author | nipkow |
Sun, 12 Jul 2009 10:14:51 +0200 | |
changeset 31992 | f8aed98faae7 |
parent 31916 | f3227bb306a4 |
child 31993 | 2ce88db62a84 |
permissions | -rw-r--r-- |
12396 | 1 |
(* Title: HOL/Finite_Set.thy |
2 |
Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel |
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with contributions by Jeremy Avigad |
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*) |
5 |
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header {* Finite sets *} |
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15131 | 8 |
theory Finite_Set |
29609 | 9 |
imports Nat Product_Type Power |
15131 | 10 |
begin |
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|
15392 | 12 |
subsection {* Definition and basic properties *} |
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|
23736 | 14 |
inductive finite :: "'a set => bool" |
22262 | 15 |
where |
16 |
emptyI [simp, intro!]: "finite {}" |
|
17 |
| insertI [simp, intro!]: "finite A ==> finite (insert a A)" |
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12396 | 18 |
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13737 | 19 |
lemma ex_new_if_finite: -- "does not depend on def of finite at all" |
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assumes "\<not> finite (UNIV :: 'a set)" and "finite A" |
21 |
shows "\<exists>a::'a. a \<notin> A" |
|
22 |
proof - |
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28823 | 23 |
from assms have "A \<noteq> UNIV" by blast |
14661 | 24 |
thus ?thesis by blast |
25 |
qed |
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12396 | 26 |
|
22262 | 27 |
lemma finite_induct [case_names empty insert, induct set: finite]: |
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"finite F ==> |
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P {} ==> (!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)) ==> P F" |
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-- {* Discharging @{text "x \<notin> F"} entails extra work. *} |
31 |
proof - |
|
13421 | 32 |
assume "P {}" and |
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insert: "!!x F. finite F ==> x \<notin> F ==> P F ==> P (insert x F)" |
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assume "finite F" |
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thus "P F" |
|
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proof induct |
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23389 | 37 |
show "P {}" by fact |
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fix x F assume F: "finite F" and P: "P F" |
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show "P (insert x F)" |
40 |
proof cases |
|
41 |
assume "x \<in> F" |
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42 |
hence "insert x F = F" by (rule insert_absorb) |
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43 |
with P show ?thesis by (simp only:) |
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44 |
next |
|
45 |
assume "x \<notin> F" |
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from F this P show ?thesis by (rule insert) |
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47 |
qed |
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48 |
qed |
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49 |
qed |
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50 |
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15484 | 51 |
lemma finite_ne_induct[case_names singleton insert, consumes 2]: |
52 |
assumes fin: "finite F" shows "F \<noteq> {} \<Longrightarrow> |
|
53 |
\<lbrakk> \<And>x. P{x}; |
|
54 |
\<And>x F. \<lbrakk> finite F; F \<noteq> {}; x \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert x F) \<rbrakk> |
|
55 |
\<Longrightarrow> P F" |
|
56 |
using fin |
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57 |
proof induct |
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58 |
case empty thus ?case by simp |
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59 |
next |
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case (insert x F) |
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61 |
show ?case |
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proof cases |
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23389 | 63 |
assume "F = {}" |
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thus ?thesis using `P {x}` by simp |
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15484 | 65 |
next |
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assume "F \<noteq> {}" |
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thus ?thesis using insert by blast |
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15484 | 68 |
qed |
69 |
qed |
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70 |
||
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lemma finite_subset_induct [consumes 2, case_names empty insert]: |
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assumes "finite F" and "F \<subseteq> A" |
73 |
and empty: "P {}" |
|
74 |
and insert: "!!a F. finite F ==> a \<in> A ==> a \<notin> F ==> P F ==> P (insert a F)" |
|
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shows "P F" |
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12396 | 76 |
proof - |
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from `finite F` and `F \<subseteq> A` |
78 |
show ?thesis |
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12396 | 79 |
proof induct |
23389 | 80 |
show "P {}" by fact |
81 |
next |
|
82 |
fix x F |
|
83 |
assume "finite F" and "x \<notin> F" and |
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P: "F \<subseteq> A ==> P F" and i: "insert x F \<subseteq> A" |
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12396 | 85 |
show "P (insert x F)" |
86 |
proof (rule insert) |
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87 |
from i show "x \<in> A" by blast |
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88 |
from i have "F \<subseteq> A" by blast |
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with P show "P F" . |
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23389 | 90 |
show "finite F" by fact |
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show "x \<notin> F" by fact |
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qed |
93 |
qed |
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qed |
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text{* A finite choice principle. Does not need the SOME choice operator. *} |
97 |
lemma finite_set_choice: |
|
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"finite A \<Longrightarrow> ALL x:A. (EX y. P x y) \<Longrightarrow> EX f. ALL x:A. P x (f x)" |
|
99 |
proof (induct set: finite) |
|
100 |
case empty thus ?case by simp |
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101 |
next |
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102 |
case (insert a A) |
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then obtain f b where f: "ALL x:A. P x (f x)" and ab: "P a b" by auto |
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show ?case (is "EX f. ?P f") |
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proof |
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106 |
show "?P(%x. if x = a then b else f x)" using f ab by auto |
|
107 |
qed |
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108 |
qed |
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109 |
||
23878 | 110 |
|
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text{* Finite sets are the images of initial segments of natural numbers: *} |
112 |
||
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lemma finite_imp_nat_seg_image_inj_on: |
114 |
assumes fin: "finite A" |
|
115 |
shows "\<exists> (n::nat) f. A = f ` {i. i<n} & inj_on f {i. i<n}" |
|
15392 | 116 |
using fin |
117 |
proof induct |
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118 |
case empty |
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15510 | 119 |
show ?case |
120 |
proof show "\<exists>f. {} = f ` {i::nat. i < 0} & inj_on f {i. i<0}" by simp |
|
121 |
qed |
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15392 | 122 |
next |
123 |
case (insert a A) |
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23389 | 124 |
have notinA: "a \<notin> A" by fact |
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from insert.hyps obtain n f |
126 |
where "A = f ` {i::nat. i < n}" "inj_on f {i. i < n}" by blast |
|
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hence "insert a A = f(n:=a) ` {i. i < Suc n}" |
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"inj_on (f(n:=a)) {i. i < Suc n}" using notinA |
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by (auto simp add: image_def Ball_def inj_on_def less_Suc_eq) |
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15392 | 130 |
thus ?case by blast |
131 |
qed |
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132 |
||
133 |
lemma nat_seg_image_imp_finite: |
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134 |
"!!f A. A = f ` {i::nat. i<n} \<Longrightarrow> finite A" |
|
135 |
proof (induct n) |
|
136 |
case 0 thus ?case by simp |
|
137 |
next |
|
138 |
case (Suc n) |
|
139 |
let ?B = "f ` {i. i < n}" |
|
140 |
have finB: "finite ?B" by(rule Suc.hyps[OF refl]) |
|
141 |
show ?case |
|
142 |
proof cases |
|
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assume "\<exists>k<n. f n = f k" |
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hence "A = ?B" using Suc.prems by(auto simp:less_Suc_eq) |
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thus ?thesis using finB by simp |
|
146 |
next |
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147 |
assume "\<not>(\<exists> k<n. f n = f k)" |
|
148 |
hence "A = insert (f n) ?B" using Suc.prems by(auto simp:less_Suc_eq) |
|
149 |
thus ?thesis using finB by simp |
|
150 |
qed |
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151 |
qed |
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152 |
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153 |
lemma finite_conv_nat_seg_image: |
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"finite A = (\<exists> (n::nat) f. A = f ` {i::nat. i<n})" |
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15510 | 155 |
by(blast intro: nat_seg_image_imp_finite dest: finite_imp_nat_seg_image_inj_on) |
15392 | 156 |
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29920 | 157 |
lemma finite_Collect_less_nat[iff]: "finite{n::nat. n<k}" |
158 |
by(fastsimp simp: finite_conv_nat_seg_image) |
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159 |
||
26441 | 160 |
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15392 | 161 |
subsubsection{* Finiteness and set theoretic constructions *} |
162 |
||
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lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)" |
29901 | 164 |
by (induct set: finite) simp_all |
12396 | 165 |
|
166 |
lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A" |
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-- {* Every subset of a finite set is finite. *} |
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168 |
proof - |
|
169 |
assume "finite B" |
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170 |
thus "!!A. A \<subseteq> B ==> finite A" |
|
171 |
proof induct |
|
172 |
case empty |
|
173 |
thus ?case by simp |
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174 |
next |
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case (insert x F A) |
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have A: "A \<subseteq> insert x F" and r: "A - {x} \<subseteq> F ==> finite (A - {x})" by fact+ |
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show "finite A" |
178 |
proof cases |
|
179 |
assume x: "x \<in> A" |
|
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with A have "A - {x} \<subseteq> F" by (simp add: subset_insert_iff) |
|
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with r have "finite (A - {x})" . |
|
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hence "finite (insert x (A - {x}))" .. |
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23389 | 183 |
also have "insert x (A - {x}) = A" using x by (rule insert_Diff) |
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finally show ?thesis . |
185 |
next |
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23389 | 186 |
show "A \<subseteq> F ==> ?thesis" by fact |
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assume "x \<notin> A" |
188 |
with A show "A \<subseteq> F" by (simp add: subset_insert_iff) |
|
189 |
qed |
|
190 |
qed |
|
191 |
qed |
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192 |
||
193 |
lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)" |
|
29901 | 194 |
by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI) |
195 |
||
29916 | 196 |
lemma finite_Collect_disjI[simp]: |
29901 | 197 |
"finite{x. P x | Q x} = (finite{x. P x} & finite{x. Q x})" |
198 |
by(simp add:Collect_disj_eq) |
|
12396 | 199 |
|
200 |
lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)" |
|
201 |
-- {* The converse obviously fails. *} |
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29901 | 202 |
by (blast intro: finite_subset) |
203 |
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29916 | 204 |
lemma finite_Collect_conjI [simp, intro]: |
29901 | 205 |
"finite{x. P x} | finite{x. Q x} ==> finite{x. P x & Q x}" |
206 |
-- {* The converse obviously fails. *} |
|
207 |
by(simp add:Collect_conj_eq) |
|
12396 | 208 |
|
29920 | 209 |
lemma finite_Collect_le_nat[iff]: "finite{n::nat. n<=k}" |
210 |
by(simp add: le_eq_less_or_eq) |
|
211 |
||
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lemma finite_insert [simp]: "finite (insert a A) = finite A" |
213 |
apply (subst insert_is_Un) |
|
14208 | 214 |
apply (simp only: finite_Un, blast) |
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done |
216 |
||
15281 | 217 |
lemma finite_Union[simp, intro]: |
218 |
"\<lbrakk> finite A; !!M. M \<in> A \<Longrightarrow> finite M \<rbrakk> \<Longrightarrow> finite(\<Union>A)" |
|
219 |
by (induct rule:finite_induct) simp_all |
|
220 |
||
31992 | 221 |
lemma finite_Inter[intro]: "EX A:M. finite(A) \<Longrightarrow> finite(Inter M)" |
222 |
by (blast intro: Inter_lower finite_subset) |
|
223 |
||
224 |
lemma finite_INT[intro]: "EX x:I. finite(A x) \<Longrightarrow> finite(INT x:I. A x)" |
|
225 |
by (blast intro: INT_lower finite_subset) |
|
226 |
||
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lemma finite_empty_induct: |
23389 | 228 |
assumes "finite A" |
229 |
and "P A" |
|
230 |
and "!!a A. finite A ==> a:A ==> P A ==> P (A - {a})" |
|
231 |
shows "P {}" |
|
12396 | 232 |
proof - |
233 |
have "P (A - A)" |
|
234 |
proof - |
|
23389 | 235 |
{ |
236 |
fix c b :: "'a set" |
|
237 |
assume c: "finite c" and b: "finite b" |
|
238 |
and P1: "P b" and P2: "!!x y. finite y ==> x \<in> y ==> P y ==> P (y - {x})" |
|
239 |
have "c \<subseteq> b ==> P (b - c)" |
|
240 |
using c |
|
241 |
proof induct |
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242 |
case empty |
|
243 |
from P1 show ?case by simp |
|
244 |
next |
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case (insert x F) |
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246 |
have "P (b - F - {x})" |
|
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proof (rule P2) |
|
248 |
from _ b show "finite (b - F)" by (rule finite_subset) blast |
|
249 |
from insert show "x \<in> b - F" by simp |
|
250 |
from insert show "P (b - F)" by simp |
|
251 |
qed |
|
252 |
also have "b - F - {x} = b - insert x F" by (rule Diff_insert [symmetric]) |
|
253 |
finally show ?case . |
|
12396 | 254 |
qed |
23389 | 255 |
} |
256 |
then show ?thesis by this (simp_all add: assms) |
|
12396 | 257 |
qed |
23389 | 258 |
then show ?thesis by simp |
12396 | 259 |
qed |
260 |
||
29901 | 261 |
lemma finite_Diff [simp]: "finite A ==> finite (A - B)" |
262 |
by (rule Diff_subset [THEN finite_subset]) |
|
263 |
||
264 |
lemma finite_Diff2 [simp]: |
|
265 |
assumes "finite B" shows "finite (A - B) = finite A" |
|
266 |
proof - |
|
267 |
have "finite A \<longleftrightarrow> finite((A-B) Un (A Int B))" by(simp add: Un_Diff_Int) |
|
268 |
also have "\<dots> \<longleftrightarrow> finite(A-B)" using `finite B` by(simp) |
|
269 |
finally show ?thesis .. |
|
270 |
qed |
|
271 |
||
272 |
lemma finite_compl[simp]: |
|
273 |
"finite(A::'a set) \<Longrightarrow> finite(-A) = finite(UNIV::'a set)" |
|
274 |
by(simp add:Compl_eq_Diff_UNIV) |
|
12396 | 275 |
|
29916 | 276 |
lemma finite_Collect_not[simp]: |
29903 | 277 |
"finite{x::'a. P x} \<Longrightarrow> finite{x. ~P x} = finite(UNIV::'a set)" |
278 |
by(simp add:Collect_neg_eq) |
|
279 |
||
12396 | 280 |
lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)" |
281 |
apply (subst Diff_insert) |
|
282 |
apply (case_tac "a : A - B") |
|
283 |
apply (rule finite_insert [symmetric, THEN trans]) |
|
14208 | 284 |
apply (subst insert_Diff, simp_all) |
12396 | 285 |
done |
286 |
||
287 |
||
15392 | 288 |
text {* Image and Inverse Image over Finite Sets *} |
13825 | 289 |
|
290 |
lemma finite_imageI[simp]: "finite F ==> finite (h ` F)" |
|
291 |
-- {* The image of a finite set is finite. *} |
|
22262 | 292 |
by (induct set: finite) simp_all |
13825 | 293 |
|
31768 | 294 |
lemma finite_image_set [simp]: |
295 |
"finite {x. P x} \<Longrightarrow> finite { f x | x. P x }" |
|
296 |
by (simp add: image_Collect [symmetric]) |
|
297 |
||
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lemma finite_surj: "finite A ==> B <= f ` A ==> finite B" |
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299 |
apply (frule finite_imageI) |
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300 |
apply (erule finite_subset, assumption) |
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301 |
done |
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302 |
|
13825 | 303 |
lemma finite_range_imageI: |
304 |
"finite (range g) ==> finite (range (%x. f (g x)))" |
|
27418 | 305 |
apply (drule finite_imageI, simp add: range_composition) |
13825 | 306 |
done |
307 |
||
12396 | 308 |
lemma finite_imageD: "finite (f`A) ==> inj_on f A ==> finite A" |
309 |
proof - |
|
310 |
have aux: "!!A. finite (A - {}) = finite A" by simp |
|
311 |
fix B :: "'a set" |
|
312 |
assume "finite B" |
|
313 |
thus "!!A. f`A = B ==> inj_on f A ==> finite A" |
|
314 |
apply induct |
|
315 |
apply simp |
|
316 |
apply (subgoal_tac "EX y:A. f y = x & F = f ` (A - {y})") |
|
317 |
apply clarify |
|
318 |
apply (simp (no_asm_use) add: inj_on_def) |
|
14208 | 319 |
apply (blast dest!: aux [THEN iffD1], atomize) |
12396 | 320 |
apply (erule_tac V = "ALL A. ?PP (A)" in thin_rl) |
14208 | 321 |
apply (frule subsetD [OF equalityD2 insertI1], clarify) |
12396 | 322 |
apply (rule_tac x = xa in bexI) |
323 |
apply (simp_all add: inj_on_image_set_diff) |
|
324 |
done |
|
325 |
qed (rule refl) |
|
326 |
||
327 |
||
13825 | 328 |
lemma inj_vimage_singleton: "inj f ==> f-`{a} \<subseteq> {THE x. f x = a}" |
329 |
-- {* The inverse image of a singleton under an injective function |
|
330 |
is included in a singleton. *} |
|
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|
331 |
apply (auto simp add: inj_on_def) |
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|
332 |
apply (blast intro: the_equality [symmetric]) |
13825 | 333 |
done |
334 |
||
335 |
lemma finite_vimageI: "[|finite F; inj h|] ==> finite (h -` F)" |
|
336 |
-- {* The inverse image of a finite set under an injective function |
|
337 |
is finite. *} |
|
22262 | 338 |
apply (induct set: finite) |
21575 | 339 |
apply simp_all |
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|
340 |
apply (subst vimage_insert) |
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paulson
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|
341 |
apply (simp add: finite_Un finite_subset [OF inj_vimage_singleton]) |
13825 | 342 |
done |
343 |
||
344 |
||
15392 | 345 |
text {* The finite UNION of finite sets *} |
12396 | 346 |
|
347 |
lemma finite_UN_I: "finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (UN a:A. B a)" |
|
22262 | 348 |
by (induct set: finite) simp_all |
12396 | 349 |
|
350 |
text {* |
|
351 |
Strengthen RHS to |
|
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|
352 |
@{prop "((ALL x:A. finite (B x)) & finite {x. x:A & B x \<noteq> {}})"}? |
12396 | 353 |
|
354 |
We'd need to prove |
|
14430
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paulson
parents:
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|
355 |
@{prop "finite C ==> ALL A B. (UNION A B) <= C --> finite {x. x:A & B x \<noteq> {}}"} |
12396 | 356 |
by induction. *} |
357 |
||
29918 | 358 |
lemma finite_UN [simp]: |
359 |
"finite A ==> finite (UNION A B) = (ALL x:A. finite (B x))" |
|
360 |
by (blast intro: finite_UN_I finite_subset) |
|
12396 | 361 |
|
29920 | 362 |
lemma finite_Collect_bex[simp]: "finite A \<Longrightarrow> |
363 |
finite{x. EX y:A. Q x y} = (ALL y:A. finite{x. Q x y})" |
|
364 |
apply(subgoal_tac "{x. EX y:A. Q x y} = UNION A (%y. {x. Q x y})") |
|
365 |
apply auto |
|
366 |
done |
|
367 |
||
368 |
lemma finite_Collect_bounded_ex[simp]: "finite{y. P y} \<Longrightarrow> |
|
369 |
finite{x. EX y. P y & Q x y} = (ALL y. P y \<longrightarrow> finite{x. Q x y})" |
|
370 |
apply(subgoal_tac "{x. EX y. P y & Q x y} = UNION {y. P y} (%y. {x. Q x y})") |
|
371 |
apply auto |
|
372 |
done |
|
373 |
||
374 |
||
17022 | 375 |
lemma finite_Plus: "[| finite A; finite B |] ==> finite (A <+> B)" |
376 |
by (simp add: Plus_def) |
|
377 |
||
31080 | 378 |
lemma finite_PlusD: |
379 |
fixes A :: "'a set" and B :: "'b set" |
|
380 |
assumes fin: "finite (A <+> B)" |
|
381 |
shows "finite A" "finite B" |
|
382 |
proof - |
|
383 |
have "Inl ` A \<subseteq> A <+> B" by auto |
|
384 |
hence "finite (Inl ` A :: ('a + 'b) set)" using fin by(rule finite_subset) |
|
385 |
thus "finite A" by(rule finite_imageD)(auto intro: inj_onI) |
|
386 |
next |
|
387 |
have "Inr ` B \<subseteq> A <+> B" by auto |
|
388 |
hence "finite (Inr ` B :: ('a + 'b) set)" using fin by(rule finite_subset) |
|
389 |
thus "finite B" by(rule finite_imageD)(auto intro: inj_onI) |
|
390 |
qed |
|
391 |
||
392 |
lemma finite_Plus_iff[simp]: "finite (A <+> B) \<longleftrightarrow> finite A \<and> finite B" |
|
393 |
by(auto intro: finite_PlusD finite_Plus) |
|
394 |
||
395 |
lemma finite_Plus_UNIV_iff[simp]: |
|
396 |
"finite (UNIV :: ('a + 'b) set) = |
|
397 |
(finite (UNIV :: 'a set) & finite (UNIV :: 'b set))" |
|
398 |
by(subst UNIV_Plus_UNIV[symmetric])(rule finite_Plus_iff) |
|
399 |
||
400 |
||
15392 | 401 |
text {* Sigma of finite sets *} |
12396 | 402 |
|
403 |
lemma finite_SigmaI [simp]: |
|
404 |
"finite A ==> (!!a. a:A ==> finite (B a)) ==> finite (SIGMA a:A. B a)" |
|
405 |
by (unfold Sigma_def) (blast intro!: finite_UN_I) |
|
406 |
||
15402 | 407 |
lemma finite_cartesian_product: "[| finite A; finite B |] ==> |
408 |
finite (A <*> B)" |
|
409 |
by (rule finite_SigmaI) |
|
410 |
||
12396 | 411 |
lemma finite_Prod_UNIV: |
412 |
"finite (UNIV::'a set) ==> finite (UNIV::'b set) ==> finite (UNIV::('a * 'b) set)" |
|
413 |
apply (subgoal_tac "(UNIV:: ('a * 'b) set) = Sigma UNIV (%x. UNIV)") |
|
414 |
apply (erule ssubst) |
|
14208 | 415 |
apply (erule finite_SigmaI, auto) |
12396 | 416 |
done |
417 |
||
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
418 |
lemma finite_cartesian_productD1: |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
419 |
"[| finite (A <*> B); B \<noteq> {} |] ==> finite A" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
420 |
apply (auto simp add: finite_conv_nat_seg_image) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
421 |
apply (drule_tac x=n in spec) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
422 |
apply (drule_tac x="fst o f" in spec) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
423 |
apply (auto simp add: o_def) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
424 |
prefer 2 apply (force dest!: equalityD2) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
425 |
apply (drule equalityD1) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
426 |
apply (rename_tac y x) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
427 |
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
428 |
prefer 2 apply force |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
429 |
apply clarify |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
430 |
apply (rule_tac x=k in image_eqI, auto) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
431 |
done |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
432 |
|
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
433 |
lemma finite_cartesian_productD2: |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
434 |
"[| finite (A <*> B); A \<noteq> {} |] ==> finite B" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
435 |
apply (auto simp add: finite_conv_nat_seg_image) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
436 |
apply (drule_tac x=n in spec) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
437 |
apply (drule_tac x="snd o f" in spec) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
438 |
apply (auto simp add: o_def) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
439 |
prefer 2 apply (force dest!: equalityD2) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
440 |
apply (drule equalityD1) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
441 |
apply (rename_tac x y) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
442 |
apply (subgoal_tac "\<exists>k. k<n & f k = (x,y)") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
443 |
prefer 2 apply force |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
444 |
apply clarify |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
445 |
apply (rule_tac x=k in image_eqI, auto) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
446 |
done |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
447 |
|
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
448 |
|
15392 | 449 |
text {* The powerset of a finite set *} |
12396 | 450 |
|
451 |
lemma finite_Pow_iff [iff]: "finite (Pow A) = finite A" |
|
452 |
proof |
|
453 |
assume "finite (Pow A)" |
|
454 |
with _ have "finite ((%x. {x}) ` A)" by (rule finite_subset) blast |
|
455 |
thus "finite A" by (rule finite_imageD [unfolded inj_on_def]) simp |
|
456 |
next |
|
457 |
assume "finite A" |
|
458 |
thus "finite (Pow A)" |
|
459 |
by induct (simp_all add: finite_UnI finite_imageI Pow_insert) |
|
460 |
qed |
|
461 |
||
29916 | 462 |
lemma finite_Collect_subsets[simp,intro]: "finite A \<Longrightarrow> finite{B. B \<subseteq> A}" |
463 |
by(simp add: Pow_def[symmetric]) |
|
15392 | 464 |
|
29918 | 465 |
|
15392 | 466 |
lemma finite_UnionD: "finite(\<Union>A) \<Longrightarrow> finite A" |
467 |
by(blast intro: finite_subset[OF subset_Pow_Union]) |
|
468 |
||
469 |
||
31441 | 470 |
lemma finite_subset_image: |
471 |
assumes "finite B" |
|
472 |
shows "B \<subseteq> f ` A \<Longrightarrow> \<exists>C\<subseteq>A. finite C \<and> B = f ` C" |
|
473 |
using assms proof(induct) |
|
474 |
case empty thus ?case by simp |
|
475 |
next |
|
476 |
case insert thus ?case |
|
477 |
by (clarsimp simp del: image_insert simp add: image_insert[symmetric]) |
|
478 |
blast |
|
479 |
qed |
|
480 |
||
481 |
||
26441 | 482 |
subsection {* Class @{text finite} *} |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
483 |
|
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
484 |
setup {* Sign.add_path "finite" *} -- {*FIXME: name tweaking*} |
29797 | 485 |
class finite = |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
486 |
assumes finite_UNIV: "finite (UNIV \<Colon> 'a set)" |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
487 |
setup {* Sign.parent_path *} |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
488 |
hide const finite |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
489 |
|
27430 | 490 |
context finite |
491 |
begin |
|
492 |
||
493 |
lemma finite [simp]: "finite (A \<Colon> 'a set)" |
|
26441 | 494 |
by (rule subset_UNIV finite_UNIV finite_subset)+ |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
495 |
|
27430 | 496 |
end |
497 |
||
26146 | 498 |
lemma UNIV_unit [noatp]: |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
499 |
"UNIV = {()}" by auto |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
500 |
|
26146 | 501 |
instance unit :: finite |
502 |
by default (simp add: UNIV_unit) |
|
503 |
||
504 |
lemma UNIV_bool [noatp]: |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
505 |
"UNIV = {False, True}" by auto |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
506 |
|
26146 | 507 |
instance bool :: finite |
508 |
by default (simp add: UNIV_bool) |
|
509 |
||
510 |
instance * :: (finite, finite) finite |
|
511 |
by default (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite) |
|
512 |
||
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
513 |
lemma inj_graph: "inj (%f. {(x, y). y = f x})" |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
514 |
by (rule inj_onI, auto simp add: expand_set_eq expand_fun_eq) |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
515 |
|
26146 | 516 |
instance "fun" :: (finite, finite) finite |
517 |
proof |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
518 |
show "finite (UNIV :: ('a => 'b) set)" |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
519 |
proof (rule finite_imageD) |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
520 |
let ?graph = "%f::'a => 'b. {(x, y). y = f x}" |
26792 | 521 |
have "range ?graph \<subseteq> Pow UNIV" by simp |
522 |
moreover have "finite (Pow (UNIV :: ('a * 'b) set))" |
|
523 |
by (simp only: finite_Pow_iff finite) |
|
524 |
ultimately show "finite (range ?graph)" |
|
525 |
by (rule finite_subset) |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
526 |
show "inj ?graph" by (rule inj_graph) |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
527 |
qed |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
528 |
qed |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
529 |
|
27981 | 530 |
instance "+" :: (finite, finite) finite |
531 |
by default (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite) |
|
532 |
||
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
533 |
|
15392 | 534 |
subsection {* A fold functional for finite sets *} |
535 |
||
536 |
text {* The intended behaviour is |
|
31916
f3227bb306a4
recovered subscripts, which were lost in b41d61c768e2 (due to Emacs accident?);
wenzelm
parents:
31907
diff
changeset
|
537 |
@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"} |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
538 |
if @{text f} is ``left-commutative'': |
15392 | 539 |
*} |
540 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
541 |
locale fun_left_comm = |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
542 |
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
543 |
assumes fun_left_comm: "f x (f y z) = f y (f x z)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
544 |
begin |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
545 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
546 |
text{* On a functional level it looks much nicer: *} |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
547 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
548 |
lemma fun_comp_comm: "f x \<circ> f y = f y \<circ> f x" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
549 |
by (simp add: fun_left_comm expand_fun_eq) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
550 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
551 |
end |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
552 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
553 |
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
554 |
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b where |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
555 |
emptyI [intro]: "fold_graph f z {} z" | |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
556 |
insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
557 |
\<Longrightarrow> fold_graph f z (insert x A) (f x y)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
558 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
559 |
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
560 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
561 |
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" where |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
562 |
[code del]: "fold f z A = (THE y. fold_graph f z A y)" |
15392 | 563 |
|
15498 | 564 |
text{*A tempting alternative for the definiens is |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
565 |
@{term "if finite A then THE y. fold_graph f z A y else e"}. |
15498 | 566 |
It allows the removal of finiteness assumptions from the theorems |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
567 |
@{text fold_comm}, @{text fold_reindex} and @{text fold_distrib}. |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
568 |
The proofs become ugly. It is not worth the effort. (???) *} |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
569 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
570 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
571 |
lemma Diff1_fold_graph: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
572 |
"fold_graph f z (A - {x}) y \<Longrightarrow> x \<in> A \<Longrightarrow> fold_graph f z A (f x y)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
573 |
by (erule insert_Diff [THEN subst], rule fold_graph.intros, auto) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
574 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
575 |
lemma fold_graph_imp_finite: "fold_graph f z A x \<Longrightarrow> finite A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
576 |
by (induct set: fold_graph) auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
577 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
578 |
lemma finite_imp_fold_graph: "finite A \<Longrightarrow> \<exists>x. fold_graph f z A x" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
579 |
by (induct set: finite) auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
580 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
581 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
582 |
subsubsection{*From @{const fold_graph} to @{term fold}*} |
15392 | 583 |
|
15510 | 584 |
lemma image_less_Suc: "h ` {i. i < Suc m} = insert (h m) (h ` {i. i < m})" |
19868 | 585 |
by (auto simp add: less_Suc_eq) |
15510 | 586 |
|
587 |
lemma insert_image_inj_on_eq: |
|
588 |
"[|insert (h m) A = h ` {i. i < Suc m}; h m \<notin> A; |
|
589 |
inj_on h {i. i < Suc m}|] |
|
590 |
==> A = h ` {i. i < m}" |
|
591 |
apply (auto simp add: image_less_Suc inj_on_def) |
|
592 |
apply (blast intro: less_trans) |
|
593 |
done |
|
594 |
||
595 |
lemma insert_inj_onE: |
|
596 |
assumes aA: "insert a A = h`{i::nat. i<n}" and anot: "a \<notin> A" |
|
597 |
and inj_on: "inj_on h {i::nat. i<n}" |
|
598 |
shows "\<exists>hm m. inj_on hm {i::nat. i<m} & A = hm ` {i. i<m} & m < n" |
|
599 |
proof (cases n) |
|
600 |
case 0 thus ?thesis using aA by auto |
|
601 |
next |
|
602 |
case (Suc m) |
|
23389 | 603 |
have nSuc: "n = Suc m" by fact |
15510 | 604 |
have mlessn: "m<n" by (simp add: nSuc) |
15532 | 605 |
from aA obtain k where hkeq: "h k = a" and klessn: "k<n" by (blast elim!: equalityE) |
27165 | 606 |
let ?hm = "Fun.swap k m h" |
15520 | 607 |
have inj_hm: "inj_on ?hm {i. i < n}" using klessn mlessn |
608 |
by (simp add: inj_on_swap_iff inj_on) |
|
15510 | 609 |
show ?thesis |
15520 | 610 |
proof (intro exI conjI) |
611 |
show "inj_on ?hm {i. i < m}" using inj_hm |
|
15510 | 612 |
by (auto simp add: nSuc less_Suc_eq intro: subset_inj_on) |
15520 | 613 |
show "m<n" by (rule mlessn) |
614 |
show "A = ?hm ` {i. i < m}" |
|
615 |
proof (rule insert_image_inj_on_eq) |
|
27165 | 616 |
show "inj_on (Fun.swap k m h) {i. i < Suc m}" using inj_hm nSuc by simp |
15520 | 617 |
show "?hm m \<notin> A" by (simp add: swap_def hkeq anot) |
618 |
show "insert (?hm m) A = ?hm ` {i. i < Suc m}" |
|
619 |
using aA hkeq nSuc klessn |
|
620 |
by (auto simp add: swap_def image_less_Suc fun_upd_image |
|
621 |
less_Suc_eq inj_on_image_set_diff [OF inj_on]) |
|
15479 | 622 |
qed |
623 |
qed |
|
624 |
qed |
|
625 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
626 |
context fun_left_comm |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
627 |
begin |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
628 |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
629 |
lemma fold_graph_determ_aux: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
630 |
"A = h`{i::nat. i<n} \<Longrightarrow> inj_on h {i. i<n} |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
631 |
\<Longrightarrow> fold_graph f z A x \<Longrightarrow> fold_graph f z A x' |
15392 | 632 |
\<Longrightarrow> x' = x" |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
633 |
proof (induct n arbitrary: A x x' h rule: less_induct) |
15510 | 634 |
case (less n) |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
635 |
have IH: "\<And>m h A x x'. m < n \<Longrightarrow> A = h ` {i. i<m} |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
636 |
\<Longrightarrow> inj_on h {i. i<m} \<Longrightarrow> fold_graph f z A x |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
637 |
\<Longrightarrow> fold_graph f z A x' \<Longrightarrow> x' = x" by fact |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
638 |
have Afoldx: "fold_graph f z A x" and Afoldx': "fold_graph f z A x'" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
639 |
and A: "A = h`{i. i<n}" and injh: "inj_on h {i. i<n}" by fact+ |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
640 |
show ?case |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
641 |
proof (rule fold_graph.cases [OF Afoldx]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
642 |
assume "A = {}" and "x = z" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
643 |
with Afoldx' show "x' = x" by auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
644 |
next |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
645 |
fix B b u |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
646 |
assume AbB: "A = insert b B" and x: "x = f b u" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
647 |
and notinB: "b \<notin> B" and Bu: "fold_graph f z B u" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
648 |
show "x'=x" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
649 |
proof (rule fold_graph.cases [OF Afoldx']) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
650 |
assume "A = {}" and "x' = z" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
651 |
with AbB show "x' = x" by blast |
15392 | 652 |
next |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
653 |
fix C c v |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
654 |
assume AcC: "A = insert c C" and x': "x' = f c v" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
655 |
and notinC: "c \<notin> C" and Cv: "fold_graph f z C v" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
656 |
from A AbB have Beq: "insert b B = h`{i. i<n}" by simp |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
657 |
from insert_inj_onE [OF Beq notinB injh] |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
658 |
obtain hB mB where inj_onB: "inj_on hB {i. i < mB}" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
659 |
and Beq: "B = hB ` {i. i < mB}" and lessB: "mB < n" by auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
660 |
from A AcC have Ceq: "insert c C = h`{i. i<n}" by simp |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
661 |
from insert_inj_onE [OF Ceq notinC injh] |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
662 |
obtain hC mC where inj_onC: "inj_on hC {i. i < mC}" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
663 |
and Ceq: "C = hC ` {i. i < mC}" and lessC: "mC < n" by auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
664 |
show "x'=x" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
665 |
proof cases |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
666 |
assume "b=c" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
667 |
then moreover have "B = C" using AbB AcC notinB notinC by auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
668 |
ultimately show ?thesis using Bu Cv x x' IH [OF lessC Ceq inj_onC] |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
669 |
by auto |
15392 | 670 |
next |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
671 |
assume diff: "b \<noteq> c" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
672 |
let ?D = "B - {c}" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
673 |
have B: "B = insert c ?D" and C: "C = insert b ?D" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
674 |
using AbB AcC notinB notinC diff by(blast elim!:equalityE)+ |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
675 |
have "finite A" by(rule fold_graph_imp_finite [OF Afoldx]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
676 |
with AbB have "finite ?D" by simp |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
677 |
then obtain d where Dfoldd: "fold_graph f z ?D d" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
678 |
using finite_imp_fold_graph by iprover |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
679 |
moreover have cinB: "c \<in> B" using B by auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
680 |
ultimately have "fold_graph f z B (f c d)" by(rule Diff1_fold_graph) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
681 |
hence "f c d = u" by (rule IH [OF lessB Beq inj_onB Bu]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
682 |
moreover have "f b d = v" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
683 |
proof (rule IH[OF lessC Ceq inj_onC Cv]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
684 |
show "fold_graph f z C (f b d)" using C notinB Dfoldd by fastsimp |
15392 | 685 |
qed |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
686 |
ultimately show ?thesis |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
687 |
using fun_left_comm [of c b] x x' by (auto simp add: o_def) |
15392 | 688 |
qed |
689 |
qed |
|
690 |
qed |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
691 |
qed |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
692 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
693 |
lemma fold_graph_determ: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
694 |
"fold_graph f z A x \<Longrightarrow> fold_graph f z A y \<Longrightarrow> y = x" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
695 |
apply (frule fold_graph_imp_finite [THEN finite_imp_nat_seg_image_inj_on]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
696 |
apply (blast intro: fold_graph_determ_aux [rule_format]) |
15392 | 697 |
done |
698 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
699 |
lemma fold_equality: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
700 |
"fold_graph f z A y \<Longrightarrow> fold f z A = y" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
701 |
by (unfold fold_def) (blast intro: fold_graph_determ) |
15392 | 702 |
|
703 |
text{* The base case for @{text fold}: *} |
|
704 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
705 |
lemma (in -) fold_empty [simp]: "fold f z {} = z" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
706 |
by (unfold fold_def) blast |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
707 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
708 |
text{* The various recursion equations for @{const fold}: *} |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
709 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
710 |
lemma fold_insert_aux: "x \<notin> A |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
711 |
\<Longrightarrow> fold_graph f z (insert x A) v \<longleftrightarrow> |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
712 |
(\<exists>y. fold_graph f z A y \<and> v = f x y)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
713 |
apply auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
714 |
apply (rule_tac A1 = A and f1 = f in finite_imp_fold_graph [THEN exE]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
715 |
apply (fastsimp dest: fold_graph_imp_finite) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
716 |
apply (blast intro: fold_graph_determ) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
717 |
done |
15392 | 718 |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
719 |
lemma fold_insert [simp]: |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
720 |
"finite A ==> x \<notin> A ==> fold f z (insert x A) = f x (fold f z A)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
721 |
apply (simp add: fold_def fold_insert_aux) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
722 |
apply (rule the_equality) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
723 |
apply (auto intro: finite_imp_fold_graph |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
724 |
cong add: conj_cong simp add: fold_def[symmetric] fold_equality) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
725 |
done |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
726 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
727 |
lemma fold_fun_comm: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
728 |
"finite A \<Longrightarrow> f x (fold f z A) = fold f (f x z) A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
729 |
proof (induct rule: finite_induct) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
730 |
case empty then show ?case by simp |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
731 |
next |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
732 |
case (insert y A) then show ?case |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
733 |
by (simp add: fun_left_comm[of x]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
734 |
qed |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
735 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
736 |
lemma fold_insert2: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
737 |
"finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
738 |
by (simp add: fold_insert fold_fun_comm) |
15392 | 739 |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
740 |
lemma fold_rec: |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
741 |
assumes "finite A" and "x \<in> A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
742 |
shows "fold f z A = f x (fold f z (A - {x}))" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
743 |
proof - |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
744 |
have A: "A = insert x (A - {x})" using `x \<in> A` by blast |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
745 |
then have "fold f z A = fold f z (insert x (A - {x}))" by simp |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
746 |
also have "\<dots> = f x (fold f z (A - {x}))" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
747 |
by (rule fold_insert) (simp add: `finite A`)+ |
15535 | 748 |
finally show ?thesis . |
749 |
qed |
|
750 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
751 |
lemma fold_insert_remove: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
752 |
assumes "finite A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
753 |
shows "fold f z (insert x A) = f x (fold f z (A - {x}))" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
754 |
proof - |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
755 |
from `finite A` have "finite (insert x A)" by auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
756 |
moreover have "x \<in> insert x A" by auto |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
757 |
ultimately have "fold f z (insert x A) = f x (fold f z (insert x A - {x}))" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
758 |
by (rule fold_rec) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
759 |
then show ?thesis by simp |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
760 |
qed |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
761 |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
762 |
end |
15392 | 763 |
|
15480 | 764 |
text{* A simplified version for idempotent functions: *} |
765 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
766 |
locale fun_left_comm_idem = fun_left_comm + |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
767 |
assumes fun_left_idem: "f x (f x z) = f x z" |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
768 |
begin |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
769 |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
770 |
text{* The nice version: *} |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
771 |
lemma fun_comp_idem : "f x o f x = f x" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
772 |
by (simp add: fun_left_idem expand_fun_eq) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
773 |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
774 |
lemma fold_insert_idem: |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
775 |
assumes fin: "finite A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
776 |
shows "fold f z (insert x A) = f x (fold f z A)" |
15480 | 777 |
proof cases |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
778 |
assume "x \<in> A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
779 |
then obtain B where "A = insert x B" and "x \<notin> B" by (rule set_insert) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
780 |
then show ?thesis using assms by (simp add:fun_left_idem) |
15480 | 781 |
next |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
782 |
assume "x \<notin> A" then show ?thesis using assms by simp |
15480 | 783 |
qed |
784 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
785 |
declare fold_insert[simp del] fold_insert_idem[simp] |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
786 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
787 |
lemma fold_insert_idem2: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
788 |
"finite A \<Longrightarrow> fold f z (insert x A) = fold f (f x z) A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
789 |
by(simp add:fold_fun_comm) |
15484 | 790 |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
791 |
end |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
792 |
|
31992 | 793 |
context ab_semigroup_idem_mult |
794 |
begin |
|
795 |
||
796 |
lemma fun_left_comm_idem: "fun_left_comm_idem(op *)" |
|
797 |
apply unfold_locales |
|
798 |
apply (simp add: mult_ac) |
|
799 |
apply (simp add: mult_idem mult_assoc[symmetric]) |
|
800 |
done |
|
801 |
||
802 |
end |
|
803 |
||
804 |
context lower_semilattice |
|
805 |
begin |
|
806 |
||
807 |
lemma ab_semigroup_idem_mult_inf: "ab_semigroup_idem_mult inf" |
|
808 |
proof qed (rule inf_assoc inf_commute inf_idem)+ |
|
809 |
||
810 |
lemma fold_inf_insert[simp]: "finite A \<Longrightarrow> fold inf b (insert a A) = inf a (fold inf b A)" |
|
811 |
by(rule fun_left_comm_idem.fold_insert_idem[OF ab_semigroup_idem_mult.fun_left_comm_idem[OF ab_semigroup_idem_mult_inf]]) |
|
812 |
||
813 |
lemma inf_le_fold_inf: "finite A \<Longrightarrow> ALL a:A. b \<le> a \<Longrightarrow> inf b c \<le> fold inf c A" |
|
814 |
by (induct pred:finite) auto |
|
815 |
||
816 |
lemma fold_inf_le_inf: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> fold inf b A \<le> inf a b" |
|
817 |
proof(induct arbitrary: a pred:finite) |
|
818 |
case empty thus ?case by simp |
|
819 |
next |
|
820 |
case (insert x A) |
|
821 |
show ?case |
|
822 |
proof cases |
|
823 |
assume "A = {}" thus ?thesis using insert by simp |
|
824 |
next |
|
825 |
assume "A \<noteq> {}" thus ?thesis using insert by auto |
|
826 |
qed |
|
827 |
qed |
|
828 |
||
829 |
end |
|
830 |
||
831 |
context upper_semilattice |
|
832 |
begin |
|
833 |
||
834 |
lemma ab_semigroup_idem_mult_sup: "ab_semigroup_idem_mult sup" |
|
835 |
by (rule lower_semilattice.ab_semigroup_idem_mult_inf)(rule dual_lattice) |
|
836 |
||
837 |
lemma fold_sup_insert[simp]: "finite A \<Longrightarrow> fold sup b (insert a A) = sup a (fold sup b A)" |
|
838 |
by(rule lower_semilattice.fold_inf_insert)(rule dual_lattice) |
|
839 |
||
840 |
lemma fold_sup_le_sup: "finite A \<Longrightarrow> ALL a:A. a \<le> b \<Longrightarrow> fold sup c A \<le> sup b c" |
|
841 |
by(rule lower_semilattice.inf_le_fold_inf)(rule dual_lattice) |
|
842 |
||
843 |
lemma sup_le_fold_sup: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a b \<le> fold sup b A" |
|
844 |
by(rule lower_semilattice.fold_inf_le_inf)(rule dual_lattice) |
|
845 |
||
846 |
end |
|
847 |
||
848 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
849 |
subsubsection{* The derived combinator @{text fold_image} *} |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
850 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
851 |
definition fold_image :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
852 |
where "fold_image f g = fold (%x y. f (g x) y)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
853 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
854 |
lemma fold_image_empty[simp]: "fold_image f g z {} = z" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
855 |
by(simp add:fold_image_def) |
15392 | 856 |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
857 |
context ab_semigroup_mult |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
858 |
begin |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
859 |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
860 |
lemma fold_image_insert[simp]: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
861 |
assumes "finite A" and "a \<notin> A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
862 |
shows "fold_image times g z (insert a A) = g a * (fold_image times g z A)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
863 |
proof - |
29223 | 864 |
interpret I: fun_left_comm "%x y. (g x) * y" |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
865 |
by unfold_locales (simp add: mult_ac) |
31992 | 866 |
show ?thesis using assms by(simp add:fold_image_def) |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
867 |
qed |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
868 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
869 |
(* |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
870 |
lemma fold_commute: |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
871 |
"finite A ==> (!!z. x * (fold times g z A) = fold times g (x * z) A)" |
22262 | 872 |
apply (induct set: finite) |
21575 | 873 |
apply simp |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
874 |
apply (simp add: mult_left_commute [of x]) |
15392 | 875 |
done |
876 |
||
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
877 |
lemma fold_nest_Un_Int: |
15392 | 878 |
"finite A ==> finite B |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
879 |
==> fold times g (fold times g z B) A = fold times g (fold times g z (A Int B)) (A Un B)" |
22262 | 880 |
apply (induct set: finite) |
21575 | 881 |
apply simp |
15392 | 882 |
apply (simp add: fold_commute Int_insert_left insert_absorb) |
883 |
done |
|
884 |
||
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
885 |
lemma fold_nest_Un_disjoint: |
15392 | 886 |
"finite A ==> finite B ==> A Int B = {} |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
887 |
==> fold times g z (A Un B) = fold times g (fold times g z B) A" |
15392 | 888 |
by (simp add: fold_nest_Un_Int) |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
889 |
*) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
890 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
891 |
lemma fold_image_reindex: |
15487 | 892 |
assumes fin: "finite A" |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
893 |
shows "inj_on h A \<Longrightarrow> fold_image times g z (h`A) = fold_image times (g\<circ>h) z A" |
31992 | 894 |
using fin by induct auto |
15392 | 895 |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
896 |
(* |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
897 |
text{* |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
898 |
Fusion theorem, as described in Graham Hutton's paper, |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
899 |
A Tutorial on the Universality and Expressiveness of Fold, |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
900 |
JFP 9:4 (355-372), 1999. |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
901 |
*} |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
902 |
|
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
903 |
lemma fold_fusion: |
27611 | 904 |
assumes "ab_semigroup_mult g" |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
905 |
assumes fin: "finite A" |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
906 |
and hyp: "\<And>x y. h (g x y) = times x (h y)" |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
907 |
shows "h (fold g j w A) = fold times j (h w) A" |
27611 | 908 |
proof - |
29223 | 909 |
class_interpret ab_semigroup_mult [g] by fact |
27611 | 910 |
show ?thesis using fin hyp by (induct set: finite) simp_all |
911 |
qed |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
912 |
*) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
913 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
914 |
lemma fold_image_cong: |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
915 |
"finite A \<Longrightarrow> |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
916 |
(!!x. x:A ==> g x = h x) ==> fold_image times g z A = fold_image times h z A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
917 |
apply (subgoal_tac "ALL C. C <= A --> (ALL x:C. g x = h x) --> fold_image times g z C = fold_image times h z C") |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
918 |
apply simp |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
919 |
apply (erule finite_induct, simp) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
920 |
apply (simp add: subset_insert_iff, clarify) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
921 |
apply (subgoal_tac "finite C") |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
922 |
prefer 2 apply (blast dest: finite_subset [COMP swap_prems_rl]) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
923 |
apply (subgoal_tac "C = insert x (C - {x})") |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
924 |
prefer 2 apply blast |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
925 |
apply (erule ssubst) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
926 |
apply (drule spec) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
927 |
apply (erule (1) notE impE) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
928 |
apply (simp add: Ball_def del: insert_Diff_single) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
929 |
done |
15392 | 930 |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
931 |
end |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
932 |
|
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
933 |
context comm_monoid_mult |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
934 |
begin |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
935 |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
936 |
lemma fold_image_Un_Int: |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
937 |
"finite A ==> finite B ==> |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
938 |
fold_image times g 1 A * fold_image times g 1 B = |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
939 |
fold_image times g 1 (A Un B) * fold_image times g 1 (A Int B)" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
940 |
by (induct set: finite) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
941 |
(auto simp add: mult_ac insert_absorb Int_insert_left) |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
942 |
|
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
943 |
corollary fold_Un_disjoint: |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
944 |
"finite A ==> finite B ==> A Int B = {} ==> |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
945 |
fold_image times g 1 (A Un B) = |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
946 |
fold_image times g 1 A * fold_image times g 1 B" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
947 |
by (simp add: fold_image_Un_Int) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
948 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
949 |
lemma fold_image_UN_disjoint: |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
950 |
"\<lbrakk> finite I; ALL i:I. finite (A i); |
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
951 |
ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {} \<rbrakk> |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
952 |
\<Longrightarrow> fold_image times g 1 (UNION I A) = |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
953 |
fold_image times (%i. fold_image times g 1 (A i)) 1 I" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
954 |
apply (induct set: finite, simp, atomize) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
955 |
apply (subgoal_tac "ALL i:F. x \<noteq> i") |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
956 |
prefer 2 apply blast |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
957 |
apply (subgoal_tac "A x Int UNION F A = {}") |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
958 |
prefer 2 apply blast |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
959 |
apply (simp add: fold_Un_disjoint) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
960 |
done |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
961 |
|
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
962 |
lemma fold_image_Sigma: "finite A ==> ALL x:A. finite (B x) ==> |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
963 |
fold_image times (%x. fold_image times (g x) 1 (B x)) 1 A = |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
964 |
fold_image times (split g) 1 (SIGMA x:A. B x)" |
15392 | 965 |
apply (subst Sigma_def) |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
966 |
apply (subst fold_image_UN_disjoint, assumption, simp) |
15392 | 967 |
apply blast |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
968 |
apply (erule fold_image_cong) |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
969 |
apply (subst fold_image_UN_disjoint, simp, simp) |
15392 | 970 |
apply blast |
15506 | 971 |
apply simp |
15392 | 972 |
done |
973 |
||
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
974 |
lemma fold_image_distrib: "finite A \<Longrightarrow> |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
975 |
fold_image times (%x. g x * h x) 1 A = |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
976 |
fold_image times g 1 A * fold_image times h 1 A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
977 |
by (erule finite_induct) (simp_all add: mult_ac) |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
978 |
|
30260
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
979 |
lemma fold_image_related: |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
980 |
assumes Re: "R e e" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
981 |
and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 * y1) (x2 * y2)" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
982 |
and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
983 |
shows "R (fold_image (op *) h e S) (fold_image (op *) g e S)" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
984 |
using fS by (rule finite_subset_induct) (insert assms, auto) |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
985 |
|
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
986 |
lemma fold_image_eq_general: |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
987 |
assumes fS: "finite S" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
988 |
and h: "\<forall>y\<in>S'. \<exists>!x. x\<in> S \<and> h(x) = y" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
989 |
and f12: "\<forall>x\<in>S. h x \<in> S' \<and> f2(h x) = f1 x" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
990 |
shows "fold_image (op *) f1 e S = fold_image (op *) f2 e S'" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
991 |
proof- |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
992 |
from h f12 have hS: "h ` S = S'" by auto |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
993 |
{fix x y assume H: "x \<in> S" "y \<in> S" "h x = h y" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
994 |
from f12 h H have "x = y" by auto } |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
995 |
hence hinj: "inj_on h S" unfolding inj_on_def Ex1_def by blast |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
996 |
from f12 have th: "\<And>x. x \<in> S \<Longrightarrow> (f2 \<circ> h) x = f1 x" by auto |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
997 |
from hS have "fold_image (op *) f2 e S' = fold_image (op *) f2 e (h ` S)" by simp |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
998 |
also have "\<dots> = fold_image (op *) (f2 o h) e S" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
999 |
using fold_image_reindex[OF fS hinj, of f2 e] . |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1000 |
also have "\<dots> = fold_image (op *) f1 e S " using th fold_image_cong[OF fS, of "f2 o h" f1 e] |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1001 |
by blast |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1002 |
finally show ?thesis .. |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1003 |
qed |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1004 |
|
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1005 |
lemma fold_image_eq_general_inverses: |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1006 |
assumes fS: "finite S" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1007 |
and kh: "\<And>y. y \<in> T \<Longrightarrow> k y \<in> S \<and> h (k y) = y" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1008 |
and hk: "\<And>x. x \<in> S \<Longrightarrow> h x \<in> T \<and> k (h x) = x \<and> g (h x) = f x" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1009 |
shows "fold_image (op *) f e S = fold_image (op *) g e T" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1010 |
(* metis solves it, but not yet available here *) |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1011 |
apply (rule fold_image_eq_general[OF fS, of T h g f e]) |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1012 |
apply (rule ballI) |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1013 |
apply (frule kh) |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1014 |
apply (rule ex1I[]) |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1015 |
apply blast |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1016 |
apply clarsimp |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1017 |
apply (drule hk) apply simp |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1018 |
apply (rule sym) |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1019 |
apply (erule conjunct1[OF conjunct2[OF hk]]) |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1020 |
apply (rule ballI) |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1021 |
apply (drule hk) |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1022 |
apply blast |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1023 |
done |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1024 |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
1025 |
end |
22917 | 1026 |
|
15402 | 1027 |
subsection {* Generalized summation over a set *} |
1028 |
||
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30325
diff
changeset
|
1029 |
interpretation comm_monoid_add: comm_monoid_mult "0::'a::comm_monoid_add" "op +" |
28823 | 1030 |
proof qed (auto intro: add_assoc add_commute) |
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25571
diff
changeset
|
1031 |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1032 |
definition setsum :: "('a => 'b) => 'a set => 'b::comm_monoid_add" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1033 |
where "setsum f A == if finite A then fold_image (op +) f 0 A else 0" |
15402 | 1034 |
|
19535 | 1035 |
abbreviation |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21249
diff
changeset
|
1036 |
Setsum ("\<Sum>_" [1000] 999) where |
19535 | 1037 |
"\<Sum>A == setsum (%x. x) A" |
1038 |
||
15402 | 1039 |
text{* Now: lot's of fancy syntax. First, @{term "setsum (%x. e) A"} is |
1040 |
written @{text"\<Sum>x\<in>A. e"}. *} |
|
1041 |
||
1042 |
syntax |
|
17189 | 1043 |
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3SUM _:_. _)" [0, 51, 10] 10) |
15402 | 1044 |
syntax (xsymbols) |
17189 | 1045 |
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10) |
15402 | 1046 |
syntax (HTML output) |
17189 | 1047 |
"_setsum" :: "pttrn => 'a set => 'b => 'b::comm_monoid_add" ("(3\<Sum>_\<in>_. _)" [0, 51, 10] 10) |
15402 | 1048 |
|
1049 |
translations -- {* Beware of argument permutation! *} |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1050 |
"SUM i:A. b" == "CONST setsum (%i. b) A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1051 |
"\<Sum>i\<in>A. b" == "CONST setsum (%i. b) A" |
15402 | 1052 |
|
1053 |
text{* Instead of @{term"\<Sum>x\<in>{x. P}. e"} we introduce the shorter |
|
1054 |
@{text"\<Sum>x|P. e"}. *} |
|
1055 |
||
1056 |
syntax |
|
17189 | 1057 |
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3SUM _ |/ _./ _)" [0,0,10] 10) |
15402 | 1058 |
syntax (xsymbols) |
17189 | 1059 |
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10) |
15402 | 1060 |
syntax (HTML output) |
17189 | 1061 |
"_qsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a" ("(3\<Sum>_ | (_)./ _)" [0,0,10] 10) |
15402 | 1062 |
|
1063 |
translations |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1064 |
"SUM x|P. t" => "CONST setsum (%x. t) {x. P}" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1065 |
"\<Sum>x|P. t" => "CONST setsum (%x. t) {x. P}" |
15402 | 1066 |
|
1067 |
print_translation {* |
|
1068 |
let |
|
19535 | 1069 |
fun setsum_tr' [Abs(x,Tx,t), Const ("Collect",_) $ Abs(y,Ty,P)] = |
1070 |
if x<>y then raise Match |
|
1071 |
else let val x' = Syntax.mark_bound x |
|
1072 |
val t' = subst_bound(x',t) |
|
1073 |
val P' = subst_bound(x',P) |
|
1074 |
in Syntax.const "_qsetsum" $ Syntax.mark_bound x $ P' $ t' end |
|
1075 |
in [("setsum", setsum_tr')] end |
|
15402 | 1076 |
*} |
1077 |
||
19535 | 1078 |
|
15402 | 1079 |
lemma setsum_empty [simp]: "setsum f {} = 0" |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1080 |
by (simp add: setsum_def) |
15402 | 1081 |
|
1082 |
lemma setsum_insert [simp]: |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1083 |
"finite F ==> a \<notin> F ==> setsum f (insert a F) = f a + setsum f F" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1084 |
by (simp add: setsum_def) |
15402 | 1085 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1086 |
lemma setsum_infinite [simp]: "~ finite A ==> setsum f A = 0" |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1087 |
by (simp add: setsum_def) |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1088 |
|
15402 | 1089 |
lemma setsum_reindex: |
1090 |
"inj_on f B ==> setsum h (f ` B) = setsum (h \<circ> f) B" |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1091 |
by(auto simp add: setsum_def comm_monoid_add.fold_image_reindex dest!:finite_imageD) |
15402 | 1092 |
|
1093 |
lemma setsum_reindex_id: |
|
1094 |
"inj_on f B ==> setsum f B = setsum id (f ` B)" |
|
1095 |
by (auto simp add: setsum_reindex) |
|
1096 |
||
29674
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1097 |
lemma setsum_reindex_nonzero: |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1098 |
assumes fS: "finite S" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1099 |
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" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1100 |
shows "setsum h (f ` S) = setsum (h o f) S" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1101 |
using nz |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1102 |
proof(induct rule: finite_induct[OF fS]) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1103 |
case 1 thus ?case by simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1104 |
next |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1105 |
case (2 x F) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1106 |
{assume fxF: "f x \<in> f ` F" hence "\<exists>y \<in> F . f y = f x" by auto |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1107 |
then obtain y where y: "y \<in> F" "f x = f y" by auto |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1108 |
from "2.hyps" y have xy: "x \<noteq> y" by auto |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1109 |
|
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1110 |
from "2.prems"[of x y] "2.hyps" xy y have h0: "h (f x) = 0" by simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1111 |
have "setsum h (f ` insert x F) = setsum h (f ` F)" using fxF by auto |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1112 |
also have "\<dots> = setsum (h o f) (insert x F)" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1113 |
unfolding setsum_insert[OF `finite F` `x\<notin>F`] |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1114 |
using h0 |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1115 |
apply simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1116 |
apply (rule "2.hyps"(3)) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1117 |
apply (rule_tac y="y" in "2.prems") |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1118 |
apply simp_all |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1119 |
done |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1120 |
finally have ?case .} |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1121 |
moreover |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1122 |
{assume fxF: "f x \<notin> f ` F" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1123 |
have "setsum h (f ` insert x F) = h (f x) + setsum h (f ` F)" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1124 |
using fxF "2.hyps" by simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1125 |
also have "\<dots> = setsum (h o f) (insert x F)" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1126 |
unfolding setsum_insert[OF `finite F` `x\<notin>F`] |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1127 |
apply simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1128 |
apply (rule cong[OF refl[of "op + (h (f x))"]]) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1129 |
apply (rule "2.hyps"(3)) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1130 |
apply (rule_tac y="y" in "2.prems") |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1131 |
apply simp_all |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1132 |
done |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1133 |
finally have ?case .} |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1134 |
ultimately show ?case by blast |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1135 |
qed |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1136 |
|
15402 | 1137 |
lemma setsum_cong: |
1138 |
"A = B ==> (!!x. x:B ==> f x = g x) ==> setsum f A = setsum g B" |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1139 |
by(fastsimp simp: setsum_def intro: comm_monoid_add.fold_image_cong) |
15402 | 1140 |
|
16733
236dfafbeb63
linear arithmetic now takes "&" in assumptions apart.
nipkow
parents:
16632
diff
changeset
|
1141 |
lemma strong_setsum_cong[cong]: |
236dfafbeb63
linear arithmetic now takes "&" in assumptions apart.
nipkow
parents:
16632
diff
changeset
|
1142 |
"A = B ==> (!!x. x:B =simp=> f x = g x) |
236dfafbeb63
linear arithmetic now takes "&" in assumptions apart.
nipkow
parents:
16632
diff
changeset
|
1143 |
==> setsum (%x. f x) A = setsum (%x. g x) B" |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1144 |
by(fastsimp simp: simp_implies_def setsum_def intro: comm_monoid_add.fold_image_cong) |
16632
ad2895beef79
Added strong_setsum_cong and strong_setprod_cong.
berghofe
parents:
16550
diff
changeset
|
1145 |
|
15554 | 1146 |
lemma setsum_cong2: "\<lbrakk>\<And>x. x \<in> A \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setsum f A = setsum g A"; |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1147 |
by (rule setsum_cong[OF refl], auto); |
15554 | 1148 |
|
15402 | 1149 |
lemma setsum_reindex_cong: |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1150 |
"[|inj_on f A; B = f ` A; !!a. a:A \<Longrightarrow> g a = h (f a)|] |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1151 |
==> setsum h B = setsum g A" |
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1152 |
by (simp add: setsum_reindex cong: setsum_cong) |
15402 | 1153 |
|
29674
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1154 |
|
15542 | 1155 |
lemma setsum_0[simp]: "setsum (%i. 0) A = 0" |
15402 | 1156 |
apply (clarsimp simp: setsum_def) |
15765 | 1157 |
apply (erule finite_induct, auto) |
15402 | 1158 |
done |
1159 |
||
15543 | 1160 |
lemma setsum_0': "ALL a:A. f a = 0 ==> setsum f A = 0" |
1161 |
by(simp add:setsum_cong) |
|
15402 | 1162 |
|
1163 |
lemma setsum_Un_Int: "finite A ==> finite B ==> |
|
1164 |
setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B" |
|
1165 |
-- {* The reversed orientation looks more natural, but LOOPS as a simprule! *} |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1166 |
by(simp add: setsum_def comm_monoid_add.fold_image_Un_Int [symmetric]) |
15402 | 1167 |
|
1168 |
lemma setsum_Un_disjoint: "finite A ==> finite B |
|
1169 |
==> A Int B = {} ==> setsum g (A Un B) = setsum g A + setsum g B" |
|
1170 |
by (subst setsum_Un_Int [symmetric], auto) |
|
1171 |
||
29674
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1172 |
lemma setsum_mono_zero_left: |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1173 |
assumes fT: "finite T" and ST: "S \<subseteq> T" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1174 |
and z: "\<forall>i \<in> T - S. f i = 0" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1175 |
shows "setsum f S = setsum f T" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1176 |
proof- |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1177 |
have eq: "T = S \<union> (T - S)" using ST by blast |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1178 |
have d: "S \<inter> (T - S) = {}" using ST by blast |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1179 |
from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1180 |
show ?thesis |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1181 |
by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z]) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1182 |
qed |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1183 |
|
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1184 |
lemma setsum_mono_zero_right: |
30837
3d4832d9f7e4
added strong_setprod_cong[cong] (in analogy with setsum)
nipkow
parents:
30729
diff
changeset
|
1185 |
"finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. f i = 0 \<Longrightarrow> setsum f T = setsum f S" |
3d4832d9f7e4
added strong_setprod_cong[cong] (in analogy with setsum)
nipkow
parents:
30729
diff
changeset
|
1186 |
by(blast intro!: setsum_mono_zero_left[symmetric]) |
29674
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1187 |
|
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1188 |
lemma setsum_mono_zero_cong_left: |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1189 |
assumes fT: "finite T" and ST: "S \<subseteq> T" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1190 |
and z: "\<forall>i \<in> T - S. g i = 0" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1191 |
and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1192 |
shows "setsum f S = setsum g T" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1193 |
proof- |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1194 |
have eq: "T = S \<union> (T - S)" using ST by blast |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1195 |
have d: "S \<inter> (T - S) = {}" using ST by blast |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1196 |
from fT ST have f: "finite S" "finite (T - S)" by (auto intro: finite_subset) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1197 |
show ?thesis |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1198 |
using fg by (simp add: setsum_Un_disjoint[OF f d, unfolded eq[symmetric]] setsum_0'[OF z]) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1199 |
qed |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1200 |
|
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1201 |
lemma setsum_mono_zero_cong_right: |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1202 |
assumes fT: "finite T" and ST: "S \<subseteq> T" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1203 |
and z: "\<forall>i \<in> T - S. f i = 0" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1204 |
and fg: "\<And>x. x \<in> S \<Longrightarrow> f x = g x" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1205 |
shows "setsum f T = setsum g S" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1206 |
using setsum_mono_zero_cong_left[OF fT ST z] fg[symmetric] by auto |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1207 |
|
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1208 |
lemma setsum_delta: |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1209 |
assumes fS: "finite S" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1210 |
shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1211 |
proof- |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1212 |
let ?f = "(\<lambda>k. if k=a then b k else 0)" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1213 |
{assume a: "a \<notin> S" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1214 |
hence "\<forall> k\<in> S. ?f k = 0" by simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1215 |
hence ?thesis using a by simp} |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1216 |
moreover |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1217 |
{assume a: "a \<in> S" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1218 |
let ?A = "S - {a}" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1219 |
let ?B = "{a}" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1220 |
have eq: "S = ?A \<union> ?B" using a by blast |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1221 |
have dj: "?A \<inter> ?B = {}" by simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1222 |
from fS have fAB: "finite ?A" "finite ?B" by auto |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1223 |
have "setsum ?f S = setsum ?f ?A + setsum ?f ?B" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1224 |
using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]] |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1225 |
by simp |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1226 |
then have ?thesis using a by simp} |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1227 |
ultimately show ?thesis by blast |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1228 |
qed |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1229 |
lemma setsum_delta': |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1230 |
assumes fS: "finite S" shows |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1231 |
"setsum (\<lambda>k. if a = k then b k else 0) S = |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1232 |
(if a\<in> S then b a else 0)" |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1233 |
using setsum_delta[OF fS, of a b, symmetric] |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1234 |
by (auto intro: setsum_cong) |
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1235 |
|
30260
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1236 |
lemma setsum_restrict_set: |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1237 |
assumes fA: "finite A" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1238 |
shows "setsum f (A \<inter> B) = setsum (\<lambda>x. if x \<in> B then f x else 0) A" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1239 |
proof- |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1240 |
from fA have fab: "finite (A \<inter> B)" by auto |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1241 |
have aba: "A \<inter> B \<subseteq> A" by blast |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1242 |
let ?g = "\<lambda>x. if x \<in> A\<inter>B then f x else 0" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1243 |
from setsum_mono_zero_left[OF fA aba, of ?g] |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1244 |
show ?thesis by simp |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1245 |
qed |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1246 |
|
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1247 |
lemma setsum_cases: |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1248 |
assumes fA: "finite A" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1249 |
shows "setsum (\<lambda>x. if x \<in> B then f x else g x) A = |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1250 |
setsum f (A \<inter> B) + setsum g (A \<inter> - B)" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1251 |
proof- |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1252 |
have a: "A = A \<inter> B \<union> A \<inter> -B" "(A \<inter> B) \<inter> (A \<inter> -B) = {}" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1253 |
by blast+ |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1254 |
from fA |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1255 |
have f: "finite (A \<inter> B)" "finite (A \<inter> -B)" by auto |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1256 |
let ?g = "\<lambda>x. if x \<in> B then f x else g x" |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1257 |
from setsum_Un_disjoint[OF f a(2), of ?g] a(1) |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1258 |
show ?thesis by simp |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1259 |
qed |
be39acd3ac85
Added general theorems for fold_image, setsum and set_prod
chaieb
parents:
29966
diff
changeset
|
1260 |
|
29674
3857d7eba390
Added theorems setsum_reindex_nonzero, setsum_mono_zero_left, setsum_mono_zero_right, setsum_mono_zero_cong_left, setsum_mono_zero_cong_right, setsum_delta, strong_setprod_reindex_cong, setprod_delta
chaieb
parents:
29609
diff
changeset
|
1261 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1262 |
(*But we can't get rid of finite I. If infinite, although the rhs is 0, |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1263 |
the lhs need not be, since UNION I A could still be finite.*) |
15402 | 1264 |
lemma setsum_UN_disjoint: |
1265 |
"finite I ==> (ALL i:I. finite (A i)) ==> |
|
1266 |
(ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==> |
|
1267 |
setsum f (UNION I A) = (\<Sum>i\<in>I. setsum f (A i))" |
|
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1268 |
by(simp add: setsum_def comm_monoid_add.fold_image_UN_disjoint cong: setsum_cong) |
15402 | 1269 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1270 |
text{*No need to assume that @{term C} is finite. If infinite, the rhs is |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1271 |
directly 0, and @{term "Union C"} is also infinite, hence the lhs is also 0.*} |
15402 | 1272 |
lemma setsum_Union_disjoint: |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1273 |
"[| (ALL A:C. finite A); |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1274 |
(ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {}) |] |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1275 |
==> setsum f (Union C) = setsum (setsum f) C" |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1276 |
apply (cases "finite C") |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1277 |
prefer 2 apply (force dest: finite_UnionD simp add: setsum_def) |
15402 | 1278 |
apply (frule setsum_UN_disjoint [of C id f]) |
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1279 |
apply (unfold Union_def id_def, assumption+) |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1280 |
done |
15402 | 1281 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1282 |
(*But we can't get rid of finite A. If infinite, although the lhs is 0, |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1283 |
the rhs need not be, since SIGMA A B could still be finite.*) |
15402 | 1284 |
lemma setsum_Sigma: "finite A ==> ALL x:A. finite (B x) ==> |
17189 | 1285 |
(\<Sum>x\<in>A. (\<Sum>y\<in>B x. f x y)) = (\<Sum>(x,y)\<in>(SIGMA x:A. B x). f x y)" |
28853
69eb69659bf3
Added new fold operator and renamed the old oe to fold_image.
nipkow
parents:
28823
diff
changeset
|
1286 |
by(simp add:setsum_def comm_monoid_add.fold_image_Sigma split_def cong:setsum_cong) |
15402 | 1287 |
|
15409
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1288 |
text{*Here we can eliminate the finiteness assumptions, by cases.*} |
a063687d24eb
new and stronger lemmas and improved simplification for finite sets
paulson
parents:
15402
diff
changeset
|
1289 |
lemma setsum_cartesian_product: |
17189 | 1290 |
"(\<Sum>x\<in>A. (\<Sum>y\<in>B. f x y)) = (\<Sum>(x,y) \<in> A <*> B. f x y)" |
15409 |