nipkow@8924: (* Title: HOL/SetInterval.thy wenzelm@32960: Author: Tobias Nipkow wenzelm@32960: Author: Clemens Ballarin wenzelm@32960: Author: Jeremy Avigad nipkow@8924: ballarin@13735: lessThan, greaterThan, atLeast, atMost and two-sided intervals nipkow@8924: *) nipkow@8924: wenzelm@14577: header {* Set intervals *} wenzelm@14577: nipkow@15131: theory SetInterval haftmann@33318: imports Int Nat_Transfer nipkow@15131: begin nipkow@8924: nipkow@24691: context ord nipkow@24691: begin nipkow@24691: definition wenzelm@32960: lessThan :: "'a => 'a set" ("(1{..<_})") where haftmann@25062: "{.. 'a set" ("(1{.._})") where haftmann@25062: "{..u} == {x. x \ u}" nipkow@24691: nipkow@24691: definition wenzelm@32960: greaterThan :: "'a => 'a set" ("(1{_<..})") where haftmann@25062: "{l<..} == {x. l 'a set" ("(1{_..})") where haftmann@25062: "{l..} == {x. l\x}" nipkow@24691: nipkow@24691: definition haftmann@25062: greaterThanLessThan :: "'a => 'a => 'a set" ("(1{_<..<_})") where haftmann@25062: "{l<.. 'a => 'a set" ("(1{_..<_})") where haftmann@25062: "{l.. 'a => 'a set" ("(1{_<.._})") where haftmann@25062: "{l<..u} == {l<..} Int {..u}" nipkow@24691: nipkow@24691: definition haftmann@25062: atLeastAtMost :: "'a => 'a => 'a set" ("(1{_.._})") where haftmann@25062: "{l..u} == {l..} Int {..u}" nipkow@24691: nipkow@24691: end nipkow@8924: ballarin@13735: nipkow@15048: text{* A note of warning when using @{term"{.. 'a => 'b set => 'b set" ("(3UN _<=_./ _)" [0, 0, 10] 10) huffman@36364: "_UNION_less" :: "'a => 'a => 'b set => 'b set" ("(3UN _<_./ _)" [0, 0, 10] 10) huffman@36364: "_INTER_le" :: "'a => 'a => 'b set => 'b set" ("(3INT _<=_./ _)" [0, 0, 10] 10) huffman@36364: "_INTER_less" :: "'a => 'a => 'b set => 'b set" ("(3INT _<_./ _)" [0, 0, 10] 10) kleing@14418: nipkow@30372: syntax (xsymbols) huffman@36364: "_UNION_le" :: "'a => 'a => 'b set => 'b set" ("(3\ _\_./ _)" [0, 0, 10] 10) huffman@36364: "_UNION_less" :: "'a => 'a => 'b set => 'b set" ("(3\ _<_./ _)" [0, 0, 10] 10) huffman@36364: "_INTER_le" :: "'a => 'a => 'b set => 'b set" ("(3\ _\_./ _)" [0, 0, 10] 10) huffman@36364: "_INTER_less" :: "'a => 'a => 'b set => 'b set" ("(3\ _<_./ _)" [0, 0, 10] 10) kleing@14418: nipkow@30372: syntax (latex output) huffman@36364: "_UNION_le" :: "'a \ 'a => 'b set => 'b set" ("(3\(00_ \ _)/ _)" [0, 0, 10] 10) huffman@36364: "_UNION_less" :: "'a \ 'a => 'b set => 'b set" ("(3\(00_ < _)/ _)" [0, 0, 10] 10) huffman@36364: "_INTER_le" :: "'a \ 'a => 'b set => 'b set" ("(3\(00_ \ _)/ _)" [0, 0, 10] 10) huffman@36364: "_INTER_less" :: "'a \ 'a => 'b set => 'b set" ("(3\(00_ < _)/ _)" [0, 0, 10] 10) kleing@14418: kleing@14418: translations kleing@14418: "UN i<=n. A" == "UN i:{..n}. A" nipkow@15045: "UN i atLeast y) = (y \ (x::'a::order))" paulson@15418: by (blast intro: order_trans) paulson@13850: paulson@13850: lemma atLeast_eq_iff [iff]: paulson@15418: "(atLeast x = atLeast y) = (x = (y::'a::linorder))" paulson@13850: by (blast intro: order_antisym order_trans) paulson@13850: paulson@13850: lemma greaterThan_subset_iff [iff]: paulson@15418: "(greaterThan x \ greaterThan y) = (y \ (x::'a::linorder))" paulson@15418: apply (auto simp add: greaterThan_def) paulson@15418: apply (subst linorder_not_less [symmetric], blast) paulson@13850: done paulson@13850: paulson@13850: lemma greaterThan_eq_iff [iff]: paulson@15418: "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))" paulson@15418: apply (rule iffI) paulson@15418: apply (erule equalityE) haftmann@29709: apply simp_all paulson@13850: done paulson@13850: paulson@15418: lemma atMost_subset_iff [iff]: "(atMost x \ atMost y) = (x \ (y::'a::order))" paulson@13850: by (blast intro: order_trans) paulson@13850: paulson@15418: lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))" paulson@13850: by (blast intro: order_antisym order_trans) paulson@13850: paulson@13850: lemma lessThan_subset_iff [iff]: paulson@15418: "(lessThan x \ lessThan y) = (x \ (y::'a::linorder))" paulson@15418: apply (auto simp add: lessThan_def) paulson@15418: apply (subst linorder_not_less [symmetric], blast) paulson@13850: done paulson@13850: paulson@13850: lemma lessThan_eq_iff [iff]: paulson@15418: "(lessThan x = lessThan y) = (x = (y::'a::linorder))" paulson@15418: apply (rule iffI) paulson@15418: apply (erule equalityE) haftmann@29709: apply simp_all ballarin@13735: done ballarin@13735: ballarin@13735: paulson@13850: subsection {*Two-sided intervals*} ballarin@13735: nipkow@24691: context ord nipkow@24691: begin nipkow@24691: blanchet@35828: lemma greaterThanLessThan_iff [simp,no_atp]: haftmann@25062: "(i : {l<.. {a..b} = {}" nipkow@32400: by(auto simp: atLeastAtMost_def atLeast_def atMost_def) nipkow@32400: nipkow@32400: lemma atLeastatMost_empty_iff[simp]: nipkow@32400: "{a..b} = {} \ (~ a <= b)" nipkow@32400: by auto (blast intro: order_trans) nipkow@32400: nipkow@32400: lemma atLeastatMost_empty_iff2[simp]: nipkow@32400: "{} = {a..b} \ (~ a <= b)" nipkow@32400: by auto (blast intro: order_trans) nipkow@32400: nipkow@32400: lemma atLeastLessThan_empty[simp]: nipkow@32400: "b <= a \ {a.. (~ a < b)" nipkow@32400: by auto (blast intro: le_less_trans) nipkow@32400: nipkow@32400: lemma atLeastLessThan_empty_iff2[simp]: nipkow@32400: "{} = {a.. (~ a < b)" nipkow@32400: by auto (blast intro: le_less_trans) nipkow@15554: nipkow@32400: lemma greaterThanAtMost_empty[simp]: "l \ k ==> {k<..l} = {}" nipkow@17719: by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def) nipkow@17719: nipkow@32400: lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {} \ ~ k < l" nipkow@32400: by auto (blast intro: less_le_trans) nipkow@32400: nipkow@32400: lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} \ ~ k < l" nipkow@32400: by auto (blast intro: less_le_trans) nipkow@32400: haftmann@29709: lemma greaterThanLessThan_empty[simp]:"l \ k ==> {k<.. {a .. b} = {a}" by simp hoelzl@36846: nipkow@32400: lemma atLeastatMost_subset_iff[simp]: nipkow@32400: "{a..b} <= {c..d} \ (~ a <= b) | c <= a & b <= d" nipkow@32400: unfolding atLeastAtMost_def atLeast_def atMost_def nipkow@32400: by (blast intro: order_trans) nipkow@32400: nipkow@32400: lemma atLeastatMost_psubset_iff: nipkow@32400: "{a..b} < {c..d} \ nipkow@32400: ((~ a <= b) | c <= a & b <= d & (c < a | b < d)) & c <= d" nipkow@32400: by(simp add: psubset_eq expand_set_eq less_le_not_le)(blast intro: order_trans) nipkow@32400: hoelzl@36846: lemma atLeastAtMost_singleton_iff[simp]: hoelzl@36846: "{a .. b} = {c} \ a = b \ b = c" hoelzl@36846: proof hoelzl@36846: assume "{a..b} = {c}" hoelzl@36846: hence "\ (\ a \ b)" unfolding atLeastatMost_empty_iff[symmetric] by simp hoelzl@36846: moreover with `{a..b} = {c}` have "c \ a \ b \ c" by auto hoelzl@36846: ultimately show "a = b \ b = c" by auto hoelzl@36846: qed simp hoelzl@36846: nipkow@24691: end paulson@14485: nipkow@32408: lemma (in linorder) atLeastLessThan_subset_iff: nipkow@32408: "{a.. b <= a | c<=a & b<=d" nipkow@32408: apply (auto simp:subset_eq Ball_def) nipkow@32408: apply(frule_tac x=a in spec) nipkow@32408: apply(erule_tac x=d in allE) nipkow@32408: apply (simp add: less_imp_le) nipkow@32408: done nipkow@32408: nipkow@32456: subsubsection {* Intersection *} nipkow@32456: nipkow@32456: context linorder nipkow@32456: begin nipkow@32456: nipkow@32456: lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}" nipkow@32456: by auto nipkow@32456: nipkow@32456: lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}" nipkow@32456: by auto nipkow@32456: nipkow@32456: lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}" nipkow@32456: by auto nipkow@32456: nipkow@32456: lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}" nipkow@32456: by auto nipkow@32456: nipkow@32456: lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}" nipkow@32456: by auto nipkow@32456: nipkow@32456: lemma Int_atLeastLessThan[simp]: "{a.. Suc ` {.. Suc (x - 1)" by auto hoelzl@39072: with `x < Suc n` show "x = 0" by auto hoelzl@39072: qed hoelzl@39072: paulson@14485: lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k" paulson@14485: by (simp add: lessThan_def atMost_def less_Suc_eq_le) paulson@14485: paulson@14485: lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV" paulson@14485: by blast paulson@14485: paulson@15047: subsubsection {* The Constant @{term greaterThan} *} paulson@15047: paulson@14485: lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc" paulson@14485: apply (simp add: greaterThan_def) paulson@14485: apply (blast dest: gr0_conv_Suc [THEN iffD1]) paulson@14485: done paulson@14485: paulson@14485: lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}" paulson@14485: apply (simp add: greaterThan_def) paulson@14485: apply (auto elim: linorder_neqE) paulson@14485: done paulson@14485: paulson@14485: lemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}" paulson@14485: by blast paulson@14485: paulson@15047: subsubsection {* The Constant @{term atLeast} *} paulson@15047: paulson@14485: lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV" paulson@14485: by (unfold atLeast_def UNIV_def, simp) paulson@14485: paulson@14485: lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}" paulson@14485: apply (simp add: atLeast_def) paulson@14485: apply (simp add: Suc_le_eq) paulson@14485: apply (simp add: order_le_less, blast) paulson@14485: done paulson@14485: paulson@14485: lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k" paulson@14485: by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le) paulson@14485: paulson@14485: lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV" paulson@14485: by blast paulson@14485: paulson@15047: subsubsection {* The Constant @{term atMost} *} paulson@15047: paulson@14485: lemma atMost_0 [simp]: "atMost (0::nat) = {0}" paulson@14485: by (simp add: atMost_def) paulson@14485: paulson@14485: lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)" paulson@14485: apply (simp add: atMost_def) paulson@14485: apply (simp add: less_Suc_eq order_le_less, blast) paulson@14485: done paulson@14485: paulson@14485: lemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV" paulson@14485: by blast paulson@14485: paulson@15047: subsubsection {* The Constant @{term atLeastLessThan} *} paulson@15047: nipkow@28068: text{*The orientation of the following 2 rules is tricky. The lhs is nipkow@24449: defined in terms of the rhs. Hence the chosen orientation makes sense nipkow@24449: in this theory --- the reverse orientation complicates proofs (eg nipkow@24449: nontermination). But outside, when the definition of the lhs is rarely nipkow@24449: used, the opposite orientation seems preferable because it reduces a nipkow@24449: specific concept to a more general one. *} nipkow@28068: paulson@15047: lemma atLeast0LessThan: "{0::nat.. n then insert n {m.. Suc n \ {m..Suc n} = insert (Suc n) {m..n}" nipkow@15554: by (auto simp add: atLeastAtMost_def) nipkow@15554: paulson@33044: lemma atLeastLessThan_add_Un: "i \ j \ {i.. {j.. ?B" by auto nipkow@16733: next nipkow@16733: show "?B \ ?A" nipkow@16733: proof nipkow@16733: fix n assume a: "n : ?B" webertj@20217: hence "n - k : {i..j}" by auto nipkow@16733: moreover have "n = (n - k) + k" using a by auto nipkow@16733: ultimately show "n : ?A" by blast nipkow@16733: qed nipkow@16733: qed nipkow@16733: nipkow@16733: lemma image_add_atLeastLessThan: nipkow@16733: "(%n::nat. n+k) ` {i.. ?B" by auto nipkow@16733: next nipkow@16733: show "?B \ ?A" nipkow@16733: proof nipkow@16733: fix n assume a: "n : ?B" webertj@20217: hence "n - k : {i..i. i - c) ` {x ..< y} = hoelzl@37664: (if c < y then {x - c ..< y - c} else if x < y then {0} else {})" hoelzl@37664: (is "_ = ?right") hoelzl@37664: proof safe hoelzl@37664: fix a assume a: "a \ ?right" hoelzl@37664: show "a \ (\i. i - c) ` {x ..< y}" hoelzl@37664: proof cases hoelzl@37664: assume "c < y" with a show ?thesis hoelzl@37664: by (auto intro!: image_eqI[of _ _ "a + c"]) hoelzl@37664: next hoelzl@37664: assume "\ c < y" with a show ?thesis hoelzl@37664: by (auto intro!: image_eqI[of _ _ x] split: split_if_asm) hoelzl@37664: qed hoelzl@37664: qed auto hoelzl@37664: hoelzl@35580: context ordered_ab_group_add hoelzl@35580: begin hoelzl@35580: hoelzl@35580: lemma hoelzl@35580: fixes x :: 'a hoelzl@35580: shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}" hoelzl@35580: and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}" hoelzl@35580: proof safe hoelzl@35580: fix y assume "y < -x" hoelzl@35580: hence *: "x < -y" using neg_less_iff_less[of "-y" x] by simp hoelzl@35580: have "- (-y) \ uminus ` {x<..}" hoelzl@35580: by (rule imageI) (simp add: *) hoelzl@35580: thus "y \ uminus ` {x<..}" by simp hoelzl@35580: next hoelzl@35580: fix y assume "y \ -x" hoelzl@35580: have "- (-y) \ uminus ` {x..}" hoelzl@35580: by (rule imageI) (insert `y \ -x`[THEN le_imp_neg_le], simp) hoelzl@35580: thus "y \ uminus ` {x..}" by simp hoelzl@35580: qed simp_all hoelzl@35580: hoelzl@35580: lemma hoelzl@35580: fixes x :: 'a hoelzl@35580: shows image_uminus_lessThan[simp]: "uminus ` {.. finite N" nipkow@28068: apply (rule finite_subset) nipkow@28068: apply (rule_tac [2] finite_lessThan, auto) nipkow@28068: done nipkow@28068: nipkow@31044: text {* A set of natural numbers is finite iff it is bounded. *} nipkow@31044: lemma finite_nat_set_iff_bounded: nipkow@31044: "finite(N::nat set) = (EX m. ALL n:N. nnat. (!!n. n \ f n) ==> finite {n. f n \ u}" nipkow@28068: by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans) paulson@14485: nipkow@24853: text{* Any subset of an interval of natural numbers the size of the nipkow@24853: subset is exactly that interval. *} nipkow@24853: nipkow@24853: lemma subset_card_intvl_is_intvl: nipkow@24853: "A <= {k.. A = {k..i\n::nat. M i) = (\i\{1..n}. M i) \ M 0" (is "?A = ?B") nipkow@36755: proof nipkow@36755: show "?A <= ?B" nipkow@36755: proof nipkow@36755: fix x assume "x : ?A" nipkow@36755: then obtain i where i: "i\n" "x : M i" by auto nipkow@36755: show "x : ?B" nipkow@36755: proof(cases i) nipkow@36755: case 0 with i show ?thesis by simp nipkow@36755: next nipkow@36755: case (Suc j) with i show ?thesis by auto nipkow@36755: qed nipkow@36755: qed nipkow@36755: next nipkow@36755: show "?B <= ?A" by auto nipkow@36755: qed nipkow@36755: nipkow@36755: lemma UN_le_add_shift: nipkow@36755: "(\i\n::nat. M(i+k)) = (\i\{k..n+k}. M i)" (is "?A = ?B") nipkow@36755: proof nipkow@36755: show "?A <= ?B" by fastsimp nipkow@36755: next nipkow@36755: show "?B <= ?A" nipkow@36755: proof nipkow@36755: fix x assume "x : ?B" nipkow@36755: then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto nipkow@36755: hence "i-k\n & x : M((i-k)+k)" by auto nipkow@36755: thus "x : ?A" by blast nipkow@36755: qed nipkow@36755: qed nipkow@36755: paulson@32596: lemma UN_UN_finite_eq: "(\n::nat. \i\{0..n. A n)" paulson@32596: by (auto simp add: atLeast0LessThan) paulson@32596: paulson@32596: lemma UN_finite_subset: "(!!n::nat. (\i\{0.. C) \ (\n. A n) \ C" paulson@32596: by (subst UN_UN_finite_eq [symmetric]) blast paulson@32596: paulson@33044: lemma UN_finite2_subset: paulson@33044: "(!!n::nat. (\i\{0.. (\i\{0.. (\n. A n) \ (\n. B n)" paulson@33044: apply (rule UN_finite_subset) paulson@33044: apply (subst UN_UN_finite_eq [symmetric, of B]) paulson@33044: apply blast paulson@33044: done paulson@32596: paulson@32596: lemma UN_finite2_eq: paulson@33044: "(!!n::nat. (\i\{0..i\{0.. (\n. A n) = (\n. B n)" paulson@33044: apply (rule subset_antisym) paulson@33044: apply (rule UN_finite2_subset, blast) paulson@33044: apply (rule UN_finite2_subset [where k=k]) huffman@35216: apply (force simp add: atLeastLessThan_add_Un [of 0]) paulson@33044: done paulson@32596: paulson@32596: paulson@14485: subsubsection {* Cardinality *} paulson@14485: nipkow@15045: 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: apply(drule ex_bij_betw_finite_nat) nipkow@31438: apply(drule ex_bij_betw_nat_finite) nipkow@31438: apply(auto intro!:bij_betw_trans) nipkow@31438: done nipkow@31438: nipkow@31438: lemma ex_bij_betw_nat_finite_1: nipkow@31438: "finite M \ \h. bij_betw h {1 .. card M} M" nipkow@31438: by (rule finite_same_card_bij) auto nipkow@31438: nipkow@26105: paulson@14485: subsection {* Intervals of integers *} paulson@14485: nipkow@15045: lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l.. u ==> nipkow@15045: {(0::int).. u") paulson@14485: apply (subst image_atLeastZeroLessThan_int, assumption) paulson@14485: apply (rule finite_imageI) paulson@14485: apply auto paulson@14485: done paulson@14485: nipkow@15045: lemma finite_atLeastLessThan_int [iff]: "finite {l.. u") paulson@14485: apply (subst image_atLeastZeroLessThan_int, assumption) paulson@14485: apply (subst card_image) paulson@14485: apply (auto simp add: inj_on_def) paulson@14485: done paulson@14485: nipkow@15045: lemma card_atLeastLessThan_int [simp]: "card {l.. k < (i::nat)}" bulwahn@27656: proof - bulwahn@27656: have "{k. P k \ k < i} \ {.. M" bulwahn@27656: shows "card {k \ M. k < Suc i} \ 0" bulwahn@27656: proof - bulwahn@27656: from zero_in_M have "{k \ M. k < Suc i} \ {}" by auto bulwahn@27656: with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff) bulwahn@27656: qed bulwahn@27656: bulwahn@27656: lemma card_less_Suc2: "0 \ M \ card {k. Suc k \ M \ k < i} = card {k \ M. k < Suc i}" haftmann@37388: apply (rule card_bij_eq [of Suc _ _ "\x. x - Suc 0"]) bulwahn@27656: apply simp bulwahn@27656: apply fastsimp bulwahn@27656: apply auto bulwahn@27656: apply (rule inj_on_diff_nat) bulwahn@27656: apply auto bulwahn@27656: apply (case_tac x) bulwahn@27656: apply auto bulwahn@27656: apply (case_tac xa) bulwahn@27656: apply auto bulwahn@27656: apply (case_tac xa) bulwahn@27656: apply auto bulwahn@27656: done bulwahn@27656: bulwahn@27656: lemma card_less_Suc: bulwahn@27656: assumes zero_in_M: "0 \ M" bulwahn@27656: shows "Suc (card {k. Suc k \ M \ k < i}) = card {k \ M. k < Suc i}" bulwahn@27656: proof - bulwahn@27656: from assms have a: "0 \ {k \ M. k < Suc i}" by simp bulwahn@27656: hence c: "{k \ M. k < Suc i} = insert 0 ({k \ M. k < Suc i} - {0})" bulwahn@27656: by (auto simp only: insert_Diff) bulwahn@27656: have b: "{k \ M. k < Suc i} - {0} = {k \ M - {0}. k < Suc i}" by auto bulwahn@27656: 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: apply (subst card_insert) bulwahn@27656: apply simp_all bulwahn@27656: apply (subst b) bulwahn@27656: apply (subst card_less_Suc2[symmetric]) bulwahn@27656: apply simp_all bulwahn@27656: done bulwahn@27656: with c show ?thesis by simp bulwahn@27656: qed bulwahn@27656: paulson@14485: paulson@13850: subsection {*Lemmas useful with the summation operator setsum*} paulson@13850: ballarin@16102: text {* For examples, see Algebra/poly/UnivPoly2.thy *} ballarin@13735: wenzelm@14577: subsubsection {* Disjoint Unions *} ballarin@13735: wenzelm@14577: text {* Singletons and open intervals *} ballarin@13735: ballarin@13735: lemma ivl_disj_un_singleton: nipkow@15045: "{l::'a::linorder} Un {l<..} = {l..}" nipkow@15045: "{.. {l} Un {l<.. {l<.. {l} Un {l<..u} = {l..u}" nipkow@15045: "(l::'a::linorder) <= u ==> {l.. {..l} Un {l<.. {.. {..l} Un {l<..u} = {..u}" nipkow@15045: "(l::'a::linorder) <= u ==> {.. {l<..u} Un {u<..} = {l<..}" nipkow@15045: "(l::'a::linorder) < u ==> {l<.. {l..u} Un {u<..} = {l..}" nipkow@15045: "(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: "[| (l::'a::linorder) <= m; m <= u |] ==> {l.. {l..m} Un {m<..u} = {l..u}" ballarin@14398: by auto ballarin@13735: ballarin@13735: lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two ballarin@13735: wenzelm@14577: subsubsection {* Disjoint Intersections *} ballarin@13735: wenzelm@14577: text {* One- and two-sided intervals *} ballarin@13735: ballarin@13735: lemma ivl_disj_int_one: nipkow@15045: "{..l::'a::order} Int {l<.. n \ {i.. {m.. i | m \ i & j \ (n::'a::linorder))" nipkow@15542: apply(auto simp:linorder_not_le) nipkow@15542: apply(rule ccontr) nipkow@15542: apply(insert linorder_le_less_linear[of i n]) nipkow@15542: apply(clarsimp simp:linorder_not_le) nipkow@15542: apply(fastsimp) nipkow@15542: done nipkow@15542: nipkow@15041: nipkow@15042: subsection {* Summation indexed over intervals *} nipkow@15042: nipkow@15042: syntax nipkow@15042: "_from_to_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10) nipkow@15048: "_from_upto_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10) nipkow@16052: "_upt_setsum" :: "idt \ 'a \ 'b \ 'b" ("(SUM _<_./ _)" [0,0,10] 10) nipkow@16052: "_upto_setsum" :: "idt \ 'a \ 'b \ 'b" ("(SUM _<=_./ _)" [0,0,10] 10) nipkow@15042: syntax (xsymbols) nipkow@15042: "_from_to_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _.._./ _)" [0,0,0,10] 10) nipkow@15048: "_from_upto_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _..<_./ _)" [0,0,0,10] 10) nipkow@16052: "_upt_setsum" :: "idt \ 'a \ 'b \ 'b" ("(3\_<_./ _)" [0,0,10] 10) nipkow@16052: "_upto_setsum" :: "idt \ 'a \ 'b \ 'b" ("(3\_\_./ _)" [0,0,10] 10) nipkow@15042: syntax (HTML output) nipkow@15042: "_from_to_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _.._./ _)" [0,0,0,10] 10) nipkow@15048: "_from_upto_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _..<_./ _)" [0,0,0,10] 10) nipkow@16052: "_upt_setsum" :: "idt \ 'a \ 'b \ 'b" ("(3\_<_./ _)" [0,0,10] 10) nipkow@16052: "_upto_setsum" :: "idt \ 'a \ 'b \ 'b" ("(3\_\_./ _)" [0,0,10] 10) nipkow@15056: syntax (latex_sum output) nipkow@15052: "_from_to_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" nipkow@15052: ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10) nipkow@15052: "_from_upto_setsum" :: "idt \ 'a \ 'a \ 'b \ 'b" nipkow@15052: ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10) nipkow@16052: "_upt_setsum" :: "idt \ 'a \ 'b \ 'b" nipkow@16052: ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10) nipkow@15052: "_upto_setsum" :: "idt \ 'a \ 'b \ 'b" nipkow@16052: ("(3\<^raw:$\sum_{>_ \ _\<^raw:}$> _)" [0,0,10] 10) nipkow@15041: nipkow@15048: translations nipkow@28853: "\x=a..b. t" == "CONST setsum (%x. t) {a..b}" nipkow@28853: "\x=a..i\n. t" == "CONST setsum (\i. t) {..n}" nipkow@28853: "\ii. t) {..x\{a..b}. e"} & @{term"\x=a..b. e"} & @{term[mode=latex_sum]"\x=a..b. e"}\\ nipkow@15056: @{term[source]"\x\{a..x=a..x=a..x\{..b}. e"} & @{term"\x\b. e"} & @{term[mode=latex_sum]"\x\b. e"}\\ nipkow@15056: @{term[source]"\x\{..xxx::nat=0..xa = c; b = d; !!x. \ c \ x; x < d \ \ f x = g x \ \ nipkow@15542: setsum f {a..i \ Suc n. f i) = (\i \ n. f i) + f(Suc n)" nipkow@16052: by (simp add:atMost_Suc add_ac) nipkow@16052: nipkow@16041: lemma setsum_lessThan_Suc[simp]: "(\i < Suc n. f i) = (\i < n. f i) + f n" nipkow@16041: by (simp add:lessThan_Suc add_ac) nipkow@15041: nipkow@15911: lemma setsum_cl_ivl_Suc[simp]: nipkow@15561: "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))" nipkow@15561: by (auto simp:add_ac atLeastAtMostSuc_conv) nipkow@15561: nipkow@15911: lemma setsum_op_ivl_Suc[simp]: nipkow@15561: "setsum f {m.. nipkow@15561: (\i=n..m+1. f i) = (\i=n..m. f i) + f(m + 1)" nipkow@15561: by (auto simp:add_ac atLeastAtMostSuc_conv) nipkow@16041: *) nipkow@28068: nipkow@28068: lemma setsum_head: nipkow@28068: fixes n :: nat nipkow@28068: assumes mn: "m <= n" nipkow@28068: shows "(\x\{m..n}. P x) = P m + (\x\{m<..n}. P x)" (is "?lhs = ?rhs") nipkow@28068: proof - nipkow@28068: from mn nipkow@28068: have "{m..n} = {m} \ {m<..n}" nipkow@28068: by (auto intro: ivl_disj_un_singleton) nipkow@28068: hence "?lhs = (\x\{m} \ {m<..n}. P x)" nipkow@28068: by (simp add: atLeast0LessThan) nipkow@28068: also have "\ = ?rhs" by simp nipkow@28068: finally show ?thesis . nipkow@28068: qed nipkow@28068: nipkow@28068: lemma setsum_head_Suc: nipkow@28068: "m \ n \ setsum f {m..n} = f m + setsum f {Suc m..n}" nipkow@28068: by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost) nipkow@28068: nipkow@28068: lemma setsum_head_upt_Suc: nipkow@28068: "m < n \ setsum f {m.. n + 1" nipkow@31501: shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}" nipkow@31501: proof- nipkow@31501: have "{m .. n+p} = {m..n} \ {n+1..n+p}" using `m \ n+1` by auto nipkow@31501: thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint nipkow@31501: atLeastSucAtMost_greaterThanAtMost) nipkow@31501: qed nipkow@28068: nipkow@15539: lemma setsum_add_nat_ivl: "\ m \ n; n \ p \ \ nipkow@15539: setsum f {m.. 'a::ab_group_add" nipkow@15539: shows "\ m \ n; n \ p \ \ nipkow@15539: setsum f {m.. ('a::ab_group_add)" nipkow@31505: shows "setsum (\k. f k - f(k + 1)) {(m::nat) .. n} = nipkow@31505: (if m <= n then f m - f(n + 1) else 0)" nipkow@31505: by (induct n, auto simp add: algebra_simps not_le le_Suc_eq) nipkow@31505: nipkow@31509: lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def] nipkow@31509: nipkow@31509: lemma setsum_setsum_restrict: nipkow@31509: "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: by (simp add: setsum_restrict_set'[unfolded mem_def] mem_def) nipkow@31509: (rule setsum_commute) nipkow@31509: nipkow@31509: lemma setsum_image_gen: assumes fS: "finite S" nipkow@31509: shows "setsum g S = setsum (\y. setsum g {x. x \ S \ f x = y}) (f ` S)" nipkow@31509: proof- nipkow@31509: { fix x assume "x \ S" then have "{y. y\ f`S \ f x = y} = {f x}" by auto } nipkow@31509: hence "setsum g S = setsum (\x. setsum (\y. g x) {y. y\ f`S \ f x = y}) S" nipkow@31509: by simp nipkow@31509: also have "\ = setsum (\y. setsum g {x. x \ S \ f x = y}) (f ` S)" nipkow@31509: by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]]) nipkow@31509: finally show ?thesis . nipkow@31509: qed nipkow@31509: hoelzl@35171: lemma setsum_le_included: haftmann@36307: fixes f :: "'a \ 'b::ordered_comm_monoid_add" hoelzl@35171: assumes "finite s" "finite t" hoelzl@35171: and "\y\t. 0 \ g y" "(\x\s. \y\t. i y = x \ f x \ g y)" hoelzl@35171: shows "setsum f s \ setsum g t" hoelzl@35171: proof - hoelzl@35171: have "setsum f s \ setsum (\y. setsum g {x. x\t \ i x = y}) s" hoelzl@35171: proof (rule setsum_mono) hoelzl@35171: fix y assume "y \ s" hoelzl@35171: with assms obtain z where z: "z \ t" "y = i z" "f y \ g z" by auto hoelzl@35171: with assms show "f y \ setsum g {x \ t. i x = y}" (is "?A y \ ?B y") hoelzl@35171: using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro] hoelzl@35171: by (auto intro!: setsum_mono2) hoelzl@35171: qed hoelzl@35171: also have "... \ setsum (\y. setsum g {x. x\t \ i x = y}) (i ` t)" hoelzl@35171: using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg) hoelzl@35171: also have "... \ setsum g t" hoelzl@35171: using assms by (auto simp: setsum_image_gen[symmetric]) hoelzl@35171: finally show ?thesis . hoelzl@35171: qed hoelzl@35171: nipkow@31509: lemma setsum_multicount_gen: nipkow@31509: assumes "finite s" "finite t" "\j\t. (card {i\s. R i j} = k j)" nipkow@31509: shows "setsum (\i. (card {j\t. R i j})) s = setsum k t" (is "?l = ?r") nipkow@31509: proof- nipkow@31509: have "?l = setsum (\i. setsum (\x.1) {j\t. R i j}) s" by auto nipkow@31509: also have "\ = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)] nipkow@31509: using assms(3) by auto nipkow@31509: finally show ?thesis . nipkow@31509: qed nipkow@31509: nipkow@31509: lemma setsum_multicount: nipkow@31509: assumes "finite S" "finite T" "\j\T. (card {i\S. R i j} = k)" nipkow@31509: shows "setsum (\i. card {j\T. R i j}) S = k * card T" (is "?l = ?r") nipkow@31509: proof- nipkow@31509: have "?l = setsum (\i. k) T" by(rule setsum_multicount_gen)(auto simp:assms) huffman@35216: also have "\ = ?r" by(simp add: mult_commute) nipkow@31509: finally show ?thesis by auto nipkow@31509: qed nipkow@31509: nipkow@28068: nipkow@16733: subsection{* Shifting bounds *} nipkow@16733: nipkow@15539: lemma setsum_shift_bounds_nat_ivl: nipkow@15539: "setsum f {m+k.. setsum f {Suc 0..k} = setsum f {0..k}" nipkow@28068: by(simp add:setsum_head_Suc) kleing@19106: nipkow@28068: lemma setsum_shift_lb_Suc0_0_upt: nipkow@28068: "f(0::nat) = 0 \ setsum f {Suc 0.. 1" haftmann@36307: shows "(\i=0.. 0" by simp_all haftmann@36307: moreover have "(\i=0.. 0` have "(1 + y) ^ n = (y * inverse y) * (1 + y) ^ n" by simp haftmann@36350: ultimately show ?case by (simp add: field_simps divide_inverse) haftmann@36307: qed haftmann@36307: ultimately show ?thesis by simp haftmann@36307: qed haftmann@36307: ballarin@17149: kleing@19469: subsection {* The formula for arithmetic sums *} kleing@19469: kleing@19469: lemma gauss_sum: huffman@23277: "((1::'a::comm_semiring_1) + 1)*(\i\{1..n}. of_nat i) = kleing@19469: of_nat n*((of_nat n)+1)" kleing@19469: proof (induct n) kleing@19469: case 0 kleing@19469: show ?case by simp kleing@19469: next kleing@19469: case (Suc n) nipkow@29667: then show ?case by (simp add: algebra_simps) kleing@19469: qed kleing@19469: kleing@19469: theorem arith_series_general: huffman@23277: "((1::'a::comm_semiring_1) + 1) * (\i\{.. 1" kleing@19469: let ?I = "\i. of_nat i" and ?n = "of_nat n" kleing@19469: have kleing@19469: "(\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: by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def) huffman@23431: (simp add: mult_ac trans [OF add_commute of_nat_Suc [symmetric]]) nipkow@29667: finally show ?thesis by (simp add: algebra_simps) kleing@19469: next kleing@19469: assume "\(n > 1)" kleing@19469: hence "n = 1 \ n = 0" by auto nipkow@29667: thus ?thesis by (auto simp: algebra_simps) kleing@19469: qed kleing@19469: kleing@19469: lemma arith_series_nat: kleing@19469: "Suc (Suc 0) * (\i\{..i\{..i\{..i\{..nat" kleing@19022: shows kleing@19022: "\x. Q x \ P x \ kleing@19022: (\xxxxx 'a \ 'a \ 'b \ 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10) paulson@29960: "_from_upto_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10) paulson@29960: "_upt_setprod" :: "idt \ 'a \ 'b \ 'b" ("(PROD _<_./ _)" [0,0,10] 10) paulson@29960: "_upto_setprod" :: "idt \ 'a \ 'b \ 'b" ("(PROD _<=_./ _)" [0,0,10] 10) paulson@29960: syntax (xsymbols) paulson@29960: "_from_to_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _.._./ _)" [0,0,0,10] 10) paulson@29960: "_from_upto_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _..<_./ _)" [0,0,0,10] 10) paulson@29960: "_upt_setprod" :: "idt \ 'a \ 'b \ 'b" ("(3\_<_./ _)" [0,0,10] 10) paulson@29960: "_upto_setprod" :: "idt \ 'a \ 'b \ 'b" ("(3\_\_./ _)" [0,0,10] 10) paulson@29960: syntax (HTML output) paulson@29960: "_from_to_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _.._./ _)" [0,0,0,10] 10) paulson@29960: "_from_upto_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" ("(3\_ = _..<_./ _)" [0,0,0,10] 10) paulson@29960: "_upt_setprod" :: "idt \ 'a \ 'b \ 'b" ("(3\_<_./ _)" [0,0,10] 10) paulson@29960: "_upto_setprod" :: "idt \ 'a \ 'b \ 'b" ("(3\_\_./ _)" [0,0,10] 10) paulson@29960: syntax (latex_prod output) paulson@29960: "_from_to_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" paulson@29960: ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10) paulson@29960: "_from_upto_setprod" :: "idt \ 'a \ 'a \ 'b \ 'b" paulson@29960: ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10) paulson@29960: "_upt_setprod" :: "idt \ 'a \ 'b \ 'b" paulson@29960: ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10) paulson@29960: "_upto_setprod" :: "idt \ 'a \ 'b \ 'b" paulson@29960: ("(3\<^raw:$\prod_{>_ \ _\<^raw:}$> _)" [0,0,10] 10) paulson@29960: paulson@29960: translations paulson@29960: "\x=a..b. t" == "CONST setprod (%x. t) {a..b}" paulson@29960: "\x=a..i\n. t" == "CONST setprod (\i. t) {..n}" paulson@29960: "\ii. t) {..= 0 \ nat_set {x..y}" haftmann@33318: by (simp add: nat_set_def) haftmann@33318: haftmann@35644: declare transfer_morphism_nat_int[transfer add haftmann@33318: return: transfer_nat_int_set_functions haftmann@33318: transfer_nat_int_set_function_closures haftmann@33318: ] haftmann@33318: haftmann@33318: lemma transfer_int_nat_set_functions: haftmann@33318: "is_nat m \ is_nat n \ {m..n} = int ` {nat m..nat n}" haftmann@33318: by (simp only: is_nat_def transfer_nat_int_set_functions haftmann@33318: transfer_nat_int_set_function_closures haftmann@33318: transfer_nat_int_set_return_embed nat_0_le haftmann@33318: cong: transfer_nat_int_set_cong) haftmann@33318: haftmann@33318: lemma transfer_int_nat_set_function_closures: haftmann@33318: "is_nat x \ nat_set {x..y}" haftmann@33318: by (simp only: transfer_nat_int_set_function_closures is_nat_def) haftmann@33318: haftmann@35644: declare transfer_morphism_int_nat[transfer add haftmann@33318: return: transfer_int_nat_set_functions haftmann@33318: transfer_int_nat_set_function_closures haftmann@33318: ] haftmann@33318: nipkow@8924: end