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a/Admin/components/main +++ b/Admin/components/main @@ -1,33 +1,33 @@ #main components for repository clones or release bundles gnu-utils-20211030 apache-commons-20211211 bash_process-1.2.4-2 bib2xhtml-20190409 csdp-6.1.1 cvc4-1.8 e-2.6-1 flatlaf-1.6.4 idea-icons-20210508 isabelle_fonts-20211004 isabelle_setup-20211109 -jdk-17.0.1+12 +jdk-17.0.2+8 jedit-20211103 jfreechart-1.5.3 jortho-1.0-2 kodkodi-1.5.7 minisat-2.2.1-1 nunchaku-0.5 opam-2.0.7 -polyml-5.9 +polyml-test-15c840d48c9a postgresql-42.2.24 scala-2.13.5 smbc-0.4.1 spass-3.8ds-2 sqlite-jdbc-3.36.0.3 ssh-java-20190323 stack-2.7.3 vampire-4.6 verit-2021.06.2-rmx xz-java-1.9 z3-4.4.0_4.4.1 zipperposition-2.1-1 diff --git a/Admin/polyml/settings b/Admin/polyml/settings --- a/Admin/polyml/settings +++ b/Admin/polyml/settings @@ -1,19 +1,19 @@ # -*- shell-script -*- :mode=shellscript: POLYML_HOME="$COMPONENT" -ML_PLATFORM="${ISABELLE_WINDOWS_PLATFORM64:-${ISABELLE_PLATFORM64}}" +ML_PLATFORM="${ISABELLE_APPLE_PLATFORM64:-${ISABELLE_WINDOWS_PLATFORM64:-${ISABELLE_PLATFORM64}}}" if grep "ML_system_64.*=.*true" "$ISABELLE_HOME_USER/etc/preferences" >/dev/null 2>/dev/null then ML_OPTIONS="--minheap 1000" else ML_PLATFORM="${ML_PLATFORM/64/64_32}" ML_OPTIONS="--minheap 500" fi ML_SYSTEM=polyml-5.9 ML_HOME="$POLYML_HOME/$ML_PLATFORM" ML_SOURCES="$POLYML_HOME/src" ISABELLE_DOCS_EXAMPLES="$ISABELLE_DOCS_EXAMPLES:\$ML_SOURCES/ROOT.ML" diff --git a/src/Doc/Implementation/Proof.thy b/src/Doc/Implementation/Proof.thy --- a/src/Doc/Implementation/Proof.thy +++ b/src/Doc/Implementation/Proof.thy @@ -1,474 +1,483 @@ (*:maxLineLen=78:*) theory Proof imports Base begin chapter \Structured proofs\ section \Variables \label{sec:variables}\ text \ Any variable that is not explicitly bound by \\\-abstraction is considered as ``free''. Logically, free variables act like outermost universal quantification at the sequent level: \A\<^sub>1(x), \, A\<^sub>n(x) \ B(x)\ means that the result holds \<^emph>\for all\ values of \x\. Free variables for terms (not types) can be fully internalized into the logic: \\ B(x)\ and \\ \x. B(x)\ are interchangeable, provided that \x\ does not occur elsewhere in the context. Inspecting \\ \x. B(x)\ more closely, we see that inside the quantifier, \x\ is essentially ``arbitrary, but fixed'', while from outside it appears as a place-holder for instantiation (thanks to \\\ elimination). The Pure logic represents the idea of variables being either inside or outside the current scope by providing separate syntactic categories for \<^emph>\fixed variables\ (e.g.\ \x\) vs.\ \<^emph>\schematic variables\ (e.g.\ \?x\). Incidently, a universal result \\ \x. B(x)\ has the HHF normal form \\ B(?x)\, which represents its generality without requiring an explicit quantifier. The same principle works for type variables: \\ B(?\)\ represents the idea of ``\\ \\. B(\)\'' without demanding a truly polymorphic framework. \<^medskip> Additional care is required to treat type variables in a way that facilitates type-inference. In principle, term variables depend on type variables, which means that type variables would have to be declared first. For example, a raw type-theoretic framework would demand the context to be constructed in stages as follows: \\ = \: type, x: \, a: A(x\<^sub>\)\. We allow a slightly less formalistic mode of operation: term variables \x\ are fixed without specifying a type yet (essentially \<^emph>\all\ potential occurrences of some instance \x\<^sub>\\ are fixed); the first occurrence of \x\ within a specific term assigns its most general type, which is then maintained consistently in the context. The above example becomes \\ = x: term, \: type, A(x\<^sub>\)\, where type \\\ is fixed \<^emph>\after\ term \x\, and the constraint \x :: \\ is an implicit consequence of the occurrence of \x\<^sub>\\ in the subsequent proposition. This twist of dependencies is also accommodated by the reverse operation of exporting results from a context: a type variable \\\ is considered fixed as long as it occurs in some fixed term variable of the context. For example, exporting \x: term, \: type \ x\<^sub>\ \ x\<^sub>\\ produces in the first step \x: term \ x\<^sub>\ \ x\<^sub>\\ for fixed \\\, and only in the second step \\ ?x\<^sub>?\<^sub>\ \ ?x\<^sub>?\<^sub>\\ for schematic \?x\ and \?\\. The following Isar source text illustrates this scenario. \ notepad begin { fix x \ \all potential occurrences of some \x::\\ are fixed\ { have "x::'a \ x" \ \implicit type assignment by concrete occurrence\ by (rule reflexive) } thm this \ \result still with fixed type \'a\\ } thm this \ \fully general result for arbitrary \?x::?'a\\ end text \ The Isabelle/Isar proof context manages the details of term vs.\ type variables, with high-level principles for moving the frontier between fixed and schematic variables. The \add_fixes\ operation explicitly declares fixed variables; the \declare_term\ operation absorbs a term into a context by fixing new type variables and adding syntactic constraints. The \export\ operation is able to perform the main work of generalizing term and type variables as sketched above, assuming that fixing variables and terms have been declared properly. There \import\ operation makes a generalized fact a genuine part of the context, by inventing fixed variables for the schematic ones. The effect can be reversed by using \export\ later, potentially with an extended context; the result is equivalent to the original modulo renaming of schematic variables. The \focus\ operation provides a variant of \import\ for nested propositions (with explicit quantification): \\x\<^sub>1 \ x\<^sub>n. B(x\<^sub>1, \, x\<^sub>n)\ is decomposed by inventing fixed variables \x\<^sub>1, \, x\<^sub>n\ for the body. \ text %mlref \ \begin{mldecls} @{define_ML Variable.add_fixes: " string list -> Proof.context -> string list * Proof.context"} \\ @{define_ML Variable.variant_fixes: " string list -> Proof.context -> string list * Proof.context"} \\ @{define_ML Variable.declare_term: "term -> Proof.context -> Proof.context"} \\ @{define_ML Variable.declare_constraints: "term -> Proof.context -> Proof.context"} \\ @{define_ML Variable.export: "Proof.context -> Proof.context -> thm list -> thm list"} \\ @{define_ML Variable.polymorphic: "Proof.context -> term list -> term list"} \\ @{define_ML Variable.import: "bool -> thm list -> Proof.context -> ((ctyp TVars.table * cterm Vars.table) * thm list) * Proof.context"} \\ @{define_ML Variable.focus: "binding list option -> term -> Proof.context -> ((string * (string * typ)) list * term) * Proof.context"} \\ \end{mldecls} \<^descr> \<^ML>\Variable.add_fixes\~\xs ctxt\ fixes term variables \xs\, returning the resulting internal names. By default, the internal representation coincides with the external one, which also means that the given variables must not be fixed already. There is a different policy within a local proof body: the given names are just hints for newly invented Skolem variables. \<^descr> \<^ML>\Variable.variant_fixes\ is similar to \<^ML>\Variable.add_fixes\, but always produces fresh variants of the given names. \<^descr> \<^ML>\Variable.declare_term\~\t ctxt\ declares term \t\ to belong to the context. This automatically fixes new type variables, but not term variables. Syntactic constraints for type and term variables are declared uniformly, though. \<^descr> \<^ML>\Variable.declare_constraints\~\t ctxt\ declares syntactic constraints from term \t\, without making it part of the context yet. \<^descr> \<^ML>\Variable.export\~\inner outer thms\ generalizes fixed type and term variables in \thms\ according to the difference of the \inner\ and \outer\ context, following the principles sketched above. \<^descr> \<^ML>\Variable.polymorphic\~\ctxt ts\ generalizes type variables in \ts\ as far as possible, even those occurring in fixed term variables. The default policy of type-inference is to fix newly introduced type variables, which is essentially reversed with \<^ML>\Variable.polymorphic\: here the given terms are detached from the context as far as possible. \<^descr> \<^ML>\Variable.import\~\open thms ctxt\ invents fixed type and term variables for the schematic ones occurring in \thms\. The \open\ flag indicates whether the fixed names should be accessible to the user, otherwise newly introduced names are marked as ``internal'' (\secref{sec:names}). \<^descr> \<^ML>\Variable.focus\~\bindings B\ decomposes the outermost \\\ prefix of proposition \B\, using the given name bindings. \ text %mlex \ The following example shows how to work with fixed term and type parameters and with type-inference. \ ML_val \ (*static compile-time context -- for testing only*) val ctxt0 = \<^context>; (*locally fixed parameters -- no type assignment yet*) val ([x, y], ctxt1) = ctxt0 |> Variable.add_fixes ["x", "y"]; (*t1: most general fixed type; t1': most general arbitrary type*) val t1 = Syntax.read_term ctxt1 "x"; val t1' = singleton (Variable.polymorphic ctxt1) t1; (*term u enforces specific type assignment*) val u = Syntax.read_term ctxt1 "(x::nat) \ y"; (*official declaration of u -- propagates constraints etc.*) val ctxt2 = ctxt1 |> Variable.declare_term u; val t2 = Syntax.read_term ctxt2 "x"; (*x::nat is enforced*) \ text \ In the above example, the starting context is derived from the toplevel theory, which means that fixed variables are internalized literally: \x\ is mapped again to \x\, and attempting to fix it again in the subsequent context is an error. Alternatively, fixed parameters can be renamed explicitly as follows: \ ML_val \ val ctxt0 = \<^context>; val ([x1, x2, x3], ctxt1) = ctxt0 |> Variable.variant_fixes ["x", "x", "x"]; \ text \ The following ML code can now work with the invented names of \x1\, \x2\, \x3\, without depending on the details on the system policy for introducing these variants. Recall that within a proof body the system always invents fresh ``Skolem constants'', e.g.\ as follows: \ notepad begin ML_prf %"ML" \val ctxt0 = \<^context>; val ([x1], ctxt1) = ctxt0 |> Variable.add_fixes ["x"]; val ([x2], ctxt2) = ctxt1 |> Variable.add_fixes ["x"]; val ([x3], ctxt3) = ctxt2 |> Variable.add_fixes ["x"]; val ([y1, y2], ctxt4) = ctxt3 |> Variable.variant_fixes ["y", "y"];\ end text \ In this situation \<^ML>\Variable.add_fixes\ and \<^ML>\Variable.variant_fixes\ are very similar, but identical name proposals given in a row are only accepted by the second version. \ section \Assumptions \label{sec:assumptions}\ text \ An \<^emph>\assumption\ is a proposition that it is postulated in the current context. Local conclusions may use assumptions as additional facts, but this imposes implicit hypotheses that weaken the overall statement. Assumptions are restricted to fixed non-schematic statements, i.e.\ all generality needs to be expressed by explicit quantifiers. Nevertheless, the result will be in HHF normal form with outermost quantifiers stripped. For example, by assuming \\x :: \. P x\ we get \\x :: \. P x \ P ?x\ for schematic \?x\ of fixed type \\\. Local derivations accumulate more and more explicit references to hypotheses: \A\<^sub>1, \, A\<^sub>n \ B\ where \A\<^sub>1, \, A\<^sub>n\ needs to be covered by the assumptions of the current context. \<^medskip> The \add_assms\ operation augments the context by local assumptions, which are parameterized by an arbitrary \export\ rule (see below). The \export\ operation moves facts from a (larger) inner context into a (smaller) outer context, by discharging the difference of the assumptions as specified by the associated export rules. Note that the discharged portion is determined by the difference of contexts, not the facts being exported! There is a separate flag to indicate a goal context, where the result is meant to refine an enclosing sub-goal of a structured proof state. \<^medskip> The most basic export rule discharges assumptions directly by means of the \\\ introduction rule: \[ \infer[(\\\intro\)]{\\ - A \ A \ B\}{\\ \ B\} \] The variant for goal refinements marks the newly introduced premises, which causes the canonical Isar goal refinement scheme to enforce unification with local premises within the goal: \[ \infer[(\#\\intro\)]{\\ - A \ #A \ B\}{\\ \ B\} \] \<^medskip> Alternative versions of assumptions may perform arbitrary transformations on export, as long as the corresponding portion of hypotheses is removed from the given facts. For example, a local definition works by fixing \x\ and assuming \x \ t\, with the following export rule to reverse the effect: \[ \infer[(\\\expand\)]{\\ - (x \ t) \ B t\}{\\ \ B x\} \] This works, because the assumption \x \ t\ was introduced in a context with \x\ being fresh, so \x\ does not occur in \\\ here. \ text %mlref \ \begin{mldecls} @{define_ML_type Assumption.export} \\ @{define_ML Assumption.assume: "Proof.context -> cterm -> thm"} \\ @{define_ML Assumption.add_assms: "Assumption.export -> cterm list -> Proof.context -> thm list * Proof.context"} \\ @{define_ML Assumption.add_assumes: " cterm list -> Proof.context -> thm list * Proof.context"} \\ @{define_ML Assumption.export: "bool -> Proof.context -> Proof.context -> thm -> thm"} \\ + @{define_ML Assumption.export_term: "Proof.context -> Proof.context -> term -> term"} \\ \end{mldecls} - \<^descr> Type \<^ML_type>\Assumption.export\ represents arbitrary export rules, which - is any function of type \<^ML_type>\bool -> cterm list -> thm -> thm\, where - the \<^ML_type>\bool\ indicates goal mode, and the \<^ML_type>\cterm list\ + \<^descr> Type \<^ML_type>\Assumption.export\ represents export rules, as a pair of + functions \<^ML_type>\bool -> cterm list -> (thm -> thm) * (term -> term)\. + The \<^ML_type>\bool\ argument indicates goal mode, and the \<^ML_type>\cterm list\ the collection of assumptions to be discharged simultaneously. \<^descr> \<^ML>\Assumption.assume\~\ctxt A\ turns proposition \A\ into a primitive assumption \A \ A'\, where the conclusion \A'\ is in HHF normal form. \<^descr> \<^ML>\Assumption.add_assms\~\r As\ augments the context by assumptions \As\ with export rule \r\. The resulting facts are hypothetical theorems as produced by the raw \<^ML>\Assumption.assume\. - \<^descr> \<^ML>\Assumption.add_assumes\~\As\ is a special case of \<^ML>\Assumption.add_assms\ where the export rule performs \\\intro\ or + \<^descr> \<^ML>\Assumption.add_assumes\~\As\ is a special case of + \<^ML>\Assumption.add_assms\ where the export rule performs \\\intro\ or \#\\intro\, depending on goal mode. \<^descr> \<^ML>\Assumption.export\~\is_goal inner outer thm\ exports result \thm\ - from the the \inner\ context back into the \outer\ one; \is_goal = true\ - means this is a goal context. The result is in HHF normal form. Note that - \<^ML>\Proof_Context.export\ combines \<^ML>\Variable.export\ and \<^ML>\Assumption.export\ in the canonical way. + from the \inner\ context back into the \outer\ one; \is_goal = true\ means + this is a goal context. The result is in HHF normal form. Note that + \<^ML>\Proof_Context.export\ combines \<^ML>\Variable.export\ and + \<^ML>\Assumption.export\ in the canonical way. + + \<^descr> \<^ML>\Assumption.export_term\~\inner outer t\ exports term \t\ from the + \inner\ context back into the \outer\ one. This is analogous to + \<^ML>\Assumption.export\, but only takes syntactical aspects of the + context into account (such as locally specified variables as seen in + @{command define} or @{command obtain}). \ text %mlex \ The following example demonstrates how rules can be derived by building up a context of assumptions first, and exporting some local fact afterwards. We refer to \<^theory>\Pure\ equality here for testing purposes. \ ML_val \ (*static compile-time context -- for testing only*) val ctxt0 = \<^context>; val ([eq], ctxt1) = ctxt0 |> Assumption.add_assumes [\<^cprop>\x \ y\]; val eq' = Thm.symmetric eq; (*back to original context -- discharges assumption*) val r = Assumption.export false ctxt1 ctxt0 eq'; \ text \ Note that the variables of the resulting rule are not generalized. This would have required to fix them properly in the context beforehand, and export wrt.\ variables afterwards (cf.\ \<^ML>\Variable.export\ or the combined \<^ML>\Proof_Context.export\). \ section \Structured goals and results \label{sec:struct-goals}\ text \ Local results are established by monotonic reasoning from facts within a context. This allows common combinations of theorems, e.g.\ via \\/\\ elimination, resolution rules, or equational reasoning, see \secref{sec:thms}. Unaccounted context manipulations should be avoided, notably raw \\/\\ introduction or ad-hoc references to free variables or assumptions not present in the proof context. \<^medskip> The \SUBPROOF\ combinator allows to structure a tactical proof recursively by decomposing a selected sub-goal: \(\x. A(x) \ B(x)) \ \\ is turned into \B(x) \ \\ after fixing \x\ and assuming \A(x)\. This means the tactic needs to solve the conclusion, but may use the premise as a local fact, for locally fixed variables. The family of \FOCUS\ combinators is similar to \SUBPROOF\, but allows to retain schematic variables and pending subgoals in the resulting goal state. The \prove\ operation provides an interface for structured backwards reasoning under program control, with some explicit sanity checks of the result. The goal context can be augmented by additional fixed variables (cf.\ \secref{sec:variables}) and assumptions (cf.\ \secref{sec:assumptions}), which will be available as local facts during the proof and discharged into implications in the result. Type and term variables are generalized as usual, according to the context. The \obtain\ operation produces results by eliminating existing facts by means of a given tactic. This acts like a dual conclusion: the proof demonstrates that the context may be augmented by parameters and assumptions, without affecting any conclusions that do not mention these parameters. See also @{cite "isabelle-isar-ref"} for the corresponding Isar proof command @{command obtain}. Final results, which may not refer to the parameters in the conclusion, need to exported explicitly into the original context. \ text %mlref \ \begin{mldecls} @{define_ML SUBPROOF: "(Subgoal.focus -> tactic) -> Proof.context -> int -> tactic"} \\ @{define_ML Subgoal.FOCUS: "(Subgoal.focus -> tactic) -> Proof.context -> int -> tactic"} \\ @{define_ML Subgoal.FOCUS_PREMS: "(Subgoal.focus -> tactic) -> Proof.context -> int -> tactic"} \\ @{define_ML Subgoal.FOCUS_PARAMS: "(Subgoal.focus -> tactic) -> Proof.context -> int -> tactic"} \\ @{define_ML Subgoal.focus: "Proof.context -> int -> binding list option -> thm -> Subgoal.focus * thm"} \\ @{define_ML Subgoal.focus_prems: "Proof.context -> int -> binding list option -> thm -> Subgoal.focus * thm"} \\ @{define_ML Subgoal.focus_params: "Proof.context -> int -> binding list option -> thm -> Subgoal.focus * thm"} \\ \end{mldecls} \begin{mldecls} @{define_ML Goal.prove: "Proof.context -> string list -> term list -> term -> ({prems: thm list, context: Proof.context} -> tactic) -> thm"} \\ @{define_ML Goal.prove_common: "Proof.context -> int option -> string list -> term list -> term list -> ({prems: thm list, context: Proof.context} -> tactic) -> thm list"} \\ \end{mldecls} \begin{mldecls} @{define_ML Obtain.result: "(Proof.context -> tactic) -> thm list -> Proof.context -> ((string * cterm) list * thm list) * Proof.context"} \\ \end{mldecls} \<^descr> \<^ML>\SUBPROOF\~\tac ctxt i\ decomposes the structure of the specified sub-goal, producing an extended context and a reduced goal, which needs to be solved by the given tactic. All schematic parameters of the goal are imported into the context as fixed ones, which may not be instantiated in the sub-proof. \<^descr> \<^ML>\Subgoal.FOCUS\, \<^ML>\Subgoal.FOCUS_PREMS\, and \<^ML>\Subgoal.FOCUS_PARAMS\ are similar to \<^ML>\SUBPROOF\, but are slightly more flexible: only the specified parts of the subgoal are imported into the context, and the body tactic may introduce new subgoals and schematic variables. \<^descr> \<^ML>\Subgoal.focus\, \<^ML>\Subgoal.focus_prems\, \<^ML>\Subgoal.focus_params\ extract the focus information from a goal state in the same way as the corresponding tacticals above. This is occasionally useful to experiment without writing actual tactics yet. \<^descr> \<^ML>\Goal.prove\~\ctxt xs As C tac\ states goal \C\ in the context augmented by fixed variables \xs\ and assumptions \As\, and applies tactic \tac\ to solve it. The latter may depend on the local assumptions being presented as facts. The result is in HHF normal form. \<^descr> \<^ML>\Goal.prove_common\~\ctxt fork_pri\ is the common form to state and prove a simultaneous goal statement, where \<^ML>\Goal.prove\ is a convenient shorthand that is most frequently used in applications. The given list of simultaneous conclusions is encoded in the goal state by means of Pure conjunction: \<^ML>\Goal.conjunction_tac\ will turn this into a collection of individual subgoals, but note that the original multi-goal state is usually required for advanced induction. It is possible to provide an optional priority for a forked proof, typically \<^ML>\SOME ~1\, while \<^ML>\NONE\ means the proof is immediate (sequential) as for \<^ML>\Goal.prove\. Note that a forked proof does not exhibit any failures in the usual way via exceptions in ML, but accumulates error situations under the execution id of the running transaction. Thus the system is able to expose error messages ultimately to the end-user, even though the subsequent ML code misses them. \<^descr> \<^ML>\Obtain.result\~\tac thms ctxt\ eliminates the given facts using a tactic, which results in additional fixed variables and assumptions in the context. Final results need to be exported explicitly. \ text %mlex \ The following minimal example illustrates how to access the focus information of a structured goal state. \ notepad begin fix A B C :: "'a \ bool" have "\x. A x \ B x \ C x" ML_val \val {goal, context = goal_ctxt, ...} = @{Isar.goal}; val (focus as {params, asms, concl, ...}, goal') = Subgoal.focus goal_ctxt 1 (SOME [\<^binding>\x\]) goal; val [A, B] = #prems focus; val [(_, x)] = #params focus;\ sorry end text \ \<^medskip> The next example demonstrates forward-elimination in a local context, using \<^ML>\Obtain.result\. \ notepad begin assume ex: "\x. B x" ML_prf %"ML" \val ctxt0 = \<^context>; val (([(_, x)], [B]), ctxt1) = ctxt0 |> Obtain.result (fn _ => eresolve_tac ctxt0 @{thms exE} 1) [@{thm ex}];\ ML_prf %"ML" \singleton (Proof_Context.export ctxt1 ctxt0) @{thm refl};\ ML_prf %"ML" \Proof_Context.export ctxt1 ctxt0 [Thm.reflexive x] handle ERROR msg => (warning msg; []);\ end end diff --git a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy --- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy +++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy @@ -1,7594 +1,7606 @@ (* Author: John Harrison Author: Robert Himmelmann, TU Muenchen (Translation from HOL light) Huge cleanup by LCP *) section \Henstock-Kurzweil Gauge Integration in Many Dimensions\ theory Henstock_Kurzweil_Integration imports Lebesgue_Measure Tagged_Division begin lemma norm_diff2: "\y = y1 + y2; x = x1 + x2; e = e1 + e2; norm(y1 - x1) \ e1; norm(y2 - x2) \ e2\ \ norm(y-x) \ e" using norm_triangle_mono [of "y1 - x1" "e1" "y2 - x2" "e2"] by (simp add: add_diff_add) lemma setcomp_dot1: "{z. P (z \ (i,0))} = {(x,y). P(x \ i)}" by auto lemma setcomp_dot2: "{z. P (z \ (0,i))} = {(x,y). P(y \ i)}" by auto lemma Sigma_Int_Paircomp1: "(Sigma A B) \ {(x, y). P x} = Sigma (A \ {x. P x}) B" by blast lemma Sigma_Int_Paircomp2: "(Sigma A B) \ {(x, y). P y} = Sigma A (\z. B z \ {y. P y})" by blast (* END MOVE *) subsection \Content (length, area, volume...) of an interval\ abbreviation content :: "'a::euclidean_space set \ real" where "content s \ measure lborel s" lemma content_cbox_cases: "content (cbox a b) = (if \i\Basis. a\i \ b\i then prod (\i. b\i - a\i) Basis else 0)" by (simp add: measure_lborel_cbox_eq inner_diff) lemma content_cbox: "\i\Basis. a\i \ b\i \ content (cbox a b) = (\i\Basis. b\i - a\i)" unfolding content_cbox_cases by simp lemma content_cbox': "cbox a b \ {} \ content (cbox a b) = (\i\Basis. b\i - a\i)" by (simp add: box_ne_empty inner_diff) lemma content_cbox_if: "content (cbox a b) = (if cbox a b = {} then 0 else \i\Basis. b\i - a\i)" by (simp add: content_cbox') lemma content_cbox_cart: "cbox a b \ {} \ content(cbox a b) = prod (\i. b$i - a$i) UNIV" by (simp add: content_cbox_if Basis_vec_def cart_eq_inner_axis axis_eq_axis prod.UNION_disjoint) lemma content_cbox_if_cart: "content(cbox a b) = (if cbox a b = {} then 0 else prod (\i. b$i - a$i) UNIV)" by (simp add: content_cbox_cart) lemma content_division_of: assumes "K \ \" "\ division_of S" shows "content K = (\i \ Basis. interval_upperbound K \ i - interval_lowerbound K \ i)" proof - obtain a b where "K = cbox a b" using cbox_division_memE assms by metis then show ?thesis using assms by (force simp: division_of_def content_cbox') qed lemma content_real: "a \ b \ content {a..b} = b - a" by simp lemma abs_eq_content: "\y - x\ = (if x\y then content {x..y} else content {y..x})" by (auto simp: content_real) lemma content_singleton: "content {a} = 0" by simp lemma content_unit[iff]: "content (cbox 0 (One::'a::euclidean_space)) = 1" by simp lemma content_pos_le [iff]: "0 \ content X" by simp corollary\<^marker>\tag unimportant\ content_nonneg [simp]: "\ content (cbox a b) < 0" using not_le by blast lemma content_pos_lt: "\i\Basis. a\i < b\i \ 0 < content (cbox a b)" by (auto simp: less_imp_le inner_diff box_eq_empty intro!: prod_pos) lemma content_eq_0: "content (cbox a b) = 0 \ (\i\Basis. b\i \ a\i)" by (auto simp: content_cbox_cases not_le intro: less_imp_le antisym eq_refl) lemma content_eq_0_interior: "content (cbox a b) = 0 \ interior(cbox a b) = {}" unfolding content_eq_0 interior_cbox box_eq_empty by auto lemma content_pos_lt_eq: "0 < content (cbox a (b::'a::euclidean_space)) \ (\i\Basis. a\i < b\i)" by (auto simp add: content_cbox_cases less_le prod_nonneg) lemma content_empty [simp]: "content {} = 0" by simp lemma content_real_if [simp]: "content {a..b} = (if a \ b then b - a else 0)" by (simp add: content_real) lemma content_subset: "cbox a b \ cbox c d \ content (cbox a b) \ content (cbox c d)" unfolding measure_def by (intro enn2real_mono emeasure_mono) (auto simp: emeasure_lborel_cbox_eq) lemma content_lt_nz: "0 < content (cbox a b) \ content (cbox a b) \ 0" unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastforce lemma content_Pair: "content (cbox (a,c) (b,d)) = content (cbox a b) * content (cbox c d)" unfolding measure_lborel_cbox_eq Basis_prod_def apply (subst prod.union_disjoint) apply (auto simp: bex_Un ball_Un) apply (subst (1 2) prod.reindex_nontrivial) apply auto done lemma content_cbox_pair_eq0_D: "content (cbox (a,c) (b,d)) = 0 \ content (cbox a b) = 0 \ content (cbox c d) = 0" by (simp add: content_Pair) lemma content_cbox_plus: fixes x :: "'a::euclidean_space" shows "content(cbox x (x + h *\<^sub>R One)) = (if h \ 0 then h ^ DIM('a) else 0)" by (simp add: algebra_simps content_cbox_if box_eq_empty) lemma content_0_subset: "content(cbox a b) = 0 \ s \ cbox a b \ content s = 0" using emeasure_mono[of s "cbox a b" lborel] by (auto simp: measure_def enn2real_eq_0_iff emeasure_lborel_cbox_eq) lemma content_ball_pos: assumes "r > 0" shows "content (ball c r) > 0" proof - from rational_boxes[OF assms, of c] obtain a b where ab: "c \ box a b" "box a b \ ball c r" by auto from ab have "0 < content (box a b)" by (subst measure_lborel_box_eq) (auto intro!: prod_pos simp: algebra_simps box_def) have "emeasure lborel (box a b) \ emeasure lborel (ball c r)" using ab by (intro emeasure_mono) auto also have "emeasure lborel (box a b) = ennreal (content (box a b))" using emeasure_lborel_box_finite[of a b] by (intro emeasure_eq_ennreal_measure) auto also have "emeasure lborel (ball c r) = ennreal (content (ball c r))" using emeasure_lborel_ball_finite[of c r] by (intro emeasure_eq_ennreal_measure) auto finally show ?thesis using \content (box a b) > 0\ by simp qed lemma content_cball_pos: assumes "r > 0" shows "content (cball c r) > 0" proof - from rational_boxes[OF assms, of c] obtain a b where ab: "c \ box a b" "box a b \ ball c r" by auto from ab have "0 < content (box a b)" by (subst measure_lborel_box_eq) (auto intro!: prod_pos simp: algebra_simps box_def) have "emeasure lborel (box a b) \ emeasure lborel (ball c r)" using ab by (intro emeasure_mono) auto also have "\ \ emeasure lborel (cball c r)" by (intro emeasure_mono) auto also have "emeasure lborel (box a b) = ennreal (content (box a b))" using emeasure_lborel_box_finite[of a b] by (intro emeasure_eq_ennreal_measure) auto also have "emeasure lborel (cball c r) = ennreal (content (cball c r))" using emeasure_lborel_cball_finite[of c r] by (intro emeasure_eq_ennreal_measure) auto finally show ?thesis using \content (box a b) > 0\ by simp qed lemma content_split: fixes a :: "'a::euclidean_space" assumes "k \ Basis" shows "content (cbox a b) = content(cbox a b \ {x. x\k \ c}) + content(cbox a b \ {x. x\k \ c})" \ \Prove using measure theory\ proof (cases "\i\Basis. a \ i \ b \ i") case True have 1: "\X Y Z. (\i\Basis. Z i (if i = k then X else Y i)) = Z k X * (\i\Basis-{k}. Z i (Y i))" by (simp add: if_distrib prod.delta_remove assms) note simps = interval_split[OF assms] content_cbox_cases have 2: "(\i\Basis. b\i - a\i) = (\i\Basis-{k}. b\i - a\i) * (b\k - a\k)" by (metis (no_types, lifting) assms finite_Basis mult.commute prod.remove) have "\x. min (b \ k) c = max (a \ k) c \ x * (b\k - a\k) = x * (max (a \ k) c - a \ k) + x * (b \ k - max (a \ k) c)" by (auto simp add: field_simps) moreover have **: "(\i\Basis. ((\i\Basis. (if i = k then min (b \ k) c else b \ i) *\<^sub>R i) \ i - a \ i)) = (\i\Basis. (if i = k then min (b \ k) c else b \ i) - a \ i)" "(\i\Basis. b \ i - ((\i\Basis. (if i = k then max (a \ k) c else a \ i) *\<^sub>R i) \ i)) = (\i\Basis. b \ i - (if i = k then max (a \ k) c else a \ i))" by (auto intro!: prod.cong) have "\ a \ k \ c \ \ c \ b \ k \ False" unfolding not_le using True assms by auto ultimately show ?thesis using assms unfolding simps ** 1[of "\i x. b\i - x"] 1[of "\i x. x - a\i"] 2 by auto next case False then have "cbox a b = {}" unfolding box_eq_empty by (auto simp: not_le) then show ?thesis by (auto simp: not_le) qed lemma division_of_content_0: assumes "content (cbox a b) = 0" "d division_of (cbox a b)" "K \ d" shows "content K = 0" unfolding forall_in_division[OF assms(2)] by (meson assms content_0_subset division_of_def) lemma sum_content_null: assumes "content (cbox a b) = 0" and "p tagged_division_of (cbox a b)" shows "(\(x,K)\p. content K *\<^sub>R f x) = (0::'a::real_normed_vector)" proof (rule sum.neutral, rule) fix y assume y: "y \ p" obtain x K where xk: "y = (x, K)" using surj_pair[of y] by blast then obtain c d where k: "K = cbox c d" "K \ cbox a b" by (metis assms(2) tagged_division_ofD(3) tagged_division_ofD(4) y) have "(\(x',K'). content K' *\<^sub>R f x') y = content K *\<^sub>R f x" unfolding xk by auto also have "\ = 0" using assms(1) content_0_subset k(2) by auto finally show "(\(x, k). content k *\<^sub>R f x) y = 0" . qed global_interpretation sum_content: operative plus 0 content rewrites "comm_monoid_set.F plus 0 = sum" proof - interpret operative plus 0 content by standard (auto simp add: content_split [symmetric] content_eq_0_interior) show "operative plus 0 content" by standard show "comm_monoid_set.F plus 0 = sum" by (simp add: sum_def) qed lemma additive_content_division: "d division_of (cbox a b) \ sum content d = content (cbox a b)" by (fact sum_content.division) lemma additive_content_tagged_division: "d tagged_division_of (cbox a b) \ sum (\(x,l). content l) d = content (cbox a b)" by (fact sum_content.tagged_division) lemma subadditive_content_division: assumes "\ division_of S" "S \ cbox a b" shows "sum content \ \ content(cbox a b)" proof - have "\ division_of \\" "\\ \ cbox a b" using assms by auto then obtain \' where "\ \ \'" "\' division_of cbox a b" using partial_division_extend_interval by metis then have "sum content \ \ sum content \'" using sum_mono2 by blast also have "... \ content(cbox a b)" by (simp add: \\' division_of cbox a b\ additive_content_division less_eq_real_def) finally show ?thesis . qed lemma content_real_eq_0: "content {a..b::real} = 0 \ a \ b" by (metis atLeastatMost_empty_iff2 content_empty content_real diff_self eq_iff le_cases le_iff_diff_le_0) lemma property_empty_interval: "\a b. content (cbox a b) = 0 \ P (cbox a b) \ P {}" using content_empty unfolding empty_as_interval by auto lemma interval_bounds_nz_content [simp]: assumes "content (cbox a b) \ 0" shows "interval_upperbound (cbox a b) = b" and "interval_lowerbound (cbox a b) = a" by (metis assms content_empty interval_bounds')+ subsection \Gauge integral\ text \Case distinction to define it first on compact intervals first, then use a limit. This is only much later unified. In Fremlin: Measure Theory, Volume 4I this is generalized using residual sets.\ definition has_integral :: "('n::euclidean_space \ 'b::real_normed_vector) \ 'b \ 'n set \ bool" (infixr "has'_integral" 46) where "(f has_integral I) s \ (if \a b. s = cbox a b then ((\p. \(x,k)\p. content k *\<^sub>R f x) \ I) (division_filter s) else (\e>0. \B>0. \a b. ball 0 B \ cbox a b \ (\z. ((\p. \(x,k)\p. content k *\<^sub>R (if x \ s then f x else 0)) \ z) (division_filter (cbox a b)) \ norm (z - I) < e)))" lemma has_integral_cbox: "(f has_integral I) (cbox a b) \ ((\p. \(x,k)\p. content k *\<^sub>R f x) \ I) (division_filter (cbox a b))" by (auto simp add: has_integral_def) lemma has_integral: "(f has_integral y) (cbox a b) \ (\e>0. \\. gauge \ \ (\\. \ tagged_division_of (cbox a b) \ \ fine \ \ norm (sum (\(x,k). content(k) *\<^sub>R f x) \ - y) < e))" by (auto simp: dist_norm eventually_division_filter has_integral_def tendsto_iff) lemma has_integral_real: "(f has_integral y) {a..b::real} \ (\e>0. \\. gauge \ \ (\\. \ tagged_division_of {a..b} \ \ fine \ \ norm (sum (\(x,k). content(k) *\<^sub>R f x) \ - y) < e))" unfolding box_real[symmetric] by (rule has_integral) lemma has_integralD[dest]: assumes "(f has_integral y) (cbox a b)" and "e > 0" obtains \ where "gauge \" and "\\. \ tagged_division_of (cbox a b) \ \ fine \ \ norm ((\(x,k)\\. content k *\<^sub>R f x) - y) < e" using assms unfolding has_integral by auto lemma has_integral_alt: "(f has_integral y) i \ (if \a b. i = cbox a b then (f has_integral y) i else (\e>0. \B>0. \a b. ball 0 B \ cbox a b \ (\z. ((\x. if x \ i then f x else 0) has_integral z) (cbox a b) \ norm (z - y) < e)))" by (subst has_integral_def) (auto simp add: has_integral_cbox) lemma has_integral_altD: assumes "(f has_integral y) i" and "\ (\a b. i = cbox a b)" and "e>0" obtains B where "B > 0" and "\a b. ball 0 B \ cbox a b \ (\z. ((\x. if x \ i then f(x) else 0) has_integral z) (cbox a b) \ norm(z - y) < e)" using assms has_integral_alt[of f y i] by auto definition integrable_on (infixr "integrable'_on" 46) where "f integrable_on i \ (\y. (f has_integral y) i)" definition "integral i f = (SOME y. (f has_integral y) i \ \ f integrable_on i \ y=0)" lemma integrable_integral[intro]: "f integrable_on i \ (f has_integral (integral i f)) i" unfolding integrable_on_def integral_def by (metis (mono_tags, lifting) someI_ex) lemma not_integrable_integral: "\ f integrable_on i \ integral i f = 0" unfolding integrable_on_def integral_def by blast lemma has_integral_integrable[dest]: "(f has_integral i) s \ f integrable_on s" unfolding integrable_on_def by auto lemma has_integral_integral: "f integrable_on s \ (f has_integral (integral s f)) s" by auto subsection \Basic theorems about integrals\ lemma has_integral_eq_rhs: "(f has_integral j) S \ i = j \ (f has_integral i) S" by (rule forw_subst) lemma has_integral_unique_cbox: fixes f :: "'n::euclidean_space \ 'a::real_normed_vector" shows "(f has_integral k1) (cbox a b) \ (f has_integral k2) (cbox a b) \ k1 = k2" by (auto simp: has_integral_cbox intro: tendsto_unique[OF division_filter_not_empty]) lemma has_integral_unique: fixes f :: "'n::euclidean_space \ 'a::real_normed_vector" assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2" proof (rule ccontr) let ?e = "norm (k1 - k2)/2" let ?F = "(\x. if x \ i then f x else 0)" assume "k1 \ k2" then have e: "?e > 0" by auto have nonbox: "\ (\a b. i = cbox a b)" using \k1 \ k2\ assms has_integral_unique_cbox by blast obtain B1 where B1: "0 < B1" "\a b. ball 0 B1 \ cbox a b \ \z. (?F has_integral z) (cbox a b) \ norm (z - k1) < norm (k1 - k2)/2" by (rule has_integral_altD[OF assms(1) nonbox,OF e]) blast obtain B2 where B2: "0 < B2" "\a b. ball 0 B2 \ cbox a b \ \z. (?F has_integral z) (cbox a b) \ norm (z - k2) < norm (k1 - k2)/2" by (rule has_integral_altD[OF assms(2) nonbox,OF e]) blast obtain a b :: 'n where ab: "ball 0 B1 \ cbox a b" "ball 0 B2 \ cbox a b" by (metis Un_subset_iff bounded_Un bounded_ball bounded_subset_cbox_symmetric) obtain w where w: "(?F has_integral w) (cbox a b)" "norm (w - k1) < norm (k1 - k2)/2" using B1(2)[OF ab(1)] by blast obtain z where z: "(?F has_integral z) (cbox a b)" "norm (z - k2) < norm (k1 - k2)/2" using B2(2)[OF ab(2)] by blast have "z = w" using has_integral_unique_cbox[OF w(1) z(1)] by auto then have "norm (k1 - k2) \ norm (z - k2) + norm (w - k1)" using norm_triangle_ineq4 [of "k1 - w" "k2 - z"] by (auto simp add: norm_minus_commute) also have "\ < norm (k1 - k2)/2 + norm (k1 - k2)/2" by (metis add_strict_mono z(2) w(2)) finally show False by auto qed lemma integral_unique [intro]: "(f has_integral y) k \ integral k f = y" unfolding integral_def by (rule some_equality) (auto intro: has_integral_unique) lemma has_integral_iff: "(f has_integral i) S \ (f integrable_on S \ integral S f = i)" by blast lemma eq_integralD: "integral k f = y \ (f has_integral y) k \ \ f integrable_on k \ y=0" unfolding integral_def integrable_on_def apply (erule subst) apply (rule someI_ex) by blast lemma has_integral_const [intro]: fixes a b :: "'a::euclidean_space" shows "((\x. c) has_integral (content (cbox a b) *\<^sub>R c)) (cbox a b)" using eventually_division_filter_tagged_division[of "cbox a b"] additive_content_tagged_division[of _ a b] by (auto simp: has_integral_cbox split_beta' scaleR_sum_left[symmetric] elim!: eventually_mono intro!: tendsto_cong[THEN iffD1, OF _ tendsto_const]) lemma has_integral_const_real [intro]: fixes a b :: real shows "((\x. c) has_integral (content {a..b} *\<^sub>R c)) {a..b}" by (metis box_real(2) has_integral_const) lemma has_integral_integrable_integral: "(f has_integral i) s \ f integrable_on s \ integral s f = i" by blast lemma integral_const [simp]: fixes a b :: "'a::euclidean_space" shows "integral (cbox a b) (\x. c) = content (cbox a b) *\<^sub>R c" by (rule integral_unique) (rule has_integral_const) lemma integral_const_real [simp]: fixes a b :: real shows "integral {a..b} (\x. c) = content {a..b} *\<^sub>R c" by (metis box_real(2) integral_const) lemma has_integral_is_0_cbox: fixes f :: "'n::euclidean_space \ 'a::real_normed_vector" assumes "\x. x \ cbox a b \ f x = 0" shows "(f has_integral 0) (cbox a b)" unfolding has_integral_cbox using eventually_division_filter_tagged_division[of "cbox a b"] assms by (subst tendsto_cong[where g="\_. 0"]) (auto elim!: eventually_mono intro!: sum.neutral simp: tag_in_interval) lemma has_integral_is_0: fixes f :: "'n::euclidean_space \ 'a::real_normed_vector" assumes "\x. x \ S \ f x = 0" shows "(f has_integral 0) S" proof (cases "(\a b. S = cbox a b)") case True with assms has_integral_is_0_cbox show ?thesis by blast next case False have *: "(\x. if x \ S then f x else 0) = (\x. 0)" by (auto simp add: assms) show ?thesis using has_integral_is_0_cbox False by (subst has_integral_alt) (force simp add: *) qed lemma has_integral_0[simp]: "((\x::'n::euclidean_space. 0) has_integral 0) S" by (rule has_integral_is_0) auto lemma has_integral_0_eq[simp]: "((\x. 0) has_integral i) S \ i = 0" using has_integral_unique[OF has_integral_0] by auto lemma has_integral_linear_cbox: fixes f :: "'n::euclidean_space \ 'a::real_normed_vector" assumes f: "(f has_integral y) (cbox a b)" and h: "bounded_linear h" shows "((h \ f) has_integral (h y)) (cbox a b)" proof - interpret bounded_linear h using h . show ?thesis unfolding has_integral_cbox using tendsto [OF f [unfolded has_integral_cbox]] by (simp add: sum scaleR split_beta') qed lemma has_integral_linear: fixes f :: "'n::euclidean_space \ 'a::real_normed_vector" assumes f: "(f has_integral y) S" and h: "bounded_linear h" shows "((h \ f) has_integral (h y)) S" proof (cases "(\a b. S = cbox a b)") case True with f h has_integral_linear_cbox show ?thesis by blast next case False interpret bounded_linear h using h . from pos_bounded obtain B where B: "0 < B" "\x. norm (h x) \ norm x * B" by blast let ?S = "\f x. if x \ S then f x else 0" show ?thesis proof (subst has_integral_alt, clarsimp simp: False) fix e :: real assume e: "e > 0" have *: "0 < e/B" using e B(1) by simp obtain M where M: "M > 0" "\a b. ball 0 M \ cbox a b \ \z. (?S f has_integral z) (cbox a b) \ norm (z - y) < e/B" using has_integral_altD[OF f False *] by blast show "\B>0. \a b. ball 0 B \ cbox a b \ (\z. (?S(h \ f) has_integral z) (cbox a b) \ norm (z - h y) < e)" proof (rule exI, intro allI conjI impI) show "M > 0" using M by metis next fix a b::'n assume sb: "ball 0 M \ cbox a b" obtain z where z: "(?S f has_integral z) (cbox a b)" "norm (z - y) < e/B" using M(2)[OF sb] by blast have *: "?S(h \ f) = h \ ?S f" using zero by auto show "\z. (?S(h \ f) has_integral z) (cbox a b) \ norm (z - h y) < e" proof (intro exI conjI) show "(?S(h \ f) has_integral h z) (cbox a b)" by (simp add: * has_integral_linear_cbox[OF z(1) h]) show "norm (h z - h y) < e" by (metis B diff le_less_trans pos_less_divide_eq z(2)) qed qed qed qed lemma has_integral_scaleR_left: "(f has_integral y) S \ ((\x. f x *\<^sub>R c) has_integral (y *\<^sub>R c)) S" using has_integral_linear[OF _ bounded_linear_scaleR_left] by (simp add: comp_def) lemma integrable_on_scaleR_left: assumes "f integrable_on A" shows "(\x. f x *\<^sub>R y) integrable_on A" using assms has_integral_scaleR_left unfolding integrable_on_def by blast lemma has_integral_mult_left: fixes c :: "_ :: real_normed_algebra" shows "(f has_integral y) S \ ((\x. f x * c) has_integral (y * c)) S" using has_integral_linear[OF _ bounded_linear_mult_left] by (simp add: comp_def) lemma has_integral_divide: fixes c :: "_ :: real_normed_div_algebra" shows "(f has_integral y) S \ ((\x. f x / c) has_integral (y / c)) S" unfolding divide_inverse by (simp add: has_integral_mult_left) text\The case analysis eliminates the condition \<^term>\f integrable_on S\ at the cost of the type class constraint \division_ring\\ corollary integral_mult_left [simp]: fixes c:: "'a::{real_normed_algebra,division_ring}" shows "integral S (\x. f x * c) = integral S f * c" proof (cases "f integrable_on S \ c = 0") case True then show ?thesis by (force intro: has_integral_mult_left) next case False then have "\ (\x. f x * c) integrable_on S" using has_integral_mult_left [of "(\x. f x * c)" _ S "inverse c"] by (auto simp add: mult.assoc) with False show ?thesis by (simp add: not_integrable_integral) qed corollary integral_mult_right [simp]: fixes c:: "'a::{real_normed_field}" shows "integral S (\x. c * f x) = c * integral S f" by (simp add: mult.commute [of c]) corollary integral_divide [simp]: fixes z :: "'a::real_normed_field" shows "integral S (\x. f x / z) = integral S (\x. f x) / z" using integral_mult_left [of S f "inverse z"] by (simp add: divide_inverse_commute) lemma has_integral_mult_right: fixes c :: "'a :: real_normed_algebra" shows "(f has_integral y) i \ ((\x. c * f x) has_integral (c * y)) i" using has_integral_linear[OF _ bounded_linear_mult_right] by (simp add: comp_def) lemma has_integral_cmul: "(f has_integral k) S \ ((\x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) S" unfolding o_def[symmetric] by (metis has_integral_linear bounded_linear_scaleR_right) lemma has_integral_cmult_real: fixes c :: real assumes "c \ 0 \ (f has_integral x) A" shows "((\x. c * f x) has_integral c * x) A" proof (cases "c = 0") case True then show ?thesis by simp next case False from has_integral_cmul[OF assms[OF this], of c] show ?thesis unfolding real_scaleR_def . qed lemma has_integral_neg: "(f has_integral k) S \ ((\x. -(f x)) has_integral -k) S" by (drule_tac c="-1" in has_integral_cmul) auto lemma has_integral_neg_iff: "((\x. - f x) has_integral k) S \ (f has_integral - k) S" using has_integral_neg[of f "- k"] has_integral_neg[of "\x. - f x" k] by auto lemma has_integral_add_cbox: fixes f :: "'n::euclidean_space \ 'a::real_normed_vector" assumes "(f has_integral k) (cbox a b)" "(g has_integral l) (cbox a b)" shows "((\x. f x + g x) has_integral (k + l)) (cbox a b)" using assms unfolding has_integral_cbox by (simp add: split_beta' scaleR_add_right sum.distrib[abs_def] tendsto_add) lemma has_integral_add: fixes f :: "'n::euclidean_space \ 'a::real_normed_vector" assumes f: "(f has_integral k) S" and g: "(g has_integral l) S" shows "((\x. f x + g x) has_integral (k + l)) S" proof (cases "\a b. S = cbox a b") case True with has_integral_add_cbox assms show ?thesis by blast next let ?S = "\f x. if x \ S then f x else 0" case False then show ?thesis proof (subst has_integral_alt, clarsimp, goal_cases) case (1 e) then have e2: "e/2 > 0" by auto obtain Bf where "0 < Bf" and Bf: "\a b. ball 0 Bf \ cbox a b \ \z. (?S f has_integral z) (cbox a b) \ norm (z - k) < e/2" using has_integral_altD[OF f False e2] by blast obtain Bg where "0 < Bg" and Bg: "\a b. ball 0 Bg \ (cbox a b) \ \z. (?S g has_integral z) (cbox a b) \ norm (z - l) < e/2" using has_integral_altD[OF g False e2] by blast show ?case proof (rule_tac x="max Bf Bg" in exI, clarsimp simp add: max.strict_coboundedI1 \0 < Bf\) fix a b assume "ball 0 (max Bf Bg) \ cbox a (b::'n)" then have fs: "ball 0 Bf \ cbox a (b::'n)" and gs: "ball 0 Bg \ cbox a (b::'n)" by auto obtain w where w: "(?S f has_integral w) (cbox a b)" "norm (w - k) < e/2" using Bf[OF fs] by blast obtain z where z: "(?S g has_integral z) (cbox a b)" "norm (z - l) < e/2" using Bg[OF gs] by blast have *: "\x. (if x \ S then f x + g x else 0) = (?S f x) + (?S g x)" by auto show "\z. (?S(\x. f x + g x) has_integral z) (cbox a b) \ norm (z - (k + l)) < e" proof (intro exI conjI) show "(?S(\x. f x + g x) has_integral (w + z)) (cbox a b)" by (simp add: has_integral_add_cbox[OF w(1) z(1), unfolded *[symmetric]]) show "norm (w + z - (k + l)) < e" by (metis dist_norm dist_triangle_add_half w(2) z(2)) qed qed qed qed lemma has_integral_diff: "(f has_integral k) S \ (g has_integral l) S \ ((\x. f x - g x) has_integral (k - l)) S" using has_integral_add[OF _ has_integral_neg, of f k S g l] by (auto simp: algebra_simps) lemma integral_0 [simp]: "integral S (\x::'n::euclidean_space. 0::'m::real_normed_vector) = 0" by (rule integral_unique has_integral_0)+ lemma integral_add: "f integrable_on S \ g integrable_on S \ integral S (\x. f x + g x) = integral S f + integral S g" by (rule integral_unique) (metis integrable_integral has_integral_add) lemma integral_cmul [simp]: "integral S (\x. c *\<^sub>R f x) = c *\<^sub>R integral S f" proof (cases "f integrable_on S \ c = 0") case True with has_integral_cmul integrable_integral show ?thesis by fastforce next case False then have "\ (\x. c *\<^sub>R f x) integrable_on S" using has_integral_cmul [of "(\x. c *\<^sub>R f x)" _ S "inverse c"] by auto with False show ?thesis by (simp add: not_integrable_integral) qed lemma integral_mult: fixes K::real shows "f integrable_on X \ K * integral X f = integral X (\x. K * f x)" unfolding real_scaleR_def[symmetric] integral_cmul .. lemma integral_neg [simp]: "integral S (\x. - f x) = - integral S f" proof (cases "f integrable_on S") case True then show ?thesis by (simp add: has_integral_neg integrable_integral integral_unique) next case False then have "\ (\x. - f x) integrable_on S" using has_integral_neg [of "(\x. - f x)" _ S ] by auto with False show ?thesis by (simp add: not_integrable_integral) qed lemma integral_diff: "f integrable_on S \ g integrable_on S \ integral S (\x. f x - g x) = integral S f - integral S g" by (rule integral_unique) (metis integrable_integral has_integral_diff) lemma integrable_0: "(\x. 0) integrable_on S" unfolding integrable_on_def using has_integral_0 by auto lemma integrable_add: "f integrable_on S \ g integrable_on S \ (\x. f x + g x) integrable_on S" unfolding integrable_on_def by(auto intro: has_integral_add) lemma integrable_cmul: "f integrable_on S \ (\x. c *\<^sub>R f(x)) integrable_on S" unfolding integrable_on_def by(auto intro: has_integral_cmul) lemma integrable_on_scaleR_iff [simp]: fixes c :: real assumes "c \ 0" shows "(\x. c *\<^sub>R f x) integrable_on S \ f integrable_on S" using integrable_cmul[of "\x. c *\<^sub>R f x" S "1 / c"] integrable_cmul[of f S c] \c \ 0\ by auto lemma integrable_on_cmult_iff [simp]: fixes c :: real assumes "c \ 0" shows "(\x. c * f x) integrable_on S \ f integrable_on S" using integrable_on_scaleR_iff [of c f] assms by simp lemma integrable_on_cmult_left: assumes "f integrable_on S" shows "(\x. of_real c * f x) integrable_on S" using integrable_cmul[of f S "of_real c"] assms by (simp add: scaleR_conv_of_real) lemma integrable_neg: "f integrable_on S \ (\x. -f(x)) integrable_on S" unfolding integrable_on_def by(auto intro: has_integral_neg) lemma integrable_neg_iff: "(\x. -f(x)) integrable_on S \ f integrable_on S" using integrable_neg by fastforce lemma integrable_diff: "f integrable_on S \ g integrable_on S \ (\x. f x - g x) integrable_on S" unfolding integrable_on_def by(auto intro: has_integral_diff) lemma integrable_linear: "f integrable_on S \ bounded_linear h \ (h \ f) integrable_on S" unfolding integrable_on_def by(auto intro: has_integral_linear) lemma integral_linear: "f integrable_on S \ bounded_linear h \ integral S (h \ f) = h (integral S f)" by (meson has_integral_iff has_integral_linear) lemma integrable_on_cnj_iff: "(\x. cnj (f x)) integrable_on A \ f integrable_on A" using integrable_linear[OF _ bounded_linear_cnj, of f A] integrable_linear[OF _ bounded_linear_cnj, of "cnj \ f" A] by (auto simp: o_def) lemma integral_cnj: "cnj (integral A f) = integral A (\x. cnj (f x))" by (cases "f integrable_on A") (simp_all add: integral_linear[OF _ bounded_linear_cnj, symmetric] o_def integrable_on_cnj_iff not_integrable_integral) lemma integral_component_eq[simp]: fixes f :: "'n::euclidean_space \ 'm::euclidean_space" assumes "f integrable_on S" shows "integral S (\x. f x \ k) = integral S f \ k" unfolding integral_linear[OF assms(1) bounded_linear_inner_left,unfolded o_def] .. lemma has_integral_sum: assumes "finite T" and "\a. a \ T \ ((f a) has_integral (i a)) S" shows "((\x. sum (\a. f a x) T) has_integral (sum i T)) S" using assms(1) subset_refl[of T] proof (induct rule: finite_subset_induct) case empty then show ?case by auto next case (insert x F) with assms show ?case by (simp add: has_integral_add) qed lemma integral_sum: "\finite I; \a. a \ I \ f a integrable_on S\ \ integral S (\x. \a\I. f a x) = (\a\I. integral S (f a))" by (simp add: has_integral_sum integrable_integral integral_unique) lemma integrable_sum: "\finite I; \a. a \ I \ f a integrable_on S\ \ (\x. \a\I. f a x) integrable_on S" unfolding integrable_on_def using has_integral_sum[of I] by metis lemma has_integral_eq: assumes "\x. x \ s \ f x = g x" and "(f has_integral k) s" shows "(g has_integral k) s" using has_integral_diff[OF assms(2), of "\x. f x - g x" 0] using has_integral_is_0[of s "\x. f x - g x"] using assms(1) by auto lemma integrable_eq: "\f integrable_on s; \x. x \ s \ f x = g x\ \ g integrable_on s" unfolding integrable_on_def using has_integral_eq[of s f g] has_integral_eq by blast lemma has_integral_cong: assumes "\x. x \ s \ f x = g x" shows "(f has_integral i) s = (g has_integral i) s" using has_integral_eq[of s f g] has_integral_eq[of s g f] assms by auto lemma integral_cong: assumes "\x. x \ s \ f x = g x" shows "integral s f = integral s g" unfolding integral_def by (metis (full_types, opaque_lifting) assms has_integral_cong integrable_eq) lemma integrable_on_cmult_left_iff [simp]: assumes "c \ 0" shows "(\x. of_real c * f x) integrable_on s \ f integrable_on s" (is "?lhs = ?rhs") proof assume ?lhs then have "(\x. of_real (1 / c) * (of_real c * f x)) integrable_on s" using integrable_cmul[of "\x. of_real c * f x" s "1 / of_real c"] by (simp add: scaleR_conv_of_real) then have "(\x. (of_real (1 / c) * of_real c * f x)) integrable_on s" by (simp add: algebra_simps) with \c \ 0\ show ?rhs by (metis (no_types, lifting) integrable_eq mult.left_neutral nonzero_divide_eq_eq of_real_1 of_real_mult) qed (blast intro: integrable_on_cmult_left) lemma integrable_on_cmult_right: fixes f :: "_ \ 'b :: {comm_ring,real_algebra_1,real_normed_vector}" assumes "f integrable_on s" shows "(\x. f x * of_real c) integrable_on s" using integrable_on_cmult_left [OF assms] by (simp add: mult.commute) lemma integrable_on_cmult_right_iff [simp]: fixes f :: "_ \ 'b :: {comm_ring,real_algebra_1,real_normed_vector}" assumes "c \ 0" shows "(\x. f x * of_real c) integrable_on s \ f integrable_on s" using integrable_on_cmult_left_iff [OF assms] by (simp add: mult.commute) lemma integrable_on_cdivide: fixes f :: "_ \ 'b :: real_normed_field" assumes "f integrable_on s" shows "(\x. f x / of_real c) integrable_on s" by (simp add: integrable_on_cmult_right divide_inverse assms flip: of_real_inverse) lemma integrable_on_cdivide_iff [simp]: fixes f :: "_ \ 'b :: real_normed_field" assumes "c \ 0" shows "(\x. f x / of_real c) integrable_on s \ f integrable_on s" by (simp add: divide_inverse assms flip: of_real_inverse) lemma has_integral_null [intro]: "content(cbox a b) = 0 \ (f has_integral 0) (cbox a b)" unfolding has_integral_cbox using eventually_division_filter_tagged_division[of "cbox a b"] by (subst tendsto_cong[where g="\_. 0"]) (auto elim: eventually_mono intro: sum_content_null) lemma has_integral_null_real [intro]: "content {a..b::real} = 0 \ (f has_integral 0) {a..b}" by (metis box_real(2) has_integral_null) lemma has_integral_null_eq[simp]: "content (cbox a b) = 0 \ (f has_integral i) (cbox a b) \ i = 0" by (auto simp add: has_integral_null dest!: integral_unique) lemma integral_null [simp]: "content (cbox a b) = 0 \ integral (cbox a b) f = 0" by (metis has_integral_null integral_unique) lemma integrable_on_null [intro]: "content (cbox a b) = 0 \ f integrable_on (cbox a b)" by (simp add: has_integral_integrable) lemma has_integral_empty[intro]: "(f has_integral 0) {}" by (meson ex_in_conv has_integral_is_0) lemma has_integral_empty_eq[simp]: "(f has_integral i) {} \ i = 0" by (auto simp add: has_integral_empty has_integral_unique) lemma integrable_on_empty[intro]: "f integrable_on {}" unfolding integrable_on_def by auto lemma integral_empty[simp]: "integral {} f = 0" by (rule integral_unique) (rule has_integral_empty) lemma has_integral_refl[intro]: fixes a :: "'a::euclidean_space" shows "(f has_integral 0) (cbox a a)" and "(f has_integral 0) {a}" proof - show "(f has_integral 0) (cbox a a)" by (rule has_integral_null) simp then show "(f has_integral 0) {a}" by simp qed lemma integrable_on_refl[intro]: "f integrable_on cbox a a" unfolding integrable_on_def by auto lemma integral_refl [simp]: "integral (cbox a a) f = 0" by (rule integral_unique) auto lemma integral_singleton [simp]: "integral {a} f = 0" by auto lemma integral_blinfun_apply: assumes "f integrable_on s" shows "integral s (\x. blinfun_apply h (f x)) = blinfun_apply h (integral s f)" by (subst integral_linear[symmetric, OF assms blinfun.bounded_linear_right]) (simp add: o_def) lemma blinfun_apply_integral: assumes "f integrable_on s" shows "blinfun_apply (integral s f) x = integral s (\y. blinfun_apply (f y) x)" by (metis (no_types, lifting) assms blinfun.prod_left.rep_eq integral_blinfun_apply integral_cong) lemma has_integral_componentwise_iff: fixes f :: "'a :: euclidean_space \ 'b :: euclidean_space" shows "(f has_integral y) A \ (\b\Basis. ((\x. f x \ b) has_integral (y \ b)) A)" proof safe fix b :: 'b assume "(f has_integral y) A" from has_integral_linear[OF this(1) bounded_linear_inner_left, of b] show "((\x. f x \ b) has_integral (y \ b)) A" by (simp add: o_def) next assume "(\b\Basis. ((\x. f x \ b) has_integral (y \ b)) A)" hence "\b\Basis. (((\x. x *\<^sub>R b) \ (\x. f x \ b)) has_integral ((y \ b) *\<^sub>R b)) A" by (intro ballI has_integral_linear) (simp_all add: bounded_linear_scaleR_left) hence "((\x. \b\Basis. (f x \ b) *\<^sub>R b) has_integral (\b\Basis. (y \ b) *\<^sub>R b)) A" by (intro has_integral_sum) (simp_all add: o_def) thus "(f has_integral y) A" by (simp add: euclidean_representation) qed lemma has_integral_componentwise: fixes f :: "'a :: euclidean_space \ 'b :: euclidean_space" shows "(\b. b \ Basis \ ((\x. f x \ b) has_integral (y \ b)) A) \ (f has_integral y) A" by (subst has_integral_componentwise_iff) blast lemma integrable_componentwise_iff: fixes f :: "'a :: euclidean_space \ 'b :: euclidean_space" shows "f integrable_on A \ (\b\Basis. (\x. f x \ b) integrable_on A)" proof assume "f integrable_on A" then obtain y where "(f has_integral y) A" by (auto simp: integrable_on_def) hence "(\b\Basis. ((\x. f x \ b) has_integral (y \ b)) A)" by (subst (asm) has_integral_componentwise_iff) thus "(\b\Basis. (\x. f x \ b) integrable_on A)" by (auto simp: integrable_on_def) next assume "(\b\Basis. (\x. f x \ b) integrable_on A)" then obtain y where "\b\Basis. ((\x. f x \ b) has_integral y b) A" unfolding integrable_on_def by (subst (asm) bchoice_iff) blast hence "\b\Basis. (((\x. x *\<^sub>R b) \ (\x. f x \ b)) has_integral (y b *\<^sub>R b)) A" by (intro ballI has_integral_linear) (simp_all add: bounded_linear_scaleR_left) hence "((\x. \b\Basis. (f x \ b) *\<^sub>R b) has_integral (\b\Basis. y b *\<^sub>R b)) A" by (intro has_integral_sum) (simp_all add: o_def) thus "f integrable_on A" by (auto simp: integrable_on_def o_def euclidean_representation) qed lemma integrable_componentwise: fixes f :: "'a :: euclidean_space \ 'b :: euclidean_space" shows "(\b. b \ Basis \ (\x. f x \ b) integrable_on A) \ f integrable_on A" by (subst integrable_componentwise_iff) blast lemma integral_componentwise: fixes f :: "'a :: euclidean_space \ 'b :: euclidean_space" assumes "f integrable_on A" shows "integral A f = (\b\Basis. integral A (\x. (f x \ b) *\<^sub>R b))" proof - from assms have integrable: "\b\Basis. (\x. x *\<^sub>R b) \ (\x. (f x \ b)) integrable_on A" by (subst (asm) integrable_componentwise_iff, intro integrable_linear ballI) (simp_all add: bounded_linear_scaleR_left) have "integral A f = integral A (\x. \b\Basis. (f x \ b) *\<^sub>R b)" by (simp add: euclidean_representation) also from integrable have "\ = (\a\Basis. integral A (\x. (f x \ a) *\<^sub>R a))" by (subst integral_sum) (simp_all add: o_def) finally show ?thesis . qed lemma integrable_component: "f integrable_on A \ (\x. f x \ (y :: 'b :: euclidean_space)) integrable_on A" by (drule integrable_linear[OF _ bounded_linear_inner_left[of y]]) (simp add: o_def) subsection \Cauchy-type criterion for integrability\ proposition integrable_Cauchy: fixes f :: "'n::euclidean_space \ 'a::{real_normed_vector,complete_space}" shows "f integrable_on cbox a b \ (\e>0. \\. gauge \ \ (\\1 \2. \1 tagged_division_of (cbox a b) \ \ fine \1 \ \2 tagged_division_of (cbox a b) \ \ fine \2 \ norm ((\(x,K)\\1. content K *\<^sub>R f x) - (\(x,K)\\2. content K *\<^sub>R f x)) < e))" (is "?l = (\e>0. \\. ?P e \)") proof (intro iffI allI impI) assume ?l then obtain y where y: "\e. e > 0 \ \\. gauge \ \ (\\. \ tagged_division_of cbox a b \ \ fine \ \ norm ((\(x,K) \ \. content K *\<^sub>R f x) - y) < e)" by (auto simp: integrable_on_def has_integral) show "\\. ?P e \" if "e > 0" for e proof - have "e/2 > 0" using that by auto with y obtain \ where "gauge \" and \: "\\. \ tagged_division_of cbox a b \ \ fine \ \ norm ((\(x,K)\\. content K *\<^sub>R f x) - y) < e/2" by meson show ?thesis apply (rule_tac x=\ in exI, clarsimp simp: \gauge \\) by (blast intro!: \ dist_triangle_half_l[where y=y,unfolded dist_norm]) qed next assume "\e>0. \\. ?P e \" then have "\n::nat. \\. ?P (1 / (n + 1)) \" by auto then obtain \ :: "nat \ 'n \ 'n set" where \: "\m. gauge (\ m)" "\m \1 \2. \\1 tagged_division_of cbox a b; \ m fine \1; \2 tagged_division_of cbox a b; \ m fine \2\ \ norm ((\(x,K) \ \1. content K *\<^sub>R f x) - (\(x,K) \ \2. content K *\<^sub>R f x)) < 1 / (m + 1)" by metis have "gauge (\x. \{\ i x |i. i \ {0..n}})" for n using \ by (intro gauge_Inter) auto then have "\n. \p. p tagged_division_of (cbox a b) \ (\x. \{\ i x |i. i \ {0..n}}) fine p" by (meson fine_division_exists) then obtain p where p: "\z. p z tagged_division_of cbox a b" "\z. (\x. \{\ i x |i. i \ {0..z}}) fine p z" by meson have dp: "\i n. i\n \ \ i fine p n" using p unfolding fine_Inter using atLeastAtMost_iff by blast have "Cauchy (\n. sum (\(x,K). content K *\<^sub>R (f x)) (p n))" proof (rule CauchyI) fix e::real assume "0 < e" then obtain N where "N \ 0" and N: "inverse (real N) < e" using real_arch_inverse[of e] by blast show "\M. \m\M. \n\M. norm ((\(x,K) \ p m. content K *\<^sub>R f x) - (\(x,K) \ p n. content K *\<^sub>R f x)) < e" proof (intro exI allI impI) fix m n assume mn: "N \ m" "N \ n" have "norm ((\(x,K) \ p m. content K *\<^sub>R f x) - (\(x,K) \ p n. content K *\<^sub>R f x)) < 1 / (real N + 1)" by (simp add: p(1) dp mn \) also have "... < e" using N \N \ 0\ \0 < e\ by (auto simp: field_simps) finally show "norm ((\(x,K) \ p m. content K *\<^sub>R f x) - (\(x,K) \ p n. content K *\<^sub>R f x)) < e" . qed qed then obtain y where y: "\no. \n\no. norm ((\(x,K) \ p n. content K *\<^sub>R f x) - y) < r" if "r > 0" for r by (auto simp: convergent_eq_Cauchy[symmetric] dest: LIMSEQ_D) show ?l unfolding integrable_on_def has_integral proof (rule_tac x=y in exI, clarify) fix e :: real assume "e>0" then have e2: "e/2 > 0" by auto then obtain N1::nat where N1: "N1 \ 0" "inverse (real N1) < e/2" using real_arch_inverse by blast obtain N2::nat where N2: "\n. n \ N2 \ norm ((\(x,K) \ p n. content K *\<^sub>R f x) - y) < e/2" using y[OF e2] by metis show "\\. gauge \ \ (\\. \ tagged_division_of (cbox a b) \ \ fine \ \ norm ((\(x,K) \ \. content K *\<^sub>R f x) - y) < e)" proof (intro exI conjI allI impI) show "gauge (\ (N1+N2))" using \ by auto show "norm ((\(x,K) \ q. content K *\<^sub>R f x) - y) < e" if "q tagged_division_of cbox a b \ \ (N1+N2) fine q" for q proof (rule norm_triangle_half_r) have "norm ((\(x,K) \ p (N1+N2). content K *\<^sub>R f x) - (\(x,K) \ q. content K *\<^sub>R f x)) < 1 / (real (N1+N2) + 1)" by (rule \; simp add: dp p that) also have "... < e/2" using N1 \0 < e\ by (auto simp: field_simps intro: less_le_trans) finally show "norm ((\(x,K) \ p (N1+N2). content K *\<^sub>R f x) - (\(x,K) \ q. content K *\<^sub>R f x)) < e/2" . show "norm ((\(x,K) \ p (N1+N2). content K *\<^sub>R f x) - y) < e/2" using N2 le_add_same_cancel2 by blast qed qed qed qed subsection \Additivity of integral on abutting intervals\ lemma tagged_division_split_left_inj_content: assumes \: "\ tagged_division_of S" and "(x1, K1) \ \" "(x2, K2) \ \" "K1 \ K2" "K1 \ {x. x\k \ c} = K2 \ {x. x\k \ c}" "k \ Basis" shows "content (K1 \ {x. x\k \ c}) = 0" proof - from tagged_division_ofD(4)[OF \ \(x1, K1) \ \\] obtain a b where K1: "K1 = cbox a b" by auto then have "interior (K1 \ {x. x \ k \ c}) = {}" by (metis tagged_division_split_left_inj assms) then show ?thesis unfolding K1 interval_split[OF \k \ Basis\] by (auto simp: content_eq_0_interior) qed lemma tagged_division_split_right_inj_content: assumes \: "\ tagged_division_of S" and "(x1, K1) \ \" "(x2, K2) \ \" "K1 \ K2" "K1 \ {x. x\k \ c} = K2 \ {x. x\k \ c}" "k \ Basis" shows "content (K1 \ {x. x\k \ c}) = 0" proof - from tagged_division_ofD(4)[OF \ \(x1, K1) \ \\] obtain a b where K1: "K1 = cbox a b" by auto then have "interior (K1 \ {x. c \ x \ k}) = {}" by (metis tagged_division_split_right_inj assms) then show ?thesis unfolding K1 interval_split[OF \k \ Basis\] by (auto simp: content_eq_0_interior) qed proposition has_integral_split: fixes f :: "'a::euclidean_space \ 'b::real_normed_vector" assumes fi: "(f has_integral i) (cbox a b \ {x. x\k \ c})" and fj: "(f has_integral j) (cbox a b \ {x. x\k \ c})" and k: "k \ Basis" shows "(f has_integral (i + j)) (cbox a b)" unfolding has_integral proof clarify fix e::real assume "0 < e" then have e: "e/2 > 0" by auto obtain \1 where \1: "gauge \1" and \1norm: "\\. \\ tagged_division_of cbox a b \ {x. x \ k \ c}; \1 fine \\ \ norm ((\(x,K) \ \. content K *\<^sub>R f x) - i) < e/2" apply (rule has_integralD[OF fi[unfolded interval_split[OF k]] e]) apply (simp add: interval_split[symmetric] k) done obtain \2 where \2: "gauge \2" and \2norm: "\\. \\ tagged_division_of cbox a b \ {x. c \ x \ k}; \2 fine \\ \ norm ((\(x, k) \ \. content k *\<^sub>R f x) - j) < e/2" apply (rule has_integralD[OF fj[unfolded interval_split[OF k]] e]) apply (simp add: interval_split[symmetric] k) done let ?\ = "\x. if x\k = c then (\1 x \ \2 x) else ball x \x\k - c\ \ \1 x \ \2 x" have "gauge ?\" using \1 \2 unfolding gauge_def by auto then show "\\. gauge \ \ (\\. \ tagged_division_of cbox a b \ \ fine \ \ norm ((\(x, k)\\. content k *\<^sub>R f x) - (i + j)) < e)" proof (rule_tac x="?\" in exI, safe) fix p assume p: "p tagged_division_of (cbox a b)" and "?\ fine p" have ab_eqp: "cbox a b = \{K. \x. (x, K) \ p}" using p by blast have xk_le_c: "x\k \ c" if as: "(x,K) \ p" and K: "K \ {x. x\k \ c} \ {}" for x K proof (rule ccontr) assume **: "\ x \ k \ c" then have "K \ ball x \x \ k - c\" using \?\ fine p\ as by (fastforce simp: not_le algebra_simps) with K obtain y where y: "y \ ball x \x \ k - c\" "y\k \ c" by blast then have "\x \ k - y \ k\ < \x \ k - c\" using Basis_le_norm[OF k, of "x - y"] by (auto simp add: dist_norm inner_diff_left intro: le_less_trans) with y show False using ** by (auto simp add: field_simps) qed have xk_ge_c: "x\k \ c" if as: "(x,K) \ p" and K: "K \ {x. x\k \ c} \ {}" for x K proof (rule ccontr) assume **: "\ x \ k \ c" then have "K \ ball x \x \ k - c\" using \?\ fine p\ as by (fastforce simp: not_le algebra_simps) with K obtain y where y: "y \ ball x \x \ k - c\" "y\k \ c" by blast then have "\x \ k - y \ k\ < \x \ k - c\" using Basis_le_norm[OF k, of "x - y"] by (auto simp add: dist_norm inner_diff_left intro: le_less_trans) with y show False using ** by (auto simp add: field_simps) qed have fin_finite: "finite {(x,f K) | x K. (x,K) \ s \ P x K}" if "finite s" for s and f :: "'a set \ 'a set" and P :: "'a \ 'a set \ bool" proof - from that have "finite ((\(x,K). (x, f K)) ` s)" by auto then show ?thesis by (rule rev_finite_subset) auto qed { fix \ :: "'a set \ 'a set" fix i :: "'a \ 'a set" assume "i \ (\(x, k). (x, \ k)) ` p - {(x, \ k) |x k. (x, k) \ p \ \ k \ {}}" then obtain x K where xk: "i = (x, \ K)" "(x,K) \ p" "(x, \ K) \ {(x, \ K) |x K. (x,K) \ p \ \ K \ {}}" by auto have "content (\ K) = 0" using xk using content_empty by auto then have "(\(x,K). content K *\<^sub>R f x) i = 0" unfolding xk split_conv by auto } note [simp] = this have "finite p" using p by blast let ?M1 = "{(x, K \ {x. x\k \ c}) |x K. (x,K) \ p \ K \ {x. x\k \ c} \ {}}" have \1_fine: "\1 fine ?M1" using \?\ fine p\ by (fastforce simp: fine_def split: if_split_asm) have "norm ((\(x, k)\?M1. content k *\<^sub>R f x) - i) < e/2" proof (rule \1norm [OF tagged_division_ofI \1_fine]) show "finite ?M1" by (rule fin_finite) (use p in blast) show "\{k. \x. (x, k) \ ?M1} = cbox a b \ {x. x\k \ c}" by (auto simp: ab_eqp) fix x L assume xL: "(x, L) \ ?M1" then obtain x' L' where xL': "x = x'" "L = L' \ {x. x \ k \ c}" "(x', L') \ p" "L' \ {x. x \ k \ c} \ {}" by blast then obtain a' b' where ab': "L' = cbox a' b'" using p by blast show "x \ L" "L \ cbox a b \ {x. x \ k \ c}" using p xk_le_c xL' by auto show "\a b. L = cbox a b" using p xL' ab' by (auto simp add: interval_split[OF k,where c=c]) fix y R assume yR: "(y, R) \ ?M1" then obtain y' R' where yR': "y = y'" "R = R' \ {x. x \ k \ c}" "(y', R') \ p" "R' \ {x. x \ k \ c} \ {}" by blast assume as: "(x, L) \ (y, R)" show "interior L \ interior R = {}" proof (cases "L' = R' \ x' = y'") case False have "interior R' = {}" by (metis (no_types) False Pair_inject inf.idem tagged_division_ofD(5) [OF p] xL'(3) yR'(3)) then show ?thesis using yR' by simp next case True then have "L' \ R'" using as unfolding xL' yR' by auto have "interior L' \ interior R' = {}" by (metis (no_types) Pair_inject \L' \ R'\ p tagged_division_ofD(5) xL'(3) yR'(3)) then show ?thesis using xL'(2) yR'(2) by auto qed qed moreover let ?M2 = "{(x,K \ {x. x\k \ c}) |x K. (x,K) \ p \ K \ {x. x\k \ c} \ {}}" have \2_fine: "\2 fine ?M2" using \?\ fine p\ by (fastforce simp: fine_def split: if_split_asm) have "norm ((\(x, k)\?M2. content k *\<^sub>R f x) - j) < e/2" proof (rule \2norm [OF tagged_division_ofI \2_fine]) show "finite ?M2" by (rule fin_finite) (use p in blast) show "\{k. \x. (x, k) \ ?M2} = cbox a b \ {x. x\k \ c}" by (auto simp: ab_eqp) fix x L assume xL: "(x, L) \ ?M2" then obtain x' L' where xL': "x = x'" "L = L' \ {x. x \ k \ c}" "(x', L') \ p" "L' \ {x. x \ k \ c} \ {}" by blast then obtain a' b' where ab': "L' = cbox a' b'" using p by blast show "x \ L" "L \ cbox a b \ {x. x \ k \ c}" using p xk_ge_c xL' by auto show "\a b. L = cbox a b" using p xL' ab' by (auto simp add: interval_split[OF k,where c=c]) fix y R assume yR: "(y, R) \ ?M2" then obtain y' R' where yR': "y = y'" "R = R' \ {x. x \ k \ c}" "(y', R') \ p" "R' \ {x. x \ k \ c} \ {}" by blast assume as: "(x, L) \ (y, R)" show "interior L \ interior R = {}" proof (cases "L' = R' \ x' = y'") case False have "interior R' = {}" by (metis (no_types) False Pair_inject inf.idem tagged_division_ofD(5) [OF p] xL'(3) yR'(3)) then show ?thesis using yR' by simp next case True then have "L' \ R'" using as unfolding xL' yR' by auto have "interior L' \ interior R' = {}" by (metis (no_types) Pair_inject \L' \ R'\ p tagged_division_ofD(5) xL'(3) yR'(3)) then show ?thesis using xL'(2) yR'(2) by auto qed qed ultimately have "norm (((\(x,K) \ ?M1. content K *\<^sub>R f x) - i) + ((\(x,K) \ ?M2. content K *\<^sub>R f x) - j)) < e/2 + e/2" using norm_add_less by blast moreover have "((\(x,K) \ ?M1. content K *\<^sub>R f x) - i) + ((\(x,K) \ ?M2. content K *\<^sub>R f x) - j) = (\(x, ka)\p. content ka *\<^sub>R f x) - (i + j)" proof - have eq0: "\x y. x = (0::real) \ x *\<^sub>R (y::'b) = 0" by auto have cont_eq: "\g. (\(x,l). content l *\<^sub>R f x) \ (\(x,l). (x,g l)) = (\(x,l). content (g l) *\<^sub>R f x)" by auto have *: "\\ :: 'a set \ 'a set. (\(x,K)\{(x, \ K) |x K. (x,K) \ p \ \ K \ {}}. content K *\<^sub>R f x) = (\(x,K)\(\(x,K). (x, \ K)) ` p. content K *\<^sub>R f x)" by (rule sum.mono_neutral_left) (auto simp: \finite p\) have "((\(x, k)\?M1. content k *\<^sub>R f x) - i) + ((\(x, k)\?M2. content k *\<^sub>R f x) - j) = (\(x, k)\?M1. content k *\<^sub>R f x) + (\(x, k)\?M2. content k *\<^sub>R f x) - (i + j)" by auto moreover have "\ = (\(x,K) \ p. content (K \ {x. x \ k \ c}) *\<^sub>R f x) + (\(x,K) \ p. content (K \ {x. c \ x \ k}) *\<^sub>R f x) - (i + j)" unfolding * apply (subst (1 2) sum.reindex_nontrivial) apply (auto intro!: k p eq0 tagged_division_split_left_inj_content tagged_division_split_right_inj_content simp: cont_eq \finite p\) done moreover have "\x. x \ p \ (\(a,B). content (B \ {a. a \ k \ c}) *\<^sub>R f a) x + (\(a,B). content (B \ {a. c \ a \ k}) *\<^sub>R f a) x = (\(a,B). content B *\<^sub>R f a) x" proof clarify fix a B assume "(a, B) \ p" with p obtain u v where uv: "B = cbox u v" by blast then show "content (B \ {x. x \ k \ c}) *\<^sub>R f a + content (B \ {x. c \ x \ k}) *\<^sub>R f a = content B *\<^sub>R f a" by (auto simp: scaleR_left_distrib uv content_split[OF k,of u v c]) qed ultimately show ?thesis by (auto simp: sum.distrib[symmetric]) qed ultimately show "norm ((\(x, k)\p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed subsection \A sort of converse, integrability on subintervals\ lemma has_integral_separate_sides: fixes f :: "'a::euclidean_space \ 'b::real_normed_vector" assumes f: "(f has_integral i) (cbox a b)" and "e > 0" and k: "k \ Basis" obtains d where "gauge d" "\p1 p2. p1 tagged_division_of (cbox a b \ {x. x\k \ c}) \ d fine p1 \ p2 tagged_division_of (cbox a b \ {x. x\k \ c}) \ d fine p2 \ norm ((sum (\(x,k). content k *\<^sub>R f x) p1 + sum (\(x,k). content k *\<^sub>R f x) p2) - i) < e" proof - obtain \ where d: "gauge \" "\p. \p tagged_division_of cbox a b; \ fine p\ \ norm ((\(x, k)\p. content k *\<^sub>R f x) - i) < e" using has_integralD[OF f \e > 0\] by metis { fix p1 p2 assume tdiv1: "p1 tagged_division_of (cbox a b) \ {x. x \ k \ c}" and "\ fine p1" note p1=tagged_division_ofD[OF this(1)] assume tdiv2: "p2 tagged_division_of (cbox a b) \ {x. c \ x \ k}" and "\ fine p2" note p2=tagged_division_ofD[OF this(1)] note tagged_division_Un_interval[OF tdiv1 tdiv2] note p12 = tagged_division_ofD[OF this] this { fix a b assume ab: "(a, b) \ p1 \ p2" have "(a, b) \ p1" using ab by auto obtain u v where uv: "b = cbox u v" using \(a, b) \ p1\ p1(4) by moura have "b \ {x. x\k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastforce moreover have "interior {x::'a. x \ k = c} = {}" proof (rule ccontr) assume "\ ?thesis" then obtain x where x: "x \ interior {x::'a. x\k = c}" by auto then obtain \ where "0 < \" and \: "ball x \ \ {x. x \ k = c}" using mem_interior by metis have x: "x\k = c" using x interior_subset by fastforce have *: "\i. i \ Basis \ \(x - (x + (\/2) *\<^sub>R k)) \ i\ = (if i = k then \/2 else 0)" using \0 < \\ k by (auto simp: inner_simps inner_not_same_Basis) have "(\i\Basis. \(x - (x + (\/2 ) *\<^sub>R k)) \ i\) = (\i\Basis. (if i = k then \/2 else 0))" using "*" by (blast intro: sum.cong) also have "\ < \" by (subst sum.delta) (use \0 < \\ in auto) finally have "x + (\/2) *\<^sub>R k \ ball x \" unfolding mem_ball dist_norm by(rule le_less_trans[OF norm_le_l1]) then have "x + (\/2) *\<^sub>R k \ {x. x\k = c}" using \ by auto then show False using \0 < \\ x k by (auto simp: inner_simps) qed ultimately have "content b = 0" unfolding uv content_eq_0_interior using interior_mono by blast then have "content b *\<^sub>R f a = 0" by auto } then have "norm ((\(x, k)\p1. content k *\<^sub>R f x) + (\(x, k)\p2. content k *\<^sub>R f x) - i) = norm ((\(x, k)\p1 \ p2. content k *\<^sub>R f x) - i)" by (subst sum.union_inter_neutral) (auto simp: p1 p2) also have "\ < e" using d(2) p12 by (simp add: fine_Un k \\ fine p1\ \\ fine p2\) finally have "norm ((\(x, k)\p1. content k *\<^sub>R f x) + (\(x, k)\p2. content k *\<^sub>R f x) - i) < e" . } then show ?thesis using d(1) that by auto qed lemma integrable_split [intro]: fixes f :: "'a::euclidean_space \ 'b::{real_normed_vector,complete_space}" assumes f: "f integrable_on cbox a b" and k: "k \ Basis" shows "f integrable_on (cbox a b \ {x. x\k \ c})" (is ?thesis1) and "f integrable_on (cbox a b \ {x. x\k \ c})" (is ?thesis2) proof - obtain y where y: "(f has_integral y) (cbox a b)" using f by blast define a' where "a' = (\i\Basis. (if i = k then max (a\k) c else a\i)*\<^sub>R i)" define b' where "b' = (\i\Basis. (if i = k then min (b\k) c else b\i)*\<^sub>R i)" have "\d. gauge d \ (\p1 p2. p1 tagged_division_of cbox a b \ {x. x \ k \ c} \ d fine p1 \ p2 tagged_division_of cbox a b \ {x. x \ k \ c} \ d fine p2 \ norm ((\(x,K) \ p1. content K *\<^sub>R f x) - (\(x,K) \ p2. content K *\<^sub>R f x)) < e)" if "e > 0" for e proof - have "e/2 > 0" using that by auto with has_integral_separate_sides[OF y this k, of c] obtain d where "gauge d" and d: "\p1 p2. \p1 tagged_division_of cbox a b \ {x. x \ k \ c}; d fine p1; p2 tagged_division_of cbox a b \ {x. c \ x \ k}; d fine p2\ \ norm ((\(x,K)\p1. content K *\<^sub>R f x) + (\(x,K)\p2. content K *\<^sub>R f x) - y) < e/2" by metis show ?thesis proof (rule_tac x=d in exI, clarsimp simp add: \gauge d\) fix p1 p2 assume as: "p1 tagged_division_of (cbox a b) \ {x. x \ k \ c}" "d fine p1" "p2 tagged_division_of (cbox a b) \ {x. x \ k \ c}" "d fine p2" show "norm ((\(x, k)\p1. content k *\<^sub>R f x) - (\(x, k)\p2. content k *\<^sub>R f x)) < e" proof (rule fine_division_exists[OF \gauge d\, of a' b]) fix p assume "p tagged_division_of cbox a' b" "d fine p" then show ?thesis using as norm_triangle_half_l[OF d[of p1 p] d[of p2 p]] unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric] by (auto simp add: algebra_simps) qed qed qed with f show ?thesis1 by (simp add: interval_split[OF k] integrable_Cauchy) have "\d. gauge d \ (\p1 p2. p1 tagged_division_of cbox a b \ {x. x \ k \ c} \ d fine p1 \ p2 tagged_division_of cbox a b \ {x. x \ k \ c} \ d fine p2 \ norm ((\(x,K) \ p1. content K *\<^sub>R f x) - (\(x,K) \ p2. content K *\<^sub>R f x)) < e)" if "e > 0" for e proof - have "e/2 > 0" using that by auto with has_integral_separate_sides[OF y this k, of c] obtain d where "gauge d" and d: "\p1 p2. \p1 tagged_division_of cbox a b \ {x. x \ k \ c}; d fine p1; p2 tagged_division_of cbox a b \ {x. c \ x \ k}; d fine p2\ \ norm ((\(x,K)\p1. content K *\<^sub>R f x) + (\(x,K)\p2. content K *\<^sub>R f x) - y) < e/2" by metis show ?thesis proof (rule_tac x=d in exI, clarsimp simp add: \gauge d\) fix p1 p2 assume as: "p1 tagged_division_of (cbox a b) \ {x. x \ k \ c}" "d fine p1" "p2 tagged_division_of (cbox a b) \ {x. x \ k \ c}" "d fine p2" show "norm ((\(x, k)\p1. content k *\<^sub>R f x) - (\(x, k)\p2. content k *\<^sub>R f x)) < e" proof (rule fine_division_exists[OF \gauge d\, of a b']) fix p assume "p tagged_division_of cbox a b'" "d fine p" then show ?thesis using as norm_triangle_half_l[OF d[of p p1] d[of p p2]] unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric] by (auto simp add: algebra_simps) qed qed qed with f show ?thesis2 by (simp add: interval_split[OF k] integrable_Cauchy) qed lemma operative_integralI: fixes f :: "'a::euclidean_space \ 'b::banach" shows "operative (lift_option (+)) (Some 0) (\i. if f integrable_on i then Some (integral i f) else None)" proof - interpret comm_monoid "lift_option plus" "Some (0::'b)" by (rule comm_monoid_lift_option) (rule add.comm_monoid_axioms) show ?thesis proof fix a b c fix k :: 'a assume k: "k \ Basis" show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) = lift_option (+) (if f integrable_on cbox a b \ {x. x \ k \ c} then Some (integral (cbox a b \ {x. x \ k \ c}) f) else None) (if f integrable_on cbox a b \ {x. c \ x \ k} then Some (integral (cbox a b \ {x. c \ x \ k}) f) else None)" proof (cases "f integrable_on cbox a b") case True with k show ?thesis by (auto simp: integrable_split intro: integral_unique [OF has_integral_split[OF _ _ k]]) next case False have "\ (f integrable_on cbox a b \ {x. x \ k \ c}) \ \ ( f integrable_on cbox a b \ {x. c \ x \ k})" proof (rule ccontr) assume "\ ?thesis" then have "f integrable_on cbox a b" unfolding integrable_on_def apply (rule_tac x="integral (cbox a b \ {x. x \ k \ c}) f + integral (cbox a b \ {x. x \ k \ c}) f" in exI) apply (auto intro: has_integral_split[OF _ _ k]) done then show False using False by auto qed then show ?thesis using False by auto qed next fix a b :: 'a assume "box a b = {}" then show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) = Some 0" using has_integral_null_eq by (auto simp: integrable_on_null content_eq_0_interior) qed qed subsection \Bounds on the norm of Riemann sums and the integral itself\ lemma dsum_bound: assumes p: "p division_of (cbox a b)" and "norm c \ e" shows "norm (sum (\l. content l *\<^sub>R c) p) \ e * content(cbox a b)" proof - have sumeq: "(\i\p. \content i\) = sum content p" by simp have e: "0 \ e" using assms(2) norm_ge_zero order_trans by blast have "norm (sum (\l. content l *\<^sub>R c) p) \ (\i\p. norm (content i *\<^sub>R c))" using norm_sum by blast also have "... \ e * (\i\p. \content i\)" by (simp add: sum_distrib_left[symmetric] mult.commute assms(2) mult_right_mono sum_nonneg) also have "... \ e * content (cbox a b)" by (metis additive_content_division p eq_iff sumeq) finally show ?thesis . qed lemma rsum_bound: assumes p: "p tagged_division_of (cbox a b)" and "\x\cbox a b. norm (f x) \ e" shows "norm (sum (\(x,k). content k *\<^sub>R f x) p) \ e * content (cbox a b)" proof (cases "cbox a b = {}") case True show ?thesis using p unfolding True tagged_division_of_trivial by auto next case False then have e: "e \ 0" by (meson ex_in_conv assms(2) norm_ge_zero order_trans) have sum_le: "sum (content \ snd) p \ content (cbox a b)" unfolding additive_content_tagged_division[OF p, symmetric] split_def by (auto intro: eq_refl) have con: "\xk. xk \ p \ 0 \ content (snd xk)" using tagged_division_ofD(4) [OF p] content_pos_le by force have "norm (sum (\(x,k). content k *\<^sub>R f x) p) \ (\i\p. norm (case i of (x, k) \ content k *\<^sub>R f x))" by (rule norm_sum) also have "... \ e * content (cbox a b)" proof - have "\xk. xk \ p \ norm (f (fst xk)) \ e" using assms(2) p tag_in_interval by force moreover have "(\i\p. \content (snd i)\ * e) \ e * content (cbox a b)" unfolding sum_distrib_right[symmetric] using con sum_le by (auto simp: mult.commute intro: mult_left_mono [OF _ e]) ultimately show ?thesis unfolding split_def norm_scaleR by (metis (no_types, lifting) mult_left_mono[OF _ abs_ge_zero] order_trans[OF sum_mono]) qed finally show ?thesis . qed lemma rsum_diff_bound: assumes "p tagged_division_of (cbox a b)" and "\x\cbox a b. norm (f x - g x) \ e" shows "norm (sum (\(x,k). content k *\<^sub>R f x) p - sum (\(x,k). content k *\<^sub>R g x) p) \ e * content (cbox a b)" using order_trans[OF _ rsum_bound[OF assms]] by (simp add: split_def scaleR_diff_right sum_subtractf eq_refl) lemma has_integral_bound: fixes f :: "'a::euclidean_space \ 'b::real_normed_vector" assumes "0 \ B" and f: "(f has_integral i) (cbox a b)" and "\x. x\cbox a b \ norm (f x) \ B" shows "norm i \ B * content (cbox a b)" proof (rule ccontr) assume "\ ?thesis" then have "norm i - B * content (cbox a b) > 0" by auto with f[unfolded has_integral] obtain \ where "gauge \" and \: "\p. \p tagged_division_of cbox a b; \ fine p\ \ norm ((\(x, K)\p. content K *\<^sub>R f x) - i) < norm i - B * content (cbox a b)" by metis then obtain p where p: "p tagged_division_of cbox a b" and "\ fine p" using fine_division_exists by blast have "\s B. norm s \ B \ \ norm (s - i) < norm i - B" unfolding not_less by (metis diff_left_mono dist_commute dist_norm norm_triangle_ineq2 order_trans) then show False using \ [OF p \\ fine p\] rsum_bound[OF p] assms by metis qed corollary integrable_bound: fixes f :: "'a::euclidean_space \ 'b::real_normed_vector" assumes "0 \ B" and "f integrable_on (cbox a b)" and "\x. x\cbox a b \ norm (f x) \ B" shows "norm (integral (cbox a b) f) \ B * content (cbox a b)" by (metis integrable_integral has_integral_bound assms) subsection \Similar theorems about relationship among components\ lemma rsum_component_le: fixes f :: "'a::euclidean_space \ 'b::euclidean_space" assumes p: "p tagged_division_of (cbox a b)" and "\x. x \ cbox a b \ (f x)\i \ (g x)\i" shows "(\(x, K)\p. content K *\<^sub>R f x) \ i \ (\(x, K)\p. content K *\<^sub>R g x) \ i" unfolding inner_sum_left proof (rule sum_mono, clarify) fix x K assume ab: "(x, K) \ p" with p obtain u v where K: "K = cbox u v" by blast then show "(content K *\<^sub>R f x) \ i \ (content K *\<^sub>R g x) \ i" by (metis ab assms inner_scaleR_left measure_nonneg mult_left_mono tag_in_interval) qed lemma has_integral_component_le: fixes f g :: "'a::euclidean_space \ 'b::euclidean_space" assumes k: "k \ Basis" assumes "(f has_integral i) S" "(g has_integral j) S" and f_le_g: "\x. x \ S \ (f x)\k \ (g x)\k" shows "i\k \ j\k" proof - have ik_le_jk: "i\k \ j\k" if f_i: "(f has_integral i) (cbox a b)" and g_j: "(g has_integral j) (cbox a b)" and le: "\x\cbox a b. (f x)\k \ (g x)\k" for a b i and j :: 'b and f g :: "'a \ 'b" proof (rule ccontr) assume "\ ?thesis" then have *: "0 < (i\k - j\k) / 3" by auto obtain \1 where "gauge \1" and \1: "\p. \p tagged_division_of cbox a b; \1 fine p\ \ norm ((\(x, k)\p. content k *\<^sub>R f x) - i) < (i \ k - j \ k) / 3" using f_i[unfolded has_integral,rule_format,OF *] by fastforce obtain \2 where "gauge \2" and \2: "\p. \p tagged_division_of cbox a b; \2 fine p\ \ norm ((\(x, k)\p. content k *\<^sub>R g x) - j) < (i \ k - j \ k) / 3" using g_j[unfolded has_integral,rule_format,OF *] by fastforce obtain p where p: "p tagged_division_of cbox a b" and "\1 fine p" "\2 fine p" using fine_division_exists[OF gauge_Int[OF \gauge \1\ \gauge \2\], of a b] unfolding fine_Int by metis then have "\((\(x, k)\p. content k *\<^sub>R f x) - i) \ k\ < (i \ k - j \ k) / 3" "\((\(x, k)\p. content k *\<^sub>R g x) - j) \ k\ < (i \ k - j \ k) / 3" using le_less_trans[OF Basis_le_norm[OF k]] k \1 \2 by metis+ then show False unfolding inner_simps using rsum_component_le[OF p] le by (fastforce simp add: abs_real_def split: if_split_asm) qed show ?thesis proof (cases "\a b. S = cbox a b") case True with ik_le_jk assms show ?thesis by auto next case False show ?thesis proof (rule ccontr) assume "\ i\k \ j\k" then have ij: "(i\k - j\k) / 3 > 0" by auto obtain B1 where "0 < B1" and B1: "\a b. ball 0 B1 \ cbox a b \ \z. ((\x. if x \ S then f x else 0) has_integral z) (cbox a b) \ norm (z - i) < (i \ k - j \ k) / 3" using has_integral_altD[OF _ False ij] assms by blast obtain B2 where "0 < B2" and B2: "\a b. ball 0 B2 \ cbox a b \ \z. ((\x. if x \ S then g x else 0) has_integral z) (cbox a b) \ norm (z - j) < (i \ k - j \ k) / 3" using has_integral_altD[OF _ False ij] assms by blast have "bounded (ball 0 B1 \ ball (0::'a) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+ from bounded_subset_cbox_symmetric[OF this] obtain a b::'a where ab: "ball 0 B1 \ cbox a b" "ball 0 B2 \ cbox a b" by (meson Un_subset_iff) then obtain w1 w2 where int_w1: "((\x. if x \ S then f x else 0) has_integral w1) (cbox a b)" and norm_w1: "norm (w1 - i) < (i \ k - j \ k) / 3" and int_w2: "((\x. if x \ S then g x else 0) has_integral w2) (cbox a b)" and norm_w2: "norm (w2 - j) < (i \ k - j \ k) / 3" using B1 B2 by blast have *: "\w1 w2 j i::real .\w1 - i\ < (i - j) / 3 \ \w2 - j\ < (i - j) / 3 \ w1 \ w2 \ False" by (simp add: abs_real_def split: if_split_asm) have "\(w1 - i) \ k\ < (i \ k - j \ k) / 3" "\(w2 - j) \ k\ < (i \ k - j \ k) / 3" using Basis_le_norm k le_less_trans norm_w1 norm_w2 by blast+ moreover have "w1\k \ w2\k" using ik_le_jk int_w1 int_w2 f_le_g by auto ultimately show False unfolding inner_simps by(rule *) qed qed qed lemma integral_component_le: fixes g f :: "'a::euclidean_space \ 'b::euclidean_space" assumes "k \ Basis" and "f integrable_on S" "g integrable_on S" and "\x. x \ S \ (f x)\k \ (g x)\k" shows "(integral S f)\k \ (integral S g)\k" using has_integral_component_le assms by blast lemma has_integral_component_nonneg: fixes f :: "'a::euclidean_space \ 'b::euclidean_space" assumes "k \ Basis" and "(f has_integral i) S" and "\x. x \ S \ 0 \ (f x)\k" shows "0 \ i\k" using has_integral_component_le[OF assms(1) has_integral_0 assms(2)] using assms(3-) by auto lemma integral_component_nonneg: fixes f :: "'a::euclidean_space \ 'b::euclidean_space" assumes "k \ Basis" and "\x. x \ S \ 0 \ (f x)\k" shows "0 \ (integral S f)\k" proof (cases "f integrable_on S") case True show ?thesis using True assms has_integral_component_nonneg by blast next case False then show ?thesis by (simp add: not_integrable_integral) qed lemma has_integral_component_neg: fixes f :: "'a::euclidean_space \ 'b::euclidean_space" assumes "k \ Basis" and "(f has_integral i) S" and "\x. x \ S \ (f x)\k \ 0" shows "i\k \ 0" using has_integral_component_le[OF assms(1,2) has_integral_0] assms(2-) by auto lemma has_integral_component_lbound: fixes f :: "'a::euclidean_space \ 'b::euclidean_space" assumes "(f has_integral i) (cbox a b)" and "\x\cbox a b. B \ f(x)\k" and "k \ Basis" shows "B * content (cbox a b) \ i\k" using has_integral_component_le[OF assms(3) has_integral_const assms(1),of "(\i\Basis. B *\<^sub>R i)::'b"] assms(2-) by (auto simp add: field_simps) lemma has_integral_component_ubound: fixes f::"'a::euclidean_space => 'b::euclidean_space" assumes "(f has_integral i) (cbox a b)" and "\x\cbox a b. f x\k \ B" and "k \ Basis" shows "i\k \ B * content (cbox a b)" using has_integral_component_le[OF assms(3,1) has_integral_const, of "\i\Basis. B *\<^sub>R i"] assms(2-) by (auto simp add: field_simps) lemma integral_component_lbound: fixes f :: "'a::euclidean_space \ 'b::euclidean_space" assumes "f integrable_on cbox a b" and "\x\cbox a b. B \ f(x)\k" and "k \ Basis" shows "B * content (cbox a b) \ (integral(cbox a b) f)\k" using assms has_integral_component_lbound by blast lemma integral_component_lbound_real: assumes "f integrable_on {a ::real..b}" and "\x\{a..b}. B \ f(x)\k" and "k \ Basis" shows "B * content {a..b} \ (integral {a..b} f)\k" using assms by (metis box_real(2) integral_component_lbound) lemma integral_component_ubound: fixes f :: "'a::euclidean_space \ 'b::euclidean_space" assumes "f integrable_on cbox a b" and "\x\cbox a b. f x\k \ B" and "k \ Basis" shows "(integral (cbox a b) f)\k \ B * content (cbox a b)" using assms has_integral_component_ubound by blast lemma integral_component_ubound_real: fixes f :: "real \ 'a::euclidean_space" assumes "f integrable_on {a..b}" and "\x\{a..b}. f x\k \ B" and "k \ Basis" shows "(integral {a..b} f)\k \ B * content {a..b}" using assms by (metis box_real(2) integral_component_ubound) subsection \Uniform limit of integrable functions is integrable\ lemma real_arch_invD: "0 < (e::real) \ (\n::nat. n \ 0 \ 0 < inverse (real n) \ inverse (real n) < e)" by (subst(asm) real_arch_inverse) lemma integrable_uniform_limit: fixes f :: "'a::euclidean_space \ 'b::banach" assumes "\e. e > 0 \ \g. (\x\cbox a b. norm (f x - g x) \ e) \ g integrable_on cbox a b" shows "f integrable_on cbox a b" proof (cases "content (cbox a b) > 0") case False then show ?thesis using has_integral_null by (simp add: content_lt_nz integrable_on_def) next case True have "1 / (real n + 1) > 0" for n by auto then have "\g. (\x\cbox a b. norm (f x - g x) \ 1 / (real n + 1)) \ g integrable_on cbox a b" for n using assms by blast then obtain g where g_near_f: "\n x. x \ cbox a b \ norm (f x - g n x) \ 1 / (real n + 1)" and int_g: "\n. g n integrable_on cbox a b" by metis then obtain h where h: "\n. (g n has_integral h n) (cbox a b)" unfolding integrable_on_def by metis have "Cauchy h" unfolding Cauchy_def proof clarify fix e :: real assume "e>0" then have "e/4 / content (cbox a b) > 0" using True by (auto simp: field_simps) then obtain M where "M \ 0" and M: "1 / (real M) < e/4 / content (cbox a b)" by (metis inverse_eq_divide real_arch_inverse) show "\M. \m\M. \n\M. dist (h m) (h n) < e" proof (rule exI [where x=M], clarify) fix m n assume m: "M \ m" and n: "M \ n" have "e/4>0" using \e>0\ by auto then obtain gm gn where "gauge gm" "gauge gn" and gm: "\\. \ tagged_division_of cbox a b \ gm fine \ \ norm ((\(x,K) \ \. content K *\<^sub>R g m x) - h m) < e/4" and gn: "\\. \ tagged_division_of cbox a b \ gn fine \ \ norm ((\(x,K) \ \. content K *\<^sub>R g n x) - h n) < e/4" using h[unfolded has_integral] by meson then obtain \ where \: "\ tagged_division_of cbox a b" "(\x. gm x \ gn x) fine \" by (metis (full_types) fine_division_exists gauge_Int) have triangle3: "norm (i1 - i2) < e" if no: "norm(s2 - s1) \ e/2" "norm (s1 - i1) < e/4" "norm (s2 - i2) < e/4" for s1 s2 i1 and i2::'b proof - have "norm (i1 - i2) \ norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)" using norm_triangle_ineq[of "i1 - s1" "s1 - i2"] using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by (auto simp: algebra_simps) also have "\ < e" using no by (auto simp: algebra_simps norm_minus_commute) finally show ?thesis . qed have finep: "gm fine \" "gn fine \" using fine_Int \ by auto have norm_le: "norm (g n x - g m x) \ 2 / real M" if x: "x \ cbox a b" for x proof - have "norm (f x - g n x) + norm (f x - g m x) \ 1 / (real n + 1) + 1 / (real m + 1)" using g_near_f[OF x, of n] g_near_f[OF x, of m] by simp also have "\ \ 1 / (real M) + 1 / (real M)" using \M \ 0\ m n by (intro add_mono; force simp: field_split_simps) also have "\ = 2 / real M" by auto finally show "norm (g n x - g m x) \ 2 / real M" using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"] by (auto simp: algebra_simps simp add: norm_minus_commute) qed have "norm ((\(x,K) \ \. content K *\<^sub>R g n x) - (\(x,K) \ \. content K *\<^sub>R g m x)) \ 2 / real M * content (cbox a b)" by (blast intro: norm_le rsum_diff_bound[OF \(1), where e="2 / real M"]) also have "... \ e/2" using M True by (auto simp: field_simps) finally have le_e2: "norm ((\(x,K) \ \. content K *\<^sub>R g n x) - (\(x,K) \ \. content K *\<^sub>R g m x)) \ e/2" . then show "dist (h m) (h n) < e" unfolding dist_norm using gm gn \ finep by (auto intro!: triangle3) qed qed then obtain s where s: "h \ s" using convergent_eq_Cauchy[symmetric] by blast show ?thesis unfolding integrable_on_def has_integral proof (rule_tac x=s in exI, clarify) fix e::real assume e: "0 < e" then have "e/3 > 0" by auto then obtain N1 where N1: "\n\N1. norm (h n - s) < e/3" using LIMSEQ_D [OF s] by metis from e True have "e/3 / content (cbox a b) > 0" by (auto simp: field_simps) then obtain N2 :: nat where "N2 \ 0" and N2: "1 / (real N2) < e/3 / content (cbox a b)" by (metis inverse_eq_divide real_arch_inverse) obtain g' where "gauge g'" and g': "\\. \ tagged_division_of cbox a b \ g' fine \ \ norm ((\(x,K) \ \. content K *\<^sub>R g (N1 + N2) x) - h (N1 + N2)) < e/3" by (metis h has_integral \e/3 > 0\) have *: "norm (sf - s) < e" if no: "norm (sf - sg) \ e/3" "norm(h - s) < e/3" "norm (sg - h) < e/3" for sf sg h proof - have "norm (sf - s) \ norm (sf - sg) + norm (sg - h) + norm (h - s)" using norm_triangle_ineq[of "sf - sg" "sg - s"] using norm_triangle_ineq[of "sg - h" " h - s"] by (auto simp: algebra_simps) also have "\ < e" using no by (auto simp: algebra_simps norm_minus_commute) finally show ?thesis . qed { fix \ assume ptag: "\ tagged_division_of (cbox a b)" and "g' fine \" then have norm_less: "norm ((\(x,K) \ \. content K *\<^sub>R g (N1 + N2) x) - h (N1 + N2)) < e/3" using g' by blast have "content (cbox a b) < e/3 * (of_nat N2)" using \N2 \ 0\ N2 using True by (auto simp: field_split_simps) moreover have "e/3 * of_nat N2 \ e/3 * (of_nat (N1 + N2) + 1)" using \e>0\ by auto ultimately have "content (cbox a b) < e/3 * (of_nat (N1 + N2) + 1)" by linarith then have le_e3: "1 / (real (N1 + N2) + 1) * content (cbox a b) \ e/3" unfolding inverse_eq_divide by (auto simp: field_simps) have ne3: "norm (h (N1 + N2) - s) < e/3" using N1 by auto have "norm ((\(x,K) \ \. content K *\<^sub>R f x) - (\(x,K) \ \. content K *\<^sub>R g (N1 + N2) x)) \ 1 / (real (N1 + N2) + 1) * content (cbox a b)" by (blast intro: g_near_f rsum_diff_bound[OF ptag]) then have "norm ((\(x,K) \ \. content K *\<^sub>R f x) - s) < e" by (rule *[OF order_trans [OF _ le_e3] ne3 norm_less]) } then show "\d. gauge d \ (\\. \ tagged_division_of cbox a b \ d fine \ \ norm ((\(x,K) \ \. content K *\<^sub>R f x) - s) < e)" by (blast intro: g' \gauge g'\) qed qed lemmas integrable_uniform_limit_real = integrable_uniform_limit [where 'a=real, simplified] subsection \Negligible sets\ definition "negligible (s:: 'a::euclidean_space set) \ (\a b. ((indicator s :: 'a\real) has_integral 0) (cbox a b))" subsubsection \Negligibility of hyperplane\ lemma content_doublesplit: fixes a :: "'a::euclidean_space" assumes "0 < e" and k: "k \ Basis" obtains d where "0 < d" and "content (cbox a b \ {x. \x\k - c\ \ d}) < e" proof cases assume *: "a \ k \ c \ c \ b \ k \ (\j\Basis. a \ j \ b \ j)" define a' where "a' d = (\j\Basis. (if j = k then max (a\j) (c - d) else a\j) *\<^sub>R j)" for d define b' where "b' d = (\j\Basis. (if j = k then min (b\j) (c + d) else b\j) *\<^sub>R j)" for d have "((\d. \j\Basis. (b' d - a' d) \ j) \ (\j\Basis. (b' 0 - a' 0) \ j)) (at_right 0)" by (auto simp: b'_def a'_def intro!: tendsto_min tendsto_max tendsto_eq_intros) also have "(\j\Basis. (b' 0 - a' 0) \ j) = 0" using k * by (intro prod_zero bexI[OF _ k]) (auto simp: b'_def a'_def inner_diff inner_sum_left inner_not_same_Basis intro!: sum.cong) also have "((\d. \j\Basis. (b' d - a' d) \ j) \ 0) (at_right 0) = ((\d. content (cbox a b \ {x. \x\k - c\ \ d})) \ 0) (at_right 0)" proof (intro tendsto_cong eventually_at_rightI) fix d :: real assume d: "d \ {0<..<1}" have "cbox a b \ {x. \x\k - c\ \ d} = cbox (a' d) (b' d)" for d using * d k by (auto simp add: cbox_def set_eq_iff Int_def ball_conj_distrib abs_diff_le_iff a'_def b'_def) moreover have "j \ Basis \ a' d \ j \ b' d \ j" for j using * d k by (auto simp: a'_def b'_def) ultimately show "(\j\Basis. (b' d - a' d) \ j) = content (cbox a b \ {x. \x\k - c\ \ d})" by simp qed simp finally have "((\d. content (cbox a b \ {x. \x \ k - c\ \ d})) \ 0) (at_right 0)" . from order_tendstoD(2)[OF this \0] obtain d' where "0 < d'" and d': "\y. y > 0 \ y < d' \ content (cbox a b \ {x. \x \ k - c\ \ y}) < e" by (subst (asm) eventually_at_right[of _ 1]) auto show ?thesis by (rule that[of "d'/2"], insert \0 d'[of "d'/2"], auto) next assume *: "\ (a \ k \ c \ c \ b \ k \ (\j\Basis. a \ j \ b \ j))" then have "(\j\Basis. b \ j < a \ j) \ (c < a \ k \ b \ k < c)" by (auto simp: not_le) show thesis proof cases assume "\j\Basis. b \ j < a \ j" then have [simp]: "cbox a b = {}" using box_ne_empty(1)[of a b] by auto show ?thesis by (rule that[of 1]) (simp_all add: \0) next assume "\ (\j\Basis. b \ j < a \ j)" with * have "c < a \ k \ b \ k < c" by auto then show thesis proof assume c: "c < a \ k" moreover have "x \ cbox a b \ c \ x \ k" for x using k c by (auto simp: cbox_def) ultimately have "cbox a b \ {x. \x \ k - c\ \ (a \ k - c)/2} = {}" using k by (auto simp: cbox_def) with \0 c that[of "(a \ k - c)/2"] show ?thesis by auto next assume c: "b \ k < c" moreover have "x \ cbox a b \ x \ k \ c" for x using k c by (auto simp: cbox_def) ultimately have "cbox a b \ {x. \x \ k - c\ \ (c - b \ k)/2} = {}" using k by (auto simp: cbox_def) with \0 c that[of "(c - b \ k)/2"] show ?thesis by auto qed qed qed proposition negligible_standard_hyperplane[intro]: fixes k :: "'a::euclidean_space" assumes k: "k \ Basis" shows "negligible {x. x\k = c}" unfolding negligible_def has_integral proof clarsimp fix a b and e::real assume "e > 0" with k obtain d where "0 < d" and d: "content (cbox a b \ {x. \x \ k - c\ \ d}) < e" by (metis content_doublesplit) let ?i = "indicator {x::'a. x\k = c} :: 'a\real" show "\\. gauge \ \ (\\. \ tagged_division_of cbox a b \ \ fine \ \ \\(x,K) \ \. content K * ?i x\ < e)" proof (intro exI, safe) show "gauge (\x. ball x d)" using \0 < d\ by blast next fix \ assume p: "\ tagged_division_of (cbox a b)" "(\x. ball x d) fine \" have "content L = content (L \ {x. \x \ k - c\ \ d})" if "(x, L) \ \" "?i x \ 0" for x L proof - have xk: "x\k = c" using that by (simp add: indicator_def split: if_split_asm) have "L \ {x. \x \ k - c\ \ d}" proof fix y assume y: "y \ L" have "L \ ball x d" using p(2) that(1) by auto then have "norm (x - y) < d" by (simp add: dist_norm subset_iff y) then have "\(x - y) \ k\ < d" using k norm_bound_Basis_lt by blast then show "y \ {x. \x \ k - c\ \ d}" unfolding inner_simps xk by auto qed then show "content L = content (L \ {x. \x \ k - c\ \ d})" by (metis inf.orderE) qed then have *: "(\(x,K)\\. content K * ?i x) = (\(x,K)\\. content (K \ {x. \x\k - c\ \ d}) *\<^sub>R ?i x)" by (force simp add: split_paired_all intro!: sum.cong [OF refl]) note p'= tagged_division_ofD[OF p(1)] and p''=division_of_tagged_division[OF p(1)] have "(\(x,K)\\. content (K \ {x. \x \ k - c\ \ d}) * indicator {x. x \ k = c} x) < e" proof - have "(\(x,K)\\. content (K \ {x. \x \ k - c\ \ d}) * ?i x) \ (\(x,K)\\. content (K \ {x. \x \ k - c\ \ d}))" by (force simp add: indicator_def intro!: sum_mono) also have "\ < e" proof (subst sum.over_tagged_division_lemma[OF p(1)]) fix u v::'a assume "box u v = {}" then have *: "content (cbox u v) = 0" unfolding content_eq_0_interior by simp have "cbox u v \ {x. \x \ k - c\ \ d} \ cbox u v" by auto then have "content (cbox u v \ {x. \x \ k - c\ \ d}) \ content (cbox u v)" unfolding interval_doublesplit[OF k] by (rule content_subset) then show "content (cbox u v \ {x. \x \ k - c\ \ d}) = 0" unfolding * interval_doublesplit[OF k] by (blast intro: antisym) next have "(\l\snd ` \. content (l \ {x. \x \ k - c\ \ d})) = sum content ((\l. l \ {x. \x \ k - c\ \ d})`{l\snd ` \. l \ {x. \x \ k - c\ \ d} \ {}})" proof (subst (2) sum.reindex_nontrivial) fix x y assume "x \ {l \ snd ` \. l \ {x. \x \ k - c\ \ d} \ {}}" "y \ {l \ snd ` \. l \ {x. \x \ k - c\ \ d} \ {}}" "x \ y" and eq: "x \ {x. \x \ k - c\ \ d} = y \ {x. \x \ k - c\ \ d}" then obtain x' y' where "(x', x) \ \" "x \ {x. \x \ k - c\ \ d} \ {}" "(y', y) \ \" "y \ {x. \x \ k - c\ \ d} \ {}" by (auto) from p'(5)[OF \(x', x) \ \\ \(y', y) \ \\] \x \ y\ have "interior (x \ y) = {}" by auto moreover have "interior ((x \ {x. \x \ k - c\ \ d}) \ (y \ {x. \x \ k - c\ \ d})) \ interior (x \ y)" by (auto intro: interior_mono) ultimately have "interior (x \ {x. \x \ k - c\ \ d}) = {}" by (auto simp: eq) then show "content (x \ {x. \x \ k - c\ \ d}) = 0" using p'(4)[OF \(x', x) \ \\] by (auto simp: interval_doublesplit[OF k] content_eq_0_interior simp del: interior_Int) qed (insert p'(1), auto intro!: sum.mono_neutral_right) also have "\ \ norm (\l\(\l. l \ {x. \x \ k - c\ \ d})`{l\snd ` \. l \ {x. \x \ k - c\ \ d} \ {}}. content l *\<^sub>R 1::real)" by simp also have "\ \ 1 * content (cbox a b \ {x. \x \ k - c\ \ d})" using division_doublesplit[OF p'' k, unfolded interval_doublesplit[OF k]] unfolding interval_doublesplit[OF k] by (intro dsum_bound) auto also have "\ < e" using d by simp finally show "(\K\snd ` \. content (K \ {x. \x \ k - c\ \ d})) < e" . qed finally show "(\(x, K)\\. content (K \ {x. \x \ k - c\ \ d}) * ?i x) < e" . qed then show "\\(x, K)\\. content K * ?i x\ < e" unfolding * by (simp add: sum_nonneg split: prod.split) qed qed corollary negligible_standard_hyperplane_cart: fixes k :: "'a::finite" shows "negligible {x. x$k = (0::real)}" by (simp add: cart_eq_inner_axis negligible_standard_hyperplane) subsubsection \Hence the main theorem about negligible sets\ lemma has_integral_negligible_cbox: fixes f :: "'b::euclidean_space \ 'a::real_normed_vector" assumes negs: "negligible S" and 0: "\x. x \ S \ f x = 0" shows "(f has_integral 0) (cbox a b)" unfolding has_integral proof clarify fix e::real assume "e > 0" then have nn_gt0: "e/2 / ((real n+1) * (2 ^ n)) > 0" for n by simp then have "\\. gauge \ \ (\\. \ tagged_division_of cbox a b \ \ fine \ \ \\(x,K) \ \. content K *\<^sub>R indicator S x\ < e/2 / ((real n + 1) * 2 ^ n))" for n using negs [unfolded negligible_def has_integral] by auto then obtain \ where gd: "\n. gauge (\ n)" and \: "\n \. \\ tagged_division_of cbox a b; \ n fine \\ \ \\(x,K) \ \. content K *\<^sub>R indicator S x\ < e/2 / ((real n + 1) * 2 ^ n)" by metis show "\\. gauge \ \ (\\. \ tagged_division_of cbox a b \ \ fine \ \ norm ((\(x,K) \ \. content K *\<^sub>R f x) - 0) < e)" proof (intro exI, safe) show "gauge (\x. \ (nat \norm (f x)\) x)" using gd by (auto simp: gauge_def) show "norm ((\(x,K) \ \. content K *\<^sub>R f x) - 0) < e" if "\ tagged_division_of (cbox a b)" "(\x. \ (nat \norm (f x)\) x) fine \" for \ proof (cases "\ = {}") case True with \0 < e\ show ?thesis by simp next case False obtain N where "Max ((\(x, K). norm (f x)) ` \) \ real N" using real_arch_simple by blast then have N: "\x. x \ (\(x, K). norm (f x)) ` \ \ x \ real N" by (meson Max_ge that(1) dual_order.trans finite_imageI tagged_division_of_finite) have "\i. \q. q tagged_division_of (cbox a b) \ (\ i) fine q \ (\(x,K) \ \. K \ (\ i) x \ (x, K) \ q)" by (auto intro: tagged_division_finer[OF that(1) gd]) from choice[OF this] obtain q where q: "\n. q n tagged_division_of cbox a b" "\n. \ n fine q n" "\n x K. \(x, K) \ \; K \ \ n x\ \ (x, K) \ q n" by fastforce have "finite \" using that(1) by blast then have sum_le_inc: "\finite T; \x y. (x,y) \ T \ (0::real) \ g(x,y); \y. y\\ \ \x. (x,y) \ T \ f(y) \ g(x,y)\ \ sum f \ \ sum g T" for f g T by (rule sum_le_included[of \ T g snd f]; force) have "norm (\(x,K) \ \. content K *\<^sub>R f x) \ (\(x,K) \ \. norm (content K *\<^sub>R f x))" unfolding split_def by (rule norm_sum) also have "... \ (\(i, j) \ Sigma {..N + 1} q. (real i + 1) * (case j of (x, K) \ content K *\<^sub>R indicator S x))" proof (rule sum_le_inc, safe) show "finite (Sigma {..N+1} q)" by (meson finite_SigmaI finite_atMost tagged_division_of_finite q(1)) next fix x K assume xk: "(x, K) \ \" define n where "n = nat \norm (f x)\" have *: "norm (f x) \ (\(x, K). norm (f x)) ` \" using xk by auto have nfx: "real n \ norm (f x)" "norm (f x) \ real n + 1" unfolding n_def by auto then have "n \ {0..N + 1}" using N[OF *] by auto moreover have "K \ \ (nat \norm (f x)\) x" using that(2) xk by auto moreover then have "(x, K) \ q (nat \norm (f x)\)" by (simp add: q(3) xk) moreover then have "(x, K) \ q n" using n_def by blast moreover have "norm (content K *\<^sub>R f x) \ (real n + 1) * (content K * indicator S x)" proof (cases "x \ S") case False then show ?thesis by (simp add: 0) next case True have *: "content K \ 0" using tagged_division_ofD(4)[OF that(1) xk] by auto moreover have "content K * norm (f x) \ content K * (real n + 1)" by (simp add: mult_left_mono nfx(2)) ultimately show ?thesis using nfx True by (auto simp: field_simps) qed ultimately show "\y. (y, x, K) \ (Sigma {..N + 1} q) \ norm (content K *\<^sub>R f x) \ (real y + 1) * (content K *\<^sub>R indicator S x)" by force qed auto also have "... = (\i\N + 1. \j\q i. (real i + 1) * (case j of (x, K) \ content K *\<^sub>R indicator S x))" using q(1) by (intro sum_Sigma_product [symmetric]) auto also have "... \ (\i\N + 1. (real i + 1) * \\(x,K) \ q i. content K *\<^sub>R indicator S x\)" by (rule sum_mono) (simp add: sum_distrib_left [symmetric]) also have "... \ (\i\N + 1. e/2/2 ^ i)" proof (rule sum_mono) show "(real i + 1) * \\(x,K) \ q i. content K *\<^sub>R indicator S x\ \ e/2/2 ^ i" if "i \ {..N + 1}" for i using \[of "q i" i] q by (simp add: divide_simps mult.left_commute) qed also have "... = e/2 * (\i\N + 1. (1/2) ^ i)" unfolding sum_distrib_left by (metis divide_inverse inverse_eq_divide power_one_over) also have "\ < e/2 * 2" proof (rule mult_strict_left_mono) have "sum (power (1/2)) {..N + 1} = sum (power (1/2::real)) {..0 < e\ in auto) finally show ?thesis by auto qed qed qed proposition has_integral_negligible: fixes f :: "'b::euclidean_space \ 'a::real_normed_vector" assumes negs: "negligible S" and "\x. x \ (T - S) \ f x = 0" shows "(f has_integral 0) T" proof (cases "\a b. T = cbox a b") case True then have "((\x. if x \ T then f x else 0) has_integral 0) T" using assms by (auto intro!: has_integral_negligible_cbox) then show ?thesis by (rule has_integral_eq [rotated]) auto next case False let ?f = "(\x. if x \ T then f x else 0)" have "((\x. if x \ T then f x else 0) has_integral 0) T" apply (auto simp: False has_integral_alt [of ?f]) apply (rule_tac x=1 in exI, auto) apply (rule_tac x=0 in exI, simp add: has_integral_negligible_cbox [OF negs] assms) done then show ?thesis by (rule_tac f="?f" in has_integral_eq) auto qed lemma assumes "negligible S" shows integrable_negligible: "f integrable_on S" and integral_negligible: "integral S f = 0" using has_integral_negligible [OF assms] by (auto simp: has_integral_iff) lemma has_integral_spike: fixes f :: "'b::euclidean_space \ 'a::real_normed_vector" assumes "negligible S" and gf: "\x. x \ T - S \ g x = f x" and fint: "(f has_integral y) T" shows "(g has_integral y) T" proof - have *: "(g has_integral y) (cbox a b)" if "(f has_integral y) (cbox a b)" "\x \ cbox a b - S. g x = f x" for a b f and g:: "'b \ 'a" and y proof - have "((\x. f x + (g x - f x)) has_integral (y + 0)) (cbox a b)" using that by (intro has_integral_add has_integral_negligible) (auto intro!: \negligible S\) then show ?thesis by auto qed have \
: "\a b z. \\x. x \ T \ x \ S \ g x = f x; ((\x. if x \ T then f x else 0) has_integral z) (cbox a b)\ \ ((\x. if x \ T then g x else 0) has_integral z) (cbox a b)" by (auto dest!: *[where f="\x. if x\T then f x else 0" and g="\x. if x \ T then g x else 0"]) show ?thesis using fint gf apply (subst has_integral_alt) apply (subst (asm) has_integral_alt) apply (auto split: if_split_asm) apply (blast dest: *) using \
by meson qed lemma has_integral_spike_eq: assumes "negligible S" and gf: "\x. x \ T - S \ g x = f x" shows "(f has_integral y) T \ (g has_integral y) T" using has_integral_spike [OF \negligible S\] gf by metis lemma integrable_spike: assumes "f integrable_on T" "negligible S" "\x. x \ T - S \ g x = f x" shows "g integrable_on T" using assms unfolding integrable_on_def by (blast intro: has_integral_spike) lemma integral_spike: assumes "negligible S" and "\x. x \ T - S \ g x = f x" shows "integral T f = integral T g" using has_integral_spike_eq[OF assms] by (auto simp: integral_def integrable_on_def) subsection \Some other trivialities about negligible sets\ lemma negligible_subset: assumes "negligible s" "t \ s" shows "negligible t" unfolding negligible_def by (metis (no_types) Diff_iff assms contra_subsetD has_integral_negligible indicator_simps(2)) lemma negligible_diff[intro?]: assumes "negligible s" shows "negligible (s - t)" using assms by (meson Diff_subset negligible_subset) lemma negligible_Int: assumes "negligible s \ negligible t" shows "negligible (s \ t)" using assms negligible_subset by force lemma negligible_Un: assumes "negligible S" and T: "negligible T" shows "negligible (S \ T)" proof - have "(indicat_real (S \ T) has_integral 0) (cbox a b)" if S0: "(indicat_real S has_integral 0) (cbox a b)" and "(indicat_real T has_integral 0) (cbox a b)" for a b proof (subst has_integral_spike_eq[OF T]) show "indicat_real S x = indicat_real (S \ T) x" if "x \ cbox a b - T" for x using that by (simp add: indicator_def) show "(indicat_real S has_integral 0) (cbox a b)" by (simp add: S0) qed with assms show ?thesis unfolding negligible_def by blast qed lemma negligible_Un_eq[simp]: "negligible (s \ t) \ negligible s \ negligible t" using negligible_Un negligible_subset by blast lemma negligible_sing[intro]: "negligible {a::'a::euclidean_space}" using negligible_standard_hyperplane[OF SOME_Basis, of "a \ (SOME i. i \ Basis)"] negligible_subset by blast lemma negligible_insert[simp]: "negligible (insert a s) \ negligible s" by (metis insert_is_Un negligible_Un_eq negligible_sing) lemma negligible_empty[iff]: "negligible {}" using negligible_insert by blast text\Useful in this form for backchaining\ lemma empty_imp_negligible: "S = {} \ negligible S" by simp lemma negligible_finite[intro]: assumes "finite s" shows "negligible s" using assms by (induct s) auto lemma negligible_Union[intro]: assumes "finite \" and "\t. t \ \ \ negligible t" shows "negligible(\\)" using assms by induct auto lemma negligible: "negligible S \ (\T. (indicat_real S has_integral 0) T)" proof (intro iffI allI) fix T assume "negligible S" then show "(indicator S has_integral 0) T" by (meson Diff_iff has_integral_negligible indicator_simps(2)) qed (simp add: negligible_def) subsection \Finite case of the spike theorem is quite commonly needed\ lemma has_integral_spike_finite: assumes "finite S" and "\x. x \ T - S \ g x = f x" and "(f has_integral y) T" shows "(g has_integral y) T" using assms has_integral_spike negligible_finite by blast lemma has_integral_spike_finite_eq: assumes "finite S" and "\x. x \ T - S \ g x = f x" shows "((f has_integral y) T \ (g has_integral y) T)" by (metis assms has_integral_spike_finite) lemma integrable_spike_finite: assumes "finite S" and "\x. x \ T - S \ g x = f x" and "f integrable_on T" shows "g integrable_on T" using assms has_integral_spike_finite by blast lemma has_integral_bound_spike_finite: fixes f :: "'a::euclidean_space \ 'b::real_normed_vector" assumes "0 \ B" "finite S" and f: "(f has_integral i) (cbox a b)" and leB: "\x. x \ cbox a b - S \ norm (f x) \ B" shows "norm i \ B * content (cbox a b)" proof - define g where "g \ (\x. if x \ S then 0 else f x)" then have "\x. x \ cbox a b - S \ norm (g x) \ B" using leB by simp moreover have "(g has_integral i) (cbox a b)" using has_integral_spike_finite [OF \finite S\ _ f] by (simp add: g_def) ultimately show ?thesis by (simp add: \0 \ B\ g_def has_integral_bound) qed corollary has_integral_bound_real: fixes f :: "real \ 'b::real_normed_vector" assumes "0 \ B" "finite S" and "(f has_integral i) {a..b}" and "\x. x \ {a..b} - S \ norm (f x) \ B" shows "norm i \ B * content {a..b}" by (metis assms box_real(2) has_integral_bound_spike_finite) subsection \In particular, the boundary of an interval is negligible\ lemma negligible_frontier_interval: "negligible(cbox (a::'a::euclidean_space) b - box a b)" proof - let ?A = "\((\k. {x. x\k = a\k} \ {x::'a. x\k = b\k}) ` Basis)" have "negligible ?A" by (force simp add: negligible_Union[OF finite_imageI]) moreover have "cbox a b - box a b \ ?A" by (force simp add: mem_box) ultimately show ?thesis by (rule negligible_subset) qed lemma has_integral_spike_interior: assumes f: "(f has_integral y) (cbox a b)" and gf: "\x. x \ box a b \ g x = f x" shows "(g has_integral y) (cbox a b)" by (meson Diff_iff gf has_integral_spike[OF negligible_frontier_interval _ f]) lemma has_integral_spike_interior_eq: assumes "\x. x \ box a b \ g x = f x" shows "(f has_integral y) (cbox a b) \ (g has_integral y) (cbox a b)" by (metis assms has_integral_spike_interior) lemma integrable_spike_interior: assumes "\x. x \ box a b \ g x = f x" and "f integrable_on cbox a b" shows "g integrable_on cbox a b" using assms has_integral_spike_interior_eq by blast subsection \Integrability of continuous functions\ lemma operative_approximableI: fixes f :: "'b::euclidean_space \ 'a::banach" assumes "0 \ e" shows "operative conj True (\i. \g. (\x\i. norm (f x - g (x::'b)) \ e) \ g integrable_on i)" proof - interpret comm_monoid conj True by standard auto show ?thesis proof (standard, safe) fix a b :: 'b show "\g. (\x\cbox a b. norm (f x - g x) \ e) \ g integrable_on cbox a b" if "box a b = {}" for a b using assms that by (metis content_eq_0_interior integrable_on_null interior_cbox norm_zero right_minus_eq) { fix c g and k :: 'b assume fg: "\x\cbox a b. norm (f x - g x) \ e" and g: "g integrable_on cbox a b" assume k: "k \ Basis" show "\g. (\x\cbox a b \ {x. x \ k \ c}. norm (f x - g x) \ e) \ g integrable_on cbox a b \ {x. x \ k \ c}" "\g. (\x\cbox a b \ {x. c \ x \ k}. norm (f x - g x) \ e) \ g integrable_on cbox a b \ {x. c \ x \ k}" using fg g k by auto } show "\g. (\x\cbox a b. norm (f x - g x) \ e) \ g integrable_on cbox a b" if fg1: "\x\cbox a b \ {x. x \ k \ c}. norm (f x - g1 x) \ e" and g1: "g1 integrable_on cbox a b \ {x. x \ k \ c}" and fg2: "\x\cbox a b \ {x. c \ x \ k}. norm (f x - g2 x) \ e" and g2: "g2 integrable_on cbox a b \ {x. c \ x \ k}" and k: "k \ Basis" for c k g1 g2 proof - let ?g = "\x. if x\k = c then f x else if x\k \ c then g1 x else g2 x" show "\g. (\x\cbox a b. norm (f x - g x) \ e) \ g integrable_on cbox a b" proof (intro exI conjI ballI) show "norm (f x - ?g x) \ e" if "x \ cbox a b" for x by (auto simp: that assms fg1 fg2) show "?g integrable_on cbox a b" proof - have "?g integrable_on cbox a b \ {x. x \ k \ c}" "?g integrable_on cbox a b \ {x. x \ k \ c}" by(rule integrable_spike[OF _ negligible_standard_hyperplane[of k c]], use k g1 g2 in auto)+ with has_integral_split[OF _ _ k] show ?thesis unfolding integrable_on_def by blast qed qed qed qed qed lemma comm_monoid_set_F_and: "comm_monoid_set.F (\) True f s \ (finite s \ (\x\s. f x))" proof - interpret bool: comm_monoid_set \(\)\ True .. show ?thesis by (induction s rule: infinite_finite_induct) auto qed lemma approximable_on_division: fixes f :: "'b::euclidean_space \ 'a::banach" assumes "0 \ e" and d: "d division_of (cbox a b)" and f: "\i\d. \g. (\x\i. norm (f x - g x) \ e) \ g integrable_on i" obtains g where "\x\cbox a b. norm (f x - g x) \ e" "g integrable_on cbox a b" proof - interpret operative conj True "\i. \g. (\x\i. norm (f x - g (x::'b)) \ e) \ g integrable_on i" using \0 \ e\ by (rule operative_approximableI) from f local.division [OF d] that show thesis by auto qed lemma integrable_continuous: fixes f :: "'b::euclidean_space \ 'a::banach" assumes "continuous_on (cbox a b) f" shows "f integrable_on cbox a b" proof (rule integrable_uniform_limit) fix e :: real assume e: "e > 0" then obtain d where "0 < d" and d: "\x x'. \x \ cbox a b; x' \ cbox a b; dist x' x < d\ \ dist (f x') (f x) < e" using compact_uniformly_continuous[OF assms compact_cbox] unfolding uniformly_continuous_on_def by metis obtain p where ptag: "p tagged_division_of cbox a b" and finep: "(\x. ball x d) fine p" using fine_division_exists[OF gauge_ball[OF \0 < d\], of a b] . have *: "\i\snd ` p. \g. (\x\i. norm (f x - g x) \ e) \ g integrable_on i" proof (safe, unfold snd_conv) fix x l assume as: "(x, l) \ p" obtain a b where l: "l = cbox a b" using as ptag by blast then have x: "x \ cbox a b" using as ptag by auto show "\g. (\x\l. norm (f x - g x) \ e) \ g integrable_on l" proof (intro exI conjI strip) show "(\y. f x) integrable_on l" unfolding integrable_on_def l by blast next fix y assume y: "y \ l" then have "y \ ball x d" using as finep by fastforce then show "norm (f y - f x) \ e" using d x y as l by (metis dist_commute dist_norm less_imp_le mem_ball ptag subsetCE tagged_division_ofD(3)) qed qed from e have "e \ 0" by auto from approximable_on_division[OF this division_of_tagged_division[OF ptag] *] show "\g. (\x\cbox a b. norm (f x - g x) \ e) \ g integrable_on cbox a b" by metis qed lemma integrable_continuous_interval: fixes f :: "'b::ordered_euclidean_space \ 'a::banach" assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}" by (metis assms integrable_continuous interval_cbox) lemmas integrable_continuous_real = integrable_continuous_interval[where 'b=real] lemma integrable_continuous_closed_segment: fixes f :: "real \ 'a::banach" assumes "continuous_on (closed_segment a b) f" shows "f integrable_on (closed_segment a b)" using assms by (auto intro!: integrable_continuous_interval simp: closed_segment_eq_real_ivl) subsection \Specialization of additivity to one dimension\ subsection \A useful lemma allowing us to factor out the content size\ lemma has_integral_factor_content: "(f has_integral i) (cbox a b) \ (\e>0. \d. gauge d \ (\p. p tagged_division_of (cbox a b) \ d fine p \ norm (sum (\(x,k). content k *\<^sub>R f x) p - i) \ e * content (cbox a b)))" proof (cases "content (cbox a b) = 0") case True have "\e p. p tagged_division_of cbox a b \ norm ((\(x, k)\p. content k *\<^sub>R f x)) \ e * content (cbox a b)" unfolding sum_content_null[OF True] True by force moreover have "i = 0" if "\e. e > 0 \ \d. gauge d \ (\p. p tagged_division_of cbox a b \ d fine p \ norm ((\(x, k)\p. content k *\<^sub>R f x) - i) \ e * content (cbox a b))" using that [of 1] by (force simp add: True sum_content_null[OF True] intro: fine_division_exists[of _ a b]) ultimately show ?thesis unfolding has_integral_null_eq[OF True] by (force simp add: ) next case False then have F: "0 < content (cbox a b)" using zero_less_measure_iff by blast let ?P = "\e opp. \d. gauge d \ (\p. p tagged_division_of (cbox a b) \ d fine p \ opp (norm ((\(x, k)\p. content k *\<^sub>R f x) - i)) e)" show ?thesis proof (subst has_integral, safe) fix e :: real assume e: "e > 0" show "?P (e * content (cbox a b)) (\)" if \
[rule_format]: "\\>0. ?P \ (<)" using \
[of "e * content (cbox a b)"] by (meson F e less_imp_le mult_pos_pos) show "?P e (<)" if \
[rule_format]: "\\>0. ?P (\ * content (cbox a b)) (\)" using \
[of "e/2 / content (cbox a b)"] using F e by (force simp add: algebra_simps) qed qed lemma has_integral_factor_content_real: "(f has_integral i) {a..b::real} \ (\e>0. \d. gauge d \ (\p. p tagged_division_of {a..b} \ d fine p \ norm (sum (\(x,k). content k *\<^sub>R f x) p - i) \ e * content {a..b} ))" unfolding box_real[symmetric] by (rule has_integral_factor_content) subsection \Fundamental theorem of calculus\ lemma interval_bounds_real: fixes q b :: real assumes "a \ b" shows "Sup {a..b} = b" and "Inf {a..b} = a" using assms by auto theorem fundamental_theorem_of_calculus: fixes f :: "real \ 'a::banach" assumes "a \ b" and vecd: "\x. x \ {a..b} \ (f has_vector_derivative f' x) (at x within {a..b})" shows "(f' has_integral (f b - f a)) {a..b}" unfolding has_integral_factor_content box_real[symmetric] proof safe fix e :: real assume "e > 0" then have "\x. \d>0. x \ {a..b} \ (\y\{a..b}. norm (y-x) < d \ norm (f y - f x - (y-x) *\<^sub>R f' x) \ e * norm (y-x))" using vecd[unfolded has_vector_derivative_def has_derivative_within_alt] by blast then obtain d where d: "\x. 0 < d x" "\x y. \x \ {a..b}; y \ {a..b}; norm (y-x) < d x\ \ norm (f y - f x - (y-x) *\<^sub>R f' x) \ e * norm (y-x)" by metis show "\d. gauge d \ (\p. p tagged_division_of (cbox a b) \ d fine p \ norm ((\(x, k)\p. content k *\<^sub>R f' x) - (f b - f a)) \ e * content (cbox a b))" proof (rule exI, safe) show "gauge (\x. ball x (d x))" using d(1) gauge_ball_dependent by blast next fix p assume ptag: "p tagged_division_of cbox a b" and finep: "(\x. ball x (d x)) fine p" have ba: "b - a = (\(x,K)\p. Sup K - Inf K)" "f b - f a = (\(x,K)\p. f(Sup K) - f(Inf K))" using additive_tagged_division_1[where f= "\x. x"] additive_tagged_division_1[where f= f] \a \ b\ ptag by auto have "norm (\(x, K) \ p. (content K *\<^sub>R f' x) - (f (Sup K) - f (Inf K))) \ (\n\p. e * (case n of (x, k) \ Sup k - Inf k))" proof (rule sum_norm_le,safe) fix x K assume "(x, K) \ p" then have "x \ K" and kab: "K \ cbox a b" using ptag by blast+ then obtain u v where k: "K = cbox u v" and "x \ K" and kab: "K \ cbox a b" using ptag \(x, K) \ p\ by auto have "u \ v" using \x \ K\ unfolding k by auto have ball: "\y\K. y \ ball x (d x)" using finep \(x, K) \ p\ by blast have "u \ K" "v \ K" by (simp_all add: \u \ v\ k) have "norm ((v - u) *\<^sub>R f' x - (f v - f u)) = norm (f u - f x - (u - x) *\<^sub>R f' x - (f v - f x - (v - x) *\<^sub>R f' x))" by (auto simp add: algebra_simps) also have "... \ norm (f u - f x - (u - x) *\<^sub>R f' x) + norm (f v - f x - (v - x) *\<^sub>R f' x)" by (rule norm_triangle_ineq4) finally have "norm ((v - u) *\<^sub>R f' x - (f v - f u)) \ norm (f u - f x - (u - x) *\<^sub>R f' x) + norm (f v - f x - (v - x) *\<^sub>R f' x)" . also have "\ \ e * norm (u - x) + e * norm (v - x)" proof (rule add_mono) show "norm (f u - f x - (u - x) *\<^sub>R f' x) \ e * norm (u - x)" proof (rule d) show "norm (u - x) < d x" using \u \ K\ ball by (auto simp add: dist_real_def) qed (use \x \ K\ \u \ K\ kab in auto) show "norm (f v - f x - (v - x) *\<^sub>R f' x) \ e * norm (v - x)" proof (rule d) show "norm (v - x) < d x" using \v \ K\ ball by (auto simp add: dist_real_def) qed (use \x \ K\ \v \ K\ kab in auto) qed also have "\ \ e * (Sup K - Inf K)" using \x \ K\ by (auto simp: k interval_bounds_real[OF \u \ v\] field_simps) finally show "norm (content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) \ e * (Sup K - Inf K)" using \u \ v\ by (simp add: k) qed with \a \ b\ show "norm ((\(x, K)\p. content K *\<^sub>R f' x) - (f b - f a)) \ e * content (cbox a b)" by (auto simp: ba split_def sum_subtractf [symmetric] sum_distrib_left) qed qed lemma ident_has_integral: fixes a::real assumes "a \ b" shows "((\x. x) has_integral (b\<^sup>2 - a\<^sup>2)/2) {a..b}" proof - have "((\x. x) has_integral inverse 2 * b\<^sup>2 - inverse 2 * a\<^sup>2) {a..b}" unfolding power2_eq_square by (rule fundamental_theorem_of_calculus [OF assms] derivative_eq_intros | simp)+ then show ?thesis by (simp add: field_simps) qed lemma integral_ident [simp]: fixes a::real assumes "a \ b" shows "integral {a..b} (\x. x) = (if a \ b then (b\<^sup>2 - a\<^sup>2)/2 else 0)" by (metis assms ident_has_integral integral_unique) lemma ident_integrable_on: fixes a::real shows "(\x. x) integrable_on {a..b}" by (metis atLeastatMost_empty_iff integrable_on_def has_integral_empty ident_has_integral) lemma integral_sin [simp]: fixes a::real assumes "a \ b" shows "integral {a..b} sin = cos a - cos b" proof - have "(sin has_integral (- cos b - - cos a)) {a..b}" proof (rule fundamental_theorem_of_calculus) show "((\a. - cos a) has_vector_derivative sin x) (at x within {a..b})" for x unfolding has_field_derivative_iff_has_vector_derivative [symmetric] by (rule derivative_eq_intros | force)+ qed (use assms in auto) then show ?thesis by (simp add: integral_unique) qed lemma integral_cos [simp]: fixes a::real assumes "a \ b" shows "integral {a..b} cos = sin b - sin a" proof - have "(cos has_integral (sin b - sin a)) {a..b}" proof (rule fundamental_theorem_of_calculus) show "(sin has_vector_derivative cos x) (at x within {a..b})" for x unfolding has_field_derivative_iff_has_vector_derivative [symmetric] by (rule derivative_eq_intros | force)+ qed (use assms in auto) then show ?thesis by (simp add: integral_unique) qed lemma has_integral_sin_nx: "((\x. sin(real_of_int n * x)) has_integral 0) {-pi..pi}" proof (cases "n = 0") case False have "((\x. sin (n * x)) has_integral (- cos (n * pi)/n - - cos (n * - pi)/n)) {-pi..pi}" proof (rule fundamental_theorem_of_calculus) show "((\x. - cos (n * x) / n) has_vector_derivative sin (n * a)) (at a within {-pi..pi})" if "a \ {-pi..pi}" for a :: real using that False unfolding has_vector_derivative_def by (intro derivative_eq_intros | force)+ qed auto then show ?thesis by simp qed auto lemma integral_sin_nx: "integral {-pi..pi} (\x. sin(x * real_of_int n)) = 0" using has_integral_sin_nx [of n] by (force simp: mult.commute) lemma has_integral_cos_nx: "((\x. cos(real_of_int n * x)) has_integral (if n = 0 then 2 * pi else 0)) {-pi..pi}" proof (cases "n = 0") case True then show ?thesis using has_integral_const_real [of "1::real" "-pi" pi] by auto next case False have "((\x. cos (n * x)) has_integral (sin (n * pi)/n - sin (n * - pi)/n)) {-pi..pi}" proof (rule fundamental_theorem_of_calculus) show "((\x. sin (n * x) / n) has_vector_derivative cos (n * x)) (at x within {-pi..pi})" if "x \ {-pi..pi}" for x :: real using that False unfolding has_vector_derivative_def by (intro derivative_eq_intros | force)+ qed auto with False show ?thesis by (simp add: mult.commute) qed lemma integral_cos_nx: "integral {-pi..pi} (\x. cos(x * real_of_int n)) = (if n = 0 then 2 * pi else 0)" using has_integral_cos_nx [of n] by (force simp: mult.commute) subsection \Taylor series expansion\ lemma mvt_integral: fixes f::"'a::real_normed_vector\'b::banach" assumes f'[derivative_intros]: "\x. x \ S \ (f has_derivative f' x) (at x within S)" assumes line_in: "\t. t \ {0..1} \ x + t *\<^sub>R y \ S" shows "f (x + y) - f x = integral {0..1} (\t. f' (x + t *\<^sub>R y) y)" (is ?th1) proof - from assms have subset: "(\xa. x + xa *\<^sub>R y) ` {0..1} \ S" by auto note [derivative_intros] = has_derivative_subset[OF _ subset] has_derivative_in_compose[where f="(\xa. x + xa *\<^sub>R y)" and g = f] note [continuous_intros] = continuous_on_compose2[where f="(\xa. x + xa *\<^sub>R y)"] continuous_on_subset[OF _ subset] have "\t. t \ {0..1} \ ((\t. f (x + t *\<^sub>R y)) has_vector_derivative f' (x + t *\<^sub>R y) y) (at t within {0..1})" using assms by (auto simp: has_vector_derivative_def linear_cmul[OF has_derivative_linear[OF f'], symmetric] intro!: derivative_eq_intros) from fundamental_theorem_of_calculus[rule_format, OF _ this] show ?th1 by (auto intro!: integral_unique[symmetric]) qed lemma (in bounded_bilinear) sum_prod_derivatives_has_vector_derivative: assumes "p>0" and f0: "Df 0 = f" and Df: "\m t. m < p \ a \ t \ t \ b \ (Df m has_vector_derivative Df (Suc m) t) (at t within {a..b})" and g0: "Dg 0 = g" and Dg: "\m t. m < p \ a \ t \ t \ b \ (Dg m has_vector_derivative Dg (Suc m) t) (at t within {a..b})" and ivl: "a \ t" "t \ b" shows "((\t. \iR prod (Df i t) (Dg (p - Suc i) t)) has_vector_derivative prod (f t) (Dg p t) - (-1)^p *\<^sub>R prod (Df p t) (g t)) (at t within {a..b})" using assms proof cases assume p: "p \ 1" define p' where "p' = p - 2" from assms p have p': "{..i. i \ p' \ Suc (Suc p' - i) = (Suc (Suc p') - i)" by auto let ?f = "\i. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg ((p - i)) t))" have "(\iR (prod (Df i t) (Dg (Suc (p - Suc i)) t) + prod (Df (Suc i) t) (Dg (p - Suc i) t))) = (\i\(Suc p'). ?f i - ?f (Suc i))" by (auto simp: algebra_simps p'(2) numeral_2_eq_2 * lessThan_Suc_atMost) also note sum_telescope finally have "(\iR (prod (Df i t) (Dg (Suc (p - Suc i)) t) + prod (Df (Suc i) t) (Dg (p - Suc i) t))) = prod (f t) (Dg p t) - (- 1) ^ p *\<^sub>R prod (Df p t) (g t)" unfolding p'[symmetric] by (simp add: assms) thus ?thesis using assms by (auto intro!: derivative_eq_intros has_vector_derivative) qed (auto intro!: derivative_eq_intros has_vector_derivative) lemma fixes f::"real\'a::banach" assumes "p>0" and f0: "Df 0 = f" and Df: "\m t. m < p \ a \ t \ t \ b \ (Df m has_vector_derivative Df (Suc m) t) (at t within {a..b})" and ivl: "a \ b" defines "i \ \x. ((b - x) ^ (p - 1) / fact (p - 1)) *\<^sub>R Df p x" shows Taylor_has_integral: "(i has_integral f b - (\iR Df i a)) {a..b}" and Taylor_integral: "f b = (\iR Df i a) + integral {a..b} i" and Taylor_integrable: "i integrable_on {a..b}" proof goal_cases case 1 interpret bounded_bilinear "scaleR::real\'a\'a" by (rule bounded_bilinear_scaleR) define g where "g s = (b - s)^(p - 1)/fact (p - 1)" for s define Dg where [abs_def]: "Dg n s = (if n < p then (-1)^n * (b - s)^(p - 1 - n) / fact (p - 1 - n) else 0)" for n s have g0: "Dg 0 = g" using \p > 0\ by (auto simp add: Dg_def field_split_simps g_def split: if_split_asm) { fix m assume "p > Suc m" hence "p - Suc m = Suc (p - Suc (Suc m))" by auto hence "real (p - Suc m) * fact (p - Suc (Suc m)) = fact (p - Suc m)" by auto } note fact_eq = this have Dg: "\m t. m < p \ a \ t \ t \ b \ (Dg m has_vector_derivative Dg (Suc m) t) (at t within {a..b})" unfolding Dg_def by (auto intro!: derivative_eq_intros simp: has_vector_derivative_def fact_eq field_split_simps) let ?sum = "\t. \iR Dg i t *\<^sub>R Df (p - Suc i) t" from sum_prod_derivatives_has_vector_derivative[of _ Dg _ _ _ Df, OF \p > 0\ g0 Dg f0 Df] have deriv: "\t. a \ t \ t \ b \ (?sum has_vector_derivative g t *\<^sub>R Df p t - (- 1) ^ p *\<^sub>R Dg p t *\<^sub>R f t) (at t within {a..b})" by auto from fundamental_theorem_of_calculus[rule_format, OF \a \ b\ deriv] have "(i has_integral ?sum b - ?sum a) {a..b}" using atLeastatMost_empty'[simp del] by (simp add: i_def g_def Dg_def) also have one: "(- 1) ^ p' * (- 1) ^ p' = (1::real)" and "{.. {i. p = Suc i} = {p - 1}" for p' using \p > 0\ by (auto simp: power_mult_distrib[symmetric]) then have "?sum b = f b" using Suc_pred'[OF \p > 0\] by (simp add: diff_eq_eq Dg_def power_0_left le_Suc_eq if_distrib if_distribR sum.If_cases f0) also have "{..x. p - x - 1) ` {.. (\x. p - x - 1) ` {..iR Df i a)" by (rule sum.reindex_cong) (auto simp add: inj_on_def Dg_def one) finally show c: ?case . case 2 show ?case using c integral_unique by (metis (lifting) add.commute diff_eq_eq integral_unique) case 3 show ?case using c by force qed subsection \Only need trivial subintervals if the interval itself is trivial\ proposition division_of_nontrivial: fixes \ :: "'a::euclidean_space set set" assumes sdiv: "\ division_of (cbox a b)" and cont0: "content (cbox a b) \ 0" shows "{k. k \ \ \ content k \ 0} division_of (cbox a b)" using sdiv proof (induction "card \" arbitrary: \ rule: less_induct) case less note \ = division_ofD[OF less.prems] { presume *: "{k \ \. content k \ 0} \ \ \ ?case" then show ?case using less.prems by fastforce } assume noteq: "{k \ \. content k \ 0} \ \" then obtain K c d where "K \ \" and contk: "content K = 0" and keq: "K = cbox c d" using \(4) by blast then have "card \ > 0" unfolding card_gt_0_iff using less by auto then have card: "card (\ - {K}) < card \" using less \K \ \\ by (simp add: \(1)) have closed: "closed (\(\ - {K}))" using less.prems by auto have "x islimpt \(\ - {K})" if "x \ K" for x unfolding islimpt_approachable proof (intro allI impI) fix e::real assume "e > 0" obtain i where i: "c\i = d\i" "i\Basis" using contk \(3) [OF \K \ \\] unfolding box_ne_empty keq by (meson content_eq_0 dual_order.antisym) then have xi: "x\i = d\i" using \x \ K\ unfolding keq mem_box by (metis antisym) define y where "y = (\j\Basis. (if j = i then if c\i \ (a\i + b\i)/2 then c\i + min e (b\i - c\i)/2 else c\i - min e (c\i - a\i)/2 else x\j) *\<^sub>R j)" show "\x'\\(\ - {K}). x' \ x \ dist x' x < e" proof (intro bexI conjI) have "d \ cbox c d" using \(3)[OF \K \ \\] by (simp add: box_ne_empty(1) keq mem_box(2)) then have "d \ cbox a b" using \(2)[OF \K \ \\] by (auto simp: keq) then have di: "a \ i \ d \ i \ d \ i \ b \ i" using \i \ Basis\ mem_box(2) by blast then have xyi: "y\i \ x\i" unfolding y_def i xi using \e > 0\ cont0 \i \ Basis\ by (auto simp: content_eq_0 elim!: ballE[of _ _ i]) then show "y \ x" unfolding euclidean_eq_iff[where 'a='a] using i by auto have "norm (y-x) \ (\b\Basis. \(y - x) \ b\)" by (rule norm_le_l1) also have "... = \(y - x) \ i\ + (\b \ Basis - {i}. \(y - x) \ b\)" by (meson finite_Basis i(2) sum.remove) also have "... < e + sum (\i. 0) Basis" proof (rule add_less_le_mono) show "\(y-x) \ i\ < e" using di \e > 0\ y_def i xi by (auto simp: inner_simps) show "(\i\Basis - {i}. \(y-x) \ i\) \ (\i\Basis. 0)" unfolding y_def by (auto simp: inner_simps) qed finally have "norm (y-x) < e + sum (\i. 0) Basis" . then show "dist y x < e" unfolding dist_norm by auto have "y \ K" unfolding keq mem_box using i(1) i(2) xi xyi by fastforce moreover have "y \ \\" using subsetD[OF \(2)[OF \K \ \\] \x \ K\] \e > 0\ di i by (auto simp: \ mem_box y_def field_simps elim!: ballE[of _ _ i]) ultimately show "y \ \(\ - {K})" by auto qed qed then have "K \ \(\ - {K})" using closed closed_limpt by blast then have "\(\ - {K}) = cbox a b" unfolding \(6)[symmetric] by auto then have "\ - {K} division_of cbox a b" by (metis Diff_subset less.prems division_of_subset \(6)) then have "{ka \ \ - {K}. content ka \ 0} division_of (cbox a b)" using card less.hyps by blast moreover have "{ka \ \ - {K}. content ka \ 0} = {K \ \. content K \ 0}" using contk by auto ultimately show ?case by auto qed subsection \Integrability on subintervals\ lemma operative_integrableI: fixes f :: "'b::euclidean_space \ 'a::banach" assumes "0 \ e" shows "operative conj True (\i. f integrable_on i)" proof - interpret comm_monoid conj True proof qed show ?thesis proof show "\a b. box a b = {} \ (f integrable_on cbox a b) = True" by (simp add: content_eq_0_interior integrable_on_null) show "\a b c k. k \ Basis \ (f integrable_on cbox a b) \ (f integrable_on cbox a b \ {x. x \ k \ c} \ f integrable_on cbox a b \ {x. c \ x \ k})" unfolding integrable_on_def by (auto intro!: has_integral_split) qed qed lemma integrable_subinterval: fixes f :: "'b::euclidean_space \ 'a::banach" assumes f: "f integrable_on cbox a b" and cd: "cbox c d \ cbox a b" shows "f integrable_on cbox c d" proof - interpret operative conj True "\i. f integrable_on i" using order_refl by (rule operative_integrableI) show ?thesis proof (cases "cbox c d = {}") case True then show ?thesis using division [symmetric] f by (auto simp: comm_monoid_set_F_and) next case False then show ?thesis by (metis cd comm_monoid_set_F_and division division_of_finite f partial_division_extend_1) qed qed lemma integrable_subinterval_real: fixes f :: "real \ 'a::banach" assumes "f integrable_on {a..b}" and "{c..d} \ {a..b}" shows "f integrable_on {c..d}" by (metis assms box_real(2) integrable_subinterval) subsection \Combining adjacent intervals in 1 dimension\ lemma has_integral_combine: fixes a b c :: real and j :: "'a::banach" assumes "a \ c" and "c \ b" and ac: "(f has_integral i) {a..c}" and cb: "(f has_integral j) {c..b}" shows "(f has_integral (i + j)) {a..b}" proof - interpret operative_real "lift_option plus" "Some 0" "\i. if f integrable_on i then Some (integral i f) else None" using operative_integralI by (rule operative_realI) from \a \ c\ \c \ b\ ac cb coalesce_less_eq have *: "lift_option (+) (if f integrable_on {a..c} then Some (integral {a..c} f) else None) (if f integrable_on {c..b} then Some (integral {c..b} f) else None) = (if f integrable_on {a..b} then Some (integral {a..b} f) else None)" by (auto simp: split: if_split_asm) then have "f integrable_on cbox a b" using ac cb by (auto split: if_split_asm) with * show ?thesis using ac cb by (auto simp add: integrable_on_def integral_unique split: if_split_asm) qed lemma integral_combine: fixes f :: "real \ 'a::banach" assumes "a \ c" and "c \ b" and ab: "f integrable_on {a..b}" shows "integral {a..c} f + integral {c..b} f = integral {a..b} f" proof - have "(f has_integral integral {a..c} f) {a..c}" using ab \c \ b\ integrable_subinterval_real by fastforce moreover have "(f has_integral integral {c..b} f) {c..b}" using ab \a \ c\ integrable_subinterval_real by fastforce ultimately have "(f has_integral integral {a..c} f + integral {c..b} f) {a..b}" using \a \ c\ \c \ b\ has_integral_combine by blast then show ?thesis by (simp add: has_integral_integrable_integral) qed lemma integrable_combine: fixes f :: "real \ 'a::banach" assumes "a \ c" and "c \ b" and "f integrable_on {a..c}" and "f integrable_on {c..b}" shows "f integrable_on {a..b}" using assms unfolding integrable_on_def by (auto intro!: has_integral_combine) lemma integral_minus_sets: fixes f::"real \ 'a::banach" shows "c \ a \ c \ b \ f integrable_on {c .. max a b} \ integral {c .. a} f - integral {c .. b} f = (if a \ b then - integral {a .. b} f else integral {b .. a} f)" using integral_combine[of c a b f] integral_combine[of c b a f] by (auto simp: algebra_simps max_def) lemma integral_minus_sets': fixes f::"real \ 'a::banach" shows "c \ a \ c \ b \ f integrable_on {min a b .. c} \ integral {a .. c} f - integral {b .. c} f = (if a \ b then integral {a .. b} f else - integral {b .. a} f)" using integral_combine[of b a c f] integral_combine[of a b c f] by (auto simp: algebra_simps min_def) subsection \Reduce integrability to "local" integrability\ lemma integrable_on_little_subintervals: fixes f :: "'b::euclidean_space \ 'a::banach" assumes "\x\cbox a b. \d>0. \u v. x \ cbox u v \ cbox u v \ ball x d \ cbox u v \ cbox a b \ f integrable_on cbox u v" shows "f integrable_on cbox a b" proof - interpret operative conj True "\i. f integrable_on i" using order_refl by (rule operative_integrableI) have "\x. \d>0. x\cbox a b \ (\u v. x \ cbox u v \ cbox u v \ ball x d \ cbox u v \ cbox a b \ f integrable_on cbox u v)" using assms by (metis zero_less_one) then obtain d where d: "\x. 0 < d x" "\x u v. \x \ cbox a b; x \ cbox u v; cbox u v \ ball x (d x); cbox u v \ cbox a b\ \ f integrable_on cbox u v" by metis obtain p where p: "p tagged_division_of cbox a b" "(\x. ball x (d x)) fine p" using fine_division_exists[OF gauge_ball_dependent,of d a b] d(1) by blast then have sndp: "snd ` p division_of cbox a b" by (metis division_of_tagged_division) have "f integrable_on k" if "(x, k) \ p" for x k using tagged_division_ofD(2-4)[OF p(1) that] fineD[OF p(2) that] d[of x] by auto then show ?thesis unfolding division [symmetric, OF sndp] comm_monoid_set_F_and by auto qed subsection \Second FTC or existence of antiderivative\ lemma integrable_const[intro]: "(\x. c) integrable_on cbox a b" unfolding integrable_on_def by blast lemma integral_has_vector_derivative_continuous_at: fixes f :: "real \ 'a::banach" assumes f: "f integrable_on {a..b}" and x: "x \ {a..b} - S" and "finite S" and fx: "continuous (at x within ({a..b} - S)) f" shows "((\u. integral {a..u} f) has_vector_derivative f x) (at x within ({a..b} - S))" proof - let ?I = "\a b. integral {a..b} f" { fix e::real assume "e > 0" obtain d where "d>0" and d: "\x'. \x' \ {a..b} - S; \x' - x\ < d\ \ norm(f x' - f x) \ e" using \e>0\ fx by (auto simp: continuous_within_eps_delta dist_norm less_imp_le) have "norm (integral {a..y} f - integral {a..x} f - (y-x) *\<^sub>R f x) \ e * \y - x\" (is "?lhs \ ?rhs") if y: "y \ {a..b} - S" and yx: "\y - x\ < d" for y proof (cases "y < x") case False have "f integrable_on {a..y}" using f y by (simp add: integrable_subinterval_real) then have Idiff: "?I a y - ?I a x = ?I x y" using False x by (simp add: algebra_simps integral_combine) have fux_int: "((\u. f u - f x) has_integral integral {x..y} f - (y-x) *\<^sub>R f x) {x..y}" proof (rule has_integral_diff) show "(f has_integral integral {x..y} f) {x..y}" using x y by (auto intro: integrable_integral [OF integrable_subinterval_real [OF f]]) show "((\u. f x) has_integral (y - x) *\<^sub>R f x) {x..y}" using has_integral_const_real [of "f x" x y] False by simp qed have "?lhs \ e * content {x..y}" using yx False d x y \e>0\ assms by (intro has_integral_bound_real[where f="(\u. f u - f x)"]) (auto simp: Idiff fux_int) also have "... \ ?rhs" using False by auto finally show ?thesis . next case True have "f integrable_on {a..x}" using f x by (simp add: integrable_subinterval_real) then have Idiff: "?I a x - ?I a y = ?I y x" using True x y by (simp add: algebra_simps integral_combine) have fux_int: "((\u. f u - f x) has_integral integral {y..x} f - (x - y) *\<^sub>R f x) {y..x}" proof (rule has_integral_diff) show "(f has_integral integral {y..x} f) {y..x}" using x y by (auto intro: integrable_integral [OF integrable_subinterval_real [OF f]]) show "((\u. f x) has_integral (x - y) *\<^sub>R f x) {y..x}" using has_integral_const_real [of "f x" y x] True by simp qed have "norm (integral {a..x} f - integral {a..y} f - (x - y) *\<^sub>R f x) \ e * content {y..x}" using yx True d x y \e>0\ assms by (intro has_integral_bound_real[where f="(\u. f u - f x)"]) (auto simp: Idiff fux_int) also have "... \ e * \y - x\" using True by auto finally have "norm (integral {a..x} f - integral {a..y} f - (x - y) *\<^sub>R f x) \ e * \y - x\" . then show ?thesis by (simp add: algebra_simps norm_minus_commute) qed then have "\d>0. \y\{a..b} - S. \y - x\ < d \ norm (integral {a..y} f - integral {a..x} f - (y-x) *\<^sub>R f x) \ e * \y - x\" using \d>0\ by blast } then show ?thesis by (simp add: has_vector_derivative_def has_derivative_within_alt bounded_linear_scaleR_left) qed lemma integral_has_vector_derivative: fixes f :: "real \ 'a::banach" assumes "continuous_on {a..b} f" and "x \ {a..b}" shows "((\u. integral {a..u} f) has_vector_derivative f(x)) (at x within {a..b})" using assms integral_has_vector_derivative_continuous_at [OF integrable_continuous_real] by (fastforce simp: continuous_on_eq_continuous_within) lemma integral_has_real_derivative: assumes "continuous_on {a..b} g" assumes "t \ {a..b}" shows "((\x. integral {a..x} g) has_real_derivative g t) (at t within {a..b})" using integral_has_vector_derivative[of a b g t] assms by (auto simp: has_field_derivative_iff_has_vector_derivative) lemma antiderivative_continuous: fixes q b :: real assumes "continuous_on {a..b} f" obtains g where "\x. x \ {a..b} \ (g has_vector_derivative (f x::_::banach)) (at x within {a..b})" using integral_has_vector_derivative[OF assms] by auto subsection \Combined fundamental theorem of calculus\ lemma antiderivative_integral_continuous: fixes f :: "real \ 'a::banach" assumes "continuous_on {a..b} f" obtains g where "\u\{a..b}. \v \ {a..b}. u \ v \ (f has_integral (g v - g u)) {u..v}" proof - obtain g where g: "\x. x \ {a..b} \ (g has_vector_derivative f x) (at x within {a..b})" using antiderivative_continuous[OF assms] by metis have "(f has_integral g v - g u) {u..v}" if "u \ {a..b}" "v \ {a..b}" "u \ v" for u v proof - have "\x. x \ cbox u v \ (g has_vector_derivative f x) (at x within cbox u v)" by (metis atLeastAtMost_iff atLeastatMost_subset_iff box_real(2) g has_vector_derivative_within_subset subsetCE that(1,2)) then show ?thesis by (metis box_real(2) that(3) fundamental_theorem_of_calculus) qed then show ?thesis using that by blast qed subsection \General "twiddling" for interval-to-interval function image\ lemma has_integral_twiddle: assumes "0 < r" and hg: "\x. h(g x) = x" and gh: "\x. g(h x) = x" and contg: "\x. continuous (at x) g" and g: "\u v. \w z. g ` cbox u v = cbox w z" and h: "\u v. \w z. h ` cbox u v = cbox w z" and r: "\u v. content(g ` cbox u v) = r * content (cbox u v)" and intfi: "(f has_integral i) (cbox a b)" shows "((\x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` cbox a b)" proof (cases "cbox a b = {}") case True then show ?thesis using intfi by auto next case False obtain w z where wz: "h ` cbox a b = cbox w z" using h by blast have inj: "inj g" "inj h" using hg gh injI by metis+ from h obtain ha hb where h_eq: "h ` cbox a b = cbox ha hb" by blast have "\d. gauge d \ (\p. p tagged_division_of h ` cbox a b \ d fine p \ norm ((\(x, k)\p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e)" if "e > 0" for e proof - have "e * r > 0" using that \0 < r\ by simp with intfi[unfolded has_integral] obtain d where "gauge d" and d: "\p. p tagged_division_of cbox a b \ d fine p \ norm ((\(x, k)\p. content k *\<^sub>R f x) - i) < e * r" by metis define d' where "d' x = g -` d (g x)" for x show ?thesis proof (rule_tac x=d' in exI, safe) show "gauge d'" using \gauge d\ continuous_open_vimage[OF _ contg] by (auto simp: gauge_def d'_def) next fix p assume ptag: "p tagged_division_of h ` cbox a b" and finep: "d' fine p" note p = tagged_division_ofD[OF ptag] have gab: "g y \ cbox a b" if "y \ K" "(x, K) \ p" for x y K by (metis hg inj(2) inj_image_mem_iff p(3) subsetCE that that) have gimp: "(\(x,K). (g x, g ` K)) ` p tagged_division_of (cbox a b) \ d fine (\(x, k). (g x, g ` k)) ` p" unfolding tagged_division_of proof safe show "finite ((\(x, k). (g x, g ` k)) ` p)" using ptag by auto show "d fine (\(x, k). (g x, g ` k)) ` p" using finep unfolding fine_def d'_def by auto next fix x k assume xk: "(x, k) \ p" show "g x \ g ` k" using p(2)[OF xk] by auto show "\u v. g ` k = cbox u v" using p(4)[OF xk] using assms(5-6) by auto fix x' K' u assume xk': "(x', K') \ p" and u: "u \ interior (g ` k)" "u \ interior (g ` K')" have "interior k \ interior K' \ {}" proof assume "interior k \ interior K' = {}" moreover have "u \ g ` (interior k \ interior K')" using interior_image_subset[OF \inj g\ contg] u unfolding image_Int[OF inj(1)] by blast ultimately show False by blast qed then have same: "(x, k) = (x', K')" using ptag xk' xk by blast then show "g x = g x'" by auto show "g u \ g ` K'"if "u \ k" for u using that same by auto show "g u \ g ` k"if "u \ K'" for u using that same by auto next fix x assume "x \ cbox a b" then have "h x \ \{k. \x. (x, k) \ p}" using p(6) by auto then obtain X y where "h x \ X" "(y, X) \ p" by blast then show "x \ \{k. \x. (x, k) \ (\(x, k). (g x, g ` k)) ` p}" by clarsimp (metis (no_types, lifting) gh image_eqI pair_imageI) qed (use gab in auto) have *: "inj_on (\(x, k). (g x, g ` k)) p" using inj(1) unfolding inj_on_def by fastforce have "(\(x,K)\(\(y,L). (g y, g ` L)) ` p. content K *\<^sub>R f x) = (\u\p. case case u of (x,K) \ (g x, g ` K) of (y,L) \ content L *\<^sub>R f y)" by (metis (mono_tags, lifting) "*" sum.reindex_cong) also have "... = (\(x,K)\p. r *\<^sub>R content K *\<^sub>R f (g x))" using r by (auto intro!: * sum.cong simp: bij_betw_def dest!: p(4)) finally have "(\(x, K)\(\(x,K). (g x, g ` K)) ` p. content K *\<^sub>R f x) - i = r *\<^sub>R (\(x,K)\p. content K *\<^sub>R f (g x)) - i" by (simp add: scaleR_right.sum split_def) also have "\ = r *\<^sub>R ((\(x,K)\p. content K *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i)" using \0 < r\ by (auto simp: scaleR_diff_right) finally show "norm ((\(x,K)\p. content K *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e" using d[OF gimp] \0 < r\ by auto qed qed then show ?thesis by (auto simp: h_eq has_integral) qed subsection \Special case of a basic affine transformation\ lemma AE_lborel_inner_neq: assumes k: "k \ Basis" shows "AE x in lborel. x \ k \ c" proof - interpret finite_product_sigma_finite "\_. lborel" Basis proof qed simp have "emeasure lborel {x\space lborel. x \ k = c} = emeasure (\\<^sub>M j::'a\Basis. lborel) (\\<^sub>E j\Basis. if j = k then {c} else UNIV)" using k by (auto simp add: lborel_eq[where 'a='a] emeasure_distr intro!: arg_cong2[where f=emeasure]) (auto simp: space_PiM PiE_iff extensional_def split: if_split_asm) also have "\ = (\j\Basis. emeasure lborel (if j = k then {c} else UNIV))" by (intro measure_times) auto also have "\ = 0" by (intro prod_zero bexI[OF _ k]) auto finally show ?thesis by (subst AE_iff_measurable[OF _ refl]) auto qed lemma content_image_stretch_interval: fixes m :: "'a::euclidean_space \ real" defines "s f x \ (\k::'a\Basis. (f k * (x\k)) *\<^sub>R k)" shows "content (s m ` cbox a b) = \\k\Basis. m k\ * content (cbox a b)" proof cases have s[measurable]: "s f \ borel \\<^sub>M borel" for f by (auto simp: s_def[abs_def]) assume m: "\k\Basis. m k \ 0" then have s_comp_s: "s (\k. 1 / m k) \ s m = id" "s m \ s (\k. 1 / m k) = id" by (auto simp: s_def[abs_def] fun_eq_iff euclidean_representation) then have "inv (s (\k. 1 / m k)) = s m" "bij (s (\k. 1 / m k))" by (auto intro: inv_unique_comp o_bij) then have eq: "s m ` cbox a b = s (\k. 1 / m k) -` cbox a b" using bij_vimage_eq_inv_image[OF \bij (s (\k. 1 / m k))\, of "cbox a b"] by auto show ?thesis using m unfolding eq measure_def by (subst lborel_affine_euclidean[where c=m and t=0]) (simp_all add: emeasure_density measurable_sets_borel[OF s] abs_prod nn_integral_cmult s_def[symmetric] emeasure_distr vimage_comp s_comp_s enn2real_mult prod_nonneg) next assume "\ (\k\Basis. m k \ 0)" then obtain k where k: "k \ Basis" "m k = 0" by auto then have [simp]: "(\k\Basis. m k) = 0" by (intro prod_zero) auto have "emeasure lborel {x\space lborel. x \ s m ` cbox a b} = 0" proof (rule emeasure_eq_0_AE) show "AE x in lborel. x \ s m ` cbox a b" using AE_lborel_inner_neq[OF \k\Basis\] proof eventually_elim show "x \ k \ 0 \ x \ s m ` cbox a b " for x using k by (auto simp: s_def[abs_def] cbox_def) qed qed then show ?thesis by (simp add: measure_def) qed lemma interval_image_affinity_interval: "\u v. (\x. m *\<^sub>R (x::'a::euclidean_space) + c) ` cbox a b = cbox u v" unfolding image_affinity_cbox by auto lemma content_image_affinity_cbox: "content((\x::'a::euclidean_space. m *\<^sub>R x + c) ` cbox a b) = \m\ ^ DIM('a) * content (cbox a b)" (is "?l = ?r") proof (cases "cbox a b = {}") case True then show ?thesis by simp next case False show ?thesis proof (cases "m \ 0") case True with \cbox a b \ {}\ have "cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c) \ {}" by (simp add: box_ne_empty inner_left_distrib mult_left_mono) moreover from True have *: "\i. (m *\<^sub>R b + c) \ i - (m *\<^sub>R a + c) \ i = m *\<^sub>R (b-a) \ i" by (simp add: inner_simps field_simps) ultimately show ?thesis by (simp add: image_affinity_cbox True content_cbox' prod.distrib inner_diff_left) next case False with \cbox a b \ {}\ have "cbox (m *\<^sub>R b + c) (m *\<^sub>R a + c) \ {}" by (simp add: box_ne_empty inner_left_distrib mult_left_mono) moreover from False have *: "\i. (m *\<^sub>R a + c) \ i - (m *\<^sub>R b + c) \ i = (-m) *\<^sub>R (b-a) \ i" by (simp add: inner_simps field_simps) ultimately show ?thesis using False by (simp add: image_affinity_cbox content_cbox' prod.distrib[symmetric] inner_diff_left flip: prod_constant) qed qed lemma has_integral_affinity: fixes a :: "'a::euclidean_space" assumes "(f has_integral i) (cbox a b)" and "m \ 0" shows "((\x. f(m *\<^sub>R x + c)) has_integral (1 / (\m\ ^ DIM('a))) *\<^sub>R i) ((\x. (1 / m) *\<^sub>R x + -((1 / m) *\<^sub>R c)) ` cbox a b)" proof (rule has_integral_twiddle) show "\w z. (\x. (1 / m) *\<^sub>R x + - ((1 / m) *\<^sub>R c)) ` cbox u v = cbox w z" "\w z. (\x. m *\<^sub>R x + c) ` cbox u v = cbox w z" for u v using interval_image_affinity_interval by blast+ show "content ((\x. m *\<^sub>R x + c) ` cbox u v) = \m\ ^ DIM('a) * content (cbox u v)" for u v using content_image_affinity_cbox by blast qed (use assms zero_less_power in \auto simp: field_simps\) lemma integrable_affinity: assumes "f integrable_on cbox a b" and "m \ 0" shows "(\x. f(m *\<^sub>R x + c)) integrable_on ((\x. (1 / m) *\<^sub>R x + -((1/m) *\<^sub>R c)) ` cbox a b)" using has_integral_affinity assms unfolding integrable_on_def by blast lemmas has_integral_affinity01 = has_integral_affinity [of _ _ 0 "1::real", simplified] lemma integrable_on_affinity: assumes "m \ 0" "f integrable_on (cbox a b)" shows "(\x. f (m *\<^sub>R x + c)) integrable_on ((\x. (1 / m) *\<^sub>R x - ((1 / m) *\<^sub>R c)) ` cbox a b)" proof - from assms obtain I where "(f has_integral I) (cbox a b)" by (auto simp: integrable_on_def) from has_integral_affinity[OF this assms(1), of c] show ?thesis by (auto simp: integrable_on_def) qed lemma has_integral_cmul_iff: assumes "c \ 0" shows "((\x. c *\<^sub>R f x) has_integral (c *\<^sub>R I)) A \ (f has_integral I) A" using assms has_integral_cmul[of f I A c] has_integral_cmul[of "\x. c *\<^sub>R f x" "c *\<^sub>R I" A "inverse c"] by (auto simp: field_simps) lemma has_integral_affinity': fixes a :: "'a::euclidean_space" assumes "(f has_integral i) (cbox a b)" and "m > 0" shows "((\x. f(m *\<^sub>R x + c)) has_integral (i /\<^sub>R m ^ DIM('a))) (cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m))" proof (cases "cbox a b = {}") case True hence "(cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m)) = {}" using \m > 0\ unfolding box_eq_empty by (auto simp: algebra_simps) with True and assms show ?thesis by simp next case False have "((\x. f (m *\<^sub>R x + c)) has_integral (1 / \m\ ^ DIM('a)) *\<^sub>R i) ((\x. (1 / m) *\<^sub>R x + - ((1 / m) *\<^sub>R c)) ` cbox a b)" using assms by (intro has_integral_affinity) auto also have "((\x. (1 / m) *\<^sub>R x + - ((1 / m) *\<^sub>R c)) ` cbox a b) = ((\x. - ((1 / m) *\<^sub>R c) + x) ` (\x. (1 / m) *\<^sub>R x) ` cbox a b)" by (simp add: image_image algebra_simps) also have "(\x. (1 / m) *\<^sub>R x) ` cbox a b = cbox ((1 / m) *\<^sub>R a) ((1 / m) *\<^sub>R b)" using \m > 0\ False by (subst image_smult_cbox) simp_all also have "(\x. - ((1 / m) *\<^sub>R c) + x) ` \ = cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m)" by (subst cbox_translation [symmetric]) (simp add: field_simps vector_add_divide_simps) finally show ?thesis using \m > 0\ by (simp add: field_simps) qed lemma has_integral_affinity_iff: fixes f :: "'a :: euclidean_space \ 'b :: real_normed_vector" assumes "m > 0" shows "((\x. f (m *\<^sub>R x + c)) has_integral (I /\<^sub>R m ^ DIM('a))) (cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m)) \ (f has_integral I) (cbox a b)" (is "?lhs = ?rhs") proof assume ?lhs from has_integral_affinity'[OF this, of "1 / m" "-c /\<^sub>R m"] and \m > 0\ show ?rhs by (simp add: vector_add_divide_simps) (simp add: field_simps) next assume ?rhs from has_integral_affinity'[OF this, of m c] and \m > 0\ show ?lhs by simp qed subsection \Special case of stretching coordinate axes separately\ lemma has_integral_stretch: fixes f :: "'a::euclidean_space \ 'b::real_normed_vector" assumes "(f has_integral i) (cbox a b)" and "\k\Basis. m k \ 0" shows "((\x. f (\k\Basis. (m k * (x\k))*\<^sub>R k)) has_integral ((1/ \prod m Basis\) *\<^sub>R i)) ((\x. (\k\Basis. (1 / m k * (x\k))*\<^sub>R k)) ` cbox a b)" apply (rule has_integral_twiddle[where f=f]) unfolding zero_less_abs_iff content_image_stretch_interval unfolding image_stretch_interval empty_as_interval euclidean_eq_iff[where 'a='a] using assms by auto lemma has_integral_stretch_real: fixes f :: "real \ 'b::real_normed_vector" assumes "(f has_integral i) {a..b}" and "m \ 0" shows "((\x. f (m * x)) has_integral (1 / \m\) *\<^sub>R i) ((\x. x / m) ` {a..b})" using has_integral_stretch [of f i a b "\b. m"] assms by simp lemma integrable_stretch: fixes f :: "'a::euclidean_space \ 'b::real_normed_vector" assumes "f integrable_on cbox a b" and "\k\Basis. m k \ 0" shows "(\x::'a. f (\k\Basis. (m k * (x\k))*\<^sub>R k)) integrable_on ((\x. \k\Basis. (1 / m k * (x\k))*\<^sub>R k) ` cbox a b)" using assms unfolding integrable_on_def by (force dest: has_integral_stretch) lemma vec_lambda_eq_sum: "(\ k. f k (x $ k)) = (\k\Basis. (f (axis_index k) (x \ k)) *\<^sub>R k)" (is "?lhs = ?rhs") proof - have "?lhs = (\ k. f k (x \ axis k 1))" by (simp add: cart_eq_inner_axis) also have "... = (\u\UNIV. f u (x \ axis u 1) *\<^sub>R axis u 1)" by (simp add: vec_eq_iff axis_def if_distrib cong: if_cong) also have "... = ?rhs" by (simp add: Basis_vec_def UNION_singleton_eq_range sum.reindex axis_eq_axis inj_on_def) finally show ?thesis . qed lemma has_integral_stretch_cart: fixes m :: "'n::finite \ real" assumes f: "(f has_integral i) (cbox a b)" and m: "\k. m k \ 0" shows "((\x. f(\ k. m k * x$k)) has_integral i /\<^sub>R \prod m UNIV\) ((\x. \ k. x$k / m k) ` (cbox a b))" proof - have *: "\k:: real^'n \ Basis. m (axis_index k) \ 0" using axis_index by (simp add: m) have eqp: "(\k:: real^'n \ Basis. m (axis_index k)) = prod m UNIV" by (simp add: Basis_vec_def UNION_singleton_eq_range prod.reindex axis_eq_axis inj_on_def) show ?thesis using has_integral_stretch [OF f *] vec_lambda_eq_sum [where f="\i x. m i * x"] vec_lambda_eq_sum [where f="\i x. x / m i"] by (simp add: field_simps eqp) qed lemma image_stretch_interval_cart: fixes m :: "'n::finite \ real" shows "(\x. \ k. m k * x$k) ` cbox a b = (if cbox a b = {} then {} else cbox (\ k. min (m k * a$k) (m k * b$k)) (\ k. max (m k * a$k) (m k * b$k)))" proof - have *: "(\k\Basis. min (m (axis_index k) * (a \ k)) (m (axis_index k) * (b \ k)) *\<^sub>R k) = (\ k. min (m k * a $ k) (m k * b $ k))" "(\k\Basis. max (m (axis_index k) * (a \ k)) (m (axis_index k) * (b \ k)) *\<^sub>R k) = (\ k. max (m k * a $ k) (m k * b $ k))" apply (simp_all add: Basis_vec_def cart_eq_inner_axis UNION_singleton_eq_range sum.reindex axis_eq_axis inj_on_def) apply (simp_all add: vec_eq_iff axis_def if_distrib cong: if_cong) done show ?thesis by (simp add: vec_lambda_eq_sum [where f="\i x. m i * x"] image_stretch_interval eq_cbox *) qed subsection \even more special cases\ lemma uminus_interval_vector[simp]: fixes a b :: "'a::euclidean_space" shows "uminus ` cbox a b = cbox (-b) (-a)" proof - have "x \ uminus ` cbox a b" if "x \ cbox (- b) (- a)" for x proof - have "-x \ cbox a b" using that by (auto simp: mem_box) then show ?thesis by force qed then show ?thesis by (auto simp: mem_box) qed lemma has_integral_reflect_lemma[intro]: assumes "(f has_integral i) (cbox a b)" shows "((\x. f(-x)) has_integral i) (cbox (-b) (-a))" using has_integral_affinity[OF assms, of "-1" 0] by auto lemma has_integral_reflect_lemma_real[intro]: assumes "(f has_integral i) {a..b::real}" shows "((\x. f(-x)) has_integral i) {-b .. -a}" using assms unfolding box_real[symmetric] by (rule has_integral_reflect_lemma) lemma has_integral_reflect[simp]: "((\x. f (-x)) has_integral i) (cbox (-b) (-a)) \ (f has_integral i) (cbox a b)" by (auto dest: has_integral_reflect_lemma) lemma has_integral_reflect_real[simp]: fixes a b::real shows "((\x. f (-x)) has_integral i) {-b..-a} \ (f has_integral i) {a..b}" by (metis has_integral_reflect interval_cbox) lemma integrable_reflect[simp]: "(\x. f(-x)) integrable_on cbox (-b) (-a) \ f integrable_on cbox a b" unfolding integrable_on_def by auto lemma integrable_reflect_real[simp]: "(\x. f(-x)) integrable_on {-b .. -a} \ f integrable_on {a..b::real}" unfolding box_real[symmetric] by (rule integrable_reflect) lemma integral_reflect[simp]: "integral (cbox (-b) (-a)) (\x. f (-x)) = integral (cbox a b) f" unfolding integral_def by auto lemma integral_reflect_real[simp]: "integral {-b .. -a} (\x. f (-x)) = integral {a..b::real} f" unfolding box_real[symmetric] by (rule integral_reflect) subsection \Stronger form of FCT; quite a tedious proof\ lemma split_minus[simp]: "(\(x, k). f x k) x - (\(x, k). g x k) x = (\(x, k). f x k - g x k) x" by (simp add: split_def) theorem fundamental_theorem_of_calculus_interior: fixes f :: "real \ 'a::real_normed_vector" assumes "a \ b" and contf: "continuous_on {a..b} f" and derf: "\x. x \ {a <..< b} \ (f has_vector_derivative f' x) (at x)" shows "(f' has_integral (f b - f a)) {a..b}" proof (cases "a = b") case True then have *: "cbox a b = {b}" "f b - f a = 0" by (auto simp add: order_antisym) with True show ?thesis by auto next case False with \a \ b\ have ab: "a < b" by arith show ?thesis unfolding has_integral_factor_content_real proof (intro allI impI) fix e :: real assume e: "e > 0" then have eba8: "(e * (b-a)) / 8 > 0" using ab by (auto simp add: field_simps) note derf_exp = derf[unfolded has_vector_derivative_def has_derivative_at_alt, THEN conjunct2, rule_format] have bounded: "\x. x \ {a<.. bounded_linear (\u. u *\<^sub>R f' x)" by (simp add: bounded_linear_scaleR_left) have "\x \ box a b. \d>0. \y. norm (y-x) < d \ norm (f y - f x - (y-x) *\<^sub>R f' x) \ e/2 * norm (y-x)" (is "\x \ box a b. ?Q x") \\The explicit quantifier is required by the following step\ proof fix x assume "x \ box a b" with e show "?Q x" using derf_exp [of x "e/2"] by auto qed then obtain d where d: "\x. 0 < d x" "\x y. \x \ box a b; norm (y-x) < d x\ \ norm (f y - f x - (y-x) *\<^sub>R f' x) \ e/2 * norm (y-x)" unfolding bgauge_existence_lemma by metis have "bounded (f ` cbox a b)" using compact_cbox assms by (auto simp: compact_imp_bounded compact_continuous_image) then obtain B where "0 < B" and B: "\x. x \ f ` cbox a b \ norm x \ B" unfolding bounded_pos by metis obtain da where "0 < da" and da: "\c. \a \ c; {a..c} \ {a..b}; {a..c} \ ball a da\ \ norm (content {a..c} *\<^sub>R f' a - (f c - f a)) \ (e * (b-a)) / 4" proof - have "continuous (at a within {a..b}) f" using contf continuous_on_eq_continuous_within by force with eba8 obtain k where "0 < k" and k: "\x. \x \ {a..b}; 0 < norm (x-a); norm (x-a) < k\ \ norm (f x - f a) < e * (b-a) / 8" unfolding continuous_within Lim_within dist_norm by metis obtain l where l: "0 < l" "norm (l *\<^sub>R f' a) \ e * (b-a) / 8" proof (cases "f' a = 0") case True with ab e that show ?thesis by auto next case False show ?thesis proof show "norm ((e * (b - a) / 8 / norm (f' a)) *\<^sub>R f' a) \ e * (b - a) / 8" "0 < e * (b - a) / 8 / norm (f' a)" using False ab e by (auto simp add: field_simps) qed qed have "norm (content {a..c} *\<^sub>R f' a - (f c - f a)) \ e * (b-a) / 4" if "a \ c" "{a..c} \ {a..b}" and bmin: "{a..c} \ ball a (min k l)" for c proof - have minkl: "\a - x\ < min k l" if "x \ {a..c}" for x using bmin dist_real_def that by auto then have lel: "\c - a\ \ \l\" using that by force have "norm ((c - a) *\<^sub>R f' a - (f c - f a)) \ norm ((c - a) *\<^sub>R f' a) + norm (f c - f a)" by (rule norm_triangle_ineq4) also have "\ \ e * (b-a) / 8 + e * (b-a) / 8" proof (rule add_mono) have "norm ((c - a) *\<^sub>R f' a) \ norm (l *\<^sub>R f' a)" by (auto intro: mult_right_mono [OF lel]) also have "... \ e * (b-a) / 8" by (rule l) finally show "norm ((c - a) *\<^sub>R f' a) \ e * (b-a) / 8" . next have "norm (f c - f a) < e * (b-a) / 8" proof (cases "a = c") case True then show ?thesis using eba8 by auto next case False show ?thesis by (rule k) (use minkl \a \ c\ that False in auto) qed then show "norm (f c - f a) \ e * (b-a) / 8" by simp qed finally show "norm (content {a..c} *\<^sub>R f' a - (f c - f a)) \ e * (b-a) / 4" unfolding content_real[OF \a \ c\] by auto qed then show ?thesis by (rule_tac da="min k l" in that) (auto simp: l \0 < k\) qed obtain db where "0 < db" and db: "\c. \c \ b; {c..b} \ {a..b}; {c..b} \ ball b db\ \ norm (content {c..b} *\<^sub>R f' b - (f b - f c)) \ (e * (b-a)) / 4" proof - have "continuous (at b within {a..b}) f" using contf continuous_on_eq_continuous_within by force with eba8 obtain k where "0 < k" and k: "\x. \x \ {a..b}; 0 < norm(x-b); norm(x-b) < k\ \ norm (f b - f x) < e * (b-a) / 8" unfolding continuous_within Lim_within dist_norm norm_minus_commute by metis obtain l where l: "0 < l" "norm (l *\<^sub>R f' b) \ (e * (b-a)) / 8" proof (cases "f' b = 0") case True thus ?thesis using ab e that by auto next case False show ?thesis proof show "norm ((e * (b - a) / 8 / norm (f' b)) *\<^sub>R f' b) \ e * (b - a) / 8" "0 < e * (b - a) / 8 / norm (f' b)" using False ab e by (auto simp add: field_simps) qed qed have "norm (content {c..b} *\<^sub>R f' b - (f b - f c)) \ e * (b-a) / 4" if "c \ b" "{c..b} \ {a..b}" and bmin: "{c..b} \ ball b (min k l)" for c proof - have minkl: "\b - x\ < min k l" if "x \ {c..b}" for x using bmin dist_real_def that by auto then have lel: "\b - c\ \ \l\" using that by force have "norm ((b - c) *\<^sub>R f' b - (f b - f c)) \ norm ((b - c) *\<^sub>R f' b) + norm (f b - f c)" by (rule norm_triangle_ineq4) also have "\ \ e * (b-a) / 8 + e * (b-a) / 8" proof (rule add_mono) have "norm ((b - c) *\<^sub>R f' b) \ norm (l *\<^sub>R f' b)" by (auto intro: mult_right_mono [OF lel]) also have "... \ e * (b-a) / 8" by (rule l) finally show "norm ((b - c) *\<^sub>R f' b) \ e * (b-a) / 8" . next have "norm (f b - f c) < e * (b-a) / 8" proof (cases "b = c") case True with eba8 show ?thesis by auto next case False show ?thesis by (rule k) (use minkl \c \ b\ that False in auto) qed then show "norm (f b - f c) \ e * (b-a) / 8" by simp qed finally show "norm (content {c..b} *\<^sub>R f' b - (f b - f c)) \ e * (b-a) / 4" unfolding content_real[OF \c \ b\] by auto qed then show ?thesis by (rule_tac db="min k l" in that) (auto simp: l \0 < k\) qed let ?d = "(\x. ball x (if x=a then da else if x=b then db else d x))" show "\d. gauge d \ (\p. p tagged_division_of {a..b} \ d fine p \ norm ((\(x,K)\p. content K *\<^sub>R f' x) - (f b - f a)) \ e * content {a..b})" proof (rule exI, safe) show "gauge ?d" using ab \db > 0\ \da > 0\ d(1) by (auto intro: gauge_ball_dependent) next fix p assume ptag: "p tagged_division_of {a..b}" and fine: "?d fine p" let ?A = "{t. fst t \ {a, b}}" note p = tagged_division_ofD[OF ptag] have pA: "p = (p \ ?A) \ (p - ?A)" "finite (p \ ?A)" "finite (p - ?A)" "(p \ ?A) \ (p - ?A) = {}" using ptag fine by auto have le_xz: "\w x y z::real. y \ z/2 \ w - x \ z/2 \ w + y \ x + z" by arith have non: False if xk: "(x,K) \ p" and "x \ a" "x \ b" and less: "e * (Sup K - Inf K)/2 < norm (content K *\<^sub>R f' x - (f (Sup K) - f (Inf K)))" for x K proof - obtain u v where k: "K = cbox u v" using p(4) xk by blast then have "u \ v" and uv: "{u, v} \ cbox u v" using p(2)[OF xk] by auto then have result: "e * (v - u)/2 < norm ((v - u) *\<^sub>R f' x - (f v - f u))" using less[unfolded k box_real interval_bounds_real content_real] by auto then have "x \ box a b" using p(2) p(3) \x \ a\ \x \ b\ xk by fastforce with d have *: "\y. norm (y-x) < d x \ norm (f y - f x - (y-x) *\<^sub>R f' x) \ e/2 * norm (y-x)" by metis have xd: "norm (u - x) < d x" "norm (v - x) < d x" using fineD[OF fine xk] \x \ a\ \x \ b\ uv by (auto simp add: k subset_eq dist_commute dist_real_def) have "norm ((v - u) *\<^sub>R f' x - (f v - f u)) = norm ((f u - f x - (u - x) *\<^sub>R f' x) - (f v - f x - (v - x) *\<^sub>R f' x))" by (rule arg_cong[where f=norm]) (auto simp: scaleR_left.diff) also have "\ \ e/2 * norm (u - x) + e/2 * norm (v - x)" by (metis norm_triangle_le_diff add_mono * xd) also have "\ \ e/2 * norm (v - u)" using p(2)[OF xk] by (auto simp add: field_simps k) also have "\ < norm ((v - u) *\<^sub>R f' x - (f v - f u))" using result by (simp add: \u \ v\) finally have "e * (v - u)/2 < e * (v - u)/2" using uv by auto then show False by auto qed have "norm (\(x, K)\p - ?A. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) \ (\(x, K)\p - ?A. norm (content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))))" by (auto intro: sum_norm_le) also have "... \ (\n\p - ?A. e * (case n of (x, k) \ Sup k - Inf k)/2)" using non by (fastforce intro: sum_mono) finally have I: "norm (\(x, k)\p - ?A. content k *\<^sub>R f' x - (f (Sup k) - f (Inf k))) \ (\n\p - ?A. e * (case n of (x, k) \ Sup k - Inf k))/2" by (simp add: sum_divide_distrib) have II: "norm (\(x, k)\p \ ?A. content k *\<^sub>R f' x - (f (Sup k) - f (Inf k))) - (\n\p \ ?A. e * (case n of (x, k) \ Sup k - Inf k)) \ (\n\p - ?A. e * (case n of (x, k) \ Sup k - Inf k))/2" proof - have ge0: "0 \ e * (Sup k - Inf k)" if xkp: "(x, k) \ p \ ?A" for x k proof - obtain u v where uv: "k = cbox u v" by (meson Int_iff xkp p(4)) with p(2) that uv have "cbox u v \ {}" by blast then show "0 \ e * ((Sup k) - (Inf k))" unfolding uv using e by (auto simp add: field_simps) qed let ?B = "\x. {t \ p. fst t = x \ content (snd t) \ 0}" let ?C = "{t \ p. fst t \ {a, b} \ content (snd t) \ 0}" have "norm (\(x, k)\p \ {t. fst t \ {a, b}}. content k *\<^sub>R f' x - (f (Sup k) - f (Inf k))) \ e * (b-a)/2" proof - have *: "\S f e. sum f S = sum f (p \ ?C) \ norm (sum f (p \ ?C)) \ e \ norm (sum f S) \ e" by auto have 1: "content K *\<^sub>R (f' x) - (f ((Sup K)) - f ((Inf K))) = 0" if "(x,K) \ p \ {t. fst t \ {a, b}} - p \ ?C" for x K proof - have xk: "(x,K) \ p" and k0: "content K = 0" using that by auto then obtain u v where uv: "K = cbox u v" using p(4) by blast then have "u = v" using xk k0 p(2) by force then show "content K *\<^sub>R (f' x) - (f ((Sup K)) - f ((Inf K))) = 0" using xk unfolding uv by auto qed have 2: "norm(\(x,K)\p \ ?C. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) \ e * (b-a)/2" proof - have norm_le: "norm (sum f S) \ e" if "\x y. \x \ S; y \ S\ \ x = y" "\x. x \ S \ norm (f x) \ e" "e > 0" for S f and e :: real proof (cases "S = {}") case True with that show ?thesis by auto next case False then obtain x where "x \ S" by auto then have "S = {x}" using that(1) by auto then show ?thesis using \x \ S\ that(2) by auto qed have *: "p \ ?C = ?B a \ ?B b" by blast then have "norm (\(x,K)\p \ ?C. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) = norm (\(x,K) \ ?B a \ ?B b. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K)))" by simp also have "... = norm ((\(x,K) \ ?B a. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) + (\(x,K) \ ?B b. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))))" using p(1) ab e by (subst sum.union_disjoint) auto also have "... \ e * (b - a) / 4 + e * (b - a) / 4" proof (rule norm_triangle_le [OF add_mono]) have pa: "\v. k = cbox a v \ a \ v" if "(a, k) \ p" for k using p(2) p(3) p(4) that by fastforce show "norm (\(x,K) \ ?B a. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) \ e * (b - a) / 4" proof (intro norm_le; clarsimp) fix K K' assume K: "(a, K) \ p" "(a, K') \ p" and ne0: "content K \ 0" "content K' \ 0" with pa obtain v v' where v: "K = cbox a v" "a \ v" and v': "K' = cbox a v'" "a \ v'" by blast let ?v = "min v v'" have "box a ?v \ K \ K'" unfolding v v' by (auto simp add: mem_box) then have "interior (box a (min v v')) \ interior K \ interior K'" using interior_Int interior_mono by blast moreover have "(a + ?v)/2 \ box a ?v" using ne0 unfolding v v' content_eq_0 not_le by (auto simp add: mem_box) ultimately have "(a + ?v)/2 \ interior K \ interior K'" unfolding interior_open[OF open_box] by auto then show "K = K'" using p(5)[OF K] by auto next fix K assume K: "(a, K) \ p" and ne0: "content K \ 0" show "norm (content c *\<^sub>R f' a - (f (Sup c) - f (Inf c))) * 4 \ e * (b-a)" if "(a, c) \ p" and ne0: "content c \ 0" for c proof - obtain v where v: "c = cbox a v" and "a \ v" using pa[OF \(a, c) \ p\] by metis then have "a \ {a..v}" "v \ b" using p(3)[OF \(a, c) \ p\] by auto moreover have "{a..v} \ ball a da" using fineD[OF \?d fine p\ \(a, c) \ p\] by (simp add: v split: if_split_asm) ultimately show ?thesis unfolding v interval_bounds_real[OF \a \ v\] box_real using da \a \ v\ by auto qed qed (use ab e in auto) next have pb: "\v. k = cbox v b \ b \ v" if "(b, k) \ p" for k using p(2) p(3) p(4) that by fastforce show "norm (\(x,K) \ ?B b. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) \ e * (b - a) / 4" proof (intro norm_le; clarsimp) fix K K' assume K: "(b, K) \ p" "(b, K') \ p" and ne0: "content K \ 0" "content K' \ 0" with pb obtain v v' where v: "K = cbox v b" "v \ b" and v': "K' = cbox v' b" "v' \ b" by blast let ?v = "max v v'" have "box ?v b \ K \ K'" unfolding v v' by (auto simp: mem_box) then have "interior (box (max v v') b) \ interior K \ interior K'" using interior_Int interior_mono by blast moreover have " ((b + ?v)/2) \ box ?v b" using ne0 unfolding v v' content_eq_0 not_le by (auto simp: mem_box) ultimately have "((b + ?v)/2) \ interior K \ interior K'" unfolding interior_open[OF open_box] by auto then show "K = K'" using p(5)[OF K] by auto next fix K assume K: "(b, K) \ p" and ne0: "content K \ 0" show "norm (content c *\<^sub>R f' b - (f (Sup c) - f (Inf c))) * 4 \ e * (b-a)" if "(b, c) \ p" and ne0: "content c \ 0" for c proof - obtain v where v: "c = cbox v b" and "v \ b" using \(b, c) \ p\ pb by blast then have "v \ a""b \ {v.. b}" using p(3)[OF \(b, c) \ p\] by auto moreover have "{v..b} \ ball b db" using fineD[OF \?d fine p\ \(b, c) \ p\] box_real(2) v False by force ultimately show ?thesis using db v by auto qed qed (use ab e in auto) qed also have "... = e * (b-a)/2" by simp finally show "norm (\(x,k)\p \ ?C. content k *\<^sub>R f' x - (f (Sup k) - f (Inf k))) \ e * (b-a)/2" . qed show "norm (\(x, k)\p \ ?A. content k *\<^sub>R f' x - (f ((Sup k)) - f ((Inf k)))) \ e * (b-a)/2" apply (rule * [OF sum.mono_neutral_right[OF pA(2)]]) using 1 2 by (auto simp: split_paired_all) qed also have "... = (\n\p. e * (case n of (x, k) \ Sup k - Inf k))/2" unfolding sum_distrib_left[symmetric] by (subst additive_tagged_division_1[OF \a \ b\ ptag]) auto finally have norm_le: "norm (\(x,K)\p \ {t. fst t \ {a, b}}. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) \ (\n\p. e * (case n of (x, K) \ Sup K - Inf K))/2" . have le2: "\x s1 s2::real. 0 \ s1 \ x \ (s1 + s2)/2 \ x - s1 \ s2/2" by auto show ?thesis apply (rule le2 [OF sum_nonneg]) using ge0 apply force by (metis (no_types, lifting) Diff_Diff_Int Diff_subset norm_le p(1) sum.subset_diff) qed note * = additive_tagged_division_1[OF assms(1) ptag, symmetric] have "norm (\(x,K)\p \ ?A \ (p - ?A). content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) \ e * (\(x,K)\p \ ?A \ (p - ?A). Sup K - Inf K)" unfolding sum_distrib_left unfolding sum.union_disjoint[OF pA(2-)] using le_xz norm_triangle_le I II by blast then show "norm ((\(x,K)\p. content K *\<^sub>R f' x) - (f b - f a)) \ e * content {a..b}" by (simp only: content_real[OF \a \ b\] *[of "\x. x"] *[of f] sum_subtractf[symmetric] split_minus pA(1) [symmetric]) qed qed qed subsection \Stronger form with finite number of exceptional points\ lemma fundamental_theorem_of_calculus_interior_strong: fixes f :: "real \ 'a::banach" assumes "finite S" and "a \ b" "\x. x \ {a <..< b} - S \ (f has_vector_derivative f'(x)) (at x)" and "continuous_on {a .. b} f" shows "(f' has_integral (f b - f a)) {a .. b}" using assms proof (induction arbitrary: a b) case empty then show ?case using fundamental_theorem_of_calculus_interior by force next case (insert x S) show ?case proof (cases "x \ {a<..continuous_on {a..b} f\ \a < x\ \x < b\ continuous_on_subset by (force simp: intro!: insert)+ then have "(f' has_integral f x - f a + (f b - f x)) {a..b}" using \a < x\ \x < b\ has_integral_combine less_imp_le by blast then show ?thesis by simp qed qed corollary fundamental_theorem_of_calculus_strong: fixes f :: "real \ 'a::banach" assumes "finite S" and "a \ b" and vec: "\x. x \ {a..b} - S \ (f has_vector_derivative f'(x)) (at x)" and "continuous_on {a..b} f" shows "(f' has_integral (f b - f a)) {a..b}" by (rule fundamental_theorem_of_calculus_interior_strong [OF \finite S\]) (force simp: assms)+ proposition indefinite_integral_continuous_left: fixes f:: "real \ 'a::banach" assumes intf: "f integrable_on {a..b}" and "a < c" "c \ b" "e > 0" obtains d where "d > 0" and "\t. c - d < t \ t \ c \ norm (integral {a..c} f - integral {a..t} f) < e" proof - obtain w where "w > 0" and w: "\t. \c - w < t; t < c\ \ norm (f c) * norm(c - t) < e/3" proof (cases "f c = 0") case False hence e3: "0 < e/3 / norm (f c)" using \e>0\ by simp moreover have "norm (f c) * norm (c - t) < e/3" if "t < c" and "c - e/3 / norm (f c) < t" for t proof - have "norm (c - t) < e/3 / norm (f c)" using that by auto then show "norm (f c) * norm (c - t) < e/3" by (metis e3 mult.commute norm_not_less_zero pos_less_divide_eq zero_less_divide_iff) qed ultimately show ?thesis using that by auto next case True then show ?thesis using \e > 0\ that by auto qed let ?SUM = "\p. (\(x,K) \ p. content K *\<^sub>R f x)" have e3: "e/3 > 0" using \e > 0\ by auto have "f integrable_on {a..c}" using \a < c\ \c \ b\ by (auto intro: integrable_subinterval_real[OF intf]) then obtain d1 where "gauge d1" and d1: "\p. \p tagged_division_of {a..c}; d1 fine p\ \ norm (?SUM p - integral {a..c} f) < e/3" using integrable_integral has_integral_real e3 by metis define d where [abs_def]: "d x = ball x w \ d1 x" for x have "gauge d" unfolding d_def using \w > 0\ \gauge d1\ by auto then obtain k where "0 < k" and k: "ball c k \ d c" by (meson gauge_def open_contains_ball) let ?d = "min k (c - a)/2" show thesis proof (intro that[of ?d] allI impI, safe) show "?d > 0" using \0 < k\ \a < c\ by auto next fix t assume t: "c - ?d < t" "t \ c" show "norm (integral ({a..c}) f - integral ({a..t}) f) < e" proof (cases "t < c") case False with \t \ c\ show ?thesis by (simp add: \e > 0\) next case True have "f integrable_on {a..t}" using \t < c\ \c \ b\ by (auto intro: integrable_subinterval_real[OF intf]) then obtain d2 where d2: "gauge d2" "\p. p tagged_division_of {a..t} \ d2 fine p \ norm (?SUM p - integral {a..t} f) < e/3" using integrable_integral has_integral_real e3 by metis define d3 where "d3 x = (if x \ t then d1 x \ d2 x else d1 x)" for x have "gauge d3" using \gauge d1\ \gauge d2\ unfolding d3_def gauge_def by auto then obtain p where ptag: "p tagged_division_of {a..t}" and pfine: "d3 fine p" by (metis box_real(2) fine_division_exists) note p' = tagged_division_ofD[OF ptag] have pt: "(x,K)\p \ x \ t" for x K by (meson atLeastAtMost_iff p'(2) p'(3) subsetCE) with pfine have "d2 fine p" unfolding fine_def d3_def by fastforce then have d2_fin: "norm (?SUM p - integral {a..t} f) < e/3" using d2(2) ptag by auto have eqs: "{a..c} \ {x. x \ t} = {a..t}" "{a..c} \ {x. x \ t} = {t..c}" using t by (auto simp add: field_simps) have "p \ {(c, {t..c})} tagged_division_of {a..c}" proof (intro tagged_division_Un_interval_real) show "{(c, {t..c})} tagged_division_of {a..c} \ {x. t \ x \ 1}" using \t \ c\ by (auto simp: eqs tagged_division_of_self_real) qed (auto simp: eqs ptag) moreover have "d1 fine p \ {(c, {t..c})}" unfolding fine_def proof safe fix x K y assume "(x,K) \ p" and "y \ K" then show "y \ d1 x" by (metis Int_iff d3_def subsetD fineD pfine) next fix x assume "x \ {t..c}" then have "dist c x < k" using t(1) by (auto simp add: field_simps dist_real_def) with k show "x \ d1 c" unfolding d_def by auto qed ultimately have d1_fin: "norm (?SUM(p \ {(c, {t..c})}) - integral {a..c} f) < e/3" using d1 by metis have SUMEQ: "?SUM(p \ {(c, {t..c})}) = (c - t) *\<^sub>R f c + ?SUM p" proof - have "?SUM(p \ {(c, {t..c})}) = (content{t..c} *\<^sub>R f c) + ?SUM p" proof (subst sum.union_disjoint) show "p \ {(c, {t..c})} = {}" using \t < c\ pt by force qed (use p'(1) in auto) also have "... = (c - t) *\<^sub>R f c + ?SUM p" using \t \ c\ by auto finally show ?thesis . qed have "c - k < t" using \k>0\ t(1) by (auto simp add: field_simps) moreover have "k \ w" proof (rule ccontr) assume "\ k \ w" then have "c + (k + w) / 2 \ d c" by (auto simp add: field_simps not_le not_less dist_real_def d_def) then have "c + (k + w) / 2 \ ball c k" using k by blast then show False using \0 < w\ \\ k \ w\ dist_real_def by auto qed ultimately have cwt: "c - w < t" by (auto simp add: field_simps) have eq: "integral {a..c} f - integral {a..t} f = -(((c - t) *\<^sub>R f c + ?SUM p) - integral {a..c} f) + (?SUM p - integral {a..t} f) + (c - t) *\<^sub>R f c" by auto have "norm (integral {a..c} f - integral {a..t} f) < e/3 + e/3 + e/3" unfolding eq proof (intro norm_triangle_lt add_strict_mono) show "norm (- ((c - t) *\<^sub>R f c + ?SUM p - integral {a..c} f)) < e/3" by (metis SUMEQ d1_fin norm_minus_cancel) show "norm (?SUM p - integral {a..t} f) < e/3" using d2_fin by blast show "norm ((c - t) *\<^sub>R f c) < e/3" using w cwt \t < c\ by simp (simp add: field_simps) qed then show ?thesis by simp qed qed qed lemma indefinite_integral_continuous_right: fixes f :: "real \ 'a::banach" assumes "f integrable_on {a..b}" and "a \ c" and "c < b" and "e > 0" obtains d where "0 < d" and "\t. c \ t \ t < c + d \ norm (integral {a..c} f - integral {a..t} f) < e" proof - have intm: "(\x. f (- x)) integrable_on {-b .. -a}" "- b < - c" "- c \ - a" using assms by auto from indefinite_integral_continuous_left[OF intm \e>0\] obtain d where "0 < d" and d: "\t. \- c - d < t; t \ -c\ \ norm (integral {- b..- c} (\x. f (-x)) - integral {- b..t} (\x. f (-x))) < e" by metis let ?d = "min d (b - c)" show ?thesis proof (intro that[of "?d"] allI impI) show "0 < ?d" using \0 < d\ \c < b\ by auto fix t :: real assume t: "c \ t \ t < c + ?d" have *: "integral {a..c} f = integral {a..b} f - integral {c..b} f" "integral {a..t} f = integral {a..b} f - integral {t..b} f" using assms t by (auto simp: algebra_simps integral_combine) have "(- c) - d < (- t)" "- t \ - c" using t by auto from d[OF this] show "norm (integral {a..c} f - integral {a..t} f) < e" by (auto simp add: algebra_simps norm_minus_commute *) qed qed lemma indefinite_integral_continuous_1: fixes f :: "real \ 'a::banach" assumes "f integrable_on {a..b}" shows "continuous_on {a..b} (\x. integral {a..x} f)" proof - have "\d>0. \x'\{a..b}. dist x' x < d \ dist (integral {a..x'} f) (integral {a..x} f) < e" if x: "x \ {a..b}" and "e > 0" for x e :: real proof (cases "a = b") case True with that show ?thesis by force next case False with x have "a < b" by force with x consider "x = a" | "x = b" | "a < x" "x < b" by force then show ?thesis proof cases case 1 then show ?thesis by (force simp: dist_norm algebra_simps intro: indefinite_integral_continuous_right [OF assms _ \a < b\ \e > 0\]) next case 2 then show ?thesis by (force simp: dist_norm norm_minus_commute algebra_simps intro: indefinite_integral_continuous_left [OF assms \a < b\ _ \e > 0\]) next case 3 obtain d1 where "0 < d1" and d1: "\t. \x - d1 < t; t \ x\ \ norm (integral {a..x} f - integral {a..t} f) < e" using 3 by (auto intro: indefinite_integral_continuous_left [OF assms \a < x\ _ \e > 0\]) obtain d2 where "0 < d2" and d2: "\t. \x \ t; t < x + d2\ \ norm (integral {a..x} f - integral {a..t} f) < e" using 3 by (auto intro: indefinite_integral_continuous_right [OF assms _ \x < b\ \e > 0\]) show ?thesis proof (intro exI ballI conjI impI) show "0 < min d1 d2" using \0 < d1\ \0 < d2\ by simp show "dist (integral {a..y} f) (integral {a..x} f) < e" if "y \ {a..b}" "dist y x < min d1 d2" for y proof (cases "y < x") case True with that d1 show ?thesis by (auto simp: dist_commute dist_norm) next case False with that d2 show ?thesis by (auto simp: dist_commute dist_norm) qed qed qed qed then show ?thesis by (auto simp: continuous_on_iff) qed lemma indefinite_integral_continuous_1': fixes f::"real \ 'a::banach" assumes "f integrable_on {a..b}" shows "continuous_on {a..b} (\x. integral {x..b} f)" proof - have "integral {a..b} f - integral {a..x} f = integral {x..b} f" if "x \ {a..b}" for x using integral_combine[OF _ _ assms, of x] that by (auto simp: algebra_simps) with _ show ?thesis by (rule continuous_on_eq) (auto intro!: continuous_intros indefinite_integral_continuous_1 assms) qed theorem integral_has_vector_derivative': fixes f :: "real \ 'b::banach" assumes "continuous_on {a..b} f" and "x \ {a..b}" shows "((\u. integral {u..b} f) has_vector_derivative - f x) (at x within {a..b})" proof - have *: "integral {x..b} f = integral {a .. b} f - integral {a .. x} f" if "a \ x" "x \ b" for x using integral_combine[of a x b for x, OF that integrable_continuous_real[OF assms(1)]] by (simp add: algebra_simps) show ?thesis using \x \ _\ * by (rule has_vector_derivative_transform) (auto intro!: derivative_eq_intros assms integral_has_vector_derivative) qed lemma integral_has_real_derivative': assumes "continuous_on {a..b} g" assumes "t \ {a..b}" shows "((\x. integral {x..b} g) has_real_derivative -g t) (at t within {a..b})" using integral_has_vector_derivative'[OF assms] by (auto simp: has_field_derivative_iff_has_vector_derivative) subsection \This doesn't directly involve integration, but that gives an easy proof\ lemma has_derivative_zero_unique_strong_interval: fixes f :: "real \ 'a::banach" assumes "finite k" and contf: "continuous_on {a..b} f" and "f a = y" and fder: "\x. x \ {a..b} - k \ (f has_derivative (\h. 0)) (at x within {a..b})" and x: "x \ {a..b}" shows "f x = y" proof - have "a \ b" "a \ x" using assms by auto have "((\x. 0::'a) has_integral f x - f a) {a..x}" proof (rule fundamental_theorem_of_calculus_interior_strong[OF \finite k\ \a \ x\]; clarify?) have "{a..x} \ {a..b}" using x by auto then show "continuous_on {a..x} f" by (rule continuous_on_subset[OF contf]) show "(f has_vector_derivative 0) (at z)" if z: "z \ {a<.. k" for z unfolding has_vector_derivative_def proof (simp add: at_within_open[OF z, symmetric]) show "(f has_derivative (\x. 0)) (at z within {a<..Generalize a bit to any convex set\ lemma has_derivative_zero_unique_strong_convex: fixes f :: "'a::euclidean_space \ 'b::banach" assumes "convex S" "finite K" and contf: "continuous_on S f" and "c \ S" "f c = y" and derf: "\x. x \ S - K \ (f has_derivative (\h. 0)) (at x within S)" and "x \ S" shows "f x = y" proof (cases "x = c") case True with \f c = y\ show ?thesis by blast next case False let ?\ = "\u. (1 - u) *\<^sub>R c + u *\<^sub>R x" have contf': "continuous_on {0 ..1} (f \ ?\)" proof (rule continuous_intros continuous_on_subset[OF contf])+ show "(\u. (1 - u) *\<^sub>R c + u *\<^sub>R x) ` {0..1} \ S" using \convex S\ \x \ S\ \c \ S\ by (auto simp add: convex_alt algebra_simps) qed have "t = u" if "?\ t = ?\ u" for t u proof - from that have "(t - u) *\<^sub>R x = (t - u) *\<^sub>R c" by (auto simp add: algebra_simps) then show ?thesis using \x \ c\ by auto qed then have eq: "(SOME t. ?\ t = ?\ u) = u" for u by blast then have "(?\ -` K) \ (\z. SOME t. ?\ t = z) ` K" by (clarsimp simp: image_iff) (metis (no_types) eq) then have fin: "finite (?\ -` K)" by (rule finite_surj[OF \finite K\]) have derf': "((\u. f (?\ u)) has_derivative (\h. 0)) (at t within {0..1})" if "t \ {0..1} - {t. ?\ t \ K}" for t proof - have df: "(f has_derivative (\h. 0)) (at (?\ t) within ?\ ` {0..1})" using \convex S\ \x \ S\ \c \ S\ that by (auto simp add: convex_alt algebra_simps intro: has_derivative_subset [OF derf]) have "(f \ ?\ has_derivative (\x. 0) \ (\z. (0 - z *\<^sub>R c) + z *\<^sub>R x)) (at t within {0..1})" by (rule derivative_eq_intros df | simp)+ then show ?thesis unfolding o_def . qed have "(f \ ?\) 1 = y" apply (rule has_derivative_zero_unique_strong_interval[OF fin contf']) unfolding o_def using \f c = y\ derf' by auto then show ?thesis by auto qed text \Also to any open connected set with finite set of exceptions. Could generalize to locally convex set with limpt-free set of exceptions.\ lemma has_derivative_zero_unique_strong_connected: fixes f :: "'a::euclidean_space \ 'b::banach" assumes "connected S" and "open S" and "finite K" and contf: "continuous_on S f" and "c \ S" and "f c = y" and derf: "\x. x \ S - K \ (f has_derivative (\h. 0)) (at x within S)" and "x \ S" shows "f x = y" proof - have "\e>0. ball x e \ (S \ f -` {f x})" if "x \ S" for x proof - obtain e where "0 < e" and e: "ball x e \ S" using \x \ S\ \open S\ open_contains_ball by blast have "ball x e \ {u \ S. f u \ {f x}}" proof safe fix y assume y: "y \ ball x e" then show "y \ S" using e by auto show "f y = f x" proof (rule has_derivative_zero_unique_strong_convex[OF convex_ball \finite K\]) show "continuous_on (ball x e) f" using contf continuous_on_subset e by blast show "(f has_derivative (\h. 0)) (at u within ball x e)" if "u \ ball x e - K" for u by (metis Diff_iff contra_subsetD derf e has_derivative_subset that) qed (use y e \0 < e\ in auto) qed then show "\e>0. ball x e \ (S \ f -` {f x})" using \0 < e\ by blast qed then have "openin (top_of_set S) (S \ f -` {y})" by (auto intro!: open_openin_trans[OF \open S\] simp: open_contains_ball) moreover have "closedin (top_of_set S) (S \ f -` {y})" by (force intro!: continuous_closedin_preimage [OF contf]) ultimately have "(S \ f -` {y}) = {} \ (S \ f -` {y}) = S" using \connected S\ by (simp add: connected_clopen) then show ?thesis using \x \ S\ \f c = y\ \c \ S\ by auto qed lemma has_derivative_zero_connected_constant: fixes f :: "'a::euclidean_space \ 'b::banach" assumes "connected S" and "open S" and "finite k" and "continuous_on S f" and "\x\(S - k). (f has_derivative (\h. 0)) (at x within S)" obtains c where "\x. x \ S \ f(x) = c" proof (cases "S = {}") case True then show ?thesis by (metis empty_iff that) next case False then obtain c where "c \ S" by (metis equals0I) then show ?thesis by (metis has_derivative_zero_unique_strong_connected assms that) qed lemma DERIV_zero_connected_constant: fixes f :: "'a::{real_normed_field,euclidean_space} \ 'a" assumes "connected S" and "open S" and "finite K" and "continuous_on S f" and "\x\(S - K). DERIV f x :> 0" obtains c where "\x. x \ S \ f(x) = c" using has_derivative_zero_connected_constant [OF assms(1-4)] assms by (metis DERIV_const has_derivative_const Diff_iff at_within_open frechet_derivative_at has_field_derivative_def) subsection \Integrating characteristic function of an interval\ lemma has_integral_restrict_open_subinterval: fixes f :: "'a::euclidean_space \ 'b::banach" assumes intf: "(f has_integral i) (cbox c d)" and cb: "cbox c d \ cbox a b" shows "((\x. if x \ box c d then f x else 0) has_integral i) (cbox a b)" proof (cases "cbox c d = {}") case True then have "box c d = {}" by (metis bot.extremum_uniqueI box_subset_cbox) then show ?thesis using True intf by auto next case False then obtain p where pdiv: "p division_of cbox a b" and inp: "cbox c d \ p" using cb partial_division_extend_1 by blast define g where [abs_def]: "g x = (if x \box c d then f x else 0)" for x interpret operative "lift_option plus" "Some (0 :: 'b)" "\i. if g integrable_on i then Some (integral i g) else None" by (fact operative_integralI) note operat = division [OF pdiv, symmetric] show ?thesis proof (cases "(g has_integral i) (cbox a b)") case True then show ?thesis by (simp add: g_def) next case False have iterate:"F (\i. if g integrable_on i then Some (integral i g) else None) (p - {cbox c d}) = Some 0" proof (intro neutral ballI) fix x assume x: "x \ p - {cbox c d}" then have "x \ p" by auto then obtain u v where uv: "x = cbox u v" using pdiv by blast have "interior x \ interior (cbox c d) = {}" using pdiv inp x by blast then have "(g has_integral 0) x" unfolding uv using has_integral_spike_interior[where f="\x. 0"] by (metis (no_types, lifting) disjoint_iff_not_equal g_def has_integral_0_eq interior_cbox) then show "(if g integrable_on x then Some (integral x g) else None) = Some 0" by auto qed interpret comm_monoid_set "lift_option plus" "Some (0 :: 'b)" by (intro comm_monoid_set.intro comm_monoid_lift_option add.comm_monoid_axioms) have intg: "g integrable_on cbox c d" using integrable_spike_interior[where f=f] by (meson g_def has_integral_integrable intf) moreover have "integral (cbox c d) g = i" proof (rule has_integral_unique[OF has_integral_spike_interior intf]) show "\x. x \ box c d \ f x = g x" by (auto simp: g_def) show "(g has_integral integral (cbox c d) g) (cbox c d)" by (rule integrable_integral[OF intg]) qed ultimately have "F (\A. if g integrable_on A then Some (integral A g) else None) p = Some i" by (metis (full_types, lifting) division_of_finite inp iterate pdiv remove right_neutral) then have "(g has_integral i) (cbox a b)" by (metis integrable_on_def integral_unique operat option.inject option.simps(3)) with False show ?thesis by blast qed qed lemma has_integral_restrict_closed_subinterval: fixes f :: "'a::euclidean_space \ 'b::banach" assumes "(f has_integral i) (cbox c d)" and "cbox c d \ cbox a b" shows "((\x. if x \ cbox c d then f x else 0) has_integral i) (cbox a b)" proof - note has_integral_restrict_open_subinterval[OF assms] note * = has_integral_spike[OF negligible_frontier_interval _ this] show ?thesis by (rule *[of c d]) (use box_subset_cbox[of c d] in auto) qed lemma has_integral_restrict_closed_subintervals_eq: fixes f :: "'a::euclidean_space \ 'b::banach" assumes "cbox c d \ cbox a b" shows "((\x. if x \ cbox c d then f x else 0) has_integral i) (cbox a b) \ (f has_integral i) (cbox c d)" (is "?l = ?r") proof (cases "cbox c d = {}") case False let ?g = "\x. if x \ cbox c d then f x else 0" show ?thesis proof assume ?l then have "?g integrable_on cbox c d" using assms has_integral_integrable integrable_subinterval by blast then have "f integrable_on cbox c d" by (rule integrable_eq) auto moreover then have "i = integral (cbox c d) f" by (meson \((\x. if x \ cbox c d then f x else 0) has_integral i) (cbox a b)\ assms has_integral_restrict_closed_subinterval has_integral_unique integrable_integral) ultimately show ?r by auto next assume ?r then show ?l by (rule has_integral_restrict_closed_subinterval[OF _ assms]) qed qed auto text \Hence we can apply the limit process uniformly to all integrals.\ lemma has_integral': fixes f :: "'n::euclidean_space \ 'a::banach" shows "(f has_integral i) S \ (\e>0. \B>0. \a b. ball 0 B \ cbox a b \ (\z. ((\x. if x \ S then f(x) else 0) has_integral z) (cbox a b) \ norm(z - i) < e))" (is "?l \ (\e>0. ?r e)") proof (cases "\a b. S = cbox a b") case False then show ?thesis by (simp add: has_integral_alt) next case True then obtain a b where S: "S = cbox a b" by blast obtain B where " 0 < B" and B: "\x. x \ cbox a b \ norm x \ B" using bounded_cbox[unfolded bounded_pos] by blast show ?thesis proof safe fix e :: real assume ?l and "e > 0" have "((\x. if x \ S then f x else 0) has_integral i) (cbox c d)" if "ball 0 (B+1) \ cbox c d" for c d unfolding S using B that by (force intro: \?l\[unfolded S] has_integral_restrict_closed_subinterval) then show "?r e" by (meson \0 < B\ \0 < e\ add_pos_pos le_less_trans zero_less_one norm_pths(2)) next assume as: "\e>0. ?r e" then obtain C where C: "\a b. ball 0 C \ cbox a b \ \z. ((\x. if x \ S then f x else 0) has_integral z) (cbox a b)" by (meson zero_less_one) define c :: 'n where "c = (\i\Basis. (- max B C) *\<^sub>R i)" define d :: 'n where "d = (\i\Basis. max B C *\<^sub>R i)" have "c \ i \ x \ i \ x \ i \ d \ i" if "norm x \ B" "i \ Basis" for x i using that and Basis_le_norm[OF \i\Basis\, of x] by (auto simp add: field_simps sum_negf c_def d_def) then have c_d: "cbox a b \ cbox c d" by (meson B mem_box(2) subsetI) have "c \ i \ x \ i \ x \ i \ d \ i" if x: "norm (0 - x) < C" and i: "i \ Basis" for x i using Basis_le_norm[OF i, of x] x i by (auto simp: sum_negf c_def d_def) then have "ball 0 C \ cbox c d" by (auto simp: mem_box dist_norm) with C obtain y where y: "(f has_integral y) (cbox a b)" using c_d has_integral_restrict_closed_subintervals_eq S by blast have "y = i" proof (rule ccontr) assume "y \ i" then have "0 < norm (y - i)" by auto from as[rule_format,OF this] obtain C where C: "\a b. ball 0 C \ cbox a b \ \z. ((\x. if x \ S then f x else 0) has_integral z) (cbox a b) \ norm (z-i) < norm (y-i)" by auto define c :: 'n where "c = (\i\Basis. (- max B C) *\<^sub>R i)" define d :: 'n where "d = (\i\Basis. max B C *\<^sub>R i)" have "c \ i \ x \ i \ x \ i \ d \ i" if "norm x \ B" and "i \ Basis" for x i using that Basis_le_norm[of i x] by (auto simp add: field_simps sum_negf c_def d_def) then have c_d: "cbox a b \ cbox c d" by (simp add: B mem_box(2) subset_eq) have "c \ i \ x \ i \ x \ i \ d \ i" if "norm (0 - x) < C" and "i \ Basis" for x i using Basis_le_norm[of i x] that by (auto simp: sum_negf c_def d_def) then have "ball 0 C \ cbox c d" by (auto simp: mem_box dist_norm) with C obtain z where z: "(f has_integral z) (cbox a b)" "norm (z-i) < norm (y-i)" using has_integral_restrict_closed_subintervals_eq[OF c_d] S by blast moreover then have "z = y" by (blast intro: has_integral_unique[OF _ y]) ultimately show False by auto qed then show ?l using y by (auto simp: S) qed qed lemma has_integral_le: fixes f :: "'n::euclidean_space \ real" assumes fg: "(f has_integral i) S" "(g has_integral j) S" and le: "\x. x \ S \ f x \ g x" shows "i \ j" using has_integral_component_le[OF _ fg, of 1] le by auto lemma integral_le: fixes f :: "'n::euclidean_space \ real" assumes "f integrable_on S" and "g integrable_on S" and "\x. x \ S \ f x \ g x" shows "integral S f \ integral S g" by (rule has_integral_le[OF assms(1,2)[unfolded has_integral_integral] assms(3)]) lemma has_integral_nonneg: fixes f :: "'n::euclidean_space \ real" assumes "(f has_integral i) S" and "\x. x \ S \ 0 \ f x" shows "0 \ i" using has_integral_component_nonneg[of 1 f i S] unfolding o_def using assms by auto lemma integral_nonneg: fixes f :: "'n::euclidean_space \ real" assumes f: "f integrable_on S" and 0: "\x. x \ S \ 0 \ f x" shows "0 \ integral S f" by (rule has_integral_nonneg[OF f[unfolded has_integral_integral] 0]) text \Hence a general restriction property.\ lemma has_integral_restrict [simp]: fixes f :: "'a :: euclidean_space \ 'b :: banach" assumes "S \ T" shows "((\x. if x \ S then f x else 0) has_integral i) T \ (f has_integral i) S" proof - have *: "\x. (if x \ T then if x \ S then f x else 0 else 0) = (if x\S then f x else 0)" using assms by auto show ?thesis apply (subst(2) has_integral') apply (subst has_integral') apply (simp add: *) done qed corollary has_integral_restrict_UNIV: fixes f :: "'n::euclidean_space \ 'a::banach" shows "((\x. if x \ s then f x else 0) has_integral i) UNIV \ (f has_integral i) s" by auto lemma has_integral_restrict_Int: fixes f :: "'a :: euclidean_space \ 'b :: banach" shows "((\x. if x \ S then f x else 0) has_integral i) T \ (f has_integral i) (S \ T)" proof - have "((\x. if x \ T then if x \ S then f x else 0 else 0) has_integral i) UNIV = ((\x. if x \ S \ T then f x else 0) has_integral i) UNIV" by (rule has_integral_cong) auto then show ?thesis using has_integral_restrict_UNIV by fastforce qed lemma integral_restrict_Int: fixes f :: "'a :: euclidean_space \ 'b :: banach" shows "integral T (\x. if x \ S then f x else 0) = integral (S \ T) f" by (metis (no_types, lifting) has_integral_cong has_integral_restrict_Int integrable_integral integral_unique not_integrable_integral) lemma integrable_restrict_Int: fixes f :: "'a :: euclidean_space \ 'b :: banach" shows "(\x. if x \ S then f x else 0) integrable_on T \ f integrable_on (S \ T)" using has_integral_restrict_Int by fastforce lemma has_integral_on_superset: fixes f :: "'n::euclidean_space \ 'a::banach" assumes f: "(f has_integral i) S" and "\x. x \ S \ f x = 0" and "S \ T" shows "(f has_integral i) T" proof - have "(\x. if x \ S then f x else 0) = (\x. if x \ T then f x else 0)" using assms by fastforce with f show ?thesis by (simp only: has_integral_restrict_UNIV [symmetric, of f]) qed lemma integrable_on_superset: fixes f :: "'n::euclidean_space \ 'a::banach" assumes "f integrable_on S" and "\x. x \ S \ f x = 0" and "S \ t" shows "f integrable_on t" using assms unfolding integrable_on_def by (auto intro:has_integral_on_superset) lemma integral_restrict_UNIV: fixes f :: "'n::euclidean_space \ 'a::banach" shows "integral UNIV (\x. if x \ S then f x else 0) = integral S f" by (simp add: integral_restrict_Int) lemma integrable_restrict_UNIV: fixes f :: "'n::euclidean_space \ 'a::banach" shows "(\x. if x \ s then f x else 0) integrable_on UNIV \ f integrable_on s" unfolding integrable_on_def by auto lemma has_integral_subset_component_le: fixes f :: "'n::euclidean_space \ 'm::euclidean_space" assumes k: "k \ Basis" and as: "S \ T" "(f has_integral i) S" "(f has_integral j) T" "\x. x\T \ 0 \ f(x)\k" shows "i\k \ j\k" proof - have \
: "((\x. if x \ S then f x else 0) has_integral i) UNIV" "((\x. if x \ T then f x else 0) has_integral j) UNIV" by (simp_all add: assms) show ?thesis using as by (force intro!: has_integral_component_le[OF k \
]) qed subsection\Integrals on set differences\ lemma has_integral_setdiff: fixes f :: "'a::euclidean_space \ 'b::banach" assumes S: "(f has_integral i) S" and T: "(f has_integral j) T" and neg: "negligible (T - S)" shows "(f has_integral (i - j)) (S - T)" proof - show ?thesis unfolding has_integral_restrict_UNIV [symmetric, of f] proof (rule has_integral_spike [OF neg]) have eq: "(\x. (if x \ S then f x else 0) - (if x \ T then f x else 0)) = (\x. if x \ T - S then - f x else if x \ S - T then f x else 0)" by (force simp add: ) have "((\x. if x \ S then f x else 0) has_integral i) UNIV" "((\x. if x \ T then f x else 0) has_integral j) UNIV" using S T has_integral_restrict_UNIV by auto from has_integral_diff [OF this] show "((\x. if x \ T - S then - f x else if x \ S - T then f x else 0) has_integral i-j) UNIV" by (simp add: eq) qed force qed lemma integral_setdiff: fixes f :: "'a::euclidean_space \ 'b::banach" assumes "f integrable_on S" "f integrable_on T" "negligible(T - S)" shows "integral (S - T) f = integral S f - integral T f" by (rule integral_unique) (simp add: assms has_integral_setdiff integrable_integral) lemma integrable_setdiff: fixes f :: "'a::euclidean_space \ 'b::banach" assumes "(f has_integral i) S" "(f has_integral j) T" "negligible (T - S)" shows "f integrable_on (S - T)" using has_integral_setdiff [OF assms] by (simp add: has_integral_iff) lemma negligible_setdiff [simp]: "T \ S \ negligible (T - S)" by (metis Diff_eq_empty_iff negligible_empty) lemma negligible_on_intervals: "negligible s \ (\a b. negligible(s \ cbox a b))" (is "?l \ ?r") proof assume R: ?r show ?l unfolding negligible_def proof safe fix a b have "negligible (s \ cbox a b)" by (simp add: R) then show "(indicator s has_integral 0) (cbox a b)" by (meson Diff_iff Int_iff has_integral_negligible indicator_simps(2)) qed qed (simp add: negligible_Int) lemma negligible_translation: assumes "negligible S" shows "negligible ((+) c ` S)" proof - have inj: "inj ((+) c)" by simp show ?thesis using assms proof (clarsimp simp: negligible_def) fix a b assume "\x y. (indicator S has_integral 0) (cbox x y)" then have *: "(indicator S has_integral 0) (cbox (a-c) (b-c))" by (meson Diff_iff assms has_integral_negligible indicator_simps(2)) have eq: "indicator ((+) c ` S) = (\x. indicator S (x - c))" by (force simp add: indicator_def) show "(indicator ((+) c ` S) has_integral 0) (cbox a b)" using has_integral_affinity [OF *, of 1 "-c"] cbox_translation [of "c" "-c+a" "-c+b"] by (simp add: eq) (simp add: ac_simps) qed qed lemma negligible_translation_rev: assumes "negligible ((+) c ` S)" shows "negligible S" by (metis negligible_translation [OF assms, of "-c"] translation_galois) lemma negligible_atLeastAtMostI: "b \ a \ negligible {a..(b::real)}" using negligible_insert by fastforce lemma has_integral_spike_set_eq: fixes f :: "'n::euclidean_space \ 'a::banach" assumes "negligible {x \ S - T. f x \ 0}" "negligible {x \ T - S. f x \ 0}" shows "(f has_integral y) S \ (f has_integral y) T" proof - have "((\x. if x \ S then f x else 0) has_integral y) UNIV = ((\x. if x \ T then f x else 0) has_integral y) UNIV" proof (rule has_integral_spike_eq) show "negligible ({x \ S - T. f x \ 0} \ {x \ T - S. f x \ 0})" by (rule negligible_Un [OF assms]) qed auto then show ?thesis by (simp add: has_integral_restrict_UNIV) qed corollary integral_spike_set: fixes f :: "'n::euclidean_space \ 'a::banach" assumes "negligible {x \ S - T. f x \ 0}" "negligible {x \ T - S. f x \ 0}" shows "integral S f = integral T f" using has_integral_spike_set_eq [OF assms] by (metis eq_integralD integral_unique) lemma integrable_spike_set: fixes f :: "'n::euclidean_space \ 'a::banach" assumes f: "f integrable_on S" and neg: "negligible {x \ S - T. f x \ 0}" "negligible {x \ T - S. f x \ 0}" shows "f integrable_on T" using has_integral_spike_set_eq [OF neg] f by blast lemma integrable_spike_set_eq: fixes f :: "'n::euclidean_space \ 'a::banach" assumes "negligible ((S - T) \ (T - S))" shows "f integrable_on S \ f integrable_on T" by (blast intro: integrable_spike_set assms negligible_subset) lemma integrable_on_insert_iff: "f integrable_on (insert x X) \ f integrable_on X" for f::"_ \ 'a::banach" by (rule integrable_spike_set_eq) (auto simp: insert_Diff_if) lemma has_integral_interior: fixes f :: "'a :: euclidean_space \ 'b :: banach" shows "negligible(frontier S) \ (f has_integral y) (interior S) \ (f has_integral y) S" by (rule has_integral_spike_set_eq [OF empty_imp_negligible negligible_subset]) (use interior_subset in \auto simp: frontier_def closure_def\) lemma has_integral_closure: fixes f :: "'a :: euclidean_space \ 'b :: banach" shows "negligible(frontier S) \ (f has_integral y) (closure S) \ (f has_integral y) S" by (rule has_integral_spike_set_eq [OF negligible_subset empty_imp_negligible]) (auto simp: closure_Un_frontier ) lemma has_integral_open_interval: fixes f :: "'a :: euclidean_space \ 'b :: banach" shows "(f has_integral y) (box a b) \ (f has_integral y) (cbox a b)" unfolding interior_cbox [symmetric] by (metis frontier_cbox has_integral_interior negligible_frontier_interval) lemma integrable_on_open_interval: fixes f :: "'a :: euclidean_space \ 'b :: banach" shows "f integrable_on box a b \ f integrable_on cbox a b" by (simp add: has_integral_open_interval integrable_on_def) lemma integral_open_interval: fixes f :: "'a :: euclidean_space \ 'b :: banach" shows "integral(box a b) f = integral(cbox a b) f" by (metis has_integral_integrable_integral has_integral_open_interval not_integrable_integral) subsection \More lemmas that are useful later\ lemma has_integral_subset_le: fixes f :: "'n::euclidean_space \ real" assumes "s \ t" and "(f has_integral i) s" and "(f has_integral j) t" and "\x\t. 0 \ f x" shows "i \ j" using has_integral_subset_component_le[OF _ assms(1), of 1 f i j] using assms by auto lemma integral_subset_component_le: fixes f :: "'n::euclidean_space \ 'm::euclidean_space" assumes "k \ Basis" and "s \ t" and "f integrable_on s" and "f integrable_on t" and "\x \ t. 0 \ f x \ k" shows "(integral s f)\k \ (integral t f)\k" by (meson assms has_integral_subset_component_le integrable_integral) lemma integral_subset_le: fixes f :: "'n::euclidean_space \ real" assumes "s \ t" and "f integrable_on s" and "f integrable_on t" and "\x \ t. 0 \ f x" shows "integral s f \ integral t f" using assms has_integral_subset_le by blast lemma has_integral_alt': fixes f :: "'n::euclidean_space \ 'a::banach" shows "(f has_integral i) s \ (\a b. (\x. if x \ s then f x else 0) integrable_on cbox a b) \ (\e>0. \B>0. \a b. ball 0 B \ cbox a b \ norm (integral (cbox a b) (\x. if x \ s then f x else 0) - i) < e)" (is "?l = ?r") proof assume rhs: ?r show ?l proof (subst has_integral', intro allI impI) fix e::real assume "e > 0" from rhs[THEN conjunct2,rule_format,OF this] show "\B>0. \a b. ball 0 B \ cbox a b \ (\z. ((\x. if x \ s then f x else 0) has_integral z) (cbox a b) \ norm (z - i) < e)" by (simp add: has_integral_iff rhs) qed next let ?\ = "\e a b. \z. ((\x. if x \ s then f x else 0) has_integral z) (cbox a b) \ norm (z - i) < e" assume ?l then have lhs: "\B>0. \a b. ball 0 B \ cbox a b \ ?\ e a b" if "e > 0" for e using that has_integral'[of f] by auto let ?f = "\x. if x \ s then f x else 0" show ?r proof (intro conjI allI impI) fix a b :: 'n from lhs[OF zero_less_one] obtain B where "0 < B" and B: "\a b. ball 0 B \ cbox a b \ ?\ 1 a b" by blast let ?a = "\i\Basis. min (a\i) (-B) *\<^sub>R i::'n" let ?b = "\i\Basis. max (b\i) B *\<^sub>R i::'n" show "?f integrable_on cbox a b" proof (rule integrable_subinterval[of _ ?a ?b]) have "?a \ i \ x \ i \ x \ i \ ?b \ i" if "norm (0 - x) < B" "i \ Basis" for x i using Basis_le_norm[of i x] that by (auto simp add:field_simps) then have "ball 0 B \ cbox ?a ?b" by (auto simp: mem_box dist_norm) then show "?f integrable_on cbox ?a ?b" unfolding integrable_on_def using B by blast show "cbox a b \ cbox ?a ?b" by (force simp: mem_box) qed fix e :: real assume "e > 0" with lhs show "\B>0. \a b. ball 0 B \ cbox a b \ norm (integral (cbox a b) (\x. if x \ s then f x else 0) - i) < e" by (metis (no_types, lifting) has_integral_integrable_integral) qed qed subsection \Continuity of the integral (for a 1-dimensional interval)\ lemma integrable_alt: fixes f :: "'n::euclidean_space \ 'a::banach" shows "f integrable_on s \ (\a b. (\x. if x \ s then f x else 0) integrable_on cbox a b) \ (\e>0. \B>0. \a b c d. ball 0 B \ cbox a b \ ball 0 B \ cbox c d \ norm (integral (cbox a b) (\x. if x \ s then f x else 0) - integral (cbox c d) (\x. if x \ s then f x else 0)) < e)" (is "?l = ?r") proof let ?F = "\x. if x \ s then f x else 0" assume ?l then obtain y where intF: "\a b. ?F integrable_on cbox a b" and y: "\e. 0 < e \ \B>0. \a b. ball 0 B \ cbox a b \ norm (integral (cbox a b) ?F - y) < e" unfolding integrable_on_def has_integral_alt'[of f] by auto show ?r proof (intro conjI allI impI intF) fix e::real assume "e > 0" then have "e/2 > 0" by auto obtain B where "0 < B" and B: "\a b. ball 0 B \ cbox a b \ norm (integral (cbox a b) ?F - y) < e/2" using \0 < e/2\ y by blast show "\B>0. \a b c d. ball 0 B \ cbox a b \ ball 0 B \ cbox c d \ norm (integral (cbox a b) ?F - integral (cbox c d) ?F) < e" proof (intro conjI exI impI allI, rule \0 < B\) fix a b c d::'n assume sub: "ball 0 B \ cbox a b \ ball 0 B \ cbox c d" show "norm (integral (cbox a b) ?F - integral (cbox c d) ?F) < e" using sub by (auto intro: norm_triangle_half_l dest: B) qed qed next let ?F = "\x. if x \ s then f x else 0" assume rhs: ?r let ?cube = "\n. cbox (\i\Basis. - real n *\<^sub>R i::'n) (\i\Basis. real n *\<^sub>R i)" have "Cauchy (\n. integral (?cube n) ?F)" unfolding Cauchy_def proof (intro allI impI) fix e::real assume "e > 0" with rhs obtain B where "0 < B" and B: "\a b c d. ball 0 B \ cbox a b \ ball 0 B \ cbox c d \ norm (integral (cbox a b) ?F - integral (cbox c d) ?F) < e" by blast obtain N where N: "B \ real N" using real_arch_simple by blast have "ball 0 B \ ?cube n" if n: "n \ N" for n proof - have "sum ((*\<^sub>R) (- real n)) Basis \ i \ x \ i \ x \ i \ sum ((*\<^sub>R) (real n)) Basis \ i" if "norm x < B" "i \ Basis" for x i::'n using Basis_le_norm[of i x] n N that by (auto simp add: field_simps sum_negf) then show ?thesis by (auto simp: mem_box dist_norm) qed then show "\M. \m\M. \n\M. dist (integral (?cube m) ?F) (integral (?cube n) ?F) < e" by (fastforce simp add: dist_norm intro!: B) qed then obtain i where i: "(\n. integral (?cube n) ?F) \ i" using convergent_eq_Cauchy by blast have "\B>0. \a b. ball 0 B \ cbox a b \ norm (integral (cbox a b) ?F - i) < e" if "e > 0" for e proof - have *: "e/2 > 0" using that by auto then obtain N where N: "\n. N \ n \ norm (i - integral (?cube n) ?F) < e/2" using i[THEN LIMSEQ_D, simplified norm_minus_commute] by meson obtain B where "0 < B" and B: "\a b c d. \ball 0 B \ cbox a b; ball 0 B \ cbox c d\ \ norm (integral (cbox a b) ?F - integral (cbox c d) ?F) < e/2" using rhs * by meson let ?B = "max (real N) B" show ?thesis proof (intro exI conjI allI impI) show "0 < ?B" using \B > 0\ by auto fix a b :: 'n assume "ball 0 ?B \ cbox a b" moreover obtain n where n: "max (real N) B \ real n" using real_arch_simple by blast moreover have "ball 0 B \ ?cube n" proof fix x :: 'n assume x: "x \ ball 0 B" have "\norm (0 - x) < B; i \ Basis\ \ sum ((*\<^sub>R) (-n)) Basis \ i\ x \ i \ x \ i \ sum ((*\<^sub>R) n) Basis \ i" for i using Basis_le_norm[of i x] n by (auto simp add: field_simps sum_negf) then show "x \ ?cube n" using x by (auto simp: mem_box dist_norm) qed ultimately show "norm (integral (cbox a b) ?F - i) < e" using norm_triangle_half_l [OF B N] by force qed qed then show ?l unfolding integrable_on_def has_integral_alt'[of f] using rhs by blast qed lemma integrable_altD: fixes f :: "'n::euclidean_space \ 'a::banach" assumes "f integrable_on s" shows "\a b. (\x. if x \ s then f x else 0) integrable_on cbox a b" and "\e. e > 0 \ \B>0. \a b c d. ball 0 B \ cbox a b \ ball 0 B \ cbox c d \ norm (integral (cbox a b) (\x. if x \ s then f x else 0) - integral (cbox c d) (\x. if x \ s then f x else 0)) < e" using assms[unfolded integrable_alt[of f]] by auto lemma integrable_alt_subset: fixes f :: "'a::euclidean_space \ 'b::banach" shows "f integrable_on S \ (\a b. (\x. if x \ S then f x else 0) integrable_on cbox a b) \ (\e>0. \B>0. \a b c d. ball 0 B \ cbox a b \ cbox a b \ cbox c d \ norm(integral (cbox a b) (\x. if x \ S then f x else 0) - integral (cbox c d) (\x. if x \ S then f x else 0)) < e)" (is "_ = ?rhs") proof - let ?g = "\x. if x \ S then f x else 0" have "f integrable_on S \ (\a b. ?g integrable_on cbox a b) \ (\e>0. \B>0. \a b c d. ball 0 B \ cbox a b \ ball 0 B \ cbox c d \ norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e)" by (rule integrable_alt) also have "\ = ?rhs" proof - { fix e :: "real" assume e: "\e. e>0 \ \B>0. \a b c d. ball 0 B \ cbox a b \ cbox a b \ cbox c d \ norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e" and "e > 0" obtain B where "B > 0" and B: "\a b c d. \ball 0 B \ cbox a b; cbox a b \ cbox c d\ \ norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e/2" using \e > 0\ e [of "e/2"] by force have "\B>0. \a b c d. ball 0 B \ cbox a b \ ball 0 B \ cbox c d \ norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e" proof (intro exI allI conjI impI) fix a b c d :: "'a" let ?\ = "\i\Basis. max (a \ i) (c \ i) *\<^sub>R i" let ?\ = "\i\Basis. min (b \ i) (d \ i) *\<^sub>R i" show "norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e" if ball: "ball 0 B \ cbox a b \ ball 0 B \ cbox c d" proof - have B': "norm (integral (cbox a b \ cbox c d) ?g - integral (cbox x y) ?g) < e/2" if "cbox a b \ cbox c d \ cbox x y" for x y using B [of ?\ ?\ x y] ball that by (simp add: Int_interval [symmetric]) show ?thesis using B' [of a b] B' [of c d] norm_triangle_half_r by blast qed qed (use \B > 0\ in auto)} then show ?thesis by force qed finally show ?thesis . qed lemma integrable_on_subcbox: fixes f :: "'n::euclidean_space \ 'a::banach" assumes intf: "f integrable_on S" and sub: "cbox a b \ S" shows "f integrable_on cbox a b" proof - have "(\x. if x \ S then f x else 0) integrable_on cbox a b" by (simp add: intf integrable_altD(1)) then show ?thesis by (metis (mono_tags) sub integrable_restrict_Int le_inf_iff order_refl subset_antisym) qed subsection \A straddling criterion for integrability\ lemma integrable_straddle_interval: fixes f :: "'n::euclidean_space \ real" assumes "\e. e>0 \ \g h i j. (g has_integral i) (cbox a b) \ (h has_integral j) (cbox a b) \ \i - j\ < e \ (\x\cbox a b. (g x) \ f x \ f x \ h x)" shows "f integrable_on cbox a b" proof - have "\d. gauge d \ (\p1 p2. p1 tagged_division_of cbox a b \ d fine p1 \ p2 tagged_division_of cbox a b \ d fine p2 \ \(\(x,K)\p1. content K *\<^sub>R f x) - (\(x,K)\p2. content K *\<^sub>R f x)\ < e)" if "e > 0" for e proof - have e: "e/3 > 0" using that by auto then obtain g h i j where ij: "\i - j\ < e/3" and "(g has_integral i) (cbox a b)" and "(h has_integral j) (cbox a b)" and fgh: "\x. x \ cbox a b \ g x \ f x \ f x \ h x" using assms real_norm_def by metis then obtain d1 d2 where "gauge d1" "gauge d2" and d1: "\p. \p tagged_division_of cbox a b; d1 fine p\ \ \(\(x,K)\p. content K *\<^sub>R g x) - i\ < e/3" and d2: "\p. \p tagged_division_of cbox a b; d2 fine p\ \ \(\(x,K) \ p. content K *\<^sub>R h x) - j\ < e/3" by (metis e has_integral real_norm_def) have "\(\(x,K) \ p1. content K *\<^sub>R f x) - (\(x,K) \ p2. content K *\<^sub>R f x)\ < e" if p1: "p1 tagged_division_of cbox a b" and 11: "d1 fine p1" and 21: "d2 fine p1" and p2: "p2 tagged_division_of cbox a b" and 12: "d1 fine p2" and 22: "d2 fine p2" for p1 p2 proof - have *: "\g1 g2 h1 h2 f1 f2. \\g2 - i\ < e/3; \g1 - i\ < e/3; \h2 - j\ < e/3; \h1 - j\ < e/3; g1 - h2 \ f1 - f2; f1 - f2 \ h1 - g2\ \ \f1 - f2\ < e" using \e > 0\ ij by arith have 0: "(\(x, k)\p1. content k *\<^sub>R f x) - (\(x, k)\p1. content k *\<^sub>R g x) \ 0" "0 \ (\(x, k)\p2. content k *\<^sub>R h x) - (\(x, k)\p2. content k *\<^sub>R f x)" "(\(x, k)\p2. content k *\<^sub>R f x) - (\(x, k)\p2. content k *\<^sub>R g x) \ 0" "0 \ (\(x, k)\p1. content k *\<^sub>R h x) - (\(x, k)\p1. content k *\<^sub>R f x)" unfolding sum_subtractf[symmetric] apply (auto intro!: sum_nonneg) apply (meson fgh measure_nonneg mult_left_mono tag_in_interval that sum_nonneg)+ done show ?thesis proof (rule *) show "\(\(x,K) \ p2. content K *\<^sub>R g x) - i\ < e/3" by (rule d1[OF p2 12]) show "\(\(x,K) \ p1. content K *\<^sub>R g x) - i\ < e/3" by (rule d1[OF p1 11]) show "\(\(x,K) \ p2. content K *\<^sub>R h x) - j\ < e/3" by (rule d2[OF p2 22]) show "\(\(x,K) \ p1. content K *\<^sub>R h x) - j\ < e/3" by (rule d2[OF p1 21]) qed (use 0 in auto) qed then show ?thesis by (rule_tac x="\x. d1 x \ d2 x" in exI) (auto simp: fine_Int intro: \gauge d1\ \gauge d2\ d1 d2) qed then show ?thesis by (simp add: integrable_Cauchy) qed lemma integrable_straddle: fixes f :: "'n::euclidean_space \ real" assumes "\e. e>0 \ \g h i j. (g has_integral i) s \ (h has_integral j) s \ \i - j\ < e \ (\x\s. g x \ f x \ f x \ h x)" shows "f integrable_on s" proof - let ?fs = "(\x. if x \ s then f x else 0)" have "?fs integrable_on cbox a b" for a b proof (rule integrable_straddle_interval) fix e::real assume "e > 0" then have *: "e/4 > 0" by auto with assms obtain g h i j where g: "(g has_integral i) s" and h: "(h has_integral j) s" and ij: "\i - j\ < e/4" and fgh: "\x. x \ s \ g x \ f x \ f x \ h x" by metis let ?gs = "(\x. if x \ s then g x else 0)" let ?hs = "(\x. if x \ s then h x else 0)" obtain Bg where Bg: "\a b. ball 0 Bg \ cbox a b \ \integral (cbox a b) ?gs - i\ < e/4" and int_g: "\a b. ?gs integrable_on cbox a b" using g * unfolding has_integral_alt' real_norm_def by meson obtain Bh where Bh: "\a b. ball 0 Bh \ cbox a b \ \integral (cbox a b) ?hs - j\ < e/4" and int_h: "\a b. ?hs integrable_on cbox a b" using h * unfolding has_integral_alt' real_norm_def by meson define c where "c = (\i\Basis. min (a\i) (- (max Bg Bh)) *\<^sub>R i)" define d where "d = (\i\Basis. max (b\i) (max Bg Bh) *\<^sub>R i)" have "\norm (0 - x) < Bg; i \ Basis\ \ c \ i \ x \ i \ x \ i \ d \ i" for x i using Basis_le_norm[of i x] unfolding c_def d_def by auto then have ballBg: "ball 0 Bg \ cbox c d" by (auto simp: mem_box dist_norm) have "\norm (0 - x) < Bh; i \ Basis\ \ c \ i \ x \ i \ x \ i \ d \ i" for x i using Basis_le_norm[of i x] unfolding c_def d_def by auto then have ballBh: "ball 0 Bh \ cbox c d" by (auto simp: mem_box dist_norm) have ab_cd: "cbox a b \ cbox c d" by (auto simp: c_def d_def subset_box_imp) have **: "\ch cg ag ah::real. \\ah - ag\ \ \ch - cg\; \cg - i\ < e/4; \ch - j\ < e/4\ \ \ag - ah\ < e" using ij by arith show "\g h i j. (g has_integral i) (cbox a b) \ (h has_integral j) (cbox a b) \ \i - j\ < e \ (\x\cbox a b. g x \ (if x \ s then f x else 0) \ (if x \ s then f x else 0) \ h x)" proof (intro exI ballI conjI) have eq: "\x f g. (if x \ s then f x else 0) - (if x \ s then g x else 0) = (if x \ s then f x - g x else (0::real))" by auto have int_hg: "(\x. if x \ s then h x - g x else 0) integrable_on cbox a b" "(\x. if x \ s then h x - g x else 0) integrable_on cbox c d" by (metis (no_types) integrable_diff g h has_integral_integrable integrable_altD(1))+ show "(?gs has_integral integral (cbox a b) ?gs) (cbox a b)" "(?hs has_integral integral (cbox a b) ?hs) (cbox a b)" by (intro integrable_integral int_g int_h)+ then have "integral (cbox a b) ?gs \ integral (cbox a b) ?hs" using fgh by (force intro: has_integral_le) then have "0 \ integral (cbox a b) ?hs - integral (cbox a b) ?gs" by simp then have "\integral (cbox a b) ?hs - integral (cbox a b) ?gs\ \ \integral (cbox c d) ?hs - integral (cbox c d) ?gs\" apply (simp add: integral_diff [symmetric] int_g int_h) apply (subst abs_of_nonneg[OF integral_nonneg[OF integrable_diff, OF int_h int_g]]) using fgh apply (force simp: eq intro!: integral_subset_le [OF ab_cd int_hg])+ done then show "\integral (cbox a b) ?gs - integral (cbox a b) ?hs\ < e" using ** Bg ballBg Bh ballBh by blast show "\x. x \ cbox a b \ ?gs x \ ?fs x" "\x. x \ cbox a b \ ?fs x \ ?hs x" using fgh by auto qed qed then have int_f: "?fs integrable_on cbox a b" for a b by simp have "\B>0. \a b c d. ball 0 B \ cbox a b \ ball 0 B \ cbox c d \ abs (integral (cbox a b) ?fs - integral (cbox c d) ?fs) < e" if "0 < e" for e proof - have *: "e/3 > 0" using that by auto with assms obtain g h i j where g: "(g has_integral i) s" and h: "(h has_integral j) s" and ij: "\i - j\ < e/3" and fgh: "\x. x \ s \ g x \ f x \ f x \ h x" by metis let ?gs = "(\x. if x \ s then g x else 0)" let ?hs = "(\x. if x \ s then h x else 0)" obtain Bg where "Bg > 0" and Bg: "\a b. ball 0 Bg \ cbox a b \ \integral (cbox a b) ?gs - i\ < e/3" and int_g: "\a b. ?gs integrable_on cbox a b" using g * unfolding has_integral_alt' real_norm_def by meson obtain Bh where "Bh > 0" and Bh: "\a b. ball 0 Bh \ cbox a b \ \integral (cbox a b) ?hs - j\ < e/3" and int_h: "\a b. ?hs integrable_on cbox a b" using h * unfolding has_integral_alt' real_norm_def by meson { fix a b c d :: 'n assume as: "ball 0 (max Bg Bh) \ cbox a b" "ball 0 (max Bg Bh) \ cbox c d" have **: "ball 0 Bg \ ball (0::'n) (max Bg Bh)" "ball 0 Bh \ ball (0::'n) (max Bg Bh)" by auto have *: "\ga gc ha hc fa fc. \\ga - i\ < e/3; \gc - i\ < e/3; \ha - j\ < e/3; \hc - j\ < e/3; ga \ fa; fa \ ha; gc \ fc; fc \ hc\ \ \fa - fc\ < e" using ij by arith have "abs (integral (cbox a b) (\x. if x \ s then f x else 0) - integral (cbox c d) (\x. if x \ s then f x else 0)) < e" proof (rule *) show "\integral (cbox a b) ?gs - i\ < e/3" using "**" Bg as by blast show "\integral (cbox c d) ?gs - i\ < e/3" using "**" Bg as by blast show "\integral (cbox a b) ?hs - j\ < e/3" using "**" Bh as by blast show "\integral (cbox c d) ?hs - j\ < e/3" using "**" Bh as by blast qed (use int_f int_g int_h fgh in \simp_all add: integral_le\) } then show ?thesis apply (rule_tac x="max Bg Bh" in exI) using \Bg > 0\ by auto qed then show ?thesis unfolding integrable_alt[of f] real_norm_def by (blast intro: int_f) qed subsection \Adding integrals over several sets\ lemma has_integral_Un: fixes f :: "'n::euclidean_space \ 'a::banach" assumes f: "(f has_integral i) S" "(f has_integral j) T" and neg: "negligible (S \ T)" shows "(f has_integral (i + j)) (S \ T)" unfolding has_integral_restrict_UNIV[symmetric, of f] proof (rule has_integral_spike[OF neg]) let ?f = "\x. (if x \ S then f x else 0) + (if x \ T then f x else 0)" show "(?f has_integral i + j) UNIV" by (simp add: f has_integral_add) qed auto lemma integral_Un [simp]: fixes f :: "'n::euclidean_space \ 'a::banach" assumes "f integrable_on S" "f integrable_on T" "negligible (S \ T)" shows "integral (S \ T) f = integral S f + integral T f" by (simp add: has_integral_Un assms integrable_integral integral_unique) lemma integrable_Un: fixes f :: "'a::euclidean_space \ 'b :: banach" assumes "negligible (A \ B)" "f integrable_on A" "f integrable_on B" shows "f integrable_on (A \ B)" proof - from assms obtain y z where "(f has_integral y) A" "(f has_integral z) B" by (auto simp: integrable_on_def) from has_integral_Un[OF this assms(1)] show ?thesis by (auto simp: integrable_on_def) qed lemma integrable_Un': fixes f :: "'a::euclidean_space \ 'b :: banach" assumes "f integrable_on A" "f integrable_on B" "negligible (A \ B)" "C = A \ B" shows "f integrable_on C" using integrable_Un[of A B f] assms by simp -lemma has_integral_Union: +lemma has_integral_UN: fixes f :: "'n::euclidean_space \ 'a::banach" - assumes \: "finite \" - and int: "\S. S \ \ \ (f has_integral (i S)) S" - and neg: "pairwise (\S S'. negligible (S \ S')) \" - shows "(f has_integral (sum i \)) (\\)" + assumes "finite I" + and int: "\i. i \ I \ (f has_integral (g i)) (\ i)" + and neg: "pairwise (\i i'. negligible (\ i \ \ i')) I" + shows "(f has_integral (sum g I)) (\i\I. \ i)" proof - - let ?\ = "((\(a,b). a \ b) ` {(a,b). a \ \ \ b \ {y. y \ \ \ a \ y}})" - have "((\x. if x \ \\ then f x else 0) has_integral sum i \) UNIV" + let ?\ = "((\(a,b). \ a \ \ b) ` {(a,b). a \ I \ b \ I-{a}})" + have "((\x. if x \ (\i\I. \ i) then f x else 0) has_integral sum g I) UNIV" proof (rule has_integral_spike) show "negligible (\?\)" proof (rule negligible_Union) - have "finite (\ \ \)" - by (simp add: \) - moreover have "{(a, b). a \ \ \ b \ {y \ \. a \ y}} \ \ \ \" + have "finite (I \ I)" + by (simp add: \finite I\) + moreover have "{(a,b). a \ I \ b \ I-{a}} \ I \ I" by auto ultimately show "finite ?\" - by (blast intro: finite_subset[of _ "\ \ \"]) + by (simp add: finite_subset) show "\t. t \ ?\ \ negligible t" using neg unfolding pairwise_def by auto qed next - show "(if x \ \\ then f x else 0) = (\A\\. if x \ A then f x else 0)" + show "(if x \ (\i\I. \ i) then f x else 0) = (\i\I. if x \ \ i then f x else 0)" if "x \ UNIV - (\?\)" for x proof clarsimp - fix S assume "S \ \" "x \ S" - moreover then have "\b\\. x \ b \ b = S" + fix i assume i: "i \ I" "x \ \ i" + then have "\j\I. x \ \ j \ j = i" using that by blast - ultimately show "f x = (\A\\. if x \ A then f x else 0)" - by (simp add: sum.delta[OF \]) + with i show "f x = (\i\I. if x \ \ i then f x else 0)" + by (simp add: sum.delta[OF \finite I\]) qed next - show "((\x. \A\\. if x \ A then f x else 0) has_integral (\A\\. i A)) UNIV" - using int by (simp add: has_integral_restrict_UNIV has_integral_sum [OF \]) + show "((\x. (\i\I. if x \ \ i then f x else 0)) has_integral sum g I) UNIV" + using int by (simp add: has_integral_restrict_UNIV has_integral_sum [OF \finite I\]) qed then show ?thesis using has_integral_restrict_UNIV by blast qed +lemma has_integral_Union: + fixes f :: "'n::euclidean_space \ 'a::banach" + assumes "finite \" + and "\S. S \ \ \ (f has_integral (i S)) S" + and "pairwise (\S S'. negligible (S \ S')) \" + shows "(f has_integral (sum i \)) (\\)" +proof - + have "(f has_integral (sum i \)) (\S\\. S)" + by (intro has_integral_UN assms) + then show ?thesis + by force +qed text \In particular adding integrals over a division, maybe not of an interval.\ lemma has_integral_combine_division: fixes f :: "'n::euclidean_space \ 'a::banach" assumes "\ division_of S" and "\k. k \ \ \ (f has_integral (i k)) k" shows "(f has_integral (sum i \)) S" proof - note \ = division_ofD[OF assms(1)] have neg: "negligible (S \ s')" if "S \ \" "s' \ \" "S \ s'" for S s' proof - obtain a c b \ where obt: "S = cbox a b" "s' = cbox c \" by (meson \S \ \\ \s' \ \\ \(4)) from \(5)[OF that] show ?thesis unfolding obt interior_cbox by (metis (no_types, lifting) Diff_empty Int_interval box_Int_box negligible_frontier_interval) qed show ?thesis unfolding \(6)[symmetric] by (auto intro: \ neg assms has_integral_Union pairwiseI) qed lemma integral_combine_division_bottomup: fixes f :: "'n::euclidean_space \ 'a::banach" assumes "\ division_of S" "\k. k \ \ \ f integrable_on k" shows "integral S f = sum (\i. integral i f) \" by (meson assms integral_unique has_integral_combine_division has_integral_integrable_integral) lemma has_integral_combine_division_topdown: fixes f :: "'n::euclidean_space \ 'a::banach" assumes f: "f integrable_on S" and \: "\ division_of K" and "K \ S" shows "(f has_integral (sum (\i. integral i f) \)) K" proof - have "f integrable_on L" if "L \ \" for L proof - have "L \ S" using \K \ S\ \ that by blast then show "f integrable_on L" using that by (metis (no_types) f \ division_ofD(4) integrable_on_subcbox) qed then show ?thesis by (meson \ has_integral_combine_division has_integral_integrable_integral) qed lemma integral_combine_division_topdown: fixes f :: "'n::euclidean_space \ 'a::banach" assumes "f integrable_on S" and "\ division_of S" shows "integral S f = sum (\i. integral i f) \" using assms has_integral_combine_division_topdown by blast lemma integrable_combine_division: fixes f :: "'n::euclidean_space \ 'a::banach" assumes \: "\ division_of S" and f: "\i. i \ \ \ f integrable_on i" shows "f integrable_on S" using f unfolding integrable_on_def by (metis has_integral_combine_division[OF \]) lemma integrable_on_subdivision: fixes f :: "'n::euclidean_space \ 'a::banach" assumes \: "\ division_of i" and f: "f integrable_on S" and "i \ S" shows "f integrable_on i" proof - have "f integrable_on i" if "i \ \" for i proof - have "i \ S" using assms that by auto then show "f integrable_on i" using that by (metis (no_types) \ f division_ofD(4) integrable_on_subcbox) qed then show ?thesis using \ integrable_combine_division by blast qed subsection \Also tagged divisions\ lemma has_integral_combine_tagged_division: fixes f :: "'n::euclidean_space \ 'a::banach" assumes "p tagged_division_of S" and "\x k. (x,k) \ p \ (f has_integral (i k)) k" shows "(f has_integral (\(x,k)\p. i k)) S" proof - have *: "(f has_integral (\k\snd`p. integral k f)) S" proof - have "snd ` p division_of S" by (simp add: assms(1) division_of_tagged_division) with assms show ?thesis by (metis (mono_tags, lifting) has_integral_combine_division has_integral_integrable_integral imageE prod.collapse) qed also have "(\k\snd`p. integral k f) = (\(x, k)\p. integral k f)" by (intro sum.over_tagged_division_lemma[OF assms(1), symmetric] integral_null) (simp add: content_eq_0_interior) finally show ?thesis using assms by (auto simp add: has_integral_iff intro!: sum.cong) qed lemma integral_combine_tagged_division_bottomup: fixes f :: "'n::euclidean_space \ 'a::banach" assumes p: "p tagged_division_of (cbox a b)" and f: "\x k. (x,k)\p \ f integrable_on k" shows "integral (cbox a b) f = sum (\(x,k). integral k f) p" by (simp add: has_integral_combine_tagged_division[OF p] integral_unique f integrable_integral) lemma has_integral_combine_tagged_division_topdown: fixes f :: "'n::euclidean_space \ 'a::banach" assumes f: "f integrable_on cbox a b" and p: "p tagged_division_of (cbox a b)" shows "(f has_integral (sum (\(x,K). integral K f) p)) (cbox a b)" proof - have "(f has_integral integral K f) K" if "(x,K) \ p" for x K by (metis assms integrable_integral integrable_on_subcbox tagged_division_ofD(3,4) that) then show ?thesis by (simp add: has_integral_combine_tagged_division p) qed lemma integral_combine_tagged_division_topdown: fixes f :: "'n::euclidean_space \ 'a::banach" assumes "f integrable_on cbox a b" and "p tagged_division_of (cbox a b)" shows "integral (cbox a b) f = sum (\(x,k). integral k f) p" using assms by (auto intro: integral_unique [OF has_integral_combine_tagged_division_topdown]) subsection \Henstock's lemma\ lemma Henstock_lemma_part1: fixes f :: "'n::euclidean_space \ 'a::banach" assumes intf: "f integrable_on cbox a b" and "e > 0" and "gauge d" and less_e: "\p. \p tagged_division_of (cbox a b); d fine p\ \ norm (sum (\(x,K). content K *\<^sub>R f x) p - integral(cbox a b) f) < e" and p: "p tagged_partial_division_of (cbox a b)" "d fine p" shows "norm (sum (\(x,K). content K *\<^sub>R f x - integral K f) p) \ e" (is "?lhs \ e") proof (rule field_le_epsilon) fix k :: real assume "k > 0" let ?SUM = "\p. (\(x,K) \ p. content K *\<^sub>R f x)" note p' = tagged_partial_division_ofD[OF p(1)] have "\(snd ` p) \ cbox a b" using p'(3) by fastforce then obtain q where q: "snd ` p \ q" and qdiv: "q division_of cbox a b" by (meson p(1) partial_division_extend_interval partial_division_of_tagged_division) note q' = division_ofD[OF qdiv] define r where "r = q - snd ` p" have "snd ` p \ r = {}" unfolding r_def by auto have "finite r" using q' unfolding r_def by auto have "\p. p tagged_division_of i \ d fine p \ norm (?SUM p - integral i f) < k / (real (card r) + 1)" if "i\r" for i proof - have gt0: "k / (real (card r) + 1) > 0" using \k > 0\ by simp have i: "i \ q" using that unfolding r_def by auto then obtain u v where uv: "i = cbox u v" using q'(4) by blast then have "cbox u v \ cbox a b" using i q'(2) by auto then have "f integrable_on cbox u v" by (rule integrable_subinterval[OF intf]) with integrable_integral[OF this, unfolded has_integral[of f]] obtain dd where "gauge dd" and dd: "\\. \\ tagged_division_of cbox u v; dd fine \\ \ norm (?SUM \ - integral (cbox u v) f) < k / (real (card r) + 1)" using gt0 by auto with gauge_Int[OF \gauge d\ \gauge dd\] obtain qq where qq: "qq tagged_division_of cbox u v" "(\x. d x \ dd x) fine qq" using fine_division_exists by blast with dd[of qq] show ?thesis by (auto simp: fine_Int uv) qed then obtain qq where qq: "\i. i \ r \ qq i tagged_division_of i \ d fine qq i \ norm (?SUM (qq i) - integral i f) < k / (real (card r) + 1)" by metis let ?p = "p \ \(qq ` r)" have "norm (?SUM ?p - integral (cbox a b) f) < e" proof (rule less_e) show "d fine ?p" by (metis (mono_tags, opaque_lifting) qq fine_Un fine_Union imageE p(2)) note ptag = tagged_partial_division_of_Union_self[OF p(1)] have "p \ \(qq ` r) tagged_division_of \(snd ` p) \ \r" proof (rule tagged_division_Un[OF ptag tagged_division_Union [OF \finite r\]]) show "\i. i \ r \ qq i tagged_division_of i" using qq by auto show "\i1 i2. \i1 \ r; i2 \ r; i1 \ i2\ \ interior i1 \ interior i2 = {}" by (simp add: q'(5) r_def) show "interior (\(snd ` p)) \ interior (\r) = {}" proof (rule Int_interior_Union_intervals [OF \finite r\]) show "open (interior (\(snd ` p)))" by blast show "\T. T \ r \ \a b. T = cbox a b" by (simp add: q'(4) r_def) have "interior T \ interior (\(snd ` p)) = {}" if "T \ r" for T proof (rule Int_interior_Union_intervals) show "\U. U \ snd ` p \ \a b. U = cbox a b" using q q'(4) by blast show "\U. U \ snd ` p \ interior T \ interior U = {}" by (metis DiffE q q'(5) r_def subsetD that) qed (use p' in auto) then show "\T. T \ r \ interior (\(snd ` p)) \ interior T = {}" by (metis Int_commute) qed qed moreover have "\(snd ` p) \ \r = cbox a b" and "{qq i |i. i \ r} = qq ` r" using qdiv q unfolding Union_Un_distrib[symmetric] r_def by auto ultimately show "?p tagged_division_of (cbox a b)" by fastforce qed then have "norm (?SUM p + (?SUM (\(qq ` r))) - integral (cbox a b) f) < e" proof (subst sum.union_inter_neutral[symmetric, OF \finite p\], safe) show "content L *\<^sub>R f x = 0" if "(x, L) \ p" "(x, L) \ qq K" "K \ r" for x K L proof - obtain u v where uv: "L = cbox u v" using \(x,L) \ p\ p'(4) by blast have "L \ K" using qq[OF that(3)] tagged_division_ofD(3) \(x,L) \ qq K\ by metis have "L \ snd ` p" using \(x,L) \ p\ image_iff by fastforce then have "L \ q" "K \ q" "L \ K" using that(1,3) q(1) unfolding r_def by auto with q'(5) have "interior L = {}" using interior_mono[OF \L \ K\] by blast then show "content L *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[symmetric] by auto qed show "finite (\(qq ` r))" by (meson finite_UN qq \finite r\ tagged_division_of_finite) qed moreover have "content M *\<^sub>R f x = 0" if x: "(x,M) \ qq K" "(x,M) \ qq L" and KL: "qq K \ qq L" and r: "K \ r" "L \ r" for x M K L proof - note kl = tagged_division_ofD(3,4)[OF qq[THEN conjunct1]] obtain u v where uv: "M = cbox u v" using \(x, M) \ qq L\ \L \ r\ kl(2) by blast have empty: "interior (K \ L) = {}" by (metis DiffD1 interior_Int q'(5) r_def KL r) have "interior M = {}" by (metis (no_types, lifting) Int_assoc empty inf.absorb_iff2 interior_Int kl(1) subset_empty x r) then show "content M *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[symmetric] by auto qed ultimately have "norm (?SUM p + sum ?SUM (qq ` r) - integral (cbox a b) f) < e" apply (subst (asm) sum.Union_comp) using qq by (force simp: split_paired_all)+ moreover have "content M *\<^sub>R f x = 0" if "K \ r" "L \ r" "K \ L" "qq K = qq L" "(x, M) \ qq K" for K L x M using tagged_division_ofD(6) qq that by (metis (no_types, lifting)) ultimately have less_e: "norm (?SUM p + sum (?SUM \ qq) r - integral (cbox a b) f) < e" proof (subst (asm) sum.reindex_nontrivial [OF \finite r\]) qed (auto simp: split_paired_all sum.neutral) have norm_le: "norm (cp - ip) \ e + k" if "norm ((cp + cr) - i) < e" "norm (cr - ir) < k" "ip + ir = i" for ir ip i cr cp::'a proof - from that show ?thesis using norm_triangle_le[of "cp + cr - i" "- (cr - ir)"] unfolding that(3)[symmetric] norm_minus_cancel by (auto simp add: algebra_simps) qed have "?lhs = norm (?SUM p - (\(x, k)\p. integral k f))" unfolding split_def sum_subtractf .. also have "\ \ e + k" proof (rule norm_le[OF less_e]) have lessk: "k * real (card r) / (1 + real (card r)) < k" using \k>0\ by (auto simp add: field_simps) have "norm (sum (?SUM \ qq) r - (\k\r. integral k f)) \ (\x\r. k / (real (card r) + 1))" unfolding sum_subtractf[symmetric] by (force dest: qq intro!: sum_norm_le) also have "... < k" by (simp add: lessk add.commute mult.commute) finally show "norm (sum (?SUM \ qq) r - (\k\r. integral k f)) < k" . next from q(1) have [simp]: "snd ` p \ q = q" by auto have "integral l f = 0" if inp: "(x, l) \ p" "(y, m) \ p" and ne: "(x, l) \ (y, m)" and "l = m" for x l y m proof - obtain u v where uv: "l = cbox u v" using inp p'(4) by blast have "content (cbox u v) = 0" unfolding content_eq_0_interior using that p(1) uv by (auto dest: tagged_partial_division_ofD) then show ?thesis using uv by blast qed then have "(\(x, K)\p. integral K f) = (\K\snd ` p. integral K f)" apply (subst sum.reindex_nontrivial [OF \finite p\]) unfolding split_paired_all split_def by auto then show "(\(x, k)\p. integral k f) + (\k\r. integral k f) = integral (cbox a b) f" unfolding integral_combine_division_topdown[OF intf qdiv] r_def using q'(1) p'(1) sum.union_disjoint [of "snd ` p" "q - snd ` p", symmetric] by simp qed finally show "?lhs \ e + k" . qed lemma Henstock_lemma_part2: fixes f :: "'m::euclidean_space \ 'n::euclidean_space" assumes fed: "f integrable_on cbox a b" "e > 0" "gauge d" and less_e: "\\. \\ tagged_division_of (cbox a b); d fine \\ \ norm (sum (\(x,k). content k *\<^sub>R f x) \ - integral (cbox a b) f) < e" and tag: "p tagged_partial_division_of (cbox a b)" and "d fine p" shows "sum (\(x,k). norm (content k *\<^sub>R f x - integral k f)) p \ 2 * real (DIM('n)) * e" proof - have "finite p" using tag tagged_partial_division_ofD by blast then show ?thesis unfolding split_def proof (rule sum_norm_allsubsets_bound) fix Q assume Q: "Q \ p" then have fine: "d fine Q" by (simp add: \d fine p\ fine_subset) show "norm (\x\Q. content (snd x) *\<^sub>R f (fst x) - integral (snd x) f) \ e" apply (rule Henstock_lemma_part1[OF fed less_e, unfolded split_def]) using Q tag tagged_partial_division_subset by (force simp add: fine)+ qed qed lemma Henstock_lemma: fixes f :: "'m::euclidean_space \ 'n::euclidean_space" assumes intf: "f integrable_on cbox a b" and "e > 0" obtains \ where "gauge \" and "\p. \p tagged_partial_division_of (cbox a b); \ fine p\ \ sum (\(x,k). norm(content k *\<^sub>R f x - integral k f)) p < e" proof - have *: "e/(2 * (real DIM('n) + 1)) > 0" using \e > 0\ by simp with integrable_integral[OF intf, unfolded has_integral] obtain \ where "gauge \" and \: "\\. \\ tagged_division_of cbox a b; \ fine \\ \ norm ((\(x,K)\\. content K *\<^sub>R f x) - integral (cbox a b) f) < e/(2 * (real DIM('n) + 1))" by metis show thesis proof (rule that [OF \gauge \\]) fix p assume p: "p tagged_partial_division_of cbox a b" "\ fine p" have "(\(x,K)\p. norm (content K *\<^sub>R f x - integral K f)) \ 2 * real DIM('n) * (e/(2 * (real DIM('n) + 1)))" using Henstock_lemma_part2[OF intf * \gauge \\ \ p] by metis also have "... < e" using \e > 0\ by (auto simp add: field_simps) finally show "(\(x,K)\p. norm (content K *\<^sub>R f x - integral K f)) < e" . qed qed subsection \Monotone convergence (bounded interval first)\ lemma bounded_increasing_convergent: fixes f :: "nat \ real" shows "\bounded (range f); \n. f n \ f (Suc n)\ \ \l. f \ l" using Bseq_mono_convergent[of f] incseq_Suc_iff[of f] by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def) lemma monotone_convergence_interval: fixes f :: "nat \ 'n::euclidean_space \ real" assumes intf: "\k. (f k) integrable_on cbox a b" and le: "\k x. x \ cbox a b \ (f k x) \ f (Suc k) x" and fg: "\x. x \ cbox a b \ ((\k. f k x) \ g x) sequentially" and bou: "bounded (range (\k. integral (cbox a b) (f k)))" shows "g integrable_on cbox a b \ ((\k. integral (cbox a b) (f k)) \ integral (cbox a b) g) sequentially" proof (cases "content (cbox a b) = 0") case True then show ?thesis by auto next case False have fg1: "(f k x) \ (g x)" if x: "x \ cbox a b" for x k proof - have "\\<^sub>F j in sequentially. f k x \ f j x" proof (rule eventually_sequentiallyI [of k]) show "\j. k \ j \ f k x \ f j x" using le x by (force intro: transitive_stepwise_le) qed then show "f k x \ g x" using tendsto_lowerbound [OF fg] x trivial_limit_sequentially by blast qed have int_inc: "\n. integral (cbox a b) (f n) \ integral (cbox a b) (f (Suc n))" by (metis integral_le intf le) then obtain i where i: "(\k. integral (cbox a b) (f k)) \ i" using bounded_increasing_convergent bou by blast have "\k. \\<^sub>F x in sequentially. integral (cbox a b) (f k) \ integral (cbox a b) (f x)" unfolding eventually_sequentially by (force intro: transitive_stepwise_le int_inc) then have i': "\k. (integral(cbox a b) (f k)) \ i" using tendsto_le [OF trivial_limit_sequentially i] by blast have "(g has_integral i) (cbox a b)" unfolding has_integral real_norm_def proof clarify fix e::real assume e: "e > 0" have "\k. (\\. gauge \ \ (\\. \ tagged_division_of (cbox a b) \ \ fine \ \ abs ((\(x,K)\\. content K *\<^sub>R f k x) - integral (cbox a b) (f k)) < e/2 ^ (k + 2)))" using intf e by (auto simp: has_integral_integral has_integral) then obtain c where c: "\x. gauge (c x)" "\x \. \\ tagged_division_of cbox a b; c x fine \\ \ abs ((\(u,K)\\. content K *\<^sub>R f x u) - integral (cbox a b) (f x)) < e/2 ^ (x + 2)" by metis have "\r. \k\r. 0 \ i - (integral (cbox a b) (f k)) \ i - (integral (cbox a b) (f k)) < e/4" proof - have "e/4 > 0" using e by auto show ?thesis using LIMSEQ_D [OF i \e/4 > 0\] i' by auto qed then obtain r where r: "\k. r \ k \ 0 \ i - integral (cbox a b) (f k)" "\k. r \ k \ i - integral (cbox a b) (f k) < e/4" by metis have "\n\r. \k\n. 0 \ (g x) - (f k x) \ (g x) - (f k x) < e/(4 * content(cbox a b))" if "x \ cbox a b" for x proof - have "e/(4 * content (cbox a b)) > 0" by (simp add: False content_lt_nz e) with fg that LIMSEQ_D obtain N where "\n\N. norm (f n x - g x) < e/(4 * content (cbox a b))" by metis then show "\n\r. \k\n. 0 \ g x - f k x \ g x - f k x < e/(4 * content (cbox a b))" apply (rule_tac x="N + r" in exI) using fg1[OF that] by (auto simp add: field_simps) qed then obtain m where r_le_m: "\x. x \ cbox a b \ r \ m x" and m: "\x k. \x \ cbox a b; m x \ k\ \ 0 \ g x - f k x \ g x - f k x < e/(4 * content (cbox a b))" by metis define d where "d x = c (m x) x" for x show "\\. gauge \ \ (\\. \ tagged_division_of cbox a b \ \ fine \ \ abs ((\(x,K)\\. content K *\<^sub>R g x) - i) < e)" proof (rule exI, safe) show "gauge d" using c(1) unfolding gauge_def d_def by auto next fix \ assume ptag: "\ tagged_division_of (cbox a b)" and "d fine \" note p'=tagged_division_ofD[OF ptag] obtain s where s: "\x. x \ \ \ m (fst x) \ s" by (metis finite_imageI finite_nat_set_iff_bounded_le p'(1) rev_image_eqI) have *: "\a - d\ < e" if "\a - b\ \ e/4" "\b - c\ < e/2" "\c - d\ < e/4" for a b c d using that norm_triangle_lt[of "a - b" "b - c" "3* e/4"] norm_triangle_lt[of "a - b + (b - c)" "c - d" e] by (auto simp add: algebra_simps) show "\(\(x, k)\\. content k *\<^sub>R g x) - i\ < e" proof (rule *) have "\(\(x,K)\\. content K *\<^sub>R g x) - (\(x,K)\\. content K *\<^sub>R f (m x) x)\ \ (\i\\. \(case i of (x, K) \ content K *\<^sub>R g x) - (case i of (x, K) \ content K *\<^sub>R f (m x) x)\)" by (metis (mono_tags) sum_subtractf sum_abs) also have "... \ (\(x, k)\\. content k * (e/(4 * content (cbox a b))))" proof (rule sum_mono, simp add: split_paired_all) fix x K assume xk: "(x,K) \ \" with ptag have x: "x \ cbox a b" by blast then have "abs (content K * (g x - f (m x) x)) \ content K * (e/(4 * content (cbox a b)))" by (metis m[OF x] mult_nonneg_nonneg abs_of_nonneg less_eq_real_def measure_nonneg mult_left_mono order_refl) then show "\content K * g x - content K * f (m x) x\ \ content K * e/(4 * content (cbox a b))" by (simp add: algebra_simps) qed also have "... = (e/(4 * content (cbox a b))) * (\(x, k)\\. content k)" by (simp add: sum_distrib_left sum_divide_distrib split_def mult.commute) also have "... \ e/4" by (metis False additive_content_tagged_division [OF ptag] nonzero_mult_divide_mult_cancel_right order_refl times_divide_eq_left) finally show "\(\(x,K)\\. content K *\<^sub>R g x) - (\(x,K)\\. content K *\<^sub>R f (m x) x)\ \ e/4" . next have "norm ((\(x,K)\\. content K *\<^sub>R f (m x) x) - (\(x,K)\\. integral K (f (m x)))) \ norm (\j = 0..s. \(x,K)\{xk \ \. m (fst xk) = j}. content K *\<^sub>R f (m x) x - integral K (f (m x)))" apply (subst sum.group) using s by (auto simp: sum_subtractf split_def p'(1)) also have "\ < e/2" proof - have "norm (\j = 0..s. \(x, k)\{xk \ \. m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x))) \ (\i = 0..s. e/2 ^ (i + 2))" proof (rule sum_norm_le) fix t assume "t \ {0..s}" have "norm (\(x,k)\{xk \ \. m (fst xk) = t}. content k *\<^sub>R f (m x) x - integral k (f (m x))) = norm (\(x,k)\{xk \ \. m (fst xk) = t}. content k *\<^sub>R f t x - integral k (f t))" by (force intro!: sum.cong arg_cong[where f=norm]) also have "... \ e/2 ^ (t + 2)" proof (rule Henstock_lemma_part1 [OF intf]) show "{xk \ \. m (fst xk) = t} tagged_partial_division_of cbox a b" proof (rule tagged_partial_division_subset[of \]) show "\ tagged_partial_division_of cbox a b" using ptag tagged_division_of_def by blast qed auto show "c t fine {xk \ \. m (fst xk) = t}" using \d fine \\ by (auto simp: fine_def d_def) qed (use c e in auto) finally show "norm (\(x,K)\{xk \ \. m (fst xk) = t}. content K *\<^sub>R f (m x) x - integral K (f (m x))) \ e/2 ^ (t + 2)" . qed also have "... = (e/2/2) * (\i = 0..s. (1/2) ^ i)" by (simp add: sum_distrib_left field_simps) also have "\ < e/2" by (simp add: sum_gp mult_strict_left_mono[OF _ e]) finally show "norm (\j = 0..s. \(x, k)\{xk \ \. m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x))) < e/2" . qed finally show "\(\(x,K)\\. content K *\<^sub>R f (m x) x) - (\(x,K)\\. integral K (f (m x)))\ < e/2" by simp next have comb: "integral (cbox a b) (f y) = (\(x, k)\\. integral k (f y))" for y using integral_combine_tagged_division_topdown[OF intf ptag] by metis have f_le: "\y m n. \y \ cbox a b; n\m\ \ f m y \ f n y" using le by (auto intro: transitive_stepwise_le) have "(\(x, k)\\. integral k (f r)) \ (\(x, K)\\. integral K (f (m x)))" proof (rule sum_mono, simp add: split_paired_all) fix x K assume xK: "(x, K) \ \" show "integral K (f r) \ integral K (f (m x))" proof (rule integral_le) show "f r integrable_on K" by (metis integrable_on_subcbox intf p'(3) p'(4) xK) show "f (m x) integrable_on K" by (metis elementary_interval integrable_on_subdivision intf p'(3) p'(4) xK) show "f r y \ f (m x) y" if "y \ K" for y using that r_le_m[of x] p'(2-3)[OF xK] f_le by auto qed qed moreover have "(\(x, K)\\. integral K (f (m x))) \ (\(x, k)\\. integral k (f s))" proof (rule sum_mono, simp add: split_paired_all) fix x K assume xK: "(x, K) \ \" show "integral K (f (m x)) \ integral K (f s)" proof (rule integral_le) show "f (m x) integrable_on K" by (metis elementary_interval integrable_on_subdivision intf p'(3) p'(4) xK) show "f s integrable_on K" by (metis integrable_on_subcbox intf p'(3) p'(4) xK) show "f (m x) y \ f s y" if "y \ K" for y using that s xK f_le p'(3) by fastforce qed qed moreover have "0 \ i - integral (cbox a b) (f r)" "i - integral (cbox a b) (f r) < e/4" using r by auto ultimately show "\(\(x,K)\\. integral K (f (m x))) - i\ < e/4" using comb i'[of s] by auto qed qed qed with i integral_unique show ?thesis by blast qed lemma monotone_convergence_increasing: fixes f :: "nat \ 'n::euclidean_space \ real" assumes int_f: "\k. (f k) integrable_on S" and "\k x. x \ S \ (f k x) \ (f (Suc k) x)" and fg: "\x. x \ S \ ((\k. f k x) \ g x) sequentially" and bou: "bounded (range (\k. integral S (f k)))" shows "g integrable_on S \ ((\k. integral S (f k)) \ integral S g) sequentially" proof - have lem: "g integrable_on S \ ((\k. integral S (f k)) \ integral S g) sequentially" if f0: "\k x. x \ S \ 0 \ f k x" and int_f: "\k. (f k) integrable_on S" and le: "\k x. x \ S \ f k x \ f (Suc k) x" and lim: "\x. x \ S \ ((\k. f k x) \ g x) sequentially" and bou: "bounded (range(\k. integral S (f k)))" for f :: "nat \ 'n::euclidean_space \ real" and g S proof - have fg: "(f k x) \ (g x)" if "x \ S" for x k proof - have "\xa. k \ xa \ f k x \ f xa x" using le by (force intro: transitive_stepwise_le that) then show ?thesis using tendsto_lowerbound [OF lim [OF that]] eventually_sequentiallyI by force qed obtain i where i: "(\k. integral S (f k)) \ i" using bounded_increasing_convergent [OF bou] le int_f integral_le by blast have i': "(integral S (f k)) \ i" for k proof - have "\k. \x. x \ S \ \n\k. f k x \ f n x" using le by (force intro: transitive_stepwise_le) then show ?thesis using tendsto_lowerbound [OF i eventually_sequentiallyI trivial_limit_sequentially] by (meson int_f integral_le) qed let ?f = "(\k x. if x \ S then f k x else 0)" let ?g = "(\x. if x \ S then g x else 0)" have int: "?f k integrable_on cbox a b" for a b k by (simp add: int_f integrable_altD(1)) have int': "\k a b. f k integrable_on cbox a b \ S" using int by (simp add: Int_commute integrable_restrict_Int) have g: "?g integrable_on cbox a b \ (\k. integral (cbox a b) (?f k)) \ integral (cbox a b) ?g" for a b proof (rule monotone_convergence_interval) have "norm (integral (cbox a b) (?f k)) \ norm (integral S (f k))" for k proof - have "0 \ integral (cbox a b) (?f k)" by (metis (no_types) integral_nonneg Int_iff f0 inf_commute integral_restrict_Int int') moreover have "0 \ integral S (f k)" by (simp add: integral_nonneg f0 int_f) moreover have "integral (S \ cbox a b) (f k) \ integral S (f k)" by (metis f0 inf_commute int' int_f integral_subset_le le_inf_iff order_refl) ultimately show ?thesis by (simp add: integral_restrict_Int) qed moreover obtain B where "\x. x \ range (\k. integral S (f k)) \ norm x \ B" using bou unfolding bounded_iff by blast ultimately show "bounded (range (\k. integral (cbox a b) (?f k)))" unfolding bounded_iff by (blast intro: order_trans) qed (use int le lim in auto) moreover have "\B>0. \a b. ball 0 B \ cbox a b \ norm (integral (cbox a b) ?g - i) < e" if "0 < e" for e proof - have "e/4>0" using that by auto with LIMSEQ_D [OF i] obtain N where N: "\n. n \ N \ norm (integral S (f n) - i) < e/4" by metis with int_f[of N, unfolded has_integral_integral has_integral_alt'[of "f N"]] obtain B where "0 < B" and B: "\a b. ball 0 B \ cbox a b \ norm (integral (cbox a b) (?f N) - integral S (f N)) < e/4" by (meson \0 < e/4\) have "norm (integral (cbox a b) ?g - i) < e" if ab: "ball 0 B \ cbox a b" for a b proof - obtain M where M: "\n. n \ M \ abs (integral (cbox a b) (?f n) - integral (cbox a b) ?g) < e/2" using \e > 0\ g by (fastforce simp add: dest!: LIMSEQ_D [where r = "e/2"]) have *: "\\ \ g. \\\ - i\ < e/2; \\ - g\ < e/2; \ \ \; \ \ i\ \ \g - i\ < e" unfolding real_inner_1_right by arith show "norm (integral (cbox a b) ?g - i) < e" unfolding real_norm_def proof (rule *) show "\integral (cbox a b) (?f N) - i\ < e/2" proof (rule abs_triangle_half_l) show "\integral (cbox a b) (?f N) - integral S (f N)\ < e/2/2" using B[OF ab] by simp show "abs (i - integral S (f N)) < e/2/2" using N by (simp add: abs_minus_commute) qed show "\integral (cbox a b) (?f (M + N)) - integral (cbox a b) ?g\ < e/2" by (metis le_add1 M[of "M + N"]) show "integral (cbox a b) (?f N) \ integral (cbox a b) (?f (M + N))" proof (intro ballI integral_le[OF int int]) fix x assume "x \ cbox a b" have "(f m x) \ (f n x)" if "x \ S" "n \ m" for m n proof (rule transitive_stepwise_le [OF \n \ m\ order_refl]) show "\u y z. \f u x \ f y x; f y x \ f z x\ \ f u x \ f z x" using dual_order.trans by blast qed (simp add: le \x \ S\) then show "(?f N)x \ (?f (M+N))x" by auto qed have "integral (cbox a b \ S) (f (M + N)) \ integral S (f (M + N))" by (metis Int_lower1 f0 inf_commute int' int_f integral_subset_le) then have "integral (cbox a b) (?f (M + N)) \ integral S (f (M + N))" by (metis (no_types) inf_commute integral_restrict_Int) also have "... \ i" using i'[of "M + N"] by auto finally show "integral (cbox a b) (?f (M + N)) \ i" . qed qed then show ?thesis using \0 < B\ by blast qed ultimately have "(g has_integral i) S" unfolding has_integral_alt' by auto then show ?thesis using has_integral_integrable_integral i integral_unique by metis qed have sub: "\k. integral S (\x. f k x - f 0 x) = integral S (f k) - integral S (f 0)" by (simp add: integral_diff int_f) have *: "\x m n. x \ S \ n\m \ f m x \ f n x" using assms(2) by (force intro: transitive_stepwise_le) have gf: "(\x. g x - f 0 x) integrable_on S \ ((\k. integral S (\x. f (Suc k) x - f 0 x)) \ integral S (\x. g x - f 0 x)) sequentially" proof (rule lem) show "\k. (\x. f (Suc k) x - f 0 x) integrable_on S" by (simp add: integrable_diff int_f) show "(\k. f (Suc k) x - f 0 x) \ g x - f 0 x" if "x \ S" for x proof - have "(\n. f (Suc n) x) \ g x" using LIMSEQ_ignore_initial_segment[OF fg[OF \x \ S\], of 1] by simp then show ?thesis by (simp add: tendsto_diff) qed show "bounded (range (\k. integral S (\x. f (Suc k) x - f 0 x)))" proof - obtain B where B: "\k. norm (integral S (f k)) \ B" using bou by (auto simp: bounded_iff) then have "norm (integral S (\x. f (Suc k) x - f 0 x)) \ B + norm (integral S (f 0))" for k unfolding sub by (meson add_le_cancel_right norm_triangle_le_diff) then show ?thesis unfolding bounded_iff by blast qed qed (use * in auto) then have "(\x. integral S (\xa. f (Suc x) xa - f 0 xa) + integral S (f 0)) \ integral S (\x. g x - f 0 x) + integral S (f 0)" by (auto simp add: tendsto_add) moreover have "(\x. g x - f 0 x + f 0 x) integrable_on S" using gf integrable_add int_f [of 0] by metis ultimately show ?thesis by (simp add: integral_diff int_f LIMSEQ_imp_Suc sub) qed lemma has_integral_monotone_convergence_increasing: fixes f :: "nat \ 'a::euclidean_space \ real" assumes f: "\k. (f k has_integral x k) s" assumes "\k x. x \ s \ f k x \ f (Suc k) x" assumes "\x. x \ s \ (\k. f k x) \ g x" assumes "x \ x'" shows "(g has_integral x') s" proof - have x_eq: "x = (\i. integral s (f i))" by (simp add: integral_unique[OF f]) then have x: "range(\k. integral s (f k)) = range x" by auto have *: "g integrable_on s \ (\k. integral s (f k)) \ integral s g" proof (intro monotone_convergence_increasing allI ballI assms) show "bounded (range(\k. integral s (f k)))" using x convergent_imp_bounded assms by metis qed (use f in auto) then have "integral s g = x'" by (intro LIMSEQ_unique[OF _ \x \ x'\]) (simp add: x_eq) with * show ?thesis by (simp add: has_integral_integral) qed lemma monotone_convergence_decreasing: fixes f :: "nat \ 'n::euclidean_space \ real" assumes intf: "\k. (f k) integrable_on S" and le: "\k x. x \ S \ f (Suc k) x \ f k x" and fg: "\x. x \ S \ ((\k. f k x) \ g x) sequentially" and bou: "bounded (range(\k. integral S (f k)))" shows "g integrable_on S \ (\k. integral S (f k)) \ integral S g" proof - have *: "range(\k. integral S (\x. - f k x)) = (*\<^sub>R) (- 1) ` (range(\k. integral S (f k)))" by force have "(\x. - g x) integrable_on S \ (\k. integral S (\x. - f k x)) \ integral S (\x. - g x)" proof (rule monotone_convergence_increasing) show "\k. (\x. - f k x) integrable_on S" by (blast intro: integrable_neg intf) show "\k x. x \ S \ - f k x \ - f (Suc k) x" by (simp add: le) show "\x. x \ S \ (\k. - f k x) \ - g x" by (simp add: fg tendsto_minus) show "bounded (range(\k. integral S (\x. - f k x)))" using "*" bou bounded_scaling by auto qed then show ?thesis by (force dest: integrable_neg tendsto_minus) qed lemma integral_norm_bound_integral: fixes f :: "'n::euclidean_space \ 'a::banach" assumes int_f: "f integrable_on S" and int_g: "g integrable_on S" and le_g: "\x. x \ S \ norm (f x) \ g x" shows "norm (integral S f) \ integral S g" proof - have norm: "norm \ \ y + e" if "norm \ \ x" and "\x - y\ < e/2" and "norm (\ - \) < e/2" for e x y and \ \ :: 'a proof - have "norm (\ - \) < e/2" by (metis norm_minus_commute that(3)) moreover have "x \ y + e/2" using that(2) by linarith ultimately show ?thesis using that(1) le_less_trans[OF norm_triangle_sub[of \ \]] by (auto simp: less_imp_le) qed have lem: "norm (integral(cbox a b) f) \ integral (cbox a b) g" if f: "f integrable_on cbox a b" and g: "g integrable_on cbox a b" and nle: "\x. x \ cbox a b \ norm (f x) \ g x" for f :: "'n \ 'a" and g a b proof (rule field_le_epsilon) fix e :: real assume "e > 0" then have e: "e/2 > 0" by auto with integrable_integral[OF f,unfolded has_integral[of f]] obtain \ where \: "gauge \" "\\. \ tagged_division_of cbox a b \ \ fine \ \ norm ((\(x, k)\\. content k *\<^sub>R f x) - integral (cbox a b) f) < e/2" by meson moreover from integrable_integral[OF g,unfolded has_integral[of g]] e obtain \ where \: "gauge \" "\\. \ tagged_division_of cbox a b \ \ fine \ \ norm ((\(x, k)\\. content k *\<^sub>R g x) - integral (cbox a b) g) < e/2" by meson ultimately have "gauge (\x. \ x \ \ x)" using gauge_Int by blast with fine_division_exists obtain \ where p: "\ tagged_division_of cbox a b" "(\x. \ x \ \ x) fine \" by metis have "\ fine \" "\ fine \" using fine_Int p(2) by blast+ show "norm (integral (cbox a b) f) \ integral (cbox a b) g + e" proof (rule norm) have "norm (content K *\<^sub>R f x) \ content K *\<^sub>R g x" if "(x, K) \ \" for x K proof- have K: "x \ K" "K \ cbox a b" using \(x, K) \ \\ p(1) by blast+ obtain u v where "K = cbox u v" using \(x, K) \ \\ p(1) by blast moreover have "content K * norm (f x) \ content K * g x" by (meson K(1) K(2) content_pos_le mult_left_mono nle subsetD) then show ?thesis by simp qed then show "norm (\(x, k)\\. content k *\<^sub>R f x) \ (\(x, k)\\. content k *\<^sub>R g x)" by (simp add: sum_norm_le split_def) show "norm ((\(x, k)\\. content k *\<^sub>R f x) - integral (cbox a b) f) < e/2" using \\ fine \\ \ p(1) by simp show "\(\(x, k)\\. content k *\<^sub>R g x) - integral (cbox a b) g\ < e/2" using \\ fine \\ \ p(1) by simp qed qed show ?thesis proof (rule field_le_epsilon) fix e :: real assume "e > 0" then have e: "e/2 > 0" by auto let ?f = "(\x. if x \ S then f x else 0)" let ?g = "(\x. if x \ S then g x else 0)" have f: "?f integrable_on cbox a b" and g: "?g integrable_on cbox a b" for a b using int_f int_g integrable_altD by auto obtain Bf where "0 < Bf" and Bf: "\a b. ball 0 Bf \ cbox a b \ \z. (?f has_integral z) (cbox a b) \ norm (z - integral S f) < e/2" using integrable_integral [OF int_f,unfolded has_integral'[of f]] e that by blast obtain Bg where "0 < Bg" and Bg: "\a b. ball 0 Bg \ cbox a b \ \z. (?g has_integral z) (cbox a b) \ norm (z - integral S g) < e/2" using integrable_integral [OF int_g,unfolded has_integral'[of g]] e that by blast obtain a b::'n where ab: "ball 0 Bf \ ball 0 Bg \ cbox a b" using ball_max_Un by (metis bounded_ball bounded_subset_cbox_symmetric) have "ball 0 Bf \ cbox a b" using ab by auto with Bf obtain z where int_fz: "(?f has_integral z) (cbox a b)" and z: "norm (z - integral S f) < e/2" by meson have "ball 0 Bg \ cbox a b" using ab by auto with Bg obtain w where int_gw: "(?g has_integral w) (cbox a b)" and w: "norm (w - integral S g) < e/2" by meson show "norm (integral S f) \ integral S g + e" proof (rule norm) show "norm (integral (cbox a b) ?f) \ integral (cbox a b) ?g" by (simp add: le_g lem[OF f g, of a b]) show "\integral (cbox a b) ?g - integral S g\ < e/2" using int_gw integral_unique w by auto show "norm (integral (cbox a b) ?f - integral S f) < e/2" using int_fz integral_unique z by blast qed qed qed lemma continuous_on_imp_absolutely_integrable_on: fixes f::"real \ 'a::banach" shows "continuous_on {a..b} f \ norm (integral {a..b} f) \ integral {a..b} (\x. norm (f x))" by (intro integral_norm_bound_integral integrable_continuous_real continuous_on_norm) auto lemma integral_bound: fixes f::"real \ 'a::banach" assumes "a \ b" assumes "continuous_on {a .. b} f" assumes "\t. t \ {a .. b} \ norm (f t) \ B" shows "norm (integral {a .. b} f) \ B * (b - a)" proof - note continuous_on_imp_absolutely_integrable_on[OF assms(2)] also have "integral {a..b} (\x. norm (f x)) \ integral {a..b} (\_. B)" by (rule integral_le) (auto intro!: integrable_continuous_real continuous_intros assms) also have "\ = B * (b - a)" using assms by simp finally show ?thesis . qed lemma integral_norm_bound_integral_component: fixes f :: "'n::euclidean_space \ 'a::banach" fixes g :: "'n \ 'b::euclidean_space" assumes f: "f integrable_on S" and g: "g integrable_on S" and fg: "\x. x \ S \ norm(f x) \ (g x)\k" shows "norm (integral S f) \ (integral S g)\k" proof - have "norm (integral S f) \ integral S ((\x. x \ k) \ g)" using integral_norm_bound_integral[OF f integrable_linear[OF g]] by (simp add: bounded_linear_inner_left fg) then show ?thesis unfolding o_def integral_component_eq[OF g] . qed lemma has_integral_norm_bound_integral_component: fixes f :: "'n::euclidean_space \ 'a::banach" fixes g :: "'n \ 'b::euclidean_space" assumes f: "(f has_integral i) S" and g: "(g has_integral j) S" and "\x. x \ S \ norm (f x) \ (g x)\k" shows "norm i \ j\k" using integral_norm_bound_integral_component[of f S g k] unfolding integral_unique[OF f] integral_unique[OF g] using assms by auto lemma uniformly_convergent_improper_integral: fixes f :: "'b \ real \ 'a :: {banach}" assumes deriv: "\x. x \ a \ (G has_field_derivative g x) (at x within {a..})" assumes integrable: "\a' b x. x \ A \ a' \ a \ b \ a' \ f x integrable_on {a'..b}" assumes G: "convergent G" assumes le: "\y x. y \ A \ x \ a \ norm (f y x) \ g x" shows "uniformly_convergent_on A (\b x. integral {a..b} (f x))" proof (intro Cauchy_uniformly_convergent uniformly_Cauchy_onI', goal_cases) case (1 \) from G have "Cauchy G" by (auto intro!: convergent_Cauchy) with 1 obtain M where M: "dist (G (real m)) (G (real n)) < \" if "m \ M" "n \ M" for m n by (force simp: Cauchy_def) define M' where "M' = max (nat \a\) M" show ?case proof (rule exI[of _ M'], safe, goal_cases) case (1 x m n) have M': "M' \ a" "M' \ M" unfolding M'_def by linarith+ have int_g: "(g has_integral (G (real n) - G (real m))) {real m..real n}" using 1 M' by (intro fundamental_theorem_of_calculus) (auto simp: has_field_derivative_iff_has_vector_derivative [symmetric] intro!: DERIV_subset[OF deriv]) have int_f: "f x integrable_on {a'..real n}" if "a' \ a" for a' using that 1 by (cases "a' \ real n") (auto intro: integrable) have "dist (integral {a..real m} (f x)) (integral {a..real n} (f x)) = norm (integral {a..real n} (f x) - integral {a..real m} (f x))" by (simp add: dist_norm norm_minus_commute) also have "integral {a..real m} (f x) + integral {real m..real n} (f x) = integral {a..real n} (f x)" using M' and 1 by (intro integral_combine int_f) auto hence "integral {a..real n} (f x) - integral {a..real m} (f x) = integral {real m..real n} (f x)" by (simp add: algebra_simps) also have "norm \ \ integral {real m..real n} g" using le 1 M' int_f int_g by (intro integral_norm_bound_integral) auto also from int_g have "integral {real m..real n} g = G (real n) - G (real m)" by (simp add: has_integral_iff) also have "\ \ dist (G m) (G n)" by (simp add: dist_norm) also from 1 and M' have "\ < \" by (intro M) auto finally show ?case . qed qed lemma uniformly_convergent_improper_integral': fixes f :: "'b \ real \ 'a :: {banach, real_normed_algebra}" assumes deriv: "\x. x \ a \ (G has_field_derivative g x) (at x within {a..})" assumes integrable: "\a' b x. x \ A \ a' \ a \ b \ a' \ f x integrable_on {a'..b}" assumes G: "convergent G" assumes le: "eventually (\x. \y\A. norm (f y x) \ g x) at_top" shows "uniformly_convergent_on A (\b x. integral {a..b} (f x))" proof - from le obtain a'' where le: "\y x. y \ A \ x \ a'' \ norm (f y x) \ g x" by (auto simp: eventually_at_top_linorder) define a' where "a' = max a a''" have "uniformly_convergent_on A (\b x. integral {a'..real b} (f x))" proof (rule uniformly_convergent_improper_integral) fix t assume t: "t \ a'" hence "(G has_field_derivative g t) (at t within {a..})" by (intro deriv) (auto simp: a'_def) moreover have "{a'..} \ {a..}" unfolding a'_def by auto ultimately show "(G has_field_derivative g t) (at t within {a'..})" by (rule DERIV_subset) qed (insert le, auto simp: a'_def intro: integrable G) hence "uniformly_convergent_on A (\b x. integral {a..a'} (f x) + integral {a'..real b} (f x))" (is ?P) by (intro uniformly_convergent_add) auto also have "eventually (\x. \y\A. integral {a..a'} (f y) + integral {a'..x} (f y) = integral {a..x} (f y)) sequentially" by (intro eventually_mono [OF eventually_ge_at_top[of "nat \a'\"]] ballI integral_combine) (auto simp: a'_def intro: integrable) hence "?P \ ?thesis" by (intro uniformly_convergent_cong) simp_all finally show ?thesis . qed subsection \differentiation under the integral sign\ lemma integral_continuous_on_param: fixes f::"'a::topological_space \ 'b::euclidean_space \ 'c::banach" assumes cont_fx: "continuous_on (U \ cbox a b) (\(x, t). f x t)" shows "continuous_on U (\x. integral (cbox a b) (f x))" proof cases assume "content (cbox a b) \ 0" then have ne: "cbox a b \ {}" by auto note [continuous_intros] = continuous_on_compose2[OF cont_fx, where f="\y. Pair x y" for x, unfolded split_beta fst_conv snd_conv] show ?thesis unfolding continuous_on_def proof (safe intro!: tendstoI) fix e'::real and x assume "e' > 0" define e where "e = e' / (content (cbox a b) + 1)" have "e > 0" using \e' > 0\ by (auto simp: e_def intro!: divide_pos_pos add_nonneg_pos) assume "x \ U" from continuous_on_prod_compactE[OF cont_fx compact_cbox \x \ U\ \0 < e\] obtain X0 where X0: "x \ X0" "open X0" and fx_bound: "\y t. y \ X0 \ U \ t \ cbox a b \ norm (f y t - f x t) \ e" unfolding split_beta fst_conv snd_conv dist_norm by metis have "\\<^sub>F y in at x within U. y \ X0 \ U" using X0(1) X0(2) eventually_at_topological by auto then show "\\<^sub>F y in at x within U. dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) < e'" proof eventually_elim case (elim y) have "dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) = norm (integral (cbox a b) (\t. f y t - f x t))" using elim \x \ U\ unfolding dist_norm by (subst integral_diff) (auto intro!: integrable_continuous continuous_intros) also have "\ \ e * content (cbox a b)" using elim \x \ U\ by (intro integrable_bound) (auto intro!: fx_bound \x \ U \ less_imp_le[OF \0 < e\] integrable_continuous continuous_intros) also have "\ < e'" using \0 < e'\ \e > 0\ by (auto simp: e_def field_split_simps) finally show "dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) < e'" . qed qed qed (auto intro!: continuous_on_const) lemma leibniz_rule: fixes f::"'a::banach \ 'b::euclidean_space \ 'c::banach" assumes fx: "\x t. x \ U \ t \ cbox a b \ ((\x. f x t) has_derivative blinfun_apply (fx x t)) (at x within U)" assumes integrable_f2: "\x. x \ U \ f x integrable_on cbox a b" assumes cont_fx: "continuous_on (U \ (cbox a b)) (\(x, t). fx x t)" assumes [intro]: "x0 \ U" assumes "convex U" shows "((\x. integral (cbox a b) (f x)) has_derivative integral (cbox a b) (fx x0)) (at x0 within U)" (is "(?F has_derivative ?dF) _") proof cases assume "content (cbox a b) \ 0" then have ne: "cbox a b \ {}" by auto note [continuous_intros] = continuous_on_compose2[OF cont_fx, where f="\y. Pair x y" for x, unfolded split_beta fst_conv snd_conv] show ?thesis proof (intro has_derivativeI bounded_linear_scaleR_left tendstoI, fold norm_conv_dist) have cont_f1: "\t. t \ cbox a b \ continuous_on U (\x. f x t)" by (auto simp: continuous_on_eq_continuous_within intro!: has_derivative_continuous fx) note [continuous_intros] = continuous_on_compose2[OF cont_f1] fix e'::real assume "e' > 0" define e where "e = e' / (content (cbox a b) + 1)" have "e > 0" using \e' > 0\ by (auto simp: e_def intro!: divide_pos_pos add_nonneg_pos) from continuous_on_prod_compactE[OF cont_fx compact_cbox \x0 \ U\ \e > 0\] obtain X0 where X0: "x0 \ X0" "open X0" and fx_bound: "\x t. x \ X0 \ U \ t \ cbox a b \ norm (fx x t - fx x0 t) \ e" unfolding split_beta fst_conv snd_conv by (metis dist_norm) note eventually_closed_segment[OF \open X0\ \x0 \ X0\, of U] moreover have "\\<^sub>F x in at x0 within U. x \ X0" using \open X0\ \x0 \ X0\ eventually_at_topological by blast moreover have "\\<^sub>F x in at x0 within U. x \ x0" by (auto simp: eventually_at_filter) moreover have "\\<^sub>F x in at x0 within U. x \ U" by (auto simp: eventually_at_filter) ultimately show "\\<^sub>F x in at x0 within U. norm ((?F x - ?F x0 - ?dF (x - x0)) /\<^sub>R norm (x - x0)) < e'" proof eventually_elim case (elim x) from elim have "0 < norm (x - x0)" by simp have "closed_segment x0 x \ U" by (rule \convex U\[unfolded convex_contains_segment, rule_format, OF \x0 \ U\ \x \ U\]) from elim have [intro]: "x \ U" by auto have "?F x - ?F x0 - ?dF (x - x0) = integral (cbox a b) (\y. f x y - f x0 y - fx x0 y (x - x0))" (is "_ = ?id") using \x \ x0\ by (subst blinfun_apply_integral integral_diff, auto intro!: integrable_diff integrable_f2 continuous_intros intro: integrable_continuous)+ also { fix t assume t: "t \ (cbox a b)" have seg: "\t. t \ {0..1} \ x0 + t *\<^sub>R (x - x0) \ X0 \ U" using \closed_segment x0 x \ U\ \closed_segment x0 x \ X0\ by (force simp: closed_segment_def algebra_simps) from t have deriv: "((\x. f x t) has_derivative (fx y t)) (at y within X0 \ U)" if "y \ X0 \ U" for y unfolding has_vector_derivative_def[symmetric] using that \x \ X0\ by (intro has_derivative_subset[OF fx]) auto have "\x. x \ X0 \ U \ onorm (blinfun_apply (fx x t) - (fx x0 t)) \ e" using fx_bound t by (auto simp add: norm_blinfun_def fun_diff_def blinfun.bilinear_simps[symmetric]) from differentiable_bound_linearization[OF seg deriv this] X0 have "norm (f x t - f x0 t - fx x0 t (x - x0)) \ e * norm (x - x0)" by (auto simp add: ac_simps) } then have "norm ?id \ integral (cbox a b) (\_. e * norm (x - x0))" by (intro integral_norm_bound_integral) (auto intro!: continuous_intros integrable_diff integrable_f2 intro: integrable_continuous) also have "\ = content (cbox a b) * e * norm (x - x0)" by simp also have "\ < e' * norm (x - x0)" proof (intro mult_strict_right_mono[OF _ \0 < norm (x - x0)\]) show "content (cbox a b) * e < e'" using \e' > 0\ by (simp add: divide_simps e_def not_less) qed finally have "norm (?F x - ?F x0 - ?dF (x - x0)) < e' * norm (x - x0)" . then show ?case by (auto simp: divide_simps) qed qed (rule blinfun.bounded_linear_right) qed (auto intro!: derivative_eq_intros simp: blinfun.bilinear_simps) lemma has_vector_derivative_eq_has_derivative_blinfun: "(f has_vector_derivative f') (at x within U) \ (f has_derivative blinfun_scaleR_left f') (at x within U)" by (simp add: has_vector_derivative_def) lemma leibniz_rule_vector_derivative: fixes f::"real \ 'b::euclidean_space \ 'c::banach" assumes fx: "\x t. x \ U \ t \ cbox a b \ ((\x. f x t) has_vector_derivative (fx x t)) (at x within U)" assumes integrable_f2: "\x. x \ U \ (f x) integrable_on cbox a b" assumes cont_fx: "continuous_on (U \ cbox a b) (\(x, t). fx x t)" assumes U: "x0 \ U" "convex U" shows "((\x. integral (cbox a b) (f x)) has_vector_derivative integral (cbox a b) (fx x0)) (at x0 within U)" proof - note [continuous_intros] = continuous_on_compose2[OF cont_fx, where f="\y. Pair x y" for x, unfolded split_beta fst_conv snd_conv] show ?thesis unfolding has_vector_derivative_eq_has_derivative_blinfun proof (rule has_derivative_eq_rhs [OF leibniz_rule[OF _ integrable_f2 _ U]]) show "continuous_on (U \ cbox a b) (\(x, t). blinfun_scaleR_left (fx x t))" using cont_fx by (auto simp: split_beta intro!: continuous_intros) show "blinfun_apply (integral (cbox a b) (\t. blinfun_scaleR_left (fx x0 t))) = blinfun_apply (blinfun_scaleR_left (integral (cbox a b) (fx x0)))" by (subst integral_linear[symmetric]) (auto simp: has_vector_derivative_def o_def intro!: integrable_continuous U continuous_intros bounded_linear_intros) qed (use fx in \auto simp: has_vector_derivative_def\) qed lemma has_field_derivative_eq_has_derivative_blinfun: "(f has_field_derivative f') (at x within U) \ (f has_derivative blinfun_mult_right f') (at x within U)" by (simp add: has_field_derivative_def) lemma leibniz_rule_field_derivative: fixes f::"'a::{real_normed_field, banach} \ 'b::euclidean_space \ 'a" assumes fx: "\x t. x \ U \ t \ cbox a b \ ((\x. f x t) has_field_derivative fx x t) (at x within U)" assumes integrable_f2: "\x. x \ U \ (f x) integrable_on cbox a b" assumes cont_fx: "continuous_on (U \ (cbox a b)) (\(x, t). fx x t)" assumes U: "x0 \ U" "convex U" shows "((\x. integral (cbox a b) (f x)) has_field_derivative integral (cbox a b) (fx x0)) (at x0 within U)" proof - note [continuous_intros] = continuous_on_compose2[OF cont_fx, where f="\y. Pair x y" for x, unfolded split_beta fst_conv snd_conv] have *: "blinfun_mult_right (integral (cbox a b) (fx x0)) = integral (cbox a b) (\t. blinfun_mult_right (fx x0 t))" by (subst integral_linear[symmetric]) (auto simp: has_vector_derivative_def o_def intro!: integrable_continuous U continuous_intros bounded_linear_intros) show ?thesis unfolding has_field_derivative_eq_has_derivative_blinfun proof (rule has_derivative_eq_rhs [OF leibniz_rule[OF _ integrable_f2 _ U, where fx="\x t. blinfun_mult_right (fx x t)"]]) show "continuous_on (U \ cbox a b) (\(x, t). blinfun_mult_right (fx x t))" using cont_fx by (auto simp: split_beta intro!: continuous_intros) show "blinfun_apply (integral (cbox a b) (\t. blinfun_mult_right (fx x0 t))) = blinfun_apply (blinfun_mult_right (integral (cbox a b) (fx x0)))" by (subst integral_linear[symmetric]) (auto simp: has_vector_derivative_def o_def intro!: integrable_continuous U continuous_intros bounded_linear_intros) qed (use fx in \auto simp: has_field_derivative_def\) qed lemma leibniz_rule_field_differentiable: fixes f::"'a::{real_normed_field, banach} \ 'b::euclidean_space \ 'a" assumes "\x t. x \ U \ t \ cbox a b \ ((\x. f x t) has_field_derivative fx x t) (at x within U)" assumes "\x. x \ U \ (f x) integrable_on cbox a b" assumes "continuous_on (U \ (cbox a b)) (\(x, t). fx x t)" assumes "x0 \ U" "convex U" shows "(\x. integral (cbox a b) (f x)) field_differentiable at x0 within U" using leibniz_rule_field_derivative[OF assms] by (auto simp: field_differentiable_def) subsection \Exchange uniform limit and integral\ lemma uniform_limit_integral_cbox: fixes f::"'a \ 'b::euclidean_space \ 'c::banach" assumes u: "uniform_limit (cbox a b) f g F" assumes c: "\n. continuous_on (cbox a b) (f n)" assumes [simp]: "F \ bot" obtains I J where "\n. (f n has_integral I n) (cbox a b)" "(g has_integral J) (cbox a b)" "(I \ J) F" proof - have fi[simp]: "f n integrable_on (cbox a b)" for n by (auto intro!: integrable_continuous assms) then obtain I where I: "\n. (f n has_integral I n) (cbox a b)" by atomize_elim (auto simp: integrable_on_def intro!: choice) moreover have gi[simp]: "g integrable_on (cbox a b)" by (auto intro!: integrable_continuous uniform_limit_theorem[OF _ u] eventuallyI c) then obtain J where J: "(g has_integral J) (cbox a b)" by blast moreover have "(I \ J) F" proof cases assume "content (cbox a b) = 0" hence "I = (\_. 0)" "J = 0" by (auto intro!: has_integral_unique I J) thus ?thesis by simp next assume content_nonzero: "content (cbox a b) \ 0" show ?thesis proof (rule tendstoI) fix e::real assume "e > 0" define e' where "e' = e/2" with \e > 0\ have "e' > 0" by simp then have "\\<^sub>F n in F. \x\cbox a b. norm (f n x - g x) < e' / content (cbox a b)" using u content_nonzero by (auto simp: uniform_limit_iff dist_norm zero_less_measure_iff) then show "\\<^sub>F n in F. dist (I n) J < e" proof eventually_elim case (elim n) have "I n = integral (cbox a b) (f n)" "J = integral (cbox a b) g" using I[of n] J by (simp_all add: integral_unique) then have "dist (I n) J = norm (integral (cbox a b) (\x. f n x - g x))" by (simp add: integral_diff dist_norm) also have "\ \ integral (cbox a b) (\x. (e' / content (cbox a b)))" using elim by (intro integral_norm_bound_integral) (auto intro!: integrable_diff) also have "\ < e" using \0 < e\ by (simp add: e'_def) finally show ?case . qed qed qed ultimately show ?thesis .. qed lemma uniform_limit_integral: fixes f::"'a \ 'b::ordered_euclidean_space \ 'c::banach" assumes u: "uniform_limit {a..b} f g F" assumes c: "\n. continuous_on {a..b} (f n)" assumes [simp]: "F \ bot" obtains I J where "\n. (f n has_integral I n) {a..b}" "(g has_integral J) {a..b}" "(I \ J) F" by (metis interval_cbox assms uniform_limit_integral_cbox) subsection \Integration by parts\ lemma integration_by_parts_interior_strong: fixes prod :: "_ \ _ \ 'b :: banach" assumes bilinear: "bounded_bilinear (prod)" assumes s: "finite s" and le: "a \ b" assumes cont [continuous_intros]: "continuous_on {a..b} f" "continuous_on {a..b} g" assumes deriv: "\x. x\{a<.. (f has_vector_derivative f' x) (at x)" "\x. x\{a<.. (g has_vector_derivative g' x) (at x)" assumes int: "((\x. prod (f x) (g' x)) has_integral (prod (f b) (g b) - prod (f a) (g a) - y)) {a..b}" shows "((\x. prod (f' x) (g x)) has_integral y) {a..b}" proof - interpret bounded_bilinear prod by fact have "((\x. prod (f x) (g' x) + prod (f' x) (g x)) has_integral (prod (f b) (g b) - prod (f a) (g a))) {a..b}" using deriv by (intro fundamental_theorem_of_calculus_interior_strong[OF s le]) (auto intro!: continuous_intros continuous_on has_vector_derivative) from has_integral_diff[OF this int] show ?thesis by (simp add: algebra_simps) qed lemma integration_by_parts_interior: fixes prod :: "_ \ _ \ 'b :: banach" assumes "bounded_bilinear (prod)" "a \ b" "continuous_on {a..b} f" "continuous_on {a..b} g" assumes "\x. x\{a<.. (f has_vector_derivative f' x) (at x)" "\x. x\{a<.. (g has_vector_derivative g' x) (at x)" assumes "((\x. prod (f x) (g' x)) has_integral (prod (f b) (g b) - prod (f a) (g a) - y)) {a..b}" shows "((\x. prod (f' x) (g x)) has_integral y) {a..b}" by (rule integration_by_parts_interior_strong[of _ "{}" _ _ f g f' g']) (insert assms, simp_all) lemma integration_by_parts: fixes prod :: "_ \ _ \ 'b :: banach" assumes "bounded_bilinear (prod)" "a \ b" "continuous_on {a..b} f" "continuous_on {a..b} g" assumes "\x. x\{a..b} \ (f has_vector_derivative f' x) (at x)" "\x. x\{a..b} \ (g has_vector_derivative g' x) (at x)" assumes "((\x. prod (f x) (g' x)) has_integral (prod (f b) (g b) - prod (f a) (g a) - y)) {a..b}" shows "((\x. prod (f' x) (g x)) has_integral y) {a..b}" by (rule integration_by_parts_interior[of _ _ _ f g f' g']) (insert assms, simp_all) lemma integrable_by_parts_interior_strong: fixes prod :: "_ \ _ \ 'b :: banach" assumes bilinear: "bounded_bilinear (prod)" assumes s: "finite s" and le: "a \ b" assumes cont [continuous_intros]: "continuous_on {a..b} f" "continuous_on {a..b} g" assumes deriv: "\x. x\{a<.. (f has_vector_derivative f' x) (at x)" "\x. x\{a<.. (g has_vector_derivative g' x) (at x)" assumes int: "(\x. prod (f x) (g' x)) integrable_on {a..b}" shows "(\x. prod (f' x) (g x)) integrable_on {a..b}" proof - from int obtain I where "((\x. prod (f x) (g' x)) has_integral I) {a..b}" unfolding integrable_on_def by blast hence "((\x. prod (f x) (g' x)) has_integral (prod (f b) (g b) - prod (f a) (g a) - (prod (f b) (g b) - prod (f a) (g a) - I))) {a..b}" by simp from integration_by_parts_interior_strong[OF assms(1-7) this] show ?thesis unfolding integrable_on_def by blast qed lemma integrable_by_parts_interior: fixes prod :: "_ \ _ \ 'b :: banach" assumes "bounded_bilinear (prod)" "a \ b" "continuous_on {a..b} f" "continuous_on {a..b} g" assumes "\x. x\{a<.. (f has_vector_derivative f' x) (at x)" "\x. x\{a<.. (g has_vector_derivative g' x) (at x)" assumes "(\x. prod (f x) (g' x)) integrable_on {a..b}" shows "(\x. prod (f' x) (g x)) integrable_on {a..b}" by (rule integrable_by_parts_interior_strong[of _ "{}" _ _ f g f' g']) (insert assms, simp_all) lemma integrable_by_parts: fixes prod :: "_ \ _ \ 'b :: banach" assumes "bounded_bilinear (prod)" "a \ b" "continuous_on {a..b} f" "continuous_on {a..b} g" assumes "\x. x\{a..b} \ (f has_vector_derivative f' x) (at x)" "\x. x\{a..b} \ (g has_vector_derivative g' x) (at x)" assumes "(\x. prod (f x) (g' x)) integrable_on {a..b}" shows "(\x. prod (f' x) (g x)) integrable_on {a..b}" by (rule integrable_by_parts_interior_strong[of _ "{}" _ _ f g f' g']) (insert assms, simp_all) subsection \Integration by substitution\ lemma has_integral_substitution_general: fixes f :: "real \ 'a::euclidean_space" and g :: "real \ real" assumes s: "finite s" and le: "a \ b" and subset: "g ` {a..b} \ {c..d}" and f [continuous_intros]: "continuous_on {c..d} f" and g [continuous_intros]: "continuous_on {a..b} g" and deriv [derivative_intros]: "\x. x \ {a..b} - s \ (g has_field_derivative g' x) (at x within {a..b})" shows "((\x. g' x *\<^sub>R f (g x)) has_integral (integral {g a..g b} f - integral {g b..g a} f)) {a..b}" proof - let ?F = "\x. integral {c..g x} f" have cont_int: "continuous_on {a..b} ?F" by (rule continuous_on_compose2[OF _ g subset] indefinite_integral_continuous_1 f integrable_continuous_real)+ have deriv: "(((\x. integral {c..x} f) \ g) has_vector_derivative g' x *\<^sub>R f (g x)) (at x within {a..b})" if "x \ {a..b} - s" for x proof (rule has_vector_derivative_eq_rhs [OF vector_diff_chain_within refl]) show "(g has_vector_derivative g' x) (at x within {a..b})" using deriv has_field_derivative_iff_has_vector_derivative that by blast show "((\x. integral {c..x} f) has_vector_derivative f (g x)) (at (g x) within g ` {a..b})" using that le subset by (blast intro: has_vector_derivative_within_subset integral_has_vector_derivative f) qed have deriv: "(?F has_vector_derivative g' x *\<^sub>R f (g x)) (at x)" if "x \ {a..b} - (s \ {a,b})" for x using deriv[of x] that by (simp add: at_within_Icc_at o_def) have "((\x. g' x *\<^sub>R f (g x)) has_integral (?F b - ?F a)) {a..b}" using le cont_int s deriv cont_int by (intro fundamental_theorem_of_calculus_interior_strong[of "s \ {a,b}"]) simp_all also from subset have "g x \ {c..d}" if "x \ {a..b}" for x using that by blast from this[of a] this[of b] le have cd: "c \ g a" "g b \ d" "c \ g b" "g a \ d" by auto have "integral {c..g b} f - integral {c..g a} f = integral {g a..g b} f - integral {g b..g a} f" proof cases assume "g a \ g b" note le = le this from cd have "integral {c..g a} f + integral {g a..g b} f = integral {c..g b} f" by (intro integral_combine integrable_continuous_real continuous_on_subset[OF f] le) simp_all with le show ?thesis by (cases "g a = g b") (simp_all add: algebra_simps) next assume less: "\g a \ g b" then have "g a \ g b" by simp note le = le this from cd have "integral {c..g b} f + integral {g b..g a} f = integral {c..g a} f" by (intro integral_combine integrable_continuous_real continuous_on_subset[OF f] le) simp_all with less show ?thesis by (simp_all add: algebra_simps) qed finally show ?thesis . qed lemma has_integral_substitution_strong: fixes f :: "real \ 'a::euclidean_space" and g :: "real \ real" assumes s: "finite s" and le: "a \ b" "g a \ g b" and subset: "g ` {a..b} \ {c..d}" and f [continuous_intros]: "continuous_on {c..d} f" and g [continuous_intros]: "continuous_on {a..b} g" and deriv [derivative_intros]: "\x. x \ {a..b} - s \ (g has_field_derivative g' x) (at x within {a..b})" shows "((\x. g' x *\<^sub>R f (g x)) has_integral (integral {g a..g b} f)) {a..b}" using has_integral_substitution_general[OF s le(1) subset f g deriv] le(2) by (cases "g a = g b") auto lemma has_integral_substitution: fixes f :: "real \ 'a::euclidean_space" and g :: "real \ real" assumes "a \ b" "g a \ g b" "g ` {a..b} \ {c..d}" and "continuous_on {c..d} f" and "\x. x \ {a..b} \ (g has_field_derivative g' x) (at x within {a..b})" shows "((\x. g' x *\<^sub>R f (g x)) has_integral (integral {g a..g b} f)) {a..b}" by (intro has_integral_substitution_strong[of "{}" a b g c d] assms) (auto intro: DERIV_continuous_on assms) lemma integral_shift: fixes f :: "real \ 'a::euclidean_space" assumes cont: "continuous_on {a + c..b + c} f" shows "integral {a..b} (f \ (\x. x + c)) = integral {a + c..b + c} f" proof (cases "a \ b") case True have "((\x. 1 *\<^sub>R f (x + c)) has_integral integral {a+c..b+c} f) {a..b}" using True cont by (intro has_integral_substitution[where c = "a + c" and d = "b + c"]) (auto intro!: derivative_eq_intros) thus ?thesis by (simp add: has_integral_iff o_def) qed auto subsection \Compute a double integral using iterated integrals and switching the order of integration\ lemma continuous_on_imp_integrable_on_Pair1: fixes f :: "_ \ 'b::banach" assumes con: "continuous_on (cbox (a,c) (b,d)) f" and x: "x \ cbox a b" shows "(\y. f (x, y)) integrable_on (cbox c d)" proof - have "f \ (\y. (x, y)) integrable_on (cbox c d)" proof (intro integrable_continuous continuous_on_compose [OF _ continuous_on_subset [OF con]]) show "continuous_on (cbox c d) (Pair x)" by (simp add: continuous_on_Pair) show "Pair x ` cbox c d \ cbox (a,c) (b,d)" using x by blast qed then show ?thesis by (simp add: o_def) qed lemma integral_integrable_2dim: fixes f :: "('a::euclidean_space * 'b::euclidean_space) \ 'c::banach" assumes "continuous_on (cbox (a,c) (b,d)) f" shows "(\x. integral (cbox c d) (\y. f (x,y))) integrable_on cbox a b" proof (cases "content(cbox c d) = 0") case True then show ?thesis by (simp add: True integrable_const) next case False have uc: "uniformly_continuous_on (cbox (a,c) (b,d)) f" by (simp add: assms compact_cbox compact_uniformly_continuous) { fix x::'a and e::real assume x: "x \ cbox a b" and e: "0 < e" then have e2_gt: "0 < e/2 / content (cbox c d)" and e2_less: "e/2 / content (cbox c d) * content (cbox c d) < e" by (auto simp: False content_lt_nz e) then obtain dd where dd: "\x x'. \x\cbox (a, c) (b, d); x'\cbox (a, c) (b, d); norm (x' - x) < dd\ \ norm (f x' - f x) \ e/(2 * content (cbox c d))" "dd>0" using uc [unfolded uniformly_continuous_on_def, THEN spec, of "e/(2 * content (cbox c d))"] by (auto simp: dist_norm intro: less_imp_le) have "\delta>0. \x'\cbox a b. norm (x' - x) < delta \ norm (integral (cbox c d) (\u. f (x', u) - f (x, u))) < e" using dd e2_gt assms x apply (rule_tac x=dd in exI) apply clarify apply (rule le_less_trans [OF integrable_bound e2_less]) apply (auto intro: integrable_diff continuous_on_imp_integrable_on_Pair1) done } note * = this show ?thesis proof (rule integrable_continuous) show "continuous_on (cbox a b) (\x. integral (cbox c d) (\y. f (x, y)))" by (simp add: * continuous_on_iff dist_norm integral_diff [symmetric] continuous_on_imp_integrable_on_Pair1 [OF assms]) qed qed lemma integral_split: fixes f :: "'a::euclidean_space \ 'b::{real_normed_vector,complete_space}" assumes f: "f integrable_on (cbox a b)" and k: "k \ Basis" shows "integral (cbox a b) f = integral (cbox a b \ {x. x\k \ c}) f + integral (cbox a b \ {x. x\k \ c}) f" using k f by (auto simp: has_integral_integral intro: integral_unique [OF has_integral_split]) lemma integral_swap_operativeI: fixes f :: "('a::euclidean_space * 'b::euclidean_space) \ 'c::banach" assumes f: "continuous_on s f" and e: "0 < e" shows "operative conj True (\k. \a b c d. cbox (a,c) (b,d) \ k \ cbox (a,c) (b,d) \ s \ norm(integral (cbox (a,c) (b,d)) f - integral (cbox a b) (\x. integral (cbox c d) (\y. f((x,y))))) \ e * content (cbox (a,c) (b,d)))" proof (standard, auto) fix a::'a and c::'b and b::'a and d::'b and u::'a and v::'a and w::'b and z::'b assume *: "box (a, c) (b, d) = {}" and cb1: "cbox (u, w) (v, z) \ cbox (a, c) (b, d)" and cb2: "cbox (u, w) (v, z) \ s" then have c0: "content (cbox (a, c) (b, d)) = 0" using * unfolding content_eq_0_interior by simp have c0': "content (cbox (u, w) (v, z)) = 0" by (fact content_0_subset [OF c0 cb1]) show "norm (integral (cbox (u,w) (v,z)) f - integral (cbox u v) (\x. integral (cbox w z) (\y. f (x, y)))) \ e * content (cbox (u,w) (v,z))" using content_cbox_pair_eq0_D [OF c0'] by (force simp add: c0') next fix a::'a and c::'b and b::'a and d::'b and M::real and i::'a and j::'b and u::'a and v::'a and w::'b and z::'b assume ij: "(i,j) \ Basis" and n1: "\a' b' c' d'. cbox (a',c') (b',d') \ cbox (a,c) (b,d) \ cbox (a',c') (b',d') \ {x. x \ (i,j) \ M} \ cbox (a',c') (b',d') \ s \ norm (integral (cbox (a',c') (b',d')) f - integral (cbox a' b') (\x. integral (cbox c' d') (\y. f (x,y)))) \ e * content (cbox (a',c') (b',d'))" and n2: "\a' b' c' d'. cbox (a',c') (b',d') \ cbox (a,c) (b,d) \ cbox (a',c') (b',d') \ {x. M \ x \ (i,j)} \ cbox (a',c') (b',d') \ s \ norm (integral (cbox (a',c') (b',d')) f - integral (cbox a' b') (\x. integral (cbox c' d') (\y. f (x,y)))) \ e * content (cbox (a',c') (b',d'))" and subs: "cbox (u,w) (v,z) \ cbox (a,c) (b,d)" "cbox (u,w) (v,z) \ s" have fcont: "continuous_on (cbox (u, w) (v, z)) f" using assms(1) continuous_on_subset subs(2) by blast then have fint: "f integrable_on cbox (u, w) (v, z)" by (metis integrable_continuous) consider "i \ Basis" "j=0" | "j \ Basis" "i=0" using ij by (auto simp: Euclidean_Space.Basis_prod_def) then show "norm (integral (cbox (u,w) (v,z)) f - integral (cbox u v) (\x. integral (cbox w z) (\y. f (x,y)))) \ e * content (cbox (u,w) (v,z))" (is ?normle) proof cases case 1 then have e: "e * content (cbox (u, w) (v, z)) = e * (content (cbox u v \ {x. x \ i \ M}) * content (cbox w z)) + e * (content (cbox u v \ {x. M \ x \ i}) * content (cbox w z))" by (simp add: content_split [where c=M] content_Pair algebra_simps) have *: "integral (cbox u v) (\x. integral (cbox w z) (\y. f (x, y))) = integral (cbox u v \ {x. x \ i \ M}) (\x. integral (cbox w z) (\y. f (x, y))) + integral (cbox u v \ {x. M \ x \ i}) (\x. integral (cbox w z) (\y. f (x, y)))" using 1 f subs integral_integrable_2dim continuous_on_subset by (blast intro: integral_split) show ?normle apply (rule norm_diff2 [OF integral_split [where c=M, OF fint ij] * e]) using 1 subs apply (simp_all add: cbox_Pair_eq setcomp_dot1 [of "\u. M\u"] setcomp_dot1 [of "\u. u\M"] Sigma_Int_Paircomp1) apply (simp_all add: interval_split ij flip: cbox_Pair_eq content_Pair) apply (force simp flip: interval_split intro!: n1 [rule_format]) apply (force simp flip: interval_split intro!: n2 [rule_format]) done next case 2 then have e: "e * content (cbox (u, w) (v, z)) = e * (content (cbox u v) * content (cbox w z \ {x. x \ j \ M})) + e * (content (cbox u v) * content (cbox w z \ {x. M \ x \ j}))" by (simp add: content_split [where c=M] content_Pair algebra_simps) have "(\x. integral (cbox w z \ {x. x \ j \ M}) (\y. f (x, y))) integrable_on cbox u v" "(\x. integral (cbox w z \ {x. M \ x \ j}) (\y. f (x, y))) integrable_on cbox u v" using 2 subs apply (simp_all add: interval_split) apply (rule integral_integrable_2dim [OF continuous_on_subset [OF f]]; auto simp: cbox_Pair_eq interval_split [symmetric])+ done with 2 have *: "integral (cbox u v) (\x. integral (cbox w z) (\y. f (x, y))) = integral (cbox u v) (\x. integral (cbox w z \ {x. x \ j \ M}) (\y. f (x, y))) + integral (cbox u v) (\x. integral (cbox w z \ {x. M \ x \ j}) (\y. f (x, y)))" by (simp add: integral_add [symmetric] integral_split [symmetric] continuous_on_imp_integrable_on_Pair1 [OF fcont] cong: integral_cong) show ?normle apply (rule norm_diff2 [OF integral_split [where c=M, OF fint ij] * e]) using 2 subs apply (simp_all add: cbox_Pair_eq setcomp_dot2 [of "\u. M\u"] setcomp_dot2 [of "\u. u\M"] Sigma_Int_Paircomp2) apply (simp_all add: interval_split ij flip: cbox_Pair_eq content_Pair) apply (force simp flip: interval_split intro!: n1 [rule_format]) apply (force simp flip: interval_split intro!: n2 [rule_format]) done qed qed lemma integral_Pair_const: "integral (cbox (a,c) (b,d)) (\x. k) = integral (cbox a b) (\x. integral (cbox c d) (\y. k))" by (simp add: content_Pair) lemma integral_prod_continuous: fixes f :: "('a::euclidean_space * 'b::euclidean_space) \ 'c::banach" assumes "continuous_on (cbox (a, c) (b, d)) f" (is "continuous_on ?CBOX f") shows "integral (cbox (a, c) (b, d)) f = integral (cbox a b) (\x. integral (cbox c d) (\y. f (x, y)))" proof (cases "content ?CBOX = 0") case True then show ?thesis by (auto simp: content_Pair) next case False then have cbp: "content ?CBOX > 0" using content_lt_nz by blast have "norm (integral ?CBOX f - integral (cbox a b) (\x. integral (cbox c d) (\y. f (x,y)))) = 0" proof (rule dense_eq0_I, simp) fix e :: real assume "0 < e" with \content ?CBOX > 0\ have "0 < e/content ?CBOX" by simp have f_int_acbd: "f integrable_on ?CBOX" by (rule integrable_continuous [OF assms]) { fix p assume p: "p division_of ?CBOX" then have "finite p" by blast define e' where "e' = e/content ?CBOX" with \0 < e\ \0 < e/content ?CBOX\ have "0 < e'" by simp interpret operative conj True "\k. \a' b' c' d'. cbox (a', c') (b', d') \ k \ cbox (a', c') (b', d') \ ?CBOX \ norm (integral (cbox (a', c') (b', d')) f - integral (cbox a' b') (\x. integral (cbox c' d') (\y. f ((x, y))))) \ e' * content (cbox (a', c') (b', d'))" using assms \0 < e'\ by (rule integral_swap_operativeI) have "norm (integral ?CBOX f - integral (cbox a b) (\x. integral (cbox c d) (\y. f (x, y)))) \ e' * content ?CBOX" if "\t u v w z. t \ p \ cbox (u, w) (v, z) \ t \ cbox (u, w) (v, z) \ ?CBOX \ norm (integral (cbox (u, w) (v, z)) f - integral (cbox u v) (\x. integral (cbox w z) (\y. f (x, y)))) \ e' * content (cbox (u, w) (v, z))" using that division [of p "(a, c)" "(b, d)"] p \finite p\ by (auto simp add: comm_monoid_set_F_and) with False have "norm (integral ?CBOX f - integral (cbox a b) (\x. integral (cbox c d) (\y. f (x, y)))) \ e" if "\t u v w z. t \ p \ cbox (u, w) (v, z) \ t \ cbox (u, w) (v, z) \ ?CBOX \ norm (integral (cbox (u, w) (v, z)) f - integral (cbox u v) (\x. integral (cbox w z) (\y. f (x, y)))) \ e * content (cbox (u, w) (v, z)) / content ?CBOX" using that by (simp add: e'_def) } note op_acbd = this { fix k::real and \ and u::'a and v w and z::'b and t1 t2 l assume k: "0 < k" and nf: "\x y u v. \x \ cbox a b; y \ cbox c d; u \ cbox a b; v\cbox c d; norm (u-x, v-y) < k\ \ norm (f(u,v) - f(x,y)) < e/(2 * (content ?CBOX))" and p_acbd: "\ tagged_division_of cbox (a,c) (b,d)" and fine: "(\x. ball x k) fine \" "((t1,t2), l) \ \" and uwvz_sub: "cbox (u,w) (v,z) \ l" "cbox (u,w) (v,z) \ cbox (a,c) (b,d)" have t: "t1 \ cbox a b" "t2 \ cbox c d" by (meson fine p_acbd cbox_Pair_iff tag_in_interval)+ have ls: "l \ ball (t1,t2) k" using fine by (simp add: fine_def Ball_def) { fix x1 x2 assume xuvwz: "x1 \ cbox u v" "x2 \ cbox w z" then have x: "x1 \ cbox a b" "x2 \ cbox c d" using uwvz_sub by auto have "norm (x1 - t1, x2 - t2) = norm (t1 - x1, t2 - x2)" by (simp add: norm_Pair norm_minus_commute) also have "norm (t1 - x1, t2 - x2) < k" using xuvwz ls uwvz_sub unfolding ball_def by (force simp add: cbox_Pair_eq dist_norm ) finally have "norm (f (x1,x2) - f (t1,t2)) \ e/content ?CBOX/2" using nf [OF t x] by simp } note nf' = this have f_int_uwvz: "f integrable_on cbox (u,w) (v,z)" using f_int_acbd uwvz_sub integrable_on_subcbox by blast have f_int_uv: "\x. x \ cbox u v \ (\y. f (x,y)) integrable_on cbox w z" using assms continuous_on_subset uwvz_sub by (blast intro: continuous_on_imp_integrable_on_Pair1) have 1: "norm (integral (cbox (u,w) (v,z)) f - integral (cbox (u,w) (v,z)) (\x. f (t1,t2))) \ e * content (cbox (u,w) (v,z)) / content ?CBOX/2" using cbp \0 < e/content ?CBOX\ nf' apply (simp only: integral_diff [symmetric] f_int_uwvz integrable_const) apply (auto simp: integrable_diff f_int_uwvz integrable_const intro: order_trans [OF integrable_bound [of "e/content ?CBOX/2"]]) done have int_integrable: "(\x. integral (cbox w z) (\y. f (x, y))) integrable_on cbox u v" using assms integral_integrable_2dim continuous_on_subset uwvz_sub(2) by blast have normint_wz: "\x. x \ cbox u v \ norm (integral (cbox w z) (\y. f (x, y)) - integral (cbox w z) (\y. f (t1, t2))) \ e * content (cbox w z) / content (cbox (a, c) (b, d))/2" using cbp \0 < e/content ?CBOX\ nf' apply (simp only: integral_diff [symmetric] f_int_uv integrable_const) apply (auto simp: integrable_diff f_int_uv integrable_const intro: order_trans [OF integrable_bound [of "e/content ?CBOX/2"]]) done have "norm (integral (cbox u v) (\x. integral (cbox w z) (\y. f (x,y)) - integral (cbox w z) (\y. f (t1,t2)))) \ e * content (cbox w z) / content ?CBOX/2 * content (cbox u v)" using cbp \0 < e/content ?CBOX\ apply (intro integrable_bound [OF _ _ normint_wz]) apply (auto simp: field_split_simps integrable_diff int_integrable integrable_const) done also have "... \ e * content (cbox (u,w) (v,z)) / content ?CBOX/2" by (simp add: content_Pair field_split_simps) finally have 2: "norm (integral (cbox u v) (\x. integral (cbox w z) (\y. f (x,y))) - integral (cbox u v) (\x. integral (cbox w z) (\y. f (t1,t2)))) \ e * content (cbox (u,w) (v,z)) / content ?CBOX/2" by (simp only: integral_diff [symmetric] int_integrable integrable_const) have norm_xx: "\x' = y'; norm(x - x') \ e/2; norm(y - y') \ e/2\ \ norm(x - y) \ e" for x::'c and y x' y' e using norm_triangle_mono [of "x-y'" "e/2" "y'-y" "e/2"] field_sum_of_halves by (simp add: norm_minus_commute) have "norm (integral (cbox (u,w) (v,z)) f - integral (cbox u v) (\x. integral (cbox w z) (\y. f (x,y)))) \ e * content (cbox (u,w) (v,z)) / content ?CBOX" by (rule norm_xx [OF integral_Pair_const 1 2]) } note * = this have "norm (integral ?CBOX f - integral (cbox a b) (\x. integral (cbox c d) (\y. f (x,y)))) \ e" if "\x\?CBOX. \x'\?CBOX. norm (x' - x) < k \ norm (f x' - f x) < e /(2 * content (?CBOX))" "0 < k" for k proof - obtain p where ptag: "p tagged_division_of cbox (a, c) (b, d)" and fine: "(\x. ball x k) fine p" using fine_division_exists \0 < k\ by blast show ?thesis using that fine ptag \0 < k\ by (auto simp: * intro: op_acbd [OF division_of_tagged_division [OF ptag]]) qed then show "norm (integral ?CBOX f - integral (cbox a b) (\x. integral (cbox c d) (\y. f (x,y)))) \ e" using compact_uniformly_continuous [OF assms compact_cbox] apply (simp add: uniformly_continuous_on_def dist_norm) apply (drule_tac x="e/2 / content?CBOX" in spec) using cbp \0 < e\ by (auto simp: zero_less_mult_iff) qed then show ?thesis by simp qed lemma integral_swap_2dim: fixes f :: "['a::euclidean_space, 'b::euclidean_space] \ 'c::banach" assumes "continuous_on (cbox (a,c) (b,d)) (\(x,y). f x y)" shows "integral (cbox (a, c) (b, d)) (\(x, y). f x y) = integral (cbox (c, a) (d, b)) (\(x, y). f y x)" proof - have "((\(x, y). f x y) has_integral integral (cbox (c, a) (d, b)) (\(x, y). f y x)) (prod.swap ` (cbox (c, a) (d, b)))" proof (rule has_integral_twiddle [of 1 prod.swap prod.swap "\(x,y). f y x" "integral (cbox (c, a) (d, b)) (\(x, y). f y x)", simplified]) show "\u v. content (prod.swap ` cbox u v) = content (cbox u v)" by (metis content_Pair mult.commute old.prod.exhaust swap_cbox_Pair) show "((\(x, y). f y x) has_integral integral (cbox (c, a) (d, b)) (\(x, y). f y x)) (cbox (c, a) (d, b))" by (simp add: assms integrable_continuous integrable_integral swap_continuous) qed (use isCont_swap in \fastforce+\) then show ?thesis by force qed theorem integral_swap_continuous: fixes f :: "['a::euclidean_space, 'b::euclidean_space] \ 'c::banach" assumes "continuous_on (cbox (a,c) (b,d)) (\(x,y). f x y)" shows "integral (cbox a b) (\x. integral (cbox c d) (f x)) = integral (cbox c d) (\y. integral (cbox a b) (\x. f x y))" proof - have "integral (cbox a b) (\x. integral (cbox c d) (f x)) = integral (cbox (a, c) (b, d)) (\(x, y). f x y)" using integral_prod_continuous [OF assms] by auto also have "... = integral (cbox (c, a) (d, b)) (\(x, y). f y x)" by (rule integral_swap_2dim [OF assms]) also have "... = integral (cbox c d) (\y. integral (cbox a b) (\x. f x y))" using integral_prod_continuous [OF swap_continuous] assms by auto finally show ?thesis . qed subsection \Definite integrals for exponential and power function\ lemma has_integral_exp_minus_to_infinity: assumes a: "a > 0" shows "((\x::real. exp (-a*x)) has_integral exp (-a*c)/a) {c..}" proof - define f where "f = (\k x. if x \ {c..real k} then exp (-a*x) else 0)" { fix k :: nat assume k: "of_nat k \ c" from k a have "((\x. exp (-a*x)) has_integral (-exp (-a*real k)/a - (-exp (-a*c)/a))) {c..real k}" by (intro fundamental_theorem_of_calculus) (auto intro!: derivative_eq_intros simp: has_field_derivative_iff_has_vector_derivative [symmetric]) hence "(f k has_integral (exp (-a*c)/a - exp (-a*real k)/a)) {c..}" unfolding f_def by (subst has_integral_restrict) simp_all } note has_integral_f = this have [simp]: "f k = (\_. 0)" if "of_nat k < c" for k using that by (auto simp: fun_eq_iff f_def) have integral_f: "integral {c..} (f k) = (if real k \ c then exp (-a*c)/a - exp (-a*real k)/a else 0)" for k using integral_unique[OF has_integral_f[of k]] by simp have A: "(\x. exp (-a*x)) integrable_on {c..} \ (\k. integral {c..} (f k)) \ integral {c..} (\x. exp (-a*x))" proof (intro monotone_convergence_increasing allI ballI) fix k ::nat have "(\x. exp (-a*x)) integrable_on {c..of_real k}" (is ?P) unfolding f_def by (auto intro!: continuous_intros integrable_continuous_real) hence "(f k) integrable_on {c..of_real k}" by (rule integrable_eq) (simp add: f_def) then show "f k integrable_on {c..}" by (rule integrable_on_superset) (auto simp: f_def) next fix x assume x: "x \ {c..}" have "sequentially \ principal {nat \x\..}" unfolding at_top_def by (simp add: Inf_lower) also have "{nat \x\..} \ {k. x \ real k}" by auto also have "inf (principal \) (principal {k. \x \ real k}) = principal ({k. x \ real k} \ {k. \x \ real k})" by simp also have "{k. x \ real k} \ {k. \x \ real k} = {}" by blast finally have "inf sequentially (principal {k. \x \ real k}) = bot" by (simp add: inf.coboundedI1 bot_unique) with x show "(\k. f k x) \ exp (-a*x)" unfolding f_def by (intro filterlim_If) auto next have "\integral {c..} (f k)\ \ exp (-a*c)/a" for k proof (cases "c > of_nat k") case False hence "abs (integral {c..} (f k)) = abs (exp (- (a * c)) / a - exp (- (a * real k)) / a)" by (simp add: integral_f) also have "abs (exp (- (a * c)) / a - exp (- (a * real k)) / a) = exp (- (a * c)) / a - exp (- (a * real k)) / a" using False a by (intro abs_of_nonneg) (simp_all add: field_simps) also have "\ \ exp (- a * c) / a" using a by simp finally show ?thesis . qed (insert a, simp_all add: integral_f) thus "bounded (range(\k. integral {c..} (f k)))" by (intro boundedI[of _ "exp (-a*c)/a"]) auto qed (auto simp: f_def) have "(\k. exp (-a*c)/a - exp (-a * of_nat k)/a) \ exp (-a*c)/a - 0/a" by (intro tendsto_intros filterlim_compose[OF exp_at_bot] filterlim_tendsto_neg_mult_at_bot[OF tendsto_const] filterlim_real_sequentially)+ (insert a, simp_all) moreover from eventually_gt_at_top[of "nat \c\"] have "eventually (\k. of_nat k > c) sequentially" by eventually_elim linarith hence "eventually (\k. exp (-a*c)/a - exp (-a * of_nat k)/a = integral {c..} (f k)) sequentially" by eventually_elim (simp add: integral_f) ultimately have "(\k. integral {c..} (f k)) \ exp (-a*c)/a - 0/a" by (rule Lim_transform_eventually) from LIMSEQ_unique[OF conjunct2[OF A] this] have "integral {c..} (\x. exp (-a*x)) = exp (-a*c)/a" by simp with conjunct1[OF A] show ?thesis by (simp add: has_integral_integral) qed lemma integrable_on_exp_minus_to_infinity: "a > 0 \ (\x. exp (-a*x) :: real) integrable_on {c..}" using has_integral_exp_minus_to_infinity[of a c] unfolding integrable_on_def by blast lemma has_integral_powr_from_0: assumes a: "a > (-1::real)" and c: "c \ 0" shows "((\x. x powr a) has_integral (c powr (a+1) / (a+1))) {0..c}" proof (cases "c = 0") case False define f where "f = (\k x. if x \ {inverse (of_nat (Suc k))..c} then x powr a else 0)" define F where "F = (\k. if inverse (of_nat (Suc k)) \ c then c powr (a+1)/(a+1) - inverse (real (Suc k)) powr (a+1)/(a+1) else 0)" { fix k :: nat have "(f k has_integral F k) {0..c}" proof (cases "inverse (of_nat (Suc k)) \ c") case True { fix x assume x: "x \ inverse (1 + real k)" have "0 < inverse (1 + real k)" by simp also note x finally have "x > 0" . } note x = this hence "((\x. x powr a) has_integral c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) {inverse (real (Suc k))..c}" using True a by (intro fundamental_theorem_of_calculus) (auto intro!: derivative_eq_intros continuous_on_powr' continuous_on_const simp: has_field_derivative_iff_has_vector_derivative [symmetric]) with True show ?thesis unfolding f_def F_def by (subst has_integral_restrict) simp_all next case False thus ?thesis unfolding f_def F_def by (subst has_integral_restrict) auto qed } note has_integral_f = this have integral_f: "integral {0..c} (f k) = F k" for k using has_integral_f[of k] by (rule integral_unique) have A: "(\x. x powr a) integrable_on {0..c} \ (\k. integral {0..c} (f k)) \ integral {0..c} (\x. x powr a)" proof (intro monotone_convergence_increasing ballI allI) fix k from has_integral_f[of k] show "f k integrable_on {0..c}" by (auto simp: integrable_on_def) next fix k :: nat and x :: real { assume x: "inverse (real (Suc k)) \ x" have "inverse (real (Suc (Suc k))) \ inverse (real (Suc k))" by (simp add: field_simps) also note x finally have "inverse (real (Suc (Suc k))) \ x" . } thus "f k x \ f (Suc k) x" by (auto simp: f_def simp del: of_nat_Suc) next fix x assume x: "x \ {0..c}" show "(\k. f k x) \ x powr a" proof (cases "x = 0") case False with x have "x > 0" by simp from order_tendstoD(2)[OF LIMSEQ_inverse_real_of_nat this] have "eventually (\k. x powr a = f k x) sequentially" by eventually_elim (insert x, simp add: f_def) moreover have "(\_. x powr a) \ x powr a" by simp ultimately show ?thesis by (blast intro: Lim_transform_eventually) qed (simp_all add: f_def) next { fix k from a have "F k \ c powr (a + 1) / (a + 1)" by (auto simp add: F_def divide_simps) also from a have "F k \ 0" by (auto simp: F_def divide_simps simp del: of_nat_Suc intro!: powr_mono2) hence "F k = abs (F k)" by simp finally have "abs (F k) \ c powr (a + 1) / (a + 1)" . } thus "bounded (range(\k. integral {0..c} (f k)))" by (intro boundedI[of _ "c powr (a+1) / (a+1)"]) (auto simp: integral_f) qed from False c have "c > 0" by simp from order_tendstoD(2)[OF LIMSEQ_inverse_real_of_nat this] have "eventually (\k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a+1) / (a+1) = integral {0..c} (f k)) sequentially" by eventually_elim (simp add: integral_f F_def) moreover have "(\k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) \ c powr (a + 1) / (a + 1) - 0 powr (a + 1) / (a + 1)" using a by (intro tendsto_intros LIMSEQ_inverse_real_of_nat) auto hence "(\k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1)) \ c powr (a + 1) / (a + 1)" by simp ultimately have "(\k. integral {0..c} (f k)) \ c powr (a+1) / (a+1)" by (blast intro: Lim_transform_eventually) with A have "integral {0..c} (\x. x powr a) = c powr (a+1) / (a+1)" by (blast intro: LIMSEQ_unique) with A show ?thesis by (simp add: has_integral_integral) qed (simp_all add: has_integral_refl) lemma integrable_on_powr_from_0: assumes a: "a > (-1::real)" and c: "c \ 0" shows "(\x. x powr a) integrable_on {0..c}" using has_integral_powr_from_0[OF assms] unfolding integrable_on_def by blast lemma has_integral_powr_to_inf: fixes a e :: real assumes "e < -1" "a > 0" shows "((\x. x powr e) has_integral -(a powr (e + 1)) / (e + 1)) {a..}" proof - define f :: "nat \ real \ real" where "f = (\n x. if x \ {a..n} then x powr e else 0)" define F where "F = (\x. x powr (e + 1) / (e + 1))" have has_integral_f: "(f n has_integral (F n - F a)) {a..}" if n: "n \ a" for n :: nat proof - from n assms have "((\x. x powr e) has_integral (F n - F a)) {a..n}" by (intro fundamental_theorem_of_calculus) (auto intro!: derivative_eq_intros simp: has_field_derivative_iff_has_vector_derivative [symmetric] F_def) hence "(f n has_integral (F n - F a)) {a..n}" by (rule has_integral_eq [rotated]) (simp add: f_def) thus "(f n has_integral (F n - F a)) {a..}" by (rule has_integral_on_superset) (auto simp: f_def) qed have integral_f: "integral {a..} (f n) = (if n \ a then F n - F a else 0)" for n :: nat proof (cases "n \ a") case True with has_integral_f[OF this] show ?thesis by (simp add: integral_unique) next case False have "(f n has_integral 0) {a}" by (rule has_integral_refl) hence "(f n has_integral 0) {a..}" by (rule has_integral_on_superset) (insert False, simp_all add: f_def) with False show ?thesis by (simp add: integral_unique) qed have *: "(\x. x powr e) integrable_on {a..} \ (\n. integral {a..} (f n)) \ integral {a..} (\x. x powr e)" proof (intro monotone_convergence_increasing allI ballI) fix n :: nat from assms have "(\x. x powr e) integrable_on {a..n}" by (auto intro!: integrable_continuous_real continuous_intros) hence "f n integrable_on {a..n}" by (rule integrable_eq) (auto simp: f_def) thus "f n integrable_on {a..}" by (rule integrable_on_superset) (auto simp: f_def) next fix n :: nat and x :: real show "f n x \ f (Suc n) x" by (auto simp: f_def) next fix x :: real assume x: "x \ {a..}" from filterlim_real_sequentially have "eventually (\n. real n \ x) at_top" by (simp add: filterlim_at_top) with x have "eventually (\n. f n x = x powr e) at_top" by (auto elim!: eventually_mono simp: f_def) thus "(\n. f n x) \ x powr e" by (simp add: tendsto_eventually) next have "norm (integral {a..} (f n)) \ -F a" for n :: nat proof (cases "n \ a") case True with assms have "a powr (e + 1) \ n powr (e + 1)" by (intro powr_mono2') simp_all with assms show ?thesis by (auto simp: divide_simps F_def integral_f) qed (insert assms, simp add: integral_f F_def field_split_simps) thus "bounded (range(\k. integral {a..} (f k)))" unfolding bounded_iff by (intro exI[of _ "-F a"]) auto qed from filterlim_real_sequentially have "eventually (\n. real n \ a) at_top" by (simp add: filterlim_at_top) hence "eventually (\n. F n - F a = integral {a..} (f n)) at_top" by eventually_elim (simp add: integral_f) moreover have "(\n. F n - F a) \ 0 / (e + 1) - F a" unfolding F_def by (insert assms, (rule tendsto_intros filterlim_compose[OF tendsto_neg_powr] filterlim_ident filterlim_real_sequentially | simp)+) hence "(\n. F n - F a) \ -F a" by simp ultimately have "(\n. integral {a..} (f n)) \ -F a" by (blast intro: Lim_transform_eventually) from conjunct2[OF *] and this have "integral {a..} (\x. x powr e) = -F a" by (rule LIMSEQ_unique) with conjunct1[OF *] show ?thesis by (simp add: has_integral_integral F_def) qed lemma has_integral_inverse_power_to_inf: fixes a :: real and n :: nat assumes "n > 1" "a > 0" shows "((\x. 1 / x ^ n) has_integral 1 / (real (n - 1) * a ^ (n - 1))) {a..}" proof - from assms have "real_of_int (-int n) < -1" by simp note has_integral_powr_to_inf[OF this \a > 0\] also have "- (a powr (real_of_int (- int n) + 1)) / (real_of_int (- int n) + 1) = 1 / (real (n - 1) * a powr (real (n - 1)))" using assms by (simp add: field_split_simps powr_add [symmetric] of_nat_diff) also from assms have "a powr (real (n - 1)) = a ^ (n - 1)" by (intro powr_realpow) finally show ?thesis by (rule has_integral_eq [rotated]) (insert assms, simp_all add: powr_minus powr_realpow field_split_simps) qed subsubsection \Adaption to ordered Euclidean spaces and the Cartesian Euclidean space\ lemma integral_component_eq_cart[simp]: fixes f :: "'n::euclidean_space \ real^'m" assumes "f integrable_on s" shows "integral s (\x. f x $ k) = integral s f $ k" using integral_linear[OF assms(1) bounded_linear_vec_nth,unfolded o_def] . lemma content_closed_interval: fixes a :: "'a::ordered_euclidean_space" assumes "a \ b" shows "content {a..b} = (\i\Basis. b\i - a\i)" using content_cbox[of a b] assms by (simp add: cbox_interval eucl_le[where 'a='a]) lemma integrable_const_ivl[intro]: fixes a::"'a::ordered_euclidean_space" shows "(\x. c) integrable_on {a..b}" unfolding cbox_interval[symmetric] by (rule integrable_const) lemma integrable_on_subinterval: fixes f :: "'n::ordered_euclidean_space \ 'a::banach" assumes "f integrable_on S" "{a..b} \ S" shows "f integrable_on {a..b}" using integrable_on_subcbox[of f S a b] assms by (simp add: cbox_interval) end