--- a/NEWS Tue Feb 23 20:41:48 2021 +0000
+++ b/NEWS Tue Feb 23 20:41:48 2021 +0000
@@ -18,6 +18,9 @@
more small lemmas. Some theorems that were stated awkwardly before were
corrected. Minor INCOMPATIBILITY.
+* Theory "Permutation" in HOL-Library has been renamed to the more
+specific "List_Permutation".
+
*** ML ***
--- a/src/HOL/Algebra/Divisibility.thy Tue Feb 23 20:41:48 2021 +0000
+++ b/src/HOL/Algebra/Divisibility.thy Tue Feb 23 20:41:48 2021 +0000
@@ -6,7 +6,7 @@
section \<open>Divisibility in monoids and rings\<close>
theory Divisibility
- imports "HOL-Library.Permutation" Coset Group
+ imports "HOL-Library.List_Permutation" Coset Group
begin
section \<open>Factorial Monoids\<close>
--- a/src/HOL/Library/Library.thy Tue Feb 23 20:41:48 2021 +0000
+++ b/src/HOL/Library/Library.thy Tue Feb 23 20:41:48 2021 +0000
@@ -50,6 +50,7 @@
Lattice_Constructions
Linear_Temporal_Logic_on_Streams
ListVector
+ List_Permutation
Lub_Glb
Mapping
Monad_Syntax
@@ -66,7 +67,6 @@
Pattern_Aliases
Periodic_Fun
Perm
- Permutation
Permutations
Poly_Mapping
Power_By_Squaring
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/List_Permutation.thy Tue Feb 23 20:41:48 2021 +0000
@@ -0,0 +1,230 @@
+(* Title: HOL/Library/List_Permutation.thy
+ Author: Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
+*)
+
+section \<open>Permuted Lists\<close>
+
+theory List_Permutation
+imports Multiset
+begin
+
+inductive perm :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixr \<open><~~>\<close> 50)
+where
+ Nil [intro!]: "[] <~~> []"
+| swap [intro!]: "y # x # l <~~> x # y # l"
+| Cons [intro!]: "xs <~~> ys \<Longrightarrow> z # xs <~~> z # ys"
+| trans [intro]: "xs <~~> ys \<Longrightarrow> ys <~~> zs \<Longrightarrow> xs <~~> zs"
+
+proposition perm_refl [iff]: "l <~~> l"
+ by (induct l) auto
+
+
+subsection \<open>Some examples of rule induction on permutations\<close>
+
+proposition perm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"
+ by (induction "[] :: 'a list" ys pred: perm) simp_all
+
+
+text \<open>\medskip This more general theorem is easier to understand!\<close>
+
+proposition perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys"
+ by (induct pred: perm) simp_all
+
+proposition perm_sym: "xs <~~> ys \<Longrightarrow> ys <~~> xs"
+ by (induct pred: perm) auto
+
+
+subsection \<open>Ways of making new permutations\<close>
+
+text \<open>We can insert the head anywhere in the list.\<close>
+
+proposition perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
+ by (induct xs) auto
+
+proposition perm_append_swap: "xs @ ys <~~> ys @ xs"
+ by (induct xs) (auto intro: perm_append_Cons)
+
+proposition perm_append_single: "a # xs <~~> xs @ [a]"
+ by (rule perm.trans [OF _ perm_append_swap]) simp
+
+proposition perm_rev: "rev xs <~~> xs"
+ by (induct xs) (auto intro!: perm_append_single intro: perm_sym)
+
+proposition perm_append1: "xs <~~> ys \<Longrightarrow> l @ xs <~~> l @ ys"
+ by (induct l) auto
+
+proposition perm_append2: "xs <~~> ys \<Longrightarrow> xs @ l <~~> ys @ l"
+ by (blast intro!: perm_append_swap perm_append1)
+
+
+subsection \<open>Further results\<close>
+
+proposition perm_empty [iff]: "[] <~~> xs \<longleftrightarrow> xs = []"
+ by (blast intro: perm_empty_imp)
+
+proposition perm_empty2 [iff]: "xs <~~> [] \<longleftrightarrow> xs = []"
+ using perm_sym by auto
+
+proposition perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]"
+ by (induct pred: perm) auto
+
+proposition perm_sing_eq [iff]: "ys <~~> [y] \<longleftrightarrow> ys = [y]"
+ by (blast intro: perm_sing_imp)
+
+proposition perm_sing_eq2 [iff]: "[y] <~~> ys \<longleftrightarrow> ys = [y]"
+ by (blast dest: perm_sym)
+
+
+subsection \<open>Removing elements\<close>
+
+proposition perm_remove: "x \<in> set ys \<Longrightarrow> ys <~~> x # remove1 x ys"
+ by (induct ys) auto
+
+
+text \<open>\medskip Congruence rule\<close>
+
+proposition perm_remove_perm: "xs <~~> ys \<Longrightarrow> remove1 z xs <~~> remove1 z ys"
+ by (induct pred: perm) auto
+
+proposition remove_hd [simp]: "remove1 z (z # xs) = xs"
+ by auto
+
+proposition cons_perm_imp_perm: "z # xs <~~> z # ys \<Longrightarrow> xs <~~> ys"
+ by (drule perm_remove_perm [where z = z]) auto
+
+proposition cons_perm_eq [iff]: "z#xs <~~> z#ys \<longleftrightarrow> xs <~~> ys"
+ by (meson cons_perm_imp_perm perm.Cons)
+
+proposition append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys"
+ by (induct zs arbitrary: xs ys rule: rev_induct) auto
+
+proposition perm_append1_eq [iff]: "zs @ xs <~~> zs @ ys \<longleftrightarrow> xs <~~> ys"
+ by (blast intro: append_perm_imp_perm perm_append1)
+
+proposition perm_append2_eq [iff]: "xs @ zs <~~> ys @ zs \<longleftrightarrow> xs <~~> ys"
+ by (meson perm.trans perm_append1_eq perm_append_swap)
+
+theorem mset_eq_perm: "mset xs = mset ys \<longleftrightarrow> xs <~~> ys"
+proof
+ assume "mset xs = mset ys"
+ then show "xs <~~> ys"
+ proof (induction xs arbitrary: ys)
+ case (Cons x xs)
+ then have "x \<in> set ys"
+ using mset_eq_setD by fastforce
+ then show ?case
+ by (metis Cons.IH Cons.prems mset_remove1 perm.Cons perm.trans perm_remove perm_sym remove_hd)
+ qed auto
+next
+ assume "xs <~~> ys"
+ then show "mset xs = mset ys"
+ by induction (simp_all add: union_ac)
+qed
+
+proposition mset_le_perm_append: "mset xs \<subseteq># mset ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
+ apply (rule iffI)
+ apply (metis mset_append mset_eq_perm mset_subset_eq_exists_conv surjD surj_mset)
+ by (metis mset_append mset_eq_perm mset_subset_eq_exists_conv)
+
+proposition perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys"
+ by (metis mset_eq_perm mset_eq_setD)
+
+proposition perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs = distinct ys"
+ by (metis card_distinct distinct_card perm_length perm_set_eq)
+
+theorem eq_set_perm_remdups: "set xs = set ys \<Longrightarrow> remdups xs <~~> remdups ys"
+proof (induction xs arbitrary: ys rule: length_induct)
+ case (1 xs)
+ show ?case
+ proof (cases "remdups xs")
+ case Nil
+ with "1.prems" show ?thesis
+ using "1.prems" by auto
+ next
+ case (Cons x us)
+ then have "x \<in> set (remdups ys)"
+ using "1.prems" set_remdups by fastforce
+ then show ?thesis
+ using "1.prems" mset_eq_perm set_eq_iff_mset_remdups_eq by blast
+ qed
+qed
+
+proposition perm_remdups_iff_eq_set: "remdups x <~~> remdups y \<longleftrightarrow> set x = set y"
+ by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)
+
+theorem permutation_Ex_bij:
+ assumes "xs <~~> ys"
+ shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"
+ using assms
+proof induct
+ case Nil
+ then show ?case
+ unfolding bij_betw_def by simp
+next
+ case (swap y x l)
+ show ?case
+ proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI)
+ show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"
+ by (auto simp: bij_betw_def)
+ fix i
+ assume "i < length (y # x # l)"
+ show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"
+ by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)
+ qed
+next
+ case (Cons xs ys z)
+ then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}"
+ and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)"
+ by blast
+ let ?f = "\<lambda>i. case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
+ show ?case
+ proof (intro exI[of _ ?f] allI conjI impI)
+ have *: "{..<length (z#xs)} = {0} \<union> Suc ` {..<length xs}"
+ "{..<length (z#ys)} = {0} \<union> Suc ` {..<length ys}"
+ by (simp_all add: lessThan_Suc_eq_insert_0)
+ show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}"
+ unfolding *
+ proof (rule bij_betw_combine)
+ show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})"
+ using bij unfolding bij_betw_def
+ by (auto intro!: inj_onI imageI dest: inj_onD simp: image_comp comp_def)
+ qed (auto simp: bij_betw_def)
+ fix i
+ assume "i < length (z # xs)"
+ then show "(z # xs) ! i = (z # ys) ! (?f i)"
+ using perm by (cases i) auto
+ qed
+next
+ case (trans xs ys zs)
+ then obtain f g
+ where bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}"
+ and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)"
+ by blast
+ show ?case
+ proof (intro exI[of _ "g \<circ> f"] conjI allI impI)
+ show "bij_betw (g \<circ> f) {..<length xs} {..<length zs}"
+ using bij by (rule bij_betw_trans)
+ fix i
+ assume "i < length xs"
+ with bij have "f i < length ys"
+ unfolding bij_betw_def by force
+ with \<open>i < length xs\<close> show "xs ! i = zs ! (g \<circ> f) i"
+ using trans(1,3)[THEN perm_length] perm by auto
+ qed
+qed
+
+proposition perm_finite: "finite {B. B <~~> A}"
+proof (rule finite_subset[where B="{xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"])
+ show "finite {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"
+ using finite_lists_length_le by blast
+next
+ show "{B. B <~~> A} \<subseteq> {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"
+ by (clarsimp simp add: perm_length perm_set_eq)
+qed
+
+proposition perm_swap:
+ assumes "i < length xs" "j < length xs"
+ shows "xs[i := xs ! j, j := xs ! i] <~~> xs"
+ using assms by (simp add: mset_eq_perm[symmetric] mset_swap)
+
+end
--- a/src/HOL/Library/Permutation.thy Tue Feb 23 20:41:48 2021 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,230 +0,0 @@
-(* Title: HOL/Library/Permutation.thy
- Author: Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
-*)
-
-section \<open>Permutations\<close>
-
-theory Permutation
-imports Multiset
-begin
-
-inductive perm :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixr \<open><~~>\<close> 50)
-where
- Nil [intro!]: "[] <~~> []"
-| swap [intro!]: "y # x # l <~~> x # y # l"
-| Cons [intro!]: "xs <~~> ys \<Longrightarrow> z # xs <~~> z # ys"
-| trans [intro]: "xs <~~> ys \<Longrightarrow> ys <~~> zs \<Longrightarrow> xs <~~> zs"
-
-proposition perm_refl [iff]: "l <~~> l"
- by (induct l) auto
-
-
-subsection \<open>Some examples of rule induction on permutations\<close>
-
-proposition perm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"
- by (induction "[] :: 'a list" ys pred: perm) simp_all
-
-
-text \<open>\medskip This more general theorem is easier to understand!\<close>
-
-proposition perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys"
- by (induct pred: perm) simp_all
-
-proposition perm_sym: "xs <~~> ys \<Longrightarrow> ys <~~> xs"
- by (induct pred: perm) auto
-
-
-subsection \<open>Ways of making new permutations\<close>
-
-text \<open>We can insert the head anywhere in the list.\<close>
-
-proposition perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
- by (induct xs) auto
-
-proposition perm_append_swap: "xs @ ys <~~> ys @ xs"
- by (induct xs) (auto intro: perm_append_Cons)
-
-proposition perm_append_single: "a # xs <~~> xs @ [a]"
- by (rule perm.trans [OF _ perm_append_swap]) simp
-
-proposition perm_rev: "rev xs <~~> xs"
- by (induct xs) (auto intro!: perm_append_single intro: perm_sym)
-
-proposition perm_append1: "xs <~~> ys \<Longrightarrow> l @ xs <~~> l @ ys"
- by (induct l) auto
-
-proposition perm_append2: "xs <~~> ys \<Longrightarrow> xs @ l <~~> ys @ l"
- by (blast intro!: perm_append_swap perm_append1)
-
-
-subsection \<open>Further results\<close>
-
-proposition perm_empty [iff]: "[] <~~> xs \<longleftrightarrow> xs = []"
- by (blast intro: perm_empty_imp)
-
-proposition perm_empty2 [iff]: "xs <~~> [] \<longleftrightarrow> xs = []"
- using perm_sym by auto
-
-proposition perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]"
- by (induct pred: perm) auto
-
-proposition perm_sing_eq [iff]: "ys <~~> [y] \<longleftrightarrow> ys = [y]"
- by (blast intro: perm_sing_imp)
-
-proposition perm_sing_eq2 [iff]: "[y] <~~> ys \<longleftrightarrow> ys = [y]"
- by (blast dest: perm_sym)
-
-
-subsection \<open>Removing elements\<close>
-
-proposition perm_remove: "x \<in> set ys \<Longrightarrow> ys <~~> x # remove1 x ys"
- by (induct ys) auto
-
-
-text \<open>\medskip Congruence rule\<close>
-
-proposition perm_remove_perm: "xs <~~> ys \<Longrightarrow> remove1 z xs <~~> remove1 z ys"
- by (induct pred: perm) auto
-
-proposition remove_hd [simp]: "remove1 z (z # xs) = xs"
- by auto
-
-proposition cons_perm_imp_perm: "z # xs <~~> z # ys \<Longrightarrow> xs <~~> ys"
- by (drule perm_remove_perm [where z = z]) auto
-
-proposition cons_perm_eq [iff]: "z#xs <~~> z#ys \<longleftrightarrow> xs <~~> ys"
- by (meson cons_perm_imp_perm perm.Cons)
-
-proposition append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys"
- by (induct zs arbitrary: xs ys rule: rev_induct) auto
-
-proposition perm_append1_eq [iff]: "zs @ xs <~~> zs @ ys \<longleftrightarrow> xs <~~> ys"
- by (blast intro: append_perm_imp_perm perm_append1)
-
-proposition perm_append2_eq [iff]: "xs @ zs <~~> ys @ zs \<longleftrightarrow> xs <~~> ys"
- by (meson perm.trans perm_append1_eq perm_append_swap)
-
-theorem mset_eq_perm: "mset xs = mset ys \<longleftrightarrow> xs <~~> ys"
-proof
- assume "mset xs = mset ys"
- then show "xs <~~> ys"
- proof (induction xs arbitrary: ys)
- case (Cons x xs)
- then have "x \<in> set ys"
- using mset_eq_setD by fastforce
- then show ?case
- by (metis Cons.IH Cons.prems mset_remove1 perm.Cons perm.trans perm_remove perm_sym remove_hd)
- qed auto
-next
- assume "xs <~~> ys"
- then show "mset xs = mset ys"
- by induction (simp_all add: union_ac)
-qed
-
-proposition mset_le_perm_append: "mset xs \<subseteq># mset ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
- apply (rule iffI)
- apply (metis mset_append mset_eq_perm mset_subset_eq_exists_conv surjD surj_mset)
- by (metis mset_append mset_eq_perm mset_subset_eq_exists_conv)
-
-proposition perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys"
- by (metis mset_eq_perm mset_eq_setD)
-
-proposition perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs = distinct ys"
- by (metis card_distinct distinct_card perm_length perm_set_eq)
-
-theorem eq_set_perm_remdups: "set xs = set ys \<Longrightarrow> remdups xs <~~> remdups ys"
-proof (induction xs arbitrary: ys rule: length_induct)
- case (1 xs)
- show ?case
- proof (cases "remdups xs")
- case Nil
- with "1.prems" show ?thesis
- using "1.prems" by auto
- next
- case (Cons x us)
- then have "x \<in> set (remdups ys)"
- using "1.prems" set_remdups by fastforce
- then show ?thesis
- using "1.prems" mset_eq_perm set_eq_iff_mset_remdups_eq by blast
- qed
-qed
-
-proposition perm_remdups_iff_eq_set: "remdups x <~~> remdups y \<longleftrightarrow> set x = set y"
- by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)
-
-theorem permutation_Ex_bij:
- assumes "xs <~~> ys"
- shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"
- using assms
-proof induct
- case Nil
- then show ?case
- unfolding bij_betw_def by simp
-next
- case (swap y x l)
- show ?case
- proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI)
- show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"
- by (auto simp: bij_betw_def)
- fix i
- assume "i < length (y # x # l)"
- show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"
- by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)
- qed
-next
- case (Cons xs ys z)
- then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}"
- and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)"
- by blast
- let ?f = "\<lambda>i. case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
- show ?case
- proof (intro exI[of _ ?f] allI conjI impI)
- have *: "{..<length (z#xs)} = {0} \<union> Suc ` {..<length xs}"
- "{..<length (z#ys)} = {0} \<union> Suc ` {..<length ys}"
- by (simp_all add: lessThan_Suc_eq_insert_0)
- show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}"
- unfolding *
- proof (rule bij_betw_combine)
- show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})"
- using bij unfolding bij_betw_def
- by (auto intro!: inj_onI imageI dest: inj_onD simp: image_comp comp_def)
- qed (auto simp: bij_betw_def)
- fix i
- assume "i < length (z # xs)"
- then show "(z # xs) ! i = (z # ys) ! (?f i)"
- using perm by (cases i) auto
- qed
-next
- case (trans xs ys zs)
- then obtain f g
- where bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}"
- and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)"
- by blast
- show ?case
- proof (intro exI[of _ "g \<circ> f"] conjI allI impI)
- show "bij_betw (g \<circ> f) {..<length xs} {..<length zs}"
- using bij by (rule bij_betw_trans)
- fix i
- assume "i < length xs"
- with bij have "f i < length ys"
- unfolding bij_betw_def by force
- with \<open>i < length xs\<close> show "xs ! i = zs ! (g \<circ> f) i"
- using trans(1,3)[THEN perm_length] perm by auto
- qed
-qed
-
-proposition perm_finite: "finite {B. B <~~> A}"
-proof (rule finite_subset[where B="{xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"])
- show "finite {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"
- using finite_lists_length_le by blast
-next
- show "{B. B <~~> A} \<subseteq> {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"
- by (clarsimp simp add: perm_length perm_set_eq)
-qed
-
-proposition perm_swap:
- assumes "i < length xs" "j < length xs"
- shows "xs[i := xs ! j, j := xs ! i] <~~> xs"
- using assms by (simp add: mset_eq_perm[symmetric] mset_swap)
-
-end
--- a/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy Tue Feb 23 20:41:48 2021 +0000
+++ b/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy Tue Feb 23 20:41:48 2021 +0000
@@ -3,7 +3,7 @@
section \<open>Formalization of a Countermeasure by Koepf \& Duermuth 2009\<close>
theory Koepf_Duermuth_Countermeasure
- imports "HOL-Probability.Information" "HOL-Library.Permutation"
+ imports "HOL-Probability.Information" "HOL-Library.List_Permutation"
begin
lemma SIGMA_image_vimage: