huffman@47317: (*  Title:      HOL/Set_Interval.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@51334: 
nipkow@51334: Modern convention: Ixy stands for an interval where x and y
nipkow@51334: describe the lower and upper bound and x,y : {c,o,i}
nipkow@51334: where c = closed, o = open, i = infinite.
nipkow@51334: Examples: Ico = {_ ..< _} and Ici = {_ ..}
nipkow@8924: *)
nipkow@8924: 
wenzelm@60758: section \<open>Set intervals\<close>
wenzelm@14577: 
huffman@47317: theory Set_Interval
blanchet@55088: imports Lattices_Big Nat_Transfer
nipkow@15131: begin
nipkow@8924: 
nipkow@24691: context ord
nipkow@24691: begin
haftmann@44008: 
nipkow@24691: definition
wenzelm@32960:   lessThan    :: "'a => 'a set" ("(1{..<_})") where
haftmann@25062:   "{..<u} == {x. x < u}"
nipkow@24691: 
nipkow@24691: definition
wenzelm@32960:   atMost      :: "'a => 'a set" ("(1{.._})") where
haftmann@25062:   "{..u} == {x. x \<le> u}"
nipkow@24691: 
nipkow@24691: definition
wenzelm@32960:   greaterThan :: "'a => 'a set" ("(1{_<..})") where
haftmann@25062:   "{l<..} == {x. l<x}"
nipkow@24691: 
nipkow@24691: definition
wenzelm@32960:   atLeast     :: "'a => 'a set" ("(1{_..})") where
haftmann@25062:   "{l..} == {x. l\<le>x}"
nipkow@24691: 
nipkow@24691: definition
haftmann@25062:   greaterThanLessThan :: "'a => 'a => 'a set"  ("(1{_<..<_})") where
haftmann@25062:   "{l<..<u} == {l<..} Int {..<u}"
nipkow@24691: 
nipkow@24691: definition
haftmann@25062:   atLeastLessThan :: "'a => 'a => 'a set"      ("(1{_..<_})") where
haftmann@25062:   "{l..<u} == {l..} Int {..<u}"
nipkow@24691: 
nipkow@24691: definition
haftmann@25062:   greaterThanAtMost :: "'a => '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: 
wenzelm@60758: text\<open>A note of warning when using @{term"{..<n}"} on type @{typ
nipkow@15048: nat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving
wenzelm@60758: @{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well.\<close>
nipkow@15048: 
kleing@14418: syntax
huffman@36364:   "_UNION_le"   :: "'a => '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)
wenzelm@60586:   "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@60586:   "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union>_<_./ _)" [0, 0, 10] 10)
wenzelm@60586:   "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@60586:   "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter>_<_./ _)" [0, 0, 10] 10)
kleing@14418: 
nipkow@30372: syntax (latex output)
huffman@36364:   "_UNION_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ \<le> _)/ _)" [0, 0, 10] 10)
huffman@36364:   "_UNION_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Union>(00_ < _)/ _)" [0, 0, 10] 10)
huffman@36364:   "_INTER_le"   :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ \<le> _)/ _)" [0, 0, 10] 10)
huffman@36364:   "_INTER_less" :: "'a \<Rightarrow> 'a => 'b set => 'b set"       ("(3\<Inter>(00_ < _)/ _)" [0, 0, 10] 10)
kleing@14418: 
kleing@14418: translations
kleing@14418:   "UN i<=n. A"  == "UN i:{..n}. A"
nipkow@15045:   "UN i<n. A"   == "UN i:{..<n}. A"
kleing@14418:   "INT i<=n. A" == "INT i:{..n}. A"
nipkow@15045:   "INT i<n. A"  == "INT i:{..<n}. A"
kleing@14418: 
kleing@14418: 
wenzelm@60758: subsection \<open>Various equivalences\<close>
ballarin@13735: 
haftmann@25062: lemma (in ord) lessThan_iff [iff]: "(i: lessThan k) = (i<k)"
paulson@13850: by (simp add: lessThan_def)
ballarin@13735: 
paulson@15418: lemma Compl_lessThan [simp]:
ballarin@13735:     "!!k:: 'a::linorder. -lessThan k = atLeast k"
paulson@13850: apply (auto simp add: lessThan_def atLeast_def)
ballarin@13735: done
ballarin@13735: 
paulson@13850: lemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"
paulson@13850: by auto
ballarin@13735: 
haftmann@25062: lemma (in ord) greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"
paulson@13850: by (simp add: greaterThan_def)
ballarin@13735: 
paulson@15418: lemma Compl_greaterThan [simp]:
ballarin@13735:     "!!k:: 'a::linorder. -greaterThan k = atMost k"
haftmann@26072:   by (auto simp add: greaterThan_def atMost_def)
ballarin@13735: 
paulson@13850: lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
paulson@13850: apply (subst Compl_greaterThan [symmetric])
paulson@15418: apply (rule double_complement)
ballarin@13735: done
ballarin@13735: 
haftmann@25062: lemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"
paulson@13850: by (simp add: atLeast_def)
ballarin@13735: 
paulson@15418: lemma Compl_atLeast [simp]:
ballarin@13735:     "!!k:: 'a::linorder. -atLeast k = lessThan k"
haftmann@26072:   by (auto simp add: lessThan_def atLeast_def)
ballarin@13735: 
haftmann@25062: lemma (in ord) atMost_iff [iff]: "(i: atMost k) = (i<=k)"
paulson@13850: by (simp add: atMost_def)
ballarin@13735: 
paulson@14485: lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
paulson@14485: by (blast intro: order_antisym)
paulson@13850: 
hoelzl@50999: lemma (in linorder) lessThan_Int_lessThan: "{ a <..} \<inter> { b <..} = { max a b <..}"
hoelzl@50999:   by auto
hoelzl@50999: 
hoelzl@50999: lemma (in linorder) greaterThan_Int_greaterThan: "{..< a} \<inter> {..< b} = {..< min a b}"
hoelzl@50999:   by auto
paulson@13850: 
wenzelm@60758: subsection \<open>Logical Equivalences for Set Inclusion and Equality\<close>
paulson@13850: 
paulson@13850: lemma atLeast_subset_iff [iff]:
paulson@15418:      "(atLeast x \<subseteq> atLeast y) = (y \<le> (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 \<subseteq> greaterThan y) = (y \<le> (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 \<subseteq> atMost y) = (x \<le> (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 \<subseteq> lessThan y) = (x \<le> (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: 
hoelzl@40703: lemma lessThan_strict_subset_iff:
hoelzl@40703:   fixes m n :: "'a::linorder"
hoelzl@40703:   shows "{..<m} < {..<n} \<longleftrightarrow> m < n"
hoelzl@40703:   by (metis leD lessThan_subset_iff linorder_linear not_less_iff_gr_or_eq psubset_eq)
ballarin@13735: 
hoelzl@57448: lemma (in linorder) Ici_subset_Ioi_iff: "{a ..} \<subseteq> {b <..} \<longleftrightarrow> b < a"
hoelzl@57448:   by auto
hoelzl@57448: 
hoelzl@57448: lemma (in linorder) Iic_subset_Iio_iff: "{.. a} \<subseteq> {..< b} \<longleftrightarrow> a < b"
hoelzl@57448:   by auto
hoelzl@57448: 
wenzelm@60758: subsection \<open>Two-sided intervals\<close>
ballarin@13735: 
nipkow@24691: context ord
nipkow@24691: begin
nipkow@24691: 
blanchet@54147: lemma greaterThanLessThan_iff [simp]:
haftmann@25062:   "(i : {l<..<u}) = (l < i & i < u)"
ballarin@13735: by (simp add: greaterThanLessThan_def)
ballarin@13735: 
blanchet@54147: lemma atLeastLessThan_iff [simp]:
haftmann@25062:   "(i : {l..<u}) = (l <= i & i < u)"
ballarin@13735: by (simp add: atLeastLessThan_def)
ballarin@13735: 
blanchet@54147: lemma greaterThanAtMost_iff [simp]:
haftmann@25062:   "(i : {l<..u}) = (l < i & i <= u)"
ballarin@13735: by (simp add: greaterThanAtMost_def)
ballarin@13735: 
blanchet@54147: lemma atLeastAtMost_iff [simp]:
haftmann@25062:   "(i : {l..u}) = (l <= i & i <= u)"
ballarin@13735: by (simp add: atLeastAtMost_def)
ballarin@13735: 
wenzelm@60758: text \<open>The above four lemmas could be declared as iffs. Unfortunately this
haftmann@52729: breaks many proofs. Since it only helps blast, it is better to leave them
wenzelm@60758: alone.\<close>
nipkow@32436: 
hoelzl@50999: lemma greaterThanLessThan_eq: "{ a <..< b} = { a <..} \<inter> {..< b }"
hoelzl@50999:   by auto
hoelzl@50999: 
nipkow@24691: end
ballarin@13735: 
wenzelm@60758: subsubsection\<open>Emptyness, singletons, subset\<close>
nipkow@15554: 
nipkow@24691: context order
nipkow@24691: begin
nipkow@15554: 
nipkow@32400: lemma atLeastatMost_empty[simp]:
nipkow@32400:   "b < a \<Longrightarrow> {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} = {} \<longleftrightarrow> (~ a <= b)"
nipkow@32400: by auto (blast intro: order_trans)
nipkow@32400: 
nipkow@32400: lemma atLeastatMost_empty_iff2[simp]:
nipkow@32400:   "{} = {a..b} \<longleftrightarrow> (~ a <= b)"
nipkow@32400: by auto (blast intro: order_trans)
nipkow@32400: 
nipkow@32400: lemma atLeastLessThan_empty[simp]:
nipkow@32400:   "b <= a \<Longrightarrow> {a..<b} = {}"
nipkow@32400: by(auto simp: atLeastLessThan_def)
nipkow@24691: 
nipkow@32400: lemma atLeastLessThan_empty_iff[simp]:
nipkow@32400:   "{a..<b} = {} \<longleftrightarrow> (~ a < b)"
nipkow@32400: by auto (blast intro: le_less_trans)
nipkow@32400: 
nipkow@32400: lemma atLeastLessThan_empty_iff2[simp]:
nipkow@32400:   "{} = {a..<b} \<longleftrightarrow> (~ a < b)"
nipkow@32400: by auto (blast intro: le_less_trans)
nipkow@15554: 
nipkow@32400: lemma greaterThanAtMost_empty[simp]: "l \<le> 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} = {} \<longleftrightarrow> ~ k < l"
nipkow@32400: by auto (blast intro: less_le_trans)
nipkow@32400: 
nipkow@32400: lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} \<longleftrightarrow> ~ k < l"
nipkow@32400: by auto (blast intro: less_le_trans)
nipkow@32400: 
haftmann@29709: lemma greaterThanLessThan_empty[simp]:"l \<le> k ==> {k<..<l} = {}"
nipkow@17719: by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)
nipkow@17719: 
haftmann@25062: lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"
nipkow@24691: by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)
nipkow@24691: 
hoelzl@36846: lemma atLeastAtMost_singleton': "a = b \<Longrightarrow> {a .. b} = {a}" by simp
hoelzl@36846: 
nipkow@32400: lemma atLeastatMost_subset_iff[simp]:
nipkow@32400:   "{a..b} <= {c..d} \<longleftrightarrow> (~ 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} \<longleftrightarrow>
nipkow@32400:    ((~ a <= b) | c <= a & b <= d & (c < a | b < d))  &  c <= d"
nipkow@39302: by(simp add: psubset_eq set_eq_iff less_le_not_le)(blast intro: order_trans)
nipkow@32400: 
nipkow@51334: lemma Icc_eq_Icc[simp]:
nipkow@51334:   "{l..h} = {l'..h'} = (l=l' \<and> h=h' \<or> \<not> l\<le>h \<and> \<not> l'\<le>h')"
nipkow@51334: by(simp add: order_class.eq_iff)(auto intro: order_trans)
nipkow@51334: 
hoelzl@36846: lemma atLeastAtMost_singleton_iff[simp]:
hoelzl@36846:   "{a .. b} = {c} \<longleftrightarrow> a = b \<and> b = c"
hoelzl@36846: proof
hoelzl@36846:   assume "{a..b} = {c}"
wenzelm@53374:   hence *: "\<not> (\<not> a \<le> b)" unfolding atLeastatMost_empty_iff[symmetric] by simp
wenzelm@60758:   with \<open>{a..b} = {c}\<close> have "c \<le> a \<and> b \<le> c" by auto
wenzelm@53374:   with * show "a = b \<and> b = c" by auto
hoelzl@36846: qed simp
hoelzl@36846: 
nipkow@51334: lemma Icc_subset_Ici_iff[simp]:
nipkow@51334:   "{l..h} \<subseteq> {l'..} = (~ l\<le>h \<or> l\<ge>l')"
nipkow@51334: by(auto simp: subset_eq intro: order_trans)
nipkow@51334: 
nipkow@51334: lemma Icc_subset_Iic_iff[simp]:
nipkow@51334:   "{l..h} \<subseteq> {..h'} = (~ l\<le>h \<or> h\<le>h')"
nipkow@51334: by(auto simp: subset_eq intro: order_trans)
nipkow@51334: 
nipkow@51334: lemma not_Ici_eq_empty[simp]: "{l..} \<noteq> {}"
nipkow@51334: by(auto simp: set_eq_iff)
nipkow@51334: 
nipkow@51334: lemma not_Iic_eq_empty[simp]: "{..h} \<noteq> {}"
nipkow@51334: by(auto simp: set_eq_iff)
nipkow@51334: 
nipkow@51334: lemmas not_empty_eq_Ici_eq_empty[simp] = not_Ici_eq_empty[symmetric]
nipkow@51334: lemmas not_empty_eq_Iic_eq_empty[simp] = not_Iic_eq_empty[symmetric]
nipkow@51334: 
nipkow@24691: end
paulson@14485: 
nipkow@51334: context no_top
nipkow@51334: begin
nipkow@51334: 
nipkow@51334: (* also holds for no_bot but no_top should suffice *)
nipkow@51334: lemma not_UNIV_le_Icc[simp]: "\<not> UNIV \<subseteq> {l..h}"
nipkow@51334: using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
nipkow@51334: 
nipkow@51334: lemma not_UNIV_le_Iic[simp]: "\<not> UNIV \<subseteq> {..h}"
nipkow@51334: using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
nipkow@51334: 
nipkow@51334: lemma not_Ici_le_Icc[simp]: "\<not> {l..} \<subseteq> {l'..h'}"
nipkow@51334: using gt_ex[of h']
nipkow@51334: by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
nipkow@51334: 
nipkow@51334: lemma not_Ici_le_Iic[simp]: "\<not> {l..} \<subseteq> {..h'}"
nipkow@51334: using gt_ex[of h']
nipkow@51334: by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
nipkow@51334: 
nipkow@51334: end
nipkow@51334: 
nipkow@51334: context no_bot
nipkow@51334: begin
nipkow@51334: 
nipkow@51334: lemma not_UNIV_le_Ici[simp]: "\<not> UNIV \<subseteq> {l..}"
nipkow@51334: using lt_ex[of l] by(auto simp: subset_eq less_le_not_le)
nipkow@51334: 
nipkow@51334: lemma not_Iic_le_Icc[simp]: "\<not> {..h} \<subseteq> {l'..h'}"
nipkow@51334: using lt_ex[of l']
nipkow@51334: by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
nipkow@51334: 
nipkow@51334: lemma not_Iic_le_Ici[simp]: "\<not> {..h} \<subseteq> {l'..}"
nipkow@51334: using lt_ex[of l']
nipkow@51334: by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
nipkow@51334: 
nipkow@51334: end
nipkow@51334: 
nipkow@51334: 
nipkow@51334: context no_top
nipkow@51334: begin
nipkow@51334: 
nipkow@51334: (* also holds for no_bot but no_top should suffice *)
nipkow@51334: lemma not_UNIV_eq_Icc[simp]: "\<not> UNIV = {l'..h'}"
nipkow@51334: using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
nipkow@51334: 
nipkow@51334: lemmas not_Icc_eq_UNIV[simp] = not_UNIV_eq_Icc[symmetric]
nipkow@51334: 
nipkow@51334: lemma not_UNIV_eq_Iic[simp]: "\<not> UNIV = {..h'}"
nipkow@51334: using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)
nipkow@51334: 
nipkow@51334: lemmas not_Iic_eq_UNIV[simp] = not_UNIV_eq_Iic[symmetric]
nipkow@51334: 
nipkow@51334: lemma not_Icc_eq_Ici[simp]: "\<not> {l..h} = {l'..}"
nipkow@51334: unfolding atLeastAtMost_def using not_Ici_le_Iic[of l'] by blast
nipkow@51334: 
nipkow@51334: lemmas not_Ici_eq_Icc[simp] = not_Icc_eq_Ici[symmetric]
nipkow@51334: 
nipkow@51334: (* also holds for no_bot but no_top should suffice *)
nipkow@51334: lemma not_Iic_eq_Ici[simp]: "\<not> {..h} = {l'..}"
nipkow@51334: using not_Ici_le_Iic[of l' h] by blast
nipkow@51334: 
nipkow@51334: lemmas not_Ici_eq_Iic[simp] = not_Iic_eq_Ici[symmetric]
nipkow@51334: 
nipkow@51334: end
nipkow@51334: 
nipkow@51334: context no_bot
nipkow@51334: begin
nipkow@51334: 
nipkow@51334: lemma not_UNIV_eq_Ici[simp]: "\<not> UNIV = {l'..}"
nipkow@51334: using lt_ex[of l'] by(auto simp: set_eq_iff  less_le_not_le)
nipkow@51334: 
nipkow@51334: lemmas not_Ici_eq_UNIV[simp] = not_UNIV_eq_Ici[symmetric]
nipkow@51334: 
nipkow@51334: lemma not_Icc_eq_Iic[simp]: "\<not> {l..h} = {..h'}"
nipkow@51334: unfolding atLeastAtMost_def using not_Iic_le_Ici[of h'] by blast
nipkow@51334: 
nipkow@51334: lemmas not_Iic_eq_Icc[simp] = not_Icc_eq_Iic[symmetric]
nipkow@51334: 
nipkow@51334: end
nipkow@51334: 
nipkow@51334: 
hoelzl@53216: context dense_linorder
hoelzl@42891: begin
hoelzl@42891: 
hoelzl@42891: lemma greaterThanLessThan_empty_iff[simp]:
hoelzl@42891:   "{ a <..< b } = {} \<longleftrightarrow> b \<le> a"
hoelzl@42891:   using dense[of a b] by (cases "a < b") auto
hoelzl@42891: 
hoelzl@42891: lemma greaterThanLessThan_empty_iff2[simp]:
hoelzl@42891:   "{} = { a <..< b } \<longleftrightarrow> b \<le> a"
hoelzl@42891:   using dense[of a b] by (cases "a < b") auto
hoelzl@42891: 
hoelzl@42901: lemma atLeastLessThan_subseteq_atLeastAtMost_iff:
hoelzl@42901:   "{a ..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
hoelzl@42901:   using dense[of "max a d" "b"]
hoelzl@42901:   by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@42901: 
hoelzl@42901: lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:
hoelzl@42901:   "{a <.. b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
hoelzl@42901:   using dense[of "a" "min c b"]
hoelzl@42901:   by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@42901: 
hoelzl@42901: lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:
hoelzl@42901:   "{a <..< b} \<subseteq> { c .. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
hoelzl@42901:   using dense[of "a" "min c b"] dense[of "max a d" "b"]
hoelzl@42901:   by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@42901: 
hoelzl@43657: lemma atLeastAtMost_subseteq_atLeastLessThan_iff:
hoelzl@43657:   "{a .. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a \<le> b \<longrightarrow> c \<le> a \<and> b < d)"
hoelzl@43657:   using dense[of "max a d" "b"]
hoelzl@43657:   by (force simp: subset_eq Ball_def not_less[symmetric])
eberlm@61524:   
eberlm@61524: lemma greaterThanLessThan_subseteq_greaterThanLessThan:
eberlm@61524:   "{a <..< b} \<subseteq> {c <..< d} \<longleftrightarrow> (a < b \<longrightarrow> a \<ge> c \<and> b \<le> d)"
eberlm@61524:   using dense[of "a" "min c b"] dense[of "max a d" "b"]
eberlm@61524:   by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@43657: 
hoelzl@43657: lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:
hoelzl@43657:   "{a <.. b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b < d)"
hoelzl@43657:   using dense[of "a" "min c b"]
hoelzl@43657:   by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@43657: 
hoelzl@43657: lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:
hoelzl@43657:   "{a <..< b} \<subseteq> { c ..< d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
hoelzl@43657:   using dense[of "a" "min c b"] dense[of "max a d" "b"]
hoelzl@43657:   by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@43657: 
hoelzl@56328: lemma greaterThanLessThan_subseteq_greaterThanAtMost_iff:
hoelzl@56328:   "{a <..< b} \<subseteq> { c <.. d } \<longleftrightarrow> (a < b \<longrightarrow> c \<le> a \<and> b \<le> d)"
hoelzl@56328:   using dense[of "a" "min c b"] dense[of "max a d" "b"]
hoelzl@56328:   by (force simp: subset_eq Ball_def not_less[symmetric])
hoelzl@56328: 
hoelzl@42891: end
hoelzl@42891: 
hoelzl@51329: context no_top
hoelzl@51329: begin
hoelzl@51329: 
nipkow@51334: lemma greaterThan_non_empty[simp]: "{x <..} \<noteq> {}"
hoelzl@51329:   using gt_ex[of x] by auto
hoelzl@51329: 
hoelzl@51329: end
hoelzl@51329: 
hoelzl@51329: context no_bot
hoelzl@51329: begin
hoelzl@51329: 
nipkow@51334: lemma lessThan_non_empty[simp]: "{..< x} \<noteq> {}"
hoelzl@51329:   using lt_ex[of x] by auto
hoelzl@51329: 
hoelzl@51329: end
hoelzl@51329: 
nipkow@32408: lemma (in linorder) atLeastLessThan_subset_iff:
nipkow@32408:   "{a..<b} <= {c..<d} \<Longrightarrow> 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: 
hoelzl@40703: lemma atLeastLessThan_inj:
hoelzl@40703:   fixes a b c d :: "'a::linorder"
hoelzl@40703:   assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"
hoelzl@40703:   shows "a = c" "b = d"
hoelzl@40703: using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le linorder_antisym_conv2 subset_refl)+
hoelzl@40703: 
hoelzl@40703: lemma atLeastLessThan_eq_iff:
hoelzl@40703:   fixes a b c d :: "'a::linorder"
hoelzl@40703:   assumes "a < b" "c < d"
hoelzl@40703:   shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"
hoelzl@40703:   using atLeastLessThan_inj assms by auto
hoelzl@40703: 
hoelzl@57447: lemma (in linorder) Ioc_inj: "{a <.. b} = {c <.. d} \<longleftrightarrow> (b \<le> a \<and> d \<le> c) \<or> a = c \<and> b = d"
hoelzl@57447:   by (metis eq_iff greaterThanAtMost_empty_iff2 greaterThanAtMost_iff le_cases not_le)
hoelzl@57447: 
hoelzl@57447: lemma (in order) Iio_Int_singleton: "{..<k} \<inter> {x} = (if x < k then {x} else {})"
hoelzl@57447:   by auto
hoelzl@57447: 
hoelzl@57447: lemma (in linorder) Ioc_subset_iff: "{a<..b} \<subseteq> {c<..d} \<longleftrightarrow> (b \<le> a \<or> c \<le> a \<and> b \<le> d)"
hoelzl@57447:   by (auto simp: subset_eq Ball_def) (metis less_le not_less)
hoelzl@57447: 
haftmann@52729: lemma (in order_bot) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"
nipkow@51334: by (auto simp: set_eq_iff intro: le_bot)
hoelzl@51328: 
haftmann@52729: lemma (in order_top) atMost_eq_UNIV_iff: "{..x} = UNIV \<longleftrightarrow> x = top"
nipkow@51334: by (auto simp: set_eq_iff intro: top_le)
hoelzl@51328: 
nipkow@51334: lemma (in bounded_lattice) atLeastAtMost_eq_UNIV_iff:
nipkow@51334:   "{x..y} = UNIV \<longleftrightarrow> (x = bot \<and> y = top)"
nipkow@51334: by (auto simp: set_eq_iff intro: top_le le_bot)
hoelzl@51328: 
hoelzl@56949: lemma Iio_eq_empty_iff: "{..< n::'a::{linorder, order_bot}} = {} \<longleftrightarrow> n = bot"
hoelzl@56949:   by (auto simp: set_eq_iff not_less le_bot)
hoelzl@56949: 
hoelzl@56949: lemma Iio_eq_empty_iff_nat: "{..< n::nat} = {} \<longleftrightarrow> n = 0"
hoelzl@56949:   by (simp add: Iio_eq_empty_iff bot_nat_def)
hoelzl@56949: 
noschinl@58970: lemma mono_image_least:
noschinl@58970:   assumes f_mono: "mono f" and f_img: "f ` {m ..< n} = {m' ..< n'}" "m < n"
noschinl@58970:   shows "f m = m'"
noschinl@58970: proof -
noschinl@58970:   from f_img have "{m' ..< n'} \<noteq> {}"
noschinl@58970:     by (metis atLeastLessThan_empty_iff image_is_empty)
noschinl@58970:   with f_img have "m' \<in> f ` {m ..< n}" by auto
noschinl@58970:   then obtain k where "f k = m'" "m \<le> k" by auto
noschinl@58970:   moreover have "m' \<le> f m" using f_img by auto
noschinl@58970:   ultimately show "f m = m'"
noschinl@58970:     using f_mono by (auto elim: monoE[where x=m and y=k])
noschinl@58970: qed
noschinl@58970: 
hoelzl@51328: 
wenzelm@60758: subsection \<open>Infinite intervals\<close>
hoelzl@56328: 
hoelzl@56328: context dense_linorder
hoelzl@56328: begin
hoelzl@56328: 
hoelzl@56328: lemma infinite_Ioo:
hoelzl@56328:   assumes "a < b"
hoelzl@56328:   shows "\<not> finite {a<..<b}"
hoelzl@56328: proof
hoelzl@56328:   assume fin: "finite {a<..<b}"
hoelzl@56328:   moreover have ne: "{a<..<b} \<noteq> {}"
wenzelm@60758:     using \<open>a < b\<close> by auto
hoelzl@56328:   ultimately have "a < Max {a <..< b}" "Max {a <..< b} < b"
hoelzl@56328:     using Max_in[of "{a <..< b}"] by auto
hoelzl@56328:   then obtain x where "Max {a <..< b} < x" "x < b"
hoelzl@56328:     using dense[of "Max {a<..<b}" b] by auto
hoelzl@56328:   then have "x \<in> {a <..< b}"
wenzelm@60758:     using \<open>a < Max {a <..< b}\<close> by auto
hoelzl@56328:   then have "x \<le> Max {a <..< b}"
hoelzl@56328:     using fin by auto
wenzelm@60758:   with \<open>Max {a <..< b} < x\<close> show False by auto
hoelzl@56328: qed
hoelzl@56328: 
hoelzl@56328: lemma infinite_Icc: "a < b \<Longrightarrow> \<not> finite {a .. b}"
hoelzl@56328:   using greaterThanLessThan_subseteq_atLeastAtMost_iff[of a b a b] infinite_Ioo[of a b]
hoelzl@56328:   by (auto dest: finite_subset)
hoelzl@56328: 
hoelzl@56328: lemma infinite_Ico: "a < b \<Longrightarrow> \<not> finite {a ..< b}"
hoelzl@56328:   using greaterThanLessThan_subseteq_atLeastLessThan_iff[of a b a b] infinite_Ioo[of a b]
hoelzl@56328:   by (auto dest: finite_subset)
hoelzl@56328: 
hoelzl@56328: lemma infinite_Ioc: "a < b \<Longrightarrow> \<not> finite {a <.. b}"
hoelzl@56328:   using greaterThanLessThan_subseteq_greaterThanAtMost_iff[of a b a b] infinite_Ioo[of a b]
hoelzl@56328:   by (auto dest: finite_subset)
hoelzl@56328: 
hoelzl@56328: end
hoelzl@56328: 
hoelzl@56328: lemma infinite_Iio: "\<not> finite {..< a :: 'a :: {no_bot, linorder}}"
hoelzl@56328: proof
hoelzl@56328:   assume "finite {..< a}"
hoelzl@56328:   then have *: "\<And>x. x < a \<Longrightarrow> Min {..< a} \<le> x"
hoelzl@56328:     by auto
hoelzl@56328:   obtain x where "x < a"
hoelzl@56328:     using lt_ex by auto
hoelzl@56328: 
hoelzl@56328:   obtain y where "y < Min {..< a}"
hoelzl@56328:     using lt_ex by auto
hoelzl@56328:   also have "Min {..< a} \<le> x"
wenzelm@60758:     using \<open>x < a\<close> by fact
wenzelm@60758:   also note \<open>x < a\<close>
hoelzl@56328:   finally have "Min {..< a} \<le> y"
hoelzl@56328:     by fact
wenzelm@60758:   with \<open>y < Min {..< a}\<close> show False by auto
hoelzl@56328: qed
hoelzl@56328: 
hoelzl@56328: lemma infinite_Iic: "\<not> finite {.. a :: 'a :: {no_bot, linorder}}"
hoelzl@56328:   using infinite_Iio[of a] finite_subset[of "{..< a}" "{.. a}"]
hoelzl@56328:   by (auto simp: subset_eq less_imp_le)
hoelzl@56328: 
hoelzl@56328: lemma infinite_Ioi: "\<not> finite {a :: 'a :: {no_top, linorder} <..}"
hoelzl@56328: proof
hoelzl@56328:   assume "finite {a <..}"
hoelzl@56328:   then have *: "\<And>x. a < x \<Longrightarrow> x \<le> Max {a <..}"
hoelzl@56328:     by auto
hoelzl@56328: 
hoelzl@56328:   obtain y where "Max {a <..} < y"
hoelzl@56328:     using gt_ex by auto
hoelzl@56328: 
hoelzl@56328:   obtain x where "a < x"
hoelzl@56328:     using gt_ex by auto
hoelzl@56328:   also then have "x \<le> Max {a <..}"
hoelzl@56328:     by fact
wenzelm@60758:   also note \<open>Max {a <..} < y\<close>
hoelzl@56328:   finally have "y \<le> Max { a <..}"
hoelzl@56328:     by fact
wenzelm@60758:   with \<open>Max {a <..} < y\<close> show False by auto
hoelzl@56328: qed
hoelzl@56328: 
hoelzl@56328: lemma infinite_Ici: "\<not> finite {a :: 'a :: {no_top, linorder} ..}"
hoelzl@56328:   using infinite_Ioi[of a] finite_subset[of "{a <..}" "{a ..}"]
hoelzl@56328:   by (auto simp: subset_eq less_imp_le)
hoelzl@56328: 
wenzelm@60758: subsubsection \<open>Intersection\<close>
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..<b} Int {c..<d} = {max a c ..< min b d}"
nipkow@32456: by auto
nipkow@32456: 
nipkow@32456: lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
nipkow@32456: by auto
nipkow@32456: 
nipkow@32456: lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
nipkow@32456: by auto
nipkow@32456: 
hoelzl@50417: lemma Int_atMost[simp]: "{..a} \<inter> {..b} = {.. min a b}"
hoelzl@50417:   by (auto simp: min_def)
hoelzl@50417: 
hoelzl@57447: lemma Ioc_disjoint: "{a<..b} \<inter> {c<..d} = {} \<longleftrightarrow> b \<le> a \<or> d \<le> c \<or> b \<le> c \<or> d \<le> a"
hoelzl@57447:   using assms by auto
hoelzl@57447: 
nipkow@32456: end
nipkow@32456: 
hoelzl@51329: context complete_lattice
hoelzl@51329: begin
hoelzl@51329: 
hoelzl@51329: lemma
hoelzl@51329:   shows Sup_atLeast[simp]: "Sup {x ..} = top"
hoelzl@51329:     and Sup_greaterThanAtLeast[simp]: "x < top \<Longrightarrow> Sup {x <..} = top"
hoelzl@51329:     and Sup_atMost[simp]: "Sup {.. y} = y"
hoelzl@51329:     and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"
hoelzl@51329:     and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"
hoelzl@51329:   by (auto intro!: Sup_eqI)
hoelzl@51329: 
hoelzl@51329: lemma
hoelzl@51329:   shows Inf_atMost[simp]: "Inf {.. x} = bot"
hoelzl@51329:     and Inf_atMostLessThan[simp]: "top < x \<Longrightarrow> Inf {..< x} = bot"
hoelzl@51329:     and Inf_atLeast[simp]: "Inf {x ..} = x"
hoelzl@51329:     and Inf_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Inf { x .. y} = x"
hoelzl@51329:     and Inf_atLeastLessThan[simp]: "x < y \<Longrightarrow> Inf { x ..< y} = x"
hoelzl@51329:   by (auto intro!: Inf_eqI)
hoelzl@51329: 
hoelzl@51329: end
hoelzl@51329: 
hoelzl@51329: lemma
hoelzl@53216:   fixes x y :: "'a :: {complete_lattice, dense_linorder}"
hoelzl@51329:   shows Sup_lessThan[simp]: "Sup {..< y} = y"
hoelzl@51329:     and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"
hoelzl@51329:     and Sup_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Sup { x <..< y} = y"
hoelzl@51329:     and Inf_greaterThan[simp]: "Inf {x <..} = x"
hoelzl@51329:     and Inf_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Inf { x <.. y} = x"
hoelzl@51329:     and Inf_greaterThanLessThan[simp]: "x < y \<Longrightarrow> Inf { x <..< y} = x"
hoelzl@51329:   by (auto intro!: Inf_eqI Sup_eqI intro: dense_le dense_le_bounded dense_ge dense_ge_bounded)
nipkow@32456: 
wenzelm@60758: subsection \<open>Intervals of natural numbers\<close>
paulson@14485: 
wenzelm@60758: subsubsection \<open>The Constant @{term lessThan}\<close>
paulson@15047: 
paulson@14485: lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
paulson@14485: by (simp add: lessThan_def)
paulson@14485: 
paulson@14485: lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
paulson@14485: by (simp add: lessThan_def less_Suc_eq, blast)
paulson@14485: 
wenzelm@60758: text \<open>The following proof is convenient in induction proofs where
hoelzl@39072: new elements get indices at the beginning. So it is used to transform
wenzelm@60758: @{term "{..<Suc n}"} to @{term "0::nat"} and @{term "{..< n}"}.\<close>
hoelzl@39072: 
hoelzl@59000: lemma zero_notin_Suc_image: "0 \<notin> Suc ` A"
hoelzl@59000:   by auto
hoelzl@59000: 
hoelzl@39072: lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"
hoelzl@59000:   by (auto simp: image_iff less_Suc_eq_0_disj)
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: 
hoelzl@59000: lemma Iic_Suc_eq_insert_0: "{.. Suc n} = insert 0 (Suc ` {.. n})"
hoelzl@59000:   unfolding lessThan_Suc_atMost[symmetric] lessThan_Suc_eq_insert_0[of "Suc n"] ..
hoelzl@59000: 
paulson@14485: lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"
paulson@14485: by blast
paulson@14485: 
wenzelm@60758: subsubsection \<open>The Constant @{term greaterThan}\<close>
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: 
wenzelm@60758: subsubsection \<open>The Constant @{term atLeast}\<close>
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: 
wenzelm@60758: subsubsection \<open>The Constant @{term atMost}\<close>
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: 
wenzelm@60758: subsubsection \<open>The Constant @{term atLeastLessThan}\<close>
paulson@15047: 
wenzelm@60758: text\<open>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
wenzelm@60758: specific concept to a more general one.\<close>
nipkow@28068: 
paulson@15047: lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"
nipkow@15042: by(simp add:lessThan_def atLeastLessThan_def)
nipkow@24449: 
nipkow@28068: lemma atLeast0AtMost: "{0..n::nat} = {..n}"
nipkow@28068: by(simp add:atMost_def atLeastAtMost_def)
nipkow@28068: 
haftmann@31998: declare atLeast0LessThan[symmetric, code_unfold]
haftmann@31998:         atLeast0AtMost[symmetric, code_unfold]
nipkow@24449: 
nipkow@24449: lemma atLeastLessThan0: "{m..<0::nat} = {}"
paulson@15047: by (simp add: atLeastLessThan_def)
nipkow@24449: 
wenzelm@60758: subsubsection \<open>Intervals of nats with @{term Suc}\<close>
paulson@15047: 
wenzelm@60758: text\<open>Not a simprule because the RHS is too messy.\<close>
paulson@15047: lemma atLeastLessThanSuc:
paulson@15047:     "{m..<Suc n} = (if m \<le> n then insert n {m..<n} else {})"
paulson@15418: by (auto simp add: atLeastLessThan_def)
paulson@15047: 
paulson@15418: lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
paulson@15047: by (auto simp add: atLeastLessThan_def)
nipkow@16041: (*
paulson@15047: lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"
paulson@15047: by (induct k, simp_all add: atLeastLessThanSuc)
paulson@15047: 
paulson@15047: lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"
paulson@15047: by (auto simp add: atLeastLessThan_def)
nipkow@16041: *)
nipkow@15045: lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
paulson@14485:   by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)
paulson@14485: 
paulson@15418: lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
paulson@15418:   by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
paulson@14485:     greaterThanAtMost_def)
paulson@14485: 
paulson@15418: lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
paulson@15418:   by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
paulson@14485:     greaterThanLessThan_def)
paulson@14485: 
nipkow@15554: lemma atLeastAtMostSuc_conv: "m \<le> Suc n \<Longrightarrow> {m..Suc n} = insert (Suc n) {m..n}"
nipkow@15554: by (auto simp add: atLeastAtMost_def)
nipkow@15554: 
noschinl@45932: lemma atLeastAtMost_insertL: "m \<le> n \<Longrightarrow> insert m {Suc m..n} = {m ..n}"
noschinl@45932: by auto
noschinl@45932: 
wenzelm@60758: text \<open>The analogous result is useful on @{typ int}:\<close>
kleing@43157: (* here, because we don't have an own int section *)
kleing@43157: lemma atLeastAtMostPlus1_int_conv:
kleing@43157:   "m <= 1+n \<Longrightarrow> {m..1+n} = insert (1+n) {m..n::int}"
kleing@43157:   by (auto intro: set_eqI)
kleing@43157: 
paulson@33044: lemma atLeastLessThan_add_Un: "i \<le> j \<Longrightarrow> {i..<j+k} = {i..<j} \<union> {j..<j+k::nat}"
paulson@33044:   apply (induct k) 
paulson@33044:   apply (simp_all add: atLeastLessThanSuc)   
paulson@33044:   done
paulson@33044: 
wenzelm@60758: subsubsection \<open>Intervals and numerals\<close>
lp15@57113: 
wenzelm@61799: lemma lessThan_nat_numeral:  \<comment>\<open>Evaluation for specific numerals\<close>
lp15@57113:   "lessThan (numeral k :: nat) = insert (pred_numeral k) (lessThan (pred_numeral k))"
lp15@57113:   by (simp add: numeral_eq_Suc lessThan_Suc)
lp15@57113: 
wenzelm@61799: lemma atMost_nat_numeral:  \<comment>\<open>Evaluation for specific numerals\<close>
lp15@57113:   "atMost (numeral k :: nat) = insert (numeral k) (atMost (pred_numeral k))"
lp15@57113:   by (simp add: numeral_eq_Suc atMost_Suc)
lp15@57113: 
wenzelm@61799: lemma atLeastLessThan_nat_numeral:  \<comment>\<open>Evaluation for specific numerals\<close>
lp15@57113:   "atLeastLessThan m (numeral k :: nat) = 
lp15@57113:      (if m \<le> (pred_numeral k) then insert (pred_numeral k) (atLeastLessThan m (pred_numeral k))
lp15@57113:                  else {})"
lp15@57113:   by (simp add: numeral_eq_Suc atLeastLessThanSuc)
lp15@57113: 
wenzelm@60758: subsubsection \<open>Image\<close>
nipkow@16733: 
lp15@60809: lemma image_add_atLeastAtMost [simp]:
lp15@60615:   fixes k ::"'a::linordered_semidom"
lp15@60615:   shows "(\<lambda>n. n+k) ` {i..j} = {i+k..j+k}" (is "?A = ?B")
nipkow@16733: proof
nipkow@16733:   show "?A \<subseteq> ?B" by auto
nipkow@16733: next
nipkow@16733:   show "?B \<subseteq> ?A"
nipkow@16733:   proof
nipkow@16733:     fix n assume a: "n : ?B"
lp15@60615:     hence "n - k : {i..j}"
lp15@60615:       by (auto simp: add_le_imp_le_diff add_le_add_imp_diff_le)
lp15@60615:     moreover have "n = (n - k) + k" using a
lp15@60615:     proof -
lp15@60615:       have "k + i \<le> n"
lp15@60615:         by (metis a add.commute atLeastAtMost_iff)
lp15@60615:       hence "k + (n - k) = n"
lp15@60615:         by (metis (no_types) ab_semigroup_add_class.add_ac(1) add_diff_cancel_left' le_add_diff_inverse)
lp15@60615:       thus ?thesis
lp15@60615:         by (simp add: add.commute)
lp15@60615:     qed
nipkow@16733:     ultimately show "n : ?A" by blast
nipkow@16733:   qed
nipkow@16733: qed
nipkow@16733: 
lp15@60809: lemma image_diff_atLeastAtMost [simp]:
lp15@60809:   fixes d::"'a::linordered_idom" shows "(op - d ` {a..b}) = {d-b..d-a}"
lp15@60809:   apply auto
lp15@60809:   apply (rule_tac x="d-x" in rev_image_eqI, auto)
lp15@60809:   done
lp15@60809: 
lp15@60809: lemma image_mult_atLeastAtMost [simp]:
lp15@60809:   fixes d::"'a::linordered_field"
lp15@60809:   assumes "d>0" shows "(op * d ` {a..b}) = {d*a..d*b}"
lp15@60809:   using assms
lp15@60809:   by (auto simp: field_simps mult_le_cancel_right intro: rev_image_eqI [where x="x/d" for x])
lp15@60809: 
lp15@60809: lemma image_affinity_atLeastAtMost:
lp15@60809:   fixes c :: "'a::linordered_field"
lp15@60809:   shows "((\<lambda>x. m*x + c) ` {a..b}) = (if {a..b}={} then {}
lp15@60809:             else if 0 \<le> m then {m*a + c .. m *b + c}
lp15@60809:             else {m*b + c .. m*a + c})"
lp15@60809:   apply (case_tac "m=0", auto simp: mult_le_cancel_left)
lp15@60809:   apply (rule_tac x="inverse m*(x-c)" in rev_image_eqI, auto simp: field_simps)
lp15@60809:   apply (rule_tac x="inverse m*(x-c)" in rev_image_eqI, auto simp: field_simps)
lp15@60809:   done
lp15@60809: 
lp15@60809: lemma image_affinity_atLeastAtMost_diff:
lp15@60809:   fixes c :: "'a::linordered_field"
lp15@60809:   shows "((\<lambda>x. m*x - c) ` {a..b}) = (if {a..b}={} then {}
lp15@60809:             else if 0 \<le> m then {m*a - c .. m*b - c}
lp15@60809:             else {m*b - c .. m*a - c})"
lp15@60809:   using image_affinity_atLeastAtMost [of m "-c" a b]
lp15@60809:   by simp
lp15@60809: 
paulson@61204: lemma image_affinity_atLeastAtMost_div:
paulson@61204:   fixes c :: "'a::linordered_field"
paulson@61204:   shows "((\<lambda>x. x/m + c) ` {a..b}) = (if {a..b}={} then {}
paulson@61204:             else if 0 \<le> m then {a/m + c .. b/m + c}
paulson@61204:             else {b/m + c .. a/m + c})"
paulson@61204:   using image_affinity_atLeastAtMost [of "inverse m" c a b]
paulson@61204:   by (simp add: field_class.field_divide_inverse algebra_simps)
paulson@61204:     
lp15@60809: lemma image_affinity_atLeastAtMost_div_diff:
lp15@60809:   fixes c :: "'a::linordered_field"
lp15@60809:   shows "((\<lambda>x. x/m - c) ` {a..b}) = (if {a..b}={} then {}
lp15@60809:             else if 0 \<le> m then {a/m - c .. b/m - c}
lp15@60809:             else {b/m - c .. a/m - c})"
lp15@60809:   using image_affinity_atLeastAtMost_diff [of "inverse m" c a b]
lp15@60809:   by (simp add: field_class.field_divide_inverse algebra_simps)
lp15@60809: 
nipkow@16733: lemma image_add_atLeastLessThan:
nipkow@16733:   "(%n::nat. n+k) ` {i..<j} = {i+k..<j+k}" (is "?A = ?B")
nipkow@16733: proof
nipkow@16733:   show "?A \<subseteq> ?B" by auto
nipkow@16733: next
nipkow@16733:   show "?B \<subseteq> ?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: corollary image_Suc_atLeastAtMost[simp]:
nipkow@16733:   "Suc ` {i..j} = {Suc i..Suc j}"
huffman@30079: using image_add_atLeastAtMost[where k="Suc 0"] by simp
nipkow@16733: 
nipkow@16733: corollary image_Suc_atLeastLessThan[simp]:
nipkow@16733:   "Suc ` {i..<j} = {Suc i..<Suc j}"
huffman@30079: using image_add_atLeastLessThan[where k="Suc 0"] by simp
nipkow@16733: 
nipkow@16733: lemma image_add_int_atLeastLessThan:
nipkow@16733:     "(%x. x + (l::int)) ` {0..<u-l} = {l..<u}"
nipkow@16733:   apply (auto simp add: image_def)
nipkow@16733:   apply (rule_tac x = "x - l" in bexI)
nipkow@16733:   apply auto
nipkow@16733:   done
nipkow@16733: 
hoelzl@37664: lemma image_minus_const_atLeastLessThan_nat:
hoelzl@37664:   fixes c :: nat
hoelzl@37664:   shows "(\<lambda>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 \<in> ?right"
hoelzl@37664:   show "a \<in> (\<lambda>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 "\<not> 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: 
Andreas@51152: lemma image_int_atLeastLessThan: "int ` {a..<b} = {int a..<int b}"
wenzelm@55143:   by (auto intro!: image_eqI [where x = "nat x" for x])
Andreas@51152: 
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) \<in> uminus ` {x<..}"
hoelzl@35580:     by (rule imageI) (simp add: *)
hoelzl@35580:   thus "y \<in> uminus ` {x<..}" by simp
hoelzl@35580: next
hoelzl@35580:   fix y assume "y \<le> -x"
hoelzl@35580:   have "- (-y) \<in> uminus ` {x..}"
wenzelm@60758:     by (rule imageI) (insert \<open>y \<le> -x\<close>[THEN le_imp_neg_le], simp)
hoelzl@35580:   thus "y \<in> 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 ` {..<x} = {-x<..}"
hoelzl@35580:   and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"
hoelzl@35580: proof -
hoelzl@35580:   have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"
hoelzl@35580:     and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all
hoelzl@35580:   thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"
hoelzl@35580:     by (simp_all add: image_image
hoelzl@35580:         del: image_uminus_greaterThan image_uminus_atLeast)
hoelzl@35580: qed
hoelzl@35580: 
hoelzl@35580: lemma
hoelzl@35580:   fixes x :: 'a
hoelzl@35580:   shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"
hoelzl@35580:   and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"
hoelzl@35580:   and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"
hoelzl@35580:   and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"
hoelzl@35580:   by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def
hoelzl@35580:       greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)
hoelzl@35580: end
nipkow@16733: 
wenzelm@60758: subsubsection \<open>Finiteness\<close>
paulson@14485: 
nipkow@15045: lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
paulson@14485:   by (induct k) (simp_all add: lessThan_Suc)
paulson@14485: 
paulson@14485: lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
paulson@14485:   by (induct k) (simp_all add: atMost_Suc)
paulson@14485: 
paulson@14485: lemma finite_greaterThanLessThan [iff]:
nipkow@15045:   fixes l :: nat shows "finite {l<..<u}"
paulson@14485: by (simp add: greaterThanLessThan_def)
paulson@14485: 
paulson@14485: lemma finite_atLeastLessThan [iff]:
nipkow@15045:   fixes l :: nat shows "finite {l..<u}"
paulson@14485: by (simp add: atLeastLessThan_def)
paulson@14485: 
paulson@14485: lemma finite_greaterThanAtMost [iff]:
nipkow@15045:   fixes l :: nat shows "finite {l<..u}"
paulson@14485: by (simp add: greaterThanAtMost_def)
paulson@14485: 
paulson@14485: lemma finite_atLeastAtMost [iff]:
paulson@14485:   fixes l :: nat shows "finite {l..u}"
paulson@14485: by (simp add: atLeastAtMost_def)
paulson@14485: 
wenzelm@60758: text \<open>A bounded set of natural numbers is finite.\<close>
paulson@14485: lemma bounded_nat_set_is_finite:
nipkow@24853:   "(ALL i:N. i < (n::nat)) ==> finite N"
nipkow@28068: apply (rule finite_subset)
nipkow@28068:  apply (rule_tac [2] finite_lessThan, auto)
nipkow@28068: done
nipkow@28068: 
wenzelm@60758: text \<open>A set of natural numbers is finite iff it is bounded.\<close>
nipkow@31044: lemma finite_nat_set_iff_bounded:
nipkow@31044:   "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")
nipkow@31044: proof
nipkow@31044:   assume f:?F  show ?B
wenzelm@60758:     using Max_ge[OF \<open>?F\<close>, simplified less_Suc_eq_le[symmetric]] by blast
nipkow@31044: next
wenzelm@60758:   assume ?B show ?F using \<open>?B\<close> by(blast intro:bounded_nat_set_is_finite)
nipkow@31044: qed
nipkow@31044: 
nipkow@31044: lemma finite_nat_set_iff_bounded_le:
nipkow@31044:   "finite(N::nat set) = (EX m. ALL n:N. n<=m)"
nipkow@31044: apply(simp add:finite_nat_set_iff_bounded)
nipkow@31044: apply(blast dest:less_imp_le_nat le_imp_less_Suc)
nipkow@31044: done
nipkow@31044: 
nipkow@28068: lemma finite_less_ub:
nipkow@28068:      "!!f::nat=>nat. (!!n. n \<le> f n) ==> finite {n. f n \<le> u}"
nipkow@28068: by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)
paulson@14485: 
hoelzl@56328: 
wenzelm@60758: text\<open>Any subset of an interval of natural numbers the size of the
wenzelm@60758: subset is exactly that interval.\<close>
nipkow@24853: 
nipkow@24853: lemma subset_card_intvl_is_intvl:
blanchet@55085:   assumes "A \<subseteq> {k..<k + card A}"
blanchet@55085:   shows "A = {k..<k + card A}"
wenzelm@53374: proof (cases "finite A")
wenzelm@53374:   case True
wenzelm@53374:   from this and assms show ?thesis
wenzelm@53374:   proof (induct A rule: finite_linorder_max_induct)
nipkow@24853:     case empty thus ?case by auto
nipkow@24853:   next
nipkow@33434:     case (insert b A)
wenzelm@53374:     hence *: "b \<notin> A" by auto
blanchet@55085:     with insert have "A <= {k..<k + card A}" and "b = k + card A"
wenzelm@53374:       by fastforce+
wenzelm@53374:     with insert * show ?case by auto
nipkow@24853:   qed
nipkow@24853: next
wenzelm@53374:   case False
wenzelm@53374:   with assms show ?thesis by simp
nipkow@24853: qed
nipkow@24853: 
nipkow@24853: 
wenzelm@60758: subsubsection \<open>Proving Inclusions and Equalities between Unions\<close>
paulson@32596: 
nipkow@36755: lemma UN_le_eq_Un0:
nipkow@36755:   "(\<Union>i\<le>n::nat. M i) = (\<Union>i\<in>{1..n}. M i) \<union> 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\<le>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:   "(\<Union>i\<le>n::nat. M(i+k)) = (\<Union>i\<in>{k..n+k}. M i)" (is "?A = ?B")
nipkow@36755: proof
nipkow@44890:   show "?A <= ?B" by fastforce
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\<le>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: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"
paulson@32596:   by (auto simp add: atLeast0LessThan) 
paulson@32596: 
paulson@32596: lemma UN_finite_subset: "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> C) \<Longrightarrow> (\<Union>n. A n) \<subseteq> C"
paulson@32596:   by (subst UN_UN_finite_eq [symmetric]) blast
paulson@32596: 
paulson@33044: lemma UN_finite2_subset: 
paulson@33044:      "(!!n::nat. (\<Union>i\<in>{0..<n}. A i) \<subseteq> (\<Union>i\<in>{0..<n+k}. B i)) \<Longrightarrow> (\<Union>n. A n) \<subseteq> (\<Union>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. (\<Union>i\<in>{0..<n}. A i) = (\<Union>i\<in>{0..<n+k}. B i)) \<Longrightarrow> (\<Union>n. A n) = (\<Union>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: 
wenzelm@60758: subsubsection \<open>Cardinality\<close>
paulson@14485: 
nipkow@15045: lemma card_lessThan [simp]: "card {..<u} = u"
paulson@15251:   by (induct u, simp_all add: lessThan_Suc)
paulson@14485: 
paulson@14485: lemma card_atMost [simp]: "card {..u} = Suc u"
paulson@14485:   by (simp add: lessThan_Suc_atMost [THEN sym])
paulson@14485: 
nipkow@15045: lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
lp15@57113: proof -
lp15@57113:   have "{l..<u} = (%x. x + l) ` {..<u-l}"
lp15@57113:     apply (auto simp add: image_def atLeastLessThan_def lessThan_def)
lp15@57113:     apply (rule_tac x = "x - l" in exI)
lp15@57113:     apply arith
lp15@57113:     done
lp15@57113:   then have "card {l..<u} = card {..<u-l}"
lp15@57113:     by (simp add: card_image inj_on_def)
lp15@57113:   then show ?thesis
lp15@57113:     by simp
lp15@57113: qed
paulson@14485: 
paulson@15418: lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
paulson@14485:   by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)
paulson@14485: 
paulson@15418: lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
paulson@14485:   by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)
paulson@14485: 
nipkow@15045: lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
paulson@14485:   by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)
paulson@14485: 
nipkow@26105: lemma ex_bij_betw_nat_finite:
nipkow@26105:   "finite M \<Longrightarrow> \<exists>h. bij_betw h {0..<card M} M"
nipkow@26105: apply(drule finite_imp_nat_seg_image_inj_on)
nipkow@26105: apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
nipkow@26105: done
nipkow@26105: 
nipkow@26105: lemma ex_bij_betw_finite_nat:
nipkow@26105:   "finite M \<Longrightarrow> \<exists>h. bij_betw h M {0..<card M}"
nipkow@26105: by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)
nipkow@26105: 
nipkow@31438: lemma finite_same_card_bij:
nipkow@31438:   "finite A \<Longrightarrow> finite B \<Longrightarrow> card A = card B \<Longrightarrow> 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 \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
nipkow@31438: by (rule finite_same_card_bij) auto
nipkow@31438: 
hoelzl@40703: lemma bij_betw_iff_card:
hoelzl@40703:   assumes FIN: "finite A" and FIN': "finite B"
hoelzl@40703:   shows BIJ: "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"
hoelzl@40703: using assms
hoelzl@40703: proof(auto simp add: bij_betw_same_card)
hoelzl@40703:   assume *: "card A = card B"
hoelzl@40703:   obtain f where "bij_betw f A {0 ..< card A}"
hoelzl@40703:   using FIN ex_bij_betw_finite_nat by blast
hoelzl@40703:   moreover obtain g where "bij_betw g {0 ..< card B} B"
hoelzl@40703:   using FIN' ex_bij_betw_nat_finite by blast
hoelzl@40703:   ultimately have "bij_betw (g o f) A B"
hoelzl@40703:   using * by (auto simp add: bij_betw_trans)
hoelzl@40703:   thus "(\<exists>f. bij_betw f A B)" by blast
hoelzl@40703: qed
hoelzl@40703: 
hoelzl@40703: lemma inj_on_iff_card_le:
hoelzl@40703:   assumes FIN: "finite A" and FIN': "finite B"
hoelzl@40703:   shows "(\<exists>f. inj_on f A \<and> f ` A \<le> B) = (card A \<le> card B)"
hoelzl@40703: proof (safe intro!: card_inj_on_le)
hoelzl@40703:   assume *: "card A \<le> card B"
hoelzl@40703:   obtain f where 1: "inj_on f A" and 2: "f ` A = {0 ..< card A}"
hoelzl@40703:   using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force
hoelzl@40703:   moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g ` {0 ..< card B} = B"
hoelzl@40703:   using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force
hoelzl@40703:   ultimately have "inj_on g (f ` A)" using subset_inj_on[of g _ "f ` A"] * by force
hoelzl@40703:   hence "inj_on (g o f) A" using 1 comp_inj_on by blast
hoelzl@40703:   moreover
hoelzl@40703:   {have "{0 ..< card A} \<le> {0 ..< card B}" using * by force
hoelzl@40703:    with 2 have "f ` A  \<le> {0 ..< card B}" by blast
hoelzl@40703:    hence "(g o f) ` A \<le> B" unfolding comp_def using 3 by force
hoelzl@40703:   }
hoelzl@40703:   ultimately show "(\<exists>f. inj_on f A \<and> f ` A \<le> B)" by blast
hoelzl@40703: qed (insert assms, auto)
nipkow@26105: 
wenzelm@60758: subsection \<open>Intervals of integers\<close>
paulson@14485: 
nipkow@15045: lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
paulson@14485:   by (auto simp add: atLeastAtMost_def atLeastLessThan_def)
paulson@14485: 
paulson@15418: lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
paulson@14485:   by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)
paulson@14485: 
paulson@15418: lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
paulson@15418:     "{l+1..<u} = {l<..<u::int}"
paulson@14485:   by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)
paulson@14485: 
wenzelm@60758: subsubsection \<open>Finiteness\<close>
paulson@14485: 
paulson@15418: lemma image_atLeastZeroLessThan_int: "0 \<le> u ==>
nipkow@15045:     {(0::int)..<u} = int ` {..<nat u}"
paulson@14485:   apply (unfold image_def lessThan_def)
paulson@14485:   apply auto
paulson@14485:   apply (rule_tac x = "nat x" in exI)
huffman@35216:   apply (auto simp add: zless_nat_eq_int_zless [THEN sym])
paulson@14485:   done
paulson@14485: 
nipkow@15045: lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
wenzelm@47988:   apply (cases "0 \<le> 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::int}"
nipkow@15045:   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485:   apply (erule subst)
paulson@14485:   apply (rule finite_imageI)
paulson@14485:   apply (rule finite_atLeastZeroLessThan_int)
nipkow@16733:   apply (rule image_add_int_atLeastLessThan)
paulson@14485:   done
paulson@14485: 
paulson@15418: lemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"
paulson@14485:   by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)
paulson@14485: 
paulson@15418: lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"
paulson@14485:   by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485: 
paulson@15418: lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"
paulson@14485:   by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485: 
nipkow@24853: 
wenzelm@60758: subsubsection \<open>Cardinality\<close>
paulson@14485: 
nipkow@15045: lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"
wenzelm@47988:   apply (cases "0 \<le> 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..<u} = nat (u - l)"
nipkow@15045:   apply (subgoal_tac "card {l..<u} = card {0..<u-l}")
paulson@14485:   apply (erule ssubst, rule card_atLeastZeroLessThan_int)
nipkow@15045:   apply (subgoal_tac "(%x. x + l) ` {0..<u-l} = {l..<u}")
paulson@14485:   apply (erule subst)
paulson@14485:   apply (rule card_image)
paulson@14485:   apply (simp add: inj_on_def)
nipkow@16733:   apply (rule image_add_int_atLeastLessThan)
paulson@14485:   done
paulson@14485: 
paulson@14485: lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
nipkow@29667: apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
nipkow@29667: apply (auto simp add: algebra_simps)
nipkow@29667: done
paulson@14485: 
paulson@15418: lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
nipkow@29667: by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
paulson@14485: 
nipkow@15045: lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
nipkow@29667: by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
paulson@14485: 
bulwahn@27656: lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
bulwahn@27656: proof -
bulwahn@27656:   have "{k. P k \<and> k < i} \<subseteq> {..<i}" by auto
bulwahn@27656:   with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)
bulwahn@27656: qed
bulwahn@27656: 
bulwahn@27656: lemma card_less:
bulwahn@27656: assumes zero_in_M: "0 \<in> M"
bulwahn@27656: shows "card {k \<in> M. k < Suc i} \<noteq> 0"
bulwahn@27656: proof -
bulwahn@27656:   from zero_in_M have "{k \<in> M. k < Suc i} \<noteq> {}" 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 \<notin> M \<Longrightarrow> card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"
haftmann@37388: apply (rule card_bij_eq [of Suc _ _ "\<lambda>x. x - Suc 0"])
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 \<in> M"
bulwahn@27656:     shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
bulwahn@27656: proof -
bulwahn@27656:   from assms have a: "0 \<in> {k \<in> M. k < Suc i}" by simp
bulwahn@27656:   hence c: "{k \<in> M. k < Suc i} = insert 0 ({k \<in> M. k < Suc i} - {0})"
bulwahn@27656:     by (auto simp only: insert_Diff)
bulwahn@27656:   have b: "{k \<in> M. k < Suc i} - {0} = {k \<in> M - {0}. k < Suc i}"  by auto
lp15@57113:   from finite_M_bounded_by_nat[of "\<lambda>x. x \<in> M" "Suc i"] 
lp15@57113:   have "Suc (card {k. Suc k \<in> M \<and> k < i}) = card (insert 0 ({k \<in> 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: 
wenzelm@60758: subsection \<open>Lemmas useful with the summation operator setsum\<close>
paulson@13850: 
wenzelm@60758: text \<open>For examples, see Algebra/poly/UnivPoly2.thy\<close>
ballarin@13735: 
wenzelm@60758: subsubsection \<open>Disjoint Unions\<close>
ballarin@13735: 
wenzelm@60758: text \<open>Singletons and open intervals\<close>
ballarin@13735: 
ballarin@13735: lemma ivl_disj_un_singleton:
nipkow@15045:   "{l::'a::linorder} Un {l<..} = {l..}"
nipkow@15045:   "{..<u} Un {u::'a::linorder} = {..u}"
nipkow@15045:   "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"
nipkow@15045:   "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"
nipkow@15045:   "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"
nipkow@15045:   "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"
ballarin@14398: by auto
ballarin@13735: 
wenzelm@60758: text \<open>One- and two-sided intervals\<close>
ballarin@13735: 
ballarin@13735: lemma ivl_disj_un_one:
nipkow@15045:   "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"
nipkow@15045:   "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"
nipkow@15045:   "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"
nipkow@15045:   "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"
nipkow@15045:   "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"
nipkow@15045:   "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"
nipkow@15045:   "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"
nipkow@15045:   "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"
ballarin@14398: by auto
ballarin@13735: 
wenzelm@60758: text \<open>Two- and two-sided intervals\<close>
ballarin@13735: 
ballarin@13735: lemma ivl_disj_un_two:
nipkow@15045:   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"
nipkow@15045:   "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"
nipkow@15045:   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"
nipkow@15045:   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"
nipkow@15045:   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"
nipkow@15045:   "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"
nipkow@15045:   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"
nipkow@15045:   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"
ballarin@14398: by auto
ballarin@13735: 
lp15@60150: lemma ivl_disj_un_two_touch:
lp15@60150:   "[| (l::'a::linorder) < m; m < u |] ==> {l<..m} Un {m..<u} = {l<..<u}"
lp15@60150:   "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m..<u} = {l..<u}"
lp15@60150:   "[| (l::'a::linorder) < m; m <= u |] ==> {l<..m} Un {m..u} = {l<..u}"
lp15@60150:   "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m..u} = {l..u}"
lp15@60150: by auto
lp15@60150: 
lp15@60150: lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two ivl_disj_un_two_touch
ballarin@13735: 
wenzelm@60758: subsubsection \<open>Disjoint Intersections\<close>
ballarin@13735: 
wenzelm@60758: text \<open>One- and two-sided intervals\<close>
ballarin@13735: 
ballarin@13735: lemma ivl_disj_int_one:
nipkow@15045:   "{..l::'a::order} Int {l<..<u} = {}"
nipkow@15045:   "{..<l} Int {l..<u} = {}"
nipkow@15045:   "{..l} Int {l<..u} = {}"
nipkow@15045:   "{..<l} Int {l..u} = {}"
nipkow@15045:   "{l<..u} Int {u<..} = {}"
nipkow@15045:   "{l<..<u} Int {u..} = {}"
nipkow@15045:   "{l..u} Int {u<..} = {}"
nipkow@15045:   "{l..<u} Int {u..} = {}"
ballarin@14398:   by auto
ballarin@13735: 
wenzelm@60758: text \<open>Two- and two-sided intervals\<close>
ballarin@13735: 
ballarin@13735: lemma ivl_disj_int_two:
nipkow@15045:   "{l::'a::order<..<m} Int {m..<u} = {}"
nipkow@15045:   "{l<..m} Int {m<..<u} = {}"
nipkow@15045:   "{l..<m} Int {m..<u} = {}"
nipkow@15045:   "{l..m} Int {m<..<u} = {}"
nipkow@15045:   "{l<..<m} Int {m..u} = {}"
nipkow@15045:   "{l<..m} Int {m<..u} = {}"
nipkow@15045:   "{l..<m} Int {m..u} = {}"
nipkow@15045:   "{l..m} Int {m<..u} = {}"
ballarin@14398:   by auto
ballarin@13735: 
nipkow@32456: lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two
ballarin@13735: 
wenzelm@60758: subsubsection \<open>Some Differences\<close>
nipkow@15542: 
nipkow@15542: lemma ivl_diff[simp]:
nipkow@15542:  "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
nipkow@15542: by(auto)
nipkow@15542: 
hoelzl@56194: lemma (in linorder) lessThan_minus_lessThan [simp]:
hoelzl@56194:   "{..< n} - {..< m} = {m ..< n}"
hoelzl@56194:   by auto
hoelzl@56194: 
paulson@60762: lemma (in linorder) atLeastAtMost_diff_ends:
paulson@60762:   "{a..b} - {a, b} = {a<..<b}"
paulson@60762:   by auto
paulson@60762: 
nipkow@15542: 
wenzelm@60758: subsubsection \<open>Some Subset Conditions\<close>
nipkow@15542: 
blanchet@54147: lemma ivl_subset [simp]:
nipkow@15542:  "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (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@44890: apply(fastforce)
nipkow@15542: done
nipkow@15542: 
nipkow@15041: 
wenzelm@60758: subsection \<open>Summation indexed over intervals\<close>
nipkow@15042: 
nipkow@15042: syntax
nipkow@15042:   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048:   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052:   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<_./ _)" [0,0,10] 10)
nipkow@16052:   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
nipkow@15042: syntax (xsymbols)
nipkow@15042:   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _.._./ _)" [0,0,0,10] 10)
nipkow@15048:   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_ = _..<_./ _)" [0,0,0,10] 10)
nipkow@16052:   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_<_./ _)" [0,0,10] 10)
nipkow@16052:   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Sum>_\<le>_./ _)" [0,0,10] 10)
nipkow@15056: syntax (latex_sum output)
nipkow@15052:   "_from_to_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052:  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@15052:   "_from_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@15052:  ("(3\<^raw:$\sum_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
nipkow@16052:   "_upt_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052:  ("(3\<^raw:$\sum_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15052:   "_upto_setsum" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
nipkow@16052:  ("(3\<^raw:$\sum_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
nipkow@15041: 
nipkow@15048: translations
nipkow@28853:   "\<Sum>x=a..b. t" == "CONST setsum (%x. t) {a..b}"
nipkow@28853:   "\<Sum>x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"
nipkow@28853:   "\<Sum>i\<le>n. t" == "CONST setsum (\<lambda>i. t) {..n}"
nipkow@28853:   "\<Sum>i<n. t" == "CONST setsum (\<lambda>i. t) {..<n}"
nipkow@15041: 
wenzelm@60758: text\<open>The above introduces some pretty alternative syntaxes for
nipkow@15056: summation over intervals:
nipkow@15052: \begin{center}
nipkow@15052: \begin{tabular}{lll}
nipkow@15056: Old & New & \LaTeX\\
nipkow@15056: @{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: @{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: @{term[source]"\<Sum>x\<in>{..b}. e"} & @{term"\<Sum>x\<le>b. e"} & @{term[mode=latex_sum]"\<Sum>x\<le>b. e"}\\
nipkow@15056: @{term[source]"\<Sum>x\<in>{..<b}. e"} & @{term"\<Sum>x<b. e"} & @{term[mode=latex_sum]"\<Sum>x<b. e"}
nipkow@15052: \end{tabular}
nipkow@15052: \end{center}
nipkow@15056: The left column shows the term before introduction of the new syntax,
nipkow@15056: the middle column shows the new (default) syntax, and the right column
nipkow@15056: shows a special syntax. The latter is only meaningful for latex output
nipkow@15056: and has to be activated explicitly by setting the print mode to
wenzelm@61799: \<open>latex_sum\<close> (e.g.\ via \<open>mode = latex_sum\<close> in
nipkow@15056: antiquotations). It is not the default \LaTeX\ output because it only
nipkow@15056: works well with italic-style formulae, not tt-style.
nipkow@15052: 
nipkow@15052: Note that for uniformity on @{typ nat} it is better to use
wenzelm@61799: @{term"\<Sum>x::nat=0..<n. e"} rather than \<open>\<Sum>x<n. e\<close>: \<open>setsum\<close> may
nipkow@15052: not provide all lemmas available for @{term"{m..<n}"} also in the
wenzelm@60758: special form for @{term"{..<n}"}.\<close>
nipkow@15052: 
wenzelm@60758: text\<open>This congruence rule should be used for sums over intervals as
haftmann@57418: the standard theorem @{text[source]setsum.cong} does not work well
nipkow@15542: with the simplifier who adds the unsimplified premise @{term"x:B"} to
wenzelm@60758: the context.\<close>
nipkow@15542: 
nipkow@15542: lemma setsum_ivl_cong:
nipkow@15542:  "\<lbrakk>a = c; b = d; !!x. \<lbrakk> c \<le> x; x < d \<rbrakk> \<Longrightarrow> f x = g x \<rbrakk> \<Longrightarrow>
nipkow@15542:  setsum f {a..<b} = setsum g {c..<d}"
haftmann@57418: by(rule setsum.cong, simp_all)
nipkow@15041: 
nipkow@16041: (* FIXME why are the following simp rules but the corresponding eqns
nipkow@16041: on intervals are not? *)
nipkow@16041: 
nipkow@16052: lemma setsum_atMost_Suc[simp]: "(\<Sum>i \<le> Suc n. f i) = (\<Sum>i \<le> n. f i) + f(Suc n)"
haftmann@57514: by (simp add:atMost_Suc ac_simps)
nipkow@16052: 
nipkow@16041: lemma setsum_lessThan_Suc[simp]: "(\<Sum>i < Suc n. f i) = (\<Sum>i < n. f i) + f n"
haftmann@57514: by (simp add:lessThan_Suc ac_simps)
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))"
haftmann@57514: by (auto simp:ac_simps atLeastAtMostSuc_conv)
nipkow@15561: 
nipkow@15911: lemma setsum_op_ivl_Suc[simp]:
nipkow@15561:   "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
haftmann@57514: by (auto simp:ac_simps atLeastLessThanSuc)
nipkow@16041: (*
nipkow@15561: lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>
nipkow@15561:     (\<Sum>i=n..m+1. f i) = (\<Sum>i=n..m. f i) + f(m + 1)"
haftmann@57514: by (auto simp:ac_simps 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 "(\<Sum>x\<in>{m..n}. P x) = P m + (\<Sum>x\<in>{m<..n}. P x)" (is "?lhs = ?rhs")
nipkow@28068: proof -
nipkow@28068:   from mn
nipkow@28068:   have "{m..n} = {m} \<union> {m<..n}"
nipkow@28068:     by (auto intro: ivl_disj_un_singleton)
nipkow@28068:   hence "?lhs = (\<Sum>x\<in>{m} \<union> {m<..n}. P x)"
nipkow@28068:     by (simp add: atLeast0LessThan)
nipkow@28068:   also have "\<dots> = ?rhs" by simp
nipkow@28068:   finally show ?thesis .
nipkow@28068: qed
nipkow@28068: 
nipkow@28068: lemma setsum_head_Suc:
nipkow@28068:   "m \<le> n \<Longrightarrow> 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 \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
huffman@30079: apply(insert setsum_head_Suc[of m "n - Suc 0" f])
nipkow@29667: apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
nipkow@28068: done
nipkow@28068: 
nipkow@31501: lemma setsum_ub_add_nat: assumes "(m::nat) \<le> n + 1"
nipkow@31501:   shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
nipkow@31501: proof-
wenzelm@60758:   have "{m .. n+p} = {m..n} \<union> {n+1..n+p}" using \<open>m \<le> n+1\<close> by auto
haftmann@57418:   thus ?thesis by (auto simp: ivl_disj_int setsum.union_disjoint
nipkow@31501:     atLeastSucAtMost_greaterThanAtMost)
nipkow@31501: qed
nipkow@28068: 
nipkow@15539: lemma setsum_add_nat_ivl: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539:   setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
haftmann@57418: by (simp add:setsum.union_disjoint[symmetric] ivl_disj_int ivl_disj_un)
nipkow@15539: 
nipkow@15539: lemma setsum_diff_nat_ivl:
nipkow@15539: fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
nipkow@15539: shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15539:   setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
nipkow@15539: using setsum_add_nat_ivl [of m n p f,symmetric]
haftmann@57514: apply (simp add: ac_simps)
nipkow@15539: done
nipkow@15539: 
nipkow@31505: lemma setsum_natinterval_difff:
nipkow@31505:   fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
nipkow@31505:   shows  "setsum (\<lambda>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: 
hoelzl@56194: lemma setsum_nat_group: "(\<Sum>m<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {..< n * k}"
hoelzl@56194:   apply (subgoal_tac "k = 0 | 0 < k", auto)
hoelzl@56194:   apply (induct "n")
haftmann@57512:   apply (simp_all add: setsum_add_nat_ivl add.commute atLeast0LessThan[symmetric])
hoelzl@56194:   done
nipkow@28068: 
lp15@60150: lemma setsum_triangle_reindex:
lp15@60150:   fixes n :: nat
lp15@60150:   shows "(\<Sum>(i,j)\<in>{(i,j). i+j < n}. f i j) = (\<Sum>k<n. \<Sum>i\<le>k. f i (k - i))"
lp15@60150:   apply (simp add: setsum.Sigma)
lp15@60150:   apply (rule setsum.reindex_bij_witness[where j="\<lambda>(i, j). (i+j, i)" and i="\<lambda>(k, i). (i, k - i)"])
lp15@60150:   apply auto
lp15@60150:   done
lp15@60150: 
lp15@60150: lemma setsum_triangle_reindex_eq:
lp15@60150:   fixes n :: nat
lp15@60150:   shows "(\<Sum>(i,j)\<in>{(i,j). i+j \<le> n}. f i j) = (\<Sum>k\<le>n. \<Sum>i\<le>k. f i (k - i))"
lp15@60150: using setsum_triangle_reindex [of f "Suc n"]
lp15@60150: by (simp only: Nat.less_Suc_eq_le lessThan_Suc_atMost)
lp15@60150: 
lp15@60162: lemma nat_diff_setsum_reindex: "(\<Sum>i<n. f (n - Suc i)) = (\<Sum>i<n. f i)"
lp15@60162:   by (rule setsum.reindex_bij_witness[where i="\<lambda>i. n - Suc i" and j="\<lambda>i. n - Suc i"]) auto
lp15@60162: 
wenzelm@60758: subsection\<open>Shifting bounds\<close>
nipkow@16733: 
nipkow@15539: lemma setsum_shift_bounds_nat_ivl:
nipkow@15539:   "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
nipkow@15539: by (induct "n", auto simp:atLeastLessThanSuc)
nipkow@15539: 
nipkow@16733: lemma setsum_shift_bounds_cl_nat_ivl:
nipkow@16733:   "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"
hoelzl@57129:   by (rule setsum.reindex_bij_witness[where i="\<lambda>i. i + k" and j="\<lambda>i. i - k"]) auto
nipkow@16733: 
nipkow@16733: corollary setsum_shift_bounds_cl_Suc_ivl:
nipkow@16733:   "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"
huffman@30079: by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
nipkow@16733: 
nipkow@16733: corollary setsum_shift_bounds_Suc_ivl:
nipkow@16733:   "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"
huffman@30079: by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])
nipkow@16733: 
nipkow@28068: lemma setsum_shift_lb_Suc0_0:
nipkow@28068:   "f(0::nat) = (0::nat) \<Longrightarrow> 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 \<Longrightarrow> setsum f {Suc 0..<k} = setsum f {0..<k}"
nipkow@28068: apply(cases k)apply simp
nipkow@28068: apply(simp add:setsum_head_upt_Suc)
nipkow@28068: done
kleing@19022: 
haftmann@52380: lemma setsum_atMost_Suc_shift:
haftmann@52380:   fixes f :: "nat \<Rightarrow> 'a::comm_monoid_add"
haftmann@52380:   shows "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
haftmann@52380: proof (induct n)
haftmann@52380:   case 0 show ?case by simp
haftmann@52380: next
haftmann@52380:   case (Suc n) note IH = this
haftmann@52380:   have "(\<Sum>i\<le>Suc (Suc n). f i) = (\<Sum>i\<le>Suc n. f i) + f (Suc (Suc n))"
haftmann@52380:     by (rule setsum_atMost_Suc)
haftmann@52380:   also have "(\<Sum>i\<le>Suc n. f i) = f 0 + (\<Sum>i\<le>n. f (Suc i))"
haftmann@52380:     by (rule IH)
haftmann@52380:   also have "f 0 + (\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) =
haftmann@52380:              f 0 + ((\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)))"
haftmann@57512:     by (rule add.assoc)
haftmann@52380:   also have "(\<Sum>i\<le>n. f (Suc i)) + f (Suc (Suc n)) = (\<Sum>i\<le>Suc n. f (Suc i))"
haftmann@52380:     by (rule setsum_atMost_Suc [symmetric])
haftmann@52380:   finally show ?case .
haftmann@52380: qed
haftmann@52380: 
lp15@56238: lemma setsum_last_plus: fixes n::nat shows "m <= n \<Longrightarrow> (\<Sum>i = m..n. f i) = f n + (\<Sum>i = m..<n. f i)"
haftmann@57512:   by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost add.commute)
lp15@56238: 
lp15@56238: lemma setsum_Suc_diff:
lp15@56238:   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
lp15@56238:   assumes "m \<le> Suc n"
lp15@56238:   shows "(\<Sum>i = m..n. f(Suc i) - f i) = f (Suc n) - f m"
lp15@56238: using assms by (induct n) (auto simp: le_Suc_eq)
lp15@55718: 
lp15@55718: lemma nested_setsum_swap:
lp15@55718:      "(\<Sum>i = 0..n. (\<Sum>j = 0..<i. a i j)) = (\<Sum>j = 0..<n. \<Sum>i = Suc j..n. a i j)"
haftmann@57418:   by (induction n) (auto simp: setsum.distrib)
lp15@55718: 
lp15@56215: lemma nested_setsum_swap':
lp15@56215:      "(\<Sum>i\<le>n. (\<Sum>j<i. a i j)) = (\<Sum>j<n. \<Sum>i = Suc j..n. a i j)"
haftmann@57418:   by (induction n) (auto simp: setsum.distrib)
lp15@56215: 
lp15@56238: lemma setsum_zero_power' [simp]:
lp15@56238:   fixes c :: "nat \<Rightarrow> 'a::field"
lp15@56238:   shows "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"
lp15@56238:   using setsum_zero_power [of "\<lambda>i. c i / d i" A]
lp15@56238:   by auto
lp15@56238: 
haftmann@52380: 
eberlm@61524: subsection \<open>Telescoping\<close>
eberlm@61524: 
eberlm@61524: lemma setsum_telescope:
eberlm@61524:   fixes f::"nat \<Rightarrow> 'a::ab_group_add"
eberlm@61524:   shows "setsum (\<lambda>i. f i - f (Suc i)) {.. i} = f 0 - f (Suc i)"
eberlm@61524:   by (induct i) simp_all
eberlm@61524: 
eberlm@61524: lemma setsum_telescope'':
eberlm@61524:   assumes "m \<le> n"
eberlm@61524:   shows   "(\<Sum>k\<in>{Suc m..n}. f k - f (k - 1)) = f n - (f m :: 'a :: ab_group_add)"
eberlm@61524:   by (rule dec_induct[OF assms]) (simp_all add: algebra_simps)
eberlm@61524: 
wenzelm@60758: subsection \<open>The formula for geometric sums\<close>
ballarin@17149: 
ballarin@17149: lemma geometric_sum:
haftmann@36307:   assumes "x \<noteq> 1"
hoelzl@56193:   shows "(\<Sum>i<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"
haftmann@36307: proof -
haftmann@36307:   from assms obtain y where "y = x - 1" and "y \<noteq> 0" by simp_all
hoelzl@56193:   moreover have "(\<Sum>i<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"
wenzelm@60758:     by (induct n) (simp_all add: power_Suc field_simps \<open>y \<noteq> 0\<close>)
haftmann@36307:   ultimately show ?thesis by simp
haftmann@36307: qed
haftmann@36307: 
lp15@60162: lemma diff_power_eq_setsum:
lp15@60162:   fixes y :: "'a::{comm_ring,monoid_mult}"
lp15@60162:   shows
lp15@60162:     "x ^ (Suc n) - y ^ (Suc n) =
lp15@60162:       (x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))"
lp15@60162: proof (induct n)
lp15@60162:   case (Suc n)
lp15@60162:   have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x^n) - y * (y * y ^ n)"
lp15@60162:     by (simp add: power_Suc)
lp15@60162:   also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x^n)"
lp15@60162:     by (simp add: power_Suc algebra_simps)
lp15@60162:   also have "... = y * ((x - y) * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
lp15@60162:     by (simp only: Suc)
lp15@60162:   also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
lp15@60162:     by (simp only: mult.left_commute)
lp15@60162:   also have "... = (x - y) * (\<Sum>p<Suc (Suc n). x ^ p * y ^ (Suc n - p))"
lp15@60162:     by (simp add: power_Suc field_simps Suc_diff_le setsum_left_distrib setsum_right_distrib)
lp15@60162:   finally show ?case .
lp15@60162: qed simp
lp15@60162: 
wenzelm@61799: corollary power_diff_sumr2: \<comment>\<open>\<open>COMPLEX_POLYFUN\<close> in HOL Light\<close>
lp15@60162:   fixes x :: "'a::{comm_ring,monoid_mult}"
lp15@60162:   shows   "x^n - y^n = (x - y) * (\<Sum>i<n. y^(n - Suc i) * x^i)"
lp15@60162: using diff_power_eq_setsum[of x "n - 1" y]
lp15@60162: by (cases "n = 0") (simp_all add: field_simps)
lp15@60162: 
lp15@60162: lemma power_diff_1_eq:
lp15@60162:   fixes x :: "'a::{comm_ring,monoid_mult}"
lp15@60162:   shows "n \<noteq> 0 \<Longrightarrow> x^n - 1 = (x - 1) * (\<Sum>i<n. (x^i))"
lp15@60162: using diff_power_eq_setsum [of x _ 1]
lp15@60162:   by (cases n) auto
lp15@60162: 
lp15@60162: lemma one_diff_power_eq':
lp15@60162:   fixes x :: "'a::{comm_ring,monoid_mult}"
lp15@60162:   shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^(n - Suc i))"
lp15@60162: using diff_power_eq_setsum [of 1 _ x]
lp15@60162:   by (cases n) auto
lp15@60162: 
lp15@60162: lemma one_diff_power_eq:
lp15@60162:   fixes x :: "'a::{comm_ring,monoid_mult}"
lp15@60162:   shows "n \<noteq> 0 \<Longrightarrow> 1 - x^n = (1 - x) * (\<Sum>i<n. x^i)"
lp15@60162: by (metis one_diff_power_eq' [of n x] nat_diff_setsum_reindex)
lp15@60162: 
ballarin@17149: 
wenzelm@60758: subsection \<open>The formula for arithmetic sums\<close>
kleing@19469: 
huffman@47222: lemma gauss_sum:
hoelzl@56193:   "(2::'a::comm_semiring_1)*(\<Sum>i\<in>{1..n}. of_nat i) = 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)
huffman@47222:   then show ?case
huffman@47222:     by (simp add: algebra_simps add: one_add_one [symmetric] del: one_add_one)
huffman@47222:       (* FIXME: make numeral cancellation simprocs work for semirings *)
kleing@19469: qed
kleing@19469: 
kleing@19469: theorem arith_series_general:
huffman@47222:   "(2::'a::comm_semiring_1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
kleing@19469:   of_nat n * (a + (a + of_nat(n - 1)*d))"
kleing@19469: proof cases
kleing@19469:   assume ngt1: "n > 1"
kleing@19469:   let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
kleing@19469:   have
kleing@19469:     "(\<Sum>i\<in>{..<n}. a+?I i*d) =
kleing@19469:      ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
haftmann@57418:     by (rule setsum.distrib)
kleing@19469:   also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
kleing@19469:   also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
huffman@30079:     unfolding One_nat_def
haftmann@57514:     by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt ac_simps)
huffman@47222:   also have "2*\<dots> = 2*?n*a + d*2*(\<Sum>i\<in>{1..<n}. ?I i)"
huffman@47222:     by (simp add: algebra_simps)
kleing@19469:   also from ngt1 have "{1..<n} = {1..n - 1}"
nipkow@28068:     by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
nipkow@28068:   also from ngt1
huffman@47222:   have "2*?n*a + d*2*(\<Sum>i\<in>{1..n - 1}. ?I i) = (2*?n*a + d*?I (n - 1)*?I n)"
haftmann@57514:     by (simp only: mult.assoc gauss_sum [of "n - 1"], unfold One_nat_def)
haftmann@57514:       (simp add:  mult.commute trans [OF add.commute of_nat_Suc [symmetric]])
huffman@47222:   finally show ?thesis
huffman@47222:     unfolding mult_2 by (simp add: algebra_simps)
kleing@19469: next
kleing@19469:   assume "\<not>(n > 1)"
kleing@19469:   hence "n = 1 \<or> n = 0" by auto
huffman@47222:   thus ?thesis by (auto simp: mult_2)
kleing@19469: qed
kleing@19469: 
kleing@19469: lemma arith_series_nat:
huffman@47222:   "(2::nat) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
kleing@19469: proof -
kleing@19469:   have
huffman@47222:     "2 * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
kleing@19469:     of_nat(n) * (a + (a + of_nat(n - 1)*d))"
kleing@19469:     by (rule arith_series_general)
huffman@30079:   thus ?thesis
huffman@35216:     unfolding One_nat_def by auto
kleing@19469: qed
kleing@19469: 
kleing@19469: lemma arith_series_int:
huffman@47222:   "2 * (\<Sum>i\<in>{..<n}. a + int i * d) = int n * (a + (a + int(n - 1)*d))"
huffman@47222:   by (fact arith_series_general) (* FIXME: duplicate *)
paulson@15418: 
hoelzl@59416: lemma sum_diff_distrib: "\<forall>x. Q x \<le> P x  \<Longrightarrow> (\<Sum>x<n. P x) - (\<Sum>x<n. Q x) = (\<Sum>x<n. P x - Q x :: nat)"
hoelzl@59416:   by (subst setsum_subtractf_nat) auto
kleing@19022: 
wenzelm@60758: subsection \<open>Products indexed over intervals\<close>
paulson@29960: 
paulson@29960: syntax
paulson@29960:   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
paulson@29960:   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960:   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<_./ _)" [0,0,10] 10)
paulson@29960:   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
paulson@29960: syntax (xsymbols)
paulson@29960:   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _.._./ _)" [0,0,0,10] 10)
paulson@29960:   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_ = _..<_./ _)" [0,0,0,10] 10)
paulson@29960:   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_<_./ _)" [0,0,10] 10)
paulson@29960:   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b" ("(3\<Prod>_\<le>_./ _)" [0,0,10] 10)
paulson@29960: syntax (latex_prod output)
paulson@29960:   "_from_to_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960:  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960:   "_from_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960:  ("(3\<^raw:$\prod_{>_ = _\<^raw:}^{<>_\<^raw:}$> _)" [0,0,0,10] 10)
paulson@29960:   "_upt_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960:  ("(3\<^raw:$\prod_{>_ < _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960:   "_upto_setprod" :: "idt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"
paulson@29960:  ("(3\<^raw:$\prod_{>_ \<le> _\<^raw:}$> _)" [0,0,10] 10)
paulson@29960: 
paulson@29960: translations
paulson@29960:   "\<Prod>x=a..b. t" == "CONST setprod (%x. t) {a..b}"
paulson@29960:   "\<Prod>x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"
paulson@29960:   "\<Prod>i\<le>n. t" == "CONST setprod (\<lambda>i. t) {..n}"
paulson@29960:   "\<Prod>i<n. t" == "CONST setprod (\<lambda>i. t) {..<n}"
paulson@29960: 
wenzelm@60758: subsection \<open>Transfer setup\<close>
haftmann@33318: 
haftmann@33318: lemma transfer_nat_int_set_functions:
haftmann@33318:     "{..n} = nat ` {0..int n}"
haftmann@33318:     "{m..n} = nat ` {int m..int n}"  (* need all variants of these! *)
haftmann@33318:   apply (auto simp add: image_def)
haftmann@33318:   apply (rule_tac x = "int x" in bexI)
haftmann@33318:   apply auto
haftmann@33318:   apply (rule_tac x = "int x" in bexI)
haftmann@33318:   apply auto
haftmann@33318:   done
haftmann@33318: 
haftmann@33318: lemma transfer_nat_int_set_function_closures:
haftmann@33318:     "x >= 0 \<Longrightarrow> 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 \<Longrightarrow> is_nat n \<Longrightarrow> {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 \<Longrightarrow> 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: 
lp15@55242: lemma setprod_int_plus_eq: "setprod int {i..i+j} =  \<Prod>{int i..int (i+j)}"
lp15@55242:   by (induct j) (auto simp add: atLeastAtMostSuc_conv atLeastAtMostPlus1_int_conv)
lp15@55242: 
lp15@55242: lemma setprod_int_eq: "setprod int {i..j} =  \<Prod>{int i..int j}"
lp15@55242: proof (cases "i \<le> j")
lp15@55242:   case True
lp15@55242:   then show ?thesis
lp15@55242:     by (metis Nat.le_iff_add setprod_int_plus_eq)
lp15@55242: next
lp15@55242:   case False
lp15@55242:   then show ?thesis
lp15@55242:     by auto
lp15@55242: qed
lp15@55242: 
eberlm@61524: 
eberlm@61524: subsection \<open>Shifting bounds\<close>
eberlm@61524: 
eberlm@61524: lemma setprod_shift_bounds_nat_ivl:
eberlm@61524:   "setprod f {m+k..<n+k} = setprod (%i. f(i + k)){m..<n::nat}"
eberlm@61524: by (induct "n", auto simp:atLeastLessThanSuc)
eberlm@61524: 
eberlm@61524: lemma setprod_shift_bounds_cl_nat_ivl:
eberlm@61524:   "setprod f {m+k..n+k} = setprod (%i. f(i + k)){m..n::nat}"
eberlm@61524:   by (rule setprod.reindex_bij_witness[where i="\<lambda>i. i + k" and j="\<lambda>i. i - k"]) auto
eberlm@61524: 
eberlm@61524: corollary setprod_shift_bounds_cl_Suc_ivl:
eberlm@61524:   "setprod f {Suc m..Suc n} = setprod (%i. f(Suc i)){m..n}"
eberlm@61524: by (simp add:setprod_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
eberlm@61524: 
eberlm@61524: corollary setprod_shift_bounds_Suc_ivl:
eberlm@61524:   "setprod f {Suc m..<Suc n} = setprod (%i. f(Suc i)){m..<n}"
eberlm@61524: by (simp add:setprod_shift_bounds_nat_ivl[where k="Suc 0", simplified])
eberlm@61524: 
eberlm@61524: lemma setprod_lessThan_Suc: "setprod f {..<Suc n} = setprod f {..<n} * f n"
eberlm@61524:   by (simp add: lessThan_Suc mult.commute)
eberlm@61524: 
eberlm@61524: lemma setprod_atLeastLessThan_Suc: "a \<le> b \<Longrightarrow> setprod f {a..<Suc b} = setprod f {a..<b} * f b"
eberlm@61524:   by (simp add: atLeastLessThanSuc mult.commute)
eberlm@61524: 
eberlm@61524: lemma setprod_nat_ivl_Suc':
eberlm@61524:   assumes "m \<le> Suc n"
eberlm@61524:   shows   "setprod f {m..Suc n} = f (Suc n) * setprod f {m..n}"
eberlm@61524: proof -
eberlm@61524:   from assms have "{m..Suc n} = insert (Suc n) {m..n}" by auto
eberlm@61524:   also have "setprod f \<dots> = f (Suc n) * setprod f {m..n}" by simp
eberlm@61524:   finally show ?thesis .
eberlm@61524: qed
eberlm@61524: 
nipkow@8924: end