diff -r 59c1bfc81d91 -r 3a4d03d1a31b src/HOL/Library/Permutation.thy --- a/src/HOL/Library/Permutation.thy Wed Aug 31 15:46:36 2005 +0200 +++ b/src/HOL/Library/Permutation.thy Wed Aug 31 15:46:37 2005 +0200 @@ -24,20 +24,20 @@ trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs" lemma perm_refl [iff]: "l <~~> l" -by (induct l, auto) + by (induct l) auto subsection {* Some examples of rule induction on permutations *} lemma xperm_empty_imp_aux: "xs <~~> ys ==> xs = [] --> ys = []" - -- {*the form of the premise lets the induction bind @{term xs} + -- {*the form of the premise lets the induction bind @{term xs} and @{term ys} *} apply (erule perm.induct) apply (simp_all (no_asm_simp)) done lemma xperm_empty_imp: "[] <~~> ys ==> ys = []" -by (insert xperm_empty_imp_aux, blast) + using xperm_empty_imp_aux by blast text {* @@ -45,16 +45,16 @@ *} lemma perm_length: "xs <~~> ys ==> length xs = length ys" -by (erule perm.induct, simp_all) + by (erule perm.induct) simp_all lemma perm_empty_imp: "[] <~~> xs ==> xs = []" -by (drule perm_length, auto) + by (drule perm_length) auto lemma perm_sym: "xs <~~> ys ==> ys <~~> xs" -by (erule perm.induct, auto) + by (erule perm.induct) auto lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys" -by (erule perm.induct, auto) + by (erule perm.induct) auto subsection {* Ways of making new permutations *} @@ -64,32 +64,34 @@ *} lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys" -by (induct xs, auto) + by (induct xs) auto lemma perm_append_swap: "xs @ ys <~~> ys @ xs" - apply (induct xs, simp_all) + apply (induct xs) + apply simp_all apply (blast intro: perm_append_Cons) done lemma perm_append_single: "a # xs <~~> xs @ [a]" - by (rule perm.trans [OF _ perm_append_swap], simp) + by (rule perm.trans [OF _ perm_append_swap]) simp lemma perm_rev: "rev xs <~~> xs" - apply (induct xs, simp_all) + apply (induct xs) + apply simp_all apply (blast intro!: perm_append_single intro: perm_sym) done lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys" -by (induct l, auto) + by (induct l) auto lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l" -by (blast intro!: perm_append_swap perm_append1) + by (blast intro!: perm_append_swap perm_append1) subsection {* Further results *} lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])" -by (blast intro: perm_empty_imp) + by (blast intro: perm_empty_imp) lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])" apply auto @@ -97,13 +99,13 @@ done lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]" -by (erule perm.induct, auto) + by (erule perm.induct) auto lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])" -by (blast intro: perm_sing_imp) + by (blast intro: perm_sing_imp) lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])" -by (blast dest: perm_sym) + by (blast dest: perm_sym) subsection {* Removing elements *} @@ -115,29 +117,31 @@ "remove x (y # ys) = (if x = y then ys else y # remove x ys)" lemma perm_remove: "x \ set ys ==> ys <~~> x # remove x ys" -by (induct ys, auto) + by (induct ys) auto lemma remove_commute: "remove x (remove y l) = remove y (remove x l)" -by (induct l, auto) + by (induct l) auto -lemma multiset_of_remove[simp]: - "multiset_of (remove a x) = multiset_of x - {#a#}" - by (induct_tac x, auto simp: multiset_eq_conv_count_eq) +lemma multiset_of_remove[simp]: + "multiset_of (remove a x) = multiset_of x - {#a#}" + apply (induct x) + apply (auto simp: multiset_eq_conv_count_eq) + done text {* \medskip Congruence rule *} lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys" -by (erule perm.induct, auto) + by (erule perm.induct) auto lemma remove_hd [simp]: "remove z (z # xs) = xs" by auto lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys" -by (drule_tac z = z in perm_remove_perm, auto) + by (drule_tac z = z in perm_remove_perm) auto lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)" -by (blast intro: cons_perm_imp_perm) + by (blast intro: cons_perm_imp_perm) lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys" apply (induct zs rule: rev_induct) @@ -146,7 +150,7 @@ done lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)" -by (blast intro: append_perm_imp_perm perm_append1) + by (blast intro: append_perm_imp_perm perm_append1) lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)" apply (safe intro!: perm_append2) @@ -157,20 +161,20 @@ done lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) " - apply (rule iffI) - apply (erule_tac [2] perm.induct, simp_all add: union_ac) - apply (erule rev_mp, rule_tac x=ys in spec) - apply (induct_tac xs, auto) - apply (erule_tac x = "remove a x" in allE, drule sym, simp) - apply (subgoal_tac "a \ set x") - apply (drule_tac z=a in perm.Cons) - apply (erule perm.trans, rule perm_sym, erule perm_remove) + apply (rule iffI) + apply (erule_tac [2] perm.induct, simp_all add: union_ac) + apply (erule rev_mp, rule_tac x=ys in spec) + apply (induct_tac xs, auto) + apply (erule_tac x = "remove a x" in allE, drule sym, simp) + apply (subgoal_tac "a \ set x") + apply (drule_tac z=a in perm.Cons) + apply (erule perm.trans, rule perm_sym, erule perm_remove) apply (drule_tac f=set_of in arg_cong, simp) done -lemma multiset_of_le_perm_append: - "(multiset_of xs \# multiset_of ys) = (\ zs. xs @ zs <~~> ys)"; - apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv) +lemma multiset_of_le_perm_append: + "(multiset_of xs \# multiset_of ys) = (\zs. xs @ zs <~~> ys)"; + apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv) apply (insert surj_multiset_of, drule surjD) apply (blast intro: sym)+ done