src/HOL/SetInterval.thy
 author hoelzl Tue May 11 19:19:45 2010 +0200 (2010-05-11) changeset 36846 0f67561ed5a6 parent 36755 d1b498f2f50b child 37388 793618618f78 permissions -rw-r--r--
 nipkow@8924  1 (* Title: HOL/SetInterval.thy  wenzelm@32960  2  Author: Tobias Nipkow  wenzelm@32960  3  Author: Clemens Ballarin  wenzelm@32960  4  Author: Jeremy Avigad  nipkow@8924  5 ballarin@13735  6 lessThan, greaterThan, atLeast, atMost and two-sided intervals  nipkow@8924  7 *)  nipkow@8924  8 wenzelm@14577  9 header {* Set intervals *}  wenzelm@14577  10 nipkow@15131  11 theory SetInterval  haftmann@33318  12 imports Int Nat_Transfer  nipkow@15131  13 begin  nipkow@8924  14 nipkow@24691  15 context ord  nipkow@24691  16 begin  nipkow@24691  17 definition  wenzelm@32960  18  lessThan :: "'a => 'a set" ("(1{..<_})") where  haftmann@25062  19  "{.. 'a set" ("(1{.._})") where  haftmann@25062  23  "{..u} == {x. x \ u}"  nipkow@24691  24 nipkow@24691  25 definition  wenzelm@32960  26  greaterThan :: "'a => 'a set" ("(1{_<..})") where  haftmann@25062  27  "{l<..} == {x. l 'a set" ("(1{_..})") where  haftmann@25062  31  "{l..} == {x. l\x}"  nipkow@24691  32 nipkow@24691  33 definition  haftmann@25062  34  greaterThanLessThan :: "'a => 'a => 'a set" ("(1{_<..<_})") where  haftmann@25062  35  "{l<.. 'a => 'a set" ("(1{_..<_})") where  haftmann@25062  39  "{l.. 'a => 'a set" ("(1{_<.._})") where  haftmann@25062  43  "{l<..u} == {l<..} Int {..u}"  nipkow@24691  44 nipkow@24691  45 definition  haftmann@25062  46  atLeastAtMost :: "'a => 'a => 'a set" ("(1{_.._})") where  haftmann@25062  47  "{l..u} == {l..} Int {..u}"  nipkow@24691  48 nipkow@24691  49 end  nipkow@8924  50 ballarin@13735  51 nipkow@15048  52 text{* A note of warning when using @{term"{.. 'a => 'b set => 'b set" ("(3UN _<=_./ _)" [0, 0, 10] 10)  huffman@36364  58  "_UNION_less" :: "'a => 'a => 'b set => 'b set" ("(3UN _<_./ _)" [0, 0, 10] 10)  huffman@36364  59  "_INTER_le" :: "'a => 'a => 'b set => 'b set" ("(3INT _<=_./ _)" [0, 0, 10] 10)  huffman@36364  60  "_INTER_less" :: "'a => 'a => 'b set => 'b set" ("(3INT _<_./ _)" [0, 0, 10] 10)  kleing@14418  61 nipkow@30372  62 syntax (xsymbols)  huffman@36364  63  "_UNION_le" :: "'a => 'a => 'b set => 'b set" ("(3\ _\_./ _)" [0, 0, 10] 10)  huffman@36364  64  "_UNION_less" :: "'a => 'a => 'b set => 'b set" ("(3\ _<_./ _)" [0, 0, 10] 10)  huffman@36364  65  "_INTER_le" :: "'a => 'a => 'b set => 'b set" ("(3\ _\_./ _)" [0, 0, 10] 10)  huffman@36364  66  "_INTER_less" :: "'a => 'a => 'b set => 'b set" ("(3\ _<_./ _)" [0, 0, 10] 10)  kleing@14418  67 nipkow@30372  68 syntax (latex output)  huffman@36364  69  "_UNION_le" :: "'a \ 'a => 'b set => 'b set" ("(3\(00_ \ _)/ _)" [0, 0, 10] 10)  huffman@36364  70  "_UNION_less" :: "'a \ 'a => 'b set => 'b set" ("(3\(00_ < _)/ _)" [0, 0, 10] 10)  huffman@36364  71  "_INTER_le" :: "'a \ 'a => 'b set => 'b set" ("(3\(00_ \ _)/ _)" [0, 0, 10] 10)  huffman@36364  72  "_INTER_less" :: "'a \ 'a => 'b set => 'b set" ("(3\(00_ < _)/ _)" [0, 0, 10] 10)  kleing@14418  73 kleing@14418  74 translations  kleing@14418  75  "UN i<=n. A" == "UN i:{..n}. A"  nipkow@15045  76  "UN i atLeast y) = (y \ (x::'a::order))"  paulson@15418  124 by (blast intro: order_trans)  paulson@13850  125 paulson@13850  126 lemma atLeast_eq_iff [iff]:  paulson@15418  127  "(atLeast x = atLeast y) = (x = (y::'a::linorder))"  paulson@13850  128 by (blast intro: order_antisym order_trans)  paulson@13850  129 paulson@13850  130 lemma greaterThan_subset_iff [iff]:  paulson@15418  131  "(greaterThan x \ greaterThan y) = (y \ (x::'a::linorder))"  paulson@15418  132 apply (auto simp add: greaterThan_def)  paulson@15418  133  apply (subst linorder_not_less [symmetric], blast)  paulson@13850  134 done  paulson@13850  135 paulson@13850  136 lemma greaterThan_eq_iff [iff]:  paulson@15418  137  "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"  paulson@15418  138 apply (rule iffI)  paulson@15418  139  apply (erule equalityE)  haftmann@29709  140  apply simp_all  paulson@13850  141 done  paulson@13850  142 paulson@15418  143 lemma atMost_subset_iff [iff]: "(atMost x \ atMost y) = (x \ (y::'a::order))"  paulson@13850  144 by (blast intro: order_trans)  paulson@13850  145 paulson@15418  146 lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"  paulson@13850  147 by (blast intro: order_antisym order_trans)  paulson@13850  148 paulson@13850  149 lemma lessThan_subset_iff [iff]:  paulson@15418  150  "(lessThan x \ lessThan y) = (x \ (y::'a::linorder))"  paulson@15418  151 apply (auto simp add: lessThan_def)  paulson@15418  152  apply (subst linorder_not_less [symmetric], blast)  paulson@13850  153 done  paulson@13850  154 paulson@13850  155 lemma lessThan_eq_iff [iff]:  paulson@15418  156  "(lessThan x = lessThan y) = (x = (y::'a::linorder))"  paulson@15418  157 apply (rule iffI)  paulson@15418  158  apply (erule equalityE)  haftmann@29709  159  apply simp_all  ballarin@13735  160 done  ballarin@13735  161 ballarin@13735  162 paulson@13850  163 subsection {*Two-sided intervals*}  ballarin@13735  164 nipkow@24691  165 context ord  nipkow@24691  166 begin  nipkow@24691  167 blanchet@35828  168 lemma greaterThanLessThan_iff [simp,no_atp]:  haftmann@25062  169  "(i : {l<.. {a..b} = {}"  nipkow@32400  197 by(auto simp: atLeastAtMost_def atLeast_def atMost_def)  nipkow@32400  198 nipkow@32400  199 lemma atLeastatMost_empty_iff[simp]:  nipkow@32400  200  "{a..b} = {} \ (~ a <= b)"  nipkow@32400  201 by auto (blast intro: order_trans)  nipkow@32400  202 nipkow@32400  203 lemma atLeastatMost_empty_iff2[simp]:  nipkow@32400  204  "{} = {a..b} \ (~ a <= b)"  nipkow@32400  205 by auto (blast intro: order_trans)  nipkow@32400  206 nipkow@32400  207 lemma atLeastLessThan_empty[simp]:  nipkow@32400  208  "b <= a \ {a.. (~ a < b)"  nipkow@32400  213 by auto (blast intro: le_less_trans)  nipkow@32400  214 nipkow@32400  215 lemma atLeastLessThan_empty_iff2[simp]:  nipkow@32400  216  "{} = {a.. (~ a < b)"  nipkow@32400  217 by auto (blast intro: le_less_trans)  nipkow@15554  218 nipkow@32400  219 lemma greaterThanAtMost_empty[simp]: "l \ k ==> {k<..l} = {}"  nipkow@17719  220 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)  nipkow@17719  221 nipkow@32400  222 lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {} \ ~ k < l"  nipkow@32400  223 by auto (blast intro: less_le_trans)  nipkow@32400  224 nipkow@32400  225 lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} \ ~ k < l"  nipkow@32400  226 by auto (blast intro: less_le_trans)  nipkow@32400  227 haftmann@29709  228 lemma greaterThanLessThan_empty[simp]:"l \ k ==> {k<.. {a .. b} = {a}" by simp  hoelzl@36846  235 nipkow@32400  236 lemma atLeastatMost_subset_iff[simp]:  nipkow@32400  237  "{a..b} <= {c..d} \ (~ a <= b) | c <= a & b <= d"  nipkow@32400  238 unfolding atLeastAtMost_def atLeast_def atMost_def  nipkow@32400  239 by (blast intro: order_trans)  nipkow@32400  240 nipkow@32400  241 lemma atLeastatMost_psubset_iff:  nipkow@32400  242  "{a..b} < {c..d} \  nipkow@32400  243  ((~ a <= b) | c <= a & b <= d & (c < a | b < d)) & c <= d"  nipkow@32400  244 by(simp add: psubset_eq expand_set_eq less_le_not_le)(blast intro: order_trans)  nipkow@32400  245 hoelzl@36846  246 lemma atLeastAtMost_singleton_iff[simp]:  hoelzl@36846  247  "{a .. b} = {c} \ a = b \ b = c"  hoelzl@36846  248 proof  hoelzl@36846  249  assume "{a..b} = {c}"  hoelzl@36846  250  hence "\ (\ a \ b)" unfolding atLeastatMost_empty_iff[symmetric] by simp  hoelzl@36846  251  moreover with {a..b} = {c} have "c \ a \ b \ c" by auto  hoelzl@36846  252  ultimately show "a = b \ b = c" by auto  hoelzl@36846  253 qed simp  hoelzl@36846  254 nipkow@24691  255 end  paulson@14485  256 nipkow@32408  257 lemma (in linorder) atLeastLessThan_subset_iff:  nipkow@32408  258  "{a.. b <= a | c<=a & b<=d"  nipkow@32408  259 apply (auto simp:subset_eq Ball_def)  nipkow@32408  260 apply(frule_tac x=a in spec)  nipkow@32408  261 apply(erule_tac x=d in allE)  nipkow@32408  262 apply (simp add: less_imp_le)  nipkow@32408  263 done  nipkow@32408  264 nipkow@32456  265 subsubsection {* Intersection *}  nipkow@32456  266 nipkow@32456  267 context linorder  nipkow@32456  268 begin  nipkow@32456  269 nipkow@32456  270 lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"  nipkow@32456  271 by auto  nipkow@32456  272 nipkow@32456  273 lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"  nipkow@32456  274 by auto  nipkow@32456  275 nipkow@32456  276 lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"  nipkow@32456  277 by auto  nipkow@32456  278 nipkow@32456  279 lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"  nipkow@32456  280 by auto  nipkow@32456  281 nipkow@32456  282 lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"  nipkow@32456  283 by auto  nipkow@32456  284 nipkow@32456  285 lemma Int_atLeastLessThan[simp]: "{a.. n then insert n {m.. Suc n \ {m..Suc n} = insert (Suc n) {m..n}"  nipkow@15554  407 by (auto simp add: atLeastAtMost_def)  nipkow@15554  408 paulson@33044  409 lemma atLeastLessThan_add_Un: "i \ j \ {i.. {j.. ?B" by auto  nipkow@16733  420 next  nipkow@16733  421  show "?B \ ?A"  nipkow@16733  422  proof  nipkow@16733  423  fix n assume a: "n : ?B"  webertj@20217  424  hence "n - k : {i..j}" by auto  nipkow@16733  425  moreover have "n = (n - k) + k" using a by auto  nipkow@16733  426  ultimately show "n : ?A" by blast  nipkow@16733  427  qed  nipkow@16733  428 qed  nipkow@16733  429 nipkow@16733  430 lemma image_add_atLeastLessThan:  nipkow@16733  431  "(%n::nat. n+k)  {i.. ?B" by auto  nipkow@16733  434 next  nipkow@16733  435  show "?B \ ?A"  nipkow@16733  436  proof  nipkow@16733  437  fix n assume a: "n : ?B"  webertj@20217  438  hence "n - k : {i.. uminus  {x<..}"  hoelzl@35580  470  by (rule imageI) (simp add: *)  hoelzl@35580  471  thus "y \ uminus  {x<..}" by simp  hoelzl@35580  472 next  hoelzl@35580  473  fix y assume "y \ -x"  hoelzl@35580  474  have "- (-y) \ uminus  {x..}"  hoelzl@35580  475  by (rule imageI) (insert y \ -x[THEN le_imp_neg_le], simp)  hoelzl@35580  476  thus "y \ uminus  {x..}" by simp  hoelzl@35580  477 qed simp_all  hoelzl@35580  478 hoelzl@35580  479 lemma  hoelzl@35580  480  fixes x :: 'a  hoelzl@35580  481  shows image_uminus_lessThan[simp]: "uminus  {.. finite N"  nipkow@28068  528 apply (rule finite_subset)  nipkow@28068  529  apply (rule_tac [2] finite_lessThan, auto)  nipkow@28068  530 done  nipkow@28068  531 nipkow@31044  532 text {* A set of natural numbers is finite iff it is bounded. *}  nipkow@31044  533 lemma finite_nat_set_iff_bounded:  nipkow@31044  534  "finite(N::nat set) = (EX m. ALL n:N. nnat. (!!n. n \ f n) ==> finite {n. f n \ u}"  nipkow@28068  550 by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)  paulson@14485  551 nipkow@24853  552 text{* Any subset of an interval of natural numbers the size of the  nipkow@24853  553 subset is exactly that interval. *}  nipkow@24853  554 nipkow@24853  555 lemma subset_card_intvl_is_intvl:  nipkow@24853  556  "A <= {k.. A = {k..i\n::nat. M i) = (\i\{1..n}. M i) \ M 0" (is "?A = ?B")  nipkow@36755  578 proof  nipkow@36755  579  show "?A <= ?B"  nipkow@36755  580  proof  nipkow@36755  581  fix x assume "x : ?A"  nipkow@36755  582  then obtain i where i: "i\n" "x : M i" by auto  nipkow@36755  583  show "x : ?B"  nipkow@36755  584  proof(cases i)  nipkow@36755  585  case 0 with i show ?thesis by simp  nipkow@36755  586  next  nipkow@36755  587  case (Suc j) with i show ?thesis by auto  nipkow@36755  588  qed  nipkow@36755  589  qed  nipkow@36755  590 next  nipkow@36755  591  show "?B <= ?A" by auto  nipkow@36755  592 qed  nipkow@36755  593 nipkow@36755  594 lemma UN_le_add_shift:  nipkow@36755  595  "(\i\n::nat. M(i+k)) = (\i\{k..n+k}. M i)" (is "?A = ?B")  nipkow@36755  596 proof  nipkow@36755  597  show "?A <= ?B" by fastsimp  nipkow@36755  598 next  nipkow@36755  599  show "?B <= ?A"  nipkow@36755  600  proof  nipkow@36755  601  fix x assume "x : ?B"  nipkow@36755  602  then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto  nipkow@36755  603  hence "i-k\n & x : M((i-k)+k)" by auto  nipkow@36755  604  thus "x : ?A" by blast  nipkow@36755  605  qed  nipkow@36755  606 qed  nipkow@36755  607 paulson@32596  608 lemma UN_UN_finite_eq: "(\n::nat. \i\{0..n. A n)"  paulson@32596  609  by (auto simp add: atLeast0LessThan)  paulson@32596  610 paulson@32596  611 lemma UN_finite_subset: "(!!n::nat. (\i\{0.. C) \ (\n. A n) \ C"  paulson@32596  612  by (subst UN_UN_finite_eq [symmetric]) blast  paulson@32596  613 paulson@33044  614 lemma UN_finite2_subset:  paulson@33044  615  "(!!n::nat. (\i\{0.. (\i\{0.. (\n. A n) \ (\n. B n)"  paulson@33044  616  apply (rule UN_finite_subset)  paulson@33044  617  apply (subst UN_UN_finite_eq [symmetric, of B])  paulson@33044  618  apply blast  paulson@33044  619  done  paulson@32596  620 paulson@32596  621 lemma UN_finite2_eq:  paulson@33044  622  "(!!n::nat. (\i\{0..i\{0.. (\n. A n) = (\n. B n)"  paulson@33044  623  apply (rule subset_antisym)  paulson@33044  624  apply (rule UN_finite2_subset, blast)  paulson@33044  625  apply (rule UN_finite2_subset [where k=k])  huffman@35216  626  apply (force simp add: atLeastLessThan_add_Un [of 0])  paulson@33044  627  done  paulson@32596  628 paulson@32596  629 paulson@14485  630 subsubsection {* Cardinality *}  paulson@14485  631 nipkow@15045  632 lemma card_lessThan [simp]: "card {.. \h. bij_betw h {0.. \h. bij_betw h M {0.. finite B \ card A = card B \ EX h. bij_betw h A B"  nipkow@31438  671 apply(drule ex_bij_betw_finite_nat)  nipkow@31438  672 apply(drule ex_bij_betw_nat_finite)  nipkow@31438  673 apply(auto intro!:bij_betw_trans)  nipkow@31438  674 done  nipkow@31438  675 nipkow@31438  676 lemma ex_bij_betw_nat_finite_1:  nipkow@31438  677  "finite M \ \h. bij_betw h {1 .. card M} M"  nipkow@31438  678 by (rule finite_same_card_bij) auto  nipkow@31438  679 nipkow@26105  680 paulson@14485  681 subsection {* Intervals of integers *}  paulson@14485  682 nipkow@15045  683 lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l.. u ==>  nipkow@15045  696  {(0::int).. u")  paulson@14485  705  apply (subst image_atLeastZeroLessThan_int, assumption)  paulson@14485  706  apply (rule finite_imageI)  paulson@14485  707  apply auto  paulson@14485  708  done  paulson@14485  709 nipkow@15045  710 lemma finite_atLeastLessThan_int [iff]: "finite {l.. u")  paulson@14485  732  apply (subst image_atLeastZeroLessThan_int, assumption)  paulson@14485  733  apply (subst card_image)  paulson@14485  734  apply (auto simp add: inj_on_def)  paulson@14485  735  done  paulson@14485  736 nipkow@15045  737 lemma card_atLeastLessThan_int [simp]: "card {l.. k < (i::nat)}"  bulwahn@27656  759 proof -  bulwahn@27656  760  have "{k. P k \ k < i} \ {.. M"  bulwahn@27656  766 shows "card {k \ M. k < Suc i} \ 0"  bulwahn@27656  767 proof -  bulwahn@27656  768  from zero_in_M have "{k \ M. k < Suc i} \ {}" by auto  bulwahn@27656  769  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)  bulwahn@27656  770 qed  bulwahn@27656  771 bulwahn@27656  772 lemma card_less_Suc2: "0 \ M \ card {k. Suc k \ M \ k < i} = card {k \ M. k < Suc i}"  huffman@30079  773 apply (rule card_bij_eq [of "Suc" _ _ "\x. x - Suc 0"])  bulwahn@27656  774 apply simp  bulwahn@27656  775 apply fastsimp  bulwahn@27656  776 apply auto  bulwahn@27656  777 apply (rule inj_on_diff_nat)  bulwahn@27656  778 apply auto  bulwahn@27656  779 apply (case_tac x)  bulwahn@27656  780 apply auto  bulwahn@27656  781 apply (case_tac xa)  bulwahn@27656  782 apply auto  bulwahn@27656  783 apply (case_tac xa)  bulwahn@27656  784 apply auto  bulwahn@27656  785 done  bulwahn@27656  786 bulwahn@27656  787 lemma card_less_Suc:  bulwahn@27656  788  assumes zero_in_M: "0 \ M"  bulwahn@27656  789  shows "Suc (card {k. Suc k \ M \ k < i}) = card {k \ M. k < Suc i}"  bulwahn@27656  790 proof -  bulwahn@27656  791  from assms have a: "0 \ {k \ M. k < Suc i}" by simp  bulwahn@27656  792  hence c: "{k \ M. k < Suc i} = insert 0 ({k \ M. k < Suc i} - {0})"  bulwahn@27656  793  by (auto simp only: insert_Diff)  bulwahn@27656  794  have b: "{k \ M. k < Suc i} - {0} = {k \ M - {0}. k < Suc i}" by auto  bulwahn@27656  795  from finite_M_bounded_by_nat[of "\x. x \ M" "Suc i"] have "Suc (card {k. Suc k \ M \ k < i}) = card (insert 0 ({k \ M. k < Suc i} - {0}))"  bulwahn@27656  796  apply (subst card_insert)  bulwahn@27656  797  apply simp_all  bulwahn@27656  798  apply (subst b)  bulwahn@27656  799  apply (subst card_less_Suc2[symmetric])  bulwahn@27656  800  apply simp_all  bulwahn@27656  801  done  bulwahn@27656  802  with c show ?thesis by simp  bulwahn@27656  803 qed  bulwahn@27656  804 paulson@14485  805 paulson@13850  806 subsection {*Lemmas useful with the summation operator setsum*}  paulson@13850  807 ballarin@16102  808 text {* For examples, see Algebra/poly/UnivPoly2.thy *}  ballarin@13735  809 wenzelm@14577  810 subsubsection {* Disjoint Unions *}  ballarin@13735  811 wenzelm@14577  812 text {* Singletons and open intervals *}  ballarin@13735  813 ballarin@13735  814 lemma ivl_disj_un_singleton:  nipkow@15045  815  "{l::'a::linorder} Un {l<..} = {l..}"  nipkow@15045  816  "{.. {l} Un {l<.. {l<.. {l} Un {l<..u} = {l..u}"  nipkow@15045  820  "(l::'a::linorder) <= u ==> {l.. {..l} Un {l<.. {.. {..l} Un {l<..u} = {..u}"  nipkow@15045  829  "(l::'a::linorder) <= u ==> {.. {l<..u} Un {u<..} = {l<..}"  nipkow@15045  831  "(l::'a::linorder) < u ==> {l<.. {l..u} Un {u<..} = {l..}"  nipkow@15045  833  "(l::'a::linorder) <= u ==> {l.. {l<.. {l<..m} Un {m<.. {l.. {l..m} Un {m<.. {l<.. {l<..m} Un {m<..u} = {l<..u}"  nipkow@15045  845  "[| (l::'a::linorder) <= m; m <= u |] ==> {l.. {l..m} Un {m<..u} = {l..u}"  ballarin@14398  847 by auto  ballarin@13735  848 ballarin@13735  849 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two  ballarin@13735  850 wenzelm@14577  851 subsubsection {* Disjoint Intersections *}  ballarin@13735  852 wenzelm@14577  853 text {* One- and two-sided intervals *}  ballarin@13735  854 ballarin@13735  855 lemma ivl_disj_int_one:  nipkow@15045  856  "{..l::'a::order} Int {l<.. n \ {i.. {m.. i | m \ i & j \ (n::'a::linorder))"  nipkow@15542  892 apply(auto simp:linorder_not_le)  nipkow@15542  893 apply(rule ccontr)  nipkow@15542  894 apply(insert linorder_le_less_linear[of i n])  nipkow@15542  895 apply(clarsimp simp:linorder_not_le)  nipkow@15542  896 apply(fastsimp)  nipkow@15542  897 done  nipkow@15542  898 nipkow@15041  899 nipkow@15042  900 subsection {* Summation indexed over intervals *}  nipkow@15042  901 nipkow@15042  902 syntax  nipkow@15042  903  "_from_to_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)  nipkow@15048  904  "_from_upto_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)  nipkow@16052  905  "_upt_setsum" :: "idt \ 'a \ 'b \ 'b" ("(SUM _<_./ _)" [0,0,10] 10)  nipkow@16052  906  "_upto_setsum" :: "idt \ 'a \ 'b \ 'b" ("(SUM _<=_./ _)" [0,0,10] 10)  nipkow@15042  907 syntax (xsymbols)  nipkow@15042  908  "_from_to_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _.._./ _)" [0,0,0,10] 10)  nipkow@15048  909  "_from_upto_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _..<_./ _)" [0,0,0,10] 10)  nipkow@16052  910  "_upt_setsum" :: "idt \ 'a \ 'b \ 'b" ("(3\_<_./ _)" [0,0,10] 10)  nipkow@16052  911  "_upto_setsum" :: "idt \ 'a \ 'b \ 'b" ("(3\_\_./ _)" [0,0,10] 10)  nipkow@15042  912 syntax (HTML output)  nipkow@15042  913  "_from_to_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _.._./ _)" [0,0,0,10] 10)  nipkow@15048  914  "_from_upto_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _..<_./ _)" [0,0,0,10] 10)  nipkow@16052  915  "_upt_setsum" :: "idt \ 'a \ 'b \ 'b" ("(3\_<_./ _)" [0,0,10] 10)  nipkow@16052  916  "_upto_setsum" :: "idt \ 'a \ 'b \ 'b" ("(3\_\_./ _)" [0,0,10] 10)  nipkow@15056  917 syntax (latex_sum output)  nipkow@15052  918  "_from_to_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b"  nipkow@15052  919  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)  nipkow@15052  920  "_from_upto_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b"  nipkow@15052  921  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)  nipkow@16052  922  "_upt_setsum" :: "idt \ 'a \ 'b \ 'b"  nipkow@16052  923  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)  nipkow@15052  924  "_upto_setsum" :: "idt \ 'a \ 'b \ 'b"  nipkow@16052  925  ("(3\<^raw:$\sum_{>_ \ _\<^raw:}$> _)" [0,0,10] 10)  nipkow@15041  926 nipkow@15048  927 translations  nipkow@28853  928  "\x=a..b. t" == "CONST setsum (%x. t) {a..b}"  nipkow@28853  929  "\x=a..i\n. t" == "CONST setsum (\i. t) {..n}"  nipkow@28853  931  "\ii. t) {..x\{a..b}. e"} & @{term"\x=a..b. e"} & @{term[mode=latex_sum]"\x=a..b. e"}\\  nipkow@15056  939 @{term[source]"\x\{a..x=a..x=a..x\{..b}. e"} & @{term"\x\b. e"} & @{term[mode=latex_sum]"\x\b. e"}\\  nipkow@15056  941 @{term[source]"\x\{..xxx::nat=0..xa = c; b = d; !!x. \ c \ x; x < d \ \ f x = g x \ \  nipkow@15542  964  setsum f {a..i \ Suc n. f i) = (\i \ n. f i) + f(Suc n)"  nipkow@16052  971 by (simp add:atMost_Suc add_ac)  nipkow@16052  972 nipkow@16041  973 lemma setsum_lessThan_Suc[simp]: "(\i < Suc n. f i) = (\i < n. f i) + f n"  nipkow@16041  974 by (simp add:lessThan_Suc add_ac)  nipkow@15041  975 nipkow@15911  976 lemma setsum_cl_ivl_Suc[simp]:  nipkow@15561  977  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"  nipkow@15561  978 by (auto simp:add_ac atLeastAtMostSuc_conv)  nipkow@15561  979 nipkow@15911  980 lemma setsum_op_ivl_Suc[simp]:  nipkow@15561  981  "setsum f {m..  nipkow@15561  985  (\i=n..m+1. f i) = (\i=n..m. f i) + f(m + 1)"  nipkow@15561  986 by (auto simp:add_ac atLeastAtMostSuc_conv)  nipkow@16041  987 *)  nipkow@28068  988 nipkow@28068  989 lemma setsum_head:  nipkow@28068  990  fixes n :: nat  nipkow@28068  991  assumes mn: "m <= n"  nipkow@28068  992  shows "(\x\{m..n}. P x) = P m + (\x\{m<..n}. P x)" (is "?lhs = ?rhs")  nipkow@28068  993 proof -  nipkow@28068  994  from mn  nipkow@28068  995  have "{m..n} = {m} \ {m<..n}"  nipkow@28068  996  by (auto intro: ivl_disj_un_singleton)  nipkow@28068  997  hence "?lhs = (\x\{m} \ {m<..n}. P x)"  nipkow@28068  998  by (simp add: atLeast0LessThan)  nipkow@28068  999  also have "\ = ?rhs" by simp  nipkow@28068  1000  finally show ?thesis .  nipkow@28068  1001 qed  nipkow@28068  1002 nipkow@28068  1003 lemma setsum_head_Suc:  nipkow@28068  1004  "m \ n \ setsum f {m..n} = f m + setsum f {Suc m..n}"  nipkow@28068  1005 by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)  nipkow@28068  1006 nipkow@28068  1007 lemma setsum_head_upt_Suc:  nipkow@28068  1008  "m < n \ setsum f {m.. n + 1"  nipkow@31501  1014  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"  nipkow@31501  1015 proof-  nipkow@31501  1016  have "{m .. n+p} = {m..n} \ {n+1..n+p}" using m \ n+1 by auto  nipkow@31501  1017  thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint  nipkow@31501  1018  atLeastSucAtMost_greaterThanAtMost)  nipkow@31501  1019 qed  nipkow@28068  1020 nipkow@15539  1021 lemma setsum_add_nat_ivl: "\ m \ n; n \ p \ \  nipkow@15539  1022  setsum f {m.. 'a::ab_group_add"  nipkow@15539  1027 shows "\ m \ n; n \ p \ \  nipkow@15539  1028  setsum f {m.. ('a::ab_group_add)"  nipkow@31505  1035  shows "setsum (\k. f k - f(k + 1)) {(m::nat) .. n} =  nipkow@31505  1036  (if m <= n then f m - f(n + 1) else 0)"  nipkow@31505  1037 by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)  nipkow@31505  1038 nipkow@31509  1039 lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]  nipkow@31509  1040 nipkow@31509  1041 lemma setsum_setsum_restrict:  nipkow@31509  1042  "finite S \ finite T \ setsum (\x. setsum (\y. f x y) {y. y\ T \ R x y}) S = setsum (\y. setsum (\x. f x y) {x. x \ S \ R x y}) T"  nipkow@31509  1043  by (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)  nipkow@31509  1044  (rule setsum_commute)  nipkow@31509  1045 nipkow@31509  1046 lemma setsum_image_gen: assumes fS: "finite S"  nipkow@31509  1047  shows "setsum g S = setsum (\y. setsum g {x. x \ S \ f x = y}) (f  S)"  nipkow@31509  1048 proof-  nipkow@31509  1049  { fix x assume "x \ S" then have "{y. y\ fS \ f x = y} = {f x}" by auto }  nipkow@31509  1050  hence "setsum g S = setsum (\x. setsum (\y. g x) {y. y\ fS \ f x = y}) S"  nipkow@31509  1051  by simp  nipkow@31509  1052  also have "\ = setsum (\y. setsum g {x. x \ S \ f x = y}) (f  S)"  nipkow@31509  1053  by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])  nipkow@31509  1054  finally show ?thesis .  nipkow@31509  1055 qed  nipkow@31509  1056 hoelzl@35171  1057 lemma setsum_le_included:  haftmann@36307  1058  fixes f :: "'a \ 'b::ordered_comm_monoid_add"  hoelzl@35171  1059  assumes "finite s" "finite t"  hoelzl@35171  1060  and "\y\t. 0 \ g y" "(\x\s. \y\t. i y = x \ f x \ g y)"  hoelzl@35171  1061  shows "setsum f s \ setsum g t"  hoelzl@35171  1062 proof -  hoelzl@35171  1063  have "setsum f s \ setsum (\y. setsum g {x. x\t \ i x = y}) s"  hoelzl@35171  1064  proof (rule setsum_mono)  hoelzl@35171  1065  fix y assume "y \ s"  hoelzl@35171  1066  with assms obtain z where z: "z \ t" "y = i z" "f y \ g z" by auto  hoelzl@35171  1067  with assms show "f y \ setsum g {x \ t. i x = y}" (is "?A y \ ?B y")  hoelzl@35171  1068  using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]  hoelzl@35171  1069  by (auto intro!: setsum_mono2)  hoelzl@35171  1070  qed  hoelzl@35171  1071  also have "... \ setsum (\y. setsum g {x. x\t \ i x = y}) (i  t)"  hoelzl@35171  1072  using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)  hoelzl@35171  1073  also have "... \ setsum g t"  hoelzl@35171  1074  using assms by (auto simp: setsum_image_gen[symmetric])  hoelzl@35171  1075  finally show ?thesis .  hoelzl@35171  1076 qed  hoelzl@35171  1077 nipkow@31509  1078 lemma setsum_multicount_gen:  nipkow@31509  1079  assumes "finite s" "finite t" "\j\t. (card {i\s. R i j} = k j)"  nipkow@31509  1080  shows "setsum (\i. (card {j\t. R i j})) s = setsum k t" (is "?l = ?r")  nipkow@31509  1081 proof-  nipkow@31509  1082  have "?l = setsum (\i. setsum (\x.1) {j\t. R i j}) s" by auto  nipkow@31509  1083  also have "\ = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]  nipkow@31509  1084  using assms(3) by auto  nipkow@31509  1085  finally show ?thesis .  nipkow@31509  1086 qed  nipkow@31509  1087 nipkow@31509  1088 lemma setsum_multicount:  nipkow@31509  1089  assumes "finite S" "finite T" "\j\T. (card {i\S. R i j} = k)"  nipkow@31509  1090  shows "setsum (\i. card {j\T. R i j}) S = k * card T" (is "?l = ?r")  nipkow@31509  1091 proof-  nipkow@31509  1092  have "?l = setsum (\i. k) T" by(rule setsum_multicount_gen)(auto simp:assms)  huffman@35216  1093  also have "\ = ?r" by(simp add: mult_commute)  nipkow@31509  1094  finally show ?thesis by auto  nipkow@31509  1095 qed  nipkow@31509  1096 nipkow@28068  1097 nipkow@16733  1098 subsection{* Shifting bounds *}  nipkow@16733  1099 nipkow@15539  1100 lemma setsum_shift_bounds_nat_ivl:  nipkow@15539  1101  "setsum f {m+k.. setsum f {Suc 0..k} = setsum f {0..k}"  nipkow@28068  1120 by(simp add:setsum_head_Suc)  kleing@19106  1121 nipkow@28068  1122 lemma setsum_shift_lb_Suc0_0_upt:  nipkow@28068  1123  "f(0::nat) = 0 \ setsum f {Suc 0.. 1"  haftmann@36307  1132  shows "(\i=0.. 0" by simp_all  haftmann@36307  1135  moreover have "(\i=0.. 0 have "(1 + y) ^ n = (y * inverse y) * (1 + y) ^ n" by simp  haftmann@36350  1141  ultimately show ?case by (simp add: field_simps divide_inverse)  haftmann@36307  1142  qed  haftmann@36307  1143  ultimately show ?thesis by simp  haftmann@36307  1144 qed  haftmann@36307  1145 ballarin@17149  1146 kleing@19469  1147 subsection {* The formula for arithmetic sums *}  kleing@19469  1148 kleing@19469  1149 lemma gauss_sum:  huffman@23277  1150  "((1::'a::comm_semiring_1) + 1)*(\i\{1..n}. of_nat i) =  kleing@19469  1151  of_nat n*((of_nat n)+1)"  kleing@19469  1152 proof (induct n)  kleing@19469  1153  case 0  kleing@19469  1154  show ?case by simp  kleing@19469  1155 next  kleing@19469  1156  case (Suc n)  nipkow@29667  1157  then show ?case by (simp add: algebra_simps)  kleing@19469  1158 qed  kleing@19469  1159 kleing@19469  1160 theorem arith_series_general:  huffman@23277  1161  "((1::'a::comm_semiring_1) + 1) * (\i\{.. 1"  kleing@19469  1165  let ?I = "\i. of_nat i" and ?n = "of_nat n"  kleing@19469  1166  have  kleing@19469  1167  "(\i\{..i\{..i\{.. = ?n*a + (\i\{.. = (?n*a + d*(\i\{1.. = (1+1)*?n*a + d*(1+1)*(\i\{1..i\{1..n - 1}. ?I i) = ((1+1)*?n*a + d*?I (n - 1)*?I n)"  huffman@30079  1180  by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)  huffman@23431  1181  (simp add: mult_ac trans [OF add_commute of_nat_Suc [symmetric]])  nipkow@29667  1182  finally show ?thesis by (simp add: algebra_simps)  kleing@19469  1183 next  kleing@19469  1184  assume "\(n > 1)"  kleing@19469  1185  hence "n = 1 \ n = 0" by auto  nipkow@29667  1186  thus ?thesis by (auto simp: algebra_simps)  kleing@19469  1187 qed  kleing@19469  1188 kleing@19469  1189 lemma arith_series_nat:  kleing@19469  1190  "Suc (Suc 0) * (\i\{..i\{..i\{..i\{..nat"  kleing@19022  1213  shows  kleing@19022  1214  "\x. Q x \ P x \  kleing@19022  1215  (\xxxxx 'a \ 'a \ 'b \ 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)  paulson@29960  1241  "_from_upto_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)  paulson@29960  1242  "_upt_setprod" :: "idt \ 'a \ 'b \ 'b" ("(PROD _<_./ _)" [0,0,10] 10)  paulson@29960  1243  "_upto_setprod" :: "idt \ 'a \ 'b \ 'b" ("(PROD _<=_./ _)" [0,0,10] 10)  paulson@29960  1244 syntax (xsymbols)  paulson@29960  1245  "_from_to_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _.._./ _)" [0,0,0,10] 10)  paulson@29960  1246  "_from_upto_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _..<_./ _)" [0,0,0,10] 10)  paulson@29960  1247  "_upt_setprod" :: "idt \ 'a \ 'b \ 'b" ("(3\_<_./ _)" [0,0,10] 10)  paulson@29960  1248  "_upto_setprod" :: "idt \ 'a \ 'b \ 'b" ("(3\_\_./ _)" [0,0,10] 10)  paulson@29960  1249 syntax (HTML output)  paulson@29960  1250  "_from_to_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _.._./ _)" [0,0,0,10] 10)  paulson@29960  1251  "_from_upto_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _..<_./ _)" [0,0,0,10] 10)  paulson@29960  1252  "_upt_setprod" :: "idt \ 'a \ 'b \ 'b" ("(3\_<_./ _)" [0,0,10] 10)  paulson@29960  1253  "_upto_setprod" :: "idt \ 'a \ 'b \ 'b" ("(3\_\_./ _)" [0,0,10] 10)  paulson@29960  1254 syntax (latex_prod output)  paulson@29960  1255  "_from_to_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b"  paulson@29960  1256  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)  paulson@29960  1257  "_from_upto_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b"  paulson@29960  1258  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)  paulson@29960  1259  "_upt_setprod" :: "idt \ 'a \ 'b \ 'b"  paulson@29960  1260  ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)  paulson@29960  1261  "_upto_setprod" :: "idt \ 'a \ 'b \ 'b"  paulson@29960  1262  ("(3\<^raw:$\prod_{>_ \ _\<^raw:}$> _)" [0,0,10] 10)  paulson@29960  1263 paulson@29960  1264 translations  paulson@29960  1265  "\x=a..b. t" == "CONST setprod (%x. t) {a..b}"  paulson@29960  1266  "\x=a..i\n. t" == "CONST setprod (\i. t) {..n}"  paulson@29960  1268  "\ii. t) {..= 0 \ nat_set {x..y}"  haftmann@33318  1284  by (simp add: nat_set_def)  haftmann@33318  1285 haftmann@35644  1286 declare transfer_morphism_nat_int[transfer add  haftmann@33318  1287  return: transfer_nat_int_set_functions  haftmann@33318  1288  transfer_nat_int_set_function_closures  haftmann@33318  1289 ]  haftmann@33318  1290 haftmann@33318  1291 lemma transfer_int_nat_set_functions:  haftmann@33318  1292  "is_nat m \ is_nat n \ {m..n} = int  {nat m..nat n}"  haftmann@33318  1293  by (simp only: is_nat_def transfer_nat_int_set_functions  haftmann@33318  1294  transfer_nat_int_set_function_closures  haftmann@33318  1295  transfer_nat_int_set_return_embed nat_0_le  haftmann@33318  1296  cong: transfer_nat_int_set_cong)  haftmann@33318  1297 haftmann@33318  1298 lemma transfer_int_nat_set_function_closures:  haftmann@33318  1299  "is_nat x \ nat_set {x..y}"  haftmann@33318  1300  by (simp only: transfer_nat_int_set_function_closures is_nat_def)  haftmann@33318  1301 haftmann@35644  1302 declare transfer_morphism_int_nat[transfer add  haftmann@33318  1303  return: transfer_int_nat_set_functions  haftmann@33318  1304  transfer_int_nat_set_function_closures  haftmann@33318  1305 ]  haftmann@33318  1306 nipkow@8924  1307 end `