src/HOL/Groups_Big.thy
 changeset 57129 7edb7550663e parent 56545 8f1e7596deb7 child 57275 0ddb5b755cdc
```     1.1 --- a/src/HOL/Groups_Big.thy	Fri May 30 12:54:42 2014 +0200
1.2 +++ b/src/HOL/Groups_Big.thy	Fri May 30 14:55:10 2014 +0200
1.3 @@ -131,27 +131,8 @@
1.4    assumes "A = B"
1.5    assumes g_h: "\<And>x. x \<in> B \<Longrightarrow> g x = h x"
1.6    shows "F g A = F h B"
1.7 -proof (cases "finite A")
1.8 -  case True
1.9 -  then have "\<And>C. C \<subseteq> A \<longrightarrow> (\<forall>x\<in>C. g x = h x) \<longrightarrow> F g C = F h C"
1.10 -  proof induct
1.11 -    case empty then show ?case by simp
1.12 -  next
1.13 -    case (insert x F) then show ?case apply -
1.14 -    apply (simp add: subset_insert_iff, clarify)
1.15 -    apply (subgoal_tac "finite C")
1.16 -      prefer 2 apply (blast dest: finite_subset [rotated])
1.17 -    apply (subgoal_tac "C = insert x (C - {x})")
1.18 -      prefer 2 apply blast
1.19 -    apply (erule ssubst)
1.20 -    apply (simp add: Ball_def del: insert_Diff_single)
1.21 -    done
1.22 -  qed
1.23 -  with `A = B` g_h show ?thesis by simp
1.24 -next
1.25 -  case False
1.26 -  with `A = B` show ?thesis by simp
1.27 -qed
1.28 +  using g_h unfolding `A = B`
1.29 +  by (induct B rule: infinite_finite_induct) auto
1.30
1.31  lemma strong_cong [cong]:
1.32    assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
1.33 @@ -206,42 +187,6 @@
1.34    shows "R (F h S) (F g S)"
1.35    using fS by (rule finite_subset_induct) (insert assms, auto)
1.36
1.37 -lemma eq_general:
1.38 -  assumes h: "\<forall>y\<in>S'. \<exists>!x. x \<in> S \<and> h x = y"
1.39 -  and f12:  "\<forall>x\<in>S. h x \<in> S' \<and> f2 (h x) = f1 x"
1.40 -  shows "F f1 S = F f2 S'"
1.41 -proof-
1.42 -  from h f12 have hS: "h ` S = S'" by blast
1.43 -  {fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y"
1.44 -    from f12 h H  have "x = y" by auto }
1.45 -  hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast
1.46 -  from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto
1.47 -  from hS have "F f2 S' = F f2 (h ` S)" by simp
1.48 -  also have "\<dots> = F (f2 o h) S" using reindex [OF hinj, of f2] .
1.49 -  also have "\<dots> = F f1 S " using th cong [of _ _ "f2 o h" f1]
1.50 -    by blast
1.51 -  finally show ?thesis ..
1.52 -qed
1.53 -
1.54 -lemma eq_general_reverses:
1.55 -  assumes kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
1.56 -  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = j x"
1.57 -  shows "F j S = F g T"
1.58 -  (* metis solves it, but not yet available here *)
1.59 -  apply (rule eq_general [of T S h g j])
1.60 -  apply (rule ballI)
1.61 -  apply (frule kh)
1.62 -  apply (rule ex1I[])
1.63 -  apply blast
1.64 -  apply clarsimp
1.65 -  apply (drule hk) apply simp
1.66 -  apply (rule sym)
1.67 -  apply (erule conjunct1[OF conjunct2[OF hk]])
1.68 -  apply (rule ballI)
1.69 -  apply (drule hk)
1.70 -  apply blast
1.71 -  done
1.72 -
1.73  lemma mono_neutral_cong_left:
1.74    assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = 1"
1.75    and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
1.76 @@ -267,6 +212,74 @@
1.77    "\<lbrakk> finite T;  S \<subseteq> T;  \<forall>i \<in> T - S. g i = 1 \<rbrakk> \<Longrightarrow> F g T = F g S"
1.78    by (blast intro!: mono_neutral_left [symmetric])
1.79
1.80 +lemma reindex_bij_betw: "bij_betw h S T \<Longrightarrow> F (\<lambda>x. g (h x)) S = F g T"
1.81 +  by (auto simp: bij_betw_def reindex)
1.82 +
1.83 +lemma reindex_bij_witness:
1.84 +  assumes witness:
1.85 +    "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"
1.86 +    "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"
1.87 +    "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"
1.88 +    "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"
1.89 +  assumes eq:
1.90 +    "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
1.91 +  shows "F g S = F h T"
1.92 +proof -
1.93 +  have "bij_betw j S T"
1.94 +    using bij_betw_byWitness[where A=S and f=j and f'=i and A'=T] witness by auto
1.95 +  moreover have "F g S = F (\<lambda>x. h (j x)) S"
1.96 +    by (intro cong) (auto simp: eq)
1.97 +  ultimately show ?thesis
1.98 +    by (simp add: reindex_bij_betw)
1.99 +qed
1.100 +
1.101 +lemma reindex_bij_betw_not_neutral:
1.102 +  assumes fin: "finite S'" "finite T'"
1.103 +  assumes bij: "bij_betw h (S - S') (T - T')"
1.104 +  assumes nn:
1.105 +    "\<And>a. a \<in> S' \<Longrightarrow> g (h a) = z"
1.106 +    "\<And>b. b \<in> T' \<Longrightarrow> g b = z"
1.107 +  shows "F (\<lambda>x. g (h x)) S = F g T"
1.108 +proof -
1.109 +  have [simp]: "finite S \<longleftrightarrow> finite T"
1.110 +    using bij_betw_finite[OF bij] fin by auto
1.111 +
1.112 +  show ?thesis
1.113 +  proof cases
1.114 +    assume "finite S"
1.115 +    with nn have "F (\<lambda>x. g (h x)) S = F (\<lambda>x. g (h x)) (S - S')"
1.116 +      by (intro mono_neutral_cong_right) auto
1.117 +    also have "\<dots> = F g (T - T')"
1.118 +      using bij by (rule reindex_bij_betw)
1.119 +    also have "\<dots> = F g T"
1.120 +      using nn `finite S` by (intro mono_neutral_cong_left) auto
1.121 +    finally show ?thesis .
1.122 +  qed simp
1.123 +qed
1.124 +
1.125 +lemma reindex_bij_witness_not_neutral:
1.126 +  assumes fin: "finite S'" "finite T'"
1.127 +  assumes witness:
1.128 +    "\<And>a. a \<in> S - S' \<Longrightarrow> i (j a) = a"
1.129 +    "\<And>a. a \<in> S - S' \<Longrightarrow> j a \<in> T - T'"
1.130 +    "\<And>b. b \<in> T - T' \<Longrightarrow> j (i b) = b"
1.131 +    "\<And>b. b \<in> T - T' \<Longrightarrow> i b \<in> S - S'"
1.132 +  assumes nn:
1.133 +    "\<And>a. a \<in> S' \<Longrightarrow> g a = z"
1.134 +    "\<And>b. b \<in> T' \<Longrightarrow> h b = z"
1.135 +  assumes eq:
1.136 +    "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a"
1.137 +  shows "F g S = F h T"
1.138 +proof -
1.139 +  have bij: "bij_betw j (S - (S' \<inter> S)) (T - (T' \<inter> T))"
1.140 +    using witness by (intro bij_betw_byWitness[where f'=i]) auto
1.141 +  have F_eq: "F g S = F (\<lambda>x. h (j x)) S"
1.142 +    by (intro cong) (auto simp: eq)
1.143 +  show ?thesis
1.144 +    unfolding F_eq using fin nn eq
1.145 +    by (intro reindex_bij_betw_not_neutral[OF _ _ bij]) auto
1.146 +qed
1.147 +
1.148  lemma delta:
1.149    assumes fS: "finite S"
1.150    shows "F (\<lambda>k. if k = a then b k else 1) S = (if a \<in> S then b a else 1)"
1.151 @@ -403,46 +416,17 @@
1.152  begin
1.153
1.154  lemma setsum_reindex_id:
1.155 -  "inj_on f B ==> setsum f B = setsum id (f ` B)"
1.156 +  "inj_on f B \<Longrightarrow> setsum f B = setsum id (f ` B)"
1.158
1.159  lemma setsum_reindex_nonzero:
1.160    assumes fS: "finite S"
1.161    and nz: "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> f x = f y \<Longrightarrow> h (f x) = 0"
1.162    shows "setsum h (f ` S) = setsum (h \<circ> f) S"
1.163 -using nz proof (induct rule: finite_induct [OF fS])
1.164 -  case 1 thus ?case by simp
1.165 -next
1.166 -  case (2 x F)
1.167 -  { assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto
1.168 -    then obtain y where y: "y \<in> F" "f x = f y" by auto
1.169 -    from "2.hyps" y have xy: "x \<noteq> y" by auto
1.170 -    from "2.prems" [of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp
1.171 -    have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto
1.172 -    also have "\<dots> = setsum (h o f) (insert x F)"
1.173 -      unfolding setsum.insert[OF `finite F` `x\<notin>F`]
1.174 -      using h0
1.175 -      apply (simp cong del: setsum.strong_cong)
1.176 -      apply (rule "2.hyps"(3))
1.177 -      apply (rule_tac y="y" in  "2.prems")
1.178 -      apply simp_all
1.179 -      done
1.180 -    finally have ?case . }
1.181 -  moreover
1.182 -  { assume fxF: "f x \<notin> f ` F"
1.183 -    have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)"
1.184 -      using fxF "2.hyps" by simp
1.185 -    also have "\<dots> = setsum (h o f) (insert x F)"
1.186 -      unfolding setsum.insert[OF `finite F` `x\<notin>F`]
1.187 -      apply (simp cong del: setsum.strong_cong)
1.188 -      apply (rule cong [OF refl [of "op + (h (f x))"]])
1.189 -      apply (rule "2.hyps"(3))
1.190 -      apply (rule_tac y="y" in  "2.prems")
1.191 -      apply simp_all
1.192 -      done
1.193 -    finally have ?case . }
1.194 -  ultimately show ?case by blast
1.195 -qed
1.196 +proof (subst setsum.reindex_bij_betw_not_neutral[symmetric])
1.197 +  show "bij_betw f (S - {x\<in>S. h (f x) = 0}) (f`S - f`{x\<in>S. h (f x) = 0})"
1.198 +    using nz by (auto intro!: inj_onI simp: bij_betw_def)
1.199 +qed (insert fS, auto)
1.200
1.201  lemma setsum_cong2:
1.202    "(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> setsum f A = setsum g A"
1.203 @@ -494,19 +478,8 @@
1.204
1.205  lemma setsum_commute:
1.206    "(\<Sum>i\<in>A. \<Sum>j\<in>B. f i j) = (\<Sum>j\<in>B. \<Sum>i\<in>A. f i j)"
1.208 -  have "(\<Sum>(x,y) \<in> A <*> B. f x y) =
1.209 -    (\<Sum>(y,x) \<in> (%(i, j). (j, i)) ` (A \<times> B). f x y)"
1.210 -    (is "?s = _")
1.211 -    apply (simp add: setsum.reindex [where h = "%(i, j). (j, i)"] swap_inj_on)
1.212 -    apply (simp add: split_def)
1.213 -    done
1.214 -  also have "... = (\<Sum>(y,x)\<in>B \<times> A. f x y)"
1.215 -    (is "_ = ?t")
1.216 -    apply (simp add: swap_product)
1.217 -    done
1.218 -  finally show "?s = ?t" .
1.219 -qed
1.220 +  unfolding setsum_cartesian_product
1.221 +  by (rule setsum.reindex_bij_witness[where i="\<lambda>(i, j). (j, i)" and j="\<lambda>(i, j). (j, i)"]) auto
1.222
1.223  lemma setsum_Plus:
1.224    fixes A :: "'a set" and B :: "'b set"
1.225 @@ -616,14 +589,6 @@
1.226    setsum f (S \<union> T) = setsum f S + setsum f T"
1.227    by (fact setsum.union_inter_neutral)
1.228
1.229 -lemma setsum_eq_general_reverses:
1.230 -  assumes fS: "finite S" and fT: "finite T"
1.231 -  and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y"
1.232 -  and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x"
1.233 -  shows "setsum f S = setsum g T"
1.234 -  using kh hk by (fact setsum.eq_general_reverses)
1.235 -
1.236 -
1.237  subsubsection {* Properties in more restricted classes of structures *}
1.238
1.239  lemma setsum_Un: "finite A ==> finite B ==>
1.240 @@ -1124,17 +1089,9 @@
1.241    by (frule setprod.reindex, simp)
1.242
1.243  lemma strong_setprod_reindex_cong:
1.244 -  assumes i: "inj_on f A"
1.245 -  and B: "B = f ` A" and eq: "\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x"
1.246 -  shows "setprod h B = setprod g A"
1.247 -proof-
1.248 -  have "setprod h B = setprod (h o f) A"
1.249 -    by (simp add: B setprod.reindex [OF i, of h])
1.250 -  then show ?thesis apply simp
1.251 -    apply (rule setprod.cong)
1.252 -    apply simp
1.253 -    by (simp add: eq)
1.254 -qed
1.255 +  "inj_on f A \<Longrightarrow> B = f ` A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> g x = (h \<circ> f) x) \<Longrightarrow> setprod h B = setprod g A"
1.256 +  by (subst setprod.reindex_bij_betw[symmetric, where h=f])
1.257 +     (auto simp: bij_betw_def)
1.258
1.259  lemma setprod_Union_disjoint:
1.260    assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A Int B = {}"
```