(* Title: HOL/Library/Extended_Nonnegative_Real.thy
Author: Johannes Hölzl
*)
subsection \<open>The type of non-negative extended real numbers\<close>
theory Extended_Nonnegative_Real
imports Extended_Real Indicator_Function
begin
lemma ereal_ineq_diff_add:
assumes "b \ (-\::ereal)" "a \ b"
shows "a = b + (a-b)"
by (metis add.commute assms ereal_eq_minus_iff ereal_minus_le_iff ereal_plus_eq_PInfty)
lemma Limsup_const_add:
fixes c :: "'a::{complete_linorder, linorder_topology, topological_monoid_add, ordered_ab_semigroup_add}"
shows "F \ bot \ Limsup F (\x. c + f x) = c + Limsup F f"
by (rule Limsup_compose_continuous_mono)
(auto intro!: monoI add_mono continuous_on_add continuous_on_id continuous_on_const)
lemma Liminf_const_add:
fixes c :: "'a::{complete_linorder, linorder_topology, topological_monoid_add, ordered_ab_semigroup_add}"
shows "F \ bot \ Liminf F (\x. c + f x) = c + Liminf F f"
by (rule Liminf_compose_continuous_mono)
(auto intro!: monoI add_mono continuous_on_add continuous_on_id continuous_on_const)
lemma Liminf_add_const:
fixes c :: "'a::{complete_linorder, linorder_topology, topological_monoid_add, ordered_ab_semigroup_add}"
shows "F \ bot \ Liminf F (\x. f x + c) = Liminf F f + c"
by (rule Liminf_compose_continuous_mono)
(auto intro!: monoI add_mono continuous_on_add continuous_on_id continuous_on_const)
lemma sums_offset:
fixes f g :: "nat \ 'a :: {t2_space, topological_comm_monoid_add}"
assumes "(\n. f (n + i)) sums l" shows "f sums (l + (\j
proof -
have "(\k. (\nj l + (\j
using assms by (auto intro!: tendsto_add simp: sums_def)
moreover
{ fix k :: nat
have "(\jj=i..j=0..
by (subst sum.union_disjoint[symmetric]) (auto intro!: sum.cong)
also have "(\j=i..j\(\n. n + i)`{0..
unfolding image_add_atLeastLessThan by simp
finally have "(\jnj
by (auto simp: inj_on_def atLeast0LessThan sum.reindex) }
ultimately have "(\k. (\n l + (\j
by simp
then show ?thesis
unfolding sums_def by (rule LIMSEQ_offset)
qed
lemma suminf_offset:
fixes f g :: "nat \ 'a :: {t2_space, topological_comm_monoid_add}"
shows "summable (\j. f (j + i)) \ suminf f = (\j. f (j + i)) + (\j
by (intro sums_unique[symmetric] sums_offset summable_sums)
lemma eventually_at_left_1: "(\z::real. 0 < z \ z < 1 \ P z) \ eventually P (at_left 1)"
by (subst eventually_at_left[of 0]) (auto intro: exI[of _ 0])
lemma mult_eq_1:
fixes a b :: "'a :: {ordered_semiring, comm_monoid_mult}"
shows "0 \ a \ a \ 1 \ b \ 1 \ a * b = 1 \ (a = 1 \ b = 1)"
by (metis mult.left_neutral eq_iff mult.commute mult_right_mono)
lemma ereal_add_diff_cancel:
fixes a b :: ereal
shows "\b\ \ \ \ (a + b) - b = a"
by (cases a b rule: ereal2_cases) auto
lemma add_top:
fixes x :: "'a::{order_top, ordered_comm_monoid_add}"
shows "0 \ x \ x + top = top"
by (intro top_le add_increasing order_refl)
lemma top_add:
fixes x :: "'a::{order_top, ordered_comm_monoid_add}"
shows "0 \ x \ top + x = top"
by (intro top_le add_increasing2 order_refl)
lemma le_lfp: "mono f \ x \ lfp f \ f x \ lfp f"
by (subst lfp_unfold) (auto dest: monoD)
lemma lfp_transfer:
assumes \<alpha>: "sup_continuous \<alpha>" and f: "sup_continuous f" and mg: "mono g"
assumes bot: "\ bot \ lfp g" and eq: "\x. x \ lfp f \ \ (f x) = g (\ x)"
shows "\ (lfp f) = lfp g"
proof (rule antisym)
note mf = sup_continuous_mono[OF f]
have f_le_lfp: "(f ^^ i) bot \ lfp f" for i
by (induction i) (auto intro: le_lfp mf)
have "\ ((f ^^ i) bot) \ lfp g" for i
by (induction i) (auto simp: bot eq f_le_lfp intro!: le_lfp mg)
then show "\ (lfp f) \ lfp g"
unfolding sup_continuous_lfp[OF f]
by (subst \<alpha>[THEN sup_continuousD])
(auto intro!: mono_funpow sup_continuous_mono[OF f] SUP_least)
show "lfp g \ \ (lfp f)"
by (rule lfp_lowerbound) (simp add: eq[symmetric] lfp_fixpoint[OF mf])
qed
lemma sup_continuous_applyD: "sup_continuous f \ sup_continuous (\x. f x h)"
using sup_continuous_apply[THEN sup_continuous_compose] .
lemma sup_continuous_SUP[order_continuous_intros]:
fixes M :: "_ \ _ \ 'a::complete_lattice"
assumes M: "\i. i \ I \ sup_continuous (M i)"
shows "sup_continuous (SUP i\I. M i)"
unfolding sup_continuous_def by (auto simp add: sup_continuousD [OF M] image_comp intro: SUP_commute)
lemma sup_continuous_apply_SUP[order_continuous_intros]:
fixes M :: "_ \ _ \ 'a::complete_lattice"
shows "(\i. i \ I \ sup_continuous (M i)) \ sup_continuous (\x. SUP i\I. M i x)"
unfolding SUP_apply[symmetric] by (rule sup_continuous_SUP)
lemma sup_continuous_lfp'[order_continuous_intros]:
assumes 1: "sup_continuous f"
assumes 2: "\g. sup_continuous g \ sup_continuous (f g)"
shows "sup_continuous (lfp f)"
proof -
have "sup_continuous ((f ^^ i) bot)" for i
proof (induction i)
case (Suc i) then show ?case
by (auto intro!: 2)
qed (simp add: bot_fun_def sup_continuous_const)
then show ?thesis
unfolding sup_continuous_lfp[OF 1] by (intro order_continuous_intros)
qed
lemma sup_continuous_lfp''[order_continuous_intros]:
assumes 1: "\s. sup_continuous (f s)"
assumes 2: "\g. sup_continuous g \ sup_continuous (\s. f s (g s))"
shows "sup_continuous (\x. lfp (f x))"
proof -
have "sup_continuous (\x. (f x ^^ i) bot)" for i
proof (induction i)
case (Suc i) then show ?case
by (auto intro!: 2)
qed (simp add: bot_fun_def sup_continuous_const)
then show ?thesis
unfolding sup_continuous_lfp[OF 1] by (intro order_continuous_intros)
qed
lemma mono_INF_fun:
"(\x y. mono (F x y)) \ mono (\z x. INF y \ X x. F x y z :: 'a :: complete_lattice)"
by (auto intro!: INF_mono[OF bexI] simp: le_fun_def mono_def)
lemma continuous_on_cmult_ereal:
"\c::ereal\ \ \ \ continuous_on A f \ continuous_on A (\x. c * f x)"
using tendsto_cmult_ereal[of c f "f x" "at x within A" for x]
by (auto simp: continuous_on_def simp del: tendsto_cmult_ereal)
lemma real_of_nat_Sup:
assumes "A \ {}" "bdd_above A"
shows "of_nat (Sup A) = (SUP a\A. of_nat a :: real)"
proof (intro antisym)
show "(SUP a\A. of_nat a::real) \ of_nat (Sup A)"
using assms by (intro cSUP_least of_nat_mono) (auto intro: cSup_upper)
have "Sup A \ A"
using assms by (auto simp: Sup_nat_def bdd_above_nat)
then show "of_nat (Sup A) \ (SUP a\A. of_nat a::real)"
by (intro cSUP_upper bdd_above_image_mono assms) (auto simp: mono_def)
qed
lemma (in complete_lattice) SUP_sup_const1:
"I \ {} \ (SUP i\I. sup c (f i)) = sup c (SUP i\I. f i)"
using SUP_sup_distrib[of "\_. c" I f] by simp
lemma (in complete_lattice) SUP_sup_const2:
"I \ {} \ (SUP i\I. sup (f i) c) = sup (SUP i\I. f i) c"
using SUP_sup_distrib[of f I "\_. c"] by simp
lemma one_less_of_natD:
assumes "(1::'a::linordered_semidom) < of_nat n" shows "1 < n"
by (cases n) (use assms in auto)
subsection \<open>Defining the extended non-negative reals\<close>
text \<open>Basic definitions and type class setup\<close>
typedef ennreal = "{x :: ereal. 0 \ x}"
morphisms enn2ereal e2ennreal'
by auto
definition "e2ennreal x = e2ennreal' (max 0 x)"
lemma enn2ereal_range: "e2ennreal ` {0..} = UNIV"
proof -
have "\y\0. x = e2ennreal y" for x
by (cases x) (auto simp: e2ennreal_def max_absorb2)
then show ?thesis
by (auto simp: image_iff Bex_def)
qed
lemma type_definition_ennreal': "type_definition enn2ereal e2ennreal {x. 0 \ x}"
using type_definition_ennreal
by (auto simp: type_definition_def e2ennreal_def max_absorb2)
setup_lifting type_definition_ennreal'
declare [[coercion e2ennreal]]
instantiation ennreal :: complete_linorder
begin
lift_definition top_ennreal :: ennreal is top by (rule top_greatest)
lift_definition bot_ennreal :: ennreal is 0 by (rule order_refl)
lift_definition sup_ennreal :: "ennreal \ ennreal \ ennreal" is sup by (rule le_supI1)
lift_definition inf_ennreal :: "ennreal \ ennreal \ ennreal" is inf by (rule le_infI)
lift_definition Inf_ennreal :: "ennreal set \ ennreal" is "Inf"
by (rule Inf_greatest)
lift_definition Sup_ennreal :: "ennreal set \ ennreal" is "sup 0 \ Sup"
by auto
lift_definition less_eq_ennreal :: "ennreal \ ennreal \ bool" is "(\)" .
lift_definition less_ennreal :: "ennreal \ ennreal \ bool" is "(<)" .
instance
by standard
(transfer ; auto simp: Inf_lower Inf_greatest Sup_upper Sup_least le_max_iff_disj max.absorb1)+
end
lemma pcr_ennreal_enn2ereal[simp]: "pcr_ennreal (enn2ereal x) x"
by (simp add: ennreal.pcr_cr_eq cr_ennreal_def)
lemma rel_fun_eq_pcr_ennreal: "rel_fun (=) pcr_ennreal f g \ f = enn2ereal \ g"
by (auto simp: rel_fun_def ennreal.pcr_cr_eq cr_ennreal_def)
instantiation ennreal :: infinity
begin
definition infinity_ennreal :: ennreal
where
[simp]: "\ = (top::ennreal)"
instance ..
end
instantiation ennreal :: "{semiring_1_no_zero_divisors, comm_semiring_1}"
begin
lift_definition one_ennreal :: ennreal is 1 by simp
lift_definition zero_ennreal :: ennreal is 0 by simp
lift_definition plus_ennreal :: "ennreal \ ennreal \ ennreal" is "(+)" by simp
lift_definition times_ennreal :: "ennreal \ ennreal \ ennreal" is "(*)" by simp
instance
by standard (transfer; auto simp: field_simps ereal_right_distrib)+
end
instantiation ennreal :: minus
begin
lift_definition minus_ennreal :: "ennreal \ ennreal \ ennreal" is "\a b. max 0 (a - b)"
by simp
instance ..
end
instance ennreal :: numeral ..
instantiation ennreal :: inverse
begin
lift_definition inverse_ennreal :: "ennreal \ ennreal" is inverse
by (rule inverse_ereal_ge0I)
definition divide_ennreal :: "ennreal \ ennreal \ ennreal"
where "x div y = x * inverse (y :: ennreal)"
instance ..
end
lemma ennreal_zero_less_one: "0 < (1::ennreal)" \<comment> \<open>TODO: remove\<close>
by transfer auto
instance ennreal :: dioid
proof (standard; transfer)
fix a b :: ereal assume "0 \ a" "0 \ b" then show "(a \ b) = (\c\Collect ((\) 0). b = a + c)"
unfolding ereal_ex_split Bex_def
by (cases a b rule: ereal2_cases) (auto intro!: exI[of _ "real_of_ereal (b - a)"])
qed
instance ennreal :: ordered_comm_semiring
by standard
(transfer ; auto intro: add_mono mult_mono mult_ac ereal_left_distrib ereal_mult_left_mono)+
instance ennreal :: linordered_nonzero_semiring
proof
fix a b::ennreal
show "a < b \ a + 1 < b + 1"
by transfer (simp add: add_right_mono ereal_add_cancel_right less_le)
qed (transfer; simp)
instance ennreal :: strict_ordered_ab_semigroup_add
proof
fix a b c d :: ennreal show "a < b \ c < d \ a + c < b + d"
by transfer (auto intro!: ereal_add_strict_mono)
qed
declare [[coercion "of_nat :: nat \ ennreal"]]
lemma e2ennreal_neg: "x \ 0 \ e2ennreal x = 0"
unfolding zero_ennreal_def e2ennreal_def by (simp add: max_absorb1)
lemma e2ennreal_mono: "x \ y \ e2ennreal x \ e2ennreal y"
by (cases "0 \ x" "0 \ y" rule: bool.exhaust[case_product bool.exhaust])
(auto simp: e2ennreal_neg less_eq_ennreal.abs_eq eq_onp_def)
lemma enn2ereal_nonneg[simp]: "0 \ enn2ereal x"
using ennreal.enn2ereal[of x] by simp
lemma ereal_ennreal_cases:
obtains b where "0 \ a" "a = enn2ereal b" | "a < 0"
using e2ennreal'_inverse[of a, symmetric] by (cases "0 \ a") (auto intro: enn2ereal_nonneg)
lemma rel_fun_liminf[transfer_rule]: "rel_fun (rel_fun (=) pcr_ennreal) pcr_ennreal liminf liminf"
proof -
have "rel_fun (rel_fun (=) pcr_ennreal) pcr_ennreal (\x. sup 0 (liminf x)) liminf"
unfolding liminf_SUP_INF[abs_def] by (transfer_prover_start, transfer_step+; simp)
then show ?thesis
apply (subst (asm) (2) rel_fun_def)
apply (subst (2) rel_fun_def)
apply (auto simp: comp_def max.absorb2 Liminf_bounded rel_fun_eq_pcr_ennreal)
done
qed
lemma rel_fun_limsup[transfer_rule]: "rel_fun (rel_fun (=) pcr_ennreal) pcr_ennreal limsup limsup"
proof -
have "rel_fun (rel_fun (=) pcr_ennreal) pcr_ennreal (\x. INF n. sup 0 (SUP i\{n..}. x i)) limsup"
unfolding limsup_INF_SUP[abs_def] by (transfer_prover_start, transfer_step+; simp)
then show ?thesis
unfolding limsup_INF_SUP[abs_def]
apply (subst (asm) (2) rel_fun_def)
apply (subst (2) rel_fun_def)
apply (auto simp: comp_def max.absorb2 Sup_upper2 rel_fun_eq_pcr_ennreal)
apply (subst (asm) max.absorb2)
apply (auto intro: SUP_upper2)
done
qed
lemma sum_enn2ereal[simp]: "(\i. i \ I \ 0 \ f i) \ (\i\I. enn2ereal (f i)) = enn2ereal (sum f I)"
by (induction I rule: infinite_finite_induct) (auto simp: sum_nonneg zero_ennreal.rep_eq plus_ennreal.rep_eq)
lemma transfer_e2ennreal_sum [transfer_rule]:
"rel_fun (rel_fun (=) pcr_ennreal) (rel_fun (=) pcr_ennreal) sum sum"
by (auto intro!: rel_funI simp: rel_fun_eq_pcr_ennreal comp_def)
lemma enn2ereal_of_nat[simp]: "enn2ereal (of_nat n) = ereal n"
by (induction n) (auto simp: zero_ennreal.rep_eq one_ennreal.rep_eq plus_ennreal.rep_eq)
lemma enn2ereal_numeral[simp]: "enn2ereal (numeral a) = numeral a"
by (metis enn2ereal_of_nat numeral_eq_ereal of_nat_numeral)
lemma transfer_numeral[transfer_rule]: "pcr_ennreal (numeral a) (numeral a)"
unfolding cr_ennreal_def pcr_ennreal_def by auto
subsection \<open>Cancellation simprocs\<close>
lemma ennreal_add_left_cancel: "a + b = a + c \ a = (\::ennreal) \ b = c"
unfolding infinity_ennreal_def by transfer (simp add: top_ereal_def ereal_add_cancel_left)
lemma ennreal_add_left_cancel_le: "a + b \ a + c \ a = (\::ennreal) \ b \ c"
unfolding infinity_ennreal_def by transfer (simp add: ereal_add_le_add_iff top_ereal_def disj_commute)
lemma ereal_add_left_cancel_less:
fixes a b c :: ereal
shows "0 \ a \ 0 \ b \ a + b < a + c \ a \ \ \ b < c"
by (cases a b c rule: ereal3_cases) auto
lemma ennreal_add_left_cancel_less: "a + b < a + c \ a \ (\::ennreal) \ b < c"
unfolding infinity_ennreal_def
by transfer (simp add: top_ereal_def ereal_add_left_cancel_less)
ML \<open>
structure Cancel_Ennreal_Common =
struct
(* copied from src/HOL/Tools/nat_numeral_simprocs.ML *)
fun find_first_t _ _ [] = raise TERM("find_first_t", [])
| find_first_t past u (t::terms) =
if u aconv t then (rev past @ terms)
else find_first_t (t::past) u terms
fun dest_summing (Const (\<^const_name>\<open>Groups.plus\<close>, _) $ t $ u, ts) =
dest_summing (t, dest_summing (u, ts))
| dest_summing (t, ts) = t :: ts
val mk_sum = Arith_Data.long_mk_sum
fun dest_sum t = dest_summing (t, [])
val find_first = find_first_t []
val trans_tac = Numeral_Simprocs.trans_tac
val norm_ss =
simpset_of (put_simpset HOL_basic_ss \<^context>
addsimps @{thms ac_simps add_0_left add_0_right})
fun norm_tac ctxt = ALLGOALS (simp_tac (put_simpset norm_ss ctxt))
fun simplify_meta_eq ctxt cancel_th th =
Arith_Data.simplify_meta_eq [] ctxt
([th, cancel_th] MRS trans)
fun mk_eq (a, b) = HOLogic.mk_Trueprop (HOLogic.mk_eq (a, b))
end
structure Eq_Ennreal_Cancel = ExtractCommonTermFun
(open Cancel_Ennreal_Common
val mk_bal = HOLogic.mk_eq
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>HOL.eq\<close> \<^typ>\<open>ennreal\<close>
fun simp_conv _ _ = SOME @{thm ennreal_add_left_cancel}
)
structure Le_Ennreal_Cancel = ExtractCommonTermFun
(open Cancel_Ennreal_Common
val mk_bal = HOLogic.mk_binrel \<^const_name>\<open>Orderings.less_eq\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Orderings.less_eq\<close> \<^typ>\<open>ennreal\<close>
fun simp_conv _ _ = SOME @{thm ennreal_add_left_cancel_le}
)
structure Less_Ennreal_Cancel = ExtractCommonTermFun
(open Cancel_Ennreal_Common
val mk_bal = HOLogic.mk_binrel \<^const_name>\<open>Orderings.less\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Orderings.less\<close> \<^typ>\<open>ennreal\<close>
fun simp_conv _ _ = SOME @{thm ennreal_add_left_cancel_less}
)
\<close>
simproc_setup ennreal_eq_cancel
("(l::ennreal) + m = n" | "(l::ennreal) = m + n") =
\<open>fn phi => fn ctxt => fn ct => Eq_Ennreal_Cancel.proc ctxt (Thm.term_of ct)\<close>
simproc_setup ennreal_le_cancel
("(l::ennreal) + m \ n" | "(l::ennreal) \ m + n") =
\<open>fn phi => fn ctxt => fn ct => Le_Ennreal_Cancel.proc ctxt (Thm.term_of ct)\<close>
simproc_setup ennreal_less_cancel
("(l::ennreal) + m < n" | "(l::ennreal) < m + n") =
\<open>fn phi => fn ctxt => fn ct => Less_Ennreal_Cancel.proc ctxt (Thm.term_of ct)\<close>
subsection \<open>Order with top\<close>
lemma ennreal_zero_less_top[simp]: "0 < (top::ennreal)"
by transfer (simp add: top_ereal_def)
lemma ennreal_one_less_top[simp]: "1 < (top::ennreal)"
by transfer (simp add: top_ereal_def)
lemma ennreal_zero_neq_top[simp]: "0 \ (top::ennreal)"
by transfer (simp add: top_ereal_def)
lemma ennreal_top_neq_zero[simp]: "(top::ennreal) \ 0"
by transfer (simp add: top_ereal_def)
lemma ennreal_top_neq_one[simp]: "top \ (1::ennreal)"
by transfer (simp add: top_ereal_def one_ereal_def flip: ereal_max)
lemma ennreal_one_neq_top[simp]: "1 \ (top::ennreal)"
by transfer (simp add: top_ereal_def one_ereal_def flip: ereal_max)
lemma ennreal_add_less_top[simp]:
fixes a b :: ennreal
shows "a + b < top \ a < top \ b < top"
by transfer (auto simp: top_ereal_def)
lemma ennreal_add_eq_top[simp]:
fixes a b :: ennreal
shows "a + b = top \ a = top \ b = top"
by transfer (auto simp: top_ereal_def)
lemma ennreal_sum_less_top[simp]:
fixes f :: "'a \ ennreal"
shows "finite I \ (\i\I. f i) < top \ (\i\I. f i < top)"
by (induction I rule: finite_induct) auto
lemma ennreal_sum_eq_top[simp]:
fixes f :: "'a \ ennreal"
shows "finite I \ (\i\I. f i) = top \ (\i\I. f i = top)"
by (induction I rule: finite_induct) auto
lemma ennreal_mult_eq_top_iff:
fixes a b :: ennreal
shows "a * b = top \ (a = top \ b \ 0) \ (b = top \ a \ 0)"
by transfer (auto simp: top_ereal_def)
lemma ennreal_top_eq_mult_iff:
fixes a b :: ennreal
shows "top = a * b \ (a = top \ b \ 0) \ (b = top \ a \ 0)"
using ennreal_mult_eq_top_iff[of a b] by auto
lemma ennreal_mult_less_top:
fixes a b :: ennreal
shows "a * b < top \ (a = 0 \ b = 0 \ (a < top \ b < top))"
by transfer (auto simp add: top_ereal_def)
lemma top_power_ennreal: "top ^ n = (if n = 0 then 1 else top :: ennreal)"
by (induction n) (simp_all add: ennreal_mult_eq_top_iff)
lemma ennreal_prod_eq_0[simp]:
fixes f :: "'a \ ennreal"
shows "(prod f A = 0) = (finite A \ (\i\A. f i = 0))"
by (induction A rule: infinite_finite_induct) auto
lemma ennreal_prod_eq_top:
fixes f :: "'a \ ennreal"
shows "(\i\I. f i) = top \ (finite I \ ((\i\I. f i \ 0) \ (\i\I. f i = top)))"
by (induction I rule: infinite_finite_induct) (auto simp: ennreal_mult_eq_top_iff)
lemma ennreal_top_mult: "top * a = (if a = 0 then 0 else top :: ennreal)"
by (simp add: ennreal_mult_eq_top_iff)
lemma ennreal_mult_top: "a * top = (if a = 0 then 0 else top :: ennreal)"
by (simp add: ennreal_mult_eq_top_iff)
lemma enn2ereal_eq_top_iff[simp]: "enn2ereal x = \ \ x = top"
by transfer (simp add: top_ereal_def)
lemma enn2ereal_top[simp]: "enn2ereal top = \"
by transfer (simp add: top_ereal_def)
lemma e2ennreal_infty[simp]: "e2ennreal \ = top"
by (simp add: top_ennreal.abs_eq top_ereal_def)
lemma ennreal_top_minus[simp]: "top - x = (top::ennreal)"
by transfer (auto simp: top_ereal_def max_def)
lemma minus_top_ennreal: "x - top = (if x = top then top else 0:: ennreal)"
by transfer (use ereal_eq_minus_iff top_ereal_def in force)
lemma bot_ennreal: "bot = (0::ennreal)"
by transfer rule
lemma ennreal_of_nat_neq_top[simp]: "of_nat i \ (top::ennreal)"
by (induction i) auto
lemma numeral_eq_of_nat: "(numeral a::ennreal) = of_nat (numeral a)"
by simp
lemma of_nat_less_top: "of_nat i < (top::ennreal)"
using less_le_trans[of "of_nat i" "of_nat (Suc i)" "top::ennreal"]
by simp
lemma top_neq_numeral[simp]: "top \ (numeral i::ennreal)"
using of_nat_less_top[of "numeral i"] by simp
lemma ennreal_numeral_less_top[simp]: "numeral i < (top::ennreal)"
using of_nat_less_top[of "numeral i"] by simp
lemma ennreal_add_bot[simp]: "bot + x = (x::ennreal)"
by transfer simp
instance ennreal :: semiring_char_0
proof (standard, safe intro!: linorder_injI)
have *: "1 + of_nat k \ (0::ennreal)" for k
using add_pos_nonneg[OF zero_less_one, of "of_nat k :: ennreal"] by auto
fix x y :: nat assume "x < y" "of_nat x = (of_nat y::ennreal)" then show False
by (auto simp add: less_iff_Suc_add *)
qed
subsection \<open>Arithmetic\<close>
lemma ennreal_minus_zero[simp]: "a - (0::ennreal) = a"
by transfer (auto simp: max_def)
lemma ennreal_add_diff_cancel_right[simp]:
fixes x y z :: ennreal shows "y \ top \ (x + y) - y = x"
by transfer (metis ereal_eq_minus_iff max_absorb2 not_MInfty_nonneg top_ereal_def)
lemma ennreal_add_diff_cancel_left[simp]:
fixes x y z :: ennreal shows "y \ top \ (y + x) - y = x"
by (simp add: add.commute)
lemma
fixes a b :: ennreal
shows "a - b = 0 \ a \ b"
by transfer (metis ereal_diff_gr0 le_cases max.absorb2 not_less)
lemma ennreal_minus_cancel:
fixes a b c :: ennreal
shows "c \ top \ a \ c \ b \ c \ c - a = c - b \ a = b"
apply transfer
subgoal for a b c
by (cases a b c rule: ereal3_cases)
(auto simp: top_ereal_def max_def split: if_splits)
done
lemma sup_const_add_ennreal:
fixes a b c :: "ennreal"
shows "sup (c + a) (c + b) = c + sup a b"
by transfer (metis add_left_mono le_cases sup.absorb2 sup.orderE)
lemma ennreal_diff_add_assoc:
fixes a b c :: ennreal
shows "a \ b \ c + b - a = c + (b - a)"
apply transfer
subgoal for a b c
by (cases a b c rule: ereal3_cases) (auto simp: field_simps max_absorb2)
done
lemma mult_divide_eq_ennreal:
fixes a b :: ennreal
shows "b \ 0 \ b \ top \ (a * b) / b = a"
unfolding divide_ennreal_def
apply transfer
apply (subst mult.assoc)
apply (simp add: top_ereal_def flip: divide_ereal_def)
done
lemma divide_mult_eq: "a \ 0 \ a \ \ \ x * a / (b * a) = x / (b::ennreal)"
unfolding divide_ennreal_def infinity_ennreal_def
apply transfer
subgoal for a b c
apply (cases a b c rule: ereal3_cases)
apply (auto simp: top_ereal_def)
done
done
lemma ennreal_mult_divide_eq:
fixes a b :: ennreal
shows "b \ 0 \ b \ top \ (a * b) / b = a"
unfolding divide_ennreal_def
apply transfer
apply (subst mult.assoc)
apply (simp add: top_ereal_def flip: divide_ereal_def)
done
lemma ennreal_add_diff_cancel:
fixes a b :: ennreal
shows "b \ \ \ (a + b) - b = a"
unfolding infinity_ennreal_def
by transfer (simp add: max_absorb2 top_ereal_def ereal_add_diff_cancel)
lemma ennreal_minus_eq_0:
"a - b = 0 \ a \ (b::ennreal)"
by transfer (metis ereal_diff_gr0 le_cases max.absorb2 not_less)
lemma ennreal_mono_minus_cancel:
fixes a b c :: ennreal
shows "a - b \ a - c \ a < top \ b \ a \ c \ a \ c \ b"
by transfer
(auto simp add: max.absorb2 ereal_diff_positive top_ereal_def dest: ereal_mono_minus_cancel)
lemma ennreal_mono_minus:
fixes a b c :: ennreal
shows "c \ b \ a - b \ a - c"
by transfer (meson ereal_minus_mono max.mono order_refl)
lemma ennreal_minus_pos_iff:
fixes a b :: ennreal
shows "a < top \ b < top \ 0 < a - b \ b < a"
by transfer (use add.left_neutral ereal_minus_le_iff less_irrefl not_less in fastforce)
lemma ennreal_inverse_top[simp]: "inverse top = (0::ennreal)"
by transfer (simp add: top_ereal_def ereal_inverse_eq_0)
lemma ennreal_inverse_zero[simp]: "inverse 0 = (top::ennreal)"
by transfer (simp add: top_ereal_def ereal_inverse_eq_0)
lemma ennreal_top_divide: "top / (x::ennreal) = (if x = top then 0 else top)"
unfolding divide_ennreal_def
by transfer (simp add: top_ereal_def ereal_inverse_eq_0 ereal_0_gt_inverse)
lemma ennreal_zero_divide[simp]: "0 / (x::ennreal) = 0"
by (simp add: divide_ennreal_def)
lemma ennreal_divide_zero[simp]: "x / (0::ennreal) = (if x = 0 then 0 else top)"
by (simp add: divide_ennreal_def ennreal_mult_top)
lemma ennreal_divide_top[simp]: "x / (top::ennreal) = 0"
by (simp add: divide_ennreal_def ennreal_top_mult)
lemma ennreal_times_divide: "a * (b / c) = a * b / (c::ennreal)"
unfolding divide_ennreal_def
by transfer (simp add: divide_ereal_def[symmetric] ereal_times_divide_eq)
lemma ennreal_zero_less_divide: "0 < a / b \ (0 < a \ b < (top::ennreal))"
unfolding divide_ennreal_def
by transfer (auto simp: ereal_zero_less_0_iff top_ereal_def ereal_0_gt_inverse)
lemma divide_right_mono_ennreal:
fixes a b c :: ennreal
shows "a \ b \ a / c \ b / c"
unfolding divide_ennreal_def by (intro mult_mono) auto
lemma ennreal_mult_strict_right_mono: "(a::ennreal) < c \ 0 < b \ b < top \ a * b < c * b"
by transfer (auto intro!: ereal_mult_strict_right_mono)
lemma ennreal_indicator_less[simp]:
"indicator A x \ (indicator B x::ennreal) \ (x \ A \ x \ B)"
by (simp add: indicator_def not_le)
lemma ennreal_inverse_positive: "0 < inverse x \ (x::ennreal) \ top"
by transfer (simp add: ereal_0_gt_inverse top_ereal_def)
lemma ennreal_inverse_mult': "((0 < b \ a < top) \ (0 < a \ b < top)) \ inverse (a * b::ennreal) = inverse a * inverse b"
apply transfer
subgoal for a b
by (cases a b rule: ereal2_cases) (auto simp: top_ereal_def)
done
lemma ennreal_inverse_mult: "a < top \ b < top \ inverse (a * b::ennreal) = inverse a * inverse b"
apply transfer
subgoal for a b
by (cases a b rule: ereal2_cases) (auto simp: top_ereal_def)
done
lemma ennreal_inverse_1[simp]: "inverse (1::ennreal) = 1"
by transfer simp
lemma ennreal_inverse_eq_0_iff[simp]: "inverse (a::ennreal) = 0 \ a = top"
by transfer (simp add: ereal_inverse_eq_0 top_ereal_def)
lemma ennreal_inverse_eq_top_iff[simp]: "inverse (a::ennreal) = top \ a = 0"
by transfer (simp add: top_ereal_def)
lemma ennreal_divide_eq_0_iff[simp]: "(a::ennreal) / b = 0 \ (a = 0 \ b = top)"
by (simp add: divide_ennreal_def)
lemma ennreal_divide_eq_top_iff: "(a::ennreal) / b = top \ ((a \ 0 \ b = 0) \ (a = top \ b \ top))"
by (auto simp add: divide_ennreal_def ennreal_mult_eq_top_iff)
lemma one_divide_one_divide_ennreal[simp]: "1 / (1 / c) = (c::ennreal)"
including ennreal.lifting
unfolding divide_ennreal_def
by transfer auto
lemma ennreal_mult_left_cong:
"((a::ennreal) \ 0 \ b = c) \ a * b = a * c"
by (cases "a = 0") simp_all
lemma ennreal_mult_right_cong:
"((a::ennreal) \ 0 \ b = c) \ b * a = c * a"
by (cases "a = 0") simp_all
lemma ennreal_zero_less_mult_iff: "0 < a * b \ 0 < a \ 0 < (b::ennreal)"
by transfer (auto simp add: ereal_zero_less_0_iff le_less)
lemma less_diff_eq_ennreal:
fixes a b c :: ennreal
shows "b < top \ c < top \ a < b - c \ a + c < b"
apply transfer
subgoal for a b c
by (cases a b c rule: ereal3_cases) (auto split: split_max)
done
lemma diff_add_cancel_ennreal:
fixes a b :: ennreal shows "a \ b \ b - a + a = b"
unfolding infinity_ennreal_def
by transfer (metis (no_types) add.commute ereal_diff_positive ereal_ineq_diff_add max_def not_MInfty_nonneg)
lemma ennreal_diff_self[simp]: "a \ top \ a - a = (0::ennreal)"
by transfer (simp add: top_ereal_def)
lemma ennreal_minus_mono:
fixes a b c :: ennreal
shows "a \ c \ d \ b \ a - b \ c - d"
by transfer (meson ereal_minus_mono max.mono order_refl)
lemma ennreal_minus_eq_top[simp]: "a - (b::ennreal) = top \ a = top"
by transfer (auto simp: top_ereal_def max.absorb2 ereal_minus_eq_PInfty_iff split: split_max)
lemma ennreal_divide_self[simp]: "a \ 0 \ a < top \ a / a = (1::ennreal)"
unfolding divide_ennreal_def
apply transfer
subgoal for a
by (cases a) (auto simp: top_ereal_def)
done
subsection \<open>Coercion from \<^typ>\<open>real\<close> to \<^typ>\<open>ennreal\<close>\<close>
lift_definition ennreal :: "real \ ennreal" is "sup 0 \ ereal"
by simp
declare [[coercion ennreal]]
lemma ennreal_cong: "x = y \ ennreal x = ennreal y" by simp
lemma ennreal_cases[cases type: ennreal]:
fixes x :: ennreal
obtains (real) r :: real where "0 \ r" "x = ennreal r" | (top) "x = top"
apply transfer
subgoal for x thesis
by (cases x) (auto simp: max.absorb2 top_ereal_def)
done
lemmas ennreal2_cases = ennreal_cases[case_product ennreal_cases]
lemmas ennreal3_cases = ennreal_cases[case_product ennreal2_cases]
lemma ennreal_neq_top[simp]: "ennreal r \ top"
by transfer (simp add: top_ereal_def zero_ereal_def flip: ereal_max)
lemma top_neq_ennreal[simp]: "top \ ennreal r"
using ennreal_neq_top[of r] by (auto simp del: ennreal_neq_top)
lemma ennreal_less_top[simp]: "ennreal x < top"
by transfer (simp add: top_ereal_def max_def)
lemma ennreal_neg: "x \ 0 \ ennreal x = 0"
by transfer (simp add: max.absorb1)
lemma ennreal_inj[simp]:
"0 \ a \ 0 \ b \ ennreal a = ennreal b \ a = b"
by (transfer fixing: a b) (auto simp: max_absorb2)
lemma ennreal_le_iff[simp]: "0 \ y \ ennreal x \ ennreal y \ x \ y"
by (auto simp: ennreal_def zero_ereal_def less_eq_ennreal.abs_eq eq_onp_def split: split_max)
lemma le_ennreal_iff: "0 \ r \ x \ ennreal r \ (\q\0. x = ennreal q \ q \ r)"
by (cases x) (auto simp: top_unique)
lemma ennreal_less_iff: "0 \ r \ ennreal r < ennreal q \ r < q"
unfolding not_le[symmetric] by auto
lemma ennreal_eq_zero_iff[simp]: "0 \ x \ ennreal x = 0 \ x = 0"
by transfer (auto simp: max_absorb2)
lemma ennreal_less_zero_iff[simp]: "0 < ennreal x \ 0 < x"
by transfer (auto simp: max_def)
lemma ennreal_lessI: "0 < q \ r < q \ ennreal r < ennreal q"
by (cases "0 \ r") (auto simp: ennreal_less_iff ennreal_neg)
lemma ennreal_leI: "x \ y \ ennreal x \ ennreal y"
by (cases "0 \ y") (auto simp: ennreal_neg)
lemma enn2ereal_ennreal[simp]: "0 \ x \ enn2ereal (ennreal x) = x"
by transfer (simp add: max_absorb2)
lemma e2ennreal_enn2ereal[simp]: "e2ennreal (enn2ereal x) = x"
by (simp add: e2ennreal_def max_absorb2 ennreal.enn2ereal_inverse)
lemma enn2ereal_e2ennreal: "x \ 0 \ enn2ereal (e2ennreal x) = x"
by (metis e2ennreal_enn2ereal ereal_ennreal_cases not_le)
lemma e2ennreal_ereal [simp]: "e2ennreal (ereal x) = ennreal x"
by (metis e2ennreal_def enn2ereal_inverse ennreal.rep_eq sup_ereal_def)
lemma ennreal_0[simp]: "ennreal 0 = 0"
by (simp add: ennreal_def max.absorb1 zero_ennreal.abs_eq)
lemma ennreal_1[simp]: "ennreal 1 = 1"
by transfer (simp add: max_absorb2)
lemma ennreal_eq_0_iff: "ennreal x = 0 \ x \ 0"
by (cases "0 \ x") (auto simp: ennreal_neg)
lemma ennreal_le_iff2: "ennreal x \ ennreal y \ ((0 \ y \ x \ y) \ (x \ 0 \ y \ 0))"
by (cases "0 \ y") (auto simp: ennreal_eq_0_iff ennreal_neg)
lemma ennreal_eq_1[simp]: "ennreal x = 1 \ x = 1"
by (cases "0 \ x") (auto simp: ennreal_neg simp flip: ennreal_1)
lemma ennreal_le_1[simp]: "ennreal x \ 1 \ x \ 1"
by (cases "0 \ x") (auto simp: ennreal_neg simp flip: ennreal_1)
lemma ennreal_ge_1[simp]: "ennreal x \ 1 \ x \ 1"
by (cases "0 \ x") (auto simp: ennreal_neg simp flip: ennreal_1)
lemma one_less_ennreal[simp]: "1 < ennreal x \ 1 < x"
by transfer (auto simp: max.absorb2 less_max_iff_disj)
lemma ennreal_plus[simp]:
"0 \ a \ 0 \ b \ ennreal (a + b) = ennreal a + ennreal b"
by (transfer fixing: a b) (auto simp: max_absorb2)
lemma add_mono_ennreal: "x < ennreal y \ x' < ennreal y' \ x + x' < ennreal (y + y')"
by (metis (full_types) add_strict_mono ennreal_less_zero_iff ennreal_plus less_le not_less zero_le)
lemma sum_ennreal[simp]: "(\i. i \ I \ 0 \ f i) \ (\i\I. ennreal (f i)) = ennreal (sum f I)"
by (induction I rule: infinite_finite_induct) (auto simp: sum_nonneg)
lemma sum_list_ennreal[simp]:
assumes "\x. x \ set xs \ f x \ 0"
shows "sum_list (map (\x. ennreal (f x)) xs) = ennreal (sum_list (map f xs))"
using assms
proof (induction xs)
case (Cons x xs)
from Cons have "(\x\x # xs. ennreal (f x)) = ennreal (f x) + ennreal (sum_list (map f xs))"
by simp
also from Cons.prems have "\ = ennreal (f x + sum_list (map f xs))"
by (intro ennreal_plus [symmetric] sum_list_nonneg) auto
finally show ?case by simp
qed simp_all
lemma ennreal_of_nat_eq_real_of_nat: "of_nat i = ennreal (of_nat i)"
by (induction i) simp_all
lemma of_nat_le_ennreal_iff[simp]: "0 \ r \ of_nat i \ ennreal r \ of_nat i \ r"
by (simp add: ennreal_of_nat_eq_real_of_nat)
lemma ennreal_le_of_nat_iff[simp]: "ennreal r \ of_nat i \ r \ of_nat i"
by (simp add: ennreal_of_nat_eq_real_of_nat)
lemma ennreal_indicator: "ennreal (indicator A x) = indicator A x"
by (auto split: split_indicator)
lemma ennreal_numeral[simp]: "ennreal (numeral n) = numeral n"
using ennreal_of_nat_eq_real_of_nat[of "numeral n"] by simp
lemma ennreal_less_numeral_iff [simp]: "ennreal n < numeral w \ n < numeral w"
by (metis ennreal_less_iff ennreal_numeral less_le not_less zero_less_numeral)
lemma numeral_less_ennreal_iff [simp]: "numeral w < ennreal n \ numeral w < n"
using ennreal_less_iff zero_le_numeral by fastforce
lemma numeral_le_ennreal_iff [simp]: "numeral n \ ennreal m \ numeral n \ m"
by (metis not_le ennreal_less_numeral_iff)
lemma min_ennreal: "0 \ x \ 0 \ y \ min (ennreal x) (ennreal y) = ennreal (min x y)"
by (auto split: split_min)
lemma ennreal_half[simp]: "ennreal (1/2) = inverse 2"
by transfer (simp add: max.absorb2)
lemma ennreal_minus: "0 \ q \ ennreal r - ennreal q = ennreal (r - q)"
by transfer
(simp add: max.absorb2 zero_ereal_def flip: ereal_max)
lemma ennreal_minus_top[simp]: "ennreal a - top = 0"
by (simp add: minus_top_ennreal)
lemma e2eenreal_enn2ereal_diff [simp]:
"e2ennreal(enn2ereal x - enn2ereal y) = x - y" for x y
by (cases x, cases y, auto simp add: ennreal_minus e2ennreal_neg)
lemma ennreal_mult: "0 \ a \ 0 \ b \ ennreal (a * b) = ennreal a * ennreal b"
by transfer (simp add: max_absorb2)
lemma ennreal_mult': "0 \ a \ ennreal (a * b) = ennreal a * ennreal b"
by (cases "0 \ b") (auto simp: ennreal_mult ennreal_neg mult_nonneg_nonpos)
lemma indicator_mult_ennreal: "indicator A x * ennreal r = ennreal (indicator A x * r)"
by (simp split: split_indicator)
lemma ennreal_mult'': "0 \ b \ ennreal (a * b) = ennreal a * ennreal b"
by (cases "0 \ a") (auto simp: ennreal_mult ennreal_neg mult_nonpos_nonneg)
lemma numeral_mult_ennreal: "0 \ x \ numeral b * ennreal x = ennreal (numeral b * x)"
by (simp add: ennreal_mult)
lemma ennreal_power: "0 \ r \ ennreal r ^ n = ennreal (r ^ n)"
by (induction n) (auto simp: ennreal_mult)
lemma power_eq_top_ennreal: "x ^ n = top \ (n \ 0 \ (x::ennreal) = top)"
by (cases x rule: ennreal_cases)
(auto simp: ennreal_power top_power_ennreal)
lemma inverse_ennreal: "0 < r \ inverse (ennreal r) = ennreal (inverse r)"
by transfer (simp add: max.absorb2)
lemma divide_ennreal: "0 \ r \ 0 < q \ ennreal r / ennreal q = ennreal (r / q)"
by (simp add: divide_ennreal_def inverse_ennreal ennreal_mult[symmetric] inverse_eq_divide)
lemma ennreal_inverse_power: "inverse (x ^ n :: ennreal) = inverse x ^ n"
proof (cases x rule: ennreal_cases)
case top with power_eq_top_ennreal[of x n] show ?thesis
by (cases "n = 0") auto
next
case (real r) then show ?thesis
proof cases
assume "x = 0" then show ?thesis
using power_eq_top_ennreal[of top "n - 1"]
by (cases n) (auto simp: ennreal_top_mult)
next
assume "x \ 0"
with real have "0 < r" by auto
with real show ?thesis
by (induction n)
(auto simp add: ennreal_power ennreal_mult[symmetric] inverse_ennreal)
qed
qed
lemma ennreal_divide_numeral: "0 \ x \ ennreal x / numeral b = ennreal (x / numeral b)"
by (subst divide_ennreal[symmetric]) auto
lemma prod_ennreal: "(\i. i \ A \ 0 \ f i) \ (\i\A. ennreal (f i)) = ennreal (prod f A)"
by (induction A rule: infinite_finite_induct)
(auto simp: ennreal_mult prod_nonneg)
lemma mult_right_ennreal_cancel: "a * ennreal c = b * ennreal c \ (a = b \ c \ 0)"
proof (cases "0 \ c")
case True
then show ?thesis
by (metis ennreal_eq_0_iff ennreal_mult_right_cong ennreal_neq_top mult_divide_eq_ennreal)
qed (use ennreal_neg in auto)
lemma ennreal_le_epsilon:
"(\e::real. y < top \ 0 < e \ x \ y + ennreal e) \ x \ y"
apply (cases y rule: ennreal_cases)
apply (cases x rule: ennreal_cases)
apply (auto simp flip: ennreal_plus simp add: top_unique intro: zero_less_one field_le_epsilon)
done
lemma ennreal_rat_dense:
fixes x y :: ennreal
shows "x < y \ \r::rat. x < real_of_rat r \ real_of_rat r < y"
proof transfer
fix x y :: ereal assume xy: "0 \ x" "0 \ y" "x < y"
moreover
from ereal_dense3[OF \<open>x < y\<close>]
obtain r where r: "x < ereal (real_of_rat r)" "ereal (real_of_rat r) < y"
by auto
then have "0 \ r"
using le_less_trans[OF \<open>0 \<le> x\<close> \<open>x < ereal (real_of_rat r)\<close>] by auto
with r show "\r. x < (sup 0 \ ereal) (real_of_rat r) \ (sup 0 \ ereal) (real_of_rat r) < y"
by (intro exI[of _ r]) (auto simp: max_absorb2)
qed
lemma ennreal_Ex_less_of_nat: "(x::ennreal) < top \ \n. x < of_nat n"
by (cases x rule: ennreal_cases)
(auto simp: ennreal_of_nat_eq_real_of_nat ennreal_less_iff reals_Archimedean2)
subsection \<open>Coercion from \<^typ>\<open>ennreal\<close> to \<^typ>\<open>real\<close>\<close>
definition "enn2real x = real_of_ereal (enn2ereal x)"
lemma enn2real_nonneg[simp]: "0 \ enn2real x"
by (auto simp: enn2real_def intro!: real_of_ereal_pos enn2ereal_nonneg)
lemma enn2real_mono: "a \ b \ b < top \ enn2real a \ enn2real b"
by (auto simp add: enn2real_def less_eq_ennreal.rep_eq intro!: real_of_ereal_positive_mono enn2ereal_nonneg)
lemma enn2real_of_nat[simp]: "enn2real (of_nat n) = n"
by (auto simp: enn2real_def)
lemma enn2real_ennreal[simp]: "0 \ r \ enn2real (ennreal r) = r"
by (simp add: enn2real_def)
lemma ennreal_enn2real[simp]: "r < top \ ennreal (enn2real r) = r"
by (cases r rule: ennreal_cases) auto
lemma real_of_ereal_enn2ereal[simp]: "real_of_ereal (enn2ereal x) = enn2real x"
by (simp add: enn2real_def)
lemma enn2real_top[simp]: "enn2real top = 0"
unfolding enn2real_def top_ennreal.rep_eq top_ereal_def by simp
lemma enn2real_0[simp]: "enn2real 0 = 0"
unfolding enn2real_def zero_ennreal.rep_eq by simp
lemma enn2real_1[simp]: "enn2real 1 = 1"
unfolding enn2real_def one_ennreal.rep_eq by simp
lemma enn2real_numeral[simp]: "enn2real (numeral n) = (numeral n)"
unfolding enn2real_def by simp
lemma enn2real_mult: "enn2real (a * b) = enn2real a * enn2real b"
unfolding enn2real_def
by (simp del: real_of_ereal_enn2ereal add: times_ennreal.rep_eq)
lemma enn2real_leI: "0 \ B \ x \ ennreal B \ enn2real x \ B"
by (cases x rule: ennreal_cases) (auto simp: top_unique)
lemma enn2real_positive_iff: "0 < enn2real x \ (0 < x \ x < top)"
by (cases x rule: ennreal_cases) auto
lemma enn2real_eq_posreal_iff[simp]: "c > 0 \ enn2real x = c \ x = c"
by (cases x) auto
lemma ennreal_enn2real_if: "ennreal (enn2real r) = (if r = top then 0 else r)"
by(auto intro!: ennreal_enn2real simp add: less_top)
subsection \<open>Coercion from \<^typ>\<open>enat\<close> to \<^typ>\<open>ennreal\<close>\<close>
definition ennreal_of_enat :: "enat \ ennreal"
where
"ennreal_of_enat n = (case n of \ \ top | enat n \ of_nat n)"
declare [[coercion ennreal_of_enat]]
declare [[coercion "of_nat :: nat \ ennreal"]]
lemma ennreal_of_enat_infty[simp]: "ennreal_of_enat \ = \"
by (simp add: ennreal_of_enat_def)
lemma ennreal_of_enat_enat[simp]: "ennreal_of_enat (enat n) = of_nat n"
by (simp add: ennreal_of_enat_def)
lemma ennreal_of_enat_0[simp]: "ennreal_of_enat 0 = 0"
using ennreal_of_enat_enat[of 0] unfolding enat_0 by simp
lemma ennreal_of_enat_1[simp]: "ennreal_of_enat 1 = 1"
using ennreal_of_enat_enat[of 1] unfolding enat_1 by simp
lemma ennreal_top_neq_of_nat[simp]: "(top::ennreal) \ of_nat i"
using ennreal_of_nat_neq_top[of i] by metis
lemma ennreal_of_enat_inj[simp]: "ennreal_of_enat i = ennreal_of_enat j \ i = j"
by (cases i j rule: enat.exhaust[case_product enat.exhaust]) auto
lemma ennreal_of_enat_le_iff[simp]: "ennreal_of_enat m \ ennreal_of_enat n \ m \ n"
by (auto simp: ennreal_of_enat_def top_unique split: enat.split)
lemma of_nat_less_ennreal_of_nat[simp]: "of_nat n \ ennreal_of_enat x \ of_nat n \ x"
by (cases x) (auto simp: of_nat_eq_enat)
lemma ennreal_of_enat_Sup: "ennreal_of_enat (Sup X) = (SUP x\X. ennreal_of_enat x)"
proof -
have "ennreal_of_enat (Sup X) \ (SUP x \ X. ennreal_of_enat x)"
unfolding Sup_enat_def
proof (clarsimp, intro conjI impI)
fix x assume "finite X" "X \ {}"
then show "ennreal_of_enat (Max X) \ (SUP x \ X. ennreal_of_enat x)"
by (intro SUP_upper Max_in)
next
assume "infinite X" "X \ {}"
have "\y\X. r < ennreal_of_enat y" if r: "r < top" for r
proof -
from ennreal_Ex_less_of_nat[OF r] guess n .. note n = this
have "\ (X \ enat ` {.. n})"
using \<open>infinite X\<close> by (auto dest: finite_subset)
then obtain x where x: "x \ X" "x \ enat ` {..n}"
by blast
then have "of_nat n \ x"
by (cases x) (auto simp: of_nat_eq_enat)
with x show ?thesis
by (auto intro!: bexI[of _ x] less_le_trans[OF n])
qed
then have "(SUP x \ X. ennreal_of_enat x) = top"
by simp
then show "top \ (SUP x \ X. ennreal_of_enat x)"
unfolding top_unique by simp
qed
then show ?thesis
by (auto intro!: antisym Sup_least intro: Sup_upper)
qed
lemma ennreal_of_enat_eSuc[simp]: "ennreal_of_enat (eSuc x) = 1 + ennreal_of_enat x"
by (cases x) (auto simp: eSuc_enat)
subsection \<open>Topology on \<^typ>\<open>ennreal\<close>\<close>
lemma enn2ereal_Iio: "enn2ereal -` {.. a then {..< e2ennreal a} else {})"
using enn2ereal_nonneg
by (cases a rule: ereal_ennreal_cases)
(auto simp add: vimage_def set_eq_iff ennreal.enn2ereal_inverse less_ennreal.rep_eq e2ennreal_def max_absorb2
simp del: enn2ereal_nonneg
intro: le_less_trans less_imp_le)
lemma enn2ereal_Ioi: "enn2ereal -` {a <..} = (if 0 \ a then {e2ennreal a <..} else UNIV)"
by (cases a rule: ereal_ennreal_cases)
(auto simp add: vimage_def set_eq_iff ennreal.enn2ereal_inverse less_ennreal.rep_eq e2ennreal_def max_absorb2
intro: less_le_trans)
instantiation ennreal :: linear_continuum_topology
begin
definition open_ennreal :: "ennreal set \ bool"
where "(open :: ennreal set \ bool) = generate_topology (range lessThan \ range greaterThan)"
instance
proof
show "\a b::ennreal. a \ b"
using zero_neq_one by (intro exI)
show "\x y::ennreal. x < y \ \z>x. z < y"
proof transfer
fix x y :: ereal assume "0 \ x" and *: "x < y"
moreover from dense[OF *] guess z ..
ultimately show "\z\Collect ((\) 0). x < z \ z < y"
by (intro bexI[of _ z]) auto
qed
qed (rule open_ennreal_def)
end
lemma continuous_on_e2ennreal: "continuous_on A e2ennreal"
proof (rule continuous_on_subset)
show "continuous_on ({0..} \ {..0}) e2ennreal"
proof (rule continuous_on_closed_Un)
show "continuous_on {0 ..} e2ennreal"
by (rule continuous_onI_mono)
(auto simp add: less_eq_ennreal.abs_eq eq_onp_def enn2ereal_range)
show "continuous_on {.. 0} e2ennreal"
by (subst continuous_on_cong[OF refl, of _ _ "\_. 0"])
(auto simp add: e2ennreal_neg continuous_on_const)
qed auto
show "A \ {0..} \ {..0::ereal}"
by auto
qed
lemma continuous_at_e2ennreal: "continuous (at x within A) e2ennreal"
by (rule continuous_on_imp_continuous_within[OF continuous_on_e2ennreal, of _ UNIV]) auto
lemma continuous_on_enn2ereal: "continuous_on UNIV enn2ereal"
by (rule continuous_on_generate_topology[OF open_generated_order])
(auto simp add: enn2ereal_Iio enn2ereal_Ioi)
lemma continuous_at_enn2ereal: "continuous (at x within A) enn2ereal"
by (rule continuous_on_imp_continuous_within[OF continuous_on_enn2ereal]) auto
lemma sup_continuous_e2ennreal[order_continuous_intros]:
assumes f: "sup_continuous f" shows "sup_continuous (\x. e2ennreal (f x))"
proof (rule sup_continuous_compose[OF _ f])
show "sup_continuous e2ennreal"
by (simp add: continuous_at_e2ennreal continuous_at_left_imp_sup_continuous e2ennreal_mono mono_def)
qed
lemma sup_continuous_enn2ereal[order_continuous_intros]:
assumes f: "sup_continuous f" shows "sup_continuous (\x. enn2ereal (f x))"
proof (rule sup_continuous_compose[OF _ f])
show "sup_continuous enn2ereal"
by (simp add: continuous_at_enn2ereal continuous_at_left_imp_sup_continuous less_eq_ennreal.rep_eq mono_def)
qed
lemma sup_continuous_mult_left_ennreal':
fixes c :: "ennreal"
shows "sup_continuous (\x. c * x)"
unfolding sup_continuous_def
by transfer (auto simp: SUP_ereal_mult_left max.absorb2 SUP_upper2)
lemma sup_continuous_mult_left_ennreal[order_continuous_intros]:
"sup_continuous f \ sup_continuous (\x. c * f x :: ennreal)"
by (rule sup_continuous_compose[OF sup_continuous_mult_left_ennreal'])
lemma sup_continuous_mult_right_ennreal[order_continuous_intros]:
"sup_continuous f \ sup_continuous (\x. f x * c :: ennreal)"
using sup_continuous_mult_left_ennreal[of f c] by (simp add: mult.commute)
lemma sup_continuous_divide_ennreal[order_continuous_intros]:
fixes f g :: "'a::complete_lattice \ ennreal"
shows "sup_continuous f \ sup_continuous (\x. f x / c)"
unfolding divide_ennreal_def by (rule sup_continuous_mult_right_ennreal)
lemma transfer_enn2ereal_continuous_on [transfer_rule]:
"rel_fun (=) (rel_fun (rel_fun (=) pcr_ennreal) (=)) continuous_on continuous_on"
proof -
have "continuous_on A f" if "continuous_on A (\x. enn2ereal (f x))" for A and f :: "'a \ ennreal"
using continuous_on_compose2[OF continuous_on_e2ennreal[of "{0..}"] that]
by (auto simp: ennreal.enn2ereal_inverse subset_eq e2ennreal_def max_absorb2)
moreover
have "continuous_on A (\x. enn2ereal (f x))" if "continuous_on A f" for A and f :: "'a \ ennreal"
using continuous_on_compose2[OF continuous_on_enn2ereal that] by auto
ultimately
show ?thesis
by (auto simp add: rel_fun_def ennreal.pcr_cr_eq cr_ennreal_def)
qed
lemma transfer_sup_continuous[transfer_rule]:
"(rel_fun (rel_fun (=) pcr_ennreal) (=)) sup_continuous sup_continuous"
proof (safe intro!: rel_funI dest!: rel_fun_eq_pcr_ennreal[THEN iffD1])
show "sup_continuous (enn2ereal \ f) \ sup_continuous f" for f :: "'a \ _"
using sup_continuous_e2ennreal[of "enn2ereal \ f"] by simp
show "sup_continuous f \ sup_continuous (enn2ereal \ f)" for f :: "'a \ _"
using sup_continuous_enn2ereal[of f] by (simp add: comp_def)
qed
lemma continuous_on_ennreal[tendsto_intros]:
"continuous_on A f \ continuous_on A (\x. ennreal (f x))"
by transfer (auto intro!: continuous_on_max continuous_on_const continuous_on_ereal)
lemma tendsto_ennrealD:
assumes lim: "((\x. ennreal (f x)) \ ennreal x) F"
assumes *: "\\<^sub>F x in F. 0 \ f x" and x: "0 \ x"
shows "(f \ x) F"
proof -
have "((\x. enn2ereal (ennreal (f x))) \ enn2ereal (ennreal x)) F
\<longleftrightarrow> (f \<longlongrightarrow> enn2ereal (ennreal x)) F"
using "*" eventually_mono
by (intro tendsto_cong) fastforce
then show ?thesis
using assms(1) continuous_at_enn2ereal isCont_tendsto_compose x by fastforce
qed
lemma tendsto_ennreal_iff[simp]:
"\\<^sub>F x in F. 0 \ f x \ 0 \ x \ ((\x. ennreal (f x)) \ ennreal x) F \ (f \ x) F"
by (auto dest: tendsto_ennrealD)
(auto simp: ennreal_def
intro!: continuous_on_tendsto_compose[OF continuous_on_e2ennreal[of UNIV]] tendsto_max)
lemma tendsto_enn2ereal_iff[simp]: "((\i. enn2ereal (f i)) \ enn2ereal x) F \ (f \ x) F"
using continuous_on_enn2ereal[THEN continuous_on_tendsto_compose, of f x F]
continuous_on_e2ennreal[THEN continuous_on_tendsto_compose, of "\x. enn2ereal (f x)" "enn2ereal x" F UNIV]
by auto
lemma ennreal_tendsto_0_iff: "(\n. f n \ 0) \ ((\n. ennreal (f n)) \ 0) \ (f \ 0)"
by (metis (mono_tags) ennreal_0 eventuallyI order_refl tendsto_ennreal_iff)
lemma continuous_on_add_ennreal:
fixes f g :: "'a::topological_space \ ennreal"
shows "continuous_on A f \ continuous_on A g \ continuous_on A (\x. f x + g x)"
by (transfer fixing: A) (auto intro!: tendsto_add_ereal_nonneg simp: continuous_on_def)
lemma continuous_on_inverse_ennreal[continuous_intros]:
fixes f :: "'a::topological_space \ ennreal"
shows "continuous_on A f \ continuous_on A (\x. inverse (f x))"
proof (transfer fixing: A)
show "pred_fun top ((\) 0) f \ continuous_on A (\x. inverse (f x))" if "continuous_on A f"
for f :: "'a \ ereal"
using continuous_on_compose2[OF continuous_on_inverse_ereal that] by (auto simp: subset_eq)
qed
instance ennreal :: topological_comm_monoid_add
proof
show "((\x. fst x + snd x) \ a + b) (nhds a \\<^sub>F nhds b)" for a b :: ennreal
using continuous_on_add_ennreal[of UNIV fst snd]
using tendsto_at_iff_tendsto_nhds[symmetric, of "\x::(ennreal \ ennreal). fst x + snd x"]
by (auto simp: continuous_on_eq_continuous_at)
(simp add: isCont_def nhds_prod[symmetric])
qed
lemma sup_continuous_add_ennreal[order_continuous_intros]:
fixes f g :: "'a::complete_lattice \ ennreal"
shows "sup_continuous f \ sup_continuous g \ sup_continuous (\x. f x + g x)"
by transfer (auto intro!: sup_continuous_add)
lemma ennreal_suminf_lessD: "(\i. f i :: ennreal) < x \ f i < x"
using le_less_trans[OF sum_le_suminf[OF summableI, of "{i}" f]] by simp
lemma sums_ennreal[simp]: "(\i. 0 \ f i) \ 0 \ x \ (\i. ennreal (f i)) sums ennreal x \ f sums x"
unfolding sums_def by (simp add: always_eventually sum_nonneg)
lemma summable_suminf_not_top: "(\i. 0 \ f i) \ (\i. ennreal (f i)) \ top \ summable f"
using summable_sums[OF summableI, of "\i. ennreal (f i)"]
by (cases "\i. ennreal (f i)" rule: ennreal_cases)
(auto simp: summable_def)
lemma suminf_ennreal[simp]:
"(\i. 0 \ f i) \ (\i. ennreal (f i)) \ top \ (\i. ennreal (f i)) = ennreal (\i. f i)"
by (rule sums_unique[symmetric]) (simp add: summable_suminf_not_top suminf_nonneg summable_sums)
lemma sums_enn2ereal[simp]: "(\i. enn2ereal (f i)) sums enn2ereal x \ f sums x"
unfolding sums_def by (simp add: always_eventually sum_nonneg)
lemma suminf_enn2ereal[simp]: "(\i. enn2ereal (f i)) = enn2ereal (suminf f)"
by (rule sums_unique[symmetric]) (simp add: summable_sums)
lemma transfer_e2ennreal_suminf [transfer_rule]: "rel_fun (rel_fun (=) pcr_ennreal) pcr_ennreal suminf suminf"
by (auto simp: rel_funI rel_fun_eq_pcr_ennreal comp_def)
lemma ennreal_suminf_cmult[simp]: "(\i. r * f i) = r * (\i. f i::ennreal)"
by transfer (auto intro!: suminf_cmult_ereal)
lemma ennreal_suminf_multc[simp]: "(\i. f i * r) = (\i. f i::ennreal) * r"
using ennreal_suminf_cmult[of r f] by (simp add: ac_simps)
lemma ennreal_suminf_divide[simp]: "(\i. f i / r) = (\i. f i::ennreal) / r"
by (simp add: divide_ennreal_def)
lemma ennreal_suminf_neq_top: "summable f \ (\i. 0 \ f i) \ (\i. ennreal (f i)) \ top"
using sums_ennreal[of f "suminf f"]
by (simp add: suminf_nonneg flip: sums_unique summable_sums_iff del: sums_ennreal)
lemma suminf_ennreal_eq:
"(\i. 0 \ f i) \ f sums x \ (\i. ennreal (f i)) = ennreal x"
using suminf_nonneg[of f] sums_unique[of f x]
by (intro sums_unique[symmetric]) (auto simp: summable_sums_iff)
lemma ennreal_suminf_bound_add:
fixes f :: "nat \ ennreal"
shows "(\N. (\n x) \ suminf f + y \ x"
by transfer (auto intro!: suminf_bound_add)
lemma ennreal_suminf_SUP_eq_directed:
fixes f :: "'a \ nat \ ennreal"
assumes *: "\N i j. i \ I \ j \ I \ finite N \ \k\I. \n\N. f i n \ f k n \ f j n \ f k n"
shows "(\n. SUP i\I. f i n) = (SUP i\I. \n. f i n)"
proof cases
assume "I \ {}"
then obtain i where "i \ I" by auto
from * show ?thesis
by (transfer fixing: I)
(auto simp: max_absorb2 SUP_upper2[OF \<open>i \<in> I\<close>] suminf_nonneg summable_ereal_pos \<open>I \<noteq> {}\<close>
intro!: suminf_SUP_eq_directed)
qed (simp add: bot_ennreal)
lemma INF_ennreal_add_const:
fixes f g :: "nat \ ennreal"
shows "(INF i. f i + c) = (INF i. f i) + c"
using continuous_at_Inf_mono[of "\x. x + c" "f`UNIV"]
using continuous_add[of "at_right (Inf (range f))", of "\x. x" "\x. c"]
by (auto simp: mono_def image_comp)
lemma INF_ennreal_const_add:
fixes f g :: "nat \ ennreal"
shows "(INF i. c + f i) = c + (INF i. f i)"
using INF_ennreal_add_const[of f c] by (simp add: ac_simps)
lemma SUP_mult_left_ennreal: "c * (SUP i\I. f i) = (SUP i\I. c * f i ::ennreal)"
proof cases
assume "I \ {}" then show ?thesis
by transfer (auto simp add: SUP_ereal_mult_left max_absorb2 SUP_upper2)
qed (simp add: bot_ennreal)
lemma SUP_mult_right_ennreal: "(SUP i\I. f i) * c = (SUP i\I. f i * c ::ennreal)"
using SUP_mult_left_ennreal by (simp add: mult.commute)
lemma SUP_divide_ennreal: "(SUP i\I. f i) / c = (SUP i\I. f i / c ::ennreal)"
using SUP_mult_right_ennreal by (simp add: divide_ennreal_def)
lemma ennreal_SUP_of_nat_eq_top: "(SUP x. of_nat x :: ennreal) = top"
proof (intro antisym top_greatest le_SUP_iff[THEN iffD2] allI impI)
fix y :: ennreal assume "y < top"
then obtain r where "y = ennreal r"
by (cases y rule: ennreal_cases) auto
then show "\i\UNIV. y < of_nat i"
using reals_Archimedean2[of "max 1 r"] zero_less_one
by (simp add: ennreal_Ex_less_of_nat)
qed
lemma ennreal_SUP_eq_top:
fixes f :: "'a \ ennreal"
assumes "\n. \i\I. of_nat n \ f i"
shows "(SUP i \ I. f i) = top"
proof -
have "(SUP x. of_nat x :: ennreal) \ (SUP i \ I. f i)"
using assms by (auto intro!: SUP_least intro: SUP_upper2)
then show ?thesis
by (auto simp: ennreal_SUP_of_nat_eq_top top_unique)
qed
lemma ennreal_INF_const_minus:
fixes f :: "'a \ ennreal"
shows "I \ {} \ (SUP x\I. c - f x) = c - (INF x\I. f x)"
by (transfer fixing: I)
(simp add: sup_max[symmetric] SUP_sup_const1 SUP_ereal_minus_right del: sup_ereal_def)
lemma of_nat_Sup_ennreal:
assumes "A \ {}" "bdd_above A"
shows "of_nat (Sup A) = (SUP a\A. of_nat a :: ennreal)"
proof (intro antisym)
show "(SUP a\A. of_nat a::ennreal) \ of_nat (Sup A)"
by (intro SUP_least of_nat_mono) (auto intro: cSup_upper assms)
have "Sup A \ A"
using assms by (auto simp: Sup_nat_def bdd_above_nat)
then show "of_nat (Sup A) \ (SUP a\A. of_nat a::ennreal)"
by (intro SUP_upper)
qed
lemma ennreal_tendsto_const_minus:
fixes g :: "'a \ ennreal"
assumes ae: "\\<^sub>F x in F. g x \ c"
assumes g: "((\x. c - g x) \ 0) F"
shows "(g \ c) F"
proof (cases c rule: ennreal_cases)
case top with tendsto_unique[OF _ g, of "top"] show ?thesis
by (cases "F = bot") auto
next
case (real r)
then have "\x. \q\0. g x \ c \ (g x = ennreal q \ q \ r)"
by (auto simp: le_ennreal_iff)
then obtain f where *: "0 \ f x" "g x = ennreal (f x)" "f x \ r" if "g x \ c" for x
by metis
from ae have ae2: "\\<^sub>F x in F. c - g x = ennreal (r - f x) \ f x \ r \ g x = ennreal (f x) \ 0 \ f x"
proof eventually_elim
fix x assume "g x \ c" with *[of x] \0 \ r\ show "c - g x = ennreal (r - f x) \ f x \ r \ g x = ennreal (f x) \ 0 \ f x"
by (auto simp: real ennreal_minus)
qed
with g have "((\x. ennreal (r - f x)) \ ennreal 0) F"
by (auto simp add: tendsto_cong eventually_conj_iff)
with ae2 have "((\x. r - f x) \ 0) F"
by (subst (asm) tendsto_ennreal_iff) (auto elim: eventually_mono)
then have "(f \ r) F"
by (rule Lim_transform2[OF tendsto_const])
with ae2 have "((\x. ennreal (f x)) \ ennreal r) F"
by (subst tendsto_ennreal_iff) (auto elim: eventually_mono simp: real)
with ae2 show ?thesis
by (auto simp: real tendsto_cong eventually_conj_iff)
qed
lemma ennreal_SUP_add:
fixes f g :: "nat \ ennreal"
shows "incseq f \ incseq g \ (SUP i. f i + g i) = Sup (f ` UNIV) + Sup (g ` UNIV)"
unfolding incseq_def le_fun_def
by transfer
(simp add: SUP_ereal_add incseq_def le_fun_def max_absorb2 SUP_upper2)
lemma ennreal_SUP_sum:
fixes f :: "'a \ nat \ ennreal"
shows "(\i. i \ I \ incseq (f i)) \ (SUP n. \i\I. f i n) = (\i\I. SUP n. f i n)"
unfolding incseq_def
by transfer
(simp add: SUP_ereal_sum incseq_def SUP_upper2 max_absorb2 sum_nonneg)
lemma ennreal_liminf_minus:
fixes f :: "nat \ ennreal"
shows "(\n. f n \ c) \ liminf (\n. c - f n) = c - limsup f"
apply transfer
apply (simp add: ereal_diff_positive max.absorb2 liminf_ereal_cminus)
apply (subst max.absorb2)
apply (rule ereal_diff_positive)
apply (rule Limsup_bounded)
apply auto
done
lemma ennreal_continuous_on_cmult:
"(c::ennreal) < top \ continuous_on A f \ continuous_on A (\x. c * f x)"
by (transfer fixing: A) (auto intro: continuous_on_cmult_ereal)
lemma ennreal_tendsto_cmult:
"(c::ennreal) < top \ (f \ x) F \ ((\x. c * f x) \ c * x) F"
by (rule continuous_on_tendsto_compose[where g=f, OF ennreal_continuous_on_cmult, where s=UNIV])
(auto simp: continuous_on_id)
lemma tendsto_ennrealI[intro, simp, tendsto_intros]:
"(f \ x) F \ ((\x. ennreal (f x)) \ ennreal x) F"
by (auto simp: ennreal_def
intro!: continuous_on_tendsto_compose[OF continuous_on_e2ennreal[of UNIV]] tendsto_max)
lemma tendsto_enn2erealI [tendsto_intros]:
assumes "(f \ l) F"
shows "((\i. enn2ereal(f i)) \ enn2ereal l) F"
using tendsto_enn2ereal_iff assms by auto
lemma tendsto_e2ennrealI [tendsto_intros]:
assumes "(f \ l) F"
shows "((\i. e2ennreal(f i)) \ e2ennreal l) F"
proof -
have *: "e2ennreal (max x 0) = e2ennreal x" for x
--> --------------------
--> maximum size reached
--> --------------------
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