src/HOL/Library/Permutations.thy
author wenzelm
Wed Mar 04 23:52:47 2009 +0100 (2009-03-04)
changeset 30267 171b3bd93c90
parent 30242 aea5d7fa7ef5
child 30488 5c4c3a9e9102
permissions -rw-r--r--
removed old/broken CVS Ids;
     1 (* Title:      Library/Permutations
     2    Author:     Amine Chaieb, University of Cambridge
     3 *)
     4 
     5 header {* Permutations, both general and specifically on finite sets.*}
     6 
     7 theory Permutations
     8 imports Main Finite_Cartesian_Product Parity Fact
     9 begin
    10 
    11   (* Why should I import Main just to solve the Typerep problem! *)
    12 
    13 definition permutes (infixr "permutes" 41) where
    14   "(p permutes S) \<longleftrightarrow> (\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y)"
    15 
    16 (* ------------------------------------------------------------------------- *)
    17 (* Transpositions.                                                           *)
    18 (* ------------------------------------------------------------------------- *)
    19 
    20 declare swap_self[simp]
    21 lemma swapid_sym: "Fun.swap a b id = Fun.swap b a id" 
    22   by (auto simp add: expand_fun_eq swap_def fun_upd_def)
    23 lemma swap_id_refl: "Fun.swap a a id = id" by simp
    24 lemma swap_id_sym: "Fun.swap a b id = Fun.swap b a id"
    25   by (rule ext, simp add: swap_def)
    26 lemma swap_id_idempotent[simp]: "Fun.swap a b id o Fun.swap a b id = id"
    27   by (rule ext, auto simp add: swap_def)
    28 
    29 lemma inv_unique_comp: assumes fg: "f o g = id" and gf: "g o f = id"
    30   shows "inv f = g"
    31   using fg gf inv_equality[of g f] by (auto simp add: expand_fun_eq)
    32 
    33 lemma inverse_swap_id: "inv (Fun.swap a b id) = Fun.swap a b id"
    34   by (rule inv_unique_comp, simp_all)
    35 
    36 lemma swap_id_eq: "Fun.swap a b id x = (if x = a then b else if x = b then a else x)"
    37   by (simp add: swap_def)
    38 
    39 (* ------------------------------------------------------------------------- *)
    40 (* Basic consequences of the definition.                                     *)
    41 (* ------------------------------------------------------------------------- *)
    42 
    43 lemma permutes_in_image: "p permutes S \<Longrightarrow> p x \<in> S \<longleftrightarrow> x \<in> S"
    44   unfolding permutes_def by metis
    45 
    46 lemma permutes_image: assumes pS: "p permutes S" shows "p ` S = S"
    47   using pS
    48   unfolding permutes_def 
    49   apply - 
    50   apply (rule set_ext) 
    51   apply (simp add: image_iff)
    52   apply metis
    53   done
    54 
    55 lemma permutes_inj: "p permutes S ==> inj p " 
    56   unfolding permutes_def inj_on_def by blast 
    57 
    58 lemma permutes_surj: "p permutes s ==> surj p" 
    59   unfolding permutes_def surj_def by metis 
    60 
    61 lemma permutes_inv_o: assumes pS: "p permutes S"
    62   shows " p o inv p = id"
    63   and "inv p o p = id"
    64   using permutes_inj[OF pS] permutes_surj[OF pS]
    65   unfolding inj_iff[symmetric] surj_iff[symmetric] by blast+
    66 
    67 
    68 lemma permutes_inverses: 
    69   fixes p :: "'a \<Rightarrow> 'a"
    70   assumes pS: "p permutes S"
    71   shows "p (inv p x) = x"
    72   and "inv p (p x) = x"
    73   using permutes_inv_o[OF pS, unfolded expand_fun_eq o_def] by auto
    74 
    75 lemma permutes_subset: "p permutes S \<Longrightarrow> S \<subseteq> T ==> p permutes T"
    76   unfolding permutes_def by blast
    77 
    78 lemma permutes_empty[simp]: "p permutes {} \<longleftrightarrow> p = id"
    79   unfolding expand_fun_eq permutes_def apply simp by metis 
    80 
    81 lemma permutes_sing[simp]: "p permutes {a} \<longleftrightarrow> p = id"
    82   unfolding expand_fun_eq permutes_def apply simp by metis
    83  
    84 lemma permutes_univ: "p permutes UNIV \<longleftrightarrow> (\<forall>y. \<exists>!x. p x = y)"
    85   unfolding permutes_def by simp
    86 
    87 lemma permutes_inv_eq: "p permutes S ==> inv p y = x \<longleftrightarrow> p x = y"
    88   unfolding permutes_def inv_def apply auto
    89   apply (erule allE[where x=y])
    90   apply (erule allE[where x=y])
    91   apply (rule someI_ex) apply blast
    92   apply (rule some1_equality)
    93   apply blast
    94   apply blast
    95   done
    96 
    97 lemma permutes_swap_id: "a \<in> S \<Longrightarrow> b \<in> S ==> Fun.swap a b id permutes S"
    98   unfolding permutes_def swap_def fun_upd_def  apply auto apply metis done
    99 
   100 lemma permutes_superset: "p permutes S \<Longrightarrow> (\<forall>x \<in> S - T. p x = x) \<Longrightarrow> p permutes T"
   101 apply (simp add: Ball_def permutes_def Diff_iff) by metis
   102 
   103 (* ------------------------------------------------------------------------- *)
   104 (* Group properties.                                                         *)
   105 (* ------------------------------------------------------------------------- *)
   106 
   107 lemma permutes_id: "id permutes S" unfolding permutes_def by simp 
   108 
   109 lemma permutes_compose: "p permutes S \<Longrightarrow> q permutes S ==> q o p permutes S"
   110   unfolding permutes_def o_def by metis
   111 
   112 lemma permutes_inv: assumes pS: "p permutes S" shows "inv p permutes S"
   113   using pS unfolding permutes_def permutes_inv_eq[OF pS] by metis  
   114 
   115 lemma permutes_inv_inv: assumes pS: "p permutes S" shows "inv (inv p) = p"
   116   unfolding expand_fun_eq permutes_inv_eq[OF pS] permutes_inv_eq[OF permutes_inv[OF pS]]
   117   by blast
   118 
   119 (* ------------------------------------------------------------------------- *)
   120 (* The number of permutations on a finite set.                               *)
   121 (* ------------------------------------------------------------------------- *)
   122 
   123 lemma permutes_insert_lemma: 
   124   assumes pS: "p permutes (insert a S)"
   125   shows "Fun.swap a (p a) id o p permutes S"
   126   apply (rule permutes_superset[where S = "insert a S"])
   127   apply (rule permutes_compose[OF pS])
   128   apply (rule permutes_swap_id, simp)
   129   using permutes_in_image[OF pS, of a] apply simp
   130   apply (auto simp add: Ball_def Diff_iff swap_def)
   131   done
   132 
   133 lemma permutes_insert: "{p. p permutes (insert a S)} =
   134         (\<lambda>(b,p). Fun.swap a b id o p) ` {(b,p). b \<in> insert a S \<and> p \<in> {p. p permutes S}}"
   135 proof-
   136 
   137   {fix p 
   138     {assume pS: "p permutes insert a S"
   139       let ?b = "p a"
   140       let ?q = "Fun.swap a (p a) id o p"
   141       have th0: "p = Fun.swap a ?b id o ?q" unfolding expand_fun_eq o_assoc by simp 
   142       have th1: "?b \<in> insert a S " unfolding permutes_in_image[OF pS] by simp 
   143       from permutes_insert_lemma[OF pS] th0 th1
   144       have "\<exists> b q. p = Fun.swap a b id o q \<and> b \<in> insert a S \<and> q permutes S" by blast}
   145     moreover
   146     {fix b q assume bq: "p = Fun.swap a b id o q" "b \<in> insert a S" "q permutes S"
   147       from permutes_subset[OF bq(3), of "insert a S"] 
   148       have qS: "q permutes insert a S" by auto
   149       have aS: "a \<in> insert a S" by simp
   150       from bq(1) permutes_compose[OF qS permutes_swap_id[OF aS bq(2)]]
   151       have "p permutes insert a S"  by simp }
   152     ultimately have "p permutes insert a S \<longleftrightarrow> (\<exists> b q. p = Fun.swap a b id o q \<and> b \<in> insert a S \<and> q permutes S)" by blast}
   153   thus ?thesis by auto
   154 qed
   155 
   156 lemma hassize_insert: "a \<notin> F \<Longrightarrow> insert a F hassize n \<Longrightarrow> F hassize (n - 1)"
   157   by (auto simp add: hassize_def)
   158 
   159 lemma hassize_permutations: assumes Sn: "S hassize n"
   160   shows "{p. p permutes S} hassize (fact n)"
   161 proof-
   162   from Sn have fS:"finite S" by (simp add: hassize_def)
   163 
   164   have "\<forall>n. (S hassize n) \<longrightarrow> ({p. p permutes S} hassize (fact n))"
   165   proof(rule finite_induct[where F = S])
   166     from fS show "finite S" .
   167   next
   168     show "\<forall>n. ({} hassize n) \<longrightarrow> ({p. p permutes {}} hassize fact n)"
   169       by (simp add: hassize_def permutes_empty)
   170   next
   171     fix x F 
   172     assume fF: "finite F" and xF: "x \<notin> F" 
   173       and H: "\<forall>n. (F hassize n) \<longrightarrow> ({p. p permutes F} hassize fact n)"
   174     {fix n assume H0: "insert x F hassize n"
   175       let ?xF = "{p. p permutes insert x F}"
   176       let ?pF = "{p. p permutes F}"
   177       let ?pF' = "{(b, p). b \<in> insert x F \<and> p \<in> ?pF}"
   178       let ?g = "(\<lambda>(b, p). Fun.swap x b id \<circ> p)"
   179       from permutes_insert[of x F]
   180       have xfgpF': "?xF = ?g ` ?pF'" .
   181       from hassize_insert[OF xF H0] have Fs: "F hassize (n - 1)" .
   182       from H Fs have pFs: "?pF hassize fact (n - 1)" by blast
   183       hence pF'f: "finite ?pF'" using H0 unfolding hassize_def 
   184 	apply (simp only: Collect_split Collect_mem_eq)
   185 	apply (rule finite_cartesian_product)
   186 	apply simp_all
   187 	done
   188 
   189       have ginj: "inj_on ?g ?pF'"
   190       proof-
   191 	{
   192 	  fix b p c q assume bp: "(b,p) \<in> ?pF'" and cq: "(c,q) \<in> ?pF'" 
   193 	    and eq: "?g (b,p) = ?g (c,q)"
   194 	  from bp cq have ths: "b \<in> insert x F" "c \<in> insert x F" "x \<in> insert x F" "p permutes F" "q permutes F" by auto
   195 	  from ths(4) xF eq have "b = ?g (b,p) x" unfolding permutes_def 
   196 	    by (auto simp add: swap_def fun_upd_def expand_fun_eq)
   197 	  also have "\<dots> = ?g (c,q) x" using ths(5) xF eq  
   198 	    by (auto simp add: swap_def fun_upd_def expand_fun_eq)
   199 	  also have "\<dots> = c"using ths(5) xF unfolding permutes_def
   200 	    by (auto simp add: swap_def fun_upd_def expand_fun_eq)
   201 	  finally have bc: "b = c" .
   202 	  hence "Fun.swap x b id = Fun.swap x c id" by simp
   203 	  with eq have "Fun.swap x b id o p = Fun.swap x b id o q" by simp
   204 	  hence "Fun.swap x b id o (Fun.swap x b id o p) = Fun.swap x b id o (Fun.swap x b id o q)" by simp
   205 	  hence "p = q" by (simp add: o_assoc)
   206 	  with bc have "(b,p) = (c,q)" by simp }
   207 	thus ?thesis  unfolding inj_on_def by blast
   208       qed
   209       from xF H0 have n0: "n \<noteq> 0 " by (auto simp add: hassize_def)
   210       hence "\<exists>m. n = Suc m" by arith
   211       then obtain m where n[simp]: "n = Suc m" by blast 
   212       from pFs H0 have xFc: "card ?xF = fact n" 
   213 	unfolding xfgpF' card_image[OF ginj] hassize_def
   214 	apply (simp only: Collect_split Collect_mem_eq card_cartesian_product)
   215 	by simp
   216       from finite_imageI[OF pF'f, of ?g] have xFf: "finite ?xF" unfolding xfgpF' by simp 
   217       have "?xF hassize fact n"
   218 	using xFf xFc 
   219 	unfolding hassize_def  xFf by blast }
   220     thus "\<forall>n. (insert x F hassize n) \<longrightarrow> ({p. p permutes insert x F} hassize fact n)" 
   221       by blast
   222   qed
   223   with Sn show ?thesis by blast
   224 qed
   225 
   226 lemma finite_permutations: "finite S ==> finite {p. p permutes S}"
   227   using hassize_permutations[of S] unfolding hassize_def by blast
   228 
   229 (* ------------------------------------------------------------------------- *)
   230 (* Permutations of index set for iterated operations.                        *)
   231 (* ------------------------------------------------------------------------- *)
   232 
   233 lemma (in ab_semigroup_mult) fold_image_permute: assumes fS: "finite S" and pS: "p permutes S" 
   234   shows "fold_image times f z S = fold_image times (f o p) z S"
   235   using fold_image_reindex[OF fS subset_inj_on[OF permutes_inj[OF pS], of S, simplified], of f z]
   236   unfolding permutes_image[OF pS] .
   237 lemma (in ab_semigroup_add) fold_image_permute: assumes fS: "finite S" and pS: "p permutes S" 
   238   shows "fold_image plus f z S = fold_image plus (f o p) z S"
   239 proof-
   240   interpret ab_semigroup_mult plus apply unfold_locales apply (simp add: add_assoc)
   241     apply (simp add: add_commute) done
   242   from fold_image_reindex[OF fS subset_inj_on[OF permutes_inj[OF pS], of S, simplified], of f z]
   243   show ?thesis
   244   unfolding permutes_image[OF pS] .
   245 qed
   246 
   247 lemma setsum_permute: assumes pS: "p permutes S" 
   248   shows "setsum f S = setsum (f o p) S"
   249   unfolding setsum_def using fold_image_permute[of S p f 0] pS by clarsimp
   250 
   251 lemma setsum_permute_natseg:assumes pS: "p permutes {m .. n}" 
   252   shows "setsum f {m .. n} = setsum (f o p) {m .. n}"
   253   using setsum_permute[OF pS, of f ] pS by blast 
   254 
   255 lemma setprod_permute: assumes pS: "p permutes S" 
   256   shows "setprod f S = setprod (f o p) S"
   257   unfolding setprod_def 
   258   using ab_semigroup_mult_class.fold_image_permute[of S p f 1] pS by clarsimp
   259 
   260 lemma setprod_permute_natseg:assumes pS: "p permutes {m .. n}" 
   261   shows "setprod f {m .. n} = setprod (f o p) {m .. n}"
   262   using setprod_permute[OF pS, of f ] pS by blast 
   263 
   264 (* ------------------------------------------------------------------------- *)
   265 (* Various combinations of transpositions with 2, 1 and 0 common elements.   *)
   266 (* ------------------------------------------------------------------------- *)
   267 
   268 lemma swap_id_common:" a \<noteq> c \<Longrightarrow> b \<noteq> c \<Longrightarrow>  Fun.swap a b id o Fun.swap a c id = Fun.swap b c id o Fun.swap a b id" by (simp add: expand_fun_eq swap_def)
   269 
   270 lemma swap_id_common': "~(a = b) \<Longrightarrow> ~(a = c) \<Longrightarrow> Fun.swap a c id o Fun.swap b c id = Fun.swap b c id o Fun.swap a b id" by (simp add: expand_fun_eq swap_def)
   271 
   272 lemma swap_id_independent: "~(a = c) \<Longrightarrow> ~(a = d) \<Longrightarrow> ~(b = c) \<Longrightarrow> ~(b = d) ==> Fun.swap a b id o Fun.swap c d id = Fun.swap c d id o Fun.swap a b id"
   273   by (simp add: swap_def expand_fun_eq)
   274 
   275 (* ------------------------------------------------------------------------- *)
   276 (* Permutations as transposition sequences.                                  *)
   277 (* ------------------------------------------------------------------------- *)
   278 
   279 
   280 inductive swapidseq :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool" where
   281   id[simp]: "swapidseq 0 id"
   282 | comp_Suc: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (Fun.swap a b id o p)"
   283 
   284 declare id[unfolded id_def, simp]
   285 definition "permutation p \<longleftrightarrow> (\<exists>n. swapidseq n p)"
   286 
   287 (* ------------------------------------------------------------------------- *)
   288 (* Some closure properties of the set of permutations, with lengths.         *)
   289 (* ------------------------------------------------------------------------- *)
   290 
   291 lemma permutation_id[simp]: "permutation id"unfolding permutation_def
   292   by (rule exI[where x=0], simp)
   293 declare permutation_id[unfolded id_def, simp]
   294 
   295 lemma swapidseq_swap: "swapidseq (if a = b then 0 else 1) (Fun.swap a b id)"
   296   apply clarsimp
   297   using comp_Suc[of 0 id a b] by simp
   298 
   299 lemma permutation_swap_id: "permutation (Fun.swap a b id)"
   300   apply (cases "a=b", simp_all)
   301   unfolding permutation_def using swapidseq_swap[of a b] by blast 
   302 
   303 lemma swapidseq_comp_add: "swapidseq n p \<Longrightarrow> swapidseq m q ==> swapidseq (n + m) (p o q)"
   304   proof (induct n p arbitrary: m q rule: swapidseq.induct)
   305     case (id m q) thus ?case by simp
   306   next
   307     case (comp_Suc n p a b m q) 
   308     have th: "Suc n + m = Suc (n + m)" by arith
   309     show ?case unfolding th o_assoc[symmetric] 
   310       apply (rule swapidseq.comp_Suc) using comp_Suc.hyps(2)[OF comp_Suc.prems]  comp_Suc.hyps(3) by blast+ 
   311 qed
   312 
   313 lemma permutation_compose: "permutation p \<Longrightarrow> permutation q ==> permutation(p o q)"
   314   unfolding permutation_def using swapidseq_comp_add[of _ p _ q] by metis
   315 
   316 lemma swapidseq_endswap: "swapidseq n p \<Longrightarrow> a \<noteq> b ==> swapidseq (Suc n) (p o Fun.swap a b id)"
   317   apply (induct n p rule: swapidseq.induct)
   318   using swapidseq_swap[of a b]
   319   by (auto simp add: o_assoc[symmetric] intro: swapidseq.comp_Suc)
   320 
   321 lemma swapidseq_inverse_exists: "swapidseq n p ==> \<exists>q. swapidseq n q \<and> p o q = id \<and> q o p = id"
   322 proof(induct n p rule: swapidseq.induct)
   323   case id  thus ?case by (rule exI[where x=id], simp)
   324 next 
   325   case (comp_Suc n p a b)
   326   from comp_Suc.hyps obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" by blast
   327   let ?q = "q o Fun.swap a b id"
   328   note H = comp_Suc.hyps
   329   from swapidseq_swap[of a b] H(3)  have th0: "swapidseq 1 (Fun.swap a b id)" by simp
   330   from swapidseq_comp_add[OF q(1) th0] have th1:"swapidseq (Suc n) ?q" by simp 
   331   have "Fun.swap a b id o p o ?q = Fun.swap a b id o (p o q) o Fun.swap a b id" by (simp add: o_assoc)
   332   also have "\<dots> = id" by (simp add: q(2))
   333   finally have th2: "Fun.swap a b id o p o ?q = id" .
   334   have "?q \<circ> (Fun.swap a b id \<circ> p) = q \<circ> (Fun.swap a b id o Fun.swap a b id) \<circ> p" by (simp only: o_assoc) 
   335   hence "?q \<circ> (Fun.swap a b id \<circ> p) = id" by (simp add: q(3))
   336   with th1 th2 show ?case by blast
   337 qed
   338 
   339 
   340 lemma swapidseq_inverse: assumes H: "swapidseq n p" shows "swapidseq n (inv p)"
   341   using swapidseq_inverse_exists[OF H] inv_unique_comp[of p] by auto
   342 
   343 lemma permutation_inverse: "permutation p ==> permutation (inv p)"
   344   using permutation_def swapidseq_inverse by blast
   345 
   346 (* ------------------------------------------------------------------------- *)
   347 (* The identity map only has even transposition sequences.                   *)
   348 (* ------------------------------------------------------------------------- *)
   349 
   350 lemma symmetry_lemma:"(\<And>a b c d. P a b c d ==> P a b d c) \<Longrightarrow>
   351    (\<And>a b c d. a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow> (a = c \<and> b = d \<or>  a = c \<and> b \<noteq> d \<or> a \<noteq> c \<and> b = d \<or> a \<noteq> c \<and> a \<noteq> d \<and> b \<noteq> c \<and> b \<noteq> d) ==> P a b c d)
   352    ==> (\<And>a b c d. a \<noteq> b --> c \<noteq> d \<longrightarrow>  P a b c d)" by metis
   353 
   354 lemma swap_general: "a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow> Fun.swap a b id o Fun.swap c d id = id \<or> 
   355   (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id)" 
   356 proof-
   357   assume H: "a\<noteq>b" "c\<noteq>d"
   358 have "a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow> 
   359 (  Fun.swap a b id o Fun.swap c d id = id \<or> 
   360   (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id))" 
   361   apply (rule symmetry_lemma[where a=a and b=b and c=c and d=d])
   362   apply (simp_all only: swapid_sym) 
   363   apply (case_tac "a = c \<and> b = d", clarsimp simp only: swapid_sym swap_id_idempotent)
   364   apply (case_tac "a = c \<and> b \<noteq> d")
   365   apply (rule disjI2)
   366   apply (rule_tac x="b" in exI)
   367   apply (rule_tac x="d" in exI)
   368   apply (rule_tac x="b" in exI)
   369   apply (clarsimp simp add: expand_fun_eq swap_def)
   370   apply (case_tac "a \<noteq> c \<and> b = d")
   371   apply (rule disjI2)
   372   apply (rule_tac x="c" in exI)
   373   apply (rule_tac x="d" in exI)
   374   apply (rule_tac x="c" in exI)
   375   apply (clarsimp simp add: expand_fun_eq swap_def)
   376   apply (rule disjI2)
   377   apply (rule_tac x="c" in exI)
   378   apply (rule_tac x="d" in exI)
   379   apply (rule_tac x="b" in exI)
   380   apply (clarsimp simp add: expand_fun_eq swap_def)
   381   done
   382 with H show ?thesis by metis 
   383 qed
   384 
   385 lemma swapidseq_id_iff[simp]: "swapidseq 0 p \<longleftrightarrow> p = id"
   386   using swapidseq.cases[of 0 p "p = id"]
   387   by auto
   388 
   389 lemma swapidseq_cases: "swapidseq n p \<longleftrightarrow> (n=0 \<and> p = id \<or> (\<exists>a b q m. n = Suc m \<and> p = Fun.swap a b id o q \<and> swapidseq m q \<and> a\<noteq> b))"
   390   apply (rule iffI)
   391   apply (erule swapidseq.cases[of n p])
   392   apply simp
   393   apply (rule disjI2)
   394   apply (rule_tac x= "a" in exI)
   395   apply (rule_tac x= "b" in exI)
   396   apply (rule_tac x= "pa" in exI)
   397   apply (rule_tac x= "na" in exI)
   398   apply simp
   399   apply auto
   400   apply (rule comp_Suc, simp_all)
   401   done
   402 lemma fixing_swapidseq_decrease:
   403   assumes spn: "swapidseq n p" and ab: "a\<noteq>b" and pa: "(Fun.swap a b id o p) a = a"
   404   shows "n \<noteq> 0 \<and> swapidseq (n - 1) (Fun.swap a b id o p)"
   405   using spn ab pa
   406 proof(induct n arbitrary: p a b)
   407   case 0 thus ?case by (auto simp add: swap_def fun_upd_def)
   408 next
   409   case (Suc n p a b)
   410   from Suc.prems(1) swapidseq_cases[of "Suc n" p] obtain
   411     c d q m where cdqm: "Suc n = Suc m" "p = Fun.swap c d id o q" "swapidseq m q" "c \<noteq> d" "n = m"
   412     by auto
   413   {assume H: "Fun.swap a b id o Fun.swap c d id = id"
   414     
   415     have ?case apply (simp only: cdqm o_assoc H) 
   416       by (simp add: cdqm)}
   417   moreover
   418   { fix x y z
   419     assume H: "x\<noteq>a" "y\<noteq>a" "z \<noteq>a" "x \<noteq>y" 
   420       "Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id"
   421     from H have az: "a \<noteq> z" by simp
   422 
   423     {fix h have "(Fun.swap x y id o h) a = a \<longleftrightarrow> h a = a"
   424       using H by (simp add: swap_def)}
   425     note th3 = this
   426     from cdqm(2) have "Fun.swap a b id o p = Fun.swap a b id o (Fun.swap c d id o q)" by simp
   427     hence "Fun.swap a b id o p = Fun.swap x y id o (Fun.swap a z id o q)" by (simp add: o_assoc H)
   428     hence "(Fun.swap a b id o p) a = (Fun.swap x y id o (Fun.swap a z id o q)) a" by simp
   429     hence "(Fun.swap x y id o (Fun.swap a z id o q)) a  = a" unfolding Suc by metis
   430     hence th1: "(Fun.swap a z id o q) a = a" unfolding th3 .
   431     from Suc.hyps[OF cdqm(3)[ unfolded cdqm(5)[symmetric]] az th1]
   432     have th2: "swapidseq (n - 1) (Fun.swap a z id o q)" "n \<noteq> 0" by blast+
   433     have th: "Suc n - 1 = Suc (n - 1)" using th2(2) by auto 
   434     have ?case unfolding cdqm(2) H o_assoc th
   435       apply (simp only: Suc_not_Zero simp_thms o_assoc[symmetric])
   436       apply (rule comp_Suc)
   437       using th2 H apply blast+
   438       done}
   439   ultimately show ?case using swap_general[OF Suc.prems(2) cdqm(4)] by metis 
   440 qed
   441 
   442 lemma swapidseq_identity_even: 
   443   assumes "swapidseq n (id :: 'a \<Rightarrow> 'a)" shows "even n"
   444   using `swapidseq n id`
   445 proof(induct n rule: nat_less_induct)
   446   fix n
   447   assume H: "\<forall>m<n. swapidseq m (id::'a \<Rightarrow> 'a) \<longrightarrow> even m" "swapidseq n (id :: 'a \<Rightarrow> 'a)"
   448   {assume "n = 0" hence "even n" by arith} 
   449   moreover 
   450   {fix a b :: 'a and q m
   451     assume h: "n = Suc m" "(id :: 'a \<Rightarrow> 'a) = Fun.swap a b id \<circ> q" "swapidseq m q" "a \<noteq> b"
   452     from fixing_swapidseq_decrease[OF h(3,4), unfolded h(2)[symmetric]]
   453     have m: "m \<noteq> 0" "swapidseq (m - 1) (id :: 'a \<Rightarrow> 'a)" by auto
   454     from h m have mn: "m - 1 < n" by arith
   455     from H(1)[rule_format, OF mn m(2)] h(1) m(1) have "even n" apply arith done}
   456   ultimately show "even n" using H(2)[unfolded swapidseq_cases[of n id]] by auto
   457 qed
   458 
   459 (* ------------------------------------------------------------------------- *)
   460 (* Therefore we have a welldefined notion of parity.                         *)
   461 (* ------------------------------------------------------------------------- *)
   462 
   463 definition "evenperm p = even (SOME n. swapidseq n p)"
   464 
   465 lemma swapidseq_even_even: assumes 
   466   m: "swapidseq m p" and n: "swapidseq n p"
   467   shows "even m \<longleftrightarrow> even n"
   468 proof-
   469   from swapidseq_inverse_exists[OF n]
   470   obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" by blast
   471   
   472   from swapidseq_identity_even[OF swapidseq_comp_add[OF m q(1), unfolded q]]
   473   show ?thesis by arith
   474 qed
   475 
   476 lemma evenperm_unique: assumes p: "swapidseq n p" and n:"even n = b"
   477   shows "evenperm p = b"
   478   unfolding n[symmetric] evenperm_def
   479   apply (rule swapidseq_even_even[where p = p])
   480   apply (rule someI[where x = n])
   481   using p by blast+
   482 
   483 (* ------------------------------------------------------------------------- *)
   484 (* And it has the expected composition properties.                           *)
   485 (* ------------------------------------------------------------------------- *)
   486 
   487 lemma evenperm_id[simp]: "evenperm id = True"
   488   apply (rule evenperm_unique[where n = 0]) by simp_all
   489 
   490 lemma evenperm_swap: "evenperm (Fun.swap a b id) = (a = b)"
   491 apply (rule evenperm_unique[where n="if a = b then 0 else 1"])
   492 by (simp_all add: swapidseq_swap)
   493 
   494 lemma evenperm_comp: 
   495   assumes p: "permutation p" and q:"permutation q"
   496   shows "evenperm (p o q) = (evenperm p = evenperm q)"
   497 proof-
   498   from p q obtain 
   499     n m where n: "swapidseq n p" and m: "swapidseq m q" 
   500     unfolding permutation_def by blast
   501   note nm =  swapidseq_comp_add[OF n m]
   502   have th: "even (n + m) = (even n \<longleftrightarrow> even m)" by arith
   503   from evenperm_unique[OF n refl] evenperm_unique[OF m refl]
   504     evenperm_unique[OF nm th]
   505   show ?thesis by blast
   506 qed
   507 
   508 lemma evenperm_inv: assumes p: "permutation p"
   509   shows "evenperm (inv p) = evenperm p"
   510 proof-
   511   from p obtain n where n: "swapidseq n p" unfolding permutation_def by blast
   512   from evenperm_unique[OF swapidseq_inverse[OF n] evenperm_unique[OF n refl, symmetric]]
   513   show ?thesis .
   514 qed
   515 
   516 (* ------------------------------------------------------------------------- *)
   517 (* A more abstract characterization of permutations.                         *)
   518 (* ------------------------------------------------------------------------- *)
   519 
   520 
   521 lemma bij_iff: "bij f \<longleftrightarrow> (\<forall>x. \<exists>!y. f y = x)"
   522   unfolding bij_def inj_on_def surj_def
   523   apply auto
   524   apply metis
   525   apply metis
   526   done
   527 
   528 lemma permutation_bijective: 
   529   assumes p: "permutation p" 
   530   shows "bij p"
   531 proof-
   532   from p obtain n where n: "swapidseq n p" unfolding permutation_def by blast
   533   from swapidseq_inverse_exists[OF n] obtain q where 
   534     q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" by blast
   535   thus ?thesis unfolding bij_iff  apply (auto simp add: expand_fun_eq) apply metis done
   536 qed  
   537 
   538 lemma permutation_finite_support: assumes p: "permutation p"
   539   shows "finite {x. p x \<noteq> x}"
   540 proof-
   541   from p obtain n where n: "swapidseq n p" unfolding permutation_def by blast
   542   from n show ?thesis
   543   proof(induct n p rule: swapidseq.induct)
   544     case id thus ?case by simp
   545   next
   546     case (comp_Suc n p a b)
   547     let ?S = "insert a (insert b {x. p x \<noteq> x})"
   548     from comp_Suc.hyps(2) have fS: "finite ?S" by simp
   549     from `a \<noteq> b` have th: "{x. (Fun.swap a b id o p) x \<noteq> x} \<subseteq> ?S"
   550       by (auto simp add: swap_def)
   551     from finite_subset[OF th fS] show ?case  .
   552 qed
   553 qed
   554 
   555 lemma bij_inv_eq_iff: "bij p ==> x = inv p y \<longleftrightarrow> p x = y"
   556   using surj_f_inv_f[of p] inv_f_f[of f] by (auto simp add: bij_def)
   557 
   558 lemma bij_swap_comp: 
   559   assumes bp: "bij p" shows "Fun.swap a b id o p = Fun.swap (inv p a) (inv p b) p"
   560   using surj_f_inv_f[OF bij_is_surj[OF bp]]
   561   by (simp add: expand_fun_eq swap_def bij_inv_eq_iff[OF bp])
   562 
   563 lemma bij_swap_ompose_bij: "bij p \<Longrightarrow> bij (Fun.swap a b id o p)"
   564 proof-
   565   assume H: "bij p"
   566   show ?thesis 
   567     unfolding bij_swap_comp[OF H] bij_swap_iff
   568     using H .
   569 qed
   570 
   571 lemma permutation_lemma: 
   572   assumes fS: "finite S" and p: "bij p" and pS: "\<forall>x. x\<notin> S \<longrightarrow> p x = x"
   573   shows "permutation p"
   574 using fS p pS
   575 proof(induct S arbitrary: p rule: finite_induct)
   576   case (empty p) thus ?case by simp
   577 next
   578   case (insert a F p)
   579   let ?r = "Fun.swap a (p a) id o p"
   580   let ?q = "Fun.swap a (p a) id o ?r "
   581   have raa: "?r a = a" by (simp add: swap_def)
   582   from bij_swap_ompose_bij[OF insert(4)]
   583   have br: "bij ?r"  . 
   584   
   585   from insert raa have th: "\<forall>x. x \<notin> F \<longrightarrow> ?r x = x"    
   586     apply (clarsimp simp add: swap_def)
   587     apply (erule_tac x="x" in allE)
   588     apply auto
   589     unfolding bij_iff apply metis
   590     done
   591   from insert(3)[OF br th]
   592   have rp: "permutation ?r" .
   593   have "permutation ?q" by (simp add: permutation_compose permutation_swap_id rp)
   594   thus ?case by (simp add: o_assoc)
   595 qed
   596 
   597 lemma permutation: "permutation p \<longleftrightarrow> bij p \<and> finite {x. p x \<noteq> x}" 
   598   (is "?lhs \<longleftrightarrow> ?b \<and> ?f")
   599 proof
   600   assume p: ?lhs
   601   from p permutation_bijective permutation_finite_support show "?b \<and> ?f" by auto
   602 next
   603   assume bf: "?b \<and> ?f"
   604   hence bf: "?f" "?b" by blast+
   605   from permutation_lemma[OF bf] show ?lhs by blast
   606 qed
   607 
   608 lemma permutation_inverse_works: assumes p: "permutation p"
   609   shows "inv p o p = id" "p o inv p = id"
   610 using permutation_bijective[OF p] surj_iff bij_def inj_iff by auto
   611 
   612 lemma permutation_inverse_compose:
   613   assumes p: "permutation p" and q: "permutation q"
   614   shows "inv (p o q) = inv q o inv p"
   615 proof-
   616   note ps = permutation_inverse_works[OF p]
   617   note qs = permutation_inverse_works[OF q]
   618   have "p o q o (inv q o inv p) = p o (q o inv q) o inv p" by (simp add: o_assoc)
   619   also have "\<dots> = id" by (simp add: ps qs)
   620   finally have th0: "p o q o (inv q o inv p) = id" .
   621   have "inv q o inv p o (p o q) = inv q o (inv p o p) o q" by (simp add: o_assoc)
   622   also have "\<dots> = id" by (simp add: ps qs)
   623   finally have th1: "inv q o inv p o (p o q) = id" . 
   624   from inv_unique_comp[OF th0 th1] show ?thesis .
   625 qed
   626 
   627 (* ------------------------------------------------------------------------- *)
   628 (* Relation to "permutes".                                                   *)
   629 (* ------------------------------------------------------------------------- *)
   630 
   631 lemma permutation_permutes: "permutation p \<longleftrightarrow> (\<exists>S. finite S \<and> p permutes S)"
   632 unfolding permutation permutes_def bij_iff[symmetric]
   633 apply (rule iffI, clarify)
   634 apply (rule exI[where x="{x. p x \<noteq> x}"])
   635 apply simp
   636 apply clarsimp
   637 apply (rule_tac B="S" in finite_subset)
   638 apply auto
   639 done
   640 
   641 (* ------------------------------------------------------------------------- *)
   642 (* Hence a sort of induction principle composing by swaps.                   *)
   643 (* ------------------------------------------------------------------------- *)
   644 
   645 lemma permutes_induct: "finite S \<Longrightarrow>  P id  \<Longrightarrow> (\<And> a b p. a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> P p \<Longrightarrow> P p \<Longrightarrow> permutation p ==> P (Fun.swap a b id o p))
   646          ==> (\<And>p. p permutes S ==> P p)"
   647 proof(induct S rule: finite_induct)
   648   case empty thus ?case by auto
   649 next 
   650   case (insert x F p)
   651   let ?r = "Fun.swap x (p x) id o p"
   652   let ?q = "Fun.swap x (p x) id o ?r"
   653   have qp: "?q = p" by (simp add: o_assoc)
   654   from permutes_insert_lemma[OF insert.prems(3)] insert have Pr: "P ?r" by blast
   655   from permutes_in_image[OF insert.prems(3), of x] 
   656   have pxF: "p x \<in> insert x F" by simp
   657   have xF: "x \<in> insert x F" by simp
   658   have rp: "permutation ?r"
   659     unfolding permutation_permutes using insert.hyps(1) 
   660       permutes_insert_lemma[OF insert.prems(3)] by blast
   661   from insert.prems(2)[OF xF pxF Pr Pr rp] 
   662   show ?case  unfolding qp . 
   663 qed
   664 
   665 (* ------------------------------------------------------------------------- *)
   666 (* Sign of a permutation as a real number.                                   *)
   667 (* ------------------------------------------------------------------------- *)
   668 
   669 definition "sign p = (if evenperm p then (1::int) else -1)"
   670 
   671 lemma sign_nz: "sign p \<noteq> 0" by (simp add: sign_def) 
   672 lemma sign_id: "sign id = 1" by (simp add: sign_def)
   673 lemma sign_inverse: "permutation p ==> sign (inv p) = sign p"
   674   by (simp add: sign_def evenperm_inv)
   675 lemma sign_compose: "permutation p \<Longrightarrow> permutation q ==> sign (p o q) = sign(p) * sign(q)" by (simp add: sign_def evenperm_comp)
   676 lemma sign_swap_id: "sign (Fun.swap a b id) = (if a = b then 1 else -1)"
   677   by (simp add: sign_def evenperm_swap)
   678 lemma sign_idempotent: "sign p * sign p = 1" by (simp add: sign_def)
   679 
   680 (* ------------------------------------------------------------------------- *)
   681 (* More lemmas about permutations.                                           *)
   682 (* ------------------------------------------------------------------------- *)
   683 
   684 lemma permutes_natset_le:
   685   assumes p: "p permutes (S::'a::wellorder set)" and le: "\<forall>i \<in> S.  p i <= i" shows "p = id"
   686 proof-
   687   {fix n
   688     have "p n = n" 
   689       using p le
   690     proof(induct n arbitrary: S rule: less_induct)
   691       fix n S assume H: "\<And>m S. \<lbrakk>m < n; p permutes S; \<forall>i\<in>S. p i \<le> i\<rbrakk> \<Longrightarrow> p m = m" 
   692 	"p permutes S" "\<forall>i \<in>S. p i \<le> i"
   693       {assume "n \<notin> S"
   694 	with H(2) have "p n = n" unfolding permutes_def by metis}
   695       moreover
   696       {assume ns: "n \<in> S"
   697 	from H(3)  ns have "p n < n \<or> p n = n" by auto 
   698 	moreover{assume h: "p n < n"
   699 	  from H h have "p (p n) = p n" by metis
   700 	  with permutes_inj[OF H(2)] have "p n = n" unfolding inj_on_def by blast
   701 	  with h have False by simp}
   702 	ultimately have "p n = n" by blast }
   703       ultimately show "p n = n"  by blast
   704     qed}
   705   thus ?thesis by (auto simp add: expand_fun_eq)
   706 qed
   707 
   708 lemma permutes_natset_ge:
   709   assumes p: "p permutes (S::'a::wellorder set)" and le: "\<forall>i \<in> S.  p i \<ge> i" shows "p = id"
   710 proof-
   711   {fix i assume i: "i \<in> S"
   712     from i permutes_in_image[OF permutes_inv[OF p]] have "inv p i \<in> S" by simp
   713     with le have "p (inv p i) \<ge> inv p i" by blast
   714     with permutes_inverses[OF p] have "i \<ge> inv p i" by simp}
   715   then have th: "\<forall>i\<in>S. inv p i \<le> i"  by blast
   716   from permutes_natset_le[OF permutes_inv[OF p] th] 
   717   have "inv p = inv id" by simp
   718   then show ?thesis 
   719     apply (subst permutes_inv_inv[OF p, symmetric])
   720     apply (rule inv_unique_comp)
   721     apply simp_all
   722     done
   723 qed
   724 
   725 lemma image_inverse_permutations: "{inv p |p. p permutes S} = {p. p permutes S}"
   726 apply (rule set_ext)
   727 apply auto
   728   using permutes_inv_inv permutes_inv apply auto
   729   apply (rule_tac x="inv x" in exI)
   730   apply auto
   731   done
   732 
   733 lemma image_compose_permutations_left: 
   734   assumes q: "q permutes S" shows "{q o p | p. p permutes S} = {p . p permutes S}"
   735 apply (rule set_ext)
   736 apply auto
   737 apply (rule permutes_compose)
   738 using q apply auto
   739 apply (rule_tac x = "inv q o x" in exI)
   740 by (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o)
   741 
   742 lemma image_compose_permutations_right:
   743   assumes q: "q permutes S"
   744   shows "{p o q | p. p permutes S} = {p . p permutes S}"
   745 apply (rule set_ext)
   746 apply auto
   747 apply (rule permutes_compose)
   748 using q apply auto
   749 apply (rule_tac x = "x o inv q" in exI)
   750 by (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o o_assoc[symmetric])
   751 
   752 lemma permutes_in_seg: "p permutes {1 ..n} \<Longrightarrow> i \<in> {1..n} ==> 1 <= p i \<and> p i <= n"
   753 
   754 apply (simp add: permutes_def)
   755 apply metis
   756 done
   757 
   758 term setsum
   759 lemma setsum_permutations_inverse: "setsum f {p. p permutes S} = setsum (\<lambda>p. f(inv p)) {p. p permutes S}" (is "?lhs = ?rhs")
   760 proof-
   761   let ?S = "{p . p permutes S}"
   762 have th0: "inj_on inv ?S" 
   763 proof(auto simp add: inj_on_def)
   764   fix q r
   765   assume q: "q permutes S" and r: "r permutes S" and qr: "inv q = inv r"
   766   hence "inv (inv q) = inv (inv r)" by simp
   767   with permutes_inv_inv[OF q] permutes_inv_inv[OF r]
   768   show "q = r" by metis
   769 qed
   770   have th1: "inv ` ?S = ?S" using image_inverse_permutations by blast
   771   have th2: "?rhs = setsum (f o inv) ?S" by (simp add: o_def)
   772   from setsum_reindex[OF th0, of f]  show ?thesis unfolding th1 th2 .
   773 qed
   774 
   775 lemma setum_permutations_compose_left:
   776   assumes q: "q permutes S"
   777   shows "setsum f {p. p permutes S} =
   778             setsum (\<lambda>p. f(q o p)) {p. p permutes S}" (is "?lhs = ?rhs")
   779 proof-
   780   let ?S = "{p. p permutes S}"
   781   have th0: "?rhs = setsum (f o (op o q)) ?S" by (simp add: o_def)
   782   have th1: "inj_on (op o q) ?S"
   783     apply (auto simp add: inj_on_def)
   784   proof-
   785     fix p r
   786     assume "p permutes S" and r:"r permutes S" and rp: "q \<circ> p = q \<circ> r"
   787     hence "inv q o q o p = inv q o q o r" by (simp add: o_assoc[symmetric])
   788     with permutes_inj[OF q, unfolded inj_iff]
   789 
   790     show "p = r" by simp
   791   qed
   792   have th3: "(op o q) ` ?S = ?S" using image_compose_permutations_left[OF q] by auto
   793   from setsum_reindex[OF th1, of f]
   794   show ?thesis unfolding th0 th1 th3 .
   795 qed
   796 
   797 lemma sum_permutations_compose_right:
   798   assumes q: "q permutes S"
   799   shows "setsum f {p. p permutes S} =
   800             setsum (\<lambda>p. f(p o q)) {p. p permutes S}" (is "?lhs = ?rhs")
   801 proof-
   802   let ?S = "{p. p permutes S}"
   803   have th0: "?rhs = setsum (f o (\<lambda>p. p o q)) ?S" by (simp add: o_def)
   804   have th1: "inj_on (\<lambda>p. p o q) ?S"
   805     apply (auto simp add: inj_on_def)
   806   proof-
   807     fix p r
   808     assume "p permutes S" and r:"r permutes S" and rp: "p o q = r o q"
   809     hence "p o (q o inv q)  = r o (q o inv q)" by (simp add: o_assoc)
   810     with permutes_surj[OF q, unfolded surj_iff]
   811 
   812     show "p = r" by simp
   813   qed
   814   have th3: "(\<lambda>p. p o q) ` ?S = ?S" using image_compose_permutations_right[OF q] by auto
   815   from setsum_reindex[OF th1, of f]
   816   show ?thesis unfolding th0 th1 th3 .
   817 qed
   818 
   819 (* ------------------------------------------------------------------------- *)
   820 (* Sum over a set of permutations (could generalize to iteration).           *)
   821 (* ------------------------------------------------------------------------- *)
   822 
   823 lemma setsum_over_permutations_insert:
   824   assumes fS: "finite S" and aS: "a \<notin> S"
   825   shows "setsum f {p. p permutes (insert a S)} = setsum (\<lambda>b. setsum (\<lambda>q. f (Fun.swap a b id o q)) {p. p permutes S}) (insert a S)"
   826 proof-
   827   have th0: "\<And>f a b. (\<lambda>(b,p). f (Fun.swap a b id o p)) = f o (\<lambda>(b,p). Fun.swap a b id o p)"
   828     by (simp add: expand_fun_eq)
   829   have th1: "\<And>P Q. P \<times> Q = {(a,b). a \<in> P \<and> b \<in> Q}" by blast
   830   have th2: "\<And>P Q. P \<Longrightarrow> (P \<Longrightarrow> Q) \<Longrightarrow> P \<and> Q" by blast
   831   show ?thesis 
   832     unfolding permutes_insert    
   833     unfolding setsum_cartesian_product
   834     unfolding  th1[symmetric]
   835     unfolding th0
   836   proof(rule setsum_reindex)
   837     let ?f = "(\<lambda>(b, y). Fun.swap a b id \<circ> y)"
   838     let ?P = "{p. p permutes S}"
   839     {fix b c p q assume b: "b \<in> insert a S" and c: "c \<in> insert a S" 
   840       and p: "p permutes S" and q: "q permutes S" 
   841       and eq: "Fun.swap a b id o p = Fun.swap a c id o q"
   842       from p q aS have pa: "p a = a" and qa: "q a = a"
   843 	unfolding permutes_def by metis+
   844       from eq have "(Fun.swap a b id o p) a  = (Fun.swap a c id o q) a" by simp
   845       hence bc: "b = c"
   846 	apply (simp add: permutes_def pa qa o_def fun_upd_def swap_def id_def cong del: if_weak_cong)
   847 	apply (cases "a = b", auto)
   848 	by (cases "b = c", auto)
   849       from eq[unfolded bc] have "(\<lambda>p. Fun.swap a c id o p) (Fun.swap a c id o p) = (\<lambda>p. Fun.swap a c id o p) (Fun.swap a c id o q)" by simp
   850       hence "p = q" unfolding o_assoc swap_id_idempotent
   851 	by (simp add: o_def)
   852       with bc have "b = c \<and> p = q" by blast
   853     }
   854     
   855     then show "inj_on ?f (insert a S \<times> ?P)" 
   856       unfolding inj_on_def
   857       apply clarify by metis
   858   qed
   859 qed
   860 
   861 end