac x=u in spec) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
838 |
apply (erule ssubst) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
839 |
apply (subst fun_power_lmap) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
840 |
apply (subst fun_power_lmap) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
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changeset
|
841 |
apply (rule llistD_Fun_LCons_I) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
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changeset
|
842 |
apply (rule lmap_iterates [THEN subst]) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
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changeset
|
843 |
apply (subst fun_power_Suc) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
844 |
apply (subst fun_power_Suc, blast) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
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changeset
|
845 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
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changeset
|
846 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
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changeset
|
847 |
|
13107 | 848 |
subsection{* @{text lappend} -- its two arguments cause some complications! *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
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changeset
|
849 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
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changeset
|
850 |
lemma lappend_LNil_LNil [simp]: "lappend LNil LNil = LNil" |
17841 | 851 |
apply (simp add: lappend_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
852 |
apply (rule llist_corec [THEN trans], simp) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
853 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
854 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
855 |
lemma lappend_LNil_LCons [simp]: |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
856 |
"lappend LNil (LCons l l') = LCons l (lappend LNil l')" |
17841 | 857 |
apply (simp add: lappend_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
858 |
apply (rule llist_corec [THEN trans], simp) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
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diff
changeset
|
859 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
860 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
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changeset
|
861 |
lemma lappend_LCons [simp]: |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
862 |
"lappend (LCons l l') N = LCons l (lappend l' N)" |
17841 | 863 |
apply (simp add: lappend_def) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
864 |
apply (rule llist_corec [THEN trans], simp) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
865 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
866 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
867 |
lemma lappend_LNil [simp]: "lappend LNil l = l" |
17841 | 868 |
by (rule_tac l = l in llist_fun_equalityI, simp_all) |
13075
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paulson
parents:
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changeset
|
869 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
870 |
lemma lappend_LNil2 [simp]: "lappend l LNil = l" |
17841 | 871 |
by (rule_tac l = l in llist_fun_equalityI, simp_all) |
13075
d3e1d554cd6d
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paulson
parents:
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changeset
|
872 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
873 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
874 |
text{*The infinite first argument blocks the second*} |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
875 |
lemma lappend_iterates [simp]: "lappend (iterates f x) N = iterates f x" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
876 |
apply (rule_tac r = "range (%u. (lappend (iterates f u) N,iterates f u))" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
877 |
in llist_equalityI) |
17841 | 878 |
apply (rule rangeI, safe) |
15944 | 879 |
apply (subst (1 2) iterates) |
880 |
apply simp |
|
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
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diff
changeset
|
881 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
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diff
changeset
|
882 |
|
13107 | 883 |
subsubsection{* Two proofs that @{text lmap} distributes over lappend *} |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
884 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
885 |
text{*Long proof requiring case analysis on both both arguments*} |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
886 |
lemma lmap_lappend_distrib: |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
887 |
"lmap f (lappend l n) = lappend (lmap f l) (lmap f n)" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
888 |
apply (rule_tac r = "\<Union>n. range (%l. (lmap f (lappend l n), |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
889 |
lappend (lmap f l) (lmap f n)))" |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
890 |
in llist_equalityI) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
891 |
apply (rule UN1_I) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
892 |
apply (rule rangeI, safe) |
17841 | 893 |
apply (rule_tac l = l in llistE) |
894 |
apply (rule_tac l = n in llistE, simp_all) |
|
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
895 |
apply (blast intro: llistD_Fun_LCons_I) |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
896 |
done |
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
897 |
|
13107 | 898 |
text{*Shorter proof of theorem above using @{text llist_equalityI} as strong coinduction*} |
13524 | 899 |
lemma lmap_lappend_distrib': |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
900 |
"lmap f (lappend l n) = lappend (lmap f l) (lmap f n)" |
17841 | 901 |
by (rule_tac l = l in llist_fun_equalityI, auto) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
902 |
|
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
903 |
text{*Without strong coinduction, three case analyses might be needed*} |
13524 | 904 |
lemma lappend_assoc': "lappend (lappend l1 l2) l3 = lappend l1 (lappend l2 l3)" |
17841 | 905 |
by (rule_tac l = l1 in llist_fun_equalityI, auto) |
13075
d3e1d554cd6d
conversion of some HOL/Induct proof scripts to Isar
paulson
parents:
10834
diff
changeset
|
906 |
|
3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset
|
907 |
end |