Theory HOL.Set_Interval
section ‹Set intervals›
theory Set_Interval
imports Parity
begin
lemma card_2_iff: "card S = 2 ⟷ (∃x y. S = {x,y} ∧ x ≠ y)"
by (auto simp: card_Suc_eq numeral_eq_Suc)
lemma card_2_iff': "card S = 2 ⟷ (∃x∈S. ∃y∈S. x ≠ y ∧ (∀z∈S. z = x ∨ z = y))"
by (auto simp: card_Suc_eq numeral_eq_Suc)
lemma card_3_iff: "card S = 3 ⟷ (∃x y z. S = {x,y,z} ∧ x ≠ y ∧ y ≠ z ∧ x ≠ z)"
by (fastforce simp: card_Suc_eq numeral_eq_Suc)
context ord
begin
definition
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}"
end
text‹A note of warning when using \<^term>‹{..<n}› on type \<^typ>‹nat›: 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 (ASCII)
"_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 (latex output)
"_UNION_le" :: "'a ⇒ 'a => 'b set => 'b set" ("(3⋃(‹unbreakable›_ ≤ _)/ _)" [0, 0, 10] 10)
"_UNION_less" :: "'a ⇒ 'a => 'b set => 'b set" ("(3⋃(‹unbreakable›_ < _)/ _)" [0, 0, 10] 10)
"_INTER_le" :: "'a ⇒ 'a => 'b set => 'b set" ("(3⋂(‹unbreakable›_ ≤ _)/ _)" [0, 0, 10] 10)
"_INTER_less" :: "'a ⇒ 'a => 'b set => 'b set" ("(3⋂(‹unbreakable›_ < _)/ _)" [0, 0, 10] 10)
syntax
"_UNION_le" :: "'a => 'a => 'b set => 'b set" ("(3⋃_≤_./ _)" [0, 0, 10] 10)
"_UNION_less" :: "'a => 'a => 'b set => 'b set" ("(3⋃_<_./ _)" [0, 0, 10] 10)
"_INTER_le" :: "'a => 'a => 'b set => 'b set" ("(3⋂_≤_./ _)" [0, 0, 10] 10)
"_INTER_less" :: "'a => 'a => 'b set => 'b set" ("(3⋂_<_./ _)" [0, 0, 10] 10)
translations
"⋃i≤n. A" ⇌ "⋃i∈{..n}. A"
"⋃i<n. A" ⇌ "⋃i∈{..<n}. A"
"⋂i≤n. A" ⇌ "⋂i∈{..n}. A"
"⋂i<n. A" ⇌ "⋂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"
by (auto simp add: lessThan_def atLeast_def)
lemma single_Diff_lessThan [simp]: "!!k:: 'a::preorder. {k} - lessThan k = {k}"
by auto
lemma (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"
by (metis Compl_greaterThan double_complement)
lemma (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)
lemma (in linorder) lessThan_Int_lessThan: "{ a <..} ∩ { b <..} = { max a b <..}"
by auto
lemma (in linorder) greaterThan_Int_greaterThan: "{..< a} ∩ {..< b} = {..< min a b}"
by auto
subsection ‹Logical Equivalences for Set Inclusion and Equality›
lemma atLeast_empty_triv [simp]: "{{}..} = UNIV"
by auto
lemma atMost_UNIV_triv [simp]: "{..UNIV} = UNIV"
by auto
lemma atLeast_subset_iff [iff]:
"(atLeast x ⊆ atLeast y) = (y ≤ (x::'a::preorder))"
by (blast intro: order_trans)
lemma atLeast_eq_iff [iff]:
"(atLeast x = atLeast y) = (x = (y::'a::order))"
by (blast intro: order_antisym order_trans)
lemma greaterThan_subset_iff [iff]:
"(greaterThan x ⊆ greaterThan y) = (y ≤ (x::'a::linorder))"
unfolding greaterThan_def by (auto simp: linorder_not_less [symmetric])
lemma greaterThan_eq_iff [iff]:
"(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
by (auto simp: elim!: equalityE)
lemma atMost_subset_iff [iff]: "(atMost x ⊆ atMost y) = (x ≤ (y::'a::preorder))"
by (blast intro: order_trans)
lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::order))"
by (blast intro: order_antisym order_trans)
lemma lessThan_subset_iff [iff]:
"(lessThan x ⊆ lessThan y) = (x ≤ (y::'a::linorder))"
unfolding lessThan_def by (auto simp: linorder_not_less [symmetric])
lemma lessThan_eq_iff [iff]:
"(lessThan x = lessThan y) = (x = (y::'a::linorder))"
by (auto simp: elim!: equalityE)
lemma 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)
lemma (in linorder) Ici_subset_Ioi_iff: "{a ..} ⊆ {b <..} ⟷ b < a"
by auto
lemma (in linorder) Iic_subset_Iio_iff: "{.. a} ⊆ {..< b} ⟷ a < b"
by auto
lemma (in preorder) Ioi_le_Ico: "{a <..} ⊆ {a ..}"
by (auto intro: less_imp_le)
subsection ‹Two-sided intervals›
context ord
begin
lemma greaterThanLessThan_iff [simp]: "(i ∈ {l<..<u}) = (l < i ∧ i < u)"
by (simp add: greaterThanLessThan_def)
lemma atLeastLessThan_iff [simp]: "(i ∈ {l..<u}) = (l ≤ i ∧ i < u)"
by (simp add: atLeastLessThan_def)
lemma greaterThanAtMost_iff [simp]: "(i ∈ {l<..u}) = (l < i ∧ i ≤ u)"
by (simp add: greaterThanAtMost_def)
lemma atLeastAtMost_iff [simp]: "(i ∈ {l..u}) = (l ≤ i ∧ i ≤ u)"
by (simp add: atLeastAtMost_def)
text ‹The above four lemmas could be declared as iffs. Unfortunately this
breaks many proofs. Since it only helps blast, it is better to leave them
alone.›
lemma greaterThanLessThan_eq: "{ a <..< b} = { a <..} ∩ {..< b }"
by auto
lemma (in order) atLeastLessThan_eq_atLeastAtMost_diff:
"{a..<b} = {a..b} - {b}"
by (auto simp add: atLeastLessThan_def atLeastAtMost_def)
lemma (in order) greaterThanAtMost_eq_atLeastAtMost_diff:
"{a<..b} = {a..b} - {a}"
by (auto simp add: greaterThanAtMost_def atLeastAtMost_def)
end
subsubsection‹Emptyness, singletons, subset›
context preorder
begin
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_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_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 atLeastatMost_subset_iff[simp]:
"{a..b} ≤ {c..d} ⟷ (¬ a ≤ b) ∨ c ≤ a ∧ b ≤ d"
unfolding atLeastAtMost_def atLeast_def atMost_def
by (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_subseteq_atLeastLessThan_iff:
"{a..b} ⊆ {c ..< d} ⟷ (a ≤ b ⟶ c ≤ a ∧ b < d)"
by auto (blast intro: local.order_trans local.le_less_trans elim: )+
lemma Icc_subset_Ici_iff[simp]:
"{l..h} ⊆ {l'..} = (¬ l≤h ∨ l≥l')"
by(auto simp: subset_eq intro: order_trans)
lemma Icc_subset_Iic_iff[simp]:
"{l..h} ⊆ {..h'} = (¬ l≤h ∨ h≤h')"
by(auto simp: subset_eq intro: order_trans)
lemma not_Ici_eq_empty[simp]: "{l..} ≠ {}"
by(auto simp: set_eq_iff)
lemma not_Iic_eq_empty[simp]: "{..h} ≠ {}"
by(auto simp: set_eq_iff)
lemmas not_empty_eq_Ici_eq_empty[simp] = not_Ici_eq_empty[symmetric]
lemmas not_empty_eq_Iic_eq_empty[simp] = not_Iic_eq_empty[symmetric]
end
context order
begin
lemma atLeastatMost_empty[simp]: "b < a ⟹ {a..b} = {}"
and atLeastatMost_empty'[simp]: "¬ a ≤ b ⟹ {a..b} = {}"
by(auto simp: atLeastAtMost_def atLeast_def atMost_def)
lemma atLeastLessThan_empty[simp]:
"b ≤ a ⟹ {a..<b} = {}"
by(auto simp: atLeastLessThan_def)
lemma greaterThanAtMost_empty[simp]: "l ≤ k ==> {k<..l} = {}"
by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)
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 simp
lemma Icc_eq_Icc[simp]:
"{l..h} = {l'..h'} = (l=l' ∧ h=h' ∨ ¬ l≤h ∧ ¬ l'≤h')"
by (simp add: order_class.order.eq_iff) (auto intro: order_trans)
lemma (in linorder) Ico_eq_Ico:
"{l..<h} = {l'..<h'} = (l=l' ∧ h=h' ∨ ¬ l<h ∧ ¬ l'<h')"
by (metis atLeastLessThan_empty_iff2 nle_le not_less ord.atLeastLessThan_iff)
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
with ‹{a..b} = {c}› have "c ≤ a ∧ b ≤ c" by auto
with * show "a = b ∧ b = c" by auto
qed simp
end
context no_top
begin
lemma not_UNIV_le_Icc[simp]: "¬ UNIV ⊆ {l..h}"
using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
lemma not_UNIV_le_Iic[simp]: "¬ UNIV ⊆ {..h}"
using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)
lemma not_Ici_le_Icc[simp]: "¬ {l..} ⊆ {l'..h'}"
using gt_ex[of h']
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
lemma not_Ici_le_Iic[simp]: "¬ {l..} ⊆ {..h'}"
using gt_ex[of h']
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
end
context no_bot
begin
lemma not_UNIV_le_Ici[simp]: "¬ UNIV ⊆ {l..}"
using lt_ex[of l] by(auto simp: subset_eq less_le_not_le)
lemma not_Iic_le_Icc[simp]: "¬ {..h} ⊆ {l'..h'}"
using lt_ex[of l']
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
lemma not_Iic_le_Ici[simp]: "¬ {..h} ⊆ {l'..}"
using lt_ex[of l']
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)
end
context no_top
begin
lemma not_UNIV_eq_Icc[simp]: "¬ UNIV = {l'..h'}"
using gt_ex[of h'] by(auto simp: set_eq_iff less_le_not_le)
lemmas not_Icc_eq_UNIV[simp] = not_UNIV_eq_Icc[symmetric]
lemma not_UNIV_eq_Iic[simp]: "¬ UNIV = {..h'}"
using gt_ex[of h'] by(auto simp: set_eq_iff less_le_not_le)
lemmas not_Iic_eq_UNIV[simp] = not_UNIV_eq_Iic[symmetric]
lemma not_Icc_eq_Ici[simp]: "¬ {l..h} = {l'..}"
unfolding atLeastAtMost_def using not_Ici_le_Iic[of l'] by blast
lemmas not_Ici_eq_Icc[simp] = not_Icc_eq_Ici[symmetric]
lemma not_Iic_eq_Ici[simp]: "¬ {..h} = {l'..}"
using not_Ici_le_Iic[of l' h] by blast
lemmas not_Ici_eq_Iic[simp] = not_Iic_eq_Ici[symmetric]
end
context no_bot
begin
lemma not_UNIV_eq_Ici[simp]: "¬ UNIV = {l'..}"
using lt_ex[of l'] by(auto simp: set_eq_iff less_le_not_le)
lemmas not_Ici_eq_UNIV[simp] = not_UNIV_eq_Ici[symmetric]
lemma not_Icc_eq_Iic[simp]: "¬ {l..h} = {..h'}"
unfolding atLeastAtMost_def using not_Iic_le_Ici[of h'] by blast
lemmas not_Iic_eq_Icc[simp] = not_Icc_eq_Iic[symmetric]
end
context dense_linorder
begin
lemma greaterThanLessThan_empty_iff[simp]:
"{ a <..< b } = {} ⟷ b ≤ a"
using dense[of a b] by (cases "a < b") auto
lemma greaterThanLessThan_empty_iff2[simp]:
"{} = { a <..< b } ⟷ b ≤ a"
using dense[of a b] by (cases "a < b") auto
lemma 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 greaterThanLessThan_subseteq_greaterThanLessThan:
"{a <..< b} ⊆ {c <..< d} ⟷ (a < b ⟶ a ≥ c ∧ 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 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])
lemma greaterThanLessThan_subseteq_greaterThanAtMost_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])
end
context no_top
begin
lemma greaterThan_non_empty[simp]: "{x <..} ≠ {}"
using gt_ex[of x] by auto
end
context no_bot
begin
lemma lessThan_non_empty[simp]: "{..< x} ≠ {}"
using lt_ex[of x] by auto
end
lemma (in linorder) atLeastLessThan_subset_iff:
"{a..<b} ⊆ {c..<d} ⟹ b ≤ a ∨ c≤a ∧ b≤d"
proof (cases "a < b")
case True
assume assm: "{a..<b} ⊆ {c..<d}"
then have 1: "c ≤ a ∧ a ≤ d"
using True by (auto simp add: subset_eq Ball_def)
then have 2: "b ≤ d"
using assm by (auto simp add: subset_eq)
from 1 2 show ?thesis
by simp
qed (auto)
lemma 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 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 auto
lemma (in linorder) Ioc_inj:
‹{a <.. b} = {c <.. d} ⟷ (b ≤ a ∧ d ≤ c) ∨ a = c ∧ b = d› (is ‹?P ⟷ ?Q›)
proof
assume ?Q
then show ?P
by auto
next
assume ?P
then have ‹a < x ∧ x ≤ b ⟷ c < x ∧ x ≤ d› for x
by (simp add: set_eq_iff)
from this [of a] this [of b] this [of c] this [of d] show ?Q
by auto
qed
lemma (in order) Iio_Int_singleton: "{..<k} ∩ {x} = (if x < k then {x} else {})"
by auto
lemma (in linorder) Ioc_subset_iff: "{a<..b} ⊆ {c<..d} ⟷ (b ≤ a ∨ c ≤ a ∧ b ≤ d)"
by (auto simp: subset_eq Ball_def) (metis less_le not_less)
lemma (in order_bot) atLeast_eq_UNIV_iff: "{x..} = UNIV ⟷ x = bot"
by (auto simp: set_eq_iff intro: le_bot)
lemma (in order_top) atMost_eq_UNIV_iff: "{..x} = UNIV ⟷ x = top"
by (auto simp: set_eq_iff intro: top_le)
lemma (in bounded_lattice) atLeastAtMost_eq_UNIV_iff:
"{x..y} = UNIV ⟷ (x = bot ∧ y = top)"
by (auto simp: set_eq_iff intro: top_le le_bot)
lemma Iio_eq_empty_iff: "{..< n::'a::{linorder, order_bot}} = {} ⟷ n = bot"
by (auto simp: set_eq_iff not_less le_bot)
lemma lessThan_empty_iff: "{..< n::nat} = {} ⟷ n = 0"
by (simp add: Iio_eq_empty_iff bot_nat_def)
lemma mono_image_least:
assumes f_mono: "mono f" and f_img: "f ` {m ..< n} = {m' ..< n'}" "m < n"
shows "f m = m'"
proof -
from f_img have "{m' ..< n'} ≠ {}"
by (metis atLeastLessThan_empty_iff image_is_empty)
with f_img have "m' ∈ f ` {m ..< n}" by auto
then obtain k where "f k = m'" "m ≤ k" by auto
moreover have "m' ≤ f m" using f_img by auto
ultimately show "f m = m'"
using f_mono by (auto elim: monoE[where x=m and y=k])
qed
subsection ‹Infinite intervals›
context dense_linorder
begin
lemma infinite_Ioo:
assumes "a < b"
shows "¬ finite {a<..<b}"
proof
assume fin: "finite {a<..<b}"
moreover have ne: "{a<..<b} ≠ {}"
using ‹a < b› by auto
ultimately have "a < Max {a <..< b}" "Max {a <..< b} < b"
using Max_in[of "{a <..< b}"] by auto
then obtain x where "Max {a <..< b} < x" "x < b"
using dense[of "Max {a<..<b}" b] by auto
then have "x ∈ {a <..< b}"
using ‹a < Max {a <..< b}› by auto
then have "x ≤ Max {a <..< b}"
using fin by auto
with ‹Max {a <..< b} < x› show False by auto
qed
lemma infinite_Icc: "a < b ⟹ ¬ finite {a .. b}"
using greaterThanLessThan_subseteq_atLeastAtMost_iff[of a b a b] infinite_Ioo[of a b]
by (auto dest: finite_subset)
lemma infinite_Ico: "a < b ⟹ ¬ finite {a ..< b}"
using greaterThanLessThan_subseteq_atLeastLessThan_iff[of a b a b] infinite_Ioo[of a b]
by (auto dest: finite_subset)
lemma infinite_Ioc: "a < b ⟹ ¬ finite {a <.. b}"
using greaterThanLessThan_subseteq_greaterThanAtMost_iff[of a b a b] infinite_Ioo[of a b]
by (auto dest: finite_subset)
lemma infinite_Ioo_iff [simp]: "infinite {a<..<b} ⟷ a < b"
using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ioo)
lemma infinite_Icc_iff [simp]: "infinite {a .. b} ⟷ a < b"
using not_less_iff_gr_or_eq by (fastforce simp: infinite_Icc)
lemma infinite_Ico_iff [simp]: "infinite {a..<b} ⟷ a < b"
using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ico)
lemma infinite_Ioc_iff [simp]: "infinite {a<..b} ⟷ a < b"
using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ioc)
end
lemma infinite_Iio: "¬ finite {..< a :: 'a :: {no_bot, linorder}}"
proof
assume "finite {..< a}"
then have *: "⋀x. x < a ⟹ Min {..< a} ≤ x"
by auto
obtain x where "x < a"
using lt_ex by auto
obtain y where "y < Min {..< a}"
using lt_ex by auto
also have "Min {..< a} ≤ x"
using ‹x < a› by fact
also note ‹x < a›
finally have "Min {..< a} ≤ y"
by fact
with ‹y < Min {..< a}› show False by auto
qed
lemma infinite_Iic: "¬ finite {.. a :: 'a :: {no_bot, linorder}}"
using infinite_Iio[of a] finite_subset[of "{..< a}" "{.. a}"]
by (auto simp: subset_eq less_imp_le)
lemma infinite_Ioi: "¬ finite {a :: 'a :: {no_top, linorder} <..}"
proof
assume "finite {a <..}"
then have *: "⋀x. a < x ⟹ x ≤ Max {a <..}"
by auto
obtain y where "Max {a <..} < y"
using gt_ex by auto
obtain x where x: "a < x"
using gt_ex by auto
also from x have "x ≤ Max {a <..}"
by fact
also note ‹Max {a <..} < y›
finally have "y ≤ Max { a <..}"
by fact
with ‹Max {a <..} < y› show False by auto
qed
lemma infinite_Ici: "¬ finite {a :: 'a :: {no_top, linorder} ..}"
using infinite_Ioi[of a] finite_subset[of "{a <..}" "{a ..}"]
by (auto simp: subset_eq less_imp_le)
subsubsection ‹Intersection›
context linorder
begin
lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"
by auto
lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"
by auto
lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"
by auto
lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"
by auto
lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"
by auto
lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"
by auto
lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
by auto
lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
by auto
lemma Int_atMost[simp]: "{..a} ∩ {..b} = {.. min a b}"
by (auto simp: min_def)
lemma Ioc_disjoint: "{a<..b} ∩ {c<..d} = {} ⟷ b ≤ a ∨ d ≤ c ∨ b ≤ c ∨ d ≤ a"
by auto
end
context complete_lattice
begin
lemma
shows Sup_atLeast[simp]: "Sup {x ..} = top"
and Sup_greaterThanAtLeast[simp]: "x < top ⟹ Sup {x <..} = top"
and Sup_atMost[simp]: "Sup {.. y} = y"
and Sup_atLeastAtMost[simp]: "x ≤ y ⟹ Sup { x .. y} = y"
and Sup_greaterThanAtMost[simp]: "x < y ⟹ Sup { x <.. y} = y"
by (auto intro!: Sup_eqI)
lemma
shows Inf_atMost[simp]: "Inf {.. x} = bot"
and Inf_atMostLessThan[simp]: "top < x ⟹ Inf {..< x} = bot"
and Inf_atLeast[simp]: "Inf {x ..} = x"
and Inf_atLeastAtMost[simp]: "x ≤ y ⟹ Inf { x .. y} = x"
and Inf_atLeastLessThan[simp]: "x < y ⟹ Inf { x ..< y} = x"
by (auto intro!: Inf_eqI)
end
lemma
fixes x y :: "'a :: {complete_lattice, dense_linorder}"
shows Sup_lessThan[simp]: "Sup {..< y} = y"
and Sup_atLeastLessThan[simp]: "x < y ⟹ Sup { x ..< y} = y"
and Sup_greaterThanLessThan[simp]: "x < y ⟹ Sup { x <..< y} = y"
and Inf_greaterThan[simp]: "Inf {x <..} = x"
and Inf_greaterThanAtMost[simp]: "x < y ⟹ Inf { x <.. y} = x"
and Inf_greaterThanLessThan[simp]: "x < y ⟹ Inf { x <..< y} = x"
by (auto intro!: Inf_eqI Sup_eqI intro: dense_le dense_le_bounded dense_ge dense_ge_bounded)
subsection ‹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 where
new elements get indices at the beginning. So it is used to transform
\<^term>‹{..<Suc n}› to \<^term>‹0::nat› and \<^term>‹{..< n}›.›
lemma zero_notin_Suc_image [simp]: "0 ∉ Suc ` A"
by auto
lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"
by (auto simp: image_iff less_Suc_eq_0_disj)
lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
by (simp add: lessThan_def atMost_def less_Suc_eq_le)
lemma atMost_Suc_eq_insert_0: "{.. Suc n} = insert 0 (Suc ` {.. n})"
unfolding lessThan_Suc_atMost[symmetric] lessThan_Suc_eq_insert_0[of "Suc n"] ..
lemma UN_lessThan_UNIV: "(⋃m::nat. lessThan m) = UNIV"
by blast
subsubsection ‹The Constant \<^term>‹greaterThan››
lemma greaterThan_0: "greaterThan 0 = range Suc"
unfolding greaterThan_def
by (blast dest: gr0_conv_Suc [THEN iffD1])
lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
unfolding greaterThan_def
by (auto elim: linorder_neqE)
lemma INT_greaterThan_UNIV: "(⋂m::nat. greaterThan m) = {}"
by blast
subsubsection ‹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}"
unfolding atLeast_def by (auto simp: order_le_less Suc_le_eq)
lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)
lemma UN_atLeast_UNIV: "(⋃m::nat. atLeast m) = UNIV"
by blast
subsubsection ‹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)"
unfolding atMost_def by (auto simp add: less_Suc_eq order_le_less)
lemma UN_atMost_UNIV: "(⋃m::nat. atMost m) = UNIV"
by blast
subsubsection ‹The Constant \<^term>‹atLeastLessThan››
text‹The orientation of the following 2 rules is tricky. The lhs is
defined in terms of the rhs. Hence the chosen orientation makes sense
in this theory --- the reverse orientation complicates proofs (eg
nontermination). But outside, when the definition of the lhs is rarely
used, the opposite orientation seems preferable because it reduces a
specific concept to a more general one.›
lemma atLeast0LessThan [code_abbrev]: "{0::nat..<n} = {..<n}"
by(simp add:lessThan_def atLeastLessThan_def)
lemma atLeast0AtMost [code_abbrev]: "{0..n::nat} = {..n}"
by(simp add:atMost_def atLeastAtMost_def)
lemma lessThan_atLeast0: "{..<n} = {0::nat..<n}"
by (simp add: atLeast0LessThan)
lemma atMost_atLeast0: "{..n} = {0::nat..n}"
by (simp add: atLeast0AtMost)
lemma atLeastLessThan0: "{m..<0::nat} = {}"
by (simp add: atLeastLessThan_def)
lemma atLeast0_lessThan_Suc: "{0..<Suc n} = insert n {0..<n}"
by (simp add: atLeast0LessThan lessThan_Suc)
lemma atLeast0_lessThan_Suc_eq_insert_0: "{0..<Suc n} = insert 0 (Suc ` {0..<n})"
by (simp add: atLeast0LessThan lessThan_Suc_eq_insert_0)
subsubsection ‹The Constant \<^term>‹atLeastAtMost››
lemma Icc_eq_insert_lb_nat: "m ≤ n ⟹ {m..n} = insert m {Suc m..n}"
by auto
lemma atLeast0_atMost_Suc:
"{0..Suc n} = insert (Suc n) {0..n}"
by (simp add: atLeast0AtMost atMost_Suc)
lemma atLeast0_atMost_Suc_eq_insert_0:
"{0..Suc n} = insert 0 (Suc ` {0..n})"
by (simp add: atLeast0AtMost atMost_Suc_eq_insert_0)
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 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
lemma atLeastAtMost_insertL: "m ≤ n ⟹ insert m {Suc m..n} = {m ..n}"
by auto
text ‹The analogous result is useful on \<^typ>‹int›:›
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}"
by (induct k) (simp_all add: atLeastLessThanSuc)
subsubsection ‹Intervals and numerals›
lemma lessThan_nat_numeral:
"lessThan (numeral k :: nat) = insert (pred_numeral k) (lessThan (pred_numeral k))"
by (simp add: numeral_eq_Suc lessThan_Suc)
lemma atMost_nat_numeral:
"atMost (numeral k :: nat) = insert (numeral k) (atMost (pred_numeral k))"
by (simp add: numeral_eq_Suc atMost_Suc)
lemma atLeastLessThan_nat_numeral:
"atLeastLessThan m (numeral k :: nat) =
(if m ≤ (pred_numeral k) then insert (pred_numeral k) (atLeastLessThan m (pred_numeral k))
else {})"
by (simp add: numeral_eq_Suc atLeastLessThanSuc)
subsubsection ‹Image›
context linordered_semidom
begin
lemma image_add_atLeast[simp]: "plus k ` {i..} = {k + i..}"
proof -
have "n = k + (n - k)" if "i + k ≤ n" for n
proof -
have "n = (n - (k + i)) + (k + i)" using that
by (metis add_commute le_add_diff_inverse)
then show "n = k + (n - k)"
by (metis local.add_diff_cancel_left' add_assoc add_commute)
qed
then show ?thesis
by (fastforce simp: add_le_imp_le_diff add.commute)
qed
lemma image_add_atLeastAtMost [simp]:
"plus k ` {i..j} = {i + k..j + k}" (is "?A = ?B")
proof
show "?A ⊆ ?B"
by (auto simp add: ac_simps)
next
show "?B ⊆ ?A"
proof
fix n
assume "n ∈ ?B"
then have "i ≤ n - k"
by (simp add: add_le_imp_le_diff)
have "n = n - k + k"
proof -
from ‹n ∈ ?B› have "n = n - (i + k) + (i + k)"
by simp
also have "… = n - k - i + i + k"
by (simp add: algebra_simps)
also have "… = n - k + k"
using ‹i ≤ n - k› by simp
finally show ?thesis .
qed
moreover have "n - k ∈ {i..j}"
using ‹n ∈ ?B›
by (auto simp: add_le_imp_le_diff add_le_add_imp_diff_le)
ultimately show "n ∈ ?A"
by (simp add: ac_simps)
qed
qed
lemma image_add_atLeastAtMost' [simp]:
"(λn. n + k) ` {i..j} = {i + k..j + k}"
by (simp add: add.commute [of _ k])
lemma image_add_atLeastLessThan [simp]:
"plus k ` {i..<j} = {i + k..<j + k}"
by (simp add: image_set_diff atLeastLessThan_eq_atLeastAtMost_diff ac_simps)
lemma image_add_atLeastLessThan' [simp]:
"(λn. n + k) ` {i..<j} = {i + k..<j + k}"
by (simp add: add.commute [of _ k])
lemma image_add_greaterThanAtMost[simp]: "(+) c ` {a<..b} = {c + a<..c + b}"
by (simp add: image_set_diff greaterThanAtMost_eq_atLeastAtMost_diff ac_simps)
end
context ordered_ab_group_add
begin
lemma
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 simp
next
fix y assume "y ≤ -x"
have "- (-y) ∈ uminus ` {x..}"
by (rule imageI) (use ‹y ≤ -x›[THEN le_imp_neg_le] in ‹simp›)
thus "y ∈ uminus ` {x..}" by simp
qed simp_all
lemma
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)
qed
lemma
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)
lemma image_add_atMost[simp]: "(+) c ` {..a} = {..c + a}"
by (auto intro!: image_eqI[where x="x - c" for x] simp: algebra_simps)
end
lemma image_Suc_atLeastAtMost [simp]:
"Suc ` {i..j} = {Suc i..Suc j}"
using image_add_atLeastAtMost [of 1 i j]
by (simp only: plus_1_eq_Suc) simp
lemma image_Suc_atLeastLessThan [simp]:
"Suc ` {i..<j} = {Suc i..<Suc j}"
using image_add_atLeastLessThan [of 1 i j]
by (simp only: plus_1_eq_Suc) simp
corollary image_Suc_atMost:
"Suc ` {..n} = {1..Suc n}"
by (simp add: atMost_atLeast0 atLeastLessThanSuc_atLeastAtMost)
corollary image_Suc_lessThan:
"Suc ` {..<n} = {1..n}"
by (simp add: lessThan_atLeast0 atLeastLessThanSuc_atLeastAtMost)
lemma image_diff_atLeastAtMost [simp]:
fixes d::"'a::linordered_idom" shows "((-) d ` {a..b}) = {d-b..d-a}"
proof
show "{d - b..d - a} ⊆ (-) d ` {a..b}"
proof
fix x
assume "x ∈ {d - b..d - a}"
then have "d - x ∈ {a..b}" and "x = d - (d - x)"
by auto
then show "x ∈ (-) d ` {a..b}"
by (rule rev_image_eqI)
qed
qed(auto)
lemma image_diff_atLeastLessThan [simp]:
fixes a b c::"'a::linordered_idom"
shows "(-) c ` {a..<b} = {c - b<..c - a}"
proof -
have "(-) c ` {a..<b} = (+) c ` uminus ` {a ..<b}"
unfolding image_image by simp
also have "… = {c - b<..c - a}" by simp
finally show ?thesis by simp
qed
lemma image_minus_const_greaterThanAtMost[simp]:
fixes a b c::"'a::linordered_idom"
shows "(-) c ` {a<..b} = {c - b..<c - a}"
proof -
have "(-) c ` {a<..b} = (+) c ` uminus ` {a<..b}"
unfolding image_image by simp
also have "… = {c - b..<c - a}" by simp
finally show ?thesis by simp
qed
lemma image_minus_const_atLeast[simp]:
fixes a c::"'a::linordered_idom"
shows "(-) c ` {a..} = {..c - a}"
proof -
have "(-) c ` {a..} = (+) c ` uminus ` {a ..}"
unfolding image_image by simp
also have "… = {..c - a}" by simp
finally show ?thesis by simp
qed
lemma image_minus_const_AtMost[simp]:
fixes b c::"'a::linordered_idom"
shows "(-) c ` {..b} = {c - b..}"
proof -
have "(-) c ` {..b} = (+) c ` uminus ` {..b}"
unfolding image_image by simp
also have "… = {c - b..}" by simp
finally show ?thesis by simp
qed
lemma image_minus_const_atLeastAtMost' [simp]:
"(λt. t-d)`{a..b} = {a-d..b-d}" for d::"'a::linordered_idom"
by (metis (no_types, lifting) diff_conv_add_uminus image_add_atLeastAtMost' image_cong)
context linordered_field
begin
lemma image_mult_atLeastAtMost [simp]:
"((*) d ` {a..b}) = {d*a..d*b}" if "d>0"
using that
by (auto simp: field_simps mult_le_cancel_right intro: rev_image_eqI [where x="x/d" for x])
lemma image_divide_atLeastAtMost [simp]:
"((λc. c / d) ` {a..b}) = {a/d..b/d}" if "d>0"
proof -
from that have "inverse d > 0"
by simp
with image_mult_atLeastAtMost [of "inverse d" a b]
have "(*) (inverse d) ` {a..b} = {inverse d * a..inverse d * b}"
by blast
moreover have "(*) (inverse d) = (λc. c / d)"
by (simp add: fun_eq_iff field_simps)
ultimately show ?thesis
by simp
qed
lemma image_mult_atLeastAtMost_if:
"(*) c ` {x .. y} =
(if c > 0 then {c * x .. c * y} else if x ≤ y then {c * y .. c * x} else {})"
proof (cases "c = 0 ∨ x > y")
case True
then show ?thesis
by auto
next
case False
then have "x ≤ y"
by auto
from False consider "c < 0"| "c > 0"
by (auto simp add: neq_iff)
then show ?thesis
proof cases
case 1
have "(*) c ` {x..y} = {c * y..c * x}"
proof (rule set_eqI)
fix d
from 1 have "inj (λz. z / c)"
by (auto intro: injI)
then have "d ∈ (*) c ` {x..y} ⟷ d / c ∈ (λz. z div c) ` (*) c ` {x..y}"
by (subst inj_image_mem_iff) simp_all
also have "… ⟷ d / c ∈ {x..y}"
using 1 by (simp add: image_image)
also have "… ⟷ d ∈ {c * y..c * x}"
by (auto simp add: field_simps 1)
finally show "d ∈ (*) c ` {x..y} ⟷ d ∈ {c * y..c * x}" .
qed
with ‹x ≤ y› show ?thesis
by auto
qed (simp add: mult_left_mono_neg)
qed
lemma image_mult_atLeastAtMost_if':
"(λx. x * c) ` {x..y} =
(if x ≤ y then if c > 0 then {x * c .. y * c} else {y * c .. x * c} else {})"
using image_mult_atLeastAtMost_if [of c x y] by (auto simp add: ac_simps)
lemma image_affinity_atLeastAtMost:
"((λx. m * x + c) ` {a..b}) = (if {a..b} = {} then {}
else if 0 ≤ m then {m * a + c .. m * b + c}
else {m * b + c .. m * a + c})"
proof -
have *: "(λx. m * x + c) = ((λx. x + c) ∘ (*) m)"
by (simp add: fun_eq_iff)
show ?thesis by (simp only: * image_comp [symmetric] image_mult_atLeastAtMost_if)
(auto simp add: mult_le_cancel_left)
qed
lemma image_affinity_atLeastAtMost_diff:
"((λx. m*x - c) ` {a..b}) = (if {a..b}={} then {}
else if 0 ≤ m then {m*a - c .. m*b - c}
else {m*b - c .. m*a - c})"
using image_affinity_atLeastAtMost [of m "-c" a b]
by simp
lemma image_affinity_atLeastAtMost_div:
"((λx. x/m + c) ` {a..b}) = (if {a..b}={} then {}
else if 0 ≤ m then {a/m + c .. b/m + c}
else {b/m + c .. a/m + c})"
using image_affinity_atLeastAtMost [of "inverse m" c a b]
by (simp add: field_class.field_divide_inverse algebra_simps inverse_eq_divide)
lemma image_affinity_atLeastAtMost_div_diff:
"((λx. x/m - c) ` {a..b}) = (if {a..b}={} then {}
else if 0 ≤ m then {a/m - c .. b/m - c}
else {b/m - c .. a/m - c})"
using image_affinity_atLeastAtMost_diff [of "inverse m" c a b]
by (simp add: field_class.field_divide_inverse algebra_simps inverse_eq_divide)
end
lemma atLeast1_lessThan_eq_remove0:
"{Suc 0..<n} = {..<n} - {0}"
by auto
lemma atLeast1_atMost_eq_remove0:
"{Suc 0..n} = {..n} - {0}"
by auto
lemma image_add_int_atLeastLessThan:
"(λx. x + (l::int)) ` {0..<u-l} = {l..<u}"
by safe auto
lemma 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: if_split_asm)
qed
qed auto
lemma image_int_atLeastLessThan:
"int ` {a..<b} = {int a..<int b}"
by (auto intro!: image_eqI [where x = "nat x" for x])
lemma image_int_atLeastAtMost:
"int ` {a..b} = {int a..int b}"
by (auto intro!: image_eqI [where x = "nat x" for x])
subsubsection ‹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: "(∀i∈N. i < (n::nat)) ⟹ finite N"
by (rule finite_subset [OF _ finite_lessThan]) auto
text ‹A set of natural numbers is finite iff it is bounded.›
lemma finite_nat_set_iff_bounded:
"finite(N::nat set) = (∃m. ∀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 blast
next
assume ?B show ?F using ‹?B› by(blast intro:bounded_nat_set_is_finite)
qed
lemma finite_nat_set_iff_bounded_le: "finite(N::nat set) = (∃m. ∀n∈N. n≤m)"
unfolding finite_nat_set_iff_bounded
by (blast dest:less_imp_le_nat le_imp_less_Suc)
lemma finite_less_ub:
"⋀f::nat⇒nat. (!!n. n ≤ f n) ⟹ finite {n. f n ≤ u}"
by (rule finite_subset[of _ "{..u}"])
(auto intro: order_trans)
lemma bounded_Max_nat:
fixes P :: "nat ⇒ bool"
assumes x: "P x" and M: "⋀x. P x ⟹ x ≤ M"
obtains m where "P m" "⋀x. P x ⟹ x ≤ m"
proof -
have "finite {x. P x}"
using M finite_nat_set_iff_bounded_le by auto
then have "Max {x. P x} ∈ {x. P x}"
using Max_in x by auto
then show ?thesis
by (simp add: ‹finite {x. P x}› that)
qed
text‹Any subset of an interval of natural numbers the size of the
subset is exactly that interval.›
lemma subset_card_intvl_is_intvl:
assumes "A ⊆ {k..<k + card A}"
shows "A = {k..<k + card A}"
proof (cases "finite A")
case True
from this and assms show ?thesis
proof (induct A rule: finite_linorder_max_induct)
case empty thus ?case by auto
next
case (insert b A)
hence *: "b ∉ A" by auto
with insert have "A ≤ {k..<k + card A}" and "b = k + card A"
by fastforce+
with insert * show ?case by auto
qed
next
case False
with assms show ?thesis by simp
qed
subsubsection ‹Proving Inclusions and Equalities between Unions›
lemma UN_le_eq_Un0:
"(⋃i≤n::nat. M i) = (⋃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
qed
next
show "?B ⊆ ?A" by fastforce
qed
lemma UN_le_add_shift:
"(⋃i≤n::nat. M(i+k)) = (⋃i∈{k..n+k}. M i)" (is "?A = ?B")
proof
show "?A ⊆ ?B" by fastforce
next
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
qed
qed
lemma UN_le_add_shift_strict:
"(⋃i<n::nat. M(i+k)) = (⋃i∈{k..<n+k}. M i)" (is "?A = ?B")
proof
show "?B ⊆ ?A"
proof
fix x assume "x ∈ ?B"
then obtain i where i: "i ∈ {k..<n+k}" "x ∈ M(i)" by auto
then have "i - k < n ∧ x ∈ M((i-k) + k)" by auto
then show "x ∈ ?A" using UN_le_add_shift by blast
qed
qed (fastforce)
lemma UN_UN_finite_eq: "(⋃n::nat. ⋃i∈{0..<n}. A i) = (⋃n. A n)"
by (auto simp add: atLeast0LessThan)
lemma UN_finite_subset:
"(⋀n::nat. (⋃i∈{0..<n}. A i) ⊆ C) ⟹ (⋃n. A n) ⊆ C"
by (subst UN_UN_finite_eq [symmetric]) blast
lemma UN_finite2_subset:
assumes "⋀n::nat. (⋃i∈{0..<n}. A i) ⊆ (⋃i∈{0..<n + k}. B i)"
shows "(⋃n. A n) ⊆ (⋃n. B n)"
proof (rule UN_finite_subset, rule subsetI)
fix n and a
from assms have "(⋃i∈{0..<n}. A i) ⊆ (⋃i∈{0..<n + k}. B i)" .
moreover assume "a ∈ (⋃i∈{0..<n}. A i)"
ultimately have "a ∈ (⋃i∈{0..<n + k}. B i)" by blast
then show "a ∈ (⋃i. B i)" by (auto simp add: UN_UN_finite_eq)
qed
lemma UN_finite2_eq:
assumes "(⋀n::nat. (⋃i∈{0..<n}. A i) = (⋃i∈{0..<n + k}. B i))"
shows "(⋃n. A n) = (⋃n. B n)"
proof (rule subset_antisym [OF UN_finite_subset UN_finite2_subset])
fix n
show "⋃ (A ` {0..<n}) ⊆ (⋃n. B n)"
using assms by auto
next
fix n
show "⋃ (B ` {0..<n}) ⊆ ⋃ (A ` {0..<n + k})"
using assms by (force simp add: atLeastLessThan_add_Un [of 0])+
qed
subsubsection ‹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"
proof -
have "(λx. x + l) ` {..<u - l} ⊆ {l..<u}"
by auto
moreover have "{l..<u} ⊆ (λx. x + l) ` {..<u-l}"
proof
fix x
assume *: "x ∈ {l..<u}"
then have "x - l ∈ {..< u -l}"
by auto
then have "(x - l) + l ∈ (λx. x + l) ` {..< u -l}"
by auto
then show "x ∈ (λx. x + l) ` {..<u - l}"
using * by auto
qed
ultimately have "{l..<u} = (λx. x + l) ` {..<u-l}"
by auto
then have "card {l..<u} = card {..<u-l}"
by (simp add: card_image inj_on_def)
then show ?thesis
by simp
qed
lemma 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 subset_eq_atLeast0_lessThan_finite:
fixes n :: nat
assumes "N ⊆ {0..<n}"
shows "finite N"
using assms finite_atLeastLessThan by (rule finite_subset)
lemma subset_eq_atLeast0_atMost_finite:
fixes n :: nat
assumes "N ⊆ {0..n}"
shows "finite N"
using assms finite_atLeastAtMost by (rule finite_subset)
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)
done
lemma 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 ⟹ ∃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)
done
lemma ex_bij_betw_nat_finite_1:
"finite M ⟹ ∃h. bij_betw h {1 .. card M} M"
by (rule finite_same_card_bij) auto
lemma bij_betw_iff_card:
assumes "finite A" "finite B"
shows "(∃f. bij_betw f A B) ⟷ (card A = card B)"
proof
assume "card A = card B"
moreover obtain f where "bij_betw f A {0 ..< card A}"
using assms ex_bij_betw_finite_nat by blast
moreover obtain g where "bij_betw g {0 ..< card B} B"
using assms ex_bij_betw_nat_finite by blast
ultimately have "bij_betw (g ∘ f) A B"
by (auto simp: bij_betw_trans)
thus "(∃f. bij_betw f A B)" by blast
qed (auto simp: bij_betw_same_card)
lemma subset_eq_atLeast0_lessThan_card:
fixes n :: nat
assumes "N ⊆ {0..<n}"
shows "card N ≤ n"
proof -
from assms finite_lessThan have "card N ≤ card {0..<n}"
using card_mono by blast
then show ?thesis by simp
qed
text ‹Relational version of @{thm [source] card_inj_on_le}:›
lemma card_le_if_inj_on_rel:
assumes "finite B"
"⋀a. a ∈ A ⟹ ∃b. b∈B ∧ r a b"
"⋀a1 a2 b. ⟦ a1 ∈ A; a2 ∈ A; b ∈ B; r a1 b; r a2 b ⟧ ⟹ a1 = a2"
shows "card A ≤ card B"
proof -
let ?P = "λa b. b ∈ B ∧ r a b"
let ?f = "λa. SOME b. ?P a b"
have 1: "?f ` A ⊆ B" by (auto intro: someI2_ex[OF assms(2)])
have "inj_on ?f A"
unfolding inj_on_def
proof safe
fix a1 a2 assume asms: "a1 ∈ A" "a2 ∈ A" "?f a1 = ?f a2"
have 0: "?f a1 ∈ B" using "1" ‹a1 ∈ A› by blast
have 1: "r a1 (?f a1)" using someI_ex[OF assms(2)[OF ‹a1 ∈ A›]] by blast
have 2: "r a2 (?f a1)" using someI_ex[OF assms(2)[OF ‹a2 ∈ A›]] asms(3) by auto
show "a1 = a2" using assms(3)[OF asms(1,2) 0 1 2] .
qed
with 1 show ?thesis using card_inj_on_le[of ?f A B] assms(1) by simp
qed
lemma inj_on_funpow_least:
‹inj_on (λk. (f ^^ k) s) {0..<n}›
if ‹(f ^^ n) s = s› ‹⋀m. 0 < m ⟹ m < n ⟹ (f ^^ m) s ≠ s›
proof -
{ fix k l assume A: "k < n" "l < n" "k ≠ l" "(f ^^ k) s = (f ^^ l) s"
define k' l' where "k' = min k l" and "l' = max k l"
with A have A': "k' < l'" "(f ^^ k') s = (f ^^ l') s" "l' < n"
by (auto simp: min_def max_def)
have "s = (f ^^ ((n - l') + l')) s" using that ‹l' < n› by simp
also have "… = (f ^^ (n - l')) ((f ^^ l') s)" by (simp add: funpow_add)
also have "(f ^^ l') s = (f ^^ k') s" by (simp add: A')
also have "(f ^^ (n - l')) … = (f ^^ (n - l' + k')) s" by (simp add: funpow_add)
finally have "(f ^^ (n - l' + k')) s = s" by simp
moreover have "n - l' + k' < n" "0 < n - l' + k'"using A' by linarith+
ultimately have False using that(2) by auto
}
then show ?thesis by (intro inj_onI) auto
qed
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:
assumes "0 ≤ u"
shows "{(0::int)..<u} = int ` {..<nat u}"
unfolding image_def lessThan_def
proof
show "{0..<u} ⊆ {y. ∃x∈{x. x < nat u}. y = int x}"
proof
fix x
assume "x ∈ {0..<u}"
then have "x = int (nat x)" and "nat x < nat u"
by (auto simp add: zless_nat_eq_int_zless [THEN sym])
then have "∃xa<nat u. x = int xa"
using exI[of _ "(nat x)"] by simp
then show "x ∈ {y. ∃x∈{x. x < nat u}. y = int x}"
by simp
qed
qed (auto)
lemma finite_atLeastZeroLessThan_int: "finite {(0::int)..<u}"
proof (cases "0 ≤ u")
case True
then show ?thesis
by (auto simp: image_atLeastZeroLessThan_int)
qed auto
lemma finite_atLeastLessThan_int [iff]: "finite {l..<u::int}"
by (simp only: image_add_int_atLeastLessThan [symmetric, of l] finite_imageI finite_atLeastZeroLessThan_int)
lemma 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"
proof (cases "0 ≤ u")
case True
then show ?thesis
by (auto simp: image_atLeastZeroLessThan_int card_image inj_on_def)
qed auto
lemma card_atLeastLessThan_int [simp]: "card {l..<u} = nat (u - l)"
proof -
have "card {l..<u} = card {0..<u-l}"
apply (subst image_add_int_atLeastLessThan [symmetric])
apply (rule card_image)
apply (simp add: inj_on_def)
done
then show ?thesis
by (simp add: card_atLeastZeroLessThan_int)
qed
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
apply (auto simp add: algebra_simps)
done
lemma 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)
qed
lemma 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)
qed
lemma card_less_Suc2:
assumes "0 ∉ M" shows "card {k. Suc k ∈ M ∧ k < i} = card {k ∈ M. k < Suc i}"
proof -
have *: "⟦j ∈ M; j < Suc i⟧ ⟹ j - Suc 0 < i ∧ Suc (j - Suc 0) ∈ M ∧ Suc 0 ≤ j" for j
by (cases j) (use assms in auto)
show ?thesis
proof (rule card_bij_eq)
show "inj_on Suc {k. Suc k ∈ M ∧ k < i}"
by force
show "inj_on (λx. x - Suc 0) {k ∈ M. k < Suc i}"
by (rule inj_on_diff_nat) (use * in blast)
qed (use * in auto)
qed
lemma card_less_Suc:
assumes "0 ∈ M"
shows "Suc (card {k. Suc k ∈ M ∧ k < i}) = card {k ∈ M. k < Suc i}"
proof -
have "Suc (card {k. Suc k ∈ M ∧ k < i}) = Suc (card {k. Suc k ∈ M - {0} ∧ k < i})"
by simp
also have "… = Suc (card {k ∈ M - {0}. k < Suc i})"
apply (subst card_less_Suc2)
using assms by auto
also have "… = Suc (card ({k ∈ M. k < Suc i} - {0}))"
by (force intro: arg_cong [where f=card])
also have "… = card (insert 0 ({k ∈ M. k < Suc i} - {0}))"
by (simp add: card.insert_remove)
also have "... = card {k ∈ M. k < Suc i}"
using assms
by (force simp add: intro: arg_cong [where f=card])
finally show ?thesis.
qed
lemma card_le_Suc_Max: "finite S ⟹ card S ≤ Suc (Max S)"
proof (rule classical)
assume "finite S" and "¬ Suc (Max S) ≥ card S"
then have "Suc (Max S) < card S"
by simp
with ‹finite S› have "S ⊆ {0..Max S}"
by auto
hence "card S ≤ card {0..Max S}"
by (intro card_mono; auto)
thus "card S ≤ Suc (Max S)"
by simp
qed
subsection ‹Lemmas useful with the summation operator sum›
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 auto
text ‹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 auto
text ‹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 auto
lemma ivl_disj_un_two_touch:
"[| (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 auto
lemmas ivl_disj_un = ivl_disj_un_singleton ivl_disj_un_one ivl_disj_un_two ivl_disj_un_two_touch
subsubsection ‹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 auto
text ‹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 auto
lemmas ivl_disj_int = ivl_disj_int_one ivl_disj_int_two
subsubsection ‹Some Differences›
lemma ivl_diff[simp]:
"i ≤ n ⟹ {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
by(auto)
lemma (in linorder) lessThan_minus_lessThan [simp]:
"{..< n} - {..< m} = {m ..< n}"
by auto
lemma (in linorder) atLeastAtMost_diff_ends:
"{a..b} - {a, b} = {a<..<b}"
by auto
subsubsection ‹Some Subset Conditions›
lemma ivl_subset [simp]: "({i..<j} ⊆ {m..<n}) = (j ≤ i ∨ m ≤ i ∧ j ≤ (n::'a::linorder))"
using linorder_class.le_less_linear[of i n]
by safe (force intro: leI)+
subsection ‹Generic big monoid operation over intervals›
context semiring_char_0
begin
lemma inj_on_of_nat [simp]:
"inj_on of_nat N"
by (rule inj_onI) simp
lemma bij_betw_of_nat [simp]:
"bij_betw of_nat N A ⟷ of_nat ` N = A"
by (simp add: bij_betw_def)
lemma Nats_infinite: "infinite (ℕ :: 'a set)"
by (metis Nats_def finite_imageD infinite_UNIV_char_0 inj_on_of_nat)
end
context comm_monoid_set
begin
lemma atLeastLessThan_reindex:
"F g {h m..<h n} = F (g ∘ h) {m..<n}"
if "bij_betw h {m..<n} {h m..<h n}" for m n ::nat
proof -
from that have "inj_on h {m..<n}" and "h ` {m..<n} = {h m..<h n}"
by (simp_all add: bij_betw_def)
then show ?thesis
using reindex [of h "{m..<n}" g] by simp
qed
lemma atLeastAtMost_reindex:
"F g {h m..h n} = F (g ∘ h) {m..n}"
if "bij_betw h {m..n} {h m..h n}" for m n ::nat
proof -
from that have "inj_on h {m..n}" and "h ` {m..n} = {h m..h n}"
by (simp_all add: bij_betw_def)
then show ?thesis
using reindex [of h "{m..n}" g] by simp
qed
lemma atLeastLessThan_shift_bounds:
"F g {m + k..<n + k} = F (g ∘ plus k) {m..<n}"
for m n k :: nat
using atLeastLessThan_reindex [of "plus k" m n g]
by (simp add: ac_simps)
lemma atLeastAtMost_shift_bounds:
"F g {m + k..n + k} = F (g ∘ plus k) {m..n}"
for m n k :: nat
using atLeastAtMost_reindex [of "plus k" m n g]
by (simp add: ac_simps)
lemma atLeast_Suc_lessThan_Suc_shift:
"F g {Suc m..<Suc n} = F (g ∘ Suc) {m..<n}"
using atLeastLessThan_shift_bounds [of _ _ 1]
by (simp add: plus_1_eq_Suc)
lemma atLeast_Suc_atMost_Suc_shift:
"F g {Suc m..Suc n} = F (g ∘ Suc) {m..n}"
using atLeastAtMost_shift_bounds [of _ _ 1]
by (simp add: plus_1_eq_Suc)
lemma atLeast_atMost_pred_shift:
"F (g ∘ (λn. n - Suc 0)) {Suc m..Suc n} = F g {m..n}"
unfolding atLeast_Suc_atMost_Suc_shift by simp
lemma atLeast_lessThan_pred_shift:
"F (g ∘ (λn. n - Suc 0)) {Suc m..<Suc n} = F g {m..<n}"
unfolding atLeast_Suc_lessThan_Suc_shift by simp
lemma atLeast_int_lessThan_int_shift:
"F g {int m..<int n} = F (g ∘ int) {m..<n}"
by (rule atLeastLessThan_reindex)
(simp add: image_int_atLeastLessThan)
lemma atLeast_int_atMost_int_shift:
"F g {int m..int n} = F (g ∘ int) {m..n}"
by (rule atLeastAtMost_reindex)
(simp add: image_int_atLeastAtMost)
lemma atLeast0_lessThan_Suc:
"F g {0..<Suc n} = F g {0..<n} ❙* g n"
by (simp add: atLeast0_lessThan_Suc ac_simps)
lemma atLeast0_atMost_Suc:
"F g {0..Suc n} = F g {0..n} ❙* g (Suc n)"
by (simp add: atLeast0_atMost_Suc ac_simps)
lemma atLeast0_lessThan_Suc_shift:
"F g {0..<Suc n} = g 0 ❙* F (g ∘ Suc) {0..<n}"
by (simp add: atLeast0_lessThan_Suc_eq_insert_0 atLeast_Suc_lessThan_Suc_shift)
lemma atLeast0_atMost_Suc_shift:
"F g {0..Suc n} = g 0 ❙* F (g ∘ Suc) {0..n}"
by (simp add: atLeast0_atMost_Suc_eq_insert_0 atLeast_Suc_atMost_Suc_shift)
lemma atLeast_Suc_lessThan:
"F g {m..<n} = g m ❙* F g {Suc m..<n}" if "m < n"
proof -
from that have "{m..<n} = insert m {Suc m..<n}"
by auto
then show ?thesis by simp
qed
lemma atLeast_Suc_atMost:
"F g {m..n} = g m ❙* F g {Suc m..n}" if "m ≤ n"
proof -
from that have "{m..n} = insert m {Suc m..n}"
by auto
then show ?thesis by simp
qed
lemma ivl_cong:
"a = c ⟹ b = d ⟹ (⋀x. c ≤ x ⟹ x < d ⟹ g x = h x)
⟹ F g {a..<b} = F h {c..<d}"
by (rule cong) simp_all
lemma atLeastLessThan_shift_0:
fixes m n p :: nat
shows "F g {m..<n} = F (g ∘ plus m) {0..<n - m}"
using atLeastLessThan_shift_bounds [of g 0 m "n - m"]
by (cases "m ≤ n") simp_all
lemma atLeastAtMost_shift_0:
fixes m n p :: nat
assumes "m ≤ n"
shows "F g {m..n} = F (g ∘ plus m) {0..n - m}"
using assms atLeastAtMost_shift_bounds [of g 0 m "n - m"] by simp
lemma atLeastLessThan_concat:
fixes m n p :: nat
shows "m ≤ n ⟹ n ≤ p ⟹ F g {m..<n} ❙* F g {n..<p} = F g {m..<p}"
by (simp add: union_disjoint [symmetric] ivl_disj_un)
lemma atLeastLessThan_rev:
"F g {n..<m} = F (λi. g (m + n - Suc i)) {n..<m}"
by (rule reindex_bij_witness [where i="λi. m + n - Suc i" and j="λi. m + n - Suc i"], auto)
lemma atLeastAtMost_rev:
fixes n m :: nat
shows "F g {n..m} = F (λi. g (m + n - i)) {n..m}"
by (rule reindex_bij_witness [where i="λi. m + n - i" and j="λi. m + n - i"]) auto
lemma atLeastLessThan_rev_at_least_Suc_atMost:
"F g {n..<m} = F (λi. g (m + n - i)) {Suc n..m}"
unfolding atLeastLessThan_rev [of g n m]
by (cases m) (simp_all add: atLeast_Suc_atMost_Suc_shift atLeastLessThanSuc_atLeastAtMost)
end
subsection ‹Summation indexed over intervals›
syntax (ASCII)
"_from_to_sum" :: "idt ⇒ 'a ⇒ 'a ⇒ 'b ⇒ 'b" ("(SUM _ = _.._./ _)" [0,0,0,10] 10)
"_from_upto_sum" :: "idt ⇒ 'a ⇒ 'a ⇒ 'b ⇒ 'b" ("(SUM _ = _..<_./ _)" [0,0,0,10] 10)
"_upt_sum" :: "idt ⇒ 'a ⇒ 'b ⇒ 'b" ("(SUM _<_./ _)" [0,0,10] 10)
"_upto_sum" :: "idt ⇒ 'a ⇒ 'b ⇒ 'b" ("(SUM _<=_./ _)" [0,0,10] 10)
syntax (latex_sum output)
"_from_to_sum" :: "idt ⇒ 'a ⇒ 'a ⇒ 'b ⇒ 'b"
("(3\<^latex>‹$\\sum_{›_ = _\<^latex>‹}^{›_\<^latex>‹}$› _)" [0,0,0,10] 10)
"_from_upto_sum" :: "idt ⇒ 'a ⇒ 'a ⇒ 'b ⇒ 'b"
("(3\<^latex>‹$\\sum_{›_ = _\<^latex>‹}^{<›_\<^latex>‹}$› _)" [0,0,0,10] 10)
"_upt_sum" :: "idt ⇒ 'a ⇒ 'b ⇒ 'b"
("(3\<^latex>‹$\\sum_{›_ < _\<^latex>‹}$› _)" [0,0,10] 10)
"_upto_sum" :: "idt ⇒ 'a ⇒ 'b ⇒ 'b"
("(3\<^latex>‹$\\sum_{›_ ≤ _\<^latex>‹}$› _)" [0,0,10] 10)
syntax
"_from_to_sum" :: "idt ⇒ 'a ⇒ 'a ⇒ 'b ⇒ 'b" ("(3∑_ = _.._./ _)" [0,0,0,10] 10)
"_from_upto_sum" :: "idt ⇒ 'a ⇒ 'a ⇒ 'b ⇒ 'b" ("(3∑_ = _..<_./ _)" [0,0,0,10] 10)
"_upt_sum" :: "idt ⇒ 'a ⇒ 'b ⇒ 'b" ("(3∑_<_./ _)" [0,0,10] 10)
"_upto_sum" :: "idt ⇒ 'a ⇒ 'b ⇒ 'b" ("(3∑_≤_./ _)" [0,0,10] 10)
translations
"∑x=a..b. t" == "CONST sum (λx. t) {a..b}"
"∑x=a..<b. t" == "CONST sum (λx. t) {a..<b}"
"∑i≤n. t" == "CONST sum (λi. t) {..n}"
"∑i<n. t" == "CONST sum (λi. t) {..<n}"
text‹The above introduces some pretty alternative syntaxes for
summation 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 column
shows a special syntax. The latter is only meaningful for latex output
and has to be activated explicitly by setting the print mode to
‹latex_sum› (e.g.\ via ‹mode = latex_sum› in
antiquotations). It is not the default \LaTeX\ output because it only
works 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 ‹∑x<n. e›: ‹sum› may
not provide all lemmas available for \<^term>‹{m..<n}› also in the
special form for \<^term>‹{..<n}›.›
text‹This congruence rule should be used for sums over intervals as
the standard theorem @{text[source]sum.cong} does not work well
with the simplifier who adds the unsimplified premise \<^term>‹x∈B› to
the context.›
context comm_monoid_set
begin
lemma zero_middle:
assumes "1 ≤ p" "k ≤ p"
shows "F (λj. if j < k then g j else if j = k then ❙1 else h (j - Suc 0)) {..p}
= F (λj. if j < k then g j else h j) {..p - Suc 0}" (is "?lhs = ?rhs")
proof -
have [simp]: "{..p - Suc 0} ∩ {j. j < k} = {..<k}" "{..p - Suc 0} ∩ - {j. j < k} = {k..p - Suc 0}"
using assms by auto
have "?lhs = F g {..<k} ❙* F (λj. if j = k then ❙1 else h (j - Suc 0)) {k..p}"
using union_disjoint [of "{..<k}" "{k..p}"] assms
by (simp add: ivl_disj_int_one ivl_disj_un_one)
also have "… = F g {..<k} ❙* F (λj. h (j - Suc 0)) {Suc k..p}"
by (simp add: atLeast_Suc_atMost [of k p] assms)
also have "… = F g {..<k} ❙* F h {k .. p - Suc 0}"
using reindex [of Suc "{k..p - Suc 0}"] assms by simp
also have "… = ?rhs"
by (simp add: If_cases)
finally show ?thesis .
qed
lemma atMost_Suc [simp]:
"F g {..Suc n} = F g {..n} ❙* g (Suc n)"
by (simp add: atMost_Suc ac_simps)
lemma lessThan_Suc [simp]:
"F g {..<Suc n} = F g {..<n} ❙* g n"
by (simp add: lessThan_Suc ac_simps)
lemma cl_ivl_Suc [simp]:
"F g {m..Suc n} = (if Suc n < m then ❙1 else F g {m..n} ❙* g(Suc n))"
by (auto simp: ac_simps atLeastAtMostSuc_conv)
lemma op_ivl_Suc [simp]:
"F g {m..<Suc n} = (if n < m then ❙1 else F g {m..<n} ❙* g(n))"
by (auto simp: ac_simps atLeastLessThanSuc)
lemma head:
fixes n :: nat
assumes mn: "m ≤ n"
shows "F g {m..n} = g m ❙* F g {m<..n}" (is "?lhs = ?rhs")
proof -
from mn
have "{m..n} = {m} ∪ {m<..n}"
by (auto intro: ivl_disj_un_singleton)
hence "?lhs = F g ({m} ∪ {m<..n})"
by (simp add: atLeast0LessThan)
also have "… = ?rhs" by simp
finally show ?thesis .
qed
lemma last_plus:
fixes n::nat shows "m ≤ n ⟹ F g {m..n} = g n ❙* F g {m..<n}"
by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost commute)
lemma head_if:
fixes n :: nat
shows "F g {m..n} = (if n < m then ❙1 else F g {m..<n} ❙* g(n))"
by (simp add: commute last_plus)
lemma ub_add_nat:
assumes "(m::nat) ≤ n + 1"
shows "F g {m..n + p} = F g {m..n} ❙* F g {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 union_disjoint atLeastSucAtMost_greaterThanAtMost)
qed
lemma nat_group:
fixes k::nat shows "F (λm. F g {m * k ..< m*k + k}) {..<n} = F g {..< n * k}"
proof (cases k)
case (Suc l)
then have "k > 0"
by auto
then show ?thesis
by (induct n) (simp_all add: atLeastLessThan_concat add.commute atLeast0LessThan[symmetric])
qed auto
lemma triangle_reindex:
fixes n :: nat
shows "F (λ(i,j). g i j) {(i,j). i+j < n} = F (λk. F (λi. g i (k - i)) {..k}) {..<n}"
apply (simp add: Sigma)
apply (rule reindex_bij_witness[where j="λ(i, j). (i+j, i)" and i="λ(k, i). (i, k - i)"])
apply auto
done
lemma triangle_reindex_eq:
fixes n :: nat
shows "F (λ(i,j). g i j) {(i,j). i+j ≤ n} = F (λk. F (λi. g i (k - i)) {..k}) {..n}"
using triangle_reindex [of g "Suc n"]
by (simp only: Nat.less_Suc_eq_le lessThan_Suc_atMost)
lemma nat_diff_reindex: "F (λi. g (n - Suc i)) {..<n} = F g {..<n}"
by (rule reindex_bij_witness[where i="λi. n - Suc i" and j="λi. n - Suc i"]) auto
lemma shift_bounds_nat_ivl:
"F g {m+k..<n+k} = F (λi. g(i + k)){m..<n::nat}"
by (induct "n", auto simp: atLeastLessThanSuc)
lemma shift_bounds_cl_nat_ivl:
"F g {m+k..n+k} = F (λi. g(i + k)){m..n::nat}"
by (rule reindex_bij_witness[where i="λi. i + k" and j="λi. i - k"]) auto
corollary shift_bounds_cl_Suc_ivl:
"F g {Suc m..Suc n} = F (λi. g(Suc i)){m..n}"
by (simp add: shift_bounds_cl_nat_ivl[where k="Suc 0", simplified])
corollary Suc_reindex_ivl: "m ≤ n ⟹ F g {m..n} ❙* g (Suc n) = g m ❙* F (λi. g (Suc i)) {m..n}"
by (simp add: assoc atLeast_Suc_atMost flip: shift_bounds_cl_Suc_ivl)
corollary shift_bounds_Suc_ivl:
"F g {Suc m..<Suc n} = F (λi. g(Suc i)){m..<n}"
by (simp add: shift_bounds_nat_ivl[where k="Suc 0", simplified])
lemma atMost_Suc_shift:
shows "F g {..Suc n} = g 0 ❙* F (λi. g (Suc i)) {..n}"
proof (induct n)
case 0 show ?case by simp
next
case (Suc n) note IH = this
have "F g {..Suc (Suc n)} = F g {..Suc n} ❙* g (Suc (Suc n))"
by (rule atMost_Suc)
also have "F g {..Suc n} = g 0 ❙* F (λi. g (Suc i)) {..n}"
by (rule IH)
also have "g 0 ❙* F (λi. g (Suc i)) {..n} ❙* g (Suc (Suc n)) =
g 0 ❙* (F (λi. g (Suc i)) {..n} ❙* g (Suc (Suc n)))"
by (rule assoc)
also have "F (λi. g (Suc i)) {..n} ❙* g (Suc (Suc n)) = F (λi. g (Suc i)) {..Suc n}"
by (rule atMost_Suc [symmetric])
finally show ?case .
qed
lemma lessThan_Suc_shift:
"F g {..<Suc n} = g 0 ❙* F (λi. g (Suc i)) {..<n}"
by (induction n) (simp_all add: ac_simps)
lemma atMost_shift:
"F g {..n} = g 0 ❙* F (λi. g (Suc i)) {..<n}"
by (metis atLeast0AtMost atLeast0LessThan atLeastLessThanSuc_atLeastAtMost
atLeastSucAtMost_greaterThanAtMost le0 head shift_bounds_Suc_ivl)
lemma nested_swap:
"F (λi. F (λj. a i j) {0..<i}) {0..n} = F (λj. F (λi. a i j) {Suc j..n}) {0..<n}"
by (induction n) (auto simp: distrib)
lemma nested_swap':
"F (λi. F (λj. a i j) {..<i}) {..n} = F (λj. F (λi. a i j) {Suc j..n}) {..<n}"
by (induction n) (auto simp: distrib)
lemma atLeast1_atMost_eq:
"F g {Suc 0..n} = F (λk. g (Suc k)) {..<n}"
proof -
have "F g {Suc 0..n} = F g (Suc ` {..<n})"
by (simp add: image_Suc_lessThan)
also have "… = F (λk. g (Suc k)) {..<n}"
by (simp add: reindex)
finally show ?thesis .
qed
lemma atLeastLessThan_Suc: "a ≤ b ⟹ F g {a..<Suc b} = F g {a..<b} ❙* g b"
by (simp add: atLeastLessThanSuc commute)
lemma nat_ivl_Suc':
assumes "m ≤ Suc n"
shows "F g {m..Suc n} = g (Suc n) ❙* F g {m..n}"
proof -
from assms have "{m..Suc n} = insert (Suc n) {m..n}" by auto
also have "F g … = g (Suc n) ❙* F g {m..n}" by simp
finally show ?thesis .
qed
lemma in_pairs: "F g {2*m..Suc(2*n)} = F (λi. g(2*i) ❙* g(Suc(2*i))) {m..n}"
proof (induction n)
case 0
show ?case
by (cases "m=0") auto
next
case (Suc n)
then show ?case
by (auto simp: assoc split: if_split_asm)
qed
lemma in_pairs_0: "F g {..Suc(2*n)} = F (λi. g(2*i) ❙* g(Suc(2*i))) {..n}"
using in_pairs [of _ 0 n] by (simp add: atLeast0AtMost)
end
lemma card_sum_le_nat_sum: "∑ {0..<card S} ≤ ∑ S"
proof (cases "finite S")
case True
then show ?thesis
proof (induction "card S" arbitrary: S)
case (Suc x)
then have "Max S ≥ x" using card_le_Suc_Max by fastforce
let ?S' = "S - {Max S}"
from Suc have "Max S ∈ S" by (auto intro: Max_in)
hence cards: "card S = Suc (card ?S')"
using ‹finite S› by (intro card.remove; auto)
hence "∑ {0..<card ?S'} ≤ ∑ ?S'"
using Suc by (intro Suc; auto)
hence "∑ {0..<card ?S'} + x ≤ ∑ ?S' + Max S"
using ‹Max S ≥ x› by simp
also have "... = ∑ S"
using sum.remove[OF ‹finite S› ‹Max S ∈ S›, where g="λx. x"]
by simp
finally show ?case
using cards Suc by auto
qed simp
qed simp
lemma sum_natinterval_diff:
fixes f:: "nat ⇒ ('a::ab_group_add)"
shows "sum (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 sum_diff_nat_ivl:
fixes f :: "nat ⇒ 'a::ab_group_add"
shows "⟦ m ≤ n; n ≤ p ⟧ ⟹ sum f {m..<p} - sum f {m..<n} = sum f {n..<p}"
using sum.atLeastLessThan_concat [of m n p f,symmetric]
by (simp add: ac_simps)
lemma sum_diff_distrib: "∀x. Q x ≤ P x ⟹ (∑x<n. P x) - (∑x<n. Q x) = (∑x<n. P x - Q x :: nat)"
by (subst sum_subtractf_nat) auto
subsubsection ‹Shifting bounds›
context comm_monoid_add
begin
context
fixes f :: "nat ⇒ 'a"
assumes "f 0 = 0"
begin
lemma sum_shift_lb_Suc0_0_upt:
"sum f {Suc 0..<k} = sum f {0..<k}"
proof (cases k)
case 0
then show ?thesis
by simp
next
case (Suc k)
moreover have "{0..<Suc k} = insert 0 {Suc 0..<Suc k}"
by auto
ultimately show ?thesis
using ‹f 0 = 0› by simp
qed
lemma sum_shift_lb_Suc0_0: "sum f {Suc 0..k} = sum f {0..k}"
proof (cases k)
case 0
with ‹f 0 = 0› show ?thesis
by simp
next
case (Suc k)
moreover have "{0..Suc k} = insert 0 {Suc 0..Suc k}"
by auto
ultimately show ?thesis
using ‹f 0 = 0› by simp
qed
end
end
lemma sum_Suc_diff:
fixes f :: "nat ⇒ 'a::ab_group_add"
assumes "m ≤ Suc n"
shows "(∑i = m..n. f(Suc i) - f i) = f (Suc n) - f m"
using assms by (induct n) (auto simp: le_Suc_eq)
lemma sum_Suc_diff':
fixes f :: "nat ⇒ 'a::ab_group_add"
assumes "m ≤ n"
shows "(∑i = m..<n. f (Suc i) - f i) = f n - f m"
using assms by (induct n) (auto simp: le_Suc_eq)
lemma sum_diff_split:
fixes f:: "nat ⇒ 'a::ab_group_add"
assumes "m ≤ n"
shows "(∑i≤n. f i) - (∑i<m. f i) = (∑i≤n - m. f(n - i))"
proof -
have "⋀i. i ≤ n-m ⟹ ∃k≥m. k ≤ n ∧ i = n-k"
by (metis Nat.le_diff_conv2 add.commute ‹m≤n› diff_diff_cancel diff_le_self order.trans)
then have eq: "{..n-m} = (-)n ` {m..n}"
by force
have inj: "inj_on ((-)n) {m..n}"
by (auto simp: inj_on_def)
have "(∑i≤n - m. f(n - i)) = (∑i=m..n. f i)"
by (simp add: eq sum.reindex_cong [OF inj])
also have "… = (∑i≤n. f i) - (∑i<m. f i)"
using sum_diff_nat_ivl[of 0 "m" "Suc n" f] assms
by (simp only: atLeast0AtMost atLeast0LessThan atLeastLessThanSuc_atLeastAtMost)
finally show ?thesis by metis
qed
subsubsection ‹Telescoping sums›
lemma sum_telescope:
fixes f::"nat ⇒ 'a::ab_group_add"
shows "sum (λi. f i - f (Suc i)) {.. i} = f 0 - f (Suc i)"
by (induct i) simp_all
lemma sum_telescope'':
assumes "m ≤ n"
shows "(∑k∈{Suc m..n}. f k - f (k - 1)) = f n - (f m :: 'a :: ab_group_add)"
by (rule dec_induct[OF assms]) (simp_all add: algebra_simps)
lemma sum_lessThan_telescope:
"(∑n<m. f (Suc n) - f n :: 'a :: ab_group_add) = f m - f 0"
by (induction m) (simp_all add: algebra_simps)
lemma sum_lessThan_telescope':
"(∑n<m. f n - f (Suc n) :: 'a :: ab_group_add) = f 0 - f m"
by (induction m) (simp_all add: algebra_simps)
subsubsection ‹The formula for geometric sums›
lemma sum_power2: "(∑i=0..<k. (2::nat)^i) = 2^k-1"
by (induction k) (auto simp: mult_2)
lemma geometric_sum:
assumes "x ≠ 1"
shows "(∑i<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<n. (y + 1) ^ i) = ((y + 1) ^ n - 1) / y"
by (induct n) (simp_all add: field_simps ‹y ≠ 0›)
ultimately show ?thesis by simp
qed
lemma geometric_sum_less:
assumes "0 < x" "x < 1" "finite S"
shows "(∑i∈S. x ^ i) < 1 / (1 - x::'a::linordered_field)"
proof -
define n where "n ≡ Suc (Max S)"
have "(∑i∈S. x ^ i) ≤ (∑i<n. x ^ i)"
unfolding n_def using assms by (fastforce intro!: sum_mono2 le_imp_less_Suc)
also have "… = (1 - x ^ n) / (1 - x)"
using assms by (simp add: geometric_sum field_simps)
also have "… < 1 / (1-x)"
using assms by (simp add: field_simps power_Suc_less)
finally show ?thesis .
qed
lemma diff_power_eq_sum:
fixes y :: "'a::{comm_ring,monoid_mult}"
shows
"x ^ (Suc n) - y ^ (Suc n) =
(x - y) * (∑p<Suc n. (x ^ p) * y ^ (n - p))"
proof (induct n)
case (Suc n)
have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x^n) - y * (y * y ^ n)"
by simp
also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x^n)"
by (simp add: algebra_simps)
also have "... = y * ((x - y) * (∑p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
by (simp only: Suc)
also have "... = (x - y) * (y * (∑p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
by (simp only: mult.left_commute)
also have "... = (x - y) * (∑p<Suc (Suc n). x ^ p * y ^ (Suc n - p))"
by (simp add: field_simps Suc_diff_le sum_distrib_right sum_distrib_left)
finally show ?case .
qed simp
corollary power_diff_sumr2:
fixes x :: "'a::{comm_ring,monoid_mult}"
shows "x^n - y^n = (x - y) * (∑i<n. y^(n - Suc i) * x^i)"
using diff_power_eq_sum[of x "n - 1" y]
by (cases "n = 0") (simp_all add: field_simps)
lemma power_diff_1_eq:
fixes x :: "'a::{comm_ring,monoid_mult}"
shows "x^n - 1 = (x - 1) * (∑i<n. (x^i))"
using diff_power_eq_sum [of x _ 1]
by (cases n) auto
lemma one_diff_power_eq':
fixes x :: "'a::{comm_ring,monoid_mult}"
shows "1 - x^n = (1 - x) * (∑i<n. x^(n - Suc i))"
using diff_power_eq_sum [of 1 _ x]
by (cases n) auto
lemma one_diff_power_eq:
fixes x :: "'a::{comm_ring,monoid_mult}"
shows "1 - x^n = (1 - x) * (∑i<n. x^i)"
by (metis one_diff_power_eq' sum.nat_diff_reindex)
lemma sum_gp_basic:
fixes x :: "'a::{comm_ring,monoid_mult}"
shows "(1 - x) * (∑i≤n. x^i) = 1 - x^Suc n"
by (simp only: one_diff_power_eq lessThan_Suc_atMost)
lemma sum_power_shift:
fixes x :: "'a::{comm_ring,monoid_mult}"
assumes "m ≤ n"
shows "(∑i=m..n. x^i) = x^m * (∑i≤n-m. x^i)"
proof -
have "(∑i=m..n. x^i) = x^m * (∑i=m..n. x^(i-m))"
by (simp add: sum_distrib_left power_add [symmetric])
also have "(∑i=m..n. x^(i-m)) = (∑i≤n-m. x^i)"
using ‹m ≤ n› by (intro sum.reindex_bij_witness[where j="λi. i - m" and i="λi. i + m"]) auto
finally show ?thesis .
qed
lemma sum_gp_multiplied:
fixes x :: "'a::{comm_ring,monoid_mult}"
assumes "m ≤ n"
shows "(1 - x) * (∑i=m..n. x^i) = x^m - x^Suc n"
proof -
have "(1 - x) * (∑i=m..n. x^i) = x^m * (1 - x) * (∑i≤n-m. x^i)"
by (metis mult.assoc mult.commute assms sum_power_shift)
also have "... =x^m * (1 - x^Suc(n-m))"
by (metis mult.assoc sum_gp_basic)
also have "... = x^m - x^Suc n"
using assms
by (simp add: algebra_simps) (metis le_add_diff_inverse power_add)
finally show ?thesis .
qed
lemma sum_gp:
fixes x :: "'a::{comm_ring,division_ring}"
shows "(∑i=m..n. x^i) =
(if n < m then 0
else if x = 1 then of_nat((n + 1) - m)
else (x^m - x^Suc n) / (1 - x))"
proof (cases "n < m")
case False
assume *: "¬ n < m"
then show ?thesis
proof (cases "x = 1")
case False
assume "x ≠ 1"
then have not_zero: "1 - x ≠ 0"
by auto
have "(1 - x) * (∑i=m..n. x^i) = x ^ m - x * x ^ n"
using sum_gp_multiplied [of m n x] * by auto
then have "(∑i=m..n. x^i) = (x ^ m - x * x ^ n) / (1 - x) "
using nonzero_divide_eq_eq mult.commute not_zero
by metis
then show ?thesis
by auto
qed (auto)
qed (auto)
subsubsection‹Geometric progressions›
lemma sum_gp0:
fixes x :: "'a::{comm_ring,division_ring}"
shows "(∑i≤n. x^i) = (if x = 1 then of_nat(n + 1) else (1 - x^Suc n) / (1 - x))"
using sum_gp_basic[of x n]
by (simp add: mult.commute field_split_simps)
lemma sum_power_add:
fixes x :: "'a::{comm_ring,monoid_mult}"
shows "(∑i∈I. x^(m+i)) = x^m * (∑i∈I. x^i)"
by (simp add: sum_distrib_left power_add)
lemma sum_gp_offset:
fixes x :: "'a::{comm_ring,division_ring}"
shows "(∑i=m..m+n. x^i) =
(if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
using sum_gp [of x m "m+n"]
by (auto simp: power_add algebra_simps)
lemma sum_gp_strict:
fixes x :: "'a::{comm_ring,division_ring}"
shows "(∑i<n. x^i) = (if x = 1 then of_nat n else (1 - x^n) / (1 - x))"
by (induct n) (auto simp: algebra_simps field_split_simps)
subsubsection ‹The formulae for arithmetic sums›
context comm_semiring_1
begin
lemma double_gauss_sum:
"2 * (∑i = 0..n. of_nat i) = of_nat n * (of_nat n + 1)"
by (induct n) (simp_all add: sum.atLeast0_atMost_Suc algebra_simps left_add_twice)
lemma double_gauss_sum_from_Suc_0:
"2 * (∑i = Suc 0..n. of_nat i) = of_nat n * (of_nat n + 1)"
proof -
have "sum of_nat {Suc 0..n} = sum of_nat (insert 0 {Suc 0..n})"
by simp
also have "… = sum of_nat {0..n}"
by (cases n) (simp_all add: atLeast0_atMost_Suc_eq_insert_0)
finally show ?thesis
by (simp add: double_gauss_sum)
qed
lemma double_arith_series:
"2 * (∑i = 0..n. a + of_nat i * d) = (of_nat n + 1) * (2 * a + of_nat n * d)"
proof -
have "(∑i = 0..n. a + of_nat i * d) = ((∑i = 0..n. a) + (∑i = 0..n. of_nat i * d))"
by (rule sum.distrib)
also have "… = (of_nat (Suc n) * a + d * (∑i = 0..n. of_nat i))"
by (simp add: sum_distrib_left algebra_simps)
finally show ?thesis
by (simp add: algebra_simps double_gauss_sum left_add_twice)
qed
end
context linordered_euclidean_semiring
begin
lemma gauss_sum:
"(∑i = 0..n. of_nat i) = of_nat n * (of_nat n + 1) div 2"
using double_gauss_sum [of n, symmetric] by simp
lemma gauss_sum_from_Suc_0:
"(∑i = Suc 0..n. of_nat i) = of_nat n * (of_nat n + 1) div 2"
using double_gauss_sum_from_Suc_0 [of n, symmetric] by simp
lemma arith_series:
"(∑i = 0..n. a + of_nat i * d) = (of_nat n + 1) * (2 * a + of_nat n * d) div 2"
using double_arith_series [of a d n, symmetric] by simp
end
lemma gauss_sum_nat:
"∑{0..n} = (n * Suc n) div 2"
using gauss_sum [of n, where ?'a = nat] by simp
lemma arith_series_nat:
"(∑i = 0..n. a + i * d) = Suc n * (2 * a + n * d) div 2"
using arith_series [of a d n] by simp
lemma Sum_Icc_int:
"∑{m..n} = (n * (n + 1) - m * (m - 1)) div 2"
if "m ≤ n" for m n :: int
using that proof (induct i ≡ "nat (n - m)" arbitrary: m n)
case 0
then have "m = n"
by arith
then show ?case
by (simp add: algebra_simps mult_2 [symmetric])
next
case (Suc i)
have 0: "i = nat((n-1) - m)" "m ≤ n-1" using Suc(2,3) by arith+
have "∑ {m..n} = ∑ {m..1+(n-1)}" by simp
also have "… = ∑ {m..n-1} + n" using ‹m ≤ n›
by(subst atLeastAtMostPlus1_int_conv) simp_all
also have "… = ((n-1)*(n-1+1) - m*(m-1)) div 2 + n"
by(simp add: Suc(1)[OF 0])
also have "… = ((n-1)*(n-1+1) - m*(m-1) + 2*n) div 2" by simp
also have "… = (n*(n+1) - m*(m-1)) div 2"
by (simp add: algebra_simps mult_2_right)
finally show ?case .
qed
lemma Sum_Icc_nat:
"∑{m..n} = (n * (n + 1) - m * (m - 1)) div 2" for m n :: nat
proof (cases "m ≤ n")
case True
then have *: "m * (m - 1) ≤ n * (n + 1)"
by (meson diff_le_self order_trans le_add1 mult_le_mono)
have "int (∑{m..n}) = (∑{int m..int n})"
by (simp add: sum.atLeast_int_atMost_int_shift)
also have "… = (int n * (int n + 1) - int m * (int m - 1)) div 2"
using ‹m ≤ n› by (simp add: Sum_Icc_int)
also have "… = int ((n * (n + 1) - m * (m - 1)) div 2)"
using le_square * by (simp add: algebra_simps of_nat_div of_nat_diff)
finally show ?thesis
by (simp only: of_nat_eq_iff)
next
case False
then show ?thesis
by (auto dest: less_imp_Suc_add simp add: not_le algebra_simps)
qed
lemma Sum_Ico_nat:
"∑{m..<n} = (n * (n - 1) - m * (m - 1)) div 2" for m n :: nat
by (cases n) (simp_all add: atLeastLessThanSuc_atLeastAtMost Sum_Icc_nat)
subsubsection ‹Division remainder›
lemma range_mod:
fixes n :: nat
assumes "n > 0"
shows "range (λm. m mod n) = {0..<n}" (is "?A = ?B")
proof (rule set_eqI)
fix m
show "m ∈ ?A ⟷ m ∈ ?B"
proof
assume "m ∈ ?A"
with assms show "m ∈ ?B"
by auto
next
assume "m ∈ ?B"
moreover have "m mod n ∈ ?A"
by (rule rangeI)
ultimately show "m ∈ ?A"
by simp
qed
qed
subsection ‹Products indexed over intervals›
syntax (ASCII)
"_from_to_prod" :: "idt ⇒ 'a ⇒ 'a ⇒ 'b ⇒ 'b" ("(PROD _ = _.._./ _)" [0,0,0,10] 10)
"_from_upto_prod" :: "idt ⇒ 'a ⇒ 'a ⇒ 'b ⇒ 'b" ("(PROD _ = _..<_./ _)" [0,0,0,10] 10)
"_upt_prod" :: "idt ⇒ 'a ⇒ 'b ⇒ 'b" ("(PROD _<_./ _)" [0,0,10] 10)
"_upto_prod" :: "idt ⇒ 'a ⇒ 'b ⇒ 'b" ("(PROD _<=_./ _)" [0,0,10] 10)
syntax (latex_prod output)
"_from_to_prod" :: "idt ⇒ 'a ⇒ 'a ⇒ 'b ⇒ 'b"
("(3\<^latex>‹$\\prod_{›_ = _\<^latex>‹}^{›_\<^latex>‹}$› _)" [0,0,0,10] 10)
"_from_upto_prod" :: "idt ⇒ 'a ⇒ 'a ⇒ 'b ⇒ 'b"
("(3\<^latex>‹$\\prod_{›_ = _\<^latex>‹}^{<›_\<^latex>‹}$› _)" [0,0,0,10] 10)
"_upt_prod" :: "idt ⇒ 'a ⇒ 'b ⇒ 'b"
("(3\<^latex>‹$\\prod_{›_ < _\<^latex>‹}$› _)" [0,0,10] 10)
"_upto_prod" :: "idt ⇒ 'a ⇒ 'b ⇒ 'b"
("(3\<^latex>‹$\\prod_{›_ ≤ _\<^latex>‹}$› _)" [0,0,10] 10)
syntax
"_from_to_prod" :: "idt ⇒ 'a ⇒ 'a ⇒ 'b ⇒ 'b" ("(3∏_ = _.._./ _)" [0,0,0,10] 10)
"_from_upto_prod" :: "idt ⇒ 'a ⇒ 'a ⇒ 'b ⇒ 'b" ("(3∏_ = _..<_./ _)" [0,0,0,10] 10)
"_upt_prod" :: "idt ⇒ 'a ⇒ 'b ⇒ 'b" ("(3∏_<_./ _)" [0,0,10] 10)
"_upto_prod" :: "idt ⇒ 'a ⇒ 'b ⇒ 'b" ("(3∏_≤_./ _)" [0,0,10] 10)
translations
"∏x=a..b. t" ⇌ "CONST prod (λx. t) {a..b}"
"∏x=a..<b. t" ⇌ "CONST prod (λx. t) {a..<b}"
"∏i≤n. t" ⇌ "CONST prod (λi. t) {..n}"
"∏i<n. t" ⇌ "CONST prod (λi. t) {..<n}"
lemma prod_int_plus_eq: "prod int {i..i+j} = ∏{int i..int (i+j)}"
by (induct j) (auto simp add: atLeastAtMostSuc_conv atLeastAtMostPlus1_int_conv)
lemma prod_int_eq: "prod int {i..j} = ∏{int i..int j}"
proof (cases "i ≤ j")
case True
then show ?thesis
by (metis le_iff_add prod_int_plus_eq)
next
case False
then show ?thesis
by auto
qed
subsubsection ‹Telescoping products›
lemma prod_telescope:
fixes f::"nat ⇒ 'a::field"
assumes "⋀i. i≤n ⟹ f (Suc i) ≠ 0"
shows "(∏i≤n. f i / f (Suc i)) = f 0 / f (Suc n)"
using assms by (induction n) auto
lemma prod_telescope'':
fixes f::"nat ⇒ 'a::field"
assumes "m ≤ n"
assumes "⋀i. i ∈ {m..n} ⟹ f i ≠ 0"
shows "(∏i = Suc m..n. f i / f (i - 1)) = f n / f m"
by (rule dec_induct[OF ‹m ≤ n›]) (auto simp add: assms)
lemma prod_lessThan_telescope:
fixes f::"nat ⇒ 'a::field"
assumes "⋀i. i≤n ⟹ f i ≠ 0"
shows "(∏i<n. f (Suc i) / f i) = f n / f 0"
using assms by (induction n) auto
lemma prod_lessThan_telescope':
fixes f::"nat ⇒ 'a::field"
assumes "⋀i. i≤n ⟹ f i ≠ 0"
shows "(∏i<n. f i / f (Suc i)) = f 0 / f n"
using assms by (induction n) auto
subsection ‹Efficient folding over intervals›
function fold_atLeastAtMost_nat where
[simp del]: "fold_atLeastAtMost_nat f a (b::nat) acc =
(if a > b then acc else fold_atLeastAtMost_nat f (a+1) b (f a acc))"
by pat_completeness auto
termination by (relation "measure (λ(_,a,b,_). Suc b - a)") auto
lemma fold_atLeastAtMost_nat:
assumes "comp_fun_commute f"
shows "fold_atLeastAtMost_nat f a b acc = Finite_Set.fold f acc {a..b}"
using assms
proof (induction f a b acc rule: fold_atLeastAtMost_nat.induct, goal_cases)
case (1 f a b acc)
interpret comp_fun_commute f by fact
show ?case
proof (cases "a > b")
case True
thus ?thesis by (subst fold_atLeastAtMost_nat.simps) auto
next
case False
with 1 show ?thesis
by (subst fold_atLeastAtMost_nat.simps)
(auto simp: atLeastAtMost_insertL[symmetric] fold_fun_left_comm)
qed
qed
lemma sum_atLeastAtMost_code:
"sum f {a..b} = fold_atLeastAtMost_nat (λa acc. f a + acc) a b 0"
proof -
have "comp_fun_commute (λa. (+) (f a))"
by unfold_locales (auto simp: o_def add_ac)
thus ?thesis
by (simp add: sum.eq_fold fold_atLeastAtMost_nat o_def)
qed
lemma prod_atLeastAtMost_code:
"prod f {a..b} = fold_atLeastAtMost_nat (λa acc. f a * acc) a b 1"
proof -
have "comp_fun_commute (λa. (*) (f a))"
by unfold_locales (auto simp: o_def mult_ac)
thus ?thesis
by (simp add: prod.eq_fold fold_atLeastAtMost_nat o_def)
qed
lemma pairs_le_eq_Sigma: "{(i, j). i + j ≤ m} = Sigma (atMost m) (λr. atMost (m - r))"
for m :: nat
by auto
lemma sum_up_index_split: "(∑k≤m + n. f k) = (∑k≤m. f k) + (∑k = Suc m..m + n. f k)"
by (metis atLeast0AtMost Suc_eq_plus1 le0 sum.ub_add_nat)
lemma Sigma_interval_disjoint: "(SIGMA i:A. {..v i}) ∩ (SIGMA i:A.{v i<..w}) = {}"
for w :: "'a::order"
by auto
lemma product_atMost_eq_Un: "A × {..m} = (SIGMA i:A.{..m - i}) ∪ (SIGMA i:A.{m - i<..m})"
for m :: nat
by auto
lemma polynomial_product:
fixes x :: "'a::idom"
assumes m: "⋀i. i > m ⟹ a i = 0"
and n: "⋀j. j > n ⟹ b j = 0"
shows "(∑i≤m. (a i) * x ^ i) * (∑j≤n. (b j) * x ^ j) =
(∑r≤m + n. (∑k≤r. (a k) * (b (r - k))) * x ^ r)"
proof -
have "⋀i j. ⟦m + n - i < j; a i ≠ 0⟧ ⟹ b j = 0"
by (meson le_add_diff leI le_less_trans m n)
then have §: "(∑(i,j)∈(SIGMA i:{..m+n}. {m+n - i<..m+n}). a i * x ^ i * (b j * x ^ j)) = 0"
by (clarsimp simp add: sum_Un Sigma_interval_disjoint intro!: sum.neutral)
have "(∑i≤m. (a i) * x ^ i) * (∑j≤n. (b j) * x ^ j) = (∑i≤m. ∑j≤n. (a i * x ^ i) * (b j * x ^ j))"
by (rule sum_product)
also have "… = (∑i≤m + n. ∑j≤n + m. a i * x ^ i * (b j * x ^ j))"
using assms by (auto simp: sum_up_index_split)
also have "… = (∑r≤m + n. ∑j≤m + n - r. a r * x ^ r * (b j * x ^ j))"
by (simp add: add_ac sum.Sigma product_atMost_eq_Un sum_Un Sigma_interval_disjoint §)
also have "… = (∑(i,j)∈{(i,j). i+j ≤ m+n}. (a i * x ^ i) * (b j * x ^ j))"
by (auto simp: pairs_le_eq_Sigma sum.Sigma)
also have "... = (∑k≤m + n. ∑i≤k. a i * x ^ i * (b (k - i) * x ^ (k - i)))"
by (rule sum.triangle_reindex_eq)
also have "… = (∑r≤m + n. (∑k≤r. (a k) * (b (r - k))) * x ^ r)"
by (auto simp: algebra_simps sum_distrib_left simp flip: power_add intro!: sum.cong)
finally show ?thesis .
qed
end