# Theory Set_Interval

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theory Set_Interval
imports Nat_Transfer
(*  Title:      HOL/Set_Interval.thy    Author:     Tobias Nipkow    Author:     Clemens Ballarin    Author:     Jeremy AvigadlessThan, greaterThan, atLeast, atMost and two-sided intervals*)header {* Set intervals *}theory Set_Intervalimports Int Nat_Transferbegincontext ordbegindefinition  lessThan    :: "'a => 'a set" ("(1{..<_})") where  "{..<u} == {x. x < u}"definition  atMost      :: "'a => 'a set" ("(1{.._})") where  "{..u} == {x. x ≤ u}"definition  greaterThan :: "'a => 'a set" ("(1{_<..})") where  "{l<..} == {x. l<x}"definition  atLeast     :: "'a => 'a set" ("(1{_..})") where  "{l..} == {x. l≤x}"definition  greaterThanLessThan :: "'a => 'a => 'a set"  ("(1{_<..<_})") where  "{l<..<u} == {l<..} Int {..<u}"definition  atLeastLessThan :: "'a => 'a => 'a set"      ("(1{_..<_})") where  "{l..<u} == {l..} Int {..<u}"definition  greaterThanAtMost :: "'a => 'a => 'a set"    ("(1{_<.._})") where  "{l<..u} == {l<..} Int {..u}"definition  atLeastAtMost :: "'a => 'a => 'a set"        ("(1{_.._})") where  "{l..u} == {l..} Int {..u}"endtext{* A note of warning when using @{term"{..<n}"} on type @{typnat}: it is equivalent to @{term"{0::nat..<n}"} but some lemmas involving@{term"{m..<n}"} may not exist in @{term"{..<n}"}-form as well. *}syntax  "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3UN _<=_./ _)" [0, 0, 10] 10)  "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3UN _<_./ _)" [0, 0, 10] 10)  "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" [0, 0, 10] 10)  "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3INT _<_./ _)" [0, 0, 10] 10)syntax (xsymbols)  "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _≤_./ _)" [0, 0, 10] 10)  "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union> _<_./ _)" [0, 0, 10] 10)  "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _≤_./ _)" [0, 0, 10] 10)  "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter> _<_./ _)" [0, 0, 10] 10)syntax (latex output)  "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Union>(00_ ≤ _)/ _)" [0, 0, 10] 10)  "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Union>(00_ < _)/ _)" [0, 0, 10] 10)  "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter>(00_ ≤ _)/ _)" [0, 0, 10] 10)  "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3\<Inter>(00_ < _)/ _)" [0, 0, 10] 10)translations  "UN i<=n. A"  == "UN i:{..n}. A"  "UN i<n. A"   == "UN i:{..<n}. A"  "INT i<=n. A" == "INT i:{..n}. A"  "INT i<n. A"  == "INT i:{..<n}. A"subsection {* Various equivalences *}lemma (in ord) lessThan_iff [iff]: "(i: lessThan k) = (i<k)"by (simp add: lessThan_def)lemma Compl_lessThan [simp]:    "!!k:: 'a::linorder. -lessThan k = atLeast k"apply (auto simp add: lessThan_def atLeast_def)donelemma single_Diff_lessThan [simp]: "!!k:: 'a::order. {k} - lessThan k = {k}"by autolemma (in ord) greaterThan_iff [iff]: "(i: greaterThan k) = (k<i)"by (simp add: greaterThan_def)lemma Compl_greaterThan [simp]:    "!!k:: 'a::linorder. -greaterThan k = atMost k"  by (auto simp add: greaterThan_def atMost_def)lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"apply (subst Compl_greaterThan [symmetric])apply (rule double_complement)donelemma (in ord) atLeast_iff [iff]: "(i: atLeast k) = (k<=i)"by (simp add: atLeast_def)lemma Compl_atLeast [simp]:    "!!k:: 'a::linorder. -atLeast k = lessThan k"  by (auto simp add: lessThan_def atLeast_def)lemma (in ord) atMost_iff [iff]: "(i: atMost k) = (i<=k)"by (simp add: atMost_def)lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"by (blast intro: order_antisym)subsection {* Logical Equivalences for Set Inclusion and Equality *}lemma atLeast_subset_iff [iff]:     "(atLeast x ⊆ atLeast y) = (y ≤ (x::'a::order))"by (blast intro: order_trans)lemma atLeast_eq_iff [iff]:     "(atLeast x = atLeast y) = (x = (y::'a::linorder))"by (blast intro: order_antisym order_trans)lemma greaterThan_subset_iff [iff]:     "(greaterThan x ⊆ greaterThan y) = (y ≤ (x::'a::linorder))"apply (auto simp add: greaterThan_def) apply (subst linorder_not_less [symmetric], blast)donelemma greaterThan_eq_iff [iff]:     "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"apply (rule iffI) apply (erule equalityE) apply simp_alldonelemma atMost_subset_iff [iff]: "(atMost x ⊆ atMost y) = (x ≤ (y::'a::order))"by (blast intro: order_trans)lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::linorder))"by (blast intro: order_antisym order_trans)lemma lessThan_subset_iff [iff]:     "(lessThan x ⊆ lessThan y) = (x ≤ (y::'a::linorder))"apply (auto simp add: lessThan_def) apply (subst linorder_not_less [symmetric], blast)donelemma lessThan_eq_iff [iff]:     "(lessThan x = lessThan y) = (x = (y::'a::linorder))"apply (rule iffI) apply (erule equalityE) apply simp_alldonelemma lessThan_strict_subset_iff:  fixes m n :: "'a::linorder"  shows "{..<m} < {..<n} <-> m < n"  by (metis leD lessThan_subset_iff linorder_linear not_less_iff_gr_or_eq psubset_eq)subsection {*Two-sided intervals*}context ordbeginlemma greaterThanLessThan_iff [simp,no_atp]:  "(i : {l<..<u}) = (l < i & i < u)"by (simp add: greaterThanLessThan_def)lemma atLeastLessThan_iff [simp,no_atp]:  "(i : {l..<u}) = (l <= i & i < u)"by (simp add: atLeastLessThan_def)lemma greaterThanAtMost_iff [simp,no_atp]:  "(i : {l<..u}) = (l < i & i <= u)"by (simp add: greaterThanAtMost_def)lemma atLeastAtMost_iff [simp,no_atp]:  "(i : {l..u}) = (l <= i & i <= u)"by (simp add: atLeastAtMost_def)text {* The above four lemmas could be declared as iffs. Unfortunately thisbreaks many proofs. Since it only helps blast, it is better to leave wellalone *}endsubsubsection{* Emptyness, singletons, subset *}context orderbeginlemma atLeastatMost_empty[simp]:  "b < a ==> {a..b} = {}"by(auto simp: atLeastAtMost_def atLeast_def atMost_def)lemma atLeastatMost_empty_iff[simp]:  "{a..b} = {} <-> (~ a <= b)"by auto (blast intro: order_trans)lemma atLeastatMost_empty_iff2[simp]:  "{} = {a..b} <-> (~ a <= b)"by auto (blast intro: order_trans)lemma atLeastLessThan_empty[simp]:  "b <= a ==> {a..<b} = {}"by(auto simp: atLeastLessThan_def)lemma atLeastLessThan_empty_iff[simp]:  "{a..<b} = {} <-> (~ a < b)"by auto (blast intro: le_less_trans)lemma atLeastLessThan_empty_iff2[simp]:  "{} = {a..<b} <-> (~ a < b)"by auto (blast intro: le_less_trans)lemma greaterThanAtMost_empty[simp]: "l ≤ k ==> {k<..l} = {}"by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {} <-> ~ k < l"by auto (blast intro: less_le_trans)lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l} <-> ~ k < l"by auto (blast intro: less_le_trans)lemma greaterThanLessThan_empty[simp]:"l ≤ k ==> {k<..<l} = {}"by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)lemma atLeastAtMost_singleton': "a = b ==> {a .. b} = {a}" by simplemma atLeastatMost_subset_iff[simp]:  "{a..b} <= {c..d} <-> (~ a <= b) | c <= a & b <= d"unfolding atLeastAtMost_def atLeast_def atMost_defby (blast intro: order_trans)lemma atLeastatMost_psubset_iff:  "{a..b} < {c..d} <->   ((~ a <= b) | c <= a & b <= d & (c < a | b < d))  &  c <= d"by(simp add: psubset_eq set_eq_iff less_le_not_le)(blast intro: order_trans)lemma atLeastAtMost_singleton_iff[simp]:  "{a .. b} = {c} <-> a = b ∧ b = c"proof  assume "{a..b} = {c}"  hence "¬ (¬ a ≤ b)" unfolding atLeastatMost_empty_iff[symmetric] by simp  moreover with {a..b} = {c} have "c ≤ a ∧ b ≤ c" by auto  ultimately show "a = b ∧ b = c" by autoqed simpendcontext dense_linorderbeginlemma greaterThanLessThan_empty_iff[simp]:  "{ a <..< b } = {} <-> b ≤ a"  using dense[of a b] by (cases "a < b") autolemma greaterThanLessThan_empty_iff2[simp]:  "{} = { a <..< b } <-> b ≤ a"  using dense[of a b] by (cases "a < b") autolemma atLeastLessThan_subseteq_atLeastAtMost_iff:  "{a ..< b} ⊆ { c .. d } <-> (a < b --> c ≤ a ∧ b ≤ d)"  using dense[of "max a d" "b"]  by (force simp: subset_eq Ball_def not_less[symmetric])lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:  "{a <.. b} ⊆ { c .. d } <-> (a < b --> c ≤ a ∧ b ≤ d)"  using dense[of "a" "min c b"]  by (force simp: subset_eq Ball_def not_less[symmetric])lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:  "{a <..< b} ⊆ { c .. d } <-> (a < b --> c ≤ a ∧ b ≤ d)"  using dense[of "a" "min c b"] dense[of "max a d" "b"]  by (force simp: subset_eq Ball_def not_less[symmetric])lemma atLeastAtMost_subseteq_atLeastLessThan_iff:  "{a .. b} ⊆ { c ..< d } <-> (a ≤ b --> c ≤ a ∧ b < d)"  using dense[of "max a d" "b"]  by (force simp: subset_eq Ball_def not_less[symmetric])lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:  "{a <.. b} ⊆ { c ..< d } <-> (a < b --> c ≤ a ∧ b < d)"  using dense[of "a" "min c b"]  by (force simp: subset_eq Ball_def not_less[symmetric])lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:  "{a <..< b} ⊆ { c ..< d } <-> (a < b --> c ≤ a ∧ b ≤ d)"  using dense[of "a" "min c b"] dense[of "max a d" "b"]  by (force simp: subset_eq Ball_def not_less[symmetric])endlemma (in linorder) atLeastLessThan_subset_iff:  "{a..<b} <= {c..<d} ==> b <= a | c<=a & b<=d"apply (auto simp:subset_eq Ball_def)apply(frule_tac x=a in spec)apply(erule_tac x=d in allE)apply (simp add: less_imp_le)donelemma atLeastLessThan_inj:  fixes a b c d :: "'a::linorder"  assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"  shows "a = c" "b = d"using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le linorder_antisym_conv2 subset_refl)+lemma atLeastLessThan_eq_iff:  fixes a b c d :: "'a::linorder"  assumes "a < b" "c < d"  shows "{a ..< b} = {c ..< d} <-> a = c ∧ b = d"  using atLeastLessThan_inj assms by autosubsubsection {* Intersection *}context linorderbeginlemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"by autolemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"by autolemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"by autolemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"by autolemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"by autolemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"by autolemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"by autolemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"by autolemma Int_atMost[simp]: "{..a} ∩ {..b} = {.. min a b}"  by (auto simp: min_def)endsubsection {* Intervals of natural numbers *}subsubsection {* The Constant @{term lessThan} *}lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"by (simp add: lessThan_def)lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"by (simp add: lessThan_def less_Suc_eq, blast)text {* The following proof is convenient in induction proofs wherenew elements get indices at the beginning. So it is used to transform@{term "{..<Suc n}"} to @{term "0::nat"} and @{term "{..< n}"}. *}lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc  {..<n})"proof safe  fix x assume "x < Suc n" "x ∉ Suc  {..<n}"  then have "x ≠ Suc (x - 1)" by auto  with x < Suc n show "x = 0" by autoqedlemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"by (simp add: lessThan_def atMost_def less_Suc_eq_le)lemma UN_lessThan_UNIV: "(UN m::nat. lessThan m) = UNIV"by blastsubsubsection {* The Constant @{term greaterThan} *}lemma greaterThan_0 [simp]: "greaterThan 0 = range Suc"apply (simp add: greaterThan_def)apply (blast dest: gr0_conv_Suc [THEN iffD1])donelemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"apply (simp add: greaterThan_def)apply (auto elim: linorder_neqE)donelemma INT_greaterThan_UNIV: "(INT m::nat. greaterThan m) = {}"by blastsubsubsection {* The Constant @{term atLeast} *}lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"by (unfold atLeast_def UNIV_def, simp)lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"apply (simp add: atLeast_def)apply (simp add: Suc_le_eq)apply (simp add: order_le_less, blast)donelemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)lemma UN_atLeast_UNIV: "(UN m::nat. atLeast m) = UNIV"by blastsubsubsection {* The Constant @{term atMost} *}lemma atMost_0 [simp]: "atMost (0::nat) = {0}"by (simp add: atMost_def)lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"apply (simp add: atMost_def)apply (simp add: less_Suc_eq order_le_less, blast)donelemma UN_atMost_UNIV: "(UN m::nat. atMost m) = UNIV"by blastsubsubsection {* The Constant @{term atLeastLessThan} *}text{*The orientation of the following 2 rules is tricky. The lhs isdefined in terms of the rhs.  Hence the chosen orientation makes sensein this theory --- the reverse orientation complicates proofs (egnontermination). But outside, when the definition of the lhs is rarelyused, the opposite orientation seems preferable because it reduces aspecific concept to a more general one. *}lemma atLeast0LessThan: "{0::nat..<n} = {..<n}"by(simp add:lessThan_def atLeastLessThan_def)lemma atLeast0AtMost: "{0..n::nat} = {..n}"by(simp add:atMost_def atLeastAtMost_def)declare atLeast0LessThan[symmetric, code_unfold]        atLeast0AtMost[symmetric, code_unfold]lemma atLeastLessThan0: "{m..<0::nat} = {}"by (simp add: atLeastLessThan_def)subsubsection {* Intervals of nats with @{term Suc} *}text{*Not a simprule because the RHS is too messy.*}lemma atLeastLessThanSuc:    "{m..<Suc n} = (if m ≤ n then insert n {m..<n} else {})"by (auto simp add: atLeastLessThan_def)lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"by (auto simp add: atLeastLessThan_def)(*lemma atLeast_sum_LessThan [simp]: "{m + k..<k::nat} = {}"by (induct k, simp_all add: atLeastLessThanSuc)lemma atLeastSucLessThan [simp]: "{Suc n..<n} = {}"by (auto simp add: atLeastLessThan_def)*)lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def    greaterThanAtMost_def)lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def    greaterThanLessThan_def)lemma atLeastAtMostSuc_conv: "m ≤ Suc n ==> {m..Suc n} = insert (Suc n) {m..n}"by (auto simp add: atLeastAtMost_def)lemma atLeastAtMost_insertL: "m ≤ n ==> insert m {Suc m..n} = {m ..n}"by autotext {* The analogous result is useful on @{typ int}: *}(* here, because we don't have an own int section *)lemma atLeastAtMostPlus1_int_conv:  "m <= 1+n ==> {m..1+n} = insert (1+n) {m..n::int}"  by (auto intro: set_eqI)lemma atLeastLessThan_add_Un: "i ≤ j ==> {i..<j+k} = {i..<j} ∪ {j..<j+k::nat}"  apply (induct k)   apply (simp_all add: atLeastLessThanSuc)     donesubsubsection {* Image *}lemma image_add_atLeastAtMost:  "(%n::nat. n+k)  {i..j} = {i+k..j+k}" (is "?A = ?B")proof  show "?A ⊆ ?B" by autonext  show "?B ⊆ ?A"  proof    fix n assume a: "n : ?B"    hence "n - k : {i..j}" by auto    moreover have "n = (n - k) + k" using a by auto    ultimately show "n : ?A" by blast  qedqedlemma image_add_atLeastLessThan:  "(%n::nat. n+k)  {i..<j} = {i+k..<j+k}" (is "?A = ?B")proof  show "?A ⊆ ?B" by autonext  show "?B ⊆ ?A"  proof    fix n assume a: "n : ?B"    hence "n - k : {i..<j}" by auto    moreover have "n = (n - k) + k" using a by auto    ultimately show "n : ?A" by blast  qedqedcorollary image_Suc_atLeastAtMost[simp]:  "Suc  {i..j} = {Suc i..Suc j}"using image_add_atLeastAtMost[where k="Suc 0"] by simpcorollary image_Suc_atLeastLessThan[simp]:  "Suc  {i..<j} = {Suc i..<Suc j}"using image_add_atLeastLessThan[where k="Suc 0"] by simplemma image_add_int_atLeastLessThan:    "(%x. x + (l::int))  {0..<u-l} = {l..<u}"  apply (auto simp add: image_def)  apply (rule_tac x = "x - l" in bexI)  apply auto  donelemma image_minus_const_atLeastLessThan_nat:  fixes c :: nat  shows "(λi. i - c)  {x ..< y} =      (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"    (is "_ = ?right")proof safe  fix a assume a: "a ∈ ?right"  show "a ∈ (λi. i - c)  {x ..< y}"  proof cases    assume "c < y" with a show ?thesis      by (auto intro!: image_eqI[of _ _ "a + c"])  next    assume "¬ c < y" with a show ?thesis      by (auto intro!: image_eqI[of _ _ x] split: split_if_asm)  qedqed autocontext ordered_ab_group_addbeginlemma  fixes x :: 'a  shows image_uminus_greaterThan[simp]: "uminus  {x<..} = {..<-x}"  and image_uminus_atLeast[simp]: "uminus  {x..} = {..-x}"proof safe  fix y assume "y < -x"  hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp  have "- (-y) ∈ uminus  {x<..}"    by (rule imageI) (simp add: *)  thus "y ∈ uminus  {x<..}" by simpnext  fix y assume "y ≤ -x"  have "- (-y) ∈ uminus  {x..}"    by (rule imageI) (insert y ≤ -x[THEN le_imp_neg_le], simp)  thus "y ∈ uminus  {x..}" by simpqed simp_alllemma  fixes x :: 'a  shows image_uminus_lessThan[simp]: "uminus  {..<x} = {-x<..}"  and image_uminus_atMost[simp]: "uminus  {..x} = {-x..}"proof -  have "uminus  {..<x} = uminus  uminus  {-x<..}"    and "uminus  {..x} = uminus  uminus  {-x..}" by simp_all  thus "uminus  {..<x} = {-x<..}" and "uminus  {..x} = {-x..}"    by (simp_all add: image_image        del: image_uminus_greaterThan image_uminus_atLeast)qedlemma  fixes x :: 'a  shows image_uminus_atLeastAtMost[simp]: "uminus  {x..y} = {-y..-x}"  and image_uminus_greaterThanAtMost[simp]: "uminus  {x<..y} = {-y..<-x}"  and image_uminus_atLeastLessThan[simp]: "uminus  {x..<y} = {-y<..-x}"  and image_uminus_greaterThanLessThan[simp]: "uminus  {x<..<y} = {-y<..<-x}"  by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def      greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)endsubsubsection {* Finiteness *}lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"  by (induct k) (simp_all add: lessThan_Suc)lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"  by (induct k) (simp_all add: atMost_Suc)lemma finite_greaterThanLessThan [iff]:  fixes l :: nat shows "finite {l<..<u}"by (simp add: greaterThanLessThan_def)lemma finite_atLeastLessThan [iff]:  fixes l :: nat shows "finite {l..<u}"by (simp add: atLeastLessThan_def)lemma finite_greaterThanAtMost [iff]:  fixes l :: nat shows "finite {l<..u}"by (simp add: greaterThanAtMost_def)lemma finite_atLeastAtMost [iff]:  fixes l :: nat shows "finite {l..u}"by (simp add: atLeastAtMost_def)text {* A bounded set of natural numbers is finite. *}lemma bounded_nat_set_is_finite:  "(ALL i:N. i < (n::nat)) ==> finite N"apply (rule finite_subset) apply (rule_tac [2] finite_lessThan, auto)donetext {* A set of natural numbers is finite iff it is bounded. *}lemma finite_nat_set_iff_bounded:  "finite(N::nat set) = (EX m. ALL n:N. n<m)" (is "?F = ?B")proof  assume f:?F  show ?B    using Max_ge[OF ?F, simplified less_Suc_eq_le[symmetric]] by blastnext  assume ?B show ?F using ?B by(blast intro:bounded_nat_set_is_finite)qedlemma finite_nat_set_iff_bounded_le:  "finite(N::nat set) = (EX m. ALL n:N. n<=m)"apply(simp add:finite_nat_set_iff_bounded)apply(blast dest:less_imp_le_nat le_imp_less_Suc)donelemma finite_less_ub:     "!!f::nat=>nat. (!!n. n ≤ f n) ==> finite {n. f n ≤ u}"by (rule_tac B="{..u}" in finite_subset, auto intro: order_trans)text{* Any subset of an interval of natural numbers the size of thesubset is exactly that interval. *}lemma subset_card_intvl_is_intvl:  "A <= {k..<k+card A} ==> A = {k..<k+card A}" (is "PROP ?P")proof cases  assume "finite A"  thus "PROP ?P"  proof(induct A rule:finite_linorder_max_induct)    case empty thus ?case by auto  next    case (insert b A)    moreover hence "b ~: A" by auto    moreover have "A <= {k..<k+card A}" and "b = k+card A"      using b ~: A insert by fastforce+    ultimately show ?case by auto  qednext  assume "~finite A" thus "PROP ?P" by simpqedsubsubsection {* Proving Inclusions and Equalities between Unions *}lemma UN_le_eq_Un0:  "(\<Union>i≤n::nat. M i) = (\<Union>i∈{1..n}. M i) ∪ M 0" (is "?A = ?B")proof  show "?A <= ?B"  proof    fix x assume "x : ?A"    then obtain i where i: "i≤n" "x : M i" by auto    show "x : ?B"    proof(cases i)      case 0 with i show ?thesis by simp    next      case (Suc j) with i show ?thesis by auto    qed  qednext  show "?B <= ?A" by autoqedlemma UN_le_add_shift:  "(\<Union>i≤n::nat. M(i+k)) = (\<Union>i∈{k..n+k}. M i)" (is "?A = ?B")proof  show "?A <= ?B" by fastforcenext  show "?B <= ?A"  proof    fix x assume "x : ?B"    then obtain i where i: "i : {k..n+k}" "x : M(i)" by auto    hence "i-k≤n & x : M((i-k)+k)" by auto    thus "x : ?A" by blast  qedqedlemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i∈{0..<n}. A i) = (\<Union>n. A n)"  by (auto simp add: atLeast0LessThan) lemma UN_finite_subset: "(!!n::nat. (\<Union>i∈{0..<n}. A i) ⊆ C) ==> (\<Union>n. A n) ⊆ C"  by (subst UN_UN_finite_eq [symmetric]) blastlemma UN_finite2_subset:      "(!!n::nat. (\<Union>i∈{0..<n}. A i) ⊆ (\<Union>i∈{0..<n+k}. B i)) ==> (\<Union>n. A n) ⊆ (\<Union>n. B n)"  apply (rule UN_finite_subset)  apply (subst UN_UN_finite_eq [symmetric, of B])   apply blast  donelemma UN_finite2_eq:  "(!!n::nat. (\<Union>i∈{0..<n}. A i) = (\<Union>i∈{0..<n+k}. B i)) ==> (\<Union>n. A n) = (\<Union>n. B n)"  apply (rule subset_antisym)   apply (rule UN_finite2_subset, blast) apply (rule UN_finite2_subset [where k=k]) apply (force simp add: atLeastLessThan_add_Un [of 0]) donesubsubsection {* Cardinality *}lemma card_lessThan [simp]: "card {..<u} = u"  by (induct u, simp_all add: lessThan_Suc)lemma card_atMost [simp]: "card {..u} = Suc u"  by (simp add: lessThan_Suc_atMost [THEN sym])lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"  apply (subgoal_tac "card {l..<u} = card {..<u-l}")  apply (erule ssubst, rule card_lessThan)  apply (subgoal_tac "(%x. x + l)  {..<u-l} = {l..<u}")  apply (erule subst)  apply (rule card_image)  apply (simp add: inj_on_def)  apply (auto simp add: image_def atLeastLessThan_def lessThan_def)  apply (rule_tac x = "x - l" in exI)  apply arith  donelemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)lemma ex_bij_betw_nat_finite:  "finite M ==> ∃h. bij_betw h {0..<card M} M"apply(drule finite_imp_nat_seg_image_inj_on)apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)donelemma ex_bij_betw_finite_nat:  "finite M ==> ∃h. bij_betw h M {0..<card M}"by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)lemma finite_same_card_bij:  "finite A ==> finite B ==> card A = card B ==> EX h. bij_betw h A B"apply(drule ex_bij_betw_finite_nat)apply(drule ex_bij_betw_nat_finite)apply(auto intro!:bij_betw_trans)donelemma ex_bij_betw_nat_finite_1:  "finite M ==> ∃h. bij_betw h {1 .. card M} M"by (rule finite_same_card_bij) autolemma bij_betw_iff_card:  assumes FIN: "finite A" and FIN': "finite B"  shows BIJ: "(∃f. bij_betw f A B) <-> (card A = card B)"using assmsproof(auto simp add: bij_betw_same_card)  assume *: "card A = card B"  obtain f where "bij_betw f A {0 ..< card A}"  using FIN ex_bij_betw_finite_nat by blast  moreover obtain g where "bij_betw g {0 ..< card B} B"  using FIN' ex_bij_betw_nat_finite by blast  ultimately have "bij_betw (g o f) A B"  using * by (auto simp add: bij_betw_trans)  thus "(∃f. bij_betw f A B)" by blastqedlemma inj_on_iff_card_le:  assumes FIN: "finite A" and FIN': "finite B"  shows "(∃f. inj_on f A ∧ f  A ≤ B) = (card A ≤ card B)"proof (safe intro!: card_inj_on_le)  assume *: "card A ≤ card B"  obtain f where 1: "inj_on f A" and 2: "f  A = {0 ..< card A}"  using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force  moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g  {0 ..< card B} = B"  using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force  ultimately have "inj_on g (f  A)" using subset_inj_on[of g _ "f  A"] * by force  hence "inj_on (g o f) A" using 1 comp_inj_on by blast  moreover  {have "{0 ..< card A} ≤ {0 ..< card B}" using * by force   with 2 have "f  A  ≤ {0 ..< card B}" by blast   hence "(g o f)  A ≤ B" unfolding comp_def using 3 by force  }  ultimately show "(∃f. inj_on f A ∧ f  A ≤ B)" by blastqed (insert assms, auto)subsection {* Intervals of integers *}lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)lemma atLeastPlusOneLessThan_greaterThanLessThan_int:    "{l+1..<u} = {l<..<u::int}"  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)subsubsection {* Finiteness *}lemma image_atLeastZeroLessThan_int: "0 ≤ u ==>    {(0::int)..<u} = int  {..<nat u}"  apply (unfold image_def lessThan_def)  apply auto  apply (rule_tac x = "nat x" in exI)  apply (auto simp add: zless_nat_eq_int_zless [THEN sym])  donelemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"  apply (cases "0 ≤ u")  apply (subst image_atLeastZeroLessThan_int, assumption)  apply (rule finite_imageI)  apply auto  donelemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"  apply (subgoal_tac "(%x. x + l)  {0..<u-l} = {l..<u}")  apply (erule subst)  apply (rule finite_imageI)  apply (rule finite_atLeastZeroLessThan_int)  apply (rule image_add_int_atLeastLessThan)  donelemma finite_atLeastAtMost_int [iff]: "finite {l..(u::int)}"  by (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym], simp)lemma finite_greaterThanAtMost_int [iff]: "finite {l<..(u::int)}"  by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)lemma finite_greaterThanLessThan_int [iff]: "finite {l<..<u::int}"  by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)subsubsection {* Cardinality *}lemma card_atLeastZeroLessThan_int: "card {(0::int)..<u} = nat u"  apply (cases "0 ≤ u")  apply (subst image_atLeastZeroLessThan_int, assumption)  apply (subst card_image)  apply (auto simp add: inj_on_def)  donelemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"  apply (subgoal_tac "card {l..<u} = card {0..<u-l}")  apply (erule ssubst, rule card_atLeastZeroLessThan_int)  apply (subgoal_tac "(%x. x + l)  {0..<u-l} = {l..<u}")  apply (erule subst)  apply (rule card_image)  apply (simp add: inj_on_def)  apply (rule image_add_int_atLeastLessThan)  donelemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])apply (auto simp add: algebra_simps)donelemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)lemma finite_M_bounded_by_nat: "finite {k. P k ∧ k < (i::nat)}"proof -  have "{k. P k ∧ k < i} ⊆ {..<i}" by auto  with finite_lessThan[of "i"] show ?thesis by (simp add: finite_subset)qedlemma card_less:assumes zero_in_M: "0 ∈ M"shows "card {k ∈ M. k < Suc i} ≠ 0"proof -  from zero_in_M have "{k ∈ M. k < Suc i} ≠ {}" by auto  with finite_M_bounded_by_nat show ?thesis by (auto simp add: card_eq_0_iff)qedlemma card_less_Suc2: "0 ∉ M ==> card {k. Suc k ∈ M ∧ k < i} = card {k ∈ M. k < Suc i}"apply (rule card_bij_eq [of Suc _ _ "λx. x - Suc 0"])apply simpapply fastforceapply autoapply (rule inj_on_diff_nat)apply autoapply (case_tac x)apply autoapply (case_tac xa)apply autoapply (case_tac xa)apply autodonelemma card_less_Suc:  assumes zero_in_M: "0 ∈ M"    shows "Suc (card {k. Suc k ∈ M ∧ k < i}) = card {k ∈ M. k < Suc i}"proof -  from assms have a: "0 ∈ {k ∈ M. k < Suc i}" by simp  hence c: "{k ∈ M. k < Suc i} = insert 0 ({k ∈ M. k < Suc i} - {0})"    by (auto simp only: insert_Diff)  have b: "{k ∈ M. k < Suc i} - {0} = {k ∈ M - {0}. k < Suc i}"  by auto  from finite_M_bounded_by_nat[of "λx. x ∈ M" "Suc i"] have "Suc (card {k. Suc k ∈ M ∧ k < i}) = card (insert 0 ({k ∈ M. k < Suc i} - {0}))"    apply (subst card_insert)    apply simp_all    apply (subst b)    apply (subst card_less_Suc2[symmetric])    apply simp_all    done  with c show ?thesis by simpqedsubsection {*Lemmas useful with the summation operator setsum*}text {* For examples, see Algebra/poly/UnivPoly2.thy *}subsubsection {* Disjoint Unions *}text {* Singletons and open intervals *}lemma ivl_disj_un_singleton:  "{l::'a::linorder} Un {l<..} = {l..}"  "{..<u} Un {u::'a::linorder} = {..u}"  "(l::'a::linorder) < u ==> {l} Un {l<..<u} = {l..<u}"  "(l::'a::linorder) < u ==> {l<..<u} Un {u} = {l<..u}"  "(l::'a::linorder) <= u ==> {l} Un {l<..u} = {l..u}"  "(l::'a::linorder) <= u ==> {l..<u} Un {u} = {l..u}"by autotext {* One- and two-sided intervals *}lemma ivl_disj_un_one:  "(l::'a::linorder) < u ==> {..l} Un {l<..<u} = {..<u}"  "(l::'a::linorder) <= u ==> {..<l} Un {l..<u} = {..<u}"  "(l::'a::linorder) <= u ==> {..l} Un {l<..u} = {..u}"  "(l::'a::linorder) <= u ==> {..<l} Un {l..u} = {..u}"  "(l::'a::linorder) <= u ==> {l<..u} Un {u<..} = {l<..}"  "(l::'a::linorder) < u ==> {l<..<u} Un {u..} = {l<..}"  "(l::'a::linorder) <= u ==> {l..u} Un {u<..} = {l..}"  "(l::'a::linorder) <= u ==> {l..<u} Un {u..} = {l..}"by autotext {* Two- and two-sided intervals *}lemma ivl_disj_un_two:  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..<u} = {l<..<u}"  "[| (l::'a::linorder) <= m; m < u |] ==> {l<..m} Un {m<..<u} = {l<..<u}"  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..<u} = {l..<u}"  "[| (l::'a::linorder) <= m; m < u |] ==> {l..m} Un {m<..<u} = {l..<u}"  "[| (l::'a::linorder) < m; m <= u |] ==> {l<..<m} Un {m..u} = {l<..u}"  "[| (l::'a::linorder) <= m; m <= u |] ==> {l<..m} Un {m<..u} = {l<..u}"  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..<m} Un {m..u} = {l..u}"  "[| (l::'a::linorder) <= m; m <= u |] ==> {l..m} Un {m<..u} = {l..u}"by autolemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_twosubsubsection {* Disjoint Intersections *}text {* One- and two-sided intervals *}lemma ivl_disj_int_one:  "{..l::'a::order} Int {l<..<u} = {}"  "{..<l} Int {l..<u} = {}"  "{..l} Int {l<..u} = {}"  "{..<l} Int {l..u} = {}"  "{l<..u} Int {u<..} = {}"  "{l<..<u} Int {u..} = {}"  "{l..u} Int {u<..} = {}"  "{l..<u} Int {u..} = {}"  by autotext {* Two- and two-sided intervals *}lemma ivl_disj_int_two:  "{l::'a::order<..<m} Int {m..<u} = {}"  "{l<..m} Int {m<..<u} = {}"  "{l..<m} Int {m..<u} = {}"  "{l..m} Int {m<..<u} = {}"  "{l<..<m} Int {m..u} = {}"  "{l<..m} Int {m<..u} = {}"  "{l..<m} Int {m..u} = {}"  "{l..m} Int {m<..u} = {}"  by autolemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_twosubsubsection {* Some Differences *}lemma ivl_diff[simp]: "i ≤ n ==> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"by(auto)subsubsection {* Some Subset Conditions *}lemma ivl_subset [simp,no_atp]: "({i..<j} ⊆ {m..<n}) = (j ≤ i | m ≤ i & j ≤ (n::'a::linorder))"apply(auto simp:linorder_not_le)apply(rule ccontr)apply(insert linorder_le_less_linear[of i n])apply(clarsimp simp:linorder_not_le)apply(fastforce)donesubsection {* Summation indexed over intervals *}syntax  "_from_to_setsum" :: "idt => 'a => 'a => 'b => 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)  "_from_upto_setsum" :: "idt => 'a => 'a => 'b => 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)  "_upt_setsum" :: "idt => 'a => 'b => 'b" ("(SUM _<_./ _)" [0,0,10] 10)  "_upto_setsum" :: "idt => 'a => 'b => 'b" ("(SUM _<=_./ _)" [0,0,10] 10)syntax (xsymbols)  "_from_to_setsum" :: "idt => 'a => 'a => 'b => 'b" ("(3∑_ = _.._./ _)" [0,0,0,10] 10)  "_from_upto_setsum" :: "idt => 'a => 'a => 'b => 'b" ("(3∑_ = _..<_./ _)" [0,0,0,10] 10)  "_upt_setsum" :: "idt => 'a => 'b => 'b" ("(3∑_<_./ _)" [0,0,10] 10)  "_upto_setsum" :: "idt => 'a => 'b => 'b" ("(3∑_≤_./ _)" [0,0,10] 10)syntax (HTML output)  "_from_to_setsum" :: "idt => 'a => 'a => 'b => 'b" ("(3∑_ = _.._./ _)" [0,0,0,10] 10)  "_from_upto_setsum" :: "idt => 'a => 'a => 'b => 'b" ("(3∑_ = _..<_./ _)" [0,0,0,10] 10)  "_upt_setsum" :: "idt => 'a => 'b => 'b" ("(3∑_<_./ _)" [0,0,10] 10)  "_upto_setsum" :: "idt => 'a => 'b => 'b" ("(3∑_≤_./ _)" [0,0,10] 10)syntax (latex_sum output)  "_from_to_setsum" :: "idt => 'a => 'a => 'b => 'b" ("(3$\sum_{_ = _}^{_}$ _)" [0,0,0,10] 10)  "_from_upto_setsum" :: "idt => 'a => 'a => 'b => 'b" ("(3$\sum_{_ = _}^{<_}$ _)" [0,0,0,10] 10)  "_upt_setsum" :: "idt => 'a => 'b => 'b" ("(3$\sum_{_ < _}$ _)" [0,0,10] 10)  "_upto_setsum" :: "idt => 'a => 'b => 'b" ("(3$\sum_{_ ≤ _}$ _)" [0,0,10] 10)translations  "∑x=a..b. t" == "CONST setsum (%x. t) {a..b}"  "∑x=a..<b. t" == "CONST setsum (%x. t) {a..<b}"  "∑i≤n. t" == "CONST setsum (λi. t) {..n}"  "∑i<n. t" == "CONST setsum (λi. t) {..<n}"text{* The above introduces some pretty alternative syntaxes forsummation over intervals:\begin{center}\begin{tabular}{lll}Old & New & \LaTeX\\@{term[source]"∑x∈{a..b}. e"} & @{term"∑x=a..b. e"} & @{term[mode=latex_sum]"∑x=a..b. e"}\\@{term[source]"∑x∈{a..<b}. e"} & @{term"∑x=a..<b. e"} & @{term[mode=latex_sum]"∑x=a..<b. e"}\\@{term[source]"∑x∈{..b}. e"} & @{term"∑x≤b. e"} & @{term[mode=latex_sum]"∑x≤b. e"}\\@{term[source]"∑x∈{..<b}. e"} & @{term"∑x<b. e"} & @{term[mode=latex_sum]"∑x<b. e"}\end{tabular}\end{center}The left column shows the term before introduction of the new syntax,the middle column shows the new (default) syntax, and the right columnshows a special syntax. The latter is only meaningful for latex outputand has to be activated explicitly by setting the print mode to@{text latex_sum} (e.g.\ via @{text "mode = latex_sum"} inantiquotations). It is not the default \LaTeX\ output because it onlyworks well with italic-style formulae, not tt-style.Note that for uniformity on @{typ nat} it is better to use@{term"∑x::nat=0..<n. e"} rather than @{text"∑x<n. e"}: @{text setsum} maynot provide all lemmas available for @{term"{m..<n}"} also in thespecial form for @{term"{..<n}"}. *}text{* This congruence rule should be used for sums over intervals asthe standard theorem @{text[source]setsum_cong} does not work wellwith the simplifier who adds the unsimplified premise @{term"x:B"} tothe context. *}lemma setsum_ivl_cong: "[|a = c; b = d; !!x. [| c ≤ x; x < d |] ==> f x = g x |] ==> setsum f {a..<b} = setsum g {c..<d}"by(rule setsum_cong, simp_all)(* FIXME why are the following simp rules but the corresponding eqnson intervals are not? *)lemma setsum_atMost_Suc[simp]: "(∑i ≤ Suc n. f i) = (∑i ≤ n. f i) + f(Suc n)"by (simp add:atMost_Suc add_ac)lemma setsum_lessThan_Suc[simp]: "(∑i < Suc n. f i) = (∑i < n. f i) + f n"by (simp add:lessThan_Suc add_ac)lemma setsum_cl_ivl_Suc[simp]:  "setsum f {m..Suc n} = (if Suc n < m then 0 else setsum f {m..n} + f(Suc n))"by (auto simp:add_ac atLeastAtMostSuc_conv)lemma setsum_op_ivl_Suc[simp]:  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"by (auto simp:add_ac atLeastLessThanSuc)(*lemma setsum_cl_ivl_add_one_nat: "(n::nat) <= m + 1 ==>    (∑i=n..m+1. f i) = (∑i=n..m. f i) + f(m + 1)"by (auto simp:add_ac atLeastAtMostSuc_conv)*)lemma setsum_head:  fixes n :: nat  assumes mn: "m <= n"   shows "(∑x∈{m..n}. P x) = P m + (∑x∈{m<..n}. P x)" (is "?lhs = ?rhs")proof -  from mn  have "{m..n} = {m} ∪ {m<..n}"    by (auto intro: ivl_disj_un_singleton)  hence "?lhs = (∑x∈{m} ∪ {m<..n}. P x)"    by (simp add: atLeast0LessThan)  also have "… = ?rhs" by simp  finally show ?thesis .qedlemma setsum_head_Suc:  "m ≤ n ==> setsum f {m..n} = f m + setsum f {Suc m..n}"by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)lemma setsum_head_upt_Suc:  "m < n ==> setsum f {m..<n} = f m + setsum f {Suc m..<n}"apply(insert setsum_head_Suc[of m "n - Suc 0" f])apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)donelemma setsum_ub_add_nat: assumes "(m::nat) ≤ n + 1"  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"proof-  have "{m .. n+p} = {m..n} ∪ {n+1..n+p}" using m ≤ n+1 by auto  thus ?thesis by (auto simp: ivl_disj_int setsum_Un_disjoint    atLeastSucAtMost_greaterThanAtMost)qedlemma setsum_add_nat_ivl: "[| m ≤ n; n ≤ p |] ==>  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)lemma setsum_diff_nat_ivl:fixes f :: "nat => 'a::ab_group_add"shows "[| m ≤ n; n ≤ p |] ==>  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"using setsum_add_nat_ivl [of m n p f,symmetric]apply (simp add: add_ac)donelemma setsum_natinterval_difff:  fixes f:: "nat => ('a::ab_group_add)"  shows  "setsum (λk. f k - f(k + 1)) {(m::nat) .. n} =          (if m <= n then f m - f(n + 1) else 0)"by (induct n, auto simp add: algebra_simps not_le le_Suc_eq)lemma setsum_restrict_set':  "finite A ==> setsum f {x ∈ A. x ∈ B} = (∑x∈A. if x ∈ B then f x else 0)"  by (simp add: setsum_restrict_set [symmetric] Int_def)lemma setsum_restrict_set'':  "finite A ==> setsum f {x ∈ A. P x} = (∑x∈A. if P x  then f x else 0)"  by (simp add: setsum_restrict_set' [of A f "{x. P x}", simplified mem_Collect_eq])lemma setsum_setsum_restrict:  "finite S ==> finite T ==>    setsum (λx. setsum (λy. f x y) {y. y ∈ T ∧ R x y}) S = setsum (λy. setsum (λx. f x y) {x. x ∈ S ∧ R x y}) T"  by (simp add: setsum_restrict_set'') (rule setsum_commute)lemma setsum_image_gen: assumes fS: "finite S"  shows "setsum g S = setsum (λy. setsum g {x. x ∈ S ∧ f x = y}) (f  S)"proof-  { fix x assume "x ∈ S" then have "{y. y∈ fS ∧ f x = y} = {f x}" by auto }  hence "setsum g S = setsum (λx. setsum (λy. g x) {y. y∈ fS ∧ f x = y}) S"    by simp  also have "… = setsum (λy. setsum g {x. x ∈ S ∧ f x = y}) (f  S)"    by (rule setsum_setsum_restrict[OF fS finite_imageI[OF fS]])  finally show ?thesis .qedlemma setsum_le_included:  fixes f :: "'a => 'b::ordered_comm_monoid_add"  assumes "finite s" "finite t"  and "∀y∈t. 0 ≤ g y" "(∀x∈s. ∃y∈t. i y = x ∧ f x ≤ g y)"  shows "setsum f s ≤ setsum g t"proof -  have "setsum f s ≤ setsum (λy. setsum g {x. x∈t ∧ i x = y}) s"  proof (rule setsum_mono)    fix y assume "y ∈ s"    with assms obtain z where z: "z ∈ t" "y = i z" "f y ≤ g z" by auto    with assms show "f y ≤ setsum g {x ∈ t. i x = y}" (is "?A y ≤ ?B y")      using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]      by (auto intro!: setsum_mono2)  qed  also have "... ≤ setsum (λy. setsum g {x. x∈t ∧ i x = y}) (i  t)"    using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)  also have "... ≤ setsum g t"    using assms by (auto simp: setsum_image_gen[symmetric])  finally show ?thesis .qedlemma setsum_multicount_gen:  assumes "finite s" "finite t" "∀j∈t. (card {i∈s. R i j} = k j)"  shows "setsum (λi. (card {j∈t. R i j})) s = setsum k t" (is "?l = ?r")proof-  have "?l = setsum (λi. setsum (λx.1) {j∈t. R i j}) s" by auto  also have "… = ?r" unfolding setsum_setsum_restrict[OF assms(1-2)]    using assms(3) by auto  finally show ?thesis .qedlemma setsum_multicount:  assumes "finite S" "finite T" "∀j∈T. (card {i∈S. R i j} = k)"  shows "setsum (λi. card {j∈T. R i j}) S = k * card T" (is "?l = ?r")proof-  have "?l = setsum (λi. k) T" by(rule setsum_multicount_gen)(auto simp:assms)  also have "… = ?r" by(simp add: mult_commute)  finally show ?thesis by autoqedsubsection{* Shifting bounds *}lemma setsum_shift_bounds_nat_ivl:  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"by (induct "n", auto simp:atLeastLessThanSuc)lemma setsum_shift_bounds_cl_nat_ivl:  "setsum f {m+k..n+k} = setsum (%i. f(i + k)){m..n::nat}"apply (insert setsum_reindex[OF inj_on_add_nat, where h=f and B = "{m..n}"])apply (simp add:image_add_atLeastAtMost o_def)donecorollary setsum_shift_bounds_cl_Suc_ivl:  "setsum f {Suc m..Suc n} = setsum (%i. f(Suc i)){m..n}"by (simp add:setsum_shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])corollary setsum_shift_bounds_Suc_ivl:  "setsum f {Suc m..<Suc n} = setsum (%i. f(Suc i)){m..<n}"by (simp add:setsum_shift_bounds_nat_ivl[where k="Suc 0", simplified])lemma setsum_shift_lb_Suc0_0:  "f(0::nat) = (0::nat) ==> setsum f {Suc 0..k} = setsum f {0..k}"by(simp add:setsum_head_Suc)lemma setsum_shift_lb_Suc0_0_upt:  "f(0::nat) = 0 ==> setsum f {Suc 0..<k} = setsum f {0..<k}"apply(cases k)apply simpapply(simp add:setsum_head_upt_Suc)donesubsection {* The formula for geometric sums *}lemma geometric_sum:  assumes "x ≠ 1"  shows "(∑i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::'a::field)"proof -  from assms obtain y where "y = x - 1" and "y ≠ 0" by simp_all  moreover have "(∑i=0..<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"  proof (induct n)    case 0 then show ?case by simp  next    case (Suc n)    moreover with y ≠ 0 have "(1 + y) ^ n = (y * inverse y) * (1 + y) ^ n" by simp     ultimately show ?case by (simp add: field_simps divide_inverse)  qed  ultimately show ?thesis by simpqedsubsection {* The formula for arithmetic sums *}lemma gauss_sum:  "(2::'a::comm_semiring_1)*(∑i∈{1..n}. of_nat i) =   of_nat n*((of_nat n)+1)"proof (induct n)  case 0  show ?case by simpnext  case (Suc n)  then show ?case    by (simp add: algebra_simps add: one_add_one [symmetric] del: one_add_one)      (* FIXME: make numeral cancellation simprocs work for semirings *)qedtheorem arith_series_general:  "(2::'a::comm_semiring_1) * (∑i∈{..<n}. a + of_nat i * d) =  of_nat n * (a + (a + of_nat(n - 1)*d))"proof cases  assume ngt1: "n > 1"  let ?I = "λi. of_nat i" and ?n = "of_nat n"  have    "(∑i∈{..<n}. a+?I i*d) =     ((∑i∈{..<n}. a) + (∑i∈{..<n}. ?I i*d))"    by (rule setsum_addf)  also from ngt1 have "… = ?n*a + (∑i∈{..<n}. ?I i*d)" by simp  also from ngt1 have "… = (?n*a + d*(∑i∈{1..<n}. ?I i))"    unfolding One_nat_def    by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt mult_ac)  also have "2*… = 2*?n*a + d*2*(∑i∈{1..<n}. ?I i)"    by (simp add: algebra_simps)  also from ngt1 have "{1..<n} = {1..n - 1}"    by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)  also from ngt1  have "2*?n*a + d*2*(∑i∈{1..n - 1}. ?I i) = (2*?n*a + d*?I (n - 1)*?I n)"    by (simp only: mult_ac gauss_sum [of "n - 1"], unfold One_nat_def)       (simp add:  mult_ac trans [OF add_commute of_nat_Suc [symmetric]])  finally show ?thesis    unfolding mult_2 by (simp add: algebra_simps)next  assume "¬(n > 1)"  hence "n = 1 ∨ n = 0" by auto  thus ?thesis by (auto simp: mult_2)qedlemma arith_series_nat:  "(2::nat) * (∑i∈{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"proof -  have    "2 * (∑i∈{..<n::nat}. a + of_nat(i)*d) =    of_nat(n) * (a + (a + of_nat(n - 1)*d))"    by (rule arith_series_general)  thus ?thesis    unfolding One_nat_def by autoqedlemma arith_series_int:  "2 * (∑i∈{..<n}. a + int i * d) = int n * (a + (a + int(n - 1)*d))"  by (fact arith_series_general) (* FIXME: duplicate *)lemma sum_diff_distrib:  fixes P::"nat=>nat"  shows  "∀x. Q x ≤ P x  ==>  (∑x<n. P x) - (∑x<n. Q x) = (∑x<n. P x - Q x)"proof (induct n)  case 0 show ?case by simpnext  case (Suc n)  let ?lhs = "(∑x<n. P x) - (∑x<n. Q x)"  let ?rhs = "∑x<n. P x - Q x"  from Suc have "?lhs = ?rhs" by simp  moreover  from Suc have "?lhs + P n - Q n = ?rhs + (P n - Q n)" by simp  moreover  from Suc have    "(∑x<n. P x) + P n - ((∑x<n. Q x) + Q n) = ?rhs + (P n - Q n)"    by (subst diff_diff_left[symmetric],        subst diff_add_assoc2)       (auto simp: diff_add_assoc2 intro: setsum_mono)  ultimately  show ?case by simpqedsubsection {* Products indexed over intervals *}syntax  "_from_to_setprod" :: "idt => 'a => 'a => 'b => 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)  "_from_upto_setprod" :: "idt => 'a => 'a => 'b => 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)  "_upt_setprod" :: "idt => 'a => 'b => 'b" ("(PROD _<_./ _)" [0,0,10] 10)  "_upto_setprod" :: "idt => 'a => 'b => 'b" ("(PROD _<=_./ _)" [0,0,10] 10)syntax (xsymbols)  "_from_to_setprod" :: "idt => 'a => 'a => 'b => 'b" ("(3∏_ = _.._./ _)" [0,0,0,10] 10)  "_from_upto_setprod" :: "idt => 'a => 'a => 'b => 'b" ("(3∏_ = _..<_./ _)" [0,0,0,10] 10)  "_upt_setprod" :: "idt => 'a => 'b => 'b" ("(3∏_<_./ _)" [0,0,10] 10)  "_upto_setprod" :: "idt => 'a => 'b => 'b" ("(3∏_≤_./ _)" [0,0,10] 10)syntax (HTML output)  "_from_to_setprod" :: "idt => 'a => 'a => 'b => 'b" ("(3∏_ = _.._./ _)" [0,0,0,10] 10)  "_from_upto_setprod" :: "idt => 'a => 'a => 'b => 'b" ("(3∏_ = _..<_./ _)" [0,0,0,10] 10)  "_upt_setprod" :: "idt => 'a => 'b => 'b" ("(3∏_<_./ _)" [0,0,10] 10)  "_upto_setprod" :: "idt => 'a => 'b => 'b" ("(3∏_≤_./ _)" [0,0,10] 10)syntax (latex_prod output)  "_from_to_setprod" :: "idt => 'a => 'a => 'b => 'b" ("(3$\prod_{_ = _}^{_}$ _)" [0,0,0,10] 10)  "_from_upto_setprod" :: "idt => 'a => 'a => 'b => 'b" ("(3$\prod_{_ = _}^{<_}$ _)" [0,0,0,10] 10)  "_upt_setprod" :: "idt => 'a => 'b => 'b" ("(3$\prod_{_ < _}$ _)" [0,0,10] 10)  "_upto_setprod" :: "idt => 'a => 'b => 'b" ("(3$\prod_{_ ≤ _}$ _)" [0,0,10] 10)translations  "∏x=a..b. t" == "CONST setprod (%x. t) {a..b}"  "∏x=a..<b. t" == "CONST setprod (%x. t) {a..<b}"  "∏i≤n. t" == "CONST setprod (λi. t) {..n}"  "∏i<n. t" == "CONST setprod (λi. t) {..<n}"subsection {* Transfer setup *}lemma transfer_nat_int_set_functions:    "{..n} = nat  {0..int n}"    "{m..n} = nat  {int m..int n}"  (* need all variants of these! *)  apply (auto simp add: image_def)  apply (rule_tac x = "int x" in bexI)  apply auto  apply (rule_tac x = "int x" in bexI)  apply auto  donelemma transfer_nat_int_set_function_closures:    "x >= 0 ==> nat_set {x..y}"  by (simp add: nat_set_def)declare transfer_morphism_nat_int[transfer add  return: transfer_nat_int_set_functions    transfer_nat_int_set_function_closures]lemma transfer_int_nat_set_functions:    "is_nat m ==> is_nat n ==> {m..n} = int  {nat m..nat n}"  by (simp only: is_nat_def transfer_nat_int_set_functions    transfer_nat_int_set_function_closures    transfer_nat_int_set_return_embed nat_0_le    cong: transfer_nat_int_set_cong)lemma transfer_int_nat_set_function_closures:    "is_nat x ==> nat_set {x..y}"  by (simp only: transfer_nat_int_set_function_closures is_nat_def)declare transfer_morphism_int_nat[transfer add  return: transfer_int_nat_set_functions    transfer_int_nat_set_function_closures]end`