nipkow@8924
|
1 |
(* Title: HOL/SetInterval.thy
|
nipkow@8924
|
2 |
ID: $Id$
|
ballarin@13735
|
3 |
Author: Tobias Nipkow and Clemens Ballarin
|
paulson@14485
|
4 |
Additions by Jeremy Avigad in March 2004
|
paulson@8957
|
5 |
Copyright 2000 TU Muenchen
|
nipkow@8924
|
6 |
|
ballarin@13735
|
7 |
lessThan, greaterThan, atLeast, atMost and two-sided intervals
|
nipkow@8924
|
8 |
*)
|
nipkow@8924
|
9 |
|
wenzelm@14577
|
10 |
header {* Set intervals *}
|
wenzelm@14577
|
11 |
|
nipkow@15131
|
12 |
theory SetInterval
|
haftmann@25919
|
13 |
imports Int
|
nipkow@15131
|
14 |
begin
|
nipkow@8924
|
15 |
|
nipkow@24691
|
16 |
context ord
|
nipkow@24691
|
17 |
begin
|
nipkow@24691
|
18 |
definition
|
haftmann@25062
|
19 |
lessThan :: "'a => 'a set" ("(1{..<_})") where
|
haftmann@25062
|
20 |
"{..<u} == {x. x < u}"
|
nipkow@24691
|
21 |
|
nipkow@24691
|
22 |
definition
|
haftmann@25062
|
23 |
atMost :: "'a => 'a set" ("(1{.._})") where
|
haftmann@25062
|
24 |
"{..u} == {x. x \<le> u}"
|
nipkow@24691
|
25 |
|
nipkow@24691
|
26 |
definition
|
haftmann@25062
|
27 |
greaterThan :: "'a => 'a set" ("(1{_<..})") where
|
haftmann@25062
|
28 |
"{l<..} == {x. l<x}"
|
nipkow@24691
|
29 |
|
nipkow@24691
|
30 |
definition
|
haftmann@25062
|
31 |
atLeast :: "'a => 'a set" ("(1{_..})") where
|
haftmann@25062
|
32 |
"{l..} == {x. l\<le>x}"
|
nipkow@24691
|
33 |
|
nipkow@24691
|
34 |
definition
|
haftmann@25062
|
35 |
greaterThanLessThan :: "'a => 'a => 'a set" ("(1{_<..<_})") where
|
haftmann@25062
|
36 |
"{l<..<u} == {l<..} Int {..<u}"
|
nipkow@24691
|
37 |
|
nipkow@24691
|
38 |
definition
|
haftmann@25062
|
39 |
atLeastLessThan :: "'a => 'a => 'a set" ("(1{_..<_})") where
|
haftmann@25062
|
40 |
"{l..<u} == {l..} Int {..<u}"
|
nipkow@24691
|
41 |
|
nipkow@24691
|
42 |
definition
|
haftmann@25062
|
43 |
greaterThanAtMost :: "'a => 'a => 'a set" ("(1{_<.._})") where
|
haftmann@25062
|
44 |
"{l<..u} == {l<..} Int {..u}"
|
nipkow@24691
|
45 |
|
nipkow@24691
|
46 |
definition
|
haftmann@25062
|
47 |
atLeastAtMost :: "'a => 'a => 'a set" ("(1{_.._})") where
|
haftmann@25062
|
48 |
"{l..u} == {l..} Int {..u}"
|
nipkow@24691
|
49 |
|
nipkow@24691
|
50 |
end
|
nipkow@24691
|
51 |
(*
|
nipkow@8924
|
52 |
constdefs
|
nipkow@15045
|
53 |
lessThan :: "('a::ord) => 'a set" ("(1{..<_})")
|
nipkow@15045
|
54 |
"{..<u} == {x. x<u}"
|
nipkow@8924
|
55 |
|
wenzelm@11609
|
56 |
atMost :: "('a::ord) => 'a set" ("(1{.._})")
|
wenzelm@11609
|
57 |
"{..u} == {x. x<=u}"
|
nipkow@8924
|
58 |
|
nipkow@15045
|
59 |
greaterThan :: "('a::ord) => 'a set" ("(1{_<..})")
|
nipkow@15045
|
60 |
"{l<..} == {x. l<x}"
|
nipkow@8924
|
61 |
|
wenzelm@11609
|
62 |
atLeast :: "('a::ord) => 'a set" ("(1{_..})")
|
wenzelm@11609
|
63 |
"{l..} == {x. l<=x}"
|
nipkow@8924
|
64 |
|
nipkow@15045
|
65 |
greaterThanLessThan :: "['a::ord, 'a] => 'a set" ("(1{_<..<_})")
|
nipkow@15045
|
66 |
"{l<..<u} == {l<..} Int {..<u}"
|
ballarin@13735
|
67 |
|
nipkow@15045
|
68 |
atLeastLessThan :: "['a::ord, 'a] => 'a set" ("(1{_..<_})")
|
nipkow@15045
|
69 |
"{l..<u} == {l..} Int {..<u}"
|
ballarin@13735
|
70 |
|
nipkow@15045
|
71 |
greaterThanAtMost :: "['a::ord, 'a] => 'a set" ("(1{_<.._})")
|
nipkow@15045
|
72 |
"{l<..u} == {l<..} Int {..u}"
|
ballarin@13735
|
73 |
|
ballarin@13735
|
74 |
atLeastAtMost :: "['a::ord, 'a] => 'a set" ("(1{_.._})")
|
ballarin@13735
|
75 |
"{l..u} == {l..} Int {..u}"
|
nipkow@24691
|
76 |
*)
|
ballarin@13735
|
77 |
|
nipkow@15048
|
78 |
text{* A note of warning when using @{term"{..<n}"} on type @{typ
|
nipkow@15048
|
79 |
nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
|
nipkow@15052
|
80 |
@{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}
|
nipkow@15048
|
81 |
|
kleing@14418
|
82 |
syntax
|
kleing@14418
|
83 |
"@UNION_le" :: "nat => nat => 'b set => 'b set" ("(3UN _<=_./ _)" 10)
|
kleing@14418
|
84 |
"@UNION_less" :: "nat => nat => 'b set => 'b set" ("(3UN _<_./ _)" 10)
|
kleing@14418
|
85 |
"@INTER_le" :: "nat => nat => 'b set => 'b set" ("(3INT _<=_./ _)" 10)
|
kleing@14418
|
86 |
"@INTER_less" :: "nat => nat => 'b set => 'b set" ("(3INT _<_./ _)" 10)
|
kleing@14418
|
87 |
|
kleing@14418
|
88 |
syntax (input)
|
kleing@14418
|
89 |
"@UNION_le" :: "nat => nat => 'b set => 'b set" ("(3\<Union> _\<le>_./ _)" 10)
|
kleing@14418
|
90 |
"@UNION_less" :: "nat => nat => 'b set => 'b set" ("(3\<Union> _<_./ _)" 10)
|
kleing@14418
|
91 |
"@INTER_le" :: "nat => nat => 'b set => 'b set" ("(3\<Inter> _\<le>_./ _)" 10)
|
kleing@14418
|
92 |
"@INTER_less" :: "nat => nat => 'b set => 'b set" ("(3\<Inter> _<_./ _)" 10)
|
kleing@14418
|
93 |
|
kleing@14418
|
94 |
syntax (xsymbols)
|
wenzelm@14846
|
95 |
"@UNION_le" :: "nat \<Rightarrow> nat => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
|
wenzelm@14846
|
96 |
"@UNION_less" :: "nat \<Rightarrow> nat => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
|
wenzelm@14846
|
97 |
"@INTER_le" :: "nat \<Rightarrow> nat => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_ \<le> _\<^esub>)/ _)" 10)
|
wenzelm@14846
|
98 |
"@INTER_less" :: "nat \<Rightarrow> nat => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_ < _\<^esub>)/ _)" 10)
|
kleing@14418
|
99 |
|
kleing@14418
|
100 |
translations
|
kleing@14418
|
101 |
"UN i<=n. A" == "UN i:{..n}. A"
|
nipkow@15045
|
102 |
"UN i<n. A" == "UN i:{..<n}. A"
|
kleing@14418
|
103 |
"INT i<=n. A" == "INT i:{..n}. A"
|
nipkow@15045
|
104 |
"INT i<n. A" == "INT i:{..<n}. A"
|
kleing@14418
|
105 |
|
kleing@14418
|
106 |
|
paulson@14485
|
107 |
subsection {* Various equivalences *}
|
ballarin@13735
|
108 |
|
haftmann@25062
|
109 |
lemma (in ord) lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
|
paulson@13850
|
110 |
by (simp add: lessThan_def)
|
ballarin@13735
|
111 |
|
paulson@15418
|
112 |
lemma Compl_lessThan [simp]:
|
paulson@13850
|
113 |
"!!k:: 'a::linorder. -lessThan k = atLeast k"
|
paulson@13850
|
114 |
apply (auto simp add: lessThan_def atLeast_def)
|
ballarin@13735
|
115 |
done
|
ballarin@13735
|
116 |
|
paulson@13850
|
117 |
lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
|
paulson@13850
|
118 |
by auto
|
paulson@13850
|
119 |
|
haftmann@25062
|
120 |
lemma (in ord) greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
|
paulson@13850
|
121 |
by (simp add: greaterThan_def)
|
paulson@13850
|
122 |
|
paulson@15418
|
123 |
lemma Compl_greaterThan [simp]:
|
paulson@13850
|
124 |
"!!k:: 'a::linorder. -greaterThan k = atMost k"
|
paulson@13850
|
125 |
apply (simp add: greaterThan_def atMost_def le_def, auto)
|
ballarin@13735
|
126 |
done
|
ballarin@13735
|
127 |
|
paulson@13850
|
128 |
lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
|
paulson@13850
|
129 |
apply (subst Compl_greaterThan [symmetric])
|
paulson@15418
|
130 |
apply (rule double_complement)
|
ballarin@13735
|
131 |
done
|
ballarin@13735
|
132 |
|
haftmann@25062
|
133 |
lemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
|
paulson@13850
|
134 |
by (simp add: atLeast_def)
|
paulson@13850
|
135 |
|
paulson@15418
|
136 |
lemma Compl_atLeast [simp]:
|
paulson@13850
|
137 |
"!!k:: 'a::linorder. -atLeast k = lessThan k"
|
paulson@13850
|
138 |
apply (simp add: lessThan_def atLeast_def le_def, auto)
|
ballarin@13735
|
139 |
done
|
ballarin@13735
|
140 |
|
haftmann@25062
|
141 |
lemma (in ord) atMost_iff [iff]: "(i: atMost k) = (i<=k)"
|
paulson@13850
|
142 |
by (simp add: atMost_def)
|
ballarin@13735
|
143 |
|
paulson@14485
|
144 |
lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
|
paulson@14485
|
145 |
by (blast intro: order_antisym)
|
ballarin@13735
|
146 |
|
ballarin@13735
|
147 |
|
paulson@14485
|
148 |
subsection {* Logical Equivalences for Set Inclusion and Equality *}
|
paulson@13850
|
149 |
|
paulson@13850
|
150 |
lemma atLeast_subset_iff [iff]:
|
paulson@15418
|
151 |
"(atLeast x \<subseteq> atLeast y) = (y \<le> (x::'a::order))"
|
paulson@15418
|
152 |
by (blast intro: order_trans)
|
paulson@13850
|
153 |
|
paulson@13850
|
154 |
lemma atLeast_eq_iff [iff]:
|
paulson@15418
|
155 |
"(atLeast x = atLeast y) = (x = (y::'a::linorder))"
|
paulson@13850
|
156 |
by (blast intro: order_antisym order_trans)
|
paulson@13850
|
157 |
|
paulson@13850
|
158 |
lemma greaterThan_subset_iff [iff]:
|
paulson@15418
|
159 |
"(greaterThan x \<subseteq> greaterThan y) = (y \<le> (x::'a::linorder))"
|
paulson@15418
|
160 |
apply (auto simp add: greaterThan_def)
|
paulson@15418
|
161 |
apply (subst linorder_not_less [symmetric], blast)
|
ballarin@13735
|
162 |
done
|
ballarin@13735
|
163 |
|
paulson@13850
|
164 |
lemma greaterThan_eq_iff [iff]:
|
paulson@15418
|
165 |
"(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
|
paulson@15418
|
166 |
apply (rule iffI)
|
paulson@15418
|
167 |
apply (erule equalityE)
|
paulson@15418
|
168 |
apply (simp_all add: greaterThan_subset_iff)
|
paulson@13850
|
169 |
done
|
ballarin@13735
|
170 |
|
paulson@15418
|
171 |
lemma atMost_subset_iff [iff]: "(atMost x \<subseteq> atMost y) = (x \<le> (y::'a::order))"
|
paulson@13850
|
172 |
by (blast intro: order_trans)
|
paulson@13850
|
173 |
|
paulson@15418
|
174 |
lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"
|
paulson@13850
|
175 |
by (blast intro: order_antisym order_trans)
|
paulson@13850
|
176 |
|
paulson@13850
|
177 |
lemma lessThan_subset_iff [iff]:
|
paulson@15418
|
178 |
"(lessThan x \<subseteq> lessThan y) = (x \<le> (y::'a::linorder))"
|
paulson@15418
|
179 |
apply (auto simp add: lessThan_def)
|
paulson@15418
|
180 |
apply (subst linorder_not_less [symmetric], blast)
|
paulson@13850
|
181 |
done
|
paulson@13850
|
182 |
|
paulson@13850
|
183 |
lemma lessThan_eq_iff [iff]:
|
paulson@15418
|
184 |
"(lessThan x = lessThan y) = (x = (y::'a::linorder))"
|
paulson@15418
|
185 |
apply (rule iffI)
|
paulson@15418
|
186 |
apply (erule equalityE)
|
paulson@15418
|
187 |
apply (simp_all add: lessThan_subset_iff)
|
paulson@13850
|
188 |
done
|
paulson@13850
|
189 |
|
paulson@13850
|
190 |
|
paulson@13850
|
191 |
subsection {*Two-sided intervals*}
|
ballarin@13735
|
192 |
|
nipkow@24691
|
193 |
context ord
|
nipkow@24691
|
194 |
begin
|
nipkow@24691
|
195 |
|
paulson@24286
|
196 |
lemma greaterThanLessThan_iff [simp,noatp]:
|
haftmann@25062
|
197 |
"(i : {l<..<u}) = (l < i & i < u)"
|
ballarin@13735
|
198 |
by (simp add: greaterThanLessThan_def)
|
ballarin@13735
|
199 |
|
paulson@24286
|
200 |
lemma atLeastLessThan_iff [simp,noatp]:
|
haftmann@25062
|
201 |
"(i : {l..<u}) = (l <= i & i < u)"
|
ballarin@13735
|
202 |
by (simp add: atLeastLessThan_def)
|
ballarin@13735
|
203 |
|
paulson@24286
|
204 |
lemma greaterThanAtMost_iff [simp,noatp]:
|
haftmann@25062
|
205 |
"(i : {l<..u}) = (l < i & i <= u)"
|
ballarin@13735
|
206 |
by (simp add: greaterThanAtMost_def)
|
ballarin@13735
|
207 |
|
paulson@24286
|
208 |
lemma atLeastAtMost_iff [simp,noatp]:
|
haftmann@25062
|
209 |
"(i : {l..u}) = (l <= i & i <= u)"
|
ballarin@13735
|
210 |
by (simp add: atLeastAtMost_def)
|
ballarin@13735
|
211 |
|
wenzelm@14577
|
212 |
text {* The above four lemmas could be declared as iffs.
|
wenzelm@14577
|
213 |
If we do so, a call to blast in Hyperreal/Star.ML, lemma @{text STAR_Int}
|
wenzelm@14577
|
214 |
seems to take forever (more than one hour). *}
|
nipkow@24691
|
215 |
end
|
ballarin@13735
|
216 |
|
nipkow@15554
|
217 |
subsubsection{* Emptyness and singletons *}
|
nipkow@15554
|
218 |
|
nipkow@24691
|
219 |
context order
|
nipkow@24691
|
220 |
begin
|
nipkow@15554
|
221 |
|
haftmann@25062
|
222 |
lemma atLeastAtMost_empty [simp]: "n < m ==> {m..n} = {}";
|
nipkow@24691
|
223 |
by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
|
nipkow@24691
|
224 |
|
haftmann@25062
|
225 |
lemma atLeastLessThan_empty[simp]: "n \<le> m ==> {m..<n} = {}"
|
nipkow@15554
|
226 |
by (auto simp add: atLeastLessThan_def)
|
nipkow@15554
|
227 |
|
haftmann@25062
|
228 |
lemma greaterThanAtMost_empty[simp]:"l \<le> k ==> {k<..l} = {}"
|
nipkow@17719
|
229 |
by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
|
nipkow@17719
|
230 |
|
haftmann@25062
|
231 |
lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..l} = {}"
|
nipkow@17719
|
232 |
by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
|
nipkow@17719
|
233 |
|
haftmann@25062
|
234 |
lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"
|
nipkow@24691
|
235 |
by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
|
nipkow@24691
|
236 |
|
nipkow@24691
|
237 |
end
|
paulson@14485
|
238 |
|
paulson@14485
|
239 |
subsection {* Intervals of natural numbers *}
|
paulson@14485
|
240 |
|
paulson@15047
|
241 |
subsubsection {* The Constant @{term lessThan} *}
|
paulson@15047
|
242 |
|
paulson@14485
|
243 |
lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
|
paulson@14485
|
244 |
by (simp add: lessThan_def)
|
paulson@14485
|
245 |
|
paulson@14485
|
246 |
lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
|
paulson@14485
|
247 |
by (simp add: lessThan_def less_Suc_eq, blast)
|
paulson@14485
|
248 |
|
paulson@14485
|
249 |
lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
|
paulson@14485
|
250 |
by (simp add: lessThan_def atMost_def less_Suc_eq_le)
|
paulson@14485
|
251 |
|
paulson@14485
|
252 |
lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
|
paulson@14485
|
253 |
by blast
|
paulson@14485
|
254 |
|
paulson@15047
|
255 |
subsubsection {* The Constant @{term greaterThan} *}
|
paulson@15047
|
256 |
|
paulson@14485
|
257 |
lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"
|
paulson@14485
|
258 |
apply (simp add: greaterThan_def)
|
paulson@14485
|
259 |
apply (blast dest: gr0_conv_Suc [THEN iffD1])
|
paulson@14485
|
260 |
done
|
paulson@14485
|
261 |
|
paulson@14485
|
262 |
lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
|
paulson@14485
|
263 |
apply (simp add: greaterThan_def)
|
paulson@14485
|
264 |
apply (auto elim: linorder_neqE)
|
paulson@14485
|
265 |
done
|
paulson@14485
|
266 |
|
paulson@14485
|
267 |
lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"
|
paulson@14485
|
268 |
by blast
|
paulson@14485
|
269 |
|
paulson@15047
|
270 |
subsubsection {* The Constant @{term atLeast} *}
|
paulson@15047
|
271 |
|
paulson@14485
|
272 |
lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
|
paulson@14485
|
273 |
by (unfold atLeast_def UNIV_def, simp)
|
paulson@14485
|
274 |
|
paulson@14485
|
275 |
lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
|
paulson@14485
|
276 |
apply (simp add: atLeast_def)
|
paulson@14485
|
277 |
apply (simp add: Suc_le_eq)
|
paulson@14485
|
278 |
apply (simp add: order_le_less, blast)
|
paulson@14485
|
279 |
done
|
paulson@14485
|
280 |
|
paulson@14485
|
281 |
lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
|
paulson@14485
|
282 |
by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
|
paulson@14485
|
283 |
|
paulson@14485
|
284 |
lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"
|
paulson@14485
|
285 |
by blast
|
paulson@14485
|
286 |
|
paulson@15047
|
287 |
subsubsection {* The Constant @{term atMost} *}
|
paulson@15047
|
288 |
|
paulson@14485
|
289 |
lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
|
paulson@14485
|
290 |
by (simp add: atMost_def)
|
paulson@14485
|
291 |
|
paulson@14485
|
292 |
lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
|
paulson@14485
|
293 |
apply (simp add: atMost_def)
|
paulson@14485
|
294 |
apply (simp add: less_Suc_eq order_le_less, blast)
|
paulson@14485
|
295 |
done
|
paulson@14485
|
296 |
|
paulson@14485
|
297 |
lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"
|
paulson@14485
|
298 |
by blast
|
paulson@14485
|
299 |
|
paulson@15047
|
300 |
subsubsection {* The Constant @{term atLeastLessThan} *}
|
paulson@15047
|
301 |
|
nipkow@24449
|
302 |
text{*The orientation of the following rule is tricky. The lhs is
|
nipkow@24449
|
303 |
defined in terms of the rhs. Hence the chosen orientation makes sense
|
nipkow@24449
|
304 |
in this theory --- the reverse orientation complicates proofs (eg
|
nipkow@24449
|
305 |
nontermination). But outside, when the definition of the lhs is rarely
|
nipkow@24449
|
306 |
used, the opposite orientation seems preferable because it reduces a
|
nipkow@24449
|
307 |
specific concept to a more general one. *}
|
paulson@15047
|
308 |
lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
|
nipkow@15042
|
309 |
by(simp add:lessThan_def atLeastLessThan_def)
|
nipkow@24449
|
310 |
|
nipkow@24449
|
311 |
declare atLeast0LessThan[symmetric, code unfold]
|
nipkow@24449
|
312 |
|
nipkow@24449
|
313 |
lemma atLeastLessThan0: "{m..<0::nat} = {}"
|
paulson@15047
|
314 |
by (simp add: atLeastLessThan_def)
|
nipkow@24449
|
315 |
|
paulson@15047
|
316 |
subsubsection {* Intervals of nats with @{term Suc} *}
|
paulson@15047
|
317 |
|
paulson@15047
|
318 |
text{*Not a simprule because the RHS is too messy.*}
|
paulson@15047
|
319 |
lemma atLeastLessThanSuc:
|
paulson@15047
|
320 |
"{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
|
paulson@15418
|
321 |
by (auto simp add: atLeastLessThan_def)
|
paulson@15047
|
322 |
|
paulson@15418
|
323 |
lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
|
paulson@15047
|
324 |
by (auto simp add: atLeastLessThan_def)
|
nipkow@16041
|
325 |
(*
|
paulson@15047
|
326 |
lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
|
paulson@15047
|
327 |
by (induct k, simp_all add: atLeastLessThanSuc)
|
paulson@15047
|
328 |
|
paulson@15047
|
329 |
lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
|
paulson@15047
|
330 |
by (auto simp add: atLeastLessThan_def)
|
nipkow@16041
|
331 |
*)
|
nipkow@15045
|
332 |
lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
|
paulson@14485
|
333 |
by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
|
paulson@14485
|
334 |
|
paulson@15418
|
335 |
lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
|
paulson@15418
|
336 |
by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
|
paulson@14485
|
337 |
greaterThanAtMost_def)
|
paulson@14485
|
338 |
|
paulson@15418
|
339 |
lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
|
paulson@15418
|
340 |
by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
|
paulson@14485
|
341 |
greaterThanLessThan_def)
|
paulson@14485
|
342 |
|
nipkow@15554
|
343 |
lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
|
nipkow@15554
|
344 |
by (auto simp add: atLeastAtMost_def)
|
nipkow@15554
|
345 |
|
nipkow@16733
|
346 |
subsubsection {* Image *}
|
nipkow@16733
|
347 |
|
nipkow@16733
|
348 |
lemma image_add_atLeastAtMost:
|
nipkow@16733
|
349 |
"(%n::nat. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
|
nipkow@16733
|
350 |
proof
|
nipkow@16733
|
351 |
show "?A \<subseteq> ?B" by auto
|
nipkow@16733
|
352 |
next
|
nipkow@16733
|
353 |
show "?B \<subseteq> ?A"
|
nipkow@16733
|
354 |
proof
|
nipkow@16733
|
355 |
fix n assume a: "n : ?B"
|
webertj@20217
|
356 |
hence "n - k : {i..j}" by auto
|
nipkow@16733
|
357 |
moreover have "n = (n - k) + k" using a by auto
|
nipkow@16733
|
358 |
ultimately show "n : ?A" by blast
|
nipkow@16733
|
359 |
qed
|
nipkow@16733
|
360 |
qed
|
nipkow@16733
|
361 |
|
nipkow@16733
|
362 |
lemma image_add_atLeastLessThan:
|
nipkow@16733
|
363 |
"(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
|
nipkow@16733
|
364 |
proof
|
nipkow@16733
|
365 |
show "?A \<subseteq> ?B" by auto
|
nipkow@16733
|
366 |
next
|
nipkow@16733
|
367 |
show "?B \<subseteq> ?A"
|
nipkow@16733
|
368 |
proof
|
nipkow@16733
|
369 |
fix n assume a: "n : ?B"
|
webertj@20217
|
370 |
hence "n - k : {i..<j}" by auto
|
nipkow@16733
|
371 |
moreover have "n = (n - k) + k" using a by auto
|
nipkow@16733
|
372 |
ultimately show "n : ?A" by blast
|
nipkow@16733
|
373 |
qed
|
nipkow@16733
|
374 |
qed
|
nipkow@16733
|
375 |
|
nipkow@16733
|
376 |
corollary image_Suc_atLeastAtMost[simp]:
|
nipkow@16733
|
377 |
"Suc ` {i..j} = {Suc i..Suc j}"
|
nipkow@16733
|
378 |
using image_add_atLeastAtMost[where k=1] by simp
|
nipkow@16733
|
379 |
|
nipkow@16733
|
380 |
corollary image_Suc_atLeastLessThan[simp]:
|
nipkow@16733
|
381 |
"Suc ` {i..<j} = {Suc i..<Suc j}"
|
nipkow@16733
|
382 |
using image_add_atLeastLessThan[where k=1] by simp
|
nipkow@16733
|
383 |
|
nipkow@16733
|
384 |
lemma image_add_int_atLeastLessThan:
|
nipkow@16733
|
385 |
"(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
|
nipkow@16733
|
386 |
apply (auto simp add: image_def)
|
nipkow@16733
|
387 |
apply (rule_tac x = "x - l" in bexI)
|
nipkow@16733
|
388 |
apply auto
|
nipkow@16733
|
389 |
done
|
nipkow@16733
|
390 |
|
nipkow@16733
|
391 |
|
paulson@14485
|
392 |
subsubsection {* Finiteness *}
|
paulson@14485
|
393 |
|
nipkow@15045
|
394 |
lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
|
paulson@14485
|
395 |
by (induct k) (simp_all add: lessThan_Suc)
|
paulson@14485
|
396 |
|
paulson@14485
|
397 |
lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
|
paulson@14485
|
398 |
by (induct k) (simp_all add: atMost_Suc)
|
paulson@14485
|
399 |
|
paulson@14485
|
400 |
lemma finite_greaterThanLessThan [iff]:
|
nipkow@15045
|
401 |
fixes l :: nat shows "finite {l<..<u}"
|
paulson@14485
|
402 |
by (simp add: greaterThanLessThan_def)
|
paulson@14485
|
403 |
|
paulson@14485
|
404 |
lemma finite_atLeastLessThan [iff]:
|
nipkow@15045
|
405 |
fixes l :: nat shows "finite {l..<u}"
|
paulson@14485
|
406 |
by (simp add: atLeastLessThan_def)
|
paulson@14485
|
407 |
|
paulson@14485
|
408 |
lemma finite_greaterThanAtMost [iff]:
|
nipkow@15045
|
409 |
fixes l :: nat shows "finite {l<..u}"
|
paulson@14485
|
410 |
by (simp add: greaterThanAtMost_def)
|
paulson@14485
|
411 |
|
paulson@14485
|
412 |
lemma finite_atLeastAtMost [iff]:
|
paulson@14485
|
413 |
fixes l :: nat shows "finite {l..u}"
|
paulson@14485
|
414 |
by (simp add: atLeastAtMost_def)
|
paulson@14485
|
415 |
|
paulson@14485
|
416 |
lemma bounded_nat_set_is_finite:
|
nipkow@24853
|
417 |
"(ALL i:N. i < (n::nat)) ==> finite N"
|
paulson@14485
|
418 |
-- {* A bounded set of natural numbers is finite. *}
|
paulson@14485
|
419 |
apply (rule finite_subset)
|
paulson@14485
|
420 |
apply (rule_tac [2] finite_lessThan, auto)
|
paulson@14485
|
421 |
done
|
paulson@14485
|
422 |
|
nipkow@24853
|
423 |
text{* Any subset of an interval of natural numbers the size of the
|
nipkow@24853
|
424 |
subset is exactly that interval. *}
|
nipkow@24853
|
425 |
|
nipkow@24853
|
426 |
lemma subset_card_intvl_is_intvl:
|
nipkow@24853
|
427 |
"A <= {k..<k+card A} \<Longrightarrow> A = {k..<k+card A}" (is "PROP ?P")
|
nipkow@24853
|
428 |
proof cases
|
nipkow@24853
|
429 |
assume "finite A"
|
nipkow@24853
|
430 |
thus "PROP ?P"
|
nipkow@24853
|
431 |
proof(induct A rule:finite_linorder_induct)
|
nipkow@24853
|
432 |
case empty thus ?case by auto
|
nipkow@24853
|
433 |
next
|
nipkow@24853
|
434 |
case (insert A b)
|
nipkow@24853
|
435 |
moreover hence "b ~: A" by auto
|
nipkow@24853
|
436 |
moreover have "A <= {k..<k+card A}" and "b = k+card A"
|
nipkow@24853
|
437 |
using `b ~: A` insert by fastsimp+
|
nipkow@24853
|
438 |
ultimately show ?case by auto
|
nipkow@24853
|
439 |
qed
|
nipkow@24853
|
440 |
next
|
nipkow@24853
|
441 |
assume "~finite A" thus "PROP ?P" by simp
|
nipkow@24853
|
442 |
qed
|
nipkow@24853
|
443 |
|
nipkow@24853
|
444 |
|
paulson@14485
|
445 |
subsubsection {* Cardinality *}
|
paulson@14485
|
446 |
|
nipkow@15045
|
447 |
lemma card_lessThan [simp]: "card {..<u} = u"
|
paulson@15251
|
448 |
by (induct u, simp_all add: lessThan_Suc)
|
paulson@14485
|
449 |
|
paulson@14485
|
450 |
lemma card_atMost [simp]: "card {..u} = Suc u"
|
paulson@14485
|
451 |
by (simp add: lessThan_Suc_atMost [THEN sym])
|
paulson@14485
|
452 |
|
nipkow@15045
|
453 |
lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
|
nipkow@15045
|
454 |
apply (subgoal_tac "card {l..<u} = card {..<u-l}")
|
paulson@14485
|
455 |
apply (erule ssubst, rule card_lessThan)
|
nipkow@15045
|
456 |
apply (subgoal_tac "(%x. x + l) ` {..<u-l} = {l..<u}")
|
paulson@14485
|
457 |
apply (erule subst)
|
paulson@14485
|
458 |
apply (rule card_image)
|
paulson@14485
|
459 |
apply (simp add: inj_on_def)
|
paulson@14485
|
460 |
apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
|
paulson@14485
|
461 |
apply (rule_tac x = "x - l" in exI)
|
paulson@14485
|
462 |
apply arith
|
paulson@14485
|
463 |
done
|
paulson@14485
|
464 |
|
paulson@15418
|
465 |
lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
|
paulson@14485
|
466 |
by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
|
paulson@14485
|
467 |
|
paulson@15418
|
468 |
lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
|
paulson@14485
|
469 |
by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
|
paulson@14485
|
470 |
|
nipkow@15045
|
471 |
lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
|
paulson@14485
|
472 |
by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
|
paulson@14485
|
473 |
|
paulson@14485
|
474 |
subsection {* Intervals of integers *}
|
paulson@14485
|
475 |
|
nipkow@15045
|
476 |
lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
|
paulson@14485
|
477 |
by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
|
paulson@14485
|
478 |
|
paulson@15418
|
479 |
lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
|
paulson@14485
|
480 |
by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
|
paulson@14485
|
481 |
|
paulson@15418
|
482 |
lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
|
paulson@15418
|
483 |
"{l+1..<u} = {l<..<u::int}"
|
paulson@14485
|
484 |
by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
|
paulson@14485
|
485 |
|
paulson@14485
|
486 |
subsubsection {* Finiteness *}
|
paulson@14485
|
487 |
|
paulson@15418
|
488 |
lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
|
nipkow@15045
|
489 |
{(0::int)..<u} = int ` {..<nat u}"
|
paulson@14485
|
490 |
apply (unfold image_def lessThan_def)
|
paulson@14485
|
491 |
apply auto
|
paulson@14485
|
492 |
apply (rule_tac x = "nat x" in exI)
|
paulson@14485
|
493 |
apply (auto simp add: zless_nat_conj zless_nat_eq_int_zless [THEN sym])
|
paulson@14485
|
494 |
done
|
paulson@14485
|
495 |
|
nipkow@15045
|
496 |
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
|
paulson@14485
|
497 |
apply (case_tac "0 \<le> u")
|
paulson@14485
|
498 |
apply (subst image_atLeastZeroLessThan_int, assumption)
|
paulson@14485
|
499 |
apply (rule finite_imageI)
|
paulson@14485
|
500 |
apply auto
|
paulson@14485
|
501 |
done
|
paulson@14485
|
502 |
|
nipkow@15045
|
503 |
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
|
nipkow@15045
|
504 |
apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
|
paulson@14485
|
505 |
apply (erule subst)
|
paulson@14485
|
506 |
apply (rule finite_imageI)
|
paulson@14485
|
507 |
apply (rule finite_atLeastZeroLessThan_int)
|
nipkow@16733
|
508 |
apply (rule image_add_int_atLeastLessThan)
|
paulson@14485
|
509 |
done
|
paulson@14485
|
510 |
|
paulson@15418
|
511 |
lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
|
paulson@14485
|
512 |
by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
|
paulson@14485
|
513 |
|
paulson@15418
|
514 |
lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
|
paulson@14485
|
515 |
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
|
paulson@14485
|
516 |
|
paulson@15418
|
517 |
lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
|
paulson@14485
|
518 |
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
|
paulson@14485
|
519 |
|
nipkow@24853
|
520 |
|
paulson@14485
|
521 |
subsubsection {* Cardinality *}
|
paulson@14485
|
522 |
|
nipkow@15045
|
523 |
lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
|
paulson@14485
|
524 |
apply (case_tac "0 \<le> u")
|
paulson@14485
|
525 |
apply (subst image_atLeastZeroLessThan_int, assumption)
|
paulson@14485
|
526 |
apply (subst card_image)
|
paulson@14485
|
527 |
apply (auto simp add: inj_on_def)
|
paulson@14485
|
528 |
done
|
paulson@14485
|
529 |
|
nipkow@15045
|
530 |
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
|
nipkow@15045
|
531 |
apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
|
paulson@14485
|
532 |
apply (erule ssubst, rule card_atLeastZeroLessThan_int)
|
nipkow@15045
|
533 |
apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
|
paulson@14485
|
534 |
apply (erule subst)
|
paulson@14485
|
535 |
apply (rule card_image)
|
paulson@14485
|
536 |
apply (simp add: inj_on_def)
|
nipkow@16733
|
537 |
apply (rule image_add_int_atLeastLessThan)
|
paulson@14485
|
538 |
done
|
paulson@14485
|
539 |
|
paulson@14485
|
540 |
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
|
paulson@14485
|
541 |
apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
|
paulson@14485
|
542 |
apply (auto simp add: compare_rls)
|
paulson@14485
|
543 |
done
|
paulson@14485
|
544 |
|
paulson@15418
|
545 |
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
|
paulson@14485
|
546 |
by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
|
paulson@14485
|
547 |
|
nipkow@15045
|
548 |
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
|
paulson@14485
|
549 |
by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
|
paulson@14485
|
550 |
|
paulson@14485
|
551 |
|
paulson@13850
|
552 |
subsection {*Lemmas useful with the summation operator setsum*}
|
paulson@13850
|
553 |
|
ballarin@16102
|
554 |
text {* For examples, see Algebra/poly/UnivPoly2.thy *}
|
ballarin@13735
|
555 |
|
wenzelm@14577
|
556 |
subsubsection {* Disjoint Unions *}
|
ballarin@13735
|
557 |
|
wenzelm@14577
|
558 |
text {* Singletons and open intervals *}
|
ballarin@13735
|
559 |
|
ballarin@13735
|
560 |
lemma ivl_disj_un_singleton:
|
nipkow@15045
|
561 |
"{l::'a::linorder} Un {l<..} = {l..}"
|
nipkow@15045
|
562 |
"{..<u} Un {u::'a::linorder} = {..u}"
|
nipkow@15045
|
563 |
"(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
|
nipkow@15045
|
564 |
"(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
|
nipkow@15045
|
565 |
"(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
|
nipkow@15045
|
566 |
"(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
|
ballarin@14398
|
567 |
by auto
|
ballarin@13735
|
568 |
|
wenzelm@14577
|
569 |
text {* One- and two-sided intervals *}
|
ballarin@13735
|
570 |
|
ballarin@13735
|
571 |
lemma ivl_disj_un_one:
|
nipkow@15045
|
572 |
"(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
|
nipkow@15045
|
573 |
"(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
|
nipkow@15045
|
574 |
"(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
|
nipkow@15045
|
575 |
"(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
|
nipkow@15045
|
576 |
"(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
|
nipkow@15045
|
577 |
"(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
|
nipkow@15045
|
578 |
"(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
|
nipkow@15045
|
579 |
"(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
|
ballarin@14398
|
580 |
by auto
|
ballarin@13735
|
581 |
|
wenzelm@14577
|
582 |
text {* Two- and two-sided intervals *}
|
ballarin@13735
|
583 |
|
ballarin@13735
|
584 |
lemma ivl_disj_un_two:
|
nipkow@15045
|
585 |
"[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
|
nipkow@15045
|
586 |
"[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
|
nipkow@15045
|
587 |
"[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
|
nipkow@15045
|
588 |
"[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
|
nipkow@15045
|
589 |
"[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
|
nipkow@15045
|
590 |
"[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
|
nipkow@15045
|
591 |
"[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
|
nipkow@15045
|
592 |
"[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
|
ballarin@14398
|
593 |
by auto
|
ballarin@13735
|
594 |
|
ballarin@13735
|
595 |
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two
|
ballarin@13735
|
596 |
|
wenzelm@14577
|
597 |
subsubsection {* Disjoint Intersections *}
|
ballarin@13735
|
598 |
|
wenzelm@14577
|
599 |
text {* Singletons and open intervals *}
|
ballarin@13735
|
600 |
|
ballarin@13735
|
601 |
lemma ivl_disj_int_singleton:
|
nipkow@15045
|
602 |
"{l::'a::order} Int {l<..} = {}"
|
nipkow@15045
|
603 |
"{..<u} Int {u} = {}"
|
nipkow@15045
|
604 |
"{l} Int {l<..<u} = {}"
|
nipkow@15045
|
605 |
"{l<..<u} Int {u} = {}"
|
nipkow@15045
|
606 |
"{l} Int {l<..u} = {}"
|
nipkow@15045
|
607 |
"{l..<u} Int {u} = {}"
|
ballarin@13735
|
608 |
by simp+
|
ballarin@13735
|
609 |
|
wenzelm@14577
|
610 |
text {* One- and two-sided intervals *}
|
ballarin@13735
|
611 |
|
ballarin@13735
|
612 |
lemma ivl_disj_int_one:
|
nipkow@15045
|
613 |
"{..l::'a::order} Int {l<..<u} = {}"
|
nipkow@15045
|
614 |
"{..<l} Int {l..<u} = {}"
|
nipkow@15045
|
615 |
"{..l} Int {l<..u} = {}"
|
nipkow@15045
|
616 |
"{..<l} Int {l..u} = {}"
|
nipkow@15045
|
617 |
"{l<..u} Int {u<..} = {}"
|
nipkow@15045
|
618 |
"{l<..<u} Int {u..} = {}"
|
nipkow@15045
|
619 |
"{l..u} Int {u<..} = {}"
|
nipkow@15045
|
620 |
"{l..<u} Int {u..} = {}"
|
ballarin@14398
|
621 |
by auto
|
ballarin@13735
|
622 |
|
wenzelm@14577
|
623 |
text {* Two- and two-sided intervals *}
|
ballarin@13735
|
624 |
|
ballarin@13735
|
625 |
lemma ivl_disj_int_two:
|
nipkow@15045
|
626 |
"{l::'a::order<..<m} Int {m..<u} = {}"
|
nipkow@15045
|
627 |
"{l<..m} Int {m<..<u} = {}"
|
nipkow@15045
|
628 |
"{l..<m} Int {m..<u} = {}"
|
nipkow@15045
|
629 |
"{l..m} Int {m<..<u} = {}"
|
nipkow@15045
|
630 |
"{l<..<m} Int {m..u} = {}"
|
nipkow@15045
|
631 |
"{l<..m} Int {m<..u} = {}"
|
nipkow@15045
|
632 |
"{l..<m} Int {m..u} = {}"
|
nipkow@15045
|
633 |
"{l..m} Int {m<..u} = {}"
|
ballarin@14398
|
634 |
by auto
|
ballarin@13735
|
635 |
|
ballarin@13735
|
636 |
lemmas ivl_disj_int = ivl_disj_int_singleton ivl_disj_int_one ivl_disj_int_two
|
ballarin@13735
|
637 |
|
nipkow@15542
|
638 |
subsubsection {* Some Differences *}
|
nipkow@15542
|
639 |
|
nipkow@15542
|
640 |
lemma ivl_diff[simp]:
|
nipkow@15542
|
641 |
"i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
|
nipkow@15542
|
642 |
by(auto)
|
nipkow@15542
|
643 |
|
nipkow@15542
|
644 |
|
nipkow@15542
|
645 |
subsubsection {* Some Subset Conditions *}
|
nipkow@15542
|
646 |
|
paulson@24286
|
647 |
lemma ivl_subset [simp,noatp]:
|
nipkow@15542
|
648 |
"({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
|
nipkow@15542
|
649 |
apply(auto simp:linorder_not_le)
|
nipkow@15542
|
650 |
apply(rule ccontr)
|
nipkow@15542
|
651 |
apply(insert linorder_le_less_linear[of i n])
|
nipkow@15542
|
652 |
apply(clarsimp simp:linorder_not_le)
|
nipkow@15542
|
653 |
apply(fastsimp)
|
nipkow@15542
|
654 |
done
|
nipkow@15542
|
655 |
|
nipkow@15041
|
656 |
|
nipkow@15042
|
657 |
subsection {* Summation indexed over intervals *}
|
nipkow@15042
|
658 |
|
nipkow@15042
|
659 |
syntax
|
nipkow@15042
|
660 |
"_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
|
nipkow@15048
|
661 |
"_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
|
nipkow@16052
|
662 |
"_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
|
nipkow@16052
|
663 |
"_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
|
nipkow@15042
|
664 |
syntax (xsymbols)
|
nipkow@15042
|
665 |
"_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
|
nipkow@15048
|
666 |
"_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
|
nipkow@16052
|
667 |
"_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
|
nipkow@16052
|
668 |
"_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
|
nipkow@15042
|
669 |
syntax (HTML output)
|
nipkow@15042
|
670 |
"_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
|
nipkow@15048
|
671 |
"_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
|
nipkow@16052
|
672 |
"_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
|
nipkow@16052
|
673 |
"_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
|
nipkow@15056
|
674 |
syntax (latex_sum output)
|
nipkow@15052
|
675 |
"_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
|
nipkow@15052
|
676 |
("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
|
nipkow@15052
|
677 |
"_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
|
nipkow@15052
|
678 |
("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
|
nipkow@16052
|
679 |
"_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
|
nipkow@16052
|
680 |
("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
|
nipkow@15052
|
681 |
"_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
|
nipkow@16052
|
682 |
("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
|
nipkow@15042
|
683 |
|
nipkow@15048
|
684 |
translations
|
nipkow@15048
|
685 |
"\<Sum>x=a..b. t" == "setsum (%x. t) {a..b}"
|
nipkow@15048
|
686 |
"\<Sum>x=a..<b. t" == "setsum (%x. t) {a..<b}"
|
nipkow@16052
|
687 |
"\<Sum>i\<le>n. t" == "setsum (\<lambda>i. t) {..n}"
|
nipkow@15048
|
688 |
"\<Sum>i<n. t" == "setsum (\<lambda>i. t) {..<n}"
|
nipkow@15042
|
689 |
|
nipkow@15052
|
690 |
text{* The above introduces some pretty alternative syntaxes for
|
nipkow@15056
|
691 |
summation over intervals:
|
nipkow@15052
|
692 |
\begin{center}
|
nipkow@15052
|
693 |
\begin{tabular}{lll}
|
nipkow@15056
|
694 |
Old & New & \LaTeX\\
|
nipkow@15056
|
695 |
@{term[source]"\<Sum>x\<in>{a..b}. e"} & @{term"\<Sum>x=a..b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..b. e"}\\
|
nipkow@15056
|
696 |
@{term[source]"\<Sum>x\<in>{a..<b}. e"} & @{term"\<Sum>x=a..<b. e"} & @{term[mode=latex_sum]"\<Sum>x=a..<b. e"}\\
|
nipkow@16052
|
697 |
@{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
|
nipkow@15056
|
698 |
@{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
|
nipkow@15052
|
699 |
\end{tabular}
|
nipkow@15052
|
700 |
\end{center}
|
nipkow@15056
|
701 |
The left column shows the term before introduction of the new syntax,
|
nipkow@15056
|
702 |
the middle column shows the new (default) syntax, and the right column
|
nipkow@15056
|
703 |
shows a special syntax. The latter is only meaningful for latex output
|
nipkow@15056
|
704 |
and has to be activated explicitly by setting the print mode to
|
wenzelm@21502
|
705 |
@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} in
|
nipkow@15056
|
706 |
antiquotations). It is not the default \LaTeX\ output because it only
|
nipkow@15056
|
707 |
works well with italic-style formulae, not tt-style.
|
nipkow@15052
|
708 |
|
nipkow@15052
|
709 |
Note that for uniformity on @{typ nat} it is better to use
|
nipkow@15052
|
710 |
@{term"\<Sum>x::nat=0..<n. e"} rather than @{text"\<Sum>x<n. e"}: @{text setsum} may
|
nipkow@15052
|
711 |
not provide all lemmas available for @{term"{m..<n}"} also in the
|
nipkow@15052
|
712 |
special form for @{term"{..<n}"}. *}
|
nipkow@15052
|
713 |
|
nipkow@15542
|
714 |
text{* This congruence rule should be used for sums over intervals as
|
nipkow@15542
|
715 |
the standard theorem @{text[source]setsum_cong} does not work well
|
nipkow@15542
|
716 |
with the simplifier who adds the unsimplified premise @{term"x:B"} to
|
nipkow@15542
|
717 |
the context. *}
|
nipkow@15542
|
718 |
|
nipkow@15542
|
719 |
lemma setsum_ivl_cong:
|
nipkow@15542
|
720 |
"\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
|
nipkow@15542
|
721 |
setsum f {a..<b} = setsum g {c..<d}"
|
nipkow@15542
|
722 |
by(rule setsum_cong, simp_all)
|
nipkow@15042
|
723 |
|
nipkow@16041
|
724 |
(* FIXME why are the following simp rules but the corresponding eqns
|
nipkow@16041
|
725 |
on intervals are not? *)
|
nipkow@16041
|
726 |
|
nipkow@16052
|
727 |
lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
|
nipkow@16052
|
728 |
by (simp add:atMost_Suc add_ac)
|
nipkow@16052
|
729 |
|
nipkow@16041
|
730 |
lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
|
nipkow@16041
|
731 |
by (simp add:lessThan_Suc add_ac)
|
nipkow@15041
|
732 |
|
nipkow@15911
|
733 |
lemma setsum_cl_ivl_Suc[simp]:
|
nipkow@15561
|
734 |
"setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"
|
nipkow@15561
|
735 |
by (auto simp:add_ac atLeastAtMostSuc_conv)
|
nipkow@15561
|
736 |
|
nipkow@15911
|
737 |
lemma setsum_op_ivl_Suc[simp]:
|
nipkow@15561
|
738 |
"setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
|
nipkow@15561
|
739 |
by (auto simp:add_ac atLeastLessThanSuc)
|
nipkow@16041
|
740 |
(*
|
nipkow@15561
|
741 |
lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
|
nipkow@15561
|
742 |
(\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
|
nipkow@15561
|
743 |
by (auto simp:add_ac atLeastAtMostSuc_conv)
|
nipkow@16041
|
744 |
*)
|
nipkow@15539
|
745 |
lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
|
nipkow@15539
|
746 |
setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
|
nipkow@15539
|
747 |
by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
|
nipkow@15539
|
748 |
|
nipkow@15539
|
749 |
lemma setsum_diff_nat_ivl:
|
nipkow@15539
|
750 |
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
|
nipkow@15539
|
751 |
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
|
nipkow@15539
|
752 |
setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
|
nipkow@15539
|
753 |
using setsum_add_nat_ivl [of m n p f,symmetric]
|
nipkow@15539
|
754 |
apply (simp add: add_ac)
|
nipkow@15539
|
755 |
done
|
nipkow@15539
|
756 |
|
nipkow@16733
|
757 |
subsection{* Shifting bounds *}
|
nipkow@16733
|
758 |
|
nipkow@15539
|
759 |
lemma setsum_shift_bounds_nat_ivl:
|
nipkow@15539
|
760 |
"setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
|
nipkow@15539
|
761 |
by (induct "n", auto simp:atLeastLessThanSuc)
|
nipkow@15539
|
762 |
|
nipkow@16733
|
763 |
lemma setsum_shift_bounds_cl_nat_ivl:
|
nipkow@16733
|
764 |
"setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
|
nipkow@16733
|
765 |
apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])
|
nipkow@16733
|
766 |
apply (simp add:image_add_atLeastAtMost o_def)
|
nipkow@16733
|
767 |
done
|
nipkow@16733
|
768 |
|
nipkow@16733
|
769 |
corollary setsum_shift_bounds_cl_Suc_ivl:
|
nipkow@16733
|
770 |
"setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
|
nipkow@16733
|
771 |
by (simp add:setsum_shift_bounds_cl_nat_ivl[where k=1,simplified])
|
nipkow@16733
|
772 |
|
nipkow@16733
|
773 |
corollary setsum_shift_bounds_Suc_ivl:
|
nipkow@16733
|
774 |
"setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
|
nipkow@16733
|
775 |
by (simp add:setsum_shift_bounds_nat_ivl[where k=1,simplified])
|
nipkow@16733
|
776 |
|
kleing@19106
|
777 |
lemma setsum_head:
|
kleing@19106
|
778 |
fixes n :: nat
|
kleing@19106
|
779 |
assumes mn: "m <= n"
|
kleing@19106
|
780 |
shows "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
|
kleing@19106
|
781 |
proof -
|
kleing@19106
|
782 |
from mn
|
kleing@19106
|
783 |
have "{m..n} = {m} \<union> {m<..n}"
|
kleing@19106
|
784 |
by (auto intro: ivl_disj_un_singleton)
|
kleing@19106
|
785 |
hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
|
kleing@19106
|
786 |
by (simp add: atLeast0LessThan)
|
kleing@19106
|
787 |
also have "\<dots> = ?rhs" by simp
|
kleing@19106
|
788 |
finally show ?thesis .
|
kleing@19106
|
789 |
qed
|
kleing@19106
|
790 |
|
kleing@19106
|
791 |
lemma setsum_head_upt:
|
kleing@19022
|
792 |
fixes m::nat
|
kleing@19022
|
793 |
assumes m: "0 < m"
|
kleing@19106
|
794 |
shows "(\<Sum>x<m. P x) = P 0 + (\<Sum>x\<in>{1..<m}. P x)"
|
kleing@19022
|
795 |
proof -
|
kleing@19106
|
796 |
have "(\<Sum>x<m. P x) = (\<Sum>x\<in>{0..<m}. P x)"
|
kleing@19022
|
797 |
by (simp add: atLeast0LessThan)
|
kleing@19106
|
798 |
also
|
kleing@19106
|
799 |
from m
|
kleing@19106
|
800 |
have "\<dots> = (\<Sum>x\<in>{0..m - 1}. P x)"
|
kleing@19106
|
801 |
by (cases m) (auto simp add: atLeastLessThanSuc_atLeastAtMost)
|
kleing@19106
|
802 |
also
|
kleing@19106
|
803 |
have "\<dots> = P 0 + (\<Sum>x\<in>{0<..m - 1}. P x)"
|
kleing@19106
|
804 |
by (simp add: setsum_head)
|
kleing@19106
|
805 |
also
|
kleing@19106
|
806 |
from m
|
kleing@19106
|
807 |
have "{0<..m - 1} = {1..<m}"
|
kleing@19106
|
808 |
by (cases m) (auto simp add: atLeastLessThanSuc_atLeastAtMost)
|
kleing@19106
|
809 |
finally show ?thesis .
|
kleing@19022
|
810 |
qed
|
kleing@19022
|
811 |
|
ballarin@17149
|
812 |
subsection {* The formula for geometric sums *}
|
ballarin@17149
|
813 |
|
ballarin@17149
|
814 |
lemma geometric_sum:
|
ballarin@17149
|
815 |
"x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) =
|
huffman@22713
|
816 |
(x ^ n - 1) / (x - 1::'a::{field, recpower})"
|
nipkow@23496
|
817 |
by (induct "n") (simp_all add:field_simps power_Suc)
|
ballarin@17149
|
818 |
|
kleing@19469
|
819 |
subsection {* The formula for arithmetic sums *}
|
kleing@19469
|
820 |
|
kleing@19469
|
821 |
lemma gauss_sum:
|
huffman@23277
|
822 |
"((1::'a::comm_semiring_1) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
|
kleing@19469
|
823 |
of_nat n*((of_nat n)+1)"
|
kleing@19469
|
824 |
proof (induct n)
|
kleing@19469
|
825 |
case 0
|
kleing@19469
|
826 |
show ?case by simp
|
kleing@19469
|
827 |
next
|
kleing@19469
|
828 |
case (Suc n)
|
nipkow@23477
|
829 |
then show ?case by (simp add: ring_simps)
|
kleing@19469
|
830 |
qed
|
kleing@19469
|
831 |
|
kleing@19469
|
832 |
theorem arith_series_general:
|
huffman@23277
|
833 |
"((1::'a::comm_semiring_1) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
|
kleing@19469
|
834 |
of_nat n * (a + (a + of_nat(n - 1)*d))"
|
kleing@19469
|
835 |
proof cases
|
kleing@19469
|
836 |
assume ngt1: "n > 1"
|
kleing@19469
|
837 |
let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
|
kleing@19469
|
838 |
have
|
kleing@19469
|
839 |
"(\<Sum>i\<in>{..<n}. a+?I i*d) =
|
kleing@19469
|
840 |
((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
|
kleing@19469
|
841 |
by (rule setsum_addf)
|
kleing@19469
|
842 |
also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
|
kleing@19469
|
843 |
also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
|
kleing@19469
|
844 |
by (simp add: setsum_right_distrib setsum_head_upt mult_ac)
|
kleing@19469
|
845 |
also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
|
kleing@19469
|
846 |
by (simp add: left_distrib right_distrib)
|
kleing@19469
|
847 |
also from ngt1 have "{1..<n} = {1..n - 1}"
|
kleing@19469
|
848 |
by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
|
kleing@19469
|
849 |
also from ngt1
|
kleing@19469
|
850 |
have "(1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..n - 1}. ?I i) = ((1+1)*?n*a + d*?I (n - 1)*?I n)"
|
kleing@19469
|
851 |
by (simp only: mult_ac gauss_sum [of "n - 1"])
|
huffman@23431
|
852 |
(simp add: mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
|
kleing@19469
|
853 |
finally show ?thesis by (simp add: mult_ac add_ac right_distrib)
|
kleing@19469
|
854 |
next
|
kleing@19469
|
855 |
assume "\<not>(n > 1)"
|
kleing@19469
|
856 |
hence "n = 1 \<or> n = 0" by auto
|
kleing@19469
|
857 |
thus ?thesis by (auto simp: mult_ac right_distrib)
|
kleing@19469
|
858 |
qed
|
kleing@19469
|
859 |
|
kleing@19469
|
860 |
lemma arith_series_nat:
|
kleing@19469
|
861 |
"Suc (Suc 0) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
|
kleing@19469
|
862 |
proof -
|
kleing@19469
|
863 |
have
|
kleing@19469
|
864 |
"((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
|
kleing@19469
|
865 |
of_nat(n) * (a + (a + of_nat(n - 1)*d))"
|
kleing@19469
|
866 |
by (rule arith_series_general)
|
kleing@19469
|
867 |
thus ?thesis by (auto simp add: of_nat_id)
|
kleing@19469
|
868 |
qed
|
kleing@19469
|
869 |
|
kleing@19469
|
870 |
lemma arith_series_int:
|
kleing@19469
|
871 |
"(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
|
kleing@19469
|
872 |
of_nat n * (a + (a + of_nat(n - 1)*d))"
|
kleing@19469
|
873 |
proof -
|
kleing@19469
|
874 |
have
|
kleing@19469
|
875 |
"((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
|
kleing@19469
|
876 |
of_nat(n) * (a + (a + of_nat(n - 1)*d))"
|
kleing@19469
|
877 |
by (rule arith_series_general)
|
kleing@19469
|
878 |
thus ?thesis by simp
|
kleing@19469
|
879 |
qed
|
paulson@15418
|
880 |
|
kleing@19022
|
881 |
lemma sum_diff_distrib:
|
kleing@19022
|
882 |
fixes P::"nat\<Rightarrow>nat"
|
kleing@19022
|
883 |
shows
|
kleing@19022
|
884 |
"\<forall>x. Q x \<le> P x \<Longrightarrow>
|
kleing@19022
|
885 |
(\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x)"
|
kleing@19022
|
886 |
proof (induct n)
|
kleing@19022
|
887 |
case 0 show ?case by simp
|
kleing@19022
|
888 |
next
|
kleing@19022
|
889 |
case (Suc n)
|
kleing@19022
|
890 |
|
kleing@19022
|
891 |
let ?lhs = "(\<Sum>x<n. P x) - (\<Sum>x<n. Q x)"
|
kleing@19022
|
892 |
let ?rhs = "\<Sum>x<n. P x - Q x"
|
kleing@19022
|
893 |
|
kleing@19022
|
894 |
from Suc have "?lhs = ?rhs" by simp
|
kleing@19022
|
895 |
moreover
|
kleing@19022
|
896 |
from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp
|
kleing@19022
|
897 |
moreover
|
kleing@19022
|
898 |
from Suc have
|
kleing@19022
|
899 |
"(\<Sum>x<n. P x) + P n - ((\<Sum>x<n. Q x) + Q n) = ?rhs + (P n - Q n)"
|
kleing@19022
|
900 |
by (subst diff_diff_left[symmetric],
|
kleing@19022
|
901 |
subst diff_add_assoc2)
|
kleing@19022
|
902 |
(auto simp: diff_add_assoc2 intro: setsum_mono)
|
kleing@19022
|
903 |
ultimately
|
kleing@19022
|
904 |
show ?case by simp
|
kleing@19022
|
905 |
qed
|
kleing@19022
|
906 |
|
nipkow@8924
|
907 |
end
|