# HG changeset patch # User wenzelm # Date 1125495998 -7200 # Node ID 3bdf1dfcdee43fa40786da4a5f117a0274127cbe # Parent 3a4d03d1a31bb609bc70ea351cfed745c94d9955 reactivate postfix by change of syntax; tuned presentation; diff -r 3a4d03d1a31b -r 3bdf1dfcdee4 src/HOL/Library/List_Prefix.thy --- a/src/HOL/Library/List_Prefix.thy Wed Aug 31 15:46:37 2005 +0200 +++ b/src/HOL/Library/List_Prefix.thy Wed Aug 31 15:46:38 2005 +0200 @@ -30,11 +30,11 @@ by (unfold strict_prefix_def prefix_def) blast lemma strict_prefixE' [elim?]: - "xs < ys ==> (!!z zs. ys = xs @ z # zs ==> C) ==> C" + assumes lt: "xs < ys" + and r: "!!z zs. ys = xs @ z # zs ==> C" + shows C proof - - assume r: "!!z zs. ys = xs @ z # zs ==> C" - assume "xs < ys" - then obtain us where "ys = xs @ us" and "xs \ ys" + from lt obtain us where "ys = xs @ us" and "xs \ ys" by (unfold strict_prefix_def prefix_def) blast with r show ?thesis by (auto simp add: neq_Nil_conv) qed @@ -105,7 +105,7 @@ qed lemma append_prefixD: "xs @ ys \ zs \ xs \ zs" -by(simp add:prefix_def) blast + by (auto simp add: prefix_def) theorem prefix_Cons: "(xs \ y # ys) = (xs = [] \ (\zs. xs = y # zs \ zs \ ys))" by (cases xs) (auto simp add: prefix_def) @@ -130,28 +130,27 @@ theorem prefix_length_le: "xs \ ys ==> length xs \ length ys" by (auto simp add: prefix_def) - lemma prefix_same_cases: - "\ (xs\<^isub>1::'a list) \ ys; xs\<^isub>2 \ ys \ \ xs\<^isub>1 \ xs\<^isub>2 \ xs\<^isub>2 \ xs\<^isub>1" -apply(simp add:prefix_def) -apply(erule exE)+ -apply(simp add: append_eq_append_conv_if split:if_splits) - apply(rule disjI2) - apply(rule_tac x = "drop (size xs\<^isub>2) xs\<^isub>1" in exI) - apply clarify - apply(drule sym) - apply(insert append_take_drop_id[of "length xs\<^isub>2" xs\<^isub>1]) - apply simp -apply(rule disjI1) -apply(rule_tac x = "drop (size xs\<^isub>1) xs\<^isub>2" in exI) -apply clarify -apply(insert append_take_drop_id[of "length xs\<^isub>1" xs\<^isub>2]) -apply simp -done + "(xs\<^isub>1::'a list) \ ys \ xs\<^isub>2 \ ys \ xs\<^isub>1 \ xs\<^isub>2 \ xs\<^isub>2 \ xs\<^isub>1" + apply (simp add: prefix_def) + apply (erule exE)+ + apply (simp add: append_eq_append_conv_if split: if_splits) + apply (rule disjI2) + apply (rule_tac x = "drop (size xs\<^isub>2) xs\<^isub>1" in exI) + apply clarify + apply (drule sym) + apply (insert append_take_drop_id [of "length xs\<^isub>2" xs\<^isub>1]) + apply simp + apply (rule disjI1) + apply (rule_tac x = "drop (size xs\<^isub>1) xs\<^isub>2" in exI) + apply clarify + apply (insert append_take_drop_id [of "length xs\<^isub>1" xs\<^isub>2]) + apply simp + done lemma set_mono_prefix: - "xs \ ys \ set xs \ set ys" -by(fastsimp simp add:prefix_def) + "xs \ ys \ set xs \ set ys" + by (auto simp add: prefix_def) subsection {* Parallel lists *} @@ -215,48 +214,48 @@ subsection {* Postfix order on lists *} -(* + constdefs - postfix :: "'a list => 'a list => bool" ("(_/ >= _)" [51, 50] 50) - "xs >= ys == \zs. xs = zs @ ys" + postfix :: "'a list => 'a list => bool" ("(_/ >>= _)" [51, 50] 50) + "xs >>= ys == \zs. xs = zs @ ys" -lemma postfix_refl [simp, intro!]: "xs >= xs" +lemma postfix_refl [simp, intro!]: "xs >>= xs" by (auto simp add: postfix_def) -lemma postfix_trans: "\xs >= ys; ys >= zs\ \ xs >= zs" +lemma postfix_trans: "\xs >>= ys; ys >>= zs\ \ xs >>= zs" by (auto simp add: postfix_def) -lemma postfix_antisym: "\xs >= ys; ys >= xs\ \ xs = ys" +lemma postfix_antisym: "\xs >>= ys; ys >>= xs\ \ xs = ys" by (auto simp add: postfix_def) -lemma Nil_postfix [iff]: "xs >= []" +lemma Nil_postfix [iff]: "xs >>= []" by (simp add: postfix_def) -lemma postfix_Nil [simp]: "([] >= xs) = (xs = [])" +lemma postfix_Nil [simp]: "([] >>= xs) = (xs = [])" by (auto simp add:postfix_def) -lemma postfix_ConsI: "xs >= ys \ x#xs >= ys" +lemma postfix_ConsI: "xs >>= ys \ x#xs >>= ys" by (auto simp add: postfix_def) -lemma postfix_ConsD: "xs >= y#ys \ xs >= ys" +lemma postfix_ConsD: "xs >>= y#ys \ xs >>= ys" by (auto simp add: postfix_def) -lemma postfix_appendI: "xs >= ys \ zs @ xs >= ys" +lemma postfix_appendI: "xs >>= ys \ zs @ xs >>= ys" by (auto simp add: postfix_def) -lemma postfix_appendD: "xs >= zs @ ys \ xs >= ys" +lemma postfix_appendD: "xs >>= zs @ ys \ xs >>= ys" by(auto simp add: postfix_def) lemma postfix_is_subset_lemma: "xs = zs @ ys \ set ys \ set xs" by (induct zs, auto) -lemma postfix_is_subset: "xs >= ys \ set ys \ set xs" +lemma postfix_is_subset: "xs >>= ys \ set ys \ set xs" by (unfold postfix_def, erule exE, erule postfix_is_subset_lemma) -lemma postfix_ConsD2_lemma [rule_format]: "x#xs = zs @ y#ys \ xs >= ys" +lemma postfix_ConsD2_lemma [rule_format]: "x#xs = zs @ y#ys \ xs >>= ys" by (induct zs, auto intro!: postfix_appendI postfix_ConsI) -lemma postfix_ConsD2: "x#xs >= y#ys \ xs >= ys" +lemma postfix_ConsD2: "x#xs >>= y#ys \ xs >>= ys" by (auto simp add: postfix_def dest!: postfix_ConsD2_lemma) -lemma postfix2prefix: "(xs >= ys) = (rev ys <= rev xs)" +lemma postfix2prefix: "(xs >>= ys) = (rev ys <= rev xs)" apply (unfold prefix_def postfix_def, safe) - apply (rule_tac x = "rev zs" in exI, simp) + apply (rule_tac x = "rev zs" in exI, simp) apply (rule_tac x = "rev zs" in exI) apply (rule rev_is_rev_conv [THEN iffD1], simp) done -*) + end