src/HOL/SetInterval.thy
 author paulson Wed Feb 18 11:18:01 2009 +0000 (2009-02-18) changeset 29960 9d5c6f376768 parent 29920 b95f5b8b93dd child 30079 293b896b9c25 permissions -rw-r--r--
Syntactic support for products over set intervals
 nipkow@8924  1 (* Title: HOL/SetInterval.thy  ballarin@13735  2  Author: Tobias Nipkow and Clemens Ballarin  paulson@14485  3  Additions by Jeremy Avigad in March 2004  paulson@8957  4  Copyright 2000 TU Muenchen  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@25919  12 imports Int  nipkow@15131  13 begin  nipkow@8924  14 nipkow@24691  15 context ord  nipkow@24691  16 begin  nipkow@24691  17 definition  haftmann@25062  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  haftmann@25062  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"{.. nat => 'b set => 'b set" ("(3UN _<=_./ _)" 10)  kleing@14418  58  "@UNION_less" :: "nat => nat => 'b set => 'b set" ("(3UN _<_./ _)" 10)  kleing@14418  59  "@INTER_le" :: "nat => nat => 'b set => 'b set" ("(3INT _<=_./ _)" 10)  kleing@14418  60  "@INTER_less" :: "nat => nat => 'b set => 'b set" ("(3INT _<_./ _)" 10)  kleing@14418  61 kleing@14418  62 syntax (input)  kleing@14418  63  "@UNION_le" :: "nat => nat => 'b set => 'b set" ("(3\ _\_./ _)" 10)  kleing@14418  64  "@UNION_less" :: "nat => nat => 'b set => 'b set" ("(3\ _<_./ _)" 10)  kleing@14418  65  "@INTER_le" :: "nat => nat => 'b set => 'b set" ("(3\ _\_./ _)" 10)  kleing@14418  66  "@INTER_less" :: "nat => nat => 'b set => 'b set" ("(3\ _<_./ _)" 10)  kleing@14418  67 kleing@14418  68 syntax (xsymbols)  paulson@29960  69  "@UNION_le" :: "nat \ nat => 'b set => 'b set" ("(3\(00_ \ _)/ _)" 10)  paulson@29960  70  "@UNION_less" :: "nat \ nat => 'b set => 'b set" ("(3\(00_ < _)/ _)" 10)  paulson@29960  71  "@INTER_le" :: "nat \ nat => 'b set => 'b set" ("(3\(00_ \ _)/ _)" 10)  paulson@29960  72  "@INTER_less" :: "nat \ nat => 'b set => 'b set" ("(3\(00_ < _)/ _)" 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 paulson@24286  168 lemma greaterThanLessThan_iff [simp,noatp]:  haftmann@25062  169  "(i : {l<.. {m..n} = {}";  nipkow@24691  195 by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)  nipkow@24691  196 haftmann@25062  197 lemma atLeastLessThan_empty[simp]: "n \ m ==> {m.. k ==> {k<..l} = {}"  nipkow@17719  201 by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)  nipkow@17719  202 haftmann@29709  203 lemma greaterThanLessThan_empty[simp]:"l \ k ==> {k<.. n then insert n {m.. Suc n \ {m..Suc n} = insert (Suc n) {m..n}"  nipkow@15554  321 by (auto simp add: atLeastAtMost_def)  nipkow@15554  322 nipkow@16733  323 subsubsection {* Image *}  nipkow@16733  324 nipkow@16733  325 lemma image_add_atLeastAtMost:  nipkow@16733  326  "(%n::nat. n+k)  {i..j} = {i+k..j+k}" (is "?A = ?B")  nipkow@16733  327 proof  nipkow@16733  328  show "?A \ ?B" by auto  nipkow@16733  329 next  nipkow@16733  330  show "?B \ ?A"  nipkow@16733  331  proof  nipkow@16733  332  fix n assume a: "n : ?B"  webertj@20217  333  hence "n - k : {i..j}" by auto  nipkow@16733  334  moreover have "n = (n - k) + k" using a by auto  nipkow@16733  335  ultimately show "n : ?A" by blast  nipkow@16733  336  qed  nipkow@16733  337 qed  nipkow@16733  338 nipkow@16733  339 lemma image_add_atLeastLessThan:  nipkow@16733  340  "(%n::nat. n+k)  {i.. ?B" by auto  nipkow@16733  343 next  nipkow@16733  344  show "?B \ ?A"  nipkow@16733  345  proof  nipkow@16733  346  fix n assume a: "n : ?B"  webertj@20217  347  hence "n - k : {i.. finite N"  nipkow@28068  396 apply (rule finite_subset)  nipkow@28068  397  apply (rule_tac [2] finite_lessThan, auto)  nipkow@28068  398 done  nipkow@28068  399 nipkow@28068  400 lemma finite_less_ub:  nipkow@28068  401  "!!f::nat=>nat. (!!n. n \ f n) ==> finite {n. f n \ u}"  nipkow@28068  402 by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)  paulson@14485  403 nipkow@24853  404 text{* Any subset of an interval of natural numbers the size of the  nipkow@24853  405 subset is exactly that interval. *}  nipkow@24853  406 nipkow@24853  407 lemma subset_card_intvl_is_intvl:  nipkow@24853  408  "A <= {k.. A = {k.. \h. bij_betw h {0.. \h. bij_betw h M {0.. u ==>  nipkow@15045  482  {(0::int).. u")  paulson@14485  491  apply (subst image_atLeastZeroLessThan_int, assumption)  paulson@14485  492  apply (rule finite_imageI)  paulson@14485  493  apply auto  paulson@14485  494  done  paulson@14485  495 nipkow@15045  496 lemma finite_atLeastLessThan_int [iff]: "finite {l.. u")  paulson@14485  518  apply (subst image_atLeastZeroLessThan_int, assumption)  paulson@14485  519  apply (subst card_image)  paulson@14485  520  apply (auto simp add: inj_on_def)  paulson@14485  521  done  paulson@14485  522 nipkow@15045  523 lemma card_atLeastLessThan_int [simp]: "card {l.. k < (i::nat)}"  bulwahn@27656  545 proof -  bulwahn@27656  546  have "{k. P k \ k < i} \ {.. M"  bulwahn@27656  552 shows "card {k \ M. k < Suc i} \ 0"  bulwahn@27656  553 proof -  bulwahn@27656  554  from zero_in_M have "{k \ M. k < Suc i} \ {}" by auto  bulwahn@27656  555  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)  bulwahn@27656  556 qed  bulwahn@27656  557 bulwahn@27656  558 lemma card_less_Suc2: "0 \ M \ card {k. Suc k \ M \ k < i} = card {k \ M. k < Suc i}"  bulwahn@27656  559 apply (rule card_bij_eq [of "Suc" _ _ "\x. x - 1"])  bulwahn@27656  560 apply simp  bulwahn@27656  561 apply fastsimp  bulwahn@27656  562 apply auto  bulwahn@27656  563 apply (rule inj_on_diff_nat)  bulwahn@27656  564 apply auto  bulwahn@27656  565 apply (case_tac x)  bulwahn@27656  566 apply auto  bulwahn@27656  567 apply (case_tac xa)  bulwahn@27656  568 apply auto  bulwahn@27656  569 apply (case_tac xa)  bulwahn@27656  570 apply auto  bulwahn@27656  571 done  bulwahn@27656  572 bulwahn@27656  573 lemma card_less_Suc:  bulwahn@27656  574  assumes zero_in_M: "0 \ M"  bulwahn@27656  575  shows "Suc (card {k. Suc k \ M \ k < i}) = card {k \ M. k < Suc i}"  bulwahn@27656  576 proof -  bulwahn@27656  577  from assms have a: "0 \ {k \ M. k < Suc i}" by simp  bulwahn@27656  578  hence c: "{k \ M. k < Suc i} = insert 0 ({k \ M. k < Suc i} - {0})"  bulwahn@27656  579  by (auto simp only: insert_Diff)  bulwahn@27656  580  have b: "{k \ M. k < Suc i} - {0} = {k \ M - {0}. k < Suc i}" by auto  bulwahn@27656  581  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  582  apply (subst card_insert)  bulwahn@27656  583  apply simp_all  bulwahn@27656  584  apply (subst b)  bulwahn@27656  585  apply (subst card_less_Suc2[symmetric])  bulwahn@27656  586  apply simp_all  bulwahn@27656  587  done  bulwahn@27656  588  with c show ?thesis by simp  bulwahn@27656  589 qed  bulwahn@27656  590 paulson@14485  591 paulson@13850  592 subsection {*Lemmas useful with the summation operator setsum*}  paulson@13850  593 ballarin@16102  594 text {* For examples, see Algebra/poly/UnivPoly2.thy *}  ballarin@13735  595 wenzelm@14577  596 subsubsection {* Disjoint Unions *}  ballarin@13735  597 wenzelm@14577  598 text {* Singletons and open intervals *}  ballarin@13735  599 ballarin@13735  600 lemma ivl_disj_un_singleton:  nipkow@15045  601  "{l::'a::linorder} Un {l<..} = {l..}"  nipkow@15045  602  "{.. {l} Un {l<.. {l<.. {l} Un {l<..u} = {l..u}"  nipkow@15045  606  "(l::'a::linorder) <= u ==> {l.. {..l} Un {l<.. {.. {..l} Un {l<..u} = {..u}"  nipkow@15045  615  "(l::'a::linorder) <= u ==> {.. {l<..u} Un {u<..} = {l<..}"  nipkow@15045  617  "(l::'a::linorder) < u ==> {l<.. {l..u} Un {u<..} = {l..}"  nipkow@15045  619  "(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  631  "[| (l::'a::linorder) <= m; m <= u |] ==> {l.. {l..m} Un {m<..u} = {l..u}"  ballarin@14398  633 by auto  ballarin@13735  634 ballarin@13735  635 lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two  ballarin@13735  636 wenzelm@14577  637 subsubsection {* Disjoint Intersections *}  ballarin@13735  638 wenzelm@14577  639 text {* Singletons and open intervals *}  ballarin@13735  640 ballarin@13735  641 lemma ivl_disj_int_singleton:  nipkow@15045  642  "{l::'a::order} Int {l<..} = {}"  nipkow@15045  643  "{.. n \ {i.. {m.. i | m \ i & j \ (n::'a::linorder))"  nipkow@15542  689 apply(auto simp:linorder_not_le)  nipkow@15542  690 apply(rule ccontr)  nipkow@15542  691 apply(insert linorder_le_less_linear[of i n])  nipkow@15542  692 apply(clarsimp simp:linorder_not_le)  nipkow@15542  693 apply(fastsimp)  nipkow@15542  694 done  nipkow@15542  695 nipkow@15041  696 nipkow@15042  697 subsection {* Summation indexed over intervals *}  nipkow@15042  698 nipkow@15042  699 syntax  nipkow@15042  700  "_from_to_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)  nipkow@15048  701  "_from_upto_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)  nipkow@16052  702  "_upt_setsum" :: "idt \ 'a \ 'b \ 'b" ("(SUM _<_./ _)" [0,0,10] 10)  nipkow@16052  703  "_upto_setsum" :: "idt \ 'a \ 'b \ 'b" ("(SUM _<=_./ _)" [0,0,10] 10)  nipkow@15042  704 syntax (xsymbols)  nipkow@15042  705  "_from_to_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _.._./ _)" [0,0,0,10] 10)  nipkow@15048  706  "_from_upto_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _..<_./ _)" [0,0,0,10] 10)  nipkow@16052  707  "_upt_setsum" :: "idt \ 'a \ 'b \ 'b" ("(3\_<_./ _)" [0,0,10] 10)  nipkow@16052  708  "_upto_setsum" :: "idt \ 'a \ 'b \ 'b" ("(3\_\_./ _)" [0,0,10] 10)  nipkow@15042  709 syntax (HTML output)  nipkow@15042  710  "_from_to_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _.._./ _)" [0,0,0,10] 10)  nipkow@15048  711  "_from_upto_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _..<_./ _)" [0,0,0,10] 10)  nipkow@16052  712  "_upt_setsum" :: "idt \ 'a \ 'b \ 'b" ("(3\_<_./ _)" [0,0,10] 10)  nipkow@16052  713  "_upto_setsum" :: "idt \ 'a \ 'b \ 'b" ("(3\_\_./ _)" [0,0,10] 10)  nipkow@15056  714 syntax (latex_sum output)  nipkow@15052  715  "_from_to_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b"  nipkow@15052  716  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)  nipkow@15052  717  "_from_upto_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b"  nipkow@15052  718  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)  nipkow@16052  719  "_upt_setsum" :: "idt \ 'a \ 'b \ 'b"  nipkow@16052  720  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)  nipkow@15052  721  "_upto_setsum" :: "idt \ 'a \ 'b \ 'b"  nipkow@16052  722  ("(3\<^raw:$\sum_{>_ \ _\<^raw:}$> _)" [0,0,10] 10)  nipkow@15041  723 nipkow@15048  724 translations  nipkow@28853  725  "\x=a..b. t" == "CONST setsum (%x. t) {a..b}"  nipkow@28853  726  "\x=a..i\n. t" == "CONST setsum (\i. t) {..n}"  nipkow@28853  728  "\ii. t) {..x\{a..b}. e"} & @{term"\x=a..b. e"} & @{term[mode=latex_sum]"\x=a..b. e"}\\  nipkow@15056  736 @{term[source]"\x\{a..x=a..x=a..x\{..b}. e"} & @{term"\x\b. e"} & @{term[mode=latex_sum]"\x\b. e"}\\  nipkow@15056  738 @{term[source]"\x\{..xxx::nat=0..xa = c; b = d; !!x. \ c \ x; x < d \ \ f x = g x \ \  nipkow@15542  761  setsum f {a..i \ Suc n. f i) = (\i \ n. f i) + f(Suc n)"  nipkow@16052  768 by (simp add:atMost_Suc add_ac)  nipkow@16052  769 nipkow@16041  770 lemma setsum_lessThan_Suc[simp]: "(\i < Suc n. f i) = (\i < n. f i) + f n"  nipkow@16041  771 by (simp add:lessThan_Suc add_ac)  nipkow@15041  772 nipkow@15911  773 lemma setsum_cl_ivl_Suc[simp]:  nipkow@15561  774  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"  nipkow@15561  775 by (auto simp:add_ac atLeastAtMostSuc_conv)  nipkow@15561  776 nipkow@15911  777 lemma setsum_op_ivl_Suc[simp]:  nipkow@15561  778  "setsum f {m..  nipkow@15561  782  (\i=n..m+1. f i) = (\i=n..m. f i) + f(m + 1)"  nipkow@15561  783 by (auto simp:add_ac atLeastAtMostSuc_conv)  nipkow@16041  784 *)  nipkow@28068  785 nipkow@28068  786 lemma setsum_head:  nipkow@28068  787  fixes n :: nat  nipkow@28068  788  assumes mn: "m <= n"  nipkow@28068  789  shows "(\x\{m..n}. P x) = P m + (\x\{m<..n}. P x)" (is "?lhs = ?rhs")  nipkow@28068  790 proof -  nipkow@28068  791  from mn  nipkow@28068  792  have "{m..n} = {m} \ {m<..n}"  nipkow@28068  793  by (auto intro: ivl_disj_un_singleton)  nipkow@28068  794  hence "?lhs = (\x\{m} \ {m<..n}. P x)"  nipkow@28068  795  by (simp add: atLeast0LessThan)  nipkow@28068  796  also have "\ = ?rhs" by simp  nipkow@28068  797  finally show ?thesis .  nipkow@28068  798 qed  nipkow@28068  799 nipkow@28068  800 lemma setsum_head_Suc:  nipkow@28068  801  "m \ n \ setsum f {m..n} = f m + setsum f {Suc m..n}"  nipkow@28068  802 by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)  nipkow@28068  803 nipkow@28068  804 lemma setsum_head_upt_Suc:  nipkow@28068  805  "m < n \ setsum f {m.. m \ n; n \ p \ \  nipkow@15539  812  setsum f {m.. 'a::ab_group_add"  nipkow@15539  817 shows "\ m \ n; n \ p \ \  nipkow@15539  818  setsum f {m.. setsum f {Suc 0..k} = setsum f {0..k}"  nipkow@28068  846 by(simp add:setsum_head_Suc)  kleing@19106  847 nipkow@28068  848 lemma setsum_shift_lb_Suc0_0_upt:  nipkow@28068  849  "f(0::nat) = 0 \ setsum f {Suc 0.. (\i=0..i\{1..n}. of_nat i) =  kleing@19469  865  of_nat n*((of_nat n)+1)"  kleing@19469  866 proof (induct n)  kleing@19469  867  case 0  kleing@19469  868  show ?case by simp  kleing@19469  869 next  kleing@19469  870  case (Suc n)  nipkow@29667  871  then show ?case by (simp add: algebra_simps)  kleing@19469  872 qed  kleing@19469  873 kleing@19469  874 theorem arith_series_general:  huffman@23277  875  "((1::'a::comm_semiring_1) + 1) * (\i\{.. 1"  kleing@19469  879  let ?I = "\i. of_nat i" and ?n = "of_nat n"  kleing@19469  880  have  kleing@19469  881  "(\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)"  kleing@19469  893  by (simp only: mult_ac gauss_sum [of "n - 1"])  huffman@23431  894  (simp add: mult_ac trans [OF add_commute of_nat_Suc [symmetric]])  nipkow@29667  895  finally show ?thesis by (simp add: algebra_simps)  kleing@19469  896 next  kleing@19469  897  assume "\(n > 1)"  kleing@19469  898  hence "n = 1 \ n = 0" by auto  nipkow@29667  899  thus ?thesis by (auto simp: algebra_simps)  kleing@19469  900 qed  kleing@19469  901 kleing@19469  902 lemma arith_series_nat:  kleing@19469  903  "Suc (Suc 0) * (\i\{..i\{..i\{..i\{..nat"  kleing@19022  925  shows  kleing@19022  926  "\x. Q x \ P x \  kleing@19022  927  (\xxxxx 'a \ 'a \ 'b \ 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)  paulson@29960  953  "_from_upto_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)  paulson@29960  954  "_upt_setprod" :: "idt \ 'a \ 'b \ 'b" ("(PROD _<_./ _)" [0,0,10] 10)  paulson@29960  955  "_upto_setprod" :: "idt \ 'a \ 'b \ 'b" ("(PROD _<=_./ _)" [0,0,10] 10)  paulson@29960  956 syntax (xsymbols)  paulson@29960  957  "_from_to_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _.._./ _)" [0,0,0,10] 10)  paulson@29960  958  "_from_upto_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _..<_./ _)" [0,0,0,10] 10)  paulson@29960  959  "_upt_setprod" :: "idt \ 'a \ 'b \ 'b" ("(3\_<_./ _)" [0,0,10] 10)  paulson@29960  960  "_upto_setprod" :: "idt \ 'a \ 'b \ 'b" ("(3\_\_./ _)" [0,0,10] 10)  paulson@29960  961 syntax (HTML output)  paulson@29960  962  "_from_to_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _.._./ _)" [0,0,0,10] 10)  paulson@29960  963  "_from_upto_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _..<_./ _)" [0,0,0,10] 10)  paulson@29960  964  "_upt_setprod" :: "idt \ 'a \ 'b \ 'b" ("(3\_<_./ _)" [0,0,10] 10)  paulson@29960  965  "_upto_setprod" :: "idt \ 'a \ 'b \ 'b" ("(3\_\_./ _)" [0,0,10] 10)  paulson@29960  966 syntax (latex_prod output)  paulson@29960  967  "_from_to_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b"  paulson@29960  968  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)  paulson@29960  969  "_from_upto_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b"  paulson@29960  970  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)  paulson@29960  971  "_upt_setprod" :: "idt \ 'a \ 'b \ 'b"  paulson@29960  972  ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)  paulson@29960  973  "_upto_setprod" :: "idt \ 'a \ 'b \ 'b"  paulson@29960  974  ("(3\<^raw:$\prod_{>_ \ _\<^raw:}$> _)" [0,0,10] 10)  paulson@29960  975 paulson@29960  976 translations  paulson@29960  977  "\x=a..b. t" == "CONST setprod (%x. t) {a..b}"  paulson@29960  978  "\x=a..i\n. t" == "CONST setprod (\i. t) {..n}"  paulson@29960  980 ` "\ii. t) {..